Datasets:

suolyer

/

pile_arxiv

like 3

--- abstract: 'Several results concerning existence of solutions of a quasiequilibrium problem defined on a finite dimensional space are established. The proof of the first result is based on a Michael selection theorem for lower semicontinuous set-valued maps which holds in finite dimensional spaces. Furthermore this result allows one to locate the position of a solution. Sufficient conditions, which are easier to verify, may be obtained by imposing restrictions either on the domain or on the bifunction. These facts make it possible to yield various existence results which reduce to the well known Ky Fan minimax inequality when the constraint map is constant and the quasiequilibrium problem coincides with an equilibrium problem. Lastly, a comparison with other results from the literature is discussed.' author: - Marco Castellani - Massimiliano Giuli - Massimo Pappalardo title: A Ky Fan minimax inequality for quasiequilibria on finite dimensional spaces --- Introduction ============ In [@Fa72] the author established the famous Ky Fan minimax inequality which concerns the existence of solutions for an inequality of minimax type that nowadays is called in literature “equilibrium problem”. Such a model has gained a lot interest in the last decades because it has been used in different contexts as economics, engineering, physics, chemistry and so on (see [@BiCaPaPa13] for a recent survey). In these equilibrium problems the constraint set is fixed and hence the model can not be used in many cases where the constraints depend on the current analyzed point. This more general setting was studied for the first time in the context of impulse control problem [@BeGoLi73] and it has been subsequently used by several authors for describing a lot of problems that arise in different fields: equilibrium problem in mechanics, Nash equilibrium problems, equilibria in economics, network equilibrium problems and so on. This general format, commonly called “quasiequilibrium problem”, received an increasing interest in the last years because many theoretical results developed for one of the abovementioned models can be often extended to the others through the unifying language provided by this common format. Unlike the equilibrium problems which have an extensive literature on results concerning existence of solutions, the study of quasiequilibrium problems to date is at the beginning even if the first seminal work in this area was in the seventies [@Mo76]. After that, the problem concerning existence of solutions has been developed in some papers [@AlRa16; @Au93; @AuCoIu17; @CaGi15; @CaGi16; @Cu95; @Cu97]. Most of the results require either monotonicity assumptions on the equilibrium bifunction or upper semicontinuity of the set-valued map which describes the constraint. Whereas other authors provided existence of solutions avoiding any monotonicity assumption and assuming lower semicontinuity of the constraint map and closedness of the set of its fixed points. Aim of this paper is to establish several results concerning existence of solutions of a quasiequilibrium problem defined on a finite dimensional space which comes down to the Ky Fan minimax inequality in the classical setting. Our approach is based on a Michael selection result [@Mi56] for lower semicontinuous set-valued maps. Moreover the proof of our results allow one to locate the position of a solution. The paper is organized as follows. Section 2 is devoted to recall the results about set-valued maps which are used later. In Section 3 we prove the main theorem and we furnish more tractable conditions on the equilibrium bifunction which guarantee that our result holds true. Basic concepts ============== Let $\Phi:X\rightrightarrows Y$ be a set-valued map with $X$ and $Y$ two topological spaces. The graph of $\Phi$ is the set $${\operatorname{gph}}\Phi:=\{(x,y)\in X\times Y:y\in \Phi(x)\}$$ and the lower section of $\Phi$ at $y\in Y$ is $$\Phi^{-1}(y):=\{x\in X:y\in \Phi(x)\}.$$ The map $\Phi$ is said to be lower semicontinuous at $x$ if for each open set $\Omega$ such that $\Phi(x)\cap\Omega\ne\emptyset$ there exists a neighborhood $U_x$ of $x$ such that $$\Phi(x')\cap\Omega\ne\emptyset,\qquad\forall x'\in U_x.$$ Notice that a set-valued map with open graph has open lower sections and, in turn, if it has open lower sections then it is lower semicontinuous. A fixed point of a function $\varphi:X\rightarrow X$ is a point $x\in X$ satisfying $\varphi(x)=x$. A fixed point of a set-valued map $\Phi:X\rightrightarrows X$ is a point $x\in X$ satisfying $x\in\Phi(x)$. The set of the fixed point of $\Phi$ is denoted by ${\operatorname{fix}}\Phi$. One of the most famous fixed point theorems for continuous functions was proven by Brouwer and it has been used across numerous fields of mathematics (see [@Bo85]). .3truecm [**Brouwer fixed point Theorem.**]{} * Every continuous function $\varphi$ from a nonempty convex compact subset $C\subseteq{{\mathbb R}}^n$ to $C$ itself has a fixed point.* .3truecm A selection of a set-valued map $\Phi:X\rightrightarrows Y$ is a function $\varphi:X\rightarrow Y$ that satisfies $\varphi(x)\in\Phi(x)$ for each $x\in X$. The Axiom of Choice guarantees that set-valued maps with nonempty values always admit selections, but they may have no additional useful properties. Michael [@Mi56] proved a series of theorems on the existence of continuous selections that assume the condition of lower semicontinuity of set-valued maps. We present here only one result [@Mi56 Theorem 3.1$^{\prime\prime\prime}$ (b)]. .3truecm [**Michael selection Theorem.**]{} * Every lower semicontinuous set-valued map $\Phi$ from a metric space to ${{\mathbb R}}^n$ with nonempty convex values admits a continuous selection.* .3truecm The Michael selection Theorem holds more in general when the domain of $\Phi$ is a perfectly normal space. Collecting the Brouwer fixed point Theorem and the Michael selection Theorem, we deduce the following fixed point result for lower semicontinuous set-valued maps. \[cor:fixed point\] Every lower semicontinuous set-valued map $\Phi$ from a nonempty convex compact subset $C\subseteq{{\mathbb R}}^n$ to $C$ itself with nonempty convex values has a fixed point. Notice that, unlike the famous Kakutani fixed point Theorem (see [@Bo85]) in which the closedness of ${\operatorname{gph}}\Phi$ is required, in Corollary \[cor:fixed point\] the lower semicontinuity of the set-valued map is needed. No relation exists between the two results as the following example shows. The set-valued map $\Phi:[0,3]\rightrightarrows [0,3]$ $$\Phi(x):=\left\{\begin{array}{ll} \{1\} & \mbox{ if } 0\leq x\leq 1\\ (1,2) & \mbox{ if } 1<x<2\\ \{2\} & \mbox{ if } 2\leq x\leq 3 \end{array}\right.$$ is lower semicontinuous and the nonemptiness of ${\operatorname{fix}}\Phi$ is guaranteed by Corollary \[cor:fixed point\]. Notice that ${\operatorname{fix}}\Phi=[1,2]$. Nevertheless the Kakutani fixed point Theorem does not apply since ${\operatorname{gph}}\Phi$ is not closed. On the converse, the set-valued map $\Phi:[0,3]\rightrightarrows [0,3]$ $$\Phi(x):=\left\{\begin{array}{ll} \{1\} & \mbox{ if } 0\leq x<1\\ {}[1,2] & \mbox{ if } 1\leq x\leq 2\\ \{2\} & \mbox{ if } 2<x\leq 3 \end{array}\right.$$ has closed graph and the nonemptiness of ${\operatorname{fix}}\Phi$ is guaranteed by the Kakutani fixed point Theorem. Again ${\operatorname{fix}}\Phi=[1,2]$. Since $\Phi$ is not lower semicontinuous, Corollary \[cor:fixed point\] can not be applied. We conclude this section recalling some topological notations. Given two subsets $A\subseteq C\subseteq{{\mathbb R}}^n$ we denote by ${\operatorname{int}}_C A$ and ${\operatorname{cl}}_C A$ the interior and the closure of $A$ in the relative topology of $C$ while $\partial_C A$ indicates the boundary of $A$ in $C$, i.e. $$\partial_C A:={\operatorname{cl}}_C A\setminus {\operatorname{int}}_CA = {\operatorname{cl}}_C A\cap {\operatorname{cl}}_C (C\setminus A).$$ Lastly $C$ is connected if and only if the subsets of $C$ which are both open and closed in $C$ are $C$ itself and the empty set. Existence results ================= From now on, $C\subseteq {{\mathbb R}}^n$ is a nonempty convex compact set and $f:C\times C\rightarrow {{\mathbb R}}$ is an equilibrium bifunction, that is $f(x,x)=0$ for all $x\in C$. The equilibrium problem is defined as follows: $$\label{eq:ep} \mbox{find } x\in C \mbox{ such that } f(x,y)\ge 0\mbox{ for all } y\in C.$$ Equilibrium problem has been traditionally studied assuming that $f$ is upper semicontinuous in its first argument and quasiconvex in its second one. Under such assumptions, the issue of sufficient conditions for existence of solutions of (\[eq:ep\]) was the starting point in the study of the problem. Ky Fan [@Fa72] proved a famous minimax inequality assuming compactness of $C$ and his result holds in a Hausdorff topological vector space. However, there is the possibility to slightly relax the continuity condition when the vector space is finite dimensional. The set-valued map $$\label{eq:mapF} F(x):=\{y\in C:f(x,y)<0\}$$ defined on $C$ plays a fundamental role in the formulation of our results. Clearly $F$ has open lower sections and convex values under the Ky Fan assumptions on the bifunction $f$, that is upper semicontinuity with respect to the first variable and quasiconvexity with respect to the second one. The fact that $F$ has open lower sections implies that $F$ is lower semicontinuous. If $F$ had nonempty values, Corollary \[cor:fixed point\] guarantees the existence of a fixed point of $F$. This contradicts the fact that $f(x,x)\geq 0$. Therefore there exists at least one $\bar x$ such that $F(\bar x)=\emptyset$, that is a solution of the equilibrium problem (\[eq:ep\]). The following result holds. .3truecm [**Ky Fan minimax inequality.**]{} *A solution of (\[eq:ep\]) exists whenever the set-valued map $F$ given in (\[eq:mapF\]) is lower semicontinuous and convex-valued.* .3truecm After describing this auxiliary result, we focus on the main aim of the paper. A quasiequilibrium problem is an equilibrium problem in which the constraint set is subject to modifications depending on the considered point. This format reads $$\label{eq:qep} \mbox{find } x\in K(x) \mbox{ such that } f(x,y)\ge 0\mbox{ for all } y\in K(x),$$ where $K:C\rightrightarrows C$ is a set-valued map. Our first existence result is the following. \[th:existenceQEP\] Assume that $K$ is lower semicontinuous with nonempty convex values and ${\operatorname{fix}}K$ is closed. Moreover suppose that 1. $F$ is convex-valued on ${\operatorname{fix}}K$, 2. $F$ is lower semicontinuous on ${\operatorname{fix}}K$, 3. $F\cap K$ is lower semicontinuous on $\partial_C{\operatorname{fix}}K$, where $F$ is the set-valued map given in (\[eq:mapF\]). Then the quasiequilibrium problem (\[eq:qep\]) has a solution. [**Proof.**]{} Corollary \[cor:fixed point\] ensures the nonemptiness of ${\operatorname{fix}}K$. If ${\operatorname{fix}}K=C$, the existence of solutions to the quasiequilibrium problem descends from the above mentioned Ky Fan minimax inequality. Otherwise, since ${\operatorname{fix}}K$ is closed and $\partial_C{\operatorname{fix}}K ={\operatorname{fix}}K\setminus {\operatorname{int}}_C{\operatorname{fix}}K$, the emptiness of $\partial_C{\operatorname{fix}}K$ it would be equivalent to ${\operatorname{fix}}K={\operatorname{int}}_C{\operatorname{fix}}K$. Therefore ${\operatorname{fix}}K$ would be both open and closed in $C$. Since every convex set is connected, the only nonempty open and closed subset of $C$ is $C$ itself and this contradicts the fact that ${\operatorname{fix}}K\ne C$. Assume that ${\operatorname{int}}_C{\operatorname{fix}}K\ne\emptyset$ (the case ${\operatorname{int}}_C{\operatorname{fix}}K=\emptyset$ is similar and will be shortly discussed at the end of the proof) and define $G:C\rightrightarrows C$ as follows $$G(x):=\left\{\begin{array}{ll} F(x) & \mbox{ if } x\in {\operatorname{int}}_C{\operatorname{fix}}K\\ F(x)\cap K(x) & \mbox{ if } x\in \partial_C{\operatorname{fix}}K\\ K(x) & \mbox{ if } x\notin {\operatorname{fix}}K \end{array}\right.$$ The proof is complete if we can show that $G(x)=\emptyset$ for some $x\in C$. Indeed, since $K$ has nonempty values, then $x\in{\operatorname{fix}}K$ and two cases are possible. If $x\in\partial_C{\operatorname{fix}}K$, then it solves (\[eq:qep\]); if $x\in{\operatorname{int}}_C{\operatorname{fix}}K$ then it solves (\[eq:ep\]). In both cases the quasiequilibrium problem has a solution. Assume by contradiction that $G$ has nonempty values. Next step is to prove the lower semicontinuity of $G$. Fix $x\in C$ and an open set $\Omega\subseteq{{\mathbb R}}^n$ such that $G(x)\cap\Omega\cap C\ne\emptyset$. We distinguish three cases. 1. If $x\in{\operatorname{int}}_C{\operatorname{fix}}K$, from the lower semicontinuity of $F$ there exists a neighborhood $U'_x$ such that $$F(x')\cap\Omega\cap C\ne\emptyset,\qquad\forall x'\in U'_x\cap{\operatorname{fix}}K$$ which implies $$G(x')\cap\Omega\cap C\ne\emptyset,\qquad\forall x'\in U'_x\cap{\operatorname{int}}_C{\operatorname{fix}}K.$$ Since $U'_x\cap{\operatorname{int}}_C{\operatorname{fix}}K$ is open in $C$, then $G$ is lower semicontinuous at $x$. 2. If $x\in\partial_C{\operatorname{fix}}K=\partial_C(C\setminus{\operatorname{fix}}K)$ from the lower semicontinuity of $F$, $K$ and $F\cap K$ there exist neighborhoods $U'_x$, $U''_x$ and $U'''_x$ such that $$\begin{aligned} F(x')\cap\Omega\cap C\ne\emptyset, & \qquad & \forall x'\in U'_x\cap{\operatorname{fix}}K,\\ K(x')\cap\Omega\cap C\ne\emptyset, & \qquad & \forall x'\in U''_x\cap C,\\ F(x')\cap K(x')\cap\Omega\cap C\ne\emptyset, & \qquad & \forall x'\in U'''_x\cap \partial_C{\operatorname{fix}}K.\end{aligned}$$ Then $$G(x')\cap\Omega\cap C\ne\emptyset,\qquad\forall x'\in U'_x\cap U''_x\cap U'''_x\cap C,$$ i.e. $G$ is lower semicontinuous at $x$. 3. Finally, if $x\notin{\operatorname{fix}}K$, from the lower semicontinuity of $K$ there exists a neighborhood $U'_x$ such that $$K(x')\cap\Omega\cap C\ne\emptyset,\qquad\forall x'\in U'_x\cap C.$$ Then $$G(x')\cap\Omega\cap C\ne\emptyset,\qquad\forall x'\in U'_x\cap (C\setminus{\operatorname{fix}}K).$$ Since $U'_x\cap(C\setminus{\operatorname{fix}}K)$ is open in $C$, then $G$ is lower semicontinuous at $x$. Since by assumption $G$ is also convex-valued, then all the conditions of Corollary \[cor:fixed point\] are satisfied and there exists $x\in{\operatorname{fix}}G$. Clearly $x\in{\operatorname{fix}}K$ and therefore $x\in{\operatorname{fix}}F$ which implies $f(x,x)<0$ and contradicts the assumption on $f$. The issue of ${\operatorname{int}}_C{\operatorname{fix}}K=\emptyset$ remains to be seen. In this case $\partial_C{\operatorname{fix}}K={\operatorname{cl}}_C{\operatorname{fix}}K={\operatorname{fix}}K$ and $G$ assumes the following form $$G(x):=\left\{\begin{array}{ll} F(x)\cap K(x) & \mbox{ if } x\in{\operatorname{fix}}K\\ K(x) & \mbox{ if } x\notin {\operatorname{fix}}K \end{array}\right.$$ The result is obtained by adapting the argument used before. It is clear from the proof that the assertion remains valid if $f(x,x)=0$ on $C\times C$ is replaced by the weaker $f(x,x)\ge0$ for all $x\in{\operatorname{fix}}K$. \[re:alternative\] The proof of Theorem \[th:existenceQEP\] allows to establish that a solution of (\[eq:qep\]) belongs to $$\partial_C{\operatorname{fix}}K\cup \{x\in {\operatorname{int}}_C{\operatorname{fix}}K: x \mbox{ solves } (\ref{eq:ep})\}.$$ In particular if (\[eq:ep\]) has no solution then Theorem \[th:existenceQEP\] ensures that a solution of (\[eq:qep\]) lies on the boundary of ${\operatorname{fix}}K$. \[re:fan\] By specializing to $K(x):=C$, for all $x\in C$, Theorem \[th:existenceQEP\] becomes the Ky Fan minimax inequality. Indeed ${\operatorname{fix}}K=C$ and conditions i) and ii) coincide with the assumptions in Ky Fan minimax inequality. Instead, since $\partial_C{\operatorname{fix}}K=\emptyset$, condition iii) is trivially satisfied. Theorem \[th:existenceQEP\] is strongly related to [@Cu95 Lemma 3.1]. The two sets of conditions differ only in that the lower semicontinuity of $F\cap K$ on the whole space $C$ assumed in [@Cu95 Lemma 3.1] is here replaced by the lower semicontinuity of $F$ on ${\operatorname{fix}}K$ and the lower semicontinuity of $F\cap K$ on $\partial_C {\operatorname{fix}}K$. We provide an example in which the results are not comparable to each other. Let $C:=[0,1]$ and $$f(x,y):=\left\{\begin{array}{ll} -1 & \mbox{ if } x=0 \mbox{ and } y\in(0,1]\\ 0 & \mbox{ otherwise} \end{array}\right.$$ If $K(x):=\{x\}$, for all $x\in [0,1]$, then $F\cap K=\emptyset$ is trivially lower semicontinuous and the assumptions of [@Cu95 Lemma 3.1] are satisfied. Instead $F$ is not lower semicontinuous at $0\in{\operatorname{fix}}K=[0,1]$. On the other hand if $K(x):=\{1-x\}$, for all $x\in [0,1]$, then ${\operatorname{fix}}K=\{1/2\}$, the assumptions of Theorem \[th:existenceQEP\] are trivially satisfied, but $F\cap K$ is not lower semicontinuous at $0$. It would be desirable to find more tractable conditions on $f$, disjoint from the ones assumed on $K$, which guarantee that all the assumptions i), ii) and iii) of Theorem \[th:existenceQEP\] are satisfied. Clearly the convexity of $F(x)$ can be deduced from the quasiconvexity of $f(x,\cdot)$ for all $x\in{\operatorname{fix}}K$. While the upper semicontinuity of $f(\cdot,y)$ on ${\operatorname{fix}}K$ implies that $F^{-1}(y)$ is open on ${\operatorname{fix}}K$ and hence $F$ is lower semicontinuous on ${\operatorname{fix}}K$. The last part of this section is devoted to furnish sufficient conditions for assumption iii), i.e. which guarantee the lower semicontinuity of the set-valued map $F\cap K$ on $\partial_C{\operatorname{fix}}K$. We propose two approaches. The former one consists in exploiting the following result in [@Pa91]. \[pr:lsc intersection\] Let $\Phi_1,\Phi_2:X\rightrightarrows Y$ be set-valued maps between two topological spaces. Assume that ${\operatorname{gph}}\Phi_1$ is open on $X\times Y$ and $\Phi_2$ is lower semicontinuous. Then $\Phi_1\cap\Phi_2$ is lower semicontinuous. Since $K$ is assumed to be lower semicontinuous, we investigate which assumptions ensure the open graph of $F$ given in (\[eq:mapF\]), that is the openness of the set $$\label{eq:open} \{(x,y)\in \partial_C{\operatorname{fix}}K\times C:f(x,y)< 0\}.$$ Hence, Theorem \[th:existenceQEP\] still works by using this condition instead of iii). It is interesting to compare this fact with [@Cu97 Theorem 2.1] where the openness of the set $\{(x,y)\in C\times C:f(x,y)< 0\}$ is required instead of the openness of (\[eq:open\]) and the lower semicontinuity of $F$ on ${\operatorname{fix}}K$. One should not overlook the fact that even though the results are formally similarly formulated, unlike our result, [@Cu97 Theorem 2.1] does not reduce to Ky Fan minimax inequality when $K(x)=C$, for all $x\in C$. An open graph result is [@Zh95 Proposition 2] which affirms that if $X$ is a topological space and $\Phi:X\rightrightarrows {{\mathbb R}}^n$ is a set-valued map with convex values, then $\Phi$ has open graph in $X\times{{\mathbb R}}^n$ if and only if $\Phi$ is lower semicontinuous and open valued. This fact has been used to establish the existence of continuous selections, maximal elements, and fixed points of correspondences in various economic applications. Up to translations, this result also holds when the codomain of $\Phi$ is an affine subset of ${{\mathbb R}}^n$ [@Yu98 Theorem 1.12]. We recall that an affine set of ${{\mathbb R}}^n$ is the translation of a vector subspace. Moreover, the affine hull of a set $C$ in ${{\mathbb R}}^n$, which is denoted by ${\operatorname{aff}}C$, is the smallest affine set containing $C$, or equivalently, the intersection of all affine sets containing $C$. \[th:sufficientconditions1\] Let $A\supseteq C$ be an open set on ${\operatorname{aff}}C$ and $\hat{f}:C\times A\rightarrow {{\mathbb R}}$ be a bifunction such that $\hat{f}(x,y)=f(x,y)$ for all $(x,y)\in C\times C$. Denote by $\hat{F}$ the set-valued map $$\hat{F}(x):=\{y\in A:\hat{f}(x,y)<0\}$$ defined on $C$ and assume that $K$ is lower semicontinuous with nonempty convex values and ${\operatorname{fix}}K$ is closed. Moreover suppose that 1. $\hat{F}$ is convex-valued on ${\operatorname{fix}}K$, 2. $\hat{F}$ has open lower sections on ${\operatorname{fix}}K$, 3. $\hat{F}(x)$ is open on ${\operatorname{aff}}C$ for all $x\in\partial_C{\operatorname{fix}}K$. Then the quasiequilibrium problem (\[eq:qep\]) has a solution. [**Proof.**]{} We have to show that all the assumptions of Theorem \[th:existenceQEP\] are fulfilled. Since the set-valued map $F$ given in (\[eq:mapF\]) can be expressed as $\hat F\cap C$, i) implies that $F$ is convex-valued on ${\operatorname{fix}}K$ and ii) implies that $F$ is open lower section on ${\operatorname{fix}}K$. In particular $F$ is lower semicontinuos on ${\operatorname{fix}}K$. Furthermore assumption iii) allows to apply [@Yu98 Theorem 1.12] which ensures that ${\operatorname{gph}}\hat F$ is open on $\partial_C{\operatorname{fix}}K\times {\operatorname{aff}}C$. Hence ${\operatorname{gph}}F={\operatorname{gph}}\hat F\cap (\partial_C{\operatorname{fix}}K\times C)$ is open on $\partial_C{\operatorname{fix}}K\times C$ and Proposition \[pr:lsc intersection\] guarantees that the intersection map $F\cap K$ is lower semicontinuous on $\partial_C{\operatorname{fix}}K$. The open graph result [@Zh95 Proposition 2] no longer holds when ${{\mathbb R}}^n$ (or an affine space) is replaced with an infinite dimensional Hilbert space [@Ba12]. However if $C\subset{{\mathbb R}}^n$ is a polytope, that is the convex hull of a finite set, then every $\Phi:X\rightrightarrows C$ with open lower sections and convex open values has open graph [@Bo85 Proposition 11.14]. This fact can be used for proving our next result. \[th:sufficientconditions2\] Assume that $C$ is a polytope and $K$ is lower semicontinuous with nonempty convex values and ${\operatorname{fix}}K$ is closed. Moreover suppose that 1. $F$ is convex-valued on ${\operatorname{fix}}K$, 2. $F$ has open lower sections on ${\operatorname{fix}}K$, 3. ì$F(x)$ is open on $C$ for all $x\in \partial_C {\operatorname{fix}}K$, where $F$ is the set-valued map given in (\[eq:mapF\]). Then the quasiequilibrium problem (\[eq:qep\]) has a solution. [**Proof.**]{} The set-valued map $F$ has open lower sections, convex and open values. Then its graph is open on $\partial_C {\operatorname{fix}}K\times C$ [@Bo85 Proposition 11.14] and the lower semicontinuity of $F\cap K$ follows from Proposition \[pr:lsc intersection\]. Notice that the lower semicontinuity condition ii) assumed in Theorem \[th:existenceQEP\] has been replaced in the last two results by the requirement that the lower sections are open. This is due to two different reasons. In the proof of Theorem \[th:sufficientconditions1\], in order to apply [@Yu98 Theorem 1.12] and get that ${\operatorname{gph}}\hat F$ is open, it would be enough to require the lower semicontinuity of $\hat F$. However such an assumption would not guarantee the lower semicontinuity of $F=\hat F\cap C$ which is assumption ii) in Theorem \[th:existenceQEP\]. On the other hand, assumption ii) in Theorem \[th:sufficientconditions2\] is necessary to get the openness of ${\operatorname{gph}}F$ as a consequence of [@Bo85 Proposition 11.14]. The next example shows that a set-valued map $\Phi$ acting from a topological vector space to a polytope $C$ may not have open graph and [@Bo85 Proposition 11.14] fails even if it is lower semicontinuous with convex and open values. Let $C:=\{(x,y)\in {{\mathbb R}}^2:|x|+|y|\leq 1\}$ be a closed convex set in ${{\mathbb R}}^2$. The set-valued map $\Phi:[0,1]\rightarrow C$ defined by $$\Phi(t):=\left\{\begin{array}{ll} C\setminus \{(x,y):x+y=1\} & \mbox{ if } t>0\\ C & \mbox{ if } t=0 \end{array}\right.$$ is lower semicontinuous with convex open values in $C$ but it has not open lower sections since $\phi^{-1}(0,1)=\{0\}$. Nevertheless ${\operatorname{gph}}\Phi$ is not open in $[0,1]\times C$ since the sequence $ \{(n^{-1},1-n^{-1},n^{-1})\}\in [0,1]\times C$ does not belong to ${\operatorname{gph}}\Phi$ but its limit $(0,1,0)\in{\operatorname{gph}}\Phi$. We answer in the negative the question posed in [@BePaRa76] where the authors affirm that they do not know whether [@Bo85 Proposition 11.14] can be generalized to the case where $C$ is an arbitrary convex subset of ${{\mathbb R}}^n$. This also explains why we need to extend the domain of $f(x,\cdot)$ from $C$ to an open subset of ${\operatorname{aff}}C$ in Theorem \[th:sufficientconditions1\]. \[ex:graphnotopen\] Let $C\subseteq {{\mathbb R}}^2$ be the closed unit ball. The set-valued map $\Phi:[0,1]\rightrightarrows C$ defined by $$\Phi(x):=\left\{\begin{array}{ll} C\setminus \{(\cos x,\sin x)\} & \mbox{ if } x>0\\ C & \mbox{ if } x=0 \end{array}\right.$$ has open lower sections and convex open values in $C$. Nevertheless ${\operatorname{gph}}\Phi$ is not open in $[0,1]\times C$. Indeed $(1,0)\in\Phi(0)$ and there is no neighborhood $U$ of $(1,0)$ such that $U\cap C\subseteq\Phi(x)$ for $x$ small enough. A second possible approach for the lower semicontinuity of $F\cap K$ could be to show the nonemptiness of the intersection between the interior of $F$ and $K$. Indeed [@BoGeMyOb84 Corollary 1.3.10] affirms that the set-valued map $\Phi_1\cap\Phi_2$ is lower semicontinuous on the topological space $X$ provided that $\Phi_1,\Phi_2:X\rightrightarrows C$ are convex-valued, lower semicontinuous set-valued maps and $$\label{eq:inte} \Phi_1(x)\cap\Phi_2(x)\neq \emptyset\quad\Rightarrow\quad\Phi_1(x)\cap{\operatorname{int}}\Phi_2(x)\neq \emptyset.$$ The following example shows that such result could not be guaranteed (as erroneously stated in [@Yu98 Theorem 1.13]) if the interior is replaced by the relative interior in condition (\[eq:inte\]). Given a set $C\subseteq{{\mathbb R}}^n$, we denote by ${\operatorname{ri}}C$ the relative interior of $C$, namely, ${\operatorname{ri}}C={\operatorname{int}}_{{\operatorname{aff}}C}C$. Let $C\subseteq {{\mathbb R}}^2$ be the closed unit ball and $\Phi_1:[0,1]\rightrightarrows C$ be defined as in Example \[ex:graphnotopen\]. Consider $\Phi_2:[0,1]\rightrightarrows C$ defined by $$\Phi_2(x):=\{(\cos x,\sin x)\}\qquad \forall x\in[0,1].$$ Then $\Phi_2$ is a continuous single-valued map and $\Phi_1$ is convex-valued with open lower sections. Furthermore $$\Phi_1(x)\cap\Phi_2(x)=\left\{\begin{array}{ll} \emptyset & \mbox{ if } x>0\\ \{(1,0)\} & \mbox{ if } x=0 \end{array}\right.$$ and $\Phi_1(0)\cap {\operatorname{ri}}\Phi_2(0)=C\cap \{(1,0)\}=\{(1,0)\}$. Nevertheless $\Phi_1\cap\Phi_2$ is not lower semicontinuous at $0$. Notice that $\Phi_1(x)$ is even open on $C$, for all $x\in[0,1]$. The following is a correct version of [@Yu98 Theorem 1.13]. \[pr:lsc intersection1\] Let $X$ be a topological space, $C\subseteq{{\mathbb R}}^n$ and $\Phi_1,\Phi_2:X\rightrightarrows C$ be lower semicontinuous and convex-valued. Moreover, for all $x\in X$ assume that ${\operatorname{aff}}\Phi_2(x)={\operatorname{aff}}C$ and $$\Phi_1(x)\cap\Phi_2(x)\neq \emptyset\quad \Rightarrow\quad\Phi_1(x)\cap{\operatorname{ri}}\Phi_2(x)\neq \emptyset$$ then $\Phi_1\cap \Phi_2$ is lower semicontinuous. [**Proof.**]{} By definition, up to isomorphism, there exists $m\leq n$ such that ${\operatorname{aff}}C=x_0+{{\mathbb R}}^m$, where $x_0\in C$ is arbitrarily fixed. Define $\hat\Phi_i:X\rightrightarrows {{\mathbb R}}^m$ by $\hat\Phi_i:=\Phi_i-x_0$, $i=1,2$. Then $\hat\Phi_1$ and $\hat\Phi_2$ are lower semicontinuous and convex-valued. Furthermore, since ${\operatorname{aff}}\Phi_2(x)={\operatorname{aff}}C$, then ${\operatorname{ri}}\Phi_2(x)=x_0+{\operatorname{int}}\hat\Phi_2(x)$ and $\hat\Phi_1(x)\cap {\operatorname{int}}\hat\Phi_2(x)\neq \emptyset$ whenever $\hat\Phi_1(x)\cap \hat\Phi_2(x)\neq \emptyset$. By [@BoGeMyOb84 Corollary 1.3.10] it follows that $\hat\Phi_1\cap \hat\Phi_2$ is lower semicontinuous. This means in turn that $\Phi_1\cap \Phi_2$ is lower semicontinuous. Now we are in position to prove our last existence result. \[th:sufficientconditions3\] Assume that $K$ is lower semicontinuous with nonempty convex values and ${\operatorname{fix}}K$ is closed. Moreover suppose that 1. $F$ is convex-valued on ${\operatorname{fix}}K$, 2. $F$ is lower semicontinuous on ${\operatorname{fix}}K$, 3. ${\operatorname{aff}}K(x)={\operatorname{aff}}C$, for all $x\in \partial_C {\operatorname{fix}}K$, 4. $F(x)$ is open on $C$, for all $x\in \partial_C {\operatorname{fix}}K$, where $F$ is the set-valued map given in (\[eq:mapF\]). Then the quasiequilibrium problem (\[eq:qep\]) has a solution. [**Proof.**]{} It is enough to show that assumption iii) of Theorem \[th:existenceQEP\] holds, i.e. $F\cap K$ is lower semicontinuous on $\partial_C{\operatorname{fix}}K$. Let $x\in\partial_C{\operatorname{fix}}K$ be fixed and assume that $F(x)\cap K(x)\neq\emptyset$ (otherwise the intersection is trivially lower semicontinuous at $x$). By assumption there exists an open set $\Omega\subseteq {{\mathbb R}}^n$ such that $F(x)=\Omega\cap C$. Then $$\emptyset\neq F(x)\cap K(x)=\Omega\cap C\cap K(x)=\Omega\cap K(x).$$ From [@Ro70 Corollary 6.3.2] we get $$\emptyset\neq \Omega\cap {\operatorname{ri}}K(x)=F(x)\cap {\operatorname{ri}}K(x)$$ The lower semicontinuity of $F\cap K$ at $x$ follows from Proposition \[pr:lsc intersection1\]. Now we make a comparison with an analogous result in [@Cu95]. The assumptions of Theorem \[th:sufficientconditions3\] are the same as those of [@Cu95 Theorem 3.2] except that conditions iii) and iv) must be verified for all $x\in \partial_C{\operatorname{fix}}K$ instead of for all $x\in C$. Thus, Theorem \[th:sufficientconditions3\] is clearly more general and, unlike [@Cu95 Theorem 3.2], it reduces to Ky Fan minimax inequality when the constraint set-valued map $K$ is equal to $C$. Conclusions =========== In this paper existence results for the solution of finite dimensional quasiequilibrium problems are obtained by using a Michael selection result for lower semicontinuous set-valued maps. The peculiarity of our results, which make them different from other results in the literature to the best of knowledge of the authors, is the fact that they reduce to Ky Fan minimax inequality when the constraint map is constant. Moreover we provide information regarding the position of a solution. In fact either it is a fixed point of the constraint set-valued map which solves an equilibrium problem or it lies in the boundary of the fixed points set. To know this property seems promising for the construction of solution methods. Future works could be devoted to exploit such result to propose computational techniques for solving quasiequilibrium problems. Another possible advance consists in studying conditions which permit to replace the compactness of the domain with suitable coercivity conditions on the equilibrium bifunction. [00]{} Fan K.: A minimax inequality and applications. In: Shisha O. (ed.): Inequalities III, pp. 103–113. Academic Press, New York (1972) Bigi G., Castellani M., Pappalardo M., Passacantando M.: Existence and solution methods for equilibria. European J. Oper. Res. 227, 1–11 (2013) Bensoussan A., Goursat M., Lions J.L.: Contrôle impulsionnel et inéquations quasi-variationnelles stationnaires. C.R. Acad. Sci. Paris Sér. A 276, 1279–1284 (1973) Mosco U.: Implicit variational problems and quasi variational inequalities. In: Lecture Notes in Math., vol. 543, pp. 83–156. Springer-Verlag, Berlin (1976) Alleche B., Rădulescu, V.D.: Solutions and approximate solutions of quasi-equilibrium problems in Banach spaces. J. Optim. Theory Appl. 170, 629–649 (2016) Aubin J.P.: Optima and equilibria. Springer-Verlag, Berlin (1993) Aussel D., Cotrina J., Iusem A.: Existence results for quasi-equilibrium problems. J. Convex Anal. 24, 55–66 (2017) Castellani M., Giuli M.: An existence result for quasiequilibrium problems in separable Banach spaces. J. Math. Anal. Appl. 425, 85–95 (2015) Castellani M., Giuli M.: Approximate solutions of quasiequilibrium problems in Banach spaces. J. Global Optim. 64, 615–620 (2016) Cubiotti P.: Existence of solutions for lower semicontinuous quasiequilibrium problems. Comput. Math. Appl. 30, 11–22 (1995) Cubiotti P.: Existence of Nash equilibria for generalized games without upper semicontinuity. Internat. J. Game Theory 26, 267–273 (1997) Michael E.: Continuous selections. I. Ann. of Math. 63, 361–382 (1956) Border K.C.: Fixed point theorems with applications to economics and game theory. Cambridge University Press, Cambridge (1985) Papageorgiou N.S.: On the existence of $\psi$-minimal viable solutions for a class of differential inclusions. Arch. Math. 27, 175–182 (1991) Zhou J.: On the existence of equilibrium for abstract economies, J. Math. Anal. Appl. 193, 839–858 (1995) Yuan G.X.-Z.: The study of minimax inequalities and applications to economies and variational inequalities. Memoirs of the American Mathematical Society, vol. 132. Providence, Rhode Island (1998) Bagh A.: Lower hemi-continuity, open sections, and convexity: counter examples in infinite dimensional spaces. Theoret. Econom. Lett. 2, 121–124 (2012) Bergstrom T.C., Parks R.P., Rader T.: Preferences which have open graphs. J. Math. Econom. 3, 265–268 (1976) Borisovich Y., Gel’man B.D., Myshkis A.D., Obukhovskii V.V.: Multivalued mappings. J. Soviet Math. 24, 719–791 (1984) Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

--- abstract: 'We observed atmospheric gamma-rays around 10 GeV at balloon altitudes (15$\sim$25 km) and at a mountain (2770 m a.s.l). The observed results were compared with Monte Carlo calculations to find that an interaction model (Lund Fritiof1.6) used in an old neutrino flux calculation was not good enough for describing the observed values. In stead, we found that two other nuclear interaction models, Lund Fritiof7.02 and dpmjet3.03, gave much better agreement with the observations. Our data will serve for examining nuclear interaction models and for deriving a reliable absolute atmospheric neutrino flux in the GeV region.' author: - 'K. Kasahara' - 'E. Mochizuki' - 'S. Torii' - 'T. Tamura' - 'N. Tateyama' - 'K. Yoshida' - 'T. Yamagami' - 'Y. Saito' - 'J. Nishimura' - 'H. Murakami' - 'T. Kobayashi' - 'Y. Komori' - 'M.Honda' - 'T. Ohuchi' - 'S. Midorikawa' - 'T. Yuda' bibliography: - 'betsgamma.bib' title: 'Atmospheric gamma-ray observation with the BETS detector for calibrating atmospheric neutrino flux calculations' --- Introduction ============ The discovery of evidence for neutrino oscillation by the Super Kamiokande group[@skoscillation] is based on the comparison of the observed atmospheric neutrino flux with calculated values. Although the conclusion is so derived that it would not be upset by the uncertainty of the absolute flux value, it is desirable to obtain a reliable expected neutrino flux (under no oscillation assumption) for further detailed discussions. Two major sources of uncertainty in the atmospheric neutrino flux calculation are 1) the primary cosmic-ray spectrum and 2) the propagation of cosmic rays in the atmosphere, especially, modeling of the nuclear interaction. The absolute flux calculations so far made by various groups are expected to have uncertainty of $\sim$ 30 %[@GHreview]. The primary proton and He spectra recently measured with magnet spectrometers by the BESS [@bess1ry] and AMS[@ams1ry] groups agree very well and seem reliable. Therefore, we may take that the first problem mentioned above have now been almost settled at least up to 100 GeV/n. This means that if we have a reliable atmospheric cosmic-ray flux data, we may compare it with a calculation which uses such primaries and test the validity of nuclear interaction models. For such an atmospheric cosmic-ray component, one may first raise the muon and actually some new observations have been or being tried[@capricemuon1; @capricemuon2; @bessmuonnori]. As a secondary cosmic-ray component, we focused on gamma-rays which are easy to measure with our detector. A good model should be able to explain muons and gamma-rays simultaneously. Muons are important since they are directly coupled with neutrinos, but the flux is affected somehow by the structure of the atmosphere which is usually not well known. Compared to muons, the flux of gamma-rays is substantially lower but is almost insensitive to the atmospheric structure and depends only on the total thickness to the observation height. In 1998, we performed first gamma-ray observation with our detector at Mt. Norikura (2770m a.s.l) in Japan, and also made subsequent two successful observations at balloon altitudes (15 $\sim 25$ km) in 1999 and 2000. In the present paper, we report the final results of these observations and consequences. The Detector ============ For our observation, we upgraded the BETS (Balloon-born Electron Telescope with Scintillating fibers) detector which had been developed for the observation of cosmic primary electrons in the 10 GeV region. Its details before being upgraded for gamma-ray observation is in [@betsnim] and the electron observation result is in [@betselec]. The basic performance was tested at CERN using electron, proton and pion beams of 10 to 200 GeV[@betsnim; @betscern]. Although this was undertaken before the upgrading, we can essentially use that calibration for the current observeions partly with a help of Monte Carlo simulations. Figure \[det\] shows a schematic structure of the main body of BETS. The calorimeter has 7.1 r.l lead thickness and the cross-section is 28 cm $\times$ 28 cm. The whole detector system is contained in a pressure vessel made of thin aluminum. ![Schematic illustration of the main body of the detector. S1, S2 and S3 are 1 cm thick plastic scintillators used for trigger. Each fiber has 1mm diameter. Originally nuclear emulsion plates were placed on the upper scifi’s and also inserted between the upper thin lead plates for detailed investigation of tracking capability of scifi. They are kept in the present system to have the same structure at the calibration time. The inlaid cascade shows charged particle tracks by a simulation for a 30 GeV incident proton. \[det\]](detconfigwithshower.eps){width="92mm"} R.M.S energy resolution(%) 21, 18, 15 (for $\theta\sim 15^\circ$) ---------------------------------- ------------------------------------------- S$\Omega$(cm$^2$sr) 243, 240,218 (at $\sim$20 km) R.M.S angular resolution (deg) 2.3, 1.3, 1.0 (for $\theta\sim 15^\circ$) Total number of scifi’s 10080 Weght including electronics (kg) 230 Cross-section of the main body 28cm $\times $ 28cm Thickness (Pb radiation length) 7.1 : Basic characteristics of BETS\ (triple numbers in the table are for gamma-ray energy of 5, 10, and 30 GeV, respectively) \[basicchara\] The main feature of the BETS detector is that it is a tracking calorimeter; it contains a number of sheets consisting of 1 mm diameter scintillating fibers (scifi), many of which are sandwiched between lead plates. The total number of scifi’s are 10080. The sheets are grouped into two types; one is to serve for x and the other for y position measurement. Each of them is fed to an image intensifier which in turn is connected to a CCD. Thus, the two CCD output gives us an $x-y$ image of cascade shower development and enables us to discriminate gamma-rays, electrons from other (mainly hadronic) background showers. The proton rejection power against electron is $R\sim 2\times 10^3$ (i.e, one misidentification among $R$ protons) at 10 GeV[^1] The basic characteristics of the detector are summarized in Table \[basicchara\]. ![Image of cascade shower by a proton (120 GeV,left) and an electron(10 GeV, right) obtained at CERN. \[image\]](showerimage.eps){width="85mm"} In Fig.\[image\], we show examples of the CCD image of a cascade shower for a proton incident case and for an electron incident case. Figure \[anti\] illustrates the yearly change of anti-counters. In 1998 (Mt.Norikura observation), the main change was limited to the upgrading of trigger logic. In 1999, we added 4 side anti-counters (each 15 cm $\times$ 36 cm $\times$ 1.5 cm plastic scintillator. Nine optical fibers containing wave length shifter are embedded in each scintillator and connected to a Hamamatu H6780 PMT. ![Yearly change of the anti-counters. Left: 1998. No change from original BETS except for trigger logic. Middle: 1999. 1.5 cm thick plastic scintillator side anti-counters were added. Right: 2000. The whole top view was covered by a 1 cm thick plastic sintillator. \[anti\]](yearlychange.eps){width="85mm"} In 2000, we further added an anti-counter which covers the whole top view of the detector and also improved data acquisition speed. The top anti-counter is 38 cm $\times$ 38 cm $\times$ 1 cm plastic scintillator. We also embedded optical fibers; 8 in the $x$ and another 8 in the $y$ direction, all of which were fed to an H6780. Although we could remove background showers without the anti-counters, inclined particles (mainly protons) entering from the gap between top scintillator (S1) and the main body degrades the desired gamma-ray event rate. The addition of the top anti-counter greatly helped improve this rate. We emphasize that detection of gamma-rays is easier for us than that of electrons, since, for gamma-rays, we can utilize absence of incident charge. Observations ============ Table \[sumtab\] shows the summary of the observations. Observation Mt.Norikura(1998) --------------------- -------------------- ------ ------- ------- ------- ------- ------- ------- ------- ------- Period Aug.31$\sim$Sep.18 Altitude(km) 2.77 15.3 18.5 21.2 24.7 32.3 15.3 18.3 21.4 25.1 Depth(g/cm$^2$) 737 126 74.8 48.9 28.0 9.5 128 73 45.7 25.3 Obs. hour (s) $1.33\times 10^6$ 1260 1560 2100 4878 3120 1560 2160 4320 2320 Live time (s) $9.8\times 10^5$ 504 450 414 852 498 752 928 1805 789 Live time (%) 74.0 40.0 28.8 19.7 17.5 16.0 48.2 43.0 42.6 44.2 Triggered events $1.8\times 10^6 $ 9513 11288 13361 30439 16741 18808 25795 46675 17436 $\gamma$ events $4.7\times 10^4$ 700 650 611 848 345 1300 1485 2299 740 (%) 2.5 7.3 5.7 4.6 2.8 2.0 6.9 5.8 4.9 4.2 g-low trigger S1 $< 0.5$ condition (in mip). S2 $> 2.3$ S3 $> 1.7$ - Mt. Norikura observation. Our first gamma-ray observation was performed in 1998 at Mt.Norikura Observatory of Univ. of Tokyo, Japan (2770 m a.s.l, latitude 36.1$^\circ$N, longitude 137.55$^\circ$E, magnetic cutoff rigidity $\sim$ 11.5 GV). The atmospheric pressure during the observation is shown in Fig.\[noripress\]. The average atmospheric depth is 737 g/cm$^2$. ![Pressure change during Mt. Norikura observation. The last pressure drop is due to a typhoon. The average pressure is 723 hP (737 g/cm$^2$). \[noripress\]](norikurapressure.eps){width="85mm"} - Balloon flight We had two similar balloon filights in 1999 and 2000. Since the main outcome of the data is from the latter, we briefly describe it. A balloon of 43$\times 10^3$ m$^3$ was launched at 6:30 am, 5th June, 2000 from the Sanriku balloon center of the Institute of Space and Astronautical Science, Japan (latitude 39.2$^\circ$N, longitude 141.8$^\circ$E, magnetic cutoff rigidity $\sim$ 8.9 GV) and recovered with the help of the helicopter. at 17:59 on the sea not far from the center. The flight curve shown in Fig.\[flight\] confirms that we have good level flights at 4 different heights. As compared to the 1999 flight, this flight realized a smaller dead time and higher ratio of desired gamma-ray events. ![Flight curve of the 2000 observation. Pressure (upper) and altitude (lower) as a function of time. Each arrow shows the level flight region. The pressure change at around 15.3 km is rather rapid but the gamma-ray intensity is almost constant there and the change can be neglected. \[flight\]](flightcurve.eps){width="73mm"} Event trigger ------------- The basic event trigger condition is created by signals from the three plastic scintillators (S1, S2 and S3). We show the discrimination level in terms of the minimum ionizing particle number which is defined by the peak of the energy loss distribution of cosmic-ray muons passing both S1 and S3 with inclination less than 30 degrees. We prepare a multi-trigger system by which event trigger with different conditions is possible at the same time. The major two trigger modes are the g-low and g-high. The g-low is responsible for low energy gamma-rays and all anti-counters, when available, are used as veto counters. Its condition is listed in Table \[sumtab\]. High energy gamma-rays normally produce a lot of back splash particles which hit S1 and/or anti-counters, and thus the g-low trigger is suppressed. In such a case, i.e, if we have a large S3 signal, anti-counter veto is invalidated and the S1 threshold is relaxed (The g-high condition is S1$<3.0$, S2$>5.0$ and S3$>8.1$). The branch even point of the g-low and g-high mode efficiency is at $\sim $30 GeV. Since we deal with gamma-rays mostly below 30 GeV, and also to avoid complexity, we present results only by the g-low mode. Analysis ======== Event selection --------------- Among the triggered events, we selected gamma-ray candidates by imposing the following conditions: ![(left)Energy concentration distribution at 21.4 km. (right)the same by electrons at CERN []{data-label="conc"}](Econc.eps){width="8.5cm"} 1. The estimated shower axis passes S1 and S3. The axis position in S3 must be at least 2 cm apart from the edge of S3. 2. The estimated shower axis has a zenith angle less than 30 degrees. 3. The energy concentration (see below) must be greater than 0.7. According to a simulation, only neutrons could be a background against gamma-rays and the 3rd conditions above reduces the neutron contribution to a negligible level ($<1$%). The energy concentration is defined as the fraction of scintillating fiber light intensity within 5 mm from the shower axis. Figure \[conc\] shows the concentration of analysed events together with the result of CERN data. Hadrons make a distribution with a peak at around 0.5. We see that the contribution of hadrons in our observation is negligible. Energy Determination -------------------- The energy calibration was performed in 1996 at CERN using electrons with energy 10 $\sim $ 200 GeV[@betsnim; @betscern]. There is no direct calibration for gamma-rays, but, for the present detector thickness and energy range, a M.C simulation tells us that the calibration in 1996 can be used for gamma-rays, too[^2]. Therefore, for the 1998 and 1999 observations, energy is obtained as a function of the S3 output and zenith angle using the CERN calibration. In 2000, we made some change in the electronics so the CERN calibration could not be used directly. The effect by the change was absorbed by a M.C simulation of which the validity was verified by examining the 1998 and 1999 data. We used the sum of S2 and S3 outputs below 20 GeV since the energy resolution was found to be better than using S3 only. Figure \[eresol\] shows r.m.s energy resolution. ![R.m.s energy resolution. The resolution by S2+S3 or S3 only is shown. Different symbols indicate different incident angles. We used S2+S3 below 20 GeV for the year 2000 data. \[eresol\]](Eres.eps){width="8cm"} Correction of the gamma-ray intensity ------------------------------------- The gamma-ray vertical flux is obtained from the raw $dN/dE$ by dividing it by the live time of the detector and the effective $S\Omega$ (area $\times$ solid angle). The latter is obtained by a simulation[@someganu00]. It is dependent on the observation hight and energy. A typical value at 10 GeV is 240 cm$^2$sr (see Table\[basicchara\]). The energy spectrum is further corrected by the following factors which are not taken into account in the $S\Omega$ calculation. ![(upper)Multiple incidence rate. (lower) Correction factor for year 2000 due to spillover. The flux must be lowered. For Norikura, the factor below 20 GeV is larger by 1$\sim 3$ %. []{data-label="correc"}](turehuta.eps "fig:"){width="7cm"} ![(upper)Multiple incidence rate. (lower) Correction factor for year 2000 due to spillover. The flux must be lowered. For Norikura, the factor below 20 GeV is larger by 1$\sim 3$ %. []{data-label="correc"}](ER_hosei.eps "fig:"){width="7cm"} 1. Systematic bias in our estimation of the shower axis. We underestimate the zenith angle systematically and it leads to overestimation of the intensity about 4% for the balloon and 1.8 % for Mt.Norikura observations. 2. Multiple incidence of particles. A gamma-ray is sometimes accompanied by other charged particles and they enter the detector simultaneously (within 1 ns time difference in 99.9 % cases). They are a family of particles generated by one and the same primary particle[^3]. The charged particles fire the anti-counter and the g-low trigger is inhibited. In some case, multiple gamma-rays enter the detector simultaneously. The rate is smaller than the charged particle case. However, this is judged as a hadronic shower in most of cases. The multiple incidence leads to the underestimation of gamma-ray intensity. The portion of multiple incidence is shown in Fig.\[correc\] (upper). 3. Finite energy resolution. The rapidly falling energy spectrum leads to the spillover effect. This normally leads to the overestimation of flux (Fig.\[correc\], lower). Results and comparison with calculations ======================================== The flux values are summarized in Table \[flux\]. We put only the statistical errors in the flux values, since systematic errors coming from the uncertainty of the S$\Omega$ calculation, various cuts and flux corrections are expected to be order of a few percent and much smaller than the present statistical errors. [|l|l|l|l|l|l|l|l|l|l|]{}\ & & & &\ \ 5.48 & 2.42 $\pm$ 0.37 & 5.48 & 2.11 $\pm$ 0.39 & 5.47 & 2.11 $\pm$ 0.24 & 5.47 & 1.58 $\pm$ 0.25 & 5.47 & 0.49 $\pm$ 0.14\ 6.47 & 1.18 $\pm$ 0.27 & 6.47 & 1.10 $\pm$ 0.24 & 6.47 & 1.35 $\pm$ 0.21 & 6.47 & 0.82 $\pm$ 0.18 & 6.57 & 0.19 $\pm$ 0.09\ 7.47 & 0.89 $\pm$ 0.24 & 7.47 & 0.79 $\pm$ 0.21 & 7.47 & 0.82 $\pm$ 0.16 & 7.47 & 0.66 $\pm$ 0.16 & 7.47 & 0.24 $\pm$ 0.10\ 8.48 & 0.37 $\pm$ 0.15 & 8.48 & 0.92 $\pm$ 0.20 & 8.48 & 0.51 $\pm$ 0.13 & 8.48 & 0.49 $\pm$ 0.14 & 8.48 & 0.16 $\pm$ 0.08\ 9.48 & 0.54 $\pm$ 0.17 & 9.85 & 0.46 $\pm$ 0.11 & 9.48 & 0.50 $\pm$ 0.12 & 9.48 & 0.36 $\pm$ 0.12 & 9.48 & 0.16 $\pm$ 0.08\ 10.5 & 0.17 $\pm$ 0.10 & 11.5 & 0.35 $\pm$ 0.12 & 10.5 & 0.41 $\pm$ 0.09 & 10.5 & 0.34 $\pm$ 0.12 & 12.3 & 0.13 $\pm$ 0.037\ 12.1 & 0.28 $\pm$ 0.09 & 14.0 & 0.24 $\pm$ 0.06 & 11.8 & 0.23 $\pm$ 0.069 & 12.2 & 0.21 $\pm$ 0.054 & 17.0 & 0.032 $\pm$ 0.018\ 14.0 & 0.17 $\pm$ 0.05 & 18.3 & 0.072 $\pm$ 0.030 & 14.0 & 0.16 $\pm$ 0.030 & 14.0 & 0.076 $\pm$ 0.03 & 21.7 & 0.022$\pm$ 0.015\ 18.5 & 0.12 $\pm$ 0.04 & 26.8 & 0.040 $\pm$ 0.017 & 18.4 & 0.086 $\pm$ 0.023& 17.8 & 0.078 $\pm$ 0.029 & &\ 25.5 & 0.06 $\pm$ 0.02 & & & 27.1 & 0.026 $\pm$ 0.009& 21.7 & 0.064 $\pm$ 0.026 & &\ & & & & & & 26.8 & 0.024 $\pm$ 0.012 & &\ & & & & & & 36.0 & 0.012 $\pm$ 0.008 & &\ E(GeV) Flux ($10^{-4}/$m$^2\cdot$s$\cdot$sr$\cdot$GeV) -------- ------------------------------------------------- 5.48 274 $\pm$ 13 6.47 183 $\pm$ 11 7.47 133 $\pm$ 9 8.47 87.8 $\pm$ 7.5 9.47 86.5 $\pm$ 7.5 10.5 54.1 $\pm$ 5.9 11.5 46.6 $\pm$ 5.5 12.5 38.3 $\pm$ 5.0 13.5 32.6 $\pm$ 4.6 14.5 24.2 $\pm$ 4.0 15.5 25.7 $\pm$ 4.1 17.0 11.9 $\pm$ 2.0 19.0 15.3 $\pm$ 2.3 21.0 13.1 $\pm$ 2.1 23.0 5.80 $\pm$ 1.4 26.0 5.31 $\pm$ 0.95 30.0 3.00 $\pm$ 0.72 34.0 2.30 $\pm$ 0.64 38.0 1.07 $\pm$ 0.44 45.0 1.45 $\pm$ 0.32 55.0 0.52 $\pm$ 0.20 65.0 0.22 $\pm$ 0.13 75.0 0.30 $\pm$ 0.15 85.0 0.15 $\pm$ 0.10 : Flux values at Mt. Norikura\[noriflux\] The gamma-ray energy spectra thus obtained at balloon altitudes are shown in Fig.\[balspec\] together with the expected ones calculated by the Cosmos simulation code[@cosmos]. Except for 32.3 km altitude, we can disregard the small difference of the observation depths and we combine two flight data with statistical weight, although the main contribution is from the flight in 2000. In the simulation calculation, we employed 3 different nuclear interaction models: 1) fritiof1.6[@oldfri] used in the HKKM calculation[@hkkm95], which was widely used for comparison with the Kamioka data, 2)fritiof7.02[@newfri][^4] and 3) dpmjet3.03[@dpmjet]. As the primary cosmic ray, we used the BESS result on protons and He. The CNO component is also considered[@cno]. Besides these we included electron and positron data by AMS[@amselec]. Their data in the 10 GeV region is consistent with the HEAT[@heat] and BETS[@betselec] data. Bremstrahlung gamma-rays from the primary electrons could contribute order of $\sim 10$ % at very high altitudes. At balloon altitudes, the two models, fritiof7.02 and dpmjet3.03, give almost the same results which are close to the observed data, while fritof1.6 gives clearly smaller fluxes than the observation. Figure \[norispec\] shows the result from the observation at Mt.Norikura. It should be noted that the flux by fritiof1.6 becomes higher than the ones by the other models at this altitude. From these figures, we see fritiof7.02 and dpmjet3.03 give rapider increase and faster attenuation of intensity than fritiof1.6; the tendency is very consistent with the observed data. The transition curve of the flux integrated over 6 GeV shown in Fig.\[transition\] clearly demonstrates this feature. {width="6.5cm"} {width="6.5cm"} {width="6.5cm"} {width="6.5cm"} ![Gamma-ray spectra at 5 balloon heights are compared with 3 different models. The vertical axis is Flux$\times E^2$. Except for 1999 data at 32.3 km, 1999 and 2000 flights data are combined. From top to bottom, at 25.1, 21.4, 18.3, 15.3 and 32.3 km. The spectra expected from three interaction models are drawn by solid (dpmjet3.03), dash (fritiof7.02) and dotted (fritiof1.6) lines. []{data-label="balspec"}](spectrum5.eps){width="6.5cm"} ![Gamma-ray spectrum at Mt. Norikura (2.77 km a.s.l). The vertical axis is Flux$\times E^2$. Our data is at $<$ 100 GeV. Data above 300 GeV is from emulsion chamber experiments. For the latter, see Sec.\[discuss\] []{data-label="norispec"}](norikura.eps){width="7.5cm"} ![The altitude variation of the flux integrated over 6 GeV. The dpmjet3.03 and fritiof7.02 give almost the same feature consistent with the observation while the deviation of fritiof1.6 from the data is obvious. \[transition\]](transition.eps){width="7.5cm"} Discussions\[discuss\] ====================== Comparison with other data -------------------------- We found Fritiof7.02 and dpmjet3.03 give good agreement with the observed gamma-ray data at around 10 GeV. We briefly see whether these models can interpret other observations. More detailed inspection will be done elsewhere. - Muon data by the BESS group at Mt.Norikura[@bessmuonnori]. Recently, the BESS group reported detailed muon spectrum over several hundred MeV/c. In their paper, calculations by dpmjet3.03 and fritiof1.6 are compared with the data; agreement by dpmjet3.03 is quit good at least above GeV where Fritiof7.02 also gives more or less the same flux. On the other hand, fritiof1.6 shows too high flux. These features are consisten with our present analysis. - Higher energy gamma-ray data by emulsion chamber. In Fig. \[norispec\], we inlaid an emulsion chamber data[@ecc][^5] at Mt. Norikura. Our data seems to be smoothly connected to their data as the two interaction models (Fritiof7.02 and dpmjet3.03) predict. Since the emulsion chamber data extends to the TeV region and the primary particle energy responsible for such high energy gamma-rays is much higher than 100 GeV where we have no accurate information comparable to the AMS and BESS data, it would be premature to draw a definite conclusion on the primary and interaction model separately. However, the fact that smooth extrapolation of the primary spectra as shown in Table \[extendprim\] and the interaction model, dpmjet3.03 or fritiof7.02, give a consistent result with the data, seems to indicate that such combination would provide a good estimate on other components at $\gg$ 10 GeV. ------- ----------- ------- ----------- -------- ---------   E    flux    E    flux    E   flux 92.6 0.593E-01 79.4 0.549E-02 100. 9.0E-5 108 0.388E-01 100. 3.0E-3 400. 1.8E-6 126 0.276E-01 200. 5.0E-4 2.0E3 3.5E-8 147 0.179E-01 400. 7.0E-5 2.0E4 9.3E-11 171 0.124E-01 2.0E3 9.98E-7 2.0E5 2.3E-13 200 0.836E-02 2.0E4 2.5E-9 14.0E5 1.3E-15 1100 8.29E-5 2.0E5 3.97E-12 3.0E6 1.7E-16 1.1E4 1.47E-7 4.0E5 6.1E-13 3.0E7 2.0E-19 1.1E5 2.8E-10 8.0E5 7.0E-14 3.0E8 2.2E-22 2.2E5 3.7E-11 8.0E6 8.7E-17 4.4E5 5.0E-12 8.0E8 5.3E-23 4.4E8 2.8E-21 ------- ----------- ------- ----------- -------- --------- : Primary flux assumed in the simulation above 100 GeV/n\ (E in kinetic energy per nucleon (GeV), flux in /m$^2\cdot$s$\cdot$sr$\cdot$GeV) \[extendprim\] The $x$-distributions --------------------- The two models, fritiof7.02 and dpmjet3.03, give almost the same results in the present comparison. However, if we look into the $x$-distribution of the particle production, we note some difference, especially in the proton $x$-distribution. We define the $x$ as the kinetic energy ratio of the incoming proton and a secondary particle in the laboratory frame. The $x$ distribution for $p$Air collisions at incident proton energy of 40 GeV is presented for photons (from $\pi^0$ plus $\eta$ decay) and protons in Fig.\[xdist\]. Difference of the three models seen in the photon distribution is quite similar to the one for charged pions. The $x$ region most effective to atmospheric gamma-ray flux is around 0.2$\sim$0.3 where the difference is not so large but fritiof7.02 and dpmjet3.03 have higher gamma-ray yield than fritiof1.6. ![The $x$-distribution of photons from $\pi^0$ plus $\eta$ decay (upper) and protons (lower) for $p$Air collisions at 40 GeV. The three model results are shown. []{data-label="xdist"}](gammaxdist.eps "fig:"){width="7.5cm"} ![The $x$-distribution of photons from $\pi^0$ plus $\eta$ decay (upper) and protons (lower) for $p$Air collisions at 40 GeV. The three model results are shown. []{data-label="xdist"}](protonxdist.eps "fig:"){width="7.5cm"} On the other hand, the proton $x$ distribution has larger difference among the three models (we note, however, the difference may be exaggerated than the photon case due to the scale difference). It is interesting to see that, in spite of these large differences, the final flux is not so much different each other. Our gamma-ray data prefers to rather more inelastic feature of collisions than fritiof1.6, i.e rapider increase and faster attenuation of the flux. We should compare the distribution with accelerator data; however, there is meager stuff appropriate for our purpose. One such comparison has been done in a recent review paper[@GHreview] for $p$Air collisions at 24 GeV/c incident momentum. The charged pion distribution by fritiof1.6 and dpmjet3.03 well fit to some scattered data which prevents to tell the superiority of the two. As to the proton distribution, among the three models, fritiof1.6 is rather close to the data but deviation from the data is much larger than the pion case. The proton $x$-distribution would strongly affect the atmospheric proton spectrum. We calculated proton flux at Mt.Norikura to find a flux relation such that fritiof1.6 $>$ fritiof7.02 $>$ dpmjet3.03 as expected naturally from the $x$-distributions. The maximum difference is factor $\sim 2.5$ in the energy region of 0.3 to 3 GeV. The BESS group has measured the proton spectrum at Mt. Norikura in the same energy region. Their result expected to come soon[@sanukibess] will help select a better model for the proton $x$ distribution. summary ======= - We have made successful observation of atmospheric gamma-rays at around 10 GeV at Mt.Norikura (2.77 km a.s.l) and at balloon altitudes (15 $\sim$ 25 km). - The observed gamma-ray fluxes are compared with calculations by three interaction models; it is found that fritiof1.6 employed by the HKKM calculation [@hkkm95], which was used in comparison with the Kamioka data, is not a very good model. - Other two models (fritiof7.02 and dpmjet3.03) give better results consistent with the data, which shows rapider increase and faster attenuation of the flux than fritiof1.6 predicts. - Our data has complementary feature to muon data and will serve for checking nuclear interaction models used in atmospheric neutrino calculations. We sincerely thank the team of the Sanriku Balloon Center of the Institute of Astronautical Science for their excellent service and the support of the balloon flight. We also thank the staff of the Norikra Cosmic-Ray observatory, Univ. of Tokyo. for their help. We are also indebted to S.Suzuki, P.Picchi, and L. Periale for their spport at CERN in the beam test. For the management of X5 beam line of SPS at CERN, we would like to thank L. Gatignon and the tecnical staffs. One of the authors (K.K) thanks S. Roesler for his help in implementing dpmjet3.03. This work is partly supported by Grants-in Aid for Scientific Research B (09440110), Grants-in Aid for Scientific Research on Priority Area A (12047224) and Grant-in Aid for Project Research of Shibaura Institute of Technology. [^1]: We note electron showers of 10 GeV are normally simulated by $\sim$ 30 GeV protons when the latter start cascade at a shallow depth of the detector. [^2]: If we don’t impose the trigger condition, the gamma-ray case shows a small difference from the electron case. [^3]: The chance coincidence probability of uncorrelated particles is negligibly small. [^4]: It is used at energies greater than 10 GeV. At lower energies, model is the same as fritiof1.6 [^5]: Electrons included in the original data is subtracted statistically by use of cascade theory which is accurate at high energies.

--- abstract: 'The maximal minors of a $p\times (m + p)$-matrix of univariate polynomials of degree $n$ with indeterminate coefficients are themselves polynomials of degree $np$. The subalgebra generated by their coefficients is the coordinate ring of the quantum Grassmannian, a singular compactification of the space of rational curves of degree $np$ in the Grassmannian of $p$-planes in ($m + p$)-space. These subalgebra generators are shown to form a sagbi basis. The resulting flat deformation from the quantum Grassmannian to a toric variety gives a new “Gröbner basis style” proof of the Ravi-Rosenthal-Wang formulas in quantum Schubert calculus. The coordinate ring of the quantum Grassmannian is an algebra with straightening law, which is normal, Cohen-Macaulay, Gorenstein and Koszul, and the ideal of quantum Plücker relations has a quadratic Gröbner basis. This holds more generally for skew quantum Schubert varieties. These results are well-known for the classical Schubert varieties $(n=0)$. We also show that the row-consecutive $p\times p$-minors of a generic matrix form a sagbi basis and we give a quadratic Gröbner basis for their algebraic relations.' address: - | - Frank Sottile\ Department of Mathematics\ University of Wisconsin\ Madison, Wisconsin 53706\ USA - | - Bernd Sturmfels\ Department of Mathematics\ University of California\ Berkeley, California 94720\ USA author: - Frank Sottile - Bernd Sturmfels title: A sagbi basis for the quantum Grassmannian --- [^1] Statement of the main result ============================ Let ${\mathcal M}(t)$ be the $p\times(m+p)$-matrix whose $i,j$th entry is the degree $n$ polynomial in $t$, $$x_{i,j}^{(n)} \cdot t^n \ + \, x_{i,j}^{(n-1)} \cdot t^{n-1} +\,\, \ldots \,\, + \, x_{i,j}^{(2)} \cdot t^2 \ + \, x_{i,j}^{(1)} \cdot t\ + \, x_{i,j}^{(0)} .$$ The coefficients $x_{i,j}^{(l)}$ are indeterminates. We write $k[X]$ for the polynomial ring over a field $k$ generated by these indeterminates, for $i=1,\ldots,p$, $j=1,\ldots,m+p$, and $l=0,\ldots,n$. Lexicographic order on the triples $l,i,j$ gives a total order of these variables. For example, $$x_{1,2}^{(0)}\ <\ x_{1,2}^{(1)}\ <\ x_{1,5}^{(1)}\ <\ x_{2,3}^{(1)}\ <\ x_{2,4}^{(1)}\ <\ x_{1,3}^{(2)} \ .$$ Let $\prec$ be the resulting degree reverse lexicographic term order on the polynomial ring $k[X]$. For each $\alpha\in\binom{[m+p]}{p}$ and $a\geq 0$, let $\alpha^{(a)}$ be a variable which, when $a\leq np$, formally represents the coefficient of $t^a$ in the maximal minor of ${\mathcal M}(t)$ given by the columns indexed by $\alpha_1,\alpha_2,\ldots,\alpha_p$. These variables $\alpha^{(a)}$ have a natural partial order, denoted ${\mathcal C}_{p,m}$, which is defined as follows: $$\alpha^{(a)}\ \leq\ \beta^{(b)} \quad\Longleftrightarrow\quad a\leq b \, \mbox{ and } \,\, \alpha_i\leq \beta_{b-a+i} \, \mbox{ for } i = 1,2,\ldots,p-b+a.$$ Fix $0\leq q\leq np$ and let ${\mathcal C}^q_{p,m}$ denote the truncation of the infinite poset ${\mathcal C}_{p,m}$ to the finite subset $\,\bigl\{ \, \alpha^{(a)} \,\, | \,\, \alpha\in\binom{[m+p]}{p} \mbox{ and } a\leq q \, \bigr\}$. The posets ${\mathcal C}^q_{p,m}$ are graded distributive lattices. Figure \[fig:qorder\] shows ${\mathcal C}^1_{2,3}$. $$\epsfxsize=1.6in \epsfbox{fig1.eps}$$ Let $\varphi : k[{\mathcal C}^q_{p,m}] \rightarrow k[X]$ denote the $k$-algebra homomorphism which sends the formal variable $\alpha^{(a)}$ to the coefficient of $t^a$ in the $\alpha$th maximal minor of the matrix ${\mathcal M}(t)$. For example, [ $$\begin{array}{l} \varphi \bigl( 456^{(2)} \bigr) \quad = \quad \mbox{\rm coefficient of $t^2$ in}\ \ \det\left[\begin{array}{ccc} x^{(0)}_{1,4}+x^{(1)}_{1,4}t&\ x^{(0)}_{1,5}+x^{(1)}_{1,5}t\ & x^{(0)}_{1,6}+x^{(1)}_{1,6}t\\ \rule{0pt}{17pt} x^{(0)}_{2,4}+x^{(1)}_{2,4}t&\ x^{(0)}_{2,5}+x^{(1)}_{2,5}t\ & x^{(0)}_{2,6}+x^{(1)}_{2,6}t\\ \rule{0pt}{17pt} x^{(0)}_{3,4}+x^{(1)}_{3,4}t&\ x^{(0)}_{3,5}+x^{(1)}_{3,5}t\ & x^{(0)}_{3,6}+x^{(1)}_{3,6}t \end{array}\right]\, \\ = \quad \rule{0pt}{20pt} -\underline{x^{(0)}_{3,6}x^{(1)}_{1,5}x^{(1)}_{2,4}} +x^{(0)}_{3,5}x^{(1)}_{1,6}x^{(1)}_{2,4} +x^{(0)}_{3,6}x^{(1)}_{1,4}x^{(1)}_{2,5} -x^{(0)}_{3,4}x^{(1)}_{1,6}x^{(1)}_{2,5} -x^{(0)}_{3,5}x^{(1)}_{1,4}x^{(1)}_{2,6} +x^{(0)}_{3,4}x^{(1)}_{1,5}x^{(1)}_{2,6}\\\rule{0pt}{17pt}\phantom{=} \quad +x^{(0)}_{2,6}x^{(1)}_{1,5}x^{(1)}_{3,4} -x^{(0)}_{2,5}x^{(1)}_{1,6}x^{(1)}_{3,4} -x^{(0)}_{2,6}x^{(1)}_{1,4}x^{(1)}_{3,5} +x^{(0)}_{2,4}x^{(1)}_{1,6}x^{(1)}_{3,5} +x^{(0)}_{2,5}x^{(1)}_{1,4}x^{(1)}_{3,6} -x^{(0)}_{2,4}x^{(1)}_{1,5}x^{(1)}_{3,6}\\\rule{0pt}{17pt}\phantom{=} \quad -x^{(0)}_{1,6}x^{(1)}_{2,5}x^{(1)}_{3,4} +x^{(0)}_{1,5}x^{(1)}_{2,6}x^{(1)}_{3,4} +x^{(0)}_{1,6}x^{(1)}_{2,4}x^{(1)}_{3,5} -x^{(0)}_{1,4}x^{(1)}_{2,6}x^{(1)}_{3,5} -x^{(0)}_{1,5}x^{(1)}_{2,4}x^{(1)}_{3,6} +x^{(0)}_{1,4}x^{(1)}_{2,5}x^{(1)}_{3,6}\,. \end{array}$$]{} \[issagbi\] The set of polynomials $\, \varphi(\alpha^{(a)}) \,$ as $\alpha^{(a)} $ runs over the poset ${\mathcal C}^q_{p,m} \,$ forms a sagbi basis with respect to the reverse lexicographic term order $\prec$ on $\,k[X]\,$ defined above. Our second theorem states that the subalgebra [*image*]{}$(\varphi)$ of $k[X]$ generated by this sagbi basis is an [*algebra with straightening law*]{} on the poset ${\mathcal C}^q_{p,m}$. Let $\prec$ be the degree reverse lexicographic term order on $k[{\mathcal C}^q_{p,m}]$ induced by any linear extension of the poset ${\mathcal C}^q_{p,m}$. This term order on $k[{\mathcal C}^q_{p,m}]$ and the previous term order on $k[X]$ are fixed throughout this paper. \[thm:gbasis\] The reduced Gröbner basis of the kernel of $\varphi$ consists of quadratic polynomials in $\,k[{\mathcal C}^q_{p,m}]\,$ which are indexed by pairs of incomparable variables $\gamma^{(c)},\delta^{(d)}$ in the poset $\,{\mathcal C}^{np}_{p,m}$, $$S(\gamma^{(c)},\delta^{(d)}) \quad = \quad \gamma^{(c)}\cdot\delta^{(d)}\ -\ (\gamma^{(c)}\vee\delta^{(d)})\cdot (\gamma^{(c)}\wedge\delta^{(d)}) \,\, + \,\,\hbox{lower terms in $\prec$},$$ and all lower terms $\,\lambda\beta^{(b)}\alpha^{(a)}\,$ in $\,S(\gamma^{(c)},\delta^{(d)}) \,$ satisfy $\,\beta^{(b)}<\gamma^{(c)}\wedge\delta^{(d)}$ and $\gamma^{(c)}\vee\delta^{(d)}<\alpha^{(a)}$. The join $\vee$ and meet $\wedge$ appearing in the above formula are the lattice operations in ${\mathcal C}^{np}_{p,m}$. The combinatorial structure of this distributive lattice will become clear in Section 2, when we introduce the toric variety associated with ${\mathcal C}^{np}_{p,m}$. In Section 3 we interpret the subalgebra ${\rm image}(\varphi)$ of $k[X]$ as the coordinate ring of the quantum Grassmannian. Section 4 contains the proofs of Theorems 1 and 2. These results generalize the classical sagbi basis property of maximal minors [@Sturmfels_invariant Theorem 3.2.9] and its geometric interpretation as a toric deformation [@Sturmfels_GBCP Proposition 11.10] from the case of the Grassmannian to the quantum Grassmannian. In Section 5 we discuss corollaries, applications and some open problems. One such application is that the row-consecutive $p\times p$-minors of any matrix of indeterminates form a sagbi basis. We thank Aldo Conca, Ezra Miller, and Brian Taylor for their helpful comments. The toric variety of the distributive lattice ============================================= Theorem 1 asserts that the initial algebra of our subalgebra $\,{\rm image}(\varphi)\,$ is generated by the initial monomials of its generators $\varphi(\alpha^{(a)}) $. Our first step is to identify the initial monomials. Here are two examples. The first one is the underlined monomial right before Theorem \[issagbi\]: $$\mbox{\rm in}_\prec \bigl( \varphi(456^{(2)}) \bigr) \ =\ x_{3,6}^{(0)}x_{1,5}^{(1)}x_{2,4}^{(1)}\qquad\mbox{and}\qquad \mbox{\rm in}_\prec \bigl( \varphi(2457^{(5)}) \bigr) \ =\ x_{2,7}^{(1)}x_{3,5}^{(1)}x_{4,4}^{(1)}x_{1,2}^{(2)}.$$ In general, the initial monomial of $\varphi(\alpha^{(a)}) $ is given by the following lemma: \[lemma:leadmon\] Let $\alpha\in\binom{[m+p]}{p}$ and $\,a=pl+r \,$ with integers $p > r \geq 0$. Then $$\mbox{\rm in}_\prec \bigl( \varphi(\alpha^{(a)}) \bigr) \quad =\quad x_{r+1,\alpha_p}^{(l)}x_{r+2,\alpha_{p-1}}^{(l)}\cdots x_{p,\alpha_{r+1}}^{(l)} x_{1,\alpha_r}^{(l+1)} x_{2,\alpha_{r-1}}^{(l+1)} \cdots x_{r,\alpha_1}^{(l+1)}.$$ [**Proof.** ]{} Let $x_{i_1,j_1}^{(l_1)}x_{i_2,j_2}^{(l_2)}\cdots x_{i_p,j_p}^{(l_p)}$ be a monomial which appears in $\varphi(\alpha^{(a)})$. We claim that $$\label{eq:comparison} x_{i_1,j_1}^{(l_1)}x_{i_2,j_2}^{(l_2)}\cdots x_{i_p,j_p}^{(l_p)} \quad \preceq\quad x_{r+1,\alpha_p}^{(l)}x_{r+2,\alpha_{p-1}}^{(l)}\cdots x_{p,\alpha_{r+1}}^{(l)}x_{1,\alpha_r}^{(l+1)}\cdots x_{r,\alpha_1}^{(l+1)}.$$ We may assume $\,x_{i_1,j_1}^{(l_1)} \prec x_{i_2,j_2}^{(l_2)} \prec \cdots \prec x_{i_p,j_p}^{(l_p)}\,$ and hence $l_1 \leq\cdots\leq l_p$. Since $l_1+\cdots+l_p=a$, either $ l_1 < q $, from which (\[eq:comparison\]) follows, or else $l_1=\cdots=l_{p-r}=l$ and $l_{p+1-r}=\cdots=l_p=l+1$. In the second case, as $\{i_1,\ldots,i_p\}=\{1,\ldots,p\}$ and the monomial is in order, we must have $i_1<\cdots<i_{p-r}$ and $i_{p+1-r}<\cdots<i_p$. If $i_1 \leq r$, then (\[eq:comparison\]) follows, and if $ i_1 = r+1$, then the ordered sequence $\,i_1,i_2,\ldots,i_p\,$ equals $\, r \! + \! 1,r \! + \! 2,\ldots,p,1,\ldots,r$. Among all monomials satisfying this new second case, the largest in the degree reverse lexicographic order $\prec $ has the second lower index appearing in reverse order. This completes the proof. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ We next introduce some combinatorics to help understand the poset ${\mathcal C}_{p,m}$. A [*row*]{} with shift $a$ consists of $p$ consecutive empty unit boxes shifted $a$ units to the left of a given vertical line. A [*skew shape*]{} is an array of such rows whose shifts are weakly increasing read from top to bottom. For example, the unshaded boxes in the figure on the left form a skew shape (with shifts 1,2,2, and 5) while those in the other figure do not: $$\epsfxsize=2.8in \epsfbox{fig2.eps}$$ A [*(skew) tableau*]{} $T$ is a filling of a skew shape with integers that increase across each row. When the entries lie in $[m+p]$, the $i$th row of a tableau is a sequence $\alpha_{(i)}\in\binom{[m+p]}{p}$. If $a_i$ is the shift of the $i$th row and $T$ has $j$ rows, then $T$ corresponds to a monomial $\alpha_{(1)}^{(a_1)}\alpha_{(2)}^{(a_2)}\cdots\alpha_{(j)}^{(a_j)}$. Conversely, any monomial in the variables $\alpha^{(a)}$ corresponds to a tableau. A tableau $T$ is [*standard*]{} if the entries are weakly increasing in each column, read top to bottom. Equivalently, $T$ is standard if we have $\alpha_{(1)}^{(a_1)}\leq\alpha_{(2)}^{(a_2)}\leq\cdots \leq\alpha_{(j)}^{(a_j)}$ in ${\mathcal C}_{p,m}$. For example, the following two tableaux correspond to the monomials $345^{(0)}123^{(1)}245^{(3)}$ and $135^{(0)}123^{(1)}257^{(3)}$. The first tableau is not standard and the second tableau is standard: $$\epsfxsize=2.8in \epsfbox{fig3.eps}$$ The elements of the poset ${\mathcal C}^q_{p,m}$ are represented by one-row tableaux with entries in $[m+p]$ and shift at most $q$. Two elements satisfy $\,\alpha^{(a)} \leq \beta^{(b)} \,$ if and only if the two-rowed tableau $T=\alpha^{(a)}\beta^{(b)}$ is standard. This representation implies that ${\mathcal C}^q_{p,m}$ is a distributive lattice. Indeed, the two lattice operations $\wedge$ and $\vee$ are described as follows. If a two-rowed tableau $\,T=\alpha^{(a)}\beta^{(b)}\,$ is non-standard then interchanging the entries in every column in which a violation ($\alpha_{a-b+i}<\beta_i$) occurs yields a standard tableau. The first row of this new tableau is the [*meet*]{} $\alpha^{(a)}\wedge\beta^{(b)}$ of $\alpha^{(a)}$ and $\beta^{(b)}$ in ${\mathcal C}^q_{p,m}$ and the second row is their [*join*]{} $\alpha^{(a)}\vee\beta^{(b)}$. Let $\psi : k[{\mathcal C}^{q}_{p,m}] \rightarrow k[X]$ denote the $k$-algebra homomorphism which sends the variable $\alpha^{(a)}$ to the monomial $\,\mbox{\rm in}_\prec \bigl(\varphi(\alpha^{(a)}) \bigr)$. Its kernel is a [*toric ideal*]{} (i.e. binomial prime)  in $\, k[{\mathcal C}^{q}_{p,m}]$. \[lem:toric\] The reduced Gröbner basis for the kernel of $\psi$ consists of the binomials $$\underline{\alpha^{(a)}\cdot\beta^{(b)}} \ -\ (\alpha^{(a)}\vee\beta^{(b)})\cdot(\alpha^{(a)}\wedge\beta^{(b)}),$$ where $\alpha^{(a)}, \beta^{(b)}$ runs over all incomparable pairs of ${\mathcal C}^q_{p,m} $. The initial monomial is underlined. [**Proof.**]{} This follows from Hibi’s Theorem [@Hibi] since ${\mathcal C}^q_{p,m}$ is a distributive lattice. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ \[cor:standard\] The set of standard tableaux is a $k$-basis for $\, k[{\mathcal C}^q_{p,m}] / {\rm kernel}(\psi) = {\rm image}(\psi) $. Here is a typical element in the reduced Gröbner basis of ${\rm kernel}(\psi) $ for $p=5,m=4,q=9$: $$\underline{45789^{(1)} \cdot 12356^{(3)}} \, - \, 35689^{(1)} \cdot 12457^{(3)}.$$ Note that the second monomial corresponds to a standard tableau while the first does not. We write $T^q_{p,m}$ for the projective toric variety cut out by the binomials in Lemma \[lem:toric\]. Its coordinate ring is the subalgebra $\,{\rm image}(\psi)$ of $k[X]$. The geometry of toric varieties associated with distributive lattices is discussed in [@Wagner]. The analogue to Corollary \[cor:standard\] always holds, i.e., multichains in the poset correspond to basis monomials in the coordinate ring. \[Degree\] The degree of the toric variety $T^q_{p,m}$ is the number of maximal chains in ${\mathcal C}^q_{p,m}$. The closed intervals of the poset ${\mathcal C}^q_{p,m}$ are also distributive lattices. They are denoted $$[\beta^{(b)}, \alpha^{(a)}] \quad := \quad \bigl\{ \gamma^{(c)} \in {\mathcal C}^q_{p,m} \,: \, \beta^{(b)} \leq \gamma^{(c)} \leq \alpha^{(a)} \bigr\}.$$ Proposition \[lem:toric\] and Corollaries \[cor:standard\] and \[Degree\] hold essentially verbatim for the distributive sublattice $\,[\beta^{(b)}, \alpha^{(a)}]\,$ as well. The projective toric variety associated with $\,[\beta^{(b)}, \alpha^{(a)}]\,$ is gotten from the toric variety of ${\mathcal C}^q_{p,m}$ by setting $\gamma^{(c)} = 0$ for all $ \gamma^{(c)} \not\in [\beta^{(b)}, \alpha^{(a)}]$. The degree of that variety is the number of saturated chains in ${\mathcal C}^q_{p,m}$ which start at $\beta^{(b)}$ and end at $ \alpha^{(a)}$. We close this section with an alternative proof, to be used in Section 4, for the fact that ${\mathcal C}_{p,m}$ is a distributive lattice. We claim that ${\mathcal C}_{p,m}$ is a sublattice of [*Young’s lattice*]{}. Given $\alpha^{(a)} \in {\mathcal C}_{p,m}$, write $a=pl+r$ with integers $p > r \geq 0$, and define a sequence $J(\alpha^{(a)})$ by $$\label{eq:seq_def} J(\alpha^{(a)})_i\quad :=\quad \left\{\begin{array}{ll} l(m+p)+\alpha_{r+i}&\quad\mbox{if }1 \leq i\leq p-r\\ (l+1)(m+p)+\alpha_{i-p+r}&\quad\mbox{if }p-r<i\leq p \end{array}\right.\,.$$ This gives an order-preserving bijection between the poset ${\mathcal C}_{p,m}$ and the poset of sequences $J:=j_1<j_2<\cdots<j_p$ of positive integers with $j_p-(m+p)<j_1$, and it preserves meet and join. This bijection preserves the rank function in the two distributive lattices: $$\label{Miracle} | \alpha^{(a)} | \quad := \quad a(m+p) + \sum_{j=1}^p (\alpha_j - j) \quad = \quad \sum_{i=1}^p \bigl( J(\alpha^{(a)})_i - i \,\bigr) \quad =: \quad | J(\alpha^{(a)})|.$$ The quantum Grassmannian ======================== Let ${\it Grass}_pk^{m+p}$ denote the Grassmannian of $p$-planes in the vector space $k^{m+p}$. This is a smooth projective variety of dimension $mp$. Consider the space $S^q_{p,m}$ of maps ${\mathbb P}^1\rightarrow \mbox{\it Grass}_pk^{m+p}$ of degree $q$. Such a map may be (non-uniquely) represented as the row space of a $p\times(m+p)$-matrix of polynomials in $t$ whose maximal minors have degree $q$. Results in [@Clark] imply that it suffices to consider the matrices ${\mathcal M}(t)$ in the introduction. The coefficients of these maximal minors define the [*Plücker embedding*]{} of $S^q_{p,m}$ into ${\mathbb P}(\wedge^pk^{m+p}\otimes k^{q+1})$; see [@Stromme; @Rosen94]. The [*quantum Grassmannian*]{} $K^q_{p,m}$ is the Zariski closure of $S^q_{p,m}$ in this Plücker embedding. It is an irreducible projective variety of dimension $mp+q(m+p)$. Its prime ideal is $\,{\rm kernel}(\varphi) \subset k[ {\mathcal C}^q_{p,m}]\,$ and its coordinate ring is our subalgebra $\,{\rm image}(\varphi)\subset k[X]$. The quantum Grassmannian $K^q_{p,m}$ is singular and it differs from other spaces used to study rational curves in Grassmann varieties (the quot scheme [@Stromme], the Kontsevich space of stable maps [@Kontsevich_Manin], or the set of autoregressive systems [@RR94]). Nevertheless, $K^q_{p,m}$ has been crucial in two important advances: in computing the intersection number $\,{\rm degree}(K^q_{p,m})\,$ in quantum cohomology [@RRW98], and in showing that this intersection problem can be fully solved over the real numbers [@Sottile_quantum]. Our result will give a new derivation of this intersection number. \[cor:counting\] [[@RRW98]]{} The degree of $\,K^q_{p,m} \,$ is the number of maximal chains in ${\mathcal C}^q_{p,m}$. [**Proof.** ]{} This follows immediately from Theorem 2 and Corollary \[Degree\]. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ This degree can also be computed in the small quantum cohomology ring of the Grassmannian [@Bertram]. Note that $\, {\rm degree} (K^1_{2,3}) = 55 $, by counting maximal chains in Figure \[fig:qorder\]. Ravi, Rosenthal, and Wang [@RRW98] were motivated by a problem in applied mathematics. The degree of $\,K^q_{p,m} \,$ is the number of dynamic feedback compensators that stabilize a certain linear system, in the sense of systems theory. This number can be described in classical projective geometry as follows. The Schubert subvariety of ${\it Grass}_pk^{m+p}$ consisting of $p$-planes meeting a fixed $m$-plane $L$ is a hyperplane section in the Plücker embedding of ${\it Grass}_pk^{m+p}$. Thus the set of maps $M\in S^q_{p,m}$ such that $M(t)$ meets $L$ non-trivially is a hyperplane section of $S^q_{p,m}$ in its Plücker embedding. Since $GL_{m+p}(k)$ acts transitively on ${\it Grass}_pk^{m+p}$, Kleiman’s Theorem on generic transversality implies the following statement when $k$ is algebraically closed of characteristic zero. Set $N:=mp+q(m+p)$ and suppose $t_1,\ldots,t_N\in{\mathbb P}^1$ are general points and $L_1,\ldots,L_N$ are general $m$-planes in $k^{m+p}$, then the degree of $K^q_{p,m}$ counts those maps $M$ for which $M(t_i)$ meets $L_i$ non-trivially for each $i=1,\ldots,N$. As to computing the desired maps $M$ numerically, we note that the sagbi basis in Theorem 1 and the Gröbner basis in Theorem 2 each lead to an [*optimal homotopy algorithm*]{} for finding these $\,{\rm degree}(K^q_{p,m})\,$ maps. These algorithms generalize the ones in [@HSS]. \[rem:flat\] The sagbi basis in Theorem 1 defines a flat deformation from the quantum Grassmannian $K^q_{p,m}$ to the projective toric variety $T^q_{p,m}$ associated with the poset ${\mathcal C}^q_{p,m}$. See [@CHV] for a precise algebraic discussion of such deformations, and see [@Sturmfels_GBCP Equation (11.9)] for the simplest example relevant to us, namely, $K^0_{2,3} = {\it Grass}_2 k^{5}$. The flat deformation is given algebraically by deleting all but the first two terms in the Gröbner basis elements $\, S(\gamma^{(c)},\delta^{(d)}) \,$ given in Theorem 2. Consider the deformation from $K^3_{3,3}$ to $T^3_{3,3}$. The incomparable pair $156^{(1)}$ and $234^{(2)}$ in ${\mathcal C}^3_{3,3}\,$ indexes the quadratic polynomial $\,\, S(156^{(1)},234^{(2)})\, = $ [ $$\label{BigSyzygy} \begin{array}{c} \underline{ 156^{(1)}234^{(2)} -146^{(1)}235^{(2)} }\, +145^{(1)}236^{(2)} +136^{(1)}245^{(2)} -135^{(1)}246^{(2)}\\ +134^{(1)}256^{(2)} -126^{(1)}345^{(2)} +125^{(1)}346^{(2)} -124^{(1)}356^{(2)} +123^{(1)}456^{(2)}\\ -456^{(0)}123^{(3)} +356^{(0)}124^{(3)} -346^{(0)}125^{(3)} +345^{(0)}126^{(3)} -256^{(0)}134^{(3)} \\ +246^{(0)}135^{(3)} -245^{(0)}136^{(3)} -236^{(0)}145^{(3)} +235^{(0)}146^{(3)} -234^{(0)}156^{(3)} \\ +2{\cdot}156^{(0)}234^{(3)} -2{\cdot}146^{(0)}235^{(3)} +2{\cdot}145^{(0)}236^{(3)} +2{\cdot}136^{(0)}245^{(3)} -2{\cdot}135^{(0)}246^{(3)}\,\\ +2{\cdot}134^{(0)}256^{(3)} -2{\cdot}126^{(0)}345^{(3)} +2{\cdot}125^{(0)}346^{(3)} -2{\cdot}124^{(0)}356^{(3)} +2{\cdot}123^{(0)}456^{(3)}, \end{array}$$ ]{} which vanishes on the quantum Grassmannian $K^3_{3,3}$. The underlined leading binomial vanishes on the toric variety $T^3_{3,3}$, by Proposition \[lem:toric\]. Our main technical problem, to be solved in the next section, is the reconstruction of quadrics such as (\[BigSyzygy\]) from their leading binomial. A key tool in proving Theorems 1 and 2 is the Schubert decomposition of the quantum Grassmannian $K^q_{p,m}$ indexed by ${\mathcal C}^q_{p,m}$. For $\alpha^{(a)}\in{\mathcal C}^q_{p,m}$, the [*quantum Schubert variety*]{} is $$Z_{\alpha^{(a)}}\quad :=\quad \bigl\{ \,({\gamma^{(c)}})\in K^q_{p,m} \,\mid\, {\gamma^{(c)}}=0 \, \mbox{ if } \, \gamma^{(c)}\not\leq \alpha^{(a)} \bigr\}\ .$$ More generally, for $\beta^{(b)}\leq \alpha^{(a)}$ in ${\mathcal C}^q_{p,m}$, we define the [*skew quantum Schubert variety*]{} $$Z_{\alpha^{(a)}/\beta^{(b)}}\quad :=\quad \bigl\{\, ({\gamma^{(c)}})\in K^q_{p,m} \,\mid \, {\gamma^{(c)}} = 0 \, \mbox{ if } \, \gamma^{(c)}\not\in [\beta^{(b)},\alpha^{(a)}] \bigr\}\ .$$ Among the quantum Schubert varieties of $K^q_{p,m}$ are the $K^d_{p,m}$ for $d<q$; namely, if $\delta^{(d)}$ is the supremum of ${\mathcal C}^d_{p,m}$, then $K^d_{p,m}=Z_{\delta^{(d)}}$. This allows us to deduce assertions about the general quantum Grassmannian $K^q_{p,m}$ from results about quantum Schubert varieties of $K^{pn}_{p,m}$. The quantum Schubert varieties and skew quantum Schubert varieties have rational parameterizations which are constructed as follows. Let $\alpha^{(a)}\in{\mathcal C}^{pn}_{p,m}$ and write $a=ps+r$ with integers $p>r\geq 0$. We define the matrix ${\mathcal M}_{\alpha^{(a)}}(t)$ to be the specialization of ${\mathcal M}(t)$ where $$x_{i,j}^{(l)}\ =\ 0\quad\mbox{if }\quad\left\{ \begin{array}{ll} ( l>s+1 \ \mbox{ and } \ i\leq r ) &\mbox{ or } \ \ (l=s+1\mbox{ and }j>\alpha_{r+1-i}) \ \ \mbox{or} \\ ( l>s \ \mbox{ and } \ i>r ) &\mbox{ or } \ \ (l=s\mbox{ and }j>\alpha_{p+r+1-i})\,. \end{array}\right.$$ Here we use the conventions $\alpha_\nu = 0$ if $\nu \leq 0\, $ and $\,\alpha_\nu = +\infty$ if $\nu > p$. For example, $${\mathcal M}_{235^{(2)}}(t) \quad = \quad \, \left[\begin{array}{cccccc} x_{1,1}^{(0)}+x_{1,1}^{(1)}\cdot t&x_{1,2}^{(0)}+x_{1,2}^{(1)}\cdot t &x_{1,3}^{(0)}+x_{1,3}^{(1)}\cdot t&x_{1,4}^{(0)} &\ x_{1,5}^{(0)}&\ x_{1,6}^{(0)}\\ \rule{0pt}{17pt} x_{2,1}^{(0)}+x_{2,1}^{(1)}\cdot t&x_{2,2}^{(0)}+x_{2,2}^{(1)}\cdot t &x_{2,3}^{(0)}\ \ \ &\ x_{2,4}^{(0)}\ &\ x_{2,5}^{(0)} &\ x_{2,6}^{(0)} \\ \rule{0pt}{17pt} x_{3,1}^{(0)}\ \ \ &x_{3,2}^{(0)}\ \ \ & x_{3,3}^{(0)}\ \ \ &\ x_{3,4}^{(0)}\ &\ x_{3,5}^{(0)} &\ \ 0 \rule{0pt}{16pt} \end{array}\right].$$ If we specialize the variables $x^{(l)}_{i,j}$ in ${\mathcal M}_{\alpha^{(a)}}(t)$ to field elements in $k$ in such a way that the resulting matrix over $k(t)$ has maximal row rank, then that matrix defines a map from $k$ to ${\it Grass}_pk^{m+p}$. If we extend this to ${\mathbb P}^1$, we obtain a map in $Z_{\alpha^{(a)}}$. Proposition \[ItIsDominant\] below implies that such maps constitute a dense subset of $Z_{\alpha^{(a)}}$. This means that the coefficients with respect to $t$ of the maximal minors of ${\mathcal M}_{\alpha^{(a)}}(t)$ give a rational parameterization of $Z_{\alpha^{(a)}}$. This construction extends to skew quantum Schubert varieties as follows. Given $\beta^{(b)}\leq\alpha^{(a)}$, write $b=ps+r$ with integers $p>r\geq 0$ and define the matrix ${\mathcal M}_{\alpha^{(a)}/\beta^{(b)}}(t)$ to be the specialization of ${\mathcal M}_{\alpha^{(a)}}(t)$ where $$x_{i,j}^{(l)}\ =\ 0\quad\mbox{if }\quad\left\{ \begin{array}{ll} ( l<s+1 \ \mbox{ and }\ i\leq r ) &\mbox{ or } \ \ ( l=s+1\mbox{ and }j< \beta_{r+1-i}) \ \ \mbox{ or } \\ ( l<s \ \mbox{ and }\ i>r ) & \mbox{ or } \ \ ( l=s\mbox{ and }j<\beta_{p+r+1-i}) \end{array}\right.\,.$$ The matrix ${\mathcal M}_{\alpha^{(a)}/\beta^{(b)}}(t)$ gives a rational map into $Z_{\alpha^{(a)}/\beta^{(b)}}$, which is described algebraically as follows. We define $\varphi_{\alpha^{(a)}}$ and $\varphi_{\alpha^{(a)}/\beta^{(b)}}$ to be the composition of the map $\,\varphi : k[{\mathcal C}^{np}_{p,m}] \rightarrow k[X] \,$ with the specializations to ${\mathcal M}_{\alpha^{(a)}}(t)$ and ${\mathcal M}_{\alpha^{(a)}/\beta^{(b)}}(t)$ respectively. We claim that these matrices parameterize dense subsets of the (skew) quantum Schubert varieties. \[ItIsDominant\] The kernel of $\varphi_{\alpha^{(a)}}$ is the homogeneous ideal of the quantum Schubert variety $Z_{\alpha^{(a)}}$. Likewise, the kernel of $\varphi_{\alpha^{(a)}/\beta^{(b)}}$ is the homogeneous ideal of the skew quantum Schubert variety $Z_{\alpha^{(a)}/\beta^{(b)}}$. In particular, the varieties $Z_{\alpha^{(a)}}$ and $Z_{\alpha^{(a)}/\beta^{(b)}}$ are irreducible. We postpone the proof of this proposition until the next section. Here is an example which illustrates the parameterization of skew quantum Schubert varieties for $p = m = 3$: $${\mathcal M}_{235^{(2)}/146^{(1)}}(t) \quad = \quad \left[\begin{array}{cccccc} x_{1,1}^{(1)}\cdot t&x_{1,2}^{(1)}\cdot t &x_{1,3}^{(1)}\cdot t&0&0&0\\ \rule{0pt}{17pt} x_{2,1}^{(1)}\cdot t&x_{2,2}^{(1)}\cdot t &0&0&0 &x_{2,6}^{(0)} \\ \rule{0pt}{17pt} 0&0&0&x_{3,4}^{(0)}&x_{3,5}^{(0)}& 0 \rule{0pt}{16pt} \end{array}\right]\,.$$ We evaluate the $3 \times 3$-minors of this matrix to find the $k$-algebra homomorphism $\,\varphi_{{235^{(2)}/146^{(1)}}} $. It takes polynomials in $12$ variables $\,\gamma^{(c)}\,$ to polynomials in $8$ variables $x^{(l)}_{i,j}$ as follows: $$\begin{array}{c} 146^{(1)} \mapsto -x^{(1)}_{1,1} x^{(0)}_{2,6} x^{(0)}_{3,4} \,\,, \quad 156^{(1)} \mapsto -x^{(1)}_{1,1} x^{(0)}_{2,6} x^{(0)}_{3,5} \,\, ,\quad 246^{(1)} \mapsto -x^{(1)}_{1,2} x^{(0)}_{2,6} x^{(0)}_{3,4} \,, \\\rule{0pt}{15pt} 256^{(1)} \mapsto -x^{(1)}_{1,2} x^{(0)}_{2,6} x^{(0)}_{3,5} \,\, ,\quad 346^{(1)} \mapsto -x^{(1)}_{1,3} x^{(0)}_{2,6} x^{(0)}_{3,4} \,\, ,\quad 356^{(1)} \mapsto -x^{(1)}_{1,3} x^{(0)}_{2,6} x^{(0)}_{3,5} \, , \\\rule{0pt}{15pt} 124^{(2)} \mapsto x^{(1)}_{1,1} x^{(1)}_{2,2} x^{(0)}_{3,4} - x^{(1)}_{2,1} x^{(1)}_{1,2} x^{(0)}_{3,4} \, , \quad 125^{(2)} \mapsto x^{(1)}_{1,1} x^{(1)}_{2,2} x^{(0)}_{3,5} - x^{(1)}_{2,1} x^{(1)}_{1,2} x^{(0)}_{3,5}\, , \\\rule{0pt}{15pt} 134^{(2)} \mapsto -x^{(1)}_{2,1} x^{(1)}_{1,3} x^{(0)}_{3,4} \,\, ,\qquad 135^{(2)} \mapsto -x^{(1)}_{2,1} x^{(1)}_{1,3} x^{(0)}_{3,5} \,, \\\rule{0pt}{15pt} 234^{(2)} \mapsto -x^{(1)}_{2,2} x^{(1)}_{1,3} x^{(0)}_{3,4} \,\, ,\qquad 235^{(2)} \mapsto -x^{(1)}_{2,2} x^{(1)}_{1,3} x^{(0)}_{3,5} .\end{array}\vspace{-4pt}$$ The $12$ variables $\gamma^{(c)}$ appearing on the left sides above are precisely the elements in the interval $\,\bigl[146^{(1)}, 235^{(2)} \bigr]\,$ of the distributive lattice ${\mathcal C}_{3,3}$. There are $18$ incomparable pairs in this interval, each giving a quadratic generator for the kernel of $\,\varphi_{{235^{(2)}/146^{(1)}}} $. This set of $18$ quadrics consists of $14$ binomials and four trinomials, and it equals the reduced Gröbner basis with respect to $\prec$. For example, one of the 14 binomials in this Gröbner basis is the underlined leading binomial of $\,\, S(156^{(1)},234^{(2)})\, $ in (\[BigSyzygy\]), and one of the four trinomials is $$\underline{ 346^{(1)} \cdot 125^{(2)} } \,\, - \,\, 246^{(1)} \cdot 135^{(2)} \,\, + \,\, 146^{(1)} \cdot 235^{(2)}.$$ The underlined term is an incomparable pair in $\bigl[146^{(1)}, 235^{(2)} \bigr]$, while the other two monomials are comparable pairs. Erasing the third term gives a binomial as in Proposition \[lem:toric\]. Construction of Straightening Syzygies {#sec:syzygy} ====================================== The following theorem is the technical heart of this paper. All three of Theorem \[issagbi\], Theorem \[thm:gbasis\], and Proposition \[ItIsDominant\] will be derived from Theorem \[thm:syzygy\] in the end of this section. \[thm:syzygy\] Let $\gamma^{(c)},\delta^{(d)}$ be a pair of incomparable variables in the poset ${\mathcal C}^{np}_{p,m}$. There is a quadric $S(\gamma^{(c)},\delta^{(d)})$ in the kernel of $\varphi : k[{\mathcal C}^{np}_{p,m}] \rightarrow k[X]$ whose first two monomials are $$\gamma^{(c)}\cdot\delta^{(d)}\ -\ (\gamma^{(c)}\vee\delta^{(d)}) \cdot (\gamma^{(c)}\wedge\delta^{(d)}).$$ Moreover, if $\lambda\beta^{(b)}\alpha^{(a)}$ is any non-initial monomial in $S(\gamma^{(c)},\delta^{(d)})$, then $\gamma^{(c)},\delta^{(d)}\in[\beta^{(b)},\alpha^{(a)}]$. The pair $ \beta^{(b)}\alpha^{(a)}$ in the second assertion is necessarily standard, i.e. $\beta^{(b)}<\alpha^{(a)}$. The quadrics $S(\gamma^{(c)},\delta^{(d)})$ are not constructed explicitly, but rather through an iterative procedure modeled on the [*subduction algorithm*]{} in [image]{}$(\varphi)$. A main idea is to utilize the well-known subduction process [@Sturmfels_invariant Algorithm 3.2.6] modulo the $p \times p$-minors of a generic $p \times N$-matrix. Set $N:=(n+1)(m+p)$. Let ${\mathcal N}$ be the $p\times N$-matrix whose $i,j$th entry is $x^{(l)}_{i,r}$, where $j= (m+p)l+r$ with $1\leq r\leq m+p$. If ${\mathcal N}_l$ is the submatrix of ${\mathcal N}$ consisting of the entries $x^{(l)}_{i,j}$, then ${\mathcal N}$ is the concatenation of ${\mathcal N}_0,{\mathcal N}_1,\ldots,{\mathcal N}_n$ and ${\mathcal M}(t)= {\mathcal N}_0+t{\mathcal N}_1+\cdots+t^n{\mathcal N}_n$. Sequences $\,J:j_1<\cdots<j_p\in\binom{[ N]}{p}\,$ are regarded as variables. We write $\phi(J)$ for the $J$th maximal minor of ${\mathcal N}$. [*Young’s poset*]{} on sequences $J$ is given by componentwise comparison and is graded via $\,|J| := \sum_i (j_i-i)$. The coefficient $\varphi(\alpha^{(a)})$ of $t^a$ in the $\alpha$th maximal minor of ${\mathcal M}(t)$ is an alternating sum of maximal minors of ${\mathcal N}$. The exact formula is $$\label{eq:phi_expand} \varphi(\alpha^{(a)})\quad \,\,\, =\quad \sum_{\stackrel{\mbox{\scriptsize $|J|=|\alpha^{(a)}|$}} {J\equiv\alpha\,\bmod{(m+p)}}} \!\! \epsilon_J \cdot \phi(J),$$ where $\epsilon_J$ is the sign of the permutation that orders the following sequence: $$j_1\bmod(m+p),\ j_2\bmod(m+p),\ \ldots,\ j_p\bmod(m+p).$$ The polynomial rings $k[{\mathcal C}^{np}_{p,m}]$ and $k[\binom{[N]}{p}]$ are graded with $\deg\alpha^{(a)}=|\alpha^{(a)}|$ and $\deg J=|J|$. Consider the degree-preserving $k$-algebra homomorphism $\pi:k[{\mathcal C}^{np}_{p,m}]\rightarrow k[\binom{[N]}{p}]$ defined by $$\label{eq:pidef} \pi(\alpha^{(a)})\quad \,\, \, =\quad \sum_{\stackrel{\mbox{\scriptsize $|J|=|\alpha^{(a)}|$}} {J\equiv\alpha\,\bmod{(m+p)}}} \!\! \epsilon_J \cdot J\,.$$ Lexicographic order on the sequences $J\in\binom{[ N]}{p}$ gives a linear extension of Young’s poset. In this ordering, the initial term of (\[eq:pidef\]) is the sequence $J(\alpha^{(a)})$ defined in (\[eq:seq\_def\]). This sequence is characterized by $\, {\rm in}_\prec \,\varphi(\alpha^{(a)}) \, = \,{\rm in}_\prec \, \phi(J(\alpha^{(a)}))$. It can be checked that all other terms $ \epsilon_J \cdot J \,$ appearing in (\[eq:pidef\]) satisfy $\, J_1 < J(\alpha^{(a)})_1 \,$ and $\, J_p-J_1 \ >\ m+p $. For example, for $m = 4$, $$\pi(235^{(2)}) \quad = \quad \underline{ (5, 9, 10)} - (3, 9, 12) + (3, 5, 16) + (2, 10, 12) - (2, 5, 17) + (2,3,19),$$ $$\mbox{and} \qquad {\rm in}_\prec \bigl( \varphi( \,235^{(2)} \,) \bigr) \quad = \quad {\rm in}_\prec\bigl(\phi( 5,9,10) \bigr) \quad = \quad x_{3,5}^{(0)} x_{1,3}^{(1)} x_{2,2}^{(1)} . \qquad$$ For $J\in\binom{[ N]}{p}$, let ${\mathcal N}_J$ be the specialization of ${\mathcal N}$ where in each row $i$, all entries in columns greater than $j_i$ are set to zero. Under the identification of ${\mathcal N}$ with ${\mathcal M}(t)$, we have ${\mathcal N}_{J(\alpha^{(a)})}={\mathcal M}_{\alpha^{(a)}}$. Let $\phi_J : k [\binom{[N]}{p}] \rightarrow k[X] $ denote the $k$-algebra homomorphism which maps the formal variable $I$ to the $I$th maximal minor of ${\mathcal N}_J$. Then $\phi_J(I)$ vanishes unless $I\leq J$. In particular, if $|I|=|J|$, then $\phi_J(I)$ vanishes unless $I=J$, and in that case, it is just the product of the last non-zero variables in each row of ${\mathcal N}_J$. From this it follows that $$\label{SomeIdentities} \begin{array}{rcl} & \varphi_{\alpha^{(a)}} \quad = \quad \phi_{J(\alpha^{(a)})} \circ \pi \\\rule{0pt}{15pt} & \varphi_{\alpha^{(a)}}(\alpha^{(a)}) \,\, =\,\, \phi_{J(\alpha^{(a)})}(J(\alpha^{(a)})) \,\, = \,\, {\rm in}_\prec \, \varphi(\alpha^{(a)}) \,\, = \,\, \psi(\alpha^{(a)}). \end{array}$$ In the Plücker embedding of [*Grass*]{}$_pk^{ N}$ into ${\mathbb P}(\wedge^p k^{N})$, the Schubert variety indexed by $J$ is $$\Omega_J\quad :=\quad \{y=(y_I)\in {\it Grass}_pk^{ N}\mid y_I=0\mbox{ if }I\not\leq J \}\ .$$ The homogeneous ideal ${\mathcal I}(\Omega_J)$ which defines this Schubert variety is precisely the kernel of $\phi_J$. The following identity of ideals in $k[\binom{[N]}{p}]$ follows from the classical Plücker relations: \[prop:Schubert-ideal\] For any $J\in\binom{[ N]}{p}$ we have $$\bigcap_{I<J}{\mathcal I}(\Omega_I)\quad =\quad {\mathcal I}(\Omega_J) \, + \, \left\langle\, J \,\right\rangle\,.$$ The map $\pi:k[{\mathcal C}^{np}_{p,m}]\rightarrow k[\binom{[N]}{p}]$ induces a birational isomorphism $\pi^*:{\it Grass}_pk^{ N}\dashrightarrow K^{np}_{p,m}$. From the identification of ${\mathcal M}_{\alpha^{(a)}}(t)$ with ${\mathcal N}_{J(\alpha^{(a)})}$ and Proposition \[ItIsDominant\], we will see that $\pi^*(\Omega_{J(\alpha^{(a)})})$ is a dense subset of $\, Z_{\alpha^{(a)}}$. We also consider the image under $\pi^*$ of the Schubert varieties $\Omega_J$ for $\, J<J(\alpha^{(a)})$. \[prop:O\_J-image\] If $J<J(\alpha^{(a)})$, then $$\pi^*(\Omega_J)\quad \subset\quad \bigcup_{\beta^{(b)}<\alpha^{(a)}} Z_{\beta^{(b)}}\,.$$ [**Proof.** ]{} The inclusion $\Omega_J\subset\Omega_{J(\alpha^{(a)})}$ implies $\pi^*(\Omega_J)\subset Z_{\alpha^{(a)}}$. Since $\varphi_{\alpha^{(a)}}(\alpha^{(a)})$ is the product of leading entries in the rows of ${\mathcal N}_{J(\alpha^{(a)})}$, it follows that $\varphi_{\alpha^{(a)}}(\alpha^{(a)})$ vanishes under the specialization to ${\mathcal N}_J$, and hence $\pi(\alpha^{(a)})$ vanishes on $\Omega_J$. This implies our claim because $\bigcup_{\beta^{(b)}<\alpha^{(a)}} Z_{\beta^{(b)}}$ is defined as a subvariety of $ Z_{\beta^{(b)}}$ by the vanishing of $\alpha^{(a)}$. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ For $L<J$ in Young’s poset, define ${\mathcal N}_{J/L}$ to be the specialization of ${\mathcal N}$ where in the $i$th row, only the entries in columns $l_i,l_i+1,\ldots,j_i$ are non-zero. Then ${\mathcal M}_{\alpha^{(a)}/\beta^{(b)}}(t)$ is the specialization of ${\mathcal M}(t)$ corresponding to ${\mathcal N}_{J(\alpha^{(a)})/J(\beta^{(b)})}(t)$. Define the $k$-algebra homomorphism $\,\phi_{J/I} : k [\binom{[N]}{p}] \rightarrow k[X] \,$ by evaluating the appropriate minors on ${\mathcal N}_{J/L}$. We observe that $$\label{eq:initial} \begin{array}{rcl} \varphi_{\alpha^{(a)}/\beta^{(b)}}(\alpha^{(a)}) &=& {\rm in}_\prec \, \varphi(\alpha^{(a)}) \quad = \quad \phi_{J(\alpha^{(a)})/J(\beta^{(b)})}\bigl( J(\alpha^{(a)})\bigr), \\\rule{0pt}{16pt} \varphi_{\alpha^{(a)}/\beta^{(b)}}(\beta^{(b)}) &=& {\rm in}_\prec \, \varphi(\beta^{(b)}) \quad = \quad \phi_{J(\alpha^{(a)})/J(\beta^{(b)})}\bigl( J(\beta^{(b)})\bigr). \end{array}$$ The following lemma is very useful in our proof of Theorem \[thm:syzygy\]. \[lem:factor\] Fix $\alpha^{(a)} \in {\mathcal C}^{np}_{p,m}$ and let $f\in k[{\mathcal C}^{np}_{p,m}]$ be a quadratic form of degree $d$. 1. Suppose that $\varphi_{\beta^{(b)}}(f)=0$ for all $\beta^{(b)}<\alpha^{(a)}$. Then there exist constants $\lambda_J\in k$ with $$\varphi_{\alpha^{(a)}}(f)\quad =\quad \varphi_{\alpha^{(a)}}(\alpha^{(a)}) \,\, \cdot \!\!\!\! \sum_{\stackrel{\mbox{\scriptsize $J \in \binom{[ N]}{p}$}} {|J|+|\alpha^{(a)}|=d}} \lambda_J \cdot \phi_{J(\alpha^{(a)})}(J)\,.$$ 2. Suppose $\beta^{(b)}<\alpha^{(a)}$ and $\varphi_{\alpha^{(a)}/\gamma^{(c)}}(f)=0$ for all $\beta^{(b)}<\gamma^{(c)}\leq\alpha^{(a)}$. For some $\lambda_J\in k$, $$\varphi_{\alpha^{(a)}/\beta^{(b)}}(f)\quad =\quad \varphi_{\alpha^{(a)}/\beta^{(b)}}(\beta^{(b)}) \, \cdot \!\!\!\! \sum_{\stackrel{\mbox{\scriptsize $J \in \binom{[ N]}{p}$}} {|J|+|\beta^{(b)}|=d}} \lambda_J\cdot \phi_{J(\alpha^{(a)})/J(\beta^{(b)})}(J)\,.$$ [**Proof.** ]{} We only prove part 1. The hypothesis states that $\, \phi_{J(\alpha^{(a)})}(\pi(f)) \, = \, \varphi_{\alpha^{(a)}}(f) \, $ vanishes on all matrices ${\mathcal N}_{J(\beta^{(b)})}$ for $\beta^{(b)}<\alpha^{(a)}$. Proposition \[prop:O\_J-image\] implies that $\pi(f)$ vanishes on all Schubert varieties $\Omega_J$ with $J<J(\alpha^{(a)})$. But then, using Proposition \[prop:Schubert-ideal\], $$\pi(f)\quad \in\quad \bigcap_{J<J(\alpha^{(a)})}{\mathcal I}(\Omega_J)\quad =\quad {\mathcal I}(\Omega_{J(\alpha^{(a)})}) \, + \, \left\langle \, J(\alpha^{(a)}) \, \right\rangle.$$ This means $\,\pi(f)=g + J(\alpha^{(a)})\cdot h$, where $g\in {\mathcal I}(\Omega_{J(\alpha^{(a)})})= {\rm ker} \bigl(\phi_{J(\alpha^{(a)})} \bigr) $ and $h\in k[\binom{[N]}{p}]$ is a linear form of degree $d-|\alpha^{(a)}|$. Such a linear form can be written as follows $$h \, \quad =\quad \sum_{|J|+|\alpha^{(a)}|=d} \! \! \lambda_J\,J\,.$$ By applying the map $\, \phi_{J(\alpha^{(a)})} \,$ to both sides of the equation $\,\pi(f)=g + J(\alpha^{(a)})\cdot h$, we obtain the first assertion of Lemma \[lem:factor\]. Part 2 is proved by similar arguments. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ Our proof of Theorem \[thm:syzygy\] will show that the sums in Lemma \[lem:factor\] are actually sums of terms of the form $\,\lambda_{J(\delta^{(d)})}\cdot\varphi_{\alpha^{(a)}}(\delta^{(d)})\,$ and $\,\lambda_{J(\delta^{(d)})}\cdot \varphi_{\alpha^{(a)}/\beta^{(b)}}(\delta^{(d)}) \,$ respectively. The next lemma provides the initial step in our inductive proof of Theorem \[thm:syzygy\]. \[lem:initial\] Let $\gamma^{(c)}$ and $\delta^{(d)}$ be incomparable variables in the poset ${\mathcal C}^{np}_{p,m}$ and set $\alpha^{(a)}:=\gamma^{(c)}\vee\delta^{(d)}$. Then $ \varphi_{\alpha^{(a)}}(\gamma^{(c)}\cdot\delta^{(d)}\ -\ \gamma^{(c)}\vee\delta^{(d)}\cdot\gamma^{(c)}\wedge\delta^{(d)}) =0$. [**Proof.** ]{} We prove the lemma by inductively showing that, for each $\,\beta^{(b)}\leq\alpha^{(a)}$, $$\label{eq:lt} \varphi_{\alpha^{(a)}/\beta^{(b)}} \bigl( \, \gamma^{(c)}\cdot\delta^{(d)}\ -\ \gamma^{(c)}\vee\delta^{(d)}\cdot\gamma^{(c)}\wedge\delta^{(d)} \,\bigr) \quad = \quad 0.$$ If $\beta^{(b)}\not\leq\gamma^{(c)}\wedge\delta^{(d)}$, then $\varphi_{\alpha^{(a)}/\beta^{(b)}}(\gamma^{(c)}\wedge\delta^{(d)})$ vanishes, and either $\varphi_{\alpha^{(a)}/\beta^{(b)}}(\gamma^{(c)})$ vanishes or $\varphi_{\alpha^{(a)}/\beta^{(b)}}(\delta^{(d)})$ vanishes. This implies that (\[eq:lt\]) holds. Next suppose $\beta^{(b)}=\gamma^{(c)}\wedge\delta^{(d)}$. We claim that $\varphi_{\alpha^{(a)}/\beta^{(b)}}$ maps each variable appearing in (\[eq:lt\]) to its initial term in $k[X]$. In view of Proposition \[lem:toric\], this claim implies (\[eq:lt\]). To establish this claim, we need only show that $\varphi_{\alpha^{(a)}/\beta^{(b)}}(\gamma^{(c)})= {\rm in}_\prec\varphi(\gamma^{(c)})$, as the case for $\delta^{(d)}$ is similar and that of the other terms follow from (\[eq:initial\]). Consider the expansion of $\varphi_{\alpha^{(a)}/\beta^{(b)}}(\gamma^{(c)})$ in terms of the minors $\phi(J)$ of ${\mathcal N}_{J(\alpha^{(a)})/J(\beta^{(b)})}$. First observe that the submatrix given by the columns from $J(\gamma^{(c)})$ is block anti-diagonal, with each block either upper or lower triangular along its anti-diagonal. This is because for each $i$, $J(\gamma^{(c)})_i$ is either $J(\beta^{(b)})_i$ or $J(\alpha^{(a)})_i$, and the non-zero entries in the $i$th row of ${\mathcal N}_{J(\alpha^{(a)})/J(\beta^{(b)})}$ lie between these two numbers. Thus the contribution of term $J(\gamma^{(c)})$ to $\varphi_{\alpha^{(a)}/\beta^{(b)}}(\gamma^{(c)})$ is simply ${\rm in}_\prec \, \varphi(\gamma^{(c)})$. We claim there are no other terms. If there were another term indexed by $L$, then the $L$th maximal minor of ${\mathcal N}_{J(\alpha^{(a)})/J(\beta^{(b)})}$ would be non-zero, and so $J(\beta^{(b)})\leq L\leq J(\alpha^{(a)})$. Thus $$J(\beta^{(b)})\quad =\quad J(\gamma^{(c)})\wedge J(\delta^{(d)}) \quad \leq\quad L\wedge J(\delta^{(d)}).$$ Comparing the first components of these sequences gives $\min\{J(\gamma^{(c)})_1,J(\delta^{(d)})_1\}\leq L_1$. Since $L_1<J(\gamma^{(c)})_1$, this implies $J(\delta^{(d)})_1\leq L_1$. Similarly, using $J(\alpha^{(a)})\geq L\vee J(\delta^{(d)})$, we see that $J(\delta^{(d)})_p\geq L_p$. Lastly, as $L$ is a summand in $\pi(\gamma^{(c)})$ and $L\neq J(\gamma^{(c)})$, we have $L_p-L_1> m+p$ and thus $$m+p\ \geq\ J(\delta^{(d)})_p-J(\delta^{(d)})_1\ \geq\ L_p-L_1\ >\ m+p,$$ a contradiction, which proves the claim. Thus (\[eq:lt\]) holds for $\beta^{(b)}=\gamma^{(c)}\wedge\delta^{(d)}$. Finally, let $\zeta^{(z)}<\gamma^{(c)}\wedge\delta^{(d)}$ and suppose that (\[eq:lt\]) holds for all $\beta^{(b)}$ with $\zeta^{(z)}<\beta^{(b)}\leq \alpha^{(a)}$. Then by Lemma \[lem:factor\], $$\varphi_{\alpha^{(a)}/\zeta^{(z)}}\bigl(\gamma^{(c)}\cdot\delta^{(d)}\ -\ \gamma^{(c)}\vee\delta^{(d)}\cdot\gamma^{(c)}\wedge\delta^{(d)} \bigr) \quad =\quad \varphi_{\alpha^{(a)}/\zeta^{(z)}}(\zeta^{(z)})\cdot \sum_{J}\lambda_J \cdot \phi_{J(\alpha^{(a)})/J(\zeta^{(z)})}(J)\,,$$ the sum over sequences $J$ of rank $|J|=|\gamma^{(c)}|+|\delta^{(d)}|-|\zeta^{(z)}|$. But this exceeds the rank of $\alpha^{(a)}$, since $\zeta^{(z)}<\gamma^{(c)}\wedge\delta^{(d)}$ and $|\alpha^{(a)}|+|\gamma^{(c)}\wedge\delta^{(d)}| =|\gamma^{(c)}|+|\delta^{(d)}|$. Thus the sum vanishes and so (\[eq:lt\]) holds for all $\beta^{(b)}\leq\alpha^{(a)}$, which proves the lemma. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ [**Proof of Theorem \[thm:syzygy\].** ]{} Let $\gamma^{(c)}$ and $\delta^{(d)}$ be incomparable variables in the poset ${\mathcal C}^{np}_{p,m}$. For each $\alpha^{(a)}\in{\mathcal C}^{np}_{p,m}$ we inductively construct quadratic polynomials $S_{\alpha^{(a)}}(\gamma^{(c)},\delta^{(d)})\in k[\beta^{(s)}\mid\beta^{(s)}\leq \alpha^{(a)}]$, and then show $S_{\alpha^{(a)}}(\gamma^{(c)},\delta^{(d)})$ is in the kernel of the map $\varphi_{\alpha^{(a)}}$. The case when $\alpha^{(a)}$ is the top element in the poset ${\mathcal C}^{np}_{p,m}$ proves the theorem. These polynomials have the following restriction property: If $\beta^{(b)}<\alpha^{(a)}$, then $S_{\beta^{(b)}}(\gamma^{(c)},\delta^{(d)})$ is the image of $S_{\alpha^{(a)}}(\gamma^{(c)},\delta^{(d)})$ under the map which sets variables $\zeta^{(z)}\not\leq \beta^{(b)}$ to zero. They also have the further homogeneity properties that each non-zero term $\lambda\zeta^{(z)}\beta^{(b)}$ must have $z+b=c+d$ and satisfy the multiset equality $\beta\cup\zeta=\gamma\cup\delta$, and if it is not the initial term, then $\gamma^{(c)},\delta^{(d)}\in[\zeta^{(z)},\beta^{(b)}]$. For $\alpha^{(a)}\not\geq \gamma^{(c)}\vee\delta^{(d)}$, set $S_{\alpha^{(a)}}(\gamma^{(c)},\delta^{(d)}):=0$ and if $\alpha^{(a)}=\gamma^{(c)}\vee\delta^{(d)}$, then set $$S_{\alpha^{(a)}}(\gamma^{(c)},\delta^{(d)})\quad:=\quad \gamma^{(c)}\cdot\delta^{(d)}\ -\ \gamma^{(c)}\vee\delta^{(d)}\cdot\gamma^{(c)}\wedge\delta^{(d)}.$$ These polynomials have the restriction and homogeneity properties, and, for $\alpha^{(a)}\not> \gamma^{(c)}\vee\delta^{(d)}$, we have $\varphi_{\alpha^{(a)}}(S_{\alpha^{(a)}}(\gamma^{(c)},\delta^{(d)}))=0$, by Lemma \[lem:initial\]. Let $\alpha^{(a)}>\gamma^{(c)}\vee\delta^{(d)}$ and suppose we have constructed $S_{\beta^{(b)}}(\gamma^{(c)},\delta^{(d)})$ for each $\beta^{(b)}<\alpha^{(a)}$. By the restriction property, there is a polynomial $S'\in k[\beta^{(b)}\mid \beta^{(b)}<\alpha^{(a)}]$ which restricts to $S_{\beta^{(b)}}(\gamma^{(c)},\delta^{(d)})$ for each $\beta^{(b)}<\alpha^{(a)}$. Thus $\varphi_{\beta^{(b)}}(S')=0$ for all $\beta^{(b)}<\alpha^{(a)}$. Set $e:=|\gamma^{(c)}|+|\delta^{(d)}|$, the degree of $S'$. By Lemma \[lem:factor\], $$\label{eq:sprime} \varphi_{\alpha^{(a)}}(S')\quad =\quad \varphi_{\alpha^{(a)}}(\alpha^{(a)})\cdot \sum_{|J|+|\alpha^{(a)}|=e}\lambda_J \cdot\phi_{J(\alpha^{(a)})}(J)\,.$$ If we consider the columns of ${\mathcal M}_{\alpha^{(a)}}(t)$ involved in $\varphi_{\alpha^{(a)}}(S')$, we see that this sum is further restricted to those $J$ which satisfy the multiset equality $(\gamma \cup\delta)\setminus \alpha\equiv J\mod(m+p)$, with $J\bmod(m+p)$ consisting of distinct integers, and with $J\leq J(\alpha^{(a)})$. If there are no such $J$, then $\varphi_{\alpha^{(a)}}(S')=0$ and we set $S_{\alpha^{(a)}}(\gamma^{(c)},\delta^{(d)})=S'$. Otherwise, let $z:=c+d-a$ and $\zeta:=(\gamma\cup\delta)\setminus\alpha$. Then the summands in (\[eq:sprime\]) are among those $J$ which appear in $\pi(\zeta^{(z)})$ so we have $J(\zeta^{(z)})<J(\alpha^{(a)})$ and hence $\zeta^{(z)}<\alpha^{(a)}$. Observe that $\varphi_{\alpha^{(a)}/\zeta^{(z)}}(S')= \lambda_{J(\zeta^{(z)})} \cdot \varphi_{\alpha^{(a)}/\zeta^{(z)}}(\alpha^{(a)}\zeta^{(z)})$. Define $$S_{\alpha^{(a)}}(\gamma^{(c)},\delta^{(d)})\quad :=\quad S' - \lambda_{J(\zeta^{(z)})} \alpha^{(a)}\zeta^{(z)}.$$ We claim that if $\lambda_{J(\zeta^{(z)})}\neq 0$, then $\zeta^{(z)}\leq\gamma^{(c)},\delta^{(d)}$. If not, then every term of $S'$ contains a variable $\xi^{(x)}$ with $\zeta^{(z)}\not\leq\xi^{(x)}$, and so we must have $\varphi_{\alpha^{(a)}/\zeta^{(z)}}(S')=0$, a contradiction. We complete the proof of Theorem \[thm:syzygy\] by showing that for $\beta^{(b)}\leq \alpha^{(a)}$, $$\label{eq:indstep} \varphi_{\alpha^{(a)}/\beta^{(b)}}(S_{\alpha^{(a)}} (\gamma^{(c)},\delta^{(d)}))\quad =\quad 0.$$ If $\beta^{(b)}\not\leq \zeta^{(z)}$, then $\varphi_{\alpha^{(a)}/\beta^{(b)}}(\zeta^{(z)})=0$ and so $\varphi_{\alpha^{(a)}/\beta^{(b)}}(S_{\alpha^{(a)}} (\gamma^{(c)},\delta^{(d)}))=\varphi_{\alpha^{(a)}/\beta^{(b)}}(S')$, which is zero as $\phi_{J(\alpha^{(a)})/J(\beta^{(b)})}(J)=0$ for all $J$ which appear in $\pi(\zeta^{(z)})$. By the construction of $S_{\alpha^{(a)}}(\gamma^{(c)},\delta^{(d)}))$, we also have $\varphi_{\alpha^{(a)}/\zeta^{(z)}}(S_{\alpha^{(a)}} (\gamma^{(c)},\delta^{(d)}))=0$. Let $\xi^{(x)}<\zeta^{(z)}$ and suppose (\[eq:indstep\]) holds for all $\beta^{(b)}$ with $\xi^{(x)}<\beta^{(b)}$. Then by Lemma \[lem:factor\], $$\varphi_{\alpha^{(a)}/\xi^{(x)}}(S_{\alpha^{(a)}} (\gamma^{(c)},\delta^{(d)}))\quad =\quad \varphi_{\alpha^{(a)}/\xi^{(x)}}(\xi^{(x)})\cdot \sum_{|J|+|\xi^{(x)}|=e} \lambda_J \cdot \phi_{J(\alpha^{(a)})/J(\xi^{(x)})}(J).$$ Since $|J|=e-|\xi^{(x)}|>e-|\zeta^{(z)}|=|\alpha^{(a)}|$, each term in the right hand sum is zero. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ It is now straightforward to derive all our assertions that were left unproven so far. [**Proof of Theorem \[issagbi\].**]{} Theorem \[thm:syzygy\] together with Proposition \[lem:toric\] shows that the subduction criterion for sagbi bases (see e.g. [@CHV Proposition 1.1] or [@Sturmfels_GBCP Theorem 11.4]) is satisfied. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ [**Proof of Theorem \[thm:gbasis\].** ]{} A standard fact on sagbi bases, proved in [@CHV Corollary 2.2] or in [@Sturmfels_GBCP Corollary 11.6 (1)], states that the reduced Gröbner basis for the binomial ideal ${\rm kernel}(\psi)$ lifts to a reduced Gröbner basis for the non-binomial ideal ${\rm kernel}(\varphi)$. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ [**Proof of Proposition \[ItIsDominant\].** ]{} If $\beta^{(b)}$ is the minimal element in the poset ${\mathcal C}_{p,m}$, then $\varphi_{\alpha^{(a)}}$ and $\varphi_{\alpha^{(a)}/\beta^{(b)}}$ have the same kernel, as the varieties $Z_{\alpha^{(a)}/\beta^{(b)}}$ and $Z_{\alpha^{(a)}}$ are equal. Hence it suffices to prove the second statement about $\varphi_{\alpha^{(a)}/\beta^{(b)}}$. Clearly, the kernel of $\varphi_{\alpha^{(a)}/\beta^{(b)}}$ contains the homogeneous ideal of the skew quantum Schubert variety $Z_{\alpha^{(a)}/\beta^{(b)}}$. If this containment were proper, then we would also get proper containment at the level of initial ideals with respect to the induced partial term order, which was denoted by ${\mathcal A}^T \omega$ in [@Sturmfels_GBCP Chapter 11]. But that is impossible since every binomial relation on the monomials $\,{\rm in}_\prec \, \varphi_{\alpha^{(a)}/\beta^{(b)}}( \gamma^{(c)}) = {\rm in}_\prec \, \varphi ( \gamma^{(c)}) \,$ lifts to a polynomial which vanishes on $Z_{\alpha^{(a)}/\beta^{(b)}}$, as shown in the proof Theorem \[thm:syzygy\]. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ Applications and future directions ================================== We first summarize some algebraic consequences of our main results. \[ASL\] The coordinate ring of the quantum Grassmannian $K^q_{p,m}$ is an algebra with straightening law on the distributive lattice $C^q_{p,m}$. It has a presentation by a non-commutative Gröbner basis consisting of quadratic elements, and, in particular, it is a Koszul algebra. [**Proof.**]{} The first statement follows from Theorem \[issagbi\] and the form of the syzygies $S(\gamma^{(c)},\delta^{(d)})$ of Theorem \[thm:gbasis\]. For the second statement, consider the coordinate ring of $K^q_{p,m}$ as the quotient of the free associative algebra on $C^q_{p,m}$ modulo a two-sided ideal. By [@EPS Proposition 3.2] that two-sided ideal has a quadratic Gröbner basis, obtained from lifting the Gröbner basis in Theorem \[thm:gbasis\]. For the classical Grassmannian $(n=0)$ this result appeared in [@Gr_Hu]. The Koszul property is a well-known consequence of the existence of a quadratic Gröbner basis; see e.g. [@Gr_Hu Theorem 3] for the non-commutative version which is relevant here. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ \[C-MG\] The coordinate ring of $K^q_{p,m}$ is a normal Cohen-Macaulay and Gorenstein domain. It has rational singularities if ${\rm char}(k) = 0$ and it is $F$-rational if ${\rm char}(k) > 0$. [**Proof.** ]{} By Corollary \[ASL\] and the results of [@CHV], these properties of $K^q_{p,m}$ follow from the corresponding properties of the toric variety $T^q_{p,m}$. But these were established in [@Wagner], as $T^q_{p,m}$ is the toric variety associated to the distributive lattice ${\mathcal C}^q_{p,m}$. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ We remark that both Corollary \[C-MG\] and the analog of Corollary \[ASL\] (with the poset ${\mathcal C}^q_{p,m}$ replaced by the appropriate interval) hold for the skew quantum Schubert varieties. Our next application is the sagbi property of the row-consecutive $p \times p$-minors of a matrix of indeterminates. This result is non-trivial since the set of [*all*]{} $p\times p$-minors is not a sagbi basis in general [@Sturmfels_GBCP Example 11.3]. A finite sagbi basis for the algebra of all $p \times p$-minors was found by Bruns and Conca [@Bruns_Conca]. Let ${\mathcal L}$ be the $p(n+1)\times(m+p)$-matrix whose $i,j$th entry is $x^{(l)}_{r,j}$, where $i=pl+r$. This matrix is obtained from ${\mathcal N}$ by stacking the matrices ${\mathcal N}_0, \ldots, {\mathcal N}_n$. Let $\chi:k[{\mathcal C}^{np}_{p,m}]\rightarrow k[X]$ denote the $k$-algebra homomorphism which sends the variable $\alpha^{(a)}$ to the $\alpha$th maximal minor of the submatrix of ${\mathcal L}$ consisting of rows $a+1,a+2,\ldots,a+p$. Thus the collection of polynomials $\chi(\alpha^{(a)})$ are the row-consecutive $p\times p$-minors of ${\mathcal L}$. \[row-c-sagbi\] The set $\, \bigl\{ \chi(\alpha^{(a)}) \, : \, \alpha^{(a)} \in {\mathcal C}^{np}_{p,m} \bigr\} \,$ of row-consecutive $p \times p$-minors of a generic matrix is a sagbi basis with respect to the degree reverse lexicographic term order $\prec$ on $k[X]$. Brain Taylor has pointed out this may also be deduced from Proposition 2.7.3 of his Ph.D. Thesis [@BDT_thesis]. [**Proof.** ]{} Let $\omega$ be the weight on the variables in $k[X]$ defined by $\omega(x^{(l)}_{i,j}):= -(pl+i)^2$. Then $\chi(\alpha^{(a)}) = {\rm in}_\omega \bigl( \varphi(\alpha^{(a)})\bigr)$, the initial form of $\varphi(\alpha^{(a)})$, and we have $\,{\rm in}_\prec \bigl( \chi(\alpha^{(a)}) \bigr) \,= \, {\rm in}_\prec \bigl( \varphi(\alpha^{(a)}) \bigr) \,$ for all $\,\alpha^{(a)} \in {\mathcal C}^{np}_{p,m}$. Thus image($\varphi$) and image($\chi$) have the same initial algebra, and so we deduce the sagbi property for the polynomials $\chi(\alpha^{(a)})$ from Theorem \[issagbi\]. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ Let ${\bf w}$ denote the weight on the variables ${\mathcal C}^{np}_{p,m}$ defined by ${\bf w}(\alpha^{(a)}):=-a^2$. For every incomparable pair $\gamma^{(c)},\delta^{(d)}$ in the poset ${\mathcal C}^{np}_{p,m}$, we define the quadratic polynomial $$R(\gamma^{(c)},\delta^{(d)})\quad:=\quad {\rm in}_{\bf w} \bigl( S(\gamma^{(c)},\delta^{(d)}) \bigr),$$ where $S(\gamma^{(c)},\delta^{(d)})$ is the element of the reduced Gröbner basis for the kernel of $\varphi$. For example, $\, R(156^{(1)},234^{(2)})\,$ equals the sum of the first ten terms in (\[BigSyzygy\]). The weight ${\bf w}$ is equivalent, modulo the homogeneities of $\,{\rm kernel}(\varphi)$, to the induced weight which was denoted by ${\mathcal A}^T \omega$ in [@Sturmfels_GBCP Chapter 11]. The only-if direction in [@Sturmfels_GBCP Theorem 11.4] implies \[row-c-GB\] The reduced Gröbner basis of the kernel of $\chi$ consists of the quadratic polynomials $R(\gamma^{(c)},\delta^{(d)})$ as $\gamma^{(c)},\delta^{(d)}$ run over the set of incomparable pairs in the poset ${\mathcal C}^{np}_{p,m}$. For the Plücker ideal defining the classical Grassmannian $(n=0)$, an explicit (but non-reduced) quadratic Gröbner basis is known. It appears in the work of Hodge-Pedoe [@Hodge_Pedoe] and Doubilet-Rota-Stein [@DRS], and it consists of the van der Waerden syzygies. They are discussed in Gröbner basis language in [@Sturmfels_invariant Section 3.1]. Our next aim is to introduce an analogous non-reduced Gröbner basis for the ideal $\, {\rm kernel}(\varphi)\,$ of the quantum Grassmannian. We begin by defining the skew van der Waerden syzygies for its initial ideal $$\label{bfw} \, {\rm kernel}(\chi) \quad = \quad {\rm in}_{\bf w} ({\rm kernel}(\varphi)) ,$$ which consists of the algebraic relations among the row-consecutive minors. Given a sequence of integers $\,D\,: \, 1 \leq d_1 < \cdots < d_p \leq m+p$ and any integer $0\leq a\leq np$, let $D^{(a)}$ denote $\pm \alpha^{(a)}$, where $\alpha$ is the reordering of the sequence $D$ and $\pm$ is the sign of the permutation which sorts the sequence $D$. Let $T=\alpha^{(a)}\beta^{(b)}$ with $a<b$ be a non-standard tableau and $i$ the smallest index of a violation $\beta_i<\alpha_{i-b+a}$. Define increasing sequences $$\begin{array}{c} A\quad:=\quad \alpha_1,\ldots,\alpha_{i-b+a-1} \qquad B\quad:=\quad \beta_{i+1},\ldots,\beta_p\\ C\quad:=\quad \beta_1,\ldots,\beta_i,\alpha_{i-b+a},\ldots,\alpha_p. \end{array}$$ For a subset $I\in\binom{[p+b-a+1]}{i}$, let $C_I$ be the corresponding numbers from $C$ (in order) and $C_{I^c}$ be the other numbers from $C$, also in order. Define the [*skew van der Waerden syzygy*]{} $$\label{eq:ElQuSy} W(T) \quad:=\quad \sum_{I\in\binom{[p+b-a+1]}{i}} (A, C_{I^c})^{(a)}\ \cdot (C_I,B)^{(b)}\ .$$ \[NonRedGB\] The syzygies $\, W(T) \,$ form a Gröbner basis for the kernel of $\chi$. [**Proof.** ]{} Our choice of term order implies $\,{\rm in}_\prec \bigl( W(T) \bigr) = T = \alpha^{(a)} \beta^{(b)}$. Therefore it suffices to show that $\chi(W(T))=0$. Let $Y_1,\ldots,Y_{m+p}$ be the columns of the submatrix of ${\mathcal L}$ given by its rows $a+1,\ldots,b+p$. The skew van der Waerden syzygy $\chi(W(T))$ is an anti-symmetric, multilinear form in the $p+b-a+1$ vectors $Y_{b_1},\ldots,Y_{b_{p+b-a+1}}$ in $(p+b-a)$-space. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ The non-reduced Gröbner basis in Proposition \[NonRedGB\] can be lifted to the quantum Grassmannian as follows. We define the [*quantum van der Waerden syzygy*]{} of the non-standard tableau $T$ to be the unique quadratic polynomial $V(T)$ in ${\rm kernel}(\varphi)$ which satisfies $${\rm in}_{\bf w} \bigl( V(T) \bigr ) \quad = \quad W(T),$$ and is a sum of syzygies $S(\gamma^{(c)},\delta^{(d)})$ with ${\bf w}(\gamma^{(c)}\delta^{(d)})={\bf w}(T)$. This syzygy exists by (\[bfw\]) and it is unique because the quadratic generators of the initial ideal are $k$-linearly independent, and any two such quadratic lifts of $W(T)$ in kernel$(\varphi)$ differ by terms whose weights are strictly less than ${\bf w}(T)$. For instance, the quantum van der Waerden syzygy $\, V(156^{(1)}234^{(2)})\,$ is the polynomial with $30$ terms given in (\[BigSyzygy\]). It would be desirable to find an explicit formula, perhaps in terms of the combinatorial formalism in [@DRS], for all of the skew van der Waerden syzygies $V(T)$, but at present we have no clue how to do this. .1cm The ideal of the quantum Grassmannian $K^q_{p,m}$ contains certain [*obvious relations*]{} which are derived from the Grassmannian ${\it Grass}_pk^{m+p}$. For each $\alpha\in\binom{[m+p]}{p}$ consider the polynomial $$g_\alpha(t) \quad = \quad \alpha^{(q)}\cdot t^q+ \cdots + \alpha^{(1)}\cdot t + \alpha^{(0)}\, .$$ Given any quadratic form $F(\alpha)$ in the Plücker ideal defining ${\it Grass}_pk^{m+p}$ and any $0\leq r \leq 2q$, let $F_r$ be the coefficient of $t^r$ in polynomial $F(g_\alpha(t))$. Since $F(g_\alpha(t))$ is a polynomial in $t$ which vanishes identically on $K^q_{p,m}$, each of its coefficients $F_r$ must also vanish on $K^q_{p,m}$. We call the collection of quadratic polynomials $F_r$ as $F$ ranges over a generating set for the Plücker ideal of ${\it Grass}_pk^{m+p}$ the [*obvious relations*]{}. Rosenthal [@Rosen94] showed the following. The obvious relations define $K^q_{p,m}$ set-theoretically, provided $k$ is infinite. When $p,m \leq 2$, the obvious relations coincide with the reduced Gröbner basis of Theorem \[thm:gbasis\], in particular, they generate the ideal of the quantum Grassmannian $K^q_{2,m}$. This is no longer true for $m=p=3$. There are 35 incomparable pairs in ${\mathcal C}^0_{3,3}$, and hence $35$ linearly independent quadrics in the Plücker ideal of ${\it Grass}_3k^6$. These give rise to $35(2q+1)$ linearly independent obvious relations but when $q>0$ there are $35(2q+1)+2q-1$ incomparable pairs in ${\mathcal C}^q_{3,3}$. Thus the obvious relations do not generate the homogeneous ideal of $K^q_{3,3}$. When $q=1$ or $q=2$ then the obvious relations generate the homogeneous ideal of $K^q_{3,3}$ together with an embedded component supported on the irrelevant ideal. Thus the obvious relations define $K^q_{3,3}$ scheme-theoretically, but not ideal-theoretically. It remains an open problem whether the the obvious relations define $K^q_{p,m}$ scheme-theoretically. Batyrev et.al. [@Batyrev] applied the familiar sagbi property for the Grassmannian in the construction of certain pairs of mirror 3-folds from Calabi-Yau complete intersections in Grassmannians. We are optimistic that the results in this paper will be similarly useful for researchers in the fascinating interplay of algebraic geometry and theoretical physics. The classical straightening law for the Grassmannian and its Schubert varieties were the starting point for the general [*standard monomial theory*]{} for flag varieties. For details and references we refer to the recent work on sagbi bases by Gonciulea and Lakshmibai [@GL]. Our results suggest that standard monomial theory might be extended to certain spaces of rational curves in flag varieties generalizing the quantum Grassmannian $K^q_{p,m}$. [99]{} , [ *Conifold transitions and mirror symmetry for [C]{}alabi-[Y]{}au complete intersections in [G]{}rassmannians*]{}, Nuclear Physics B, 514 (1998), pp. 640–666. , [*Quantum [S]{}chubert calculus*]{}, Adv. Math., 128 (1997), pp. 289–305 , [*KRS and powers of determinantal ideals*]{}, Compositio Mathematica, 111 (1998), pp. 111–122. , [*The consistent selection of local coordinates in linear system identification*]{}, in Proc. Joint Automatic Control Conference, 1976, pp. 576–580. , [*Sagbi bases with applications to blow-up algebras*]{}, J. reine angew. Math., 474 (1996), pp. 113–138. , [*Foundations of combinatorial theory [I]{}[X]{}: [C]{}ombinatorial methods in invariant theory*]{}, Studies in Applied Math., 53 (1974), pp. 185–216. , [*Non-commutative [G]{}r[ö]{}bner bases for commutative algebras*]{}, Proc. Amer. Math. Soc., 126 (1998), pp. 687–691. , [*Projective resolutions of straightening closed algebras generated by minors*]{}, Adv. Math., 110 (1995), pp. 314–333. , [*Distributive lattices, affine semigroup rings and algebras with straightening laws*]{}, Advanced Studies in Pure Mathematics, Vol. 11, North-Holland, 1987, pp. 93–109. , [*Methods of Algebraic Geometry*]{}, vol. II, Cambridge Univ. Press, 1952. , [*Numerical [S]{}chubert calculus*]{}, J. Symb. Comp., 26 (1998), pp. 767–788. , [*[G]{}romov-[W]{}itten classes, quantum cohomology, and enumerative geometry*]{}, Comm. Math. Phys., 164 (1994), pp. 525–562. , [*Schubert varieties, toric varieties, and ladder determinantal varieties*]{}, Ann. Inst. Fourier (Grenoble) 47 (1997), pp. 1013–1064. , [*A smooth compactification of the space of transfer functions with fixed [M]{}c[M]{}illan degree*]{}, Acta Appl. Math., 34 (1994), pp. 329–352. , [*Degree of the generalized [P]{}l[ü]{}cker embedding of a quot scheme and quantum cohomology*]{}, Math. Ann., 311 (1998), pp. 11–26. , [*On dynamic feedback compensation and compactification of systems*]{}, SIAM J. Control Optim., 32 (1994), pp. 279–296. , [*Real rational curves in [G]{}rassmannians*]{}. MSRI preprint \# 1999-025, math.AG/9904167. , [*On parameterized rational curves in [G]{}rassmann varieties*]{}, in Space Curves, F. Ghione, C. Peskine, E. Sernesi, eds., Lecture Notes in Mathematics 1266, Springer-Verlag, 1987, pp. 251–272. , [*Algorithms in Invariant Theory*]{}, Texts and Monographs in Symbolic Computation, Springer-Verlag, 1993. height 2pt depth -1.6pt width 23pt, [*Gr[ö]{}bner Bases and Convex Polytopes*]{}, vol. 8 of University Lecture Series, American Math. Soc., Providence, RI, 1996. , [*Generalized Straightening Laws for Products of Determinants*]{}, PhD thesis, MIT, 1997. , [*Singularities of toric varieties associated with finite distributive lattices*]{}, J. Alg. Combin., 5 (1996), pp. 149–165. [^1]: Second author supported in part by NSF grant DMS-9796181. Research at MSRI supported in part by NSF grant DMS-9701755

--- abstract: 'We propose new signals for the direct detection of ultralight dark matter such as the axion. Axion or axion like particle (ALP) dark matter may be thought of as a background, classical field. We consider couplings for this field which give rise to observable effects including a nuclear electric dipole moment, and axial nucleon and electron moments. These moments oscillate rapidly with frequencies accessible in the laboratory, $\sim$ kHz to GHz, given by the dark matter mass. Thus, in contrast to WIMP detection, instead of searching for the hard scattering of a single dark matter particle, we are searching for the coherent effects of the entire classical dark matter field. We calculate current bounds on such time varying moments and consider a technique utilizing NMR methods to search for the induced spin precession. The parameter space probed by these techniques is well beyond current astrophysical limits and significantly extends laboratory probes. Spin precession is one way to search for these ultralight particles, but there may well be many new types of experiments that can search for dark matter using such time-varying moments.' author: - 'Peter W. Graham' - Surjeet Rajendran title: New Observables for Direct Detection of Axion Dark Matter --- Introduction ============ The existence of dark matter is concrete evidence for physics beyond the standard model. It is reasonable to expect the dark matter to interact non-gravitationally with the standard model. The identification of such interactions may allow new probes of the Universe and unveil new structures in particle physics. The generic expectation of the existence of new physics at the weak scale lead to the Weakly Interacting Massive Particle (WIMP) hypothesis - the dark matter is a weak scale particle that interacts with the standard model with weak scale cross-sections. A variety of experimental approaches have been developed to test the WIMP hypothesis. These include techniques to observe the direct scattering of WIMP particles with nuclei [@Goodman:1984dc] and electrons [@Graham:2012su; @Essig:2011nj], the detection of cosmic rays produced from the annihilation [@Rudaz:1987ry; @Galli:2009zc; @Slatyer:2009yq] or decay [@Arvanitaki:2008hq; @Arvanitaki:2009yb] of dark matter particles as well as searches in colliders for weakly interacting particles [@Sigurdson:2004zp; @Rajaraman:2011wf; @Bai:2010hh]. These techniques have been deployed in a variety of dedicated experiments that have placed significant constraints on the parameter space of viable WIMP dark matter [@Ahmed:2011gh; @:2013doa; @Meade:2009iu; @Essig:2012yx]. Further, direct probes at the Large Hadron Collider (LHC) of frameworks such as supersymmetry that have provided the theoretical support for WIMP dark matter have placed stringent constraints on such models [@Lowette:2012uh]. In fact, these stringent constraints arise from the assumption that the new physics at the LHC always produces a metastable WIMP particle. Indeed, the bounds on these frameworks are significantly alleviated by allowing for such WIMP particles to decay rapidly within the collider [@Graham:2012th], precluding a cosmological role for them. Given our ignorance of the ultra-violet framework of particle physics and the nature of dark matter, it is important to develop techniques to search for a wide variety of interactions that could be carried by the dark matter. Ultra-light scalars such as axions ($a$) and axion-like-particles (ALPs) with masses $m_a$ significantly smaller than the weak scale are a well motivated class of dark matter candidates. These particles emerge naturally as the Goldstone bosons of global symmetries that are broken at some high scale $f_a$ [@Peccei:1977hh; @Peccei:1977ur; @Weinberg:1977ma; @Wilczek:1977pj; @Kim:1979if; @Shifman:1979if; @Dine:1981rt; @Zhitnitsky:1980he; @Khlopov:1999rs] (see section \[Sec:Overview\] for an overview of such particles). Their Goldstone nature is manifest in their derivative interactions with the standard model: $$\label{Eqn:Terms} \frac{a}{f_a} F_{\mu \nu} \tilde{F}^{\mu \nu}, \, \frac{a}{f_a} G_{\mu \nu} \tilde{G}^{\mu \nu}, \frac{\partial_{\mu} a}{f_a} \bar{\Psi}_{f} \gamma^{\mu} \gamma_5 \Psi_{f}.$$ Here, $F_{\mu \nu}$ and $G_{\mu \nu}$ represent the field strengths of electromagnetism and QCD respectively and $\Psi_f$ denotes a standard model fermion. The first of these interactions couples axions (and ALPs) to photons and is used in a variety of experiments to search for the axion. These include methods to search for the conversion of dark matter axions into photons in the presence of a background magnetic field [@Sikivie:1985yu; @Asztalos:2009yp], the detection of axions produced in the Sun [@Irastorza:2006gs] and axion-aided transport of photons through optical barriers [@Ehret:2009sq]. These experiments can search for axions with $f_a \lessapprox 10^{12}$ GeV, with limited ability to go above this scale. In this paper, we argue that the other operators in can also be used to search for axion (and ALP) dark matter. These operators are particularly useful in probing dark matter axion parameter space where $f_a \gtrapprox 10^{14}$ GeV. An ultra-light particle like the axion ($m_a \ll $ meV) can be a significant fraction of the dark matter only if it has a large number density, leading to large field occupation numbers. Consequently, axion dark matter can be viewed as a background classical field oscillating at a frequency equal to its mass [@Dine:1982ah; @Preskill:1982cy]. Conventional axion experiments search for energy deposition from this background classical field. But, a background classical field can also lead to additional physical effects such as giving rise to phase differences in local experiments. For example, gravitational wave experiments [@Abbott:2007kv; @Accadia:2012zz; @Dimopoulos:2007cj; @Dimopoulos:2008sv; @Hogan:2010fz; @Graham:2012sy] aim to detect gravitational waves through the phase differences created by the wave instead of the unobservably small rate with which a single graviton would scatter and deposit energy in a detector. Similarly, we show that the classical axion (and ALP) dark matter field through its interaction with the operators in , leads to energy (and phase) differences in atomic systems. These phase differences manifest themselves as time varying moments that can be used to search for such dark matter. Essentially, axion dark matter can be thought of as an oscillating value of the Strong CP angle $\theta_\text{QCD}$, which can lead to new ways to detect it. For example, it was pointed out in [@Graham:2011qk] that the second operator in gives rise to a time varying nuclear electric dipole moment. These moments oscillate at a frequency equal to the mass of the axion, which can span the frequency space kHz-GHz. Even though the axion arises from ultra-high energy physics, its mass is small enough to be accessible in the laboratory. These frequencies though are rapid enough that conventional laboratory searches for such moments (such as a nuclear electric dipole moment) would have a reduced sensitivity to them since such experiments (for example, [@Romalis:2001qb]) gain sensitivity through long ($\gtrapprox 1$ s) interrogation times. Further, conventional searches for time dependent moments have focussed on time variations occurring over the Hubble scale [@Hubble]. These time variations require the existence of scalar fields with masses comparable to the Hubble scale and such supremely light masses are difficult to obtain without fine-tuning [@Dvali:2001dd]. In contrast, time variations in the frequency range kHz - GHz emerge naturally in many axion (and ALP) models (see section \[Sec:Overview\]). In this paper, we will show that the time varying moments induced by the dark matter axion (and ALP) can couple to nuclear or electronic spin leading to a precession of the spin. While there may be other experimental strategies to measure these time varying moments, we highlight the technique described in [@NMR; @paper]. In this technique, the induced spin precession changes the magnetization of a sample of material, which can be observed with precision magnetometry. The signal in such an experiment benefits from the large number of spins that can be obtained in a condensed matter system and the availability of high precision SQUID and SERF magnetometers. The time varying nature of the signal can be used to devise resonant schemes that can significantly boost their detectability. Further, since this time variation occurs at a frequency set by fundamental physics, it can also be helpful in combatting systematic noise sources that are encountered in searching for a time independent moment. We begin by briefly reviewing the physics of the axion in section \[Sec:Overview\]. In this section, we discuss the theoretical origins of axions and ALPs, their parameter space, the current constraints on this parameter space and the region where they can be dark matter. We also review the salient features of axion (and ALP) dark matter. Following this review, we discuss current bounds on such moments and estimate the reach of the precision magnetometry experiments discussed above to probe this parameter space. Axions and Axion-like-Particles (ALPs) {#Sec:Overview} ====================================== Axions and ALPs are generically expected in many models of physics beyond the standard model [@Svrcek:2006yi]. They are the Goldstone bosons of global symmetries that are broken at some scale $f_a$. The Goldstone nature of their origin is manifest in the fact that all their interactions are suppressed by the scale $f_a$ and they are coupled derivatively in these interactions (for example, the operators in ). If they were pure Goldstone bosons, they would be completely massless and would not be the dark matter. However, their pure Goldstone nature could be broken leading to a generation of a mass for them. This mass generation can occur if the associated global symmetry is anomalous. For example, when this global symmetry has an anomaly with QCD, non-perturbative dynamics at the QCD scale can generate a mass $m_a \sim \frac{\Lambda_{\text{QCD}}^2}{f_a}$ or more precisely $$\label{eqn: axion mass} m_a \approx 6 \times 10^{-10} \text{ eV} \, \left(\frac{10^{16} \text{ GeV}}{f_a}\right)$$ for the Goldstone boson with about a 10% uncertainty due to QCD [@Cadamuro:2010cz]. This Goldstone boson is called the axion. While it was initially introduced to dynamically solve the strong CP problem [@Peccei:1977hh; @Peccei:1977ur], the underlying mechanism responsible for its dynamics is very general. For example, if the associated symmetry had a mixed anomaly with some other gauge group that also had strong dynamics, that Goldstone boson would also acquire a mass $\sim \frac{\Lambda_s^2}{f_a}$, where $\Lambda_s$ is the scale where the new gauge group becomes strong. A small breaking of the global symmetry is another source of mass. One source of such a small breaking could be quantum gravity which is generically expected to violate global symmetries. In this case, the Goldstone bosons would acquire a mass proportional to the breaking of this global symmetry, leading to a tiny mass for them. Goldstone bosons that do not acquire a mass from QCD are called axion-like-particles (ALPs). Thus, while the mass of the axion depends upon only one free parameter $f_a$, the mass of ALPs depends upon additional parameters. In this paper, we will be interested in ALPs whose masses are comparable to that of the axion ($\sim$ kHz - GHz). The experimental limits and signatures discussed will apply to both axions and ALPs and henceforth we will refer to them both as ALPs. There are several astrophysical bounds on the ALP operators in and they rule out ALPs with $f_a \lessapprox 10^{9}$ GeV [@Raffelt:2006cw]. Theoretical prejudice suggests that $f_a$ should lie around the fundamental scales of particle physics such as the grand-unified ($\sim 10^{16}$ GeV) or the Planck ($\sim 10^{19}$ GeV) scales, where we expect other symmetries of nature to be broken [@Svrcek:2006yi]. A weak upper bound of $f_a \lessapprox 10^{12}$ GeV can also be placed on the axion based upon its cosmological abundance [@Dine:1982ah; @Preskill:1982cy]. This bound is specific to the QCD axion and arises by assuming that the axion field is displaced $\mathcal{O}\left(1\right)$ from its minimum in the early Universe. This bound can be relaxed, allowing for larger values of $f_a$, if the field displacement was not as large. In fact, the maximum field displacement scales $ \propto \frac{1}{\sqrt{f_a}}$ implying that $f_a \approxeq 10^{16}$ GeV (the grand unified scale) would be allowed as long as the initial displacement was $\sim \mathcal{O}\left(1 \% \right)$. As argued in [@Linde:1987bx], inflationary cosmology can provide a natural mechanism that would allow a range of initial values for the axion field. As long as the scale of inflation is lower than $f_a$, the pre-inflationary space-time can have a generic inhomogeneous distribution of the axion field. Inflation can make any small part of this initial space-time into our Hubble patch, allowing for a uniform axion field value throughout our patch. The field value in our Hubble patch would be equal to the local field value of the pre-inflationary space-time that inflated to become our patch. Since a range of initial field values are scanned in the pre-inflationary space-time, the axion field can take any value in our patch. It is difficult to estimate the likelihood of any particular value in our patch since inflationary space-times do not possess a natural measure [@Linde:2007nm]. For any $f_a \gtrapprox 10^{9}$ GeV, a cosmologically viable axion field exists as long as the initial field value is appropriately small ($\propto \frac{1}{\sqrt{f_a}}$). In the canonical axion window $f_a \sim 10^{12}$ GeV, it is assumed that it is natural for the axion field to have a large $\mathcal{O}\left(1\right)$ initial displacement, forcing the scale $f_a$ to be much lighter than the fundamental scales of particle physics. But, having $f_a$ much lighter than these fundamental scales creates another hierarchy problem. The constraint [@Dine:1982ah; @Preskill:1982cy] on the cosmological abundance of the axion requires one small number - this could be the initial value of the axion field or the scale $f_a$ (measured in terms of the fundamental scales) or any combination of these. Given our ignorance of the ultraviolet structures of particle physics and the difficulties of obtaining well defined measures on initial conditions in inflationary cosmology, there is no strong reason to prefer any particular value of $f_a$. It should be noted that the details of the cosmological bound discussed above are specific to the QCD axion since this bound depends upon the details of the mass generation mechanism. For a general ALP, depending upon the mass generation mechanism, similar bounds could be placed. However, much like the case of the axion, bounds based on the cosmological abundance of ALPs are subject to similar uncertainties. It is thus important to search for ALPs over their entire range of parameter space, just above $f_a \gtrapprox 10^{9}$ GeV. The axion can constitute a significant fraction of the dark matter when $f_a \gtrapprox 10^{11}$ GeV, while for a generic ALP, the exact region where it can be the dark matter depends upon its mass generation mechanism. The phenomenology of cosmic ALP dark matter is insensitive to the details of the mechanism responsible for generating its mass. The classical field that describes ALP dark matter can be expressed as $a_0 \cos \left(m_a t\right)$. The amplitude $a_0$ is obtained by setting the energy density in the field $\frac{1}{2} m_a^2 a_0^2$ equal to the local dark matter density $\rho_{\text{dm}}\sim 0.3 \, \frac{\text{GeV}}{\text{cm}^3}$. This energy density can be understood as the result of the oscillations of the classical dark matter ALP field with amplitude $a_0$ and frequency $m_a$. The temporal coherence of these oscillations in an experiment is limited by motion through the spatial gradients of the ALP field. The gradients are set by the de Broglie wavelength of the ALP $\frac{1}{m_a v} \sim 1000 \, \text{km} \, \left(\frac{\text{MHz}}{m_a}\right)$, where $v \sim 10^{-3}$ is the galactic virial velocity of the ALP dark matter. Since the velocity between the experiment and the dark matter is also $v$, the time $\tau_a$ over which the ALP will interact coherently is at least $\tau_a \sim \frac{2 \pi}{m_a v^2} \sim 10^6 \, \frac{2 \pi}{m_a} \sim 1 \,s \, \left(\frac{\text{MHz}}{m_a}\right)$. In other words, the ALP’s frequency $m_a$ is broadened by its kinetic energy $m_a v^2$. Lighter ALPs, corresponding to larger values of $f_a$, are coherent for longer. For the rest of this paper, we will assume that ALPs constitute a significant fraction of dark matter and propose techniques utilizing their classical nature to search for them. For another proposal using the fact that the axion acts like a classical field causing an oscillating $\theta$ angle, but using a different axion coupling from us, see for example [@Hong:1991fp]. Current Searches {#sec: current searches} ================ Essentially all experiments attempting to directly detect axions or ALPs use the ALP coupling to electromagnetic fields $$\label{eqn: axion photon coupling} \mathcal{L} \ni g_{a\gamma\gamma} a F \tilde{F}$$ where $F_{\mu \nu}$ is the field strength of electromagnetism and $a$ is the ALP field. Such fields are easily manipulatable in the laboratory. This has allowed a wide range of tests from microwave cavity experiments such as ADMX, to helioscopes such as CAST, to light-through-walls experiments such as ALPS [@Asztalos:2009yp; @Irastorza:2006gs; @Ehret:2009sq]. There are also many astrophysical observations which limit this coupling of axions and ALPs to photons [@Raffelt:2006cw]. A summary of all current constraints on this parameter space is reproduced in Fig. \[Fig:photon\] from [@Ringwald:2012cu] (see also [@Hewett:2012ns; @Arias:2012az]). Although this space is well-covered by experiments and astrophysical bounds at higher mass and coupling $g_{a\gamma\gamma}$, it is challenging to search in the low mass and coupling region. A large piece of the parameter space for light ALPs is currently not reachable. Indeed, most of the masses for which the QCD axion could be dark matter are not reachable by current experiments. ![ \[Fig:photon\] Reproduced with permission from Fig. 2 of A. Ringwald [@Ringwald:2012hr], this figure is adapted from [@Hewett:2012ns; @Arias:2012az; @Cadamuro:2011fd] (see also [@Ringwald:2012cu]). ALP parameter space in axion-photon coupling (as in Eq. ) vs mass of ALP. The QCD axion is the yellow band. The width of the yellow band gives an indication of the model-dependence in this coupling, though the coupling can even be tuned to zero.](axion-photonplot.pdf){width="5"} For the QCD axion there is a constrained set of predictions for $g_{a\gamma\gamma}$ as a function of the axion decay constant $f_a$, $g_{a\gamma\gamma} \propto f_a^{-1}$. Thus the QCD likely lies in the yellow band in Fig. \[Fig:photon\]. It may be the dark matter over a wide range of masses $\text{meV} \lesssim m_a \lesssim 10^{-12}$ eV, with axion decay constant anywhere in the range $f_a \gtrsim 10^{10}$ GeV all the way up to the Planck scale $\sim 10^{19}$ GeV. The only experiment which can currently reach the QCD axion in this range is ADMX and it cannot probe the region of high $f_a \gtrsim 10^{13}$ GeV, or $m_a \lesssim 10^{-6}$ eV. Coming upgrades to ADMX, such as ADMX-HF, may probe higher masses (lower $f_a$). However, it is very challenging for microwave cavity searches to get to higher $f_a$ because these experiments search for axion to photon conversion through the coupling in Eq. . The amplitude for this process necessarily goes as the square of the coupling $\propto f_a^2$. Further, the cavity must be on resonance with the axion mass (frequency) in order to enhance the signal. This requires the cavity to be approximately the size of the axion wavelength $\sim m_a^{-1}$. For GUT scale axions $f_a \sim 10^{16}$ GeV gives $m_a^{-1} \sim 300$ m, which makes for a rather large cavity. If the cavity size cannot be increased with the wavelength, then the sensitivity of the experiment will fall off even more rapidly with increasing $f_a$. While microwave cavities make excellent axion detectors for the lower $f_a$, they are many orders of magnitude away from detecting axions with higher $f_a$. Similarly, other proposals using the coupling to electromagnetism in Eq. , e.g. the interesting, recent proposal of using a dish detector [@Horns:2012jf], may work at lower $f_a \lesssim 10^{13}$ GeV, but cannot reach higher $f_a$. Given how well-motivated axion dark matter is, it is important not to miss a such a large piece of its parameter space. It is therefore crucial to design experiments that can detect axions or ALPs with masses below $\mu$eV. This is clearly challenging using the coupling in Eq. . Axion-EDM Coupling {#Sec: EDM} ================== There are two general problems to using the axion-photon coupling Eq.  for detection of light, weakly-coupled axions. First, all experiments are measuring rates for axion to photon conversion so they go as amplitude squared $\propto g_{a\gamma\gamma}^2$. In fact, in the case of light-through-walls experiments since a photon must convert to an axion and then convert back to a photon the rate goes as $g_{a\gamma\gamma}^4$. Second, the operator in Eq.  usually suppresses the signal in a possible experiment by the ratio of the size of the experiment over the wavelength of the axion (often squared). This arises because $F \tilde{F}$ is a total derivative and therefore the operator in Eq.  can be thought of as having a derivative on the axion field. For high mass axions this is not a problem, microwave cavities can easily be the same size as the axion wavelength. But for low mass axions this is a large suppression for any laboratory sized experiment. A New Operator for Axion Detection ---------------------------------- To detect low mass axions or ALPs we must avoid these problems. We therefore propose using a different coupling instead of the one in Eq. . The QCD axion solves the strong CP problem, the problem that a nucleon electric dipole moment (EDM) would be generated by the $\theta$ parameter of QCD. This parameter arises in the Standard Model lagrangian term $\theta G \tilde{G}$, where $G$ is the QCD field strength. The QCD axion solves this problem essentially by turning $\theta$ into the dynamical axion field. Thus the QCD axion is defined by its coupling $\propto \frac{a}{f_a} G \tilde{G}$. This means the axion gives an effective $\theta$ angle. This will then give rise to an EDM for nucleons sourced by the axion. Because the axion is a dynamical field, this EDM will change in time, giving rise to unique signals. We propose to search for this time-varying EDM as a new way to detect axions or ALPs. This EDM can be expressed as the operator coupling the axion to nucleons $N$: $$\label{eqn: axion EDM coupling} \mathcal{L} \ni -\frac{i}{2} g_d \, a \, \bar{N} \sigma_{\mu \nu} \gamma_5 N F^{\mu \nu}.$$ where $g_d$ is a coupling constant we introduce. In general a new light particle or ALP could have a coupling like this as well. The nucleon EDM generated by this operator is $$\label{eqn: nucleon EDM} d_n = g_d a$$ where $a$ is the value of the local axion or ALP field at the position of the nucleus. Thus, any nucleon in the axion or ALP dark matter will acquire an EDM proportional to the dark matter field. Note that the operator in Eq.  is a non-derivative coupling for the ALP so it naturally avoids the wavelength suppression discussed above for low mass axions or ALPs. For the QCD axion the nucleon EDM created by its QCD coupling is determined in terms of the axion decay constant $f_a$: $$\label{eqn: qcd axion nucleon EDM} d_n^\text{QCD} \approx 2.4 \times 10^{-16} \frac{a}{f_a} e \cdot {\text{cm}}$$ with about a 40% uncertainty [@Pospelov:1999ha]. Thus, for the QCD axion our coupling $g_d$ is determined by the axion decay constant $f_a$ as $$\label{Eqn: QCD axion gd} g_d^\text{QCD} \approx \frac{2.4 \times 10^{-16}}{f_a} \, e \cdot {\text{cm}}\approx 5.9 \times 10^{-10} \left( \frac{m_a}{\text{eV}} \right) {\text{GeV}}^{-2}$$ where $m_a$ is the axion mass from Eq. . For an ALP of course, the coupling $g_d$ is in general arbitrary, independent of the mass of the ALP. Note that one useful feature of searching for this EDM is that it is naturally a measurement of an amplitude (or phase), not a rate. Thus the signal in an experiment will only be proportional to one power of $g_d$, or one power of $\frac{1}{f_a}$ for the QCD axion. This is in contrast to all other experiments which are measuring rates and whose signals therefore are $\propto g_d^2$ or $\propto \frac{1}{f_a^2}$. Measuring an amplitude and not a rate makes it much easier to push the sensitivity up to high $f_a$ (low axion couplings). Further, the actual size of the EDM is set by the product $g_d a$, where $a$ is the local dark matter, axion or ALP, field. As discussed in Section \[Sec:Overview\] this is approximately $a \approx a_0 \cos \left( m_a t \right)$. The amplitude of this field, $a_0$, is known if we require that this field makes up (all of) the local dark matter density $$\label{eqn: dm abundance} \rho_\text{DM} = \frac{1}{2} m_a^2 a_0^2 \approx 0.3 \frac{{\text{GeV}}}{{\text{cm}}^3}$$ since the field $a$ is essentially a free scalar field with this mass term as the leading term in its potential. This then determines the nucleon EDM generated by ALP dark matter from Eqs. and to be $$\label{eqn: ALP EDM} d_n = g_d \frac{\sqrt{2 \, \rho_\text{DM}}}{m_a} \cos \left( m_a t \right) \approx \left( 1.4 \times 10^{-25} \, e \cdot {\text{cm}}\right) \left( \frac{\text{eV} }{m_a} \right) \left( g_d \, {\text{GeV}}^2 \right) \cos \left( m_a t \right)$$ For the QCD axion, since the axion mass from Eq.  scales as $m_a \propto \frac{1}{f_a}$, taking the axion to be all of the dark matter fixes the effective $\theta$ angle of the axion to be independent of $f_a$: $$\label{qcd axion theta} \frac{a_0}{f_a} \approx 3.6 \times 10^{-19}$$ So for the QCD axion dark matter, the nucleon EDM it induces from Eq. , is actually independent of $f_a$ $$\label{eqn: qcd axion edm} d_n^\text{QCD} \approx \left( 9 \times 10^{-35} \, e \cdot {\text{cm}}\right) \cos \left( m_a t \right).$$ Thus we have found a physical effect that does not decouple as $f_a$ increases. This useful fact allows experiments searching for this EDM to probe high $f_a$ axions. Note that the EDM induced by the axion from Eq.  is small. Of course the EDM induced by a general ALP, Eq. , is arbitrary. For the QCD axion though, the EDM is about eight orders of magnitude smaller than the current bound on the static nucleon EDM, though these may improve in the future. However the reason to believe that this EDM may be detectable is that it is not a static EDM. The axion or ALP field oscillates with a frequency approximately equal to its mass Eq. . Thus the nucleon EDM from Eq.  also oscillates with this frequency. This allows a resonant enhancement of the signal in an experiment (similar to ADMX) which can be many orders of magnitude. Further, backgrounds for an oscillating signal are very different, and usually much more controllable, than backgrounds for a static signal. The frequency of the oscillation is set by high energy physics, independent of anything in the laboratory setup. Thus every nucleon bathed in axion (or this type of ALP) dark matter has an oscillating EDM with frequency set by the mass of the axion field. Further, this oscillation will be in phase for all nucleons within the axion dark matter coherence length $\sim \frac{1}{m_a v}$, see Section \[Sec:Overview\]. One immediate idea is to search for the electromagnetic radiation given off by all these nuclear “antennas." We could not find a plausibly observable signal of this radiation, using either laboratory or astrophysical (e.g. neutron star) sources. One problem is that the radiation rate is proportional to the square of the EDM, and additionally to the small frequency or mass of the axion or ALP. We have already proposed one idea using interferometry of cold molecules to detect axion dark matter using the nuclear EDM [@Graham:2011qk]. We also believe that experiments based on nuclear magnetic resonance (NMR) techniques to measure spin precession may allow detection of ALPs and even the QCD axion over many orders of magnitude in parameter space [@NMR; @paper] (see Figure 2 of [@NMR; @paper] for sensitivity estimates). A New ALP Parameter Space ------------------------- Using the EDM coupling in Eq.  naturally suggests a new parameter space in which to search for axion-like particles. The EDM coupling is naturally generated for the QCD axion. Of course, the scalar field $a$ in Eq.  does not have to be the QCD axion. If not, it would be a type of ALP. Most considerations of ALPs have focused on fields that couple to electromagnetism through the coupling in Eq. . However there is no reason that an ALP has to couple only, or at all, through the electromagnetic coupling. The space of ALPs that have EDM couplings as in Eq.  is a new parameter space that is worth exploring since this coupling motivates new experimental signatures and appears promising for axion detection. Figure \[Fig:EDM\] shows the parameter space for an ALP in the space of the EDM coupling $g_d$ in Eq.  versus the mass of the ALP. ![ \[Fig:EDM\] ALP parameter space in EDM coupling Eq.  vs mass of ALP. The green region is excluded by excess cooling in SN1987A. The blue regions are excluded by the best static nucleon EDM experiments. The purple band is the QCD axion region, with dark purple showing the most theoretically-motivated region for QCD axion dark matter. The width of the band shows the uncertainty in the calculation of the axion-induced EDM and the axion mass. The ADMX region shows the part of QCD axion parameter space which has been covered (darker blue) [@Asztalos:2009yp] or will be covered in the near future (lighter blue) [@ADMXwebpage; @snowdarktalk] by ADMX. For the static EDM and ADMX bounds we assume that the ALP makes up all of the dark matter. See also Figure 2 of [@NMR; @paper] for sensitivity of the proposed NMR experiment.](dipoleplot.pdf){width="6"} The QCD axion may lie anywhere on the purple line in Figure \[Fig:EDM\]. It lies on a line because it has only one free parameter, $f_a$, which determines its mass and $g_d$ as in Eqs.  and . The width of the QCD axion line shows an estimate of the theoretical uncertainty in the calculation of these quantities, as described above. The darker purple section of the line shows where the QCD axion may make up the dark matter. For values of $m_a \gtrsim$ meV the QCD axion cannot have enough abundance to make up all of the dark matter, as discussed in Section \[Sec:Overview\], though it may be a subdominant component. The lower edge of the dark purple region, where $m_a \approx 5 \times 10^{-13}$ eV, is where the axion decay constant is around the Planck scale, $f_a = M_\text{pl} \approx 1.2 \times 10^{19}$ GeV. It is not clear if it is possible to make models of the QCD axion with masses below this scale, since that would require $f_a > M_\text{pl}$. For that reason we keep this area light purple. It is certainly worth searching for such axions though, since although it is not obvious how to make a model of such an axion without a full understanding of quantum gravity, it may well be possible. We show the ADMX constraints as the blue region in Figure \[Fig:EDM\]. Since ADMX searches for the axion electromagnetic coupling Eq.  these constraints cannot properly be put on this figure for the QCD or EDM coupling. But ADMX can constrain most QCD axion models, where the coupling to photons is related to the coupling to gluons. So we show the ADMX constraints in Figure \[Fig:EDM\] as constraints just on the QCD axion parameter space. The darker blue represents the current bounds from ADMX [@Asztalos:2009yp] while the lighter blue represents the region ADMX will cover in the near future [@ADMXwebpage; @snowdarktalk]. The constraint from SN1987A arises from excess cooling caused by axion emission. This constraint usually arises from the other axion-nucleon coupling Eqn.  as in [@Raffelt:2006cw]. We adapted it to the higher dimension EDM operator Eqn.  by calculating the axion emission rate from the SN using the process $N + \gamma \to N + a$. We approximate this cross section as $\sigma v \approx g_d^2 T^2$ where $T \approx 30$ MeV is the temperature of the SN. As an approximation we assume the axion is produced with energy equal to the average photon energy in the SN $\approx \frac{\pi^4}{30 \zeta(3)} T \approx 2.7 \, T$. The energy lose rate per unit volume is then $\approx 2.7 \, T n_\gamma n_N \sigma v$, where $n_N \approx \frac{1.8 \times 10^{38}}{\text{cm}^3}$ is the number density of nucleons and $n_\gamma \approx \frac{2 \zeta(3)}{\pi^2} T^3$ is the number density of photons in the supernova. Dividing this by the mass density of the supernova $\rho \approx 3 \times 10^{14} \frac{\text{g}}{{\text{cm}}^3}$ gives the cooling rate per unit mass. This cooling rate is then compared to the bound $10^{19} \, \text{erg} \, \text{g}^{-1} \, \text{s}^{-1}$ from [@Raffelt:2006cw]. This gives a bound on the EDM coupling $g_d \lesssim 4 \times 10^{-9} \, {\text{GeV}}^{-2}$. This bound is shown as the green region in Figure \[Fig:EDM\]. Of course this calculation is only a rough approximation, but it is good enough for our purposes since the parameter space we are interested in extends many orders of magnitude below this bound. We leave a more precise calculation for future work. The experiments searching for (static) nuclear EDMs have drastically reduced sensitivity to an oscillating EDM of the type we are considering. These experiments gain sensitivity by integrating for relatively long periods of time compared to the period of the oscillating EDM we are considering, Eq. . Since this oscillating EDM has an average value of zero these static EDM experiments are not well-suited to searching for this signal of an ALP. The limits set by the static EDM experiments are shown in the dark blue region of Figure \[Fig:EDM\] [@Baker:2006ts; @Harris:1999jx]. These experiments have not done a specific search for an oscillating EDM so we calculated an approximate limit using their limits on the static nuclear EDM as follows. We assume these experiments gain sensitivity linearly in time over their shot time. They measure the total precession of the neutron spin when exposed to an electric field for a shot time, which is $t_{\text{shot}}= 130$ s for [@Baker:2006ts]. However since the oscillating EDM averages to zero, only the last fraction of a period of oscillation will cause a net precession. Thus these experiments lose a factor of $\sim t_{\text{shot}}m_a$ in sensitivity to an oscillating EDM. Further, the signal of this oscillating EDM will be stochastic from shot to shot and so we assume roughly that the sensitivity will not improve with the number of shots. We assume this means the static EDM experiments lose another factor of $\sim \sqrt{N_{\text{shot}}}$ in their sensitivity. The number of shots was $N_{\text{shot}}= 221600$ in [@Baker:2006ts]. So finally we estimate the limit these experiments set on the amplitude of an oscillating EDM is $\sim d_N^{\text{static}}\left( t_{\text{shot}}m_a \right) \sqrt{N_{\text{shot}}}$ where $d_N^{\text{static}}= 2.9 \times 10^{-26} \, \text{e} \cdot {\text{cm}}$ is the limit they set on the static nucleon EDM. Under the assumption that the ALP makes up all of the dark matter we can translate this to a bound on the coupling $g_d$ using Eq. . This gives a bound of $g_d \lesssim 1.8 \times 10^{19} \left( \frac{m_a}{{\text{eV}}} \right)^2 {\text{GeV}}^{-2}$, as shown in Figure \[Fig:EDM\]. This bound is only a rough approximation but it is good enough for our purpose since it is a very weak bound on these ALPs. Ideally the experiments themselves would reanalyze their data to directly search for an oscillating EDM. Since these experiments are looking for a static nucleon EDM they are simply not designed appropriately to look for the oscillating EDM signal we are discussing. The EDM coupling in Eq.  is an interesting and useful one to consider for axion or ALP detection partially because it is a non-derivative coupling. For the QCD axion this coupling arises completely naturally. For an ALP this coupling may also be natural. However it can induce a mass for the ALP through a two-loop diagram. Thus the most natural part of such ALP parameter space, Figure \[Fig:EDM\], is either below the QCD axion line or above it and within several orders of it. ALP parameter space that is many orders of magnitude above the QCD axion line may become tuned, though at some point of course is ruled out by SN1987A anyway. Of course, this concern applies to the scalar ALP model we have considered, other fields or models could change this. Having ALP dark matter with the coupling we are considering changes the status of the nucleon EDM from a fundamental constant of nature to a parameter dependent on the local field value. Thus we see that the nucleon EDM may be expected to change in time (and space), likely oscillating at high frequencies $\sim$ kHz to GHz. It is thus important to consider the limits that existing experiments put on the parameter space in Figure \[Fig:EDM\]. Further, this parameter space has not been considered before. Therefore it is also important to design experiments which are optimized to search for this signal. Beyond the cold molecule [@Graham:2011qk] and NMR techniques [@NMR; @paper] that we have considered there could be many possibilities for other experiments, for example using proton storage rings [@Semertzidis:2011qv; @Orlov:2006su; @Semertzidis:2003iq; @yannis]. Axial Nuclear Moment {#Sec: axial nuclear} ==================== The third operator in gives rise to the coupling $$\label{eqn:gaNN} \mathcal{L} \supset g_\text{aNN} \left( \partial_\mu a \right) \bar{N} \gamma^\mu \gamma_5 N$$ between the ALP and the axial nuclear current. For the QCD axion, this coupling usually exists to both protons and neutrons and is approximately $g_\text{aNN} \sim \frac{1}{f_a}$. Current bounds on this operator arise from two sources. First, this operator allows an accelerated nucleon to lose energy through ALP emission. These emissive processes are constrained by observations of the cooling rates of supernova, imposing an upper bound on $g_{aNN} \lessapprox 10^{-9} \text{ GeV}^{-1}$ [@Raffelt:2006cw]. Second, this operator leads to a force between nucleons through the exchange of ALPs. This force is spin dependent with a range $\sim m_a^{-1}$ [@Moody:1984ba]. Such spin-spin interactions have been searched for using a variety of spin polarized targets, but the limits on $g_{aNN}$ from them [@Vasilakis:2008yn] are weaker than the constraints from supernova emission [@Raffelt:2006cw; @Engel:1990zd] (see Fig. \[Fig:Nucleon\]). The above effects do not require the presence of a background ALP field. In the presence of such a field, for example as ALP dark matter, the non-relativistic limit of this operator leads to the following term in the nucleon Hamiltonian $$H_N \supset g_{aNN} \vec{\nabla} a.\vec{\sigma_N}$$ where $\sigma_N$ is the nucleon spin operator. Much like a spin precessing around a background magnetic field, this coupling causes spin precession of a nucleon around the local direction of the ALP momentum $\vec{\nabla} a$. The motion of the Earth through the galaxy leads to a relative velocity between it and the dark matter. As long as the nucleon spin is not aligned with this velocity, the spin will precess about this ALP dark matter “wind". Since the magnitude of this relative velocity is the galactic virial velocity $v \sim 10^{-3} c$, the ALP has a momentum of $\vec{\nabla} a \sim 10^{-3} \, \partial_0 a $. To leading order the ALP dark matter field is simply a free scalar field with low momentum which is oscillating in its potential so it is approximately $a \approx a_0 \cos \left( m_a t \right)$. Thus $\partial_0 a$ has magnitude $a_ 0 m_a$ and oscillates with frequency $m_a$. The effective coupling in the nucleon Hamiltonian is $$\label{eqn: nucleon precession rate from gann} H_N \supset g_{\text{aNN}} \, m_a a_0 \cos \left(m_a t\right) \, \vec{v} .\vec{\sigma_{N}}$$ The amplitude $a_0$ of the ALP field is constrained by the requirement that the energy density $\frac{1}{2} m_a^2 a_0^2$ in the ALP oscillations not exceed the local dark matter density $\rho_\text{DM} \sim 0.3 \, \frac{{\text{GeV}}}{{\text{cm}}^3}$. Hence, the maximum size of this perturbation is $$\label{eqn: numbers for nucleon precession rate from gann} \Delta E \sim g_\text{aNN} \, \sqrt{\rho_{\text{DM}}} \, v \sim 3 \times 10^{-9} \text{ s}^{-1} \, \left(\frac{g_{aNN}}{ 10^{-9} \text{ GeV}^{-1}}\right) \left(\sqrt{\frac{\rho_{\text{DM}}}{0.3 \, \frac{\text{GeV}}{\text{cm}^3}}}\right)$$ oscillating at a frequency equal to the ALP mass $m_a \sim$ kHz - GHz. The expected coherence time for this oscillation is set by the ALP coherence time $\tau_a \sim \frac{1}{m_a v^2} \sim 1 \text{ s} \, \left(\frac{\text{MHz}}{m_a}\right)$, leading to a signal bandwidth $\sim 10^{-6} m_a$. A Detection Strategy {#subsec: det strategy} -------------------- The detection of this small but time varying energy shift requires the development of new experimental techniques. While there may be many experimental avenues that could be pursued, we highlight the approach proposed in [@NMR; @paper] utilizing NMR techniques. In this approach, a sample of nuclear spin polarized material is placed with the polarization chosen along a direction that is not collinear to the relative velocity $\vec{v}$ between the Earth and the dark matter, as in Fig. \[Fig:setup\]. An axial nuclear moment in the presence of a dark matter ALP field will cause the spins to precess around this relative velocity. This precession changes the magnetization of the material and can be measured using precision magnetometers such as SQUIDs or SERFs. ![ \[Fig:setup\] Geometry of the experiment, adapted from [@NMR; @paper]. The applied magnetic field $\vec{B}_\text{ext}$ is collinear with the sample magnetization $\vec{M}$. The relative velocity $\vec{v}$ between the sample and the dark matter ALP field is in any direction that is not collinear with $\vec{M}$. The SQUID pickup loop is arranged to measure the transverse magnetization of the sample.](setup.pdf){width="3.5"} More specifically, the procedure is to polarize the nuclear spins of a sample of material in an external magnetic (${\vec{B}_\text{ext}}$) to achieve a net magnetization. When this net magnetization is not collinear with the dark matter velocity, the spins will precess around this relative velocity. Once they are no longer aligned with the external magnetic field, they will precess around both the relative velocity and this magnetic field. Equivalently, the nuclear spins precess around the relative velocity as seen in a rotating frame in which the magnetic field is eliminated. This results (as seen in the lab frame) in a magnetization at an angle to the magnetic field, which precesses around this field with the Larmor frequency. This gives rise to a transverse magnetization, which can be measured with a magnetometer such as a superconducting quantum interference device (SQUID) with a pickup loop oriented as shown in Fig. \[Fig:setup\]. The transverse magnetization rotates at the Larmor frequency set by the external magnetic field. When the ALP oscillation frequency is different from the Larmor frequency, no measurable transverse magnetization ensues. However, when the two frequencies coincide, there occurs a resonance akin to that in the usual NMR, where the spins precess around a transverse axis rotating at the Larmor frequency [@BudkerBook]. This effect enhances the precessing transverse magnetization that can be detected with the SQUID magnetometer. The magnitude of the external magnetic field (${B_\text{ext}}$) is swept to search for a resonance. At time $t=0$ the spins are prepared along ${\vec{B}_\text{ext}}$, then the magnitude of the transverse magnetization is given by $$\label{eqn: magnetization signal} M(t) \approx n p \mu \, \left(g_{\text{aNN}} \sqrt{2 \, \rho_{\text{DM}}} v\right) \, \frac{\sin \left( \left( 2 \mu {B_\text{ext}}- m_a \right) t \right)}{2 \mu {B_\text{ext}}- m_a} \sin \left( 2 \mu {B_\text{ext}}t \right),$$ where $n$ is the number density of nuclear spins, $p$ is the polarization, and $\mu$ is the nuclear magnetic dipole moment. The resonant enhancement occurs when $2 \mu {B_\text{ext}}\approx m_a$. Taking $n \sim \frac{10^{22}}{\text{cm}^3}$, $p \sim \mathcal{O}\left(1\right)$, $\mu \sim \mu_N$ (the nuclear Bohr magneton) and interrogation time $ t \sim \tau_a \sim \frac{10^{6}}{m_a}$ (the ALP coherence time), we get $$M \approx 2 \times 10^{-14} \, \text{T} \, \left(\frac{g_{\text{aNN}}}{10^{-10} \, \text{GeV}^{-1}}\right) \, \left(\frac{\text{MHz}}{m_a}\right)$$ This magnetic field is above the sensitivity of modern SQUID and atomic SERF magnetometers that typically have sensitivities $\sim 10^{-16} \frac{\text{T}}{\sqrt{\text{Hz}}}$. The axial nuclear moment oscillates at a frequency set by particle physics, independent of the experimental setup. This distinguishes the signal from many possible backgrounds. For example, control over noise sources is only required over the signal’s relatively high frequency range (kHz - MHz) and narrow bandwidth ($\sim 10^{-6} \, m_a$). Further, though the induced axial nuclear moment is small, its oscillation at laboratory frequencies enables resonant schemes that boost the signal significantly. This idea is based on and very similar to the one proposed in [@NMR; @paper] to detect the time varying EDM induced by the dark matter axion. However, in this case, since we are not searching for an EDM, we do not need a material with a large Schiff moment nor do we need to expose it to significant electric fields. The techniques described in [@NMR; @paper] to achieve large nuclear polarizations and quality factors for the NMR resonance can also be employed in this case. Further, noise sources such as the intrinsic magnetization noise of the sample and strategies to mitigate them should also be similar to the discussions of [@NMR; @paper]. ![ \[Fig:Nucleon\] ALP parameter space in pseudoscalar coupling of axion to nucleons Eqn.  vs mass of ALP. The purple line is the region in which the QCD axion may lie. The width of the purple band gives an approximation to the axion model-dependence in this coupling. The darker purple portion of the line shows the region in which the QCD axion could be all of the dark matter and have $f_a < M_\text{pl}$ as in Figure \[Fig:EDM\]. The green region is excluded by SN1987A from [@Raffelt:2006cw]. The blue region is excluded by searches for new spin dependent forces between nuclei [@Vasilakis:2008yn]. The red line is the preliminary sensitivity of an NMR style experiment using Xe, the blue line is the sensitivity using $^3\text{He}$. The dashed lines show the limit from magnetization noise for each sample. These lines assume the parameters in Table \[Tab: experiments\]. The ADMX region shows the part of QCD axion parameter space which has been covered (darker blue) [@Asztalos:2009yp] or will be covered in the near future (lighter blue) [@ADMXwebpage; @snowdarktalk] by ADMX.](nucleonplot.pdf){width="6"} Figure \[Fig:Nucleon\] shows constraints on $g_\text{aNN}$ and the potential sensitivity of our proposals. The width of the line shows axion model-dependence in the axion-nucleon coupling. The solid lines are preliminary sensitivity curves with the sensitivity limited by magnetometer noise. Both lines assume samples of volume $\left(10 \text{ cm}\right)^3$ with 100 percent nuclear polarization. Other sample parameters are described in Table \[Tab: experiments\]. The dashed lines show the limits from sample magnetization noise, so where they are higher than the corresponding solid line, they are the limit on sensitivity. The solid curves are cutoff at high frequencies by the requirement that the Larmor frequency be achievable with the assumed maximum magnetic field. ---- --------- ------------------------------------------------ ----------------- ------- -------- ---------------------------------------------- Element Density Magnetic Moment $T_2$ Max. B Magnetometer ($n$) ($\mu$) Sensitivity 1. Xe $1.3 \times 10^{22} \frac{1}{{\text{cm}}^{3}}$ $0.35 \, \mu_N$ 100 10 T $10^{-16} \frac{\text{T}}{\sqrt{\text{Hz}}}$ 2. $^3$He $2.8 \times 10^{22} \frac{1}{{\text{cm}}^{3}}$ $2.12 \, \mu_N$ 100 20 T $10^{-17} \frac{\text{T}}{\sqrt{\text{Hz}}}$ ---- --------- ------------------------------------------------ ----------------- ------- -------- ---------------------------------------------- : \[Tab: experiments\] The parameters used for the sensitivity curves shown in Figure \[Fig:Nucleon\]. The first row corresponds to the upper (red) lines in the figure while the second row is the lower (blue) lines. For the Xe experiment we used the average magnetic moment from the naturally occurring abundances of $^{129}$Xe and $^{131}$Xe. The sixth column shows the maximum magnetic field that is assumed, which is relevant only for setting the upper frequency limit on the curves. The last column shows the assumed magnetometer sensitivity. Note that there are many ways to verify a positive signal in such an experiment, the same ways as described in [@NMR; @paper]. If a positive signal is found, the scan can be stopped and that particular frequency can be explored for much more time than was needed to observe it in the scanning mode. Thus one can effectively make many measurements of the dark matter signal. This has several interesting consequences. In particular, the signal from operator of Eqn.  is proportional to the spatial derivative of the axion field, i.e. the local axion velocity. This is unlike the case for the EDM operator, Eqn.  (our NMR proposal of [@NMR; @paper]) or the photon coupling Eqn.  (used in ADMX). Hence, if the ALP signal can be observed through this operator, Eqn. , we will actually have a directional dark matter detector. One could observe simultaneously with 3 different samples with perpendicular magnetization directions (or just vary the magnetization direction using one sample). This would give us the local axion velocity. Within the axion coherence length, the wavelength $\sim \frac{1}{m_a v}$, all experiments must agree on this measured direction of the axion velocity. So this is another check on a positive signal. But it also gives much more information since it tells us about the velocity structure of the dark matter. At any one instant of time the local velocity may appear random and changes on a timescale of order the axion coherence time $\tau_a \sim \frac{1}{m_a v^2}$. However, if the signal is folded on a yearly period or a daily period, the average velocity should modulate exactly with the Earth’s velocity around the sun or rotational velocity around its axis respectively. This would be yet another check that the signal is correct. Even using the EDM coupling or the photon coupling could lead to interesting knowledge about the dark matter velocity profile including knowledge of local streams, as has been pointed out for ADMX [@Sikivie:1992bk; @Duffy:2006aa; @Hoskins:2011iv], because of the high frequency resolution. However when using the pseudoscalar nucleon coupling, Eqn. , we have something more, we have a directional detector so we learn information about the full velocity distribution of the dark matter. It is very interesting that even this experiment can get close to the QCD axion over a very large range of axion masses and further can cover a large piece of ALP parameter space. Also, very importantly, the fundamental limit from magnetization noise can be reduced by using samples with larger volumes [@NMR; @paper]. This scheme could thus potentially allow detection of the QCD axion over an interesting range of higher masses through the use of improved magnetometers. This NMR technique appears to have the capability to probe hitherto unconstrained ALP dark matter parameter space when the ALP couples to nuclear moments such as the electric dipole moment [@NMR; @paper] or the axial nuclear moment. While constraints from current laboratory experiments for these ALP induced nuclear moments are much weaker than astrophysical limits, this search for ALP dark matter probes regions well beyond these limits (see Fig. \[Fig:Nucleon\]). Axial Electron Moment {#Sec: axial electron} ===================== Much like the axial nuclear moment discussed above, ALPs can also couple to electrons through the third operator in giving rise to the interaction $$\label{eqn:gaee} \mathcal{L} \supset g_\text{aee} \, \partial_{\mu}a \left( \bar{e} \gamma_5 \gamma^{\mu} e \right).$$ This coupling is very similar to the nucleon coupling in Eqn  and leads to similar effects. The QCD axion generally has this coupling with $g_{aee} \sim \frac{1}{f_a}$, though it can be fine-tuned to zero. Astrophysics constrains $g_{aee} \lessapprox 10^{-10} \text{ GeV}^{-1}$ from bounds on the cooling of white dwarves [@Raffelt:2006cw]. This interaction also gives rise to spin dependent dipole - dipole forces between electrons. However, bounds from such searches are significantly weaker than the astrophysical limits on this coupling [@Dobrescu:2006au; @electronspin]. ![ \[Fig:Electron\] ALP parameter space in pseudoscalar coupling of axion to electrons Eqn.  vs mass of ALP. The green region is excluded by White Dwarf cooling rates from [@Raffelt:2006cw]. The blue region is excluded by searches for new spin dependent forces between electrons [@Dobrescu:2006au; @electronspin]. The region below the solid purple line shows the possible parameter space for a QCD axion, with the region bounded by darker purple lines being the region where the QCD axion could be all of dark matter and have $f_a < M_\text{pl}$. The frequency range of the QCD axion covered by ADMX is identical to the range plotted in Figure \[Fig:Nucleon\]. ](electronplot.pdf){width="6"} Similar to the axial nuclear moment, in the presence of a background dark matter ALP field, the non-relativistic limit of this operator leads to the following term in the electron Hamiltonian $$H_e \supset g_{aee} \vec{\nabla} a.\vec{\sigma_e}$$ where $\sigma_e$ is the electron spin operator. An electron spin that is not aligned with the ALP dark matter “wind" will then precess due to the coupling $$\label{eqn: electron precession rate from gaee} H_e \supset g_\text{aee} \, m_a \, a_0 \cos \left(m_a t\right) \, \vec{v}.\sigma_{e}$$ Using the constraint that the energy density in the ALP oscillations not exceed the local dark matter density, this perturbation is of size $$\label{eqn: numbers for electron precession rate from gaee} \Delta E \sim g_\text{aee} \, | \vec{\nabla} a | \sim g_\text{aee} \, v \, \sqrt{\rho_{\text{DM}}} \sim 3 \times 10^{-9} \text{ s}^{-1} \, \left(\frac{g_{aee}}{ 10^{-9} \text{ GeV}^{-1}}\right) \left(\sqrt{\frac{\rho_{\text{DM}}}{0.3 \, \frac{\text{GeV}}{\text{cm}^3}}}\right)$$ This perturbation also oscillates at a frequency equal to the ALP mass $m_a \sim$ kHz - GHz, with an expected bandwidth $\sim 10^{-6} m_a$. We have not been able to invent techniques that could probe this unconstrained parameter space of ALP dark matter. We show constraints on this coupling in Figure \[Fig:Electron\]. The solid purple line in the figure shows the largest value that $g_{\text{aee}}$ could take for the QCD axion. Since $g_{\text{aee}}$ is model dependent, it could in principle be tuned to zero, though it is generally expected to be close to the purple line. As in Figures \[Fig:EDM\] and \[Fig:Nucleon\] the darker purple portion shows the part of QCD axion parameter space where the axion may be all of the dark matter and has $f_a < M_\text{pl}$. In this figure this region is bounded by the solid dark purple on top and the dashed lines on the sides. For a general ALP, there is no such expectation and the coupling could lie anywhere on the unconstrained portion of Figure \[Fig:Electron\]. Experimental techniques to probe time varying electron axial moments could thus probe an unexplored range of ALP dark matter. Conclusions {#Sec: conclusions} =========== All previous axion detection experiments have been based on the axion-photon coupling in Equation . We have considered several new operators for axion and ALP detection in Equations , , and in Sections \[Sec: EDM\], \[Sec: axial nuclear\], and \[Sec: axial electron\]. For the QCD axion the EDM operator arises from the axion-gluon coupling $\propto \frac{a}{f_a} G \tilde{G}$. We mapped out the parameter spaces for these operators including finding the current constraints in Figures \[Fig:EDM\], \[Fig:Nucleon\], and \[Fig:Electron\]. These operators suggest new ways to search for axion and ALP dark matter. For the EDM coupling we previously proposed an experiment using cold molecules [@Graham:2011qk]. These operators suggest promising detection strategies using spin precession, NMR-based, techniques which we discuss in detail in [@NMR; @paper]. For the QCD axion, high-scale decay constants $f_a$, or masses below $\sim \mu {\text{eV}}$, make up a well-motivated part of parameter space but are very challenging to detect with current experiments. Use of these new operators may allow detection of QCD axion dark matter over a wider range of its parameter space, especially for $f_a$ near the fundamental GUT or Planck scales. In particular the EDM operator Eq.  may be the most promising. Because it is a non-derivative operator, it avoids the axion wavelength suppressions that plague the use of any other axion coupling for detecting low mass axions. We have argued that it is useful to think of ALP dark matter produced through the misalignment mechanism as a classical field with an oscillating vacuum expectation value (VEV). The interaction of a single axion or ALP particle with a detector may be too weak to observe. But thinking of the ALP as a background field motivates searching for the coherent effects of the interaction of the entire classical scalar field with the detector. For example, as we have shown, the ALP field may cause an oscillating nucleon EDM proportional to the classical VEV of the field, a collective effect of all the ALP ‘particles’ comprising the field. Or the ALP field may induce axial moments for nucleons or electrons, causing their spins to precess around the gradient of the field. The continuous, coherent nature of these effects also enable secondary tests that can confirm the ALP dark matter origin of a signal in such experiments. As pointed out in [@NMR; @paper], a signal in one sample can be correlated with another that is within the de-Broglie wavelength ($\gg 100$ m) of the ALP field. Further, a positive signal in a particular bin can be verified by tuning the experiment to that bin and spending additional time to observe the build up of the signal in that bin. Since the assumed scanning time at any particular frequency bin is rather short ($\sim 10$ s), additional time can be spent in some bins without significant loss of efficiency. As discussed earlier in Section \[Sec: axial nuclear\], the spin dependent nature of the ALP coupling to axial currents can be exploited to detect the direction of the dark matter wind for such ALPs. These effects are very different from the single, hard, particle scatterings which are used to search for WIMP dark matter. For WIMP direct detection the signal is a stochastic energy deposition event in the detector. The effects we propose searching for are not dominantly energy-deposition signals. They are the continuous, coherent effects of the entire ALP field on the sample. We considered ways to search for axion or ALP dark matter. Similarly to ADMX, such signals benefit from requiring only one insertion of the small coupling between the axion or ALP and the Standard Model fields. These couplings, $g_{a\gamma\gamma}$, $g_d$, $g_\text{aNN}$, and $g_\text{aee}$ in Equations , , , and respectively, are exceedingly small. For the QCD axion they are all $\propto f_a^{-1}$, where $f_a$ is a high scale. By contrast, light-through-walls and spin-dependent force experiments require two insertions of these couplings. In such experiments the axion or ALP must be sourced (either by the laser or the source mass) and then must interact again to be detected. The Feynman diagram would have two insertions of this operator and so the amplitude for the process is suppressed by the relevant coupling squared. The light-through-walls experiments measure a rate and so are suppressed by the coupling to the fourth power. This is why experiments searching for axion or ALP dark matter such as ADMX or through the effects we propose are sensitive to significantly smaller couplings and may even reach the QCD axion. We have proposed a new type of dark matter signal to search for: the rapid oscillation of some parameter, e.g. the nucleon EDM. In general, such a signal may arise from any type of modulus dark matter. The QCD axion provides a well-motivated example of such a modulus, but many others are possible. For example perhaps the dark matter is a modulus of electric charge, in which case the fine-structure constant would oscillate in time. Unlike the current experiments searching for time-variation of $\alpha$ which look on timescales of years or more, the most motivated variation is on much faster timescales, frequencies of $\sim$ kHz to GHz or more. Further, for definiteness here we have considered scalar fields. However it is also possible that other fields (e.g. vectors) may provide a natural realization of the experimental signatures we have considered. Although we have considered some experimental designs to detect these signals of axion and ALP dark matter, it seems likely that many other experiments are also possible. For example, some static EDM experiments may be modifiable to search for oscillating EDMs. It would be valuable to make progress covering the ALP parameter spaces of Figures \[Fig:EDM\], \[Fig:Nucleon\], and \[Fig:Electron\] and reaching towards the QCD axion. Once we start considering the parameter space for this new type of signal, it becomes clear that there is a large, new class of dark matter direct detection experiments that have not been considered before. Over the last couple decades, WIMP direct detection experiments have made tremendous progress, improving sensitivities by many orders of magnitude. A similar improvement in the search for axion dark matter may be possible with new experiments designed to search for the coherent field effects we have described. The axion is an excellent dark matter candidate. Hopefully consideration of these types of signals will open new avenues to its discovery. Acknowledgments {#acknowledgments .unnumbered} =============== We would like to thank Dmitry Budker, Hooman Davoudiasl, Savas Dimopoulos, Micah Ledbetter, Yannis Semertzidis, Alex Sushkov, and Scott Thomas for useful discussions. SR was supported by ERC grant BSMOXFORD no. 228169. [10]{} url \#1[[\#1]{}]{}urlprefix M. W. Goodman and E. Witten, Phys. Rev. D [**31**]{}, 3059 (1985). P. W. Graham, D. E. Kaplan, S. Rajendran and M. T. Walters, Phys. Dark Univ.  [**1**]{}, 32 (2012) \[arXiv:1203.2531 \[hep-ph\]\]. R. Essig, J. Mardon and T. Volansky, Phys. Rev. D [**85**]{}, 076007 (2012) \[arXiv:1108.5383 \[hep-ph\]\]. S. Rudaz and F. W. Stecker, Astrophys. J.  [**325**]{}, 16 (1988). S. Galli, F. Iocco, G. Bertone and A. Melchiorri, Phys. Rev. D [**80**]{}, 023505 (2009) \[arXiv:0905.0003 \[astro-ph.CO\]\]. T. R. Slatyer, N. Padmanabhan and D. P. Finkbeiner, Phys. Rev. D [**80**]{}, 043526 (2009) \[arXiv:0906.1197 \[astro-ph.CO\]\]. A. Arvanitaki, S. Dimopoulos, S. Dubovsky, P. W. Graham, R. Harnik and S. Rajendran, Phys. Rev. D [**79**]{}, 105022 (2009) \[arXiv:0812.2075 \[hep-ph\]\]. A. Arvanitaki, S. Dimopoulos, S. Dubovsky, P. W. Graham, R. Harnik and S. Rajendran, Phys. Rev. D [**80**]{}, 055011 (2009) \[arXiv:0904.2789 \[hep-ph\]\]. K. Sigurdson, M. Doran, A. Kurylov, R. R. Caldwell and M. Kamionkowski, Phys. Rev. D [**70**]{}, 083501 (2004) \[Erratum-ibid. D [**73**]{}, 089903 (2006)\] \[astro-ph/0406355\]. A. Rajaraman, W. Shepherd, T. M. P. Tait and A. M. Wijangco, Phys. Rev. D [**84**]{}, 095013 (2011) \[arXiv:1108.1196 \[hep-ph\]\]. Y. Bai, P. J. Fox and R. Harnik, JHEP [**1012**]{}, 048 (2010) \[arXiv:1005.3797 \[hep-ph\]\]. Z. Ahmed [*et al.*]{} \[CDMS and EDELWEISS Collaborations\], Phys. Rev. D [**84**]{}, 011102 (2011) \[arXiv:1105.3377 \[astro-ph.CO\]\]. , [*et al.*]{} \[XENON100 Collaboration\], arXiv:1301.6620 \[astro-ph.CO\]. P. Meade, M. Papucci, A. Strumia and T. Volansky, Nucl. Phys. B [**831**]{}, 178 (2010) \[arXiv:0905.0480 \[hep-ph\]\]. R. Essig, A. Manalaysay, J. Mardon, P. Sorensen and T. Volansky, Phys. Rev. Lett.  [**109**]{}, 021301 (2012) \[arXiv:1206.2644 \[astro-ph.CO\]\]. S. Lowette [*et al.*]{} \[ATLAS and CMS Collaborations\], arXiv:1205.4053 \[hep-ex\]. P. W. Graham, D. E. Kaplan, S. Rajendran and P. Saraswat, JHEP [**1207**]{}, 149 (2012) \[arXiv:1204.6038 \[hep-ph\]\]. D. Cadamuro, S. Hannestad, G. Raffelt and J. Redondo, JCAP [**1102**]{}, 003 (2011) \[arXiv:1011.3694 \[hep-ph\]\]. R. D. Peccei, H. R. Quinn, Phys. Rev. Lett.  [**38**]{}, 1440-1443 (1977). R. D. Peccei, H. R. Quinn, Phys. Rev.  [**D16**]{}, 1791-1797 (1977). S. Weinberg, Phys. Rev. Lett.  [**40**]{}, 223-226 (1978). F. Wilczek, Phys. Rev. Lett.  [**40**]{}, 279-282 (1978). J. E. Kim, Phys. Rev. Lett.  [**43**]{}, 103 (1979). M. A. Shifman, A. I. Vainshtein, V. I. Zakharov, Nucl. Phys.  [**B166**]{}, 493 (1980). M. Dine, W. Fischler, M. Srednicki, Phys. Lett.  [**B104**]{}, 199 (1981). A. R. Zhitnitsky, Sov. J. Nucl. Phys.  [**31**]{}, 529-534 (1980). M. Y. .Khlopov, Singapore, Singapore: World Scientific (1999) 577 p P. Sikivie, Phys. Rev.  [**D32**]{}, 2988 (1985). J. Hong and J. E. Kim, Phys. Lett. B [**265**]{}, 197 (1991). S. J. Asztalos [*et al.*]{} \[The ADMX Collaboration\], Phys. Rev. Lett.  [**104**]{}, 041301 (2010) \[arXiv:0910.5914 \[astro-ph.CO\]\]. I. G. Irastorza [*et al.*]{}, [*In the Proceedings of International Symposium on Detector Development for Particle, Astroparticle and Synchrotron Radiation Experiments (SNIC 2006), Menlo Park, California, 3-6 Apr 2006, pp 0036*]{}. K. Ehret [*et al.*]{} \[ALPS Collaboration\], Nucl. Instrum. Meth. A [**612**]{}, 83 (2009) \[arXiv:0905.4159 \[physics.ins-det\]\]. M. Dine, W. Fischler, Phys. Lett.  [**B120**]{}, 137-141 (1983). J. Preskill, M. B. Wise, F. Wilczek, Phys. Lett.  [**B120**]{}, 127-132 (1983). B. Abbott [*et al.*]{} \[LIGO Scientific Collaboration\], Rept. Prog. Phys.  [**72**]{}, 076901 (2009) \[arXiv:0711.3041 \[gr-qc\]\]. T. Accadia [*et al.*]{} \[VIRGO Collaboration\], JINST [**7**]{}, P03012 (2012). S. Dimopoulos, P. W. Graham, J. M. Hogan, M. A. Kasevich and S. Rajendran, Phys. Lett. B [**678**]{}, 37 (2009) \[arXiv:0712.1250 \[gr-qc\]\]. S. Dimopoulos, P. W. Graham, J. M. Hogan, M. A. Kasevich and S. Rajendran, Phys. Rev. D [**78**]{}, 122002 (2008) \[arXiv:0806.2125 \[gr-qc\]\]. J. M. Hogan, D. M. S. Johnson, S. Dickerson, T. Kovachy, A. Sugarbaker, S. -w. Chiow, P. W. Graham and M. A. Kasevich [*et al.*]{}, Gen. Rel. Grav.  [**43**]{}, 1953 (2011) \[arXiv:1009.2702 \[physics.atom-ph\]\]. P. W. Graham, J. M. Hogan, M. A. Kasevich and S. Rajendran, Phys. Rev. Lett.  [**110**]{}, 171102 (2013) \[arXiv:1206.0818 \[quant-ph\]\]. A. Ringwald, arXiv:1209.2299 \[hep-ph\]. P. W. Graham and S. Rajendran, Phys. Rev. D [**84**]{}, 055013 (2011) \[arXiv:1101.2691 \[hep-ph\]\]. M. V. Romalis, W. C. Griffith, J. P. Jacobs and E. N. Fortson, Proc. of Nucl. Physics in the 21st century, INPC 2001, Berkeley M. Takamoto, F. L. Hong, R. Higashi and H. Katori, Nature, 435 (7040), 321-324. G. R. Dvali and M. Zaldarriaga, Phys. Rev. Lett.  [**88**]{}, 091303 (2002) \[hep-ph/0108217\]. P. Svrcek and E. Witten, JHEP [**0606**]{}, 051 (2006) \[hep-th/0605206\]. G. G. Raffelt, Lect. Notes Phys.  [**741**]{}, 51 (2008) \[hep-ph/0611350\]. A. D. Linde, Phys. Lett. B [**201**]{}, 437 (1988). A. D. Linde, JCAP [**0706**]{}, 017 (2007) \[arXiv:0705.1160 \[hep-th\]\]. A. Ringwald, Phys. Dark Univ.  [**1**]{}, 116 (2012) \[arXiv:1210.5081 \[hep-ph\]\]. D. Cadamuro and J. Redondo, JCAP [**1202**]{}, 032 (2012) \[arXiv:1110.2895 \[hep-ph\]\]. J. L. Hewett, H. Weerts, R. Brock, J. N. Butler, B. C. K. Casey, J. Collar, A. de Govea and R. Essig [*et al.*]{}, arXiv:1205.2671 \[hep-ex\]. P. Arias, D. Cadamuro, M. Goodsell, J. Jaeckel, J. Redondo and A. Ringwald, JCAP [**1206**]{}, 013 (2012) \[arXiv:1201.5902 \[hep-ph\]\]. D. Horns, J. Jaeckel, A. Lindner, A. Lobanov, J. Redondo, A. Ringwald and , arXiv:1212.2970 \[hep-ph\]. M. Pospelov, A. Ritz and , Phys. Rev. Lett.  [**83**]{}, 2526 (1999) \[hep-ph/9904483\]. J. Ruz talk at SnowDark 2013. http://www.physics.utah.edu/snowpac/index.php/snowdark-2013/snowdark-2013-talks-slides ADMX website http://www.phys.washington.edu/groups/admx/home.html C. A. Baker, D. D. Doyle, P. Geltenbort, K. Green, M. G. D. van der Grinten, P. G. Harris, P. Iaydjiev and S. N. Ivanov [*et al.*]{}, Phys. Rev. Lett.  [**97**]{}, 131801 (2006) \[hep-ex/0602020\]. P. G. Harris, C. A. Baker, K. Green, P. Iaydjiev, S. Ivanov, D. J. R. May, J. M. Pendlebury and D. Shiers [*et al.*]{}, Phys. Rev. Lett.  [**82**]{}, 904 (1999). Y. K. Semertzidis \[Storage Ring EDM Collaboration\], arXiv:1110.3378 \[physics.acc-ph\]. Y. F. Orlov, W. M. Morse and Y. K. Semertzidis, Phys. Rev. Lett.  [**96**]{}, 214802 (2006) \[hep-ex/0605022\]. Y. K. Semertzidis [*et al.*]{} \[EDM Collaboration\], AIP Conf. Proc.  [**698**]{}, 200 (2004) \[hep-ex/0308063\]. Y. Semertzidis private communication. D. Budker, P. W. Graham, M. Ledbetter, S. Rajendran and A. O. Sushkov, arXiv:1306.6089 \[hep-ph\]. J. E. Moody and F. Wilczek, Phys. Rev. D [**30**]{}, 130 (1984). G. Vasilakis, J. M. Brown, T. W. Kornack and M. V. Romalis, Phys. Rev. Lett.  [**103**]{}, 261801 (2009) \[arXiv:0809.4700 \[physics.atom-ph\]\]. J. Engel, D. Seckel and A. C. Hayes, Phys. Rev. Lett.  [**65**]{}, 960 (1990). S. K. Lamoreaux, Phys. Rev. A [**66**]{}, 022109 (2002) \[nucl-ex/0109014\]. B. A. Dobrescu and I. Mocioiu, JHEP [**0611**]{}, 005 (2006) \[hep-ph/0605342\]. W. T. Ni, T. C. P.  Chui, S. S.  Pan and B. Y.  Cheng, Physica B 194-196, 153 (1994). D. Budker, D. Kimball, and D. DeMille (2008) (New York: Oxford University Press, 2008). P. Sikivie and J. R. Ipser, Phys. Lett. B [**291**]{}, 288 (1992). J. Hoskins, J. Hwang, C. Martin, P. Sikivie, N. S. Sullivan, D. B. Tanner, M. Hotz and L. JRosenberg [*et al.*]{}, Phys. Rev. D [**84**]{}, 121302 (2011) \[arXiv:1109.4128 \[astro-ph.CO\]\]. L. D. Duffy, P. Sikivie, D. B. Tanner, S. J. Asztalos, C. Hagmann, D. Kinion, L. JRosenberg and K. van Bibber [*et al.*]{}, Phys. Rev. D [**74**]{}, 012006 (2006) \[astro-ph/0603108\].

--- abstract: 'We investigate the influence of quantum fluctuations and weak disorder on the vortex dynamics in a two-dimensional superconducting Berezinskii-Kosterlitz-Thouless system. The temperature below which quantum fluctuations dominate the vortex creep is determined, and the transport in this quantum regime is described. The crossover from quantum to classical regime is discussed and the quantum correction to the classical current-voltage relation is determined. It is found that weak disorder can effectively reduce the critical current as compared to that in the clean system.' author: - 'Aleksandra Petkovi'' c$^{1,2}$, Valerii M. Vinokur$^{2}$, and Thomas Nattermann$^{1}$' date: ',' title: '<span style="font-variant:small-caps;">Transport properties of clean and disordered Josephson junction arrays</span>' --- The physics of the Berezinskii-Kosterlitz-Thouless (BKT) transition in thin superconducting films is re-emerging as one of the mainstreams in the current condensed matter physics. The interest is motivated by recent advances in studies of layered high-temperature superconductors [@Ruef-2006; @Matthey-2007], the discovery of the superconductivity at the interface between the insulating oxides [@Reyren-2007; @Caviglia-2008], new studies in thin superconducting films uncovering the role of the two-dimensional (2D) superconducting fluctuations [@Crane-2007; @Pourret-2006], and the intense developments in the physics of the superconductor-insulator transition where the BKT transition may play a major role [@Vinokur-Nature]. The predicted benchmark of the transition that serves to detect it experimentally is the change of the shape of the $I$-$V$ characteristics, $V\propto I^{1+\alpha}$ from $\alpha=0$ above the transition, $T>T_{BKT}$, to $\alpha=2(T_{BKT}/T)$ at $T\leq T_{BKT}$ [@HN; @Minnhagen]. However, experimental data on superconducting films show appreciable deviations from the theoretical predictions and are still inconclusive [@exp]. Among the possible sources of the deviation from the classic predictions, one can consider the finite size effect, [@Kogan; @GV] and effects of disorder [@Na+_95; @KoNa_96; @Giam]. Another important issue is the role of quantum effects which become crucial when the BKT transition occurs at low enough temperatures. In this Brief Report we will analyze the role of quantum effects in the BKT transition paying a special attention to the intermediate region of the interplay between thermal and quantum contributions. We will discuss the effect of disorder-generated vortices on the BKT transition, neglecting quantum fluctuations, namely the effective reduction of the critical current as compared to that in clean samples. *Model.* We choose a disordered Josephson junction array as a convenient discrete model for the 2D disordered superconducting film [@EcSc_89]. The Hamiltonian describing the system is: $$\label{eq:H} {\cal H} = \frac{1}{2} \sum_{i,j} ( C^{-1})_{i,j}\hat n_i\hat n_j -J\sum_{\langle i,j\rangle}\cos (\hat \varphi_i-\hat \varphi_{j}-A_{ij})$$ where $[\hat n_j, \hat \varphi_k]=-2e i\delta_{j,k}$. We ignore single electron tunneling and other sources of dissipation. The only non-vanishing elements of the capacitance matrix $C_{ij}$ are its diagonal elements $C_{jj} =4C$ (no summation over the repeated index) and $C_{ij}=-C$ for the nearest neighbors $i,j$, i.e., the capacitance to the ground is assumed negligible as compared to the mutual capacitances of the superconducting islands. The second sum in (\[eq:H\]) is taken over all nearest-neighbor pairs on a square lattice. Random phase shifts $A_{ij}$ result from the deviations of the flux in a distorted plaquette from an integer multiple of the flux quantum $\Phi_0=\hbar c/2e$ [@Granato+86]. In the clean classical case, i.e. for $A_{ij}=0$ and in the limit $C\to \infty$, the physics of the system can be most adequately described in terms of vortices that experience the superconducting BKT transition at the temperature $T_{BKT}\simeq \pi \widetilde{J}/2$, where $\widetilde{J}$ denotes the renormalized coupling constant. It is convenient to decompose the phase at the site $i$, as $\varphi_i = \varphi_i^{(v)}+\varphi_{i}^{(sw)}$ where $(v)$ and $(sw)$ stand for the vortex and the spin wave part, respectively. Then, the vortex Hamiltonian can be written as $$\begin{aligned} \label{eq:H_v} \mathcal{H}_{v}=&-{J\pi} \sum_{i\neq j} m_{i} m_j \ln{ \frac{| {\bf r}_{i}-{\bf r}_{j}|}{ \xi}} +\sum_i E_c m_i^2,\end{aligned}$$ where $E_c$ denotes the core energy of a vortex. The sums are taken over the sites ${\bf r}_i$ of a dual lattice; $m_i$ is the vorticity of the $i$th vortex, and we assumed that $\sum_i m_i=0$, where $\xi$ denotes the superconductor coherence length. Next we want to include quantum fluctuations. After going over to the path integral representation of the partition function and integrating out the charge degrees of freedom, the action of the Josephson junction array in the limit $E_c=e^2/2C \ll J$ assumes the form [@EcSc_89; @fazio91] $$\begin{aligned} S=\int d\tau \left(\frac{M}{2}\sum_{i}(\partial_{\tau}\mathbf r_{i})^2 +{\cal H}_v\right).\end{aligned}$$ The vectors ${\bf r}_i(\tau)$ are the world lines of the vortices and $M=h^2 C /(8 e^2 \xi^2)$. *Clean case*. We begin with the discussion of a clean case. If we apply an external transport current, it will exert the force ${\bf f}\sim {\bf j}$ on the vortices, where $\bf{j}$ is the current density [@Tinkham]. This generates an additional term $-\sum_i m_i{\bf f}\cdot{\bf r}_i$ in (\[eq:H\_v\]). In order to describe the effect of vortices on the current-voltage relation quantitatively, we consider the effect of vortices crossing the system transversely to the transport current. This motion dissipates energy. The Bardeen-Stephen flux flow resistance [@Bardeen] gives for the current-voltage ($V-I$) relation $$V=2\pi\xi^2\rho_nn_vI$$ where $n_v$ is the vortex density and $\rho_n$ is the normal state resistivity. The rate equation for the vortex density is $$\begin{aligned} \label{eq:decayrate} \partial_t n_v=\Gamma -\frac{\xi^2}{\tau_{rec}}n_v^2.\end{aligned}$$ Here $\Gamma$ denotes the rate of generation of free vortices, while the second term on the rhs of (\[eq:decayrate\]) describes their recombination, $\tau_{rec}$ denotes the recombination parameter. The steady state value $n_v=(\tau_{rec}\Gamma)^{1/2}/\xi$ of the vortex density determines the current-voltage relation. In order to determine $\Gamma$, we consider the appearance of a vortex-antivortex pair and its subsequent separation via tunneling or thermal activation under the influence of the external force $\mathbf{f}$. In the clean case this process is symmetric, i.e., the coordinates of the vortex ${{\mathbf r}}_1$ and the antivortex ${{\mathbf r}}_2$ satisfy ${\bf r}_1=-{\bf r}_2={{\mathbf r}}$ with ${\mathbf {f}\cdot \bf{r}}=f r$. The action of the vortex pair can be rewritten as $$\begin{aligned} \label{ch2_eq:action} &S=\int d\tau \left[M(\partial_{\tau}r)^2+U(r)\right ],\end{aligned}$$ where $U(r)=2 \pi J \ln{\left( \frac{2r}{\xi} \right)}-2 f r+2 E_c$. The problem effectively reduces to a single particle motion through one-dimensional potential barrier $U(r)$ [@footnote1]. The rate $\Gamma$ is given by [@Affleck81] $$\label{eq:Gamma-T} \Gamma\sim \int_{0}^{\infty} dE \;\Gamma(E)e^{-E/T},$$ where $\Gamma(E)$ denotes the zero temperature tunneling rate of a particle in the potential $U(r)$ having an energy $E$. For low temperatures and hence $E$ smaller than the barrier height $U_0=2\pi J\left[\ln(\frac{2\pi J}{f\xi})-1\right]+2E_c$, $\Gamma(E)$ in the WKB approximation is $$\begin{aligned} \Gamma(E)= e^{-4\sqrt{M}\int_{r_a(E)}^{r_b(E)}d r\sqrt{U(r)-E}/\hbar},\end{aligned}$$ where $r_{a/b}(E)$ satisfy $U(r_{a/b})=E$ (see Fig. \[fig:potential\]). ![Potential barrier for the separation of the vortex-antivortex pair.[]{data-label="fig:potential"}](potential.eps){width="0.7\columnwidth"} In the following different regimes will be considered.\ (i)At zero temperature the only contribution in Eq. (\[eq:Gamma-T\]) comes from $E=0$. The generated voltage for small currents ($f\xi/J\ll 1$) is $$\begin{aligned} \label{eq:quantum} &V\sim\Gamma^{1/2}\sim e^{-S(0,0)/2\hbar}\notag\\ &\frac{S(0,0)}{\hbar}\approx c_1\frac{\sqrt{M}(2J\pi)^{3/2}}{\hbar f}{\left(\ln{\frac{2J\pi}{f\xi}}\right)}^{3/2}.\end{aligned}$$ $c_1$ is a positive constant of the order of unity and $$\begin{aligned} \frac{S(E,T)}{\hbar}=\frac{E}{T}+4\sqrt{M}\int_{r_a(E)}^{r_b(E)}d r\frac{\sqrt{U(r)-E}}{\hbar}\end{aligned}$$ is the action of the classical path of the particle in the potential $-U(r)$ with the energy $E$ and mass $2M$. The result (\[eq:quantum\]) is in an agreement with that of Ref. [@Iengo+96] where it is obtained using the different technique [@footnote2]. We find that the result (\[eq:quantum\]) holds also at finite temperatures as long as $$\begin{aligned} \label{eq:To} T\leq T_0=\frac{1}{c_2}\frac{\hbar f}{\sqrt{\pi 2J M}}\frac{1}{\sqrt{\ln{\frac{\pi 2J}{f\xi}}}},\end{aligned}$$ where $c_2$ is positive constant of the order of unity.\ (ii) At intermediate temperatures $T_0<T<T^*$, where $$\begin{aligned} \label{eq:T^*} T^*=\frac{\hbar}{2\pi}\sqrt{\frac{-U''(r_c)}{2M}}=\frac{\hbar f}{2\pi}\sqrt{\frac{1}{M J \pi}},\end{aligned}$$ the main contribution in Eq. (\[eq:Gamma-T\]) comes from the stationary point $E_T$. Therefore, $V\sim \exp \left [ -S(E_T,T)/2\hbar \right ]$. $E_T$ depends on the temperature and is implicitly given by the equation $$\begin{aligned} \label{eq:finiteT} \frac{\hbar}{T}=2\sqrt{M}\int_{r_a(E_T)}^{r_b(E_T)}d r \frac{1}{\sqrt{U(r)-E_T}}=\tau(E_T),\end{aligned}$$ where $\tau(E)$ can be interpreted as the period of the classical motion of a particle with the mass $2M$ and energy $E$, in the potential $-U(r)$. Since $\tau(E)$ is the monotonically decreasing function of $E$ for small currents, Eq. (\[eq:finiteT\]) has the unique solution $E_T$ for every $T$ in a range $T_0\leq T\leq T^*$. We come back to the discussion of the voltage characteristic in this regime below.\ (iii)At even higher temperatures $T^*< T \leq T_{\mathrm{BKT}}$, the decay rate is dominated by $E>U_0$ [@Goldanskii; @Affleck81] and the thermally activated breaking of vortex pairs dominates the dynamics. Then, the decay rate is given by the Arrhenius law $\Gamma\sim \exp[-S_{class}/ \hbar]$ where $S_{class}=\hbar U_0/T$. The voltage-current relation reads [@HN; @Doniah+79] $$\begin{aligned} \label{ch2_eq:VIclassical} V\sim f e^{-U_0/(2T)} \sim j^{\delta(T)},\quad\quad \delta(T)=1+{\pi J}/{T}.\end{aligned}$$ Taking into account the presence of other vortices by replacing $J\to \widetilde{J}$, the coefficient assumes a universal value $\delta(T_{BKT})=3$.\ (iv) At $T>T_{\mathrm{BKT}}$ a finite density of free vortices appears in an equilibrium, and the system is characterized by a linear current-voltage relation for small enough currents. ![Dynamic phase diagram in current-temperature coordinates showing different types $V(j,T)$ dependencies for $T<T_{BKT}$. The dashed and the solid lines sketch $T_0(j)$ and $T^*(j)$, respectively. In the domain $T<T_0$ quantum tunneling of vortices dominates the vortex dynamics, while at $T>T^*$ the voltage-current characteristics is determined by the thermally activated motion. In the shaded region the quantum correction to the classical result, given by Eq. (\[eq:correction\]), applies.[]{data-label="Fig:regions"}](regimes.eps){width="0.8\linewidth"} Next, we consider crossover from the quantum- ($T\leq T_0$) to the classical regime ($T>T^*$) in more detail. Within the semiclassical approximation the decay rate is given, with the exponential accuracy, by $\Gamma\sim \exp[-S_{\mathrm{min}}/\hbar]$, where $S_{\mathrm{min}}$ is the action of the trajectory minimizing the Euclidean action of Eq. (\[ch2\_eq:action\]). For temperatures below $T_0$ the extremal action is $S_{\mathrm{min}}=S(0,0)$, in the intermediate region ($T_0<T<T^*$) the minimal action is $S_{\mathrm{min}}=S(E_T,T)$, and in the high temperature regime the trajectory extremizing the action is time independent, and therefore $S_{\mathrm{min}}=\hbar U_0/T$. We find that $S_{\mathrm{min}}$ at $T^*$ has a continuous first derivative with respect to temperature, while the second derivative has a jump: $$\begin{aligned} \label{ch2_eq:secondorder} &\frac{{{\mathrm d}}S(E_T,T)}{{{\mathrm d}}T}\Big|_{T^*}=\frac{{{\mathrm d}}S_{\mathrm{class}}}{{{\mathrm d}}T}\Big|_{T^*}\notag\\ &\frac{{{\mathrm d}}^2 S(E_T,T)}{{{\mathrm d}}T^2}\Big|_{T^*} < \frac{{{\mathrm d}}^2 S_{\mathrm{class}}}{{{\mathrm d}}T^2}\Big|_{T^*}.\end{aligned}$$ Following Ref. [@Larkin+83] we call this situation a second-order transition at the crossover point [@footnote3]. The result of Eqs. (\[ch2\_eq:secondorder\]) is a general property of a massive particle trapped in a metastable state formed by a potential $U(r)$, provided $\tau(E)$ is a monotonously decreasing function of energy [@Chudnovsky92]. Generally, in the case of a second-order transition the trajectory extremizing the action can be written as [@Larkin+83] $$\begin{aligned} r(\tau)=r_c+\sum_{n=1}^{\infty}a_n \cos{\left( \frac{2\pi T}{\hbar}n \tau\right)},\end{aligned}$$ where the coefficients $|a_n|\ll |a_1|$ ($n>1$) are small near the transition temperature $T^*$. Substituting $r(\tau)$ in Eq. (\[ch2\_eq:action\]), the action can be expanded in powers of $a_n$, yielding an effective action $ S\approx {U_0\hbar}/{T}+\alpha a_1^2+\beta a_1^4, $ where the coefficient $\alpha$ is negative below $T^*$ and vanishes at the transition temperature $T^*$ [@Larkin+83]. Then the coefficient $a_1$ can be found from the minimization of the action $S$ and the minimal action is $ S_{\mathrm{min}}=U_0\hbar/T-\alpha^2/(4\beta). $ Following Refs. [@Larkin+83], we determine the coefficients $\alpha$ and $\beta$ and find a quantum correction to the classical result of Eq. (\[ch2\_eq:VIclassical\]) $$\begin{aligned} \label{eq:correction} V&\sim j^{\delta(T)} e^{\Delta},\notag\\ \Delta & = \frac{(T^2-T^{*2})^2}{T T^{*^3}} \frac{\sqrt{M J^3} }{\hbar f} \frac{\pi^{5/2}}{1+2(1-4(T/T^*)^2)^{-1}} .\end{aligned}$$ This result is valid near the transition, for temperatures approaching $T^*$ from below, see Fig. \[Fig:regions\]. We conclude that quantum effects significantly enhance the decay rate in comparison to the classical rate for the asymptotically small currents. It would be interesting to probe the result of Eq. (\[eq:correction\]) in experiments.\ *Disordered case*. Next we include disorder into the consideration, in the limit $C\to \infty$. The phase shifts are assumed to be uncorrelated from bond to bond, and each is Gaussian distributed with the mean value and the variance $$\label{eq:A} \overline {A_{ij}} = 0,\quad\quad\overline {A_{ij}^2} =\sigma,$$ respectively. Then, an additional term $\sum _i m_iV({\bf r}_i)$ is generated in (\[eq:H\_v\]), where $V({\bf r}_i)=2\pi J\sum_j Q_j\ln(| {\bf r}_{i}-{\bf r}_{j}| / \xi)$. $Q_i=(1/2\pi)\sum_{<plaq>} A_{ij}$ are the frozen charges sitting on the dual lattice in the center of a plaquette whereas the sum is over a plaquette formed by 4 bonds. From (\[eq:A\]) follows $\overline{(V({\bf r}_i)-V({\bf r}_j))^2}=4\pi\sigma J^2\ln (| {\bf r}_{i}-{\bf r}_{j}|/ \xi)$. It was shown in Ref. [@Na+_95], that the system in the classical case, at $T = 0$ undergoes a disorder driven transition from the ‘ordered’ BKT state to a disordered phase at the critical disorder strength $\sigma_c=\pi/8$. In the ordered BKT phase vortices appear, on average, only in a form of the bound pairs. Indeed, the energy of a vortex pair with the separation $R$ and $m_1=-m_2=1$ in a clean sample is given by $2 \pi J \ln (R/ \xi)$. Since $V({\bf r}_i)$ is Gaussian distributed, the typical energy gain is $-2J \sqrt {\pi \sigma \ln(R/ \xi)}$ which is smaller by a factor $\sim (\ln (R/\xi))^{-1/2}$ than the energy cost of a pair. However, the maximum energy gain of a vortex dipole in a region of linear size $L>R$ is larger by a factor $\sqrt{2\ln N}$ than the typical energy gain, which arises from the $N$ independent realizations of the vortex positions [@KoNa_96]. The disorder potential, that one vortex-antivortex pair of size $R+dR$ feels, is uncorrelated when the pair is translated over a distance larger than $R$ [@LHT96]. Therefore, we introduce a lattice with a lattice constant $R$. Since also the correlations of disorder potential inside the cell give only subleading-order corrections [@LHT96], we estimate $N\approx (L/R)^2 (R/\xi)^2 (2\pi R/\xi) dR/\xi$. For $dR\approx R$, we get the free energy of the pair at $T=0$ $$\begin{aligned} \label{eq:pairenergy} E\approx 2\pi J \ln{\frac{R}{\xi}}\left[1-\sqrt{\frac{4\sigma}{\pi} \frac{\ln{(L R/ \xi^2)}}{\ln{(R/\xi)}}}\right].\end{aligned}$$ Thus, if $R\approx L$, the total energy of the coresponding vortex pair becomes negative and free vortices are favored by disorder provided $\sigma >\sigma_c=\pi/8$, in an agreement with the renormalization group result in Ref. [@Na+_95]. Note that strictly speaking these vortices are “pseudo-free" since despite the fact that their attraction is overruled by disorder, they remain pinned by the same disorder-induced forces. It follows from the above reasoning that even for $\sigma<\sigma_c$ some rare vortex pairs of the negative energy can appear. From (\[eq:pairenergy\]) we get that their maximal size is $R_c\approx \xi (L/\xi)^{\frac{1}{2\sigma_c/\sigma-1}}$, which reaches the size of the system for $\sigma\to \sigma_c-0$, as expected. Typically there is a single dipole of the size $R_c$ in the system. If we divide the system into $M^2$ subsystems, each part will contain a dipole of the maximum size $R_{M} \approx R_c M^{-{\frac{1}{2\sigma_c/\sigma-1}}}$. The density of dipoles of the size $R_M$ is $\xi^{-2}(R_M/ \xi)^{2(1-2\sigma_c/\sigma)}$ at $T=0$, in agreement with Ref. [@LHT96]. We further determine the critical current. If the transport current is strong enough, it will depin vortices such that the dissipation sets in. A crude estimate for the critical depinning force at $T=0$ and $\sigma<\sigma_c$ is given by $$\label{eq:criticalforce} f_c\sim \frac{J}{R_c} \sim \frac{J}{\xi}\left(\frac{L}{\xi}\right)^{\frac{-1}{2\sigma_c/\sigma-1}},$$ since smaller dipoles are depinned at larger forces. The influence of disorder on the voltage-current relation is left for further studies. *Conclusion.* We have investigated transport properties of Josephson junction arrays taking into account the influence of quantum fluctuations on the unbinding of vortex pairs for $E_c\ll J$. At sufficiently low temperatures the quantum tunneling of vortices turns out to be more probable than their thermal activation. We have derived the $V$-$I$ relation corresponding to the quantum creep of the BKT-vortices and found the range of temperatures, $0\leq T\leq T_0$, where this law is applicable. We have determined the temperature $T^*$ above which the thermally activated breaking of vortex pairs dominates the vortex nucleation. We have discussed the region of intermediate temperatures $T_0<T<T^*$ where a crossover from classical to quantum behavior occurs, and found the quantum correction to the classical result, see Eq. (\[eq:correction\]). The results are schematically summarized in Fig. \[Fig:regions\] and can be straightforwardly extended to the quantum limit $E_c\gg J$, where the transport is mediated by the motion of charges dual to the superconducting vortices, via the standard dual transformation. Moreover, in the presence of positional disorder and for $C\to \infty$, we have shown that additional vortices generated by the disorder contribute to transport, effectively reducing the critical current as compared to that in a clean system. We are delighted to thank R. Fazio and Z. Ristivojevic for useful discussion. This work was supported by the U.S. Department of Energy Office of Science through contract No. DE-AC02-06CH11357; authors like to acknowledge the support from the SFB 608 (AP and TN) and the AvH foundation (VMV). [99]{} A.Rüfenacht, *et al*, Phys. Rev. Lett., **96** 227002 (2006). D.Matthey, N.Reyren, J.-M.Triscone, and T.Schneider, Phys. Rev. Lett., **98**, 057002 (2007). N.Reyren *et al*, Science, **317**, 1196 (2007). A.D.Caviglia, *et al*, Nature **456**, 624 (2008). R.W.Crane, *et al*, Phys. Rev **B 75**, 094506 (2007). A.Pourret, *et al*, Nat. Phys. **2**, 683 (2006). V. M. Vinokur *et al*, Nature (London) **452**, 613 (2008). B.I.Halperin and D.R.Nelson, J. Low Temp. Phys. **36**, 599 (1979). P.Minnhagen, Rev. Mod. Phys. **59**, 1001 (1987). J.M.Repaci et al., Phys. Rev. B **54**, R9674 (1996); D. R. Strachan, C.J.Lobb, and R.S.Newrock, Phys. Rev. B **67**, 174517 (2003); M.M.Ozer, J.R.Thompson, and H.H.Weitering, Phys. Rev. B **74**, 235427 (2006); M.M.Ozer et al., Science **316**, 1594 (2007); A. Rüfenacht et al., Phys. Rev. Lett. **96**, 227002 (2006); F.Tafuri et al., Europhys. Lett. **73**, 948 (2006). V. G. Kogan, Phys. Rev. B **75**, 064514 (2007). A.Gurevich and V.M.Vinokur, Phys. Rev. Lett. **100**, 227007 (2008). T.Nattermann, S.Scheidl, S.E.Korshunov and M.S.Li, J. Phys. (France) I, [**5**]{}, 565 (1995). S.E.Korshunov and T.Nattermann, Phys. Rev. B [**53**]{}, 2746 (1996), Physica B [**222**]{}, 280 (1996). L.Benfatto, C.Castellani and T.Giamarchi, arxiv:0909.0479 U.Eckern and A.Schmid, Phys. Rev. B [**39**]{}, 6441 (1989). E.Granato and J.M.Kosterlitz, Phys. Rev. B [**33**]{}, 6533 (1986). R.Fazio and G.Sch" on, Phys. Rev. **B 43**, 5307 (1991). M.Tinkham, “*Introduction to Superconductivity*” Dover publications 2004. J.Bardeen and M.J.Stephen, Phys. Rev. [**140**]{} 1197A (1965). I.Affleck, Phys. Rev. Lett. **46**, 388 (1981). R.Iengo and G.Jug, Phys. Rev. B [**54**]{}, 13207 (1996). V.I.Goldanskii, Dokl. Acad. Nauk SSSR **124**, 1261 (1959) \[Sov. Phys. Dokl. **4**,74 (1959)\]. S.Doniach and B.A.Huberman, Phys. Rev. Lett. [**42**]{}, 1169 (1979). A.I.Larkin and Yu.N.Ovchinnikov, JETP Lett. [**37**]{}, 382 (1983), Sov. Phys. JETP [**59**]{}, 2 (1984). E.M.Chudnovsky, Phys. Rev. A [**46**]{}, 8011 (1992). L.-H.Tang, Phys. Rev. B [**54**]{}, 3350 (1996).

--- abstract: 'The 7.6-eV-isomer of Thorium-229 offers the opportunity to perform high resolution laser spectroscopy of a nuclear transition. We give a brief review of the investigations of this isomer. The nuclear resonance connecting ground state and isomer may be used as the reference of an optical clock of very high accuracy using trapped and laser-cooled thorium ions, or in a compact solid-state optical frequency standard of high stability.' --- [**Prospects for a Nuclear Optical Frequency Standard based on Thorium-229**]{}\ [E. Peik, K. Zimmermann, M. Okhapkin, Chr. Tamm]{} [*Physikalisch-Technische Bundesanstalt,\ Bundesallee 100, 38116 Braunschweig, Germany*]{} [**1. The low-lying isomer of $^{229}$Th**]{} Thorium-229 seems to be a unique system in nuclear physics in that it possesses the only known isomer with an excitation energy in the range of optical photon energies and in the range of outer-shell electronic transitions. $^{229}$Th is part of the decay chain of $^{233}$U and undergoes $\alpha$-decay with a halflife of 7880 years. Its energy level structure was studied by the group of C. W. Reich at the Idaho National Engineering Laboratory since the 1970s, mainly relying on spectroscopy of the $\gamma$-radiation emitted after the $\alpha$-decay of $^{233}$U [@kroger; @helmer]. It was noted that the lower energy levels belong to two rotational bands whose band heads must be very close, one being the ground state, the other the isomer. Evaluating several $\gamma$-decay cascades, the value $(3.5\pm 1.0)$ eV was obtained for the isomer energy [@helmer]. Further studies confirmed and extended the knowledge on the overall nuclear level scheme of $^{229}$Th [@gulda; @barci; @ruchowska] and also supported the value for the isomer excitation energy [@barci]. The isomer may decay to the ground state under the emission of magnetic dipole radiation with an estimated lifetime of a few 1000 s [@dykhne; @ruchowska] in an isolated nucleus. These results on the extremely low excitation energy of the isomer inspired a number of theoretical studies, investigating the decay modes of the isomer in different chemical surroundings and possible ways of exciting it with radiation (see Refs. [@matinyan; @tkalya] for reviews). On the experimental side, two false optical detections of the decay of the isomer were reported [@irwin; @richardson], but it was quickly clarified that the observed light was luminescence induced by the background of $\alpha$-radiation [@utter; @shaw]. Fluorescence experiments with radioactive samples and ultraviolet light sources may be severely affected by thermoluminescence, Cherenkov radiation etc. [@young; @peik1]. All further attempts at a direct observation of the optical transition connecting ground state and isomer failed. A reanalysis of the data presented in Ref. [@helmer] indicated a possible shift to higher energies by about +2 eV [@helene]. New experimental data on the transition energy became available only after a group at LLNL used a high resolution $\gamma$-spectrometer and measured two decay cascades very precisely. The result on the transition energy is $(7.6\pm0.5)$ eV [@beck], placing the transition in the vacuum ultraviolet at about 160 nm. Radiation at this wavelength is not transmitted through air, water or optical glasses, which explained the failure of most of the previous attempts to detect the radiation emitted in the decay of the isomer. The energy of 7.6 eV is higher than the ionization potential of the Th atom but lower than that of Th$^+$. This opens internal conversion as an alternative decay channel in neutral thorium. ![The ground state and lowest excited state of the $^{229}$Th nucleus with the level classification in the Nilsson model, radiative lifetime [@ruchowska; @dykhne] for the magnetic dipole transition, magnetic moments in nuclear magnetons and quadrupole moment of the ground state [@gerstenkorn; @dykhne].](Th-levels.eps){width="5cm"} We have extended the search for light emission from $^{229}$Th isomers produced in the $\alpha$-decay of U-233 using photodetectors with sensitivity in the VUV range. Light emission from a solid $^{233}$U-nitrate sample as well as from freshly produced $^{229}$Th recoil nuclei in a catcher foil was investigated – without seeing the expected signals [@zimmermann]. In the experiment with $^{233}$U a spectrometer was used but the noise level on the detector was too high to reach the required sensitivity. The experiment with the recoil nuclei showed negligible background and the negative outcome may be interpreted as a sign for a rapid nonradiative decay of the isomer in the solid. The expected transition wavelength is in a part of the VUV range that is accessible by frequency upconversion of narrow-bandwidth continuous and phase-coherent femtosecond laser sources. Thus it appears possible to measure the nuclear energy level splitting of $^{229}$Th with the precision afforded by high-resolution laser spectroscopy and optical frequency metrology. Figure 1 summarizes the relevant spectroscopic information that is available today. [**2. Nuclear optical clock with trapped ions**]{} Nuclear transition frequencies are much more stable against external perturbations than transition frequencies of the electron shell because the characteristic nuclear dimensions are small compared to the atomic dimension. Therefore nuclear transitions are attractive as highly accurate frequency references with small field-induced shifts [@peik2]. Apart from motional frequency shifts that can be well controlled e.g. in laser-cooled trapped ions, the interaction with ambient electric or magnetic fields usually is the dominant source of systematic uncertainty in optical frequency standards. Estimates on the magnitude of systematic frequency shifts must, however, also consider the coupling of the nuclear and electronic energy level systems through the Coulomb and hyperfine interactions. For external electric field gradients the electron shell may actually lead to an enhancement at the nucleus (Sternheimer anti-shielding). It will therefore be important to select a suitable electronic state for the nuclear excitation. In order to illustrate the role of hyperfine interactions in nuclear spectroscopy of an isolated atom or ion, let us consider the Zeeman and Stark shifts of the nuclear transition frequency. In an $LS$ coupling scheme the eigenstates of the coupled electronic and nuclear system are characterised by sets of quantum numbers $|\alpha,I;\beta,L,S,J;F,m_F\rangle$, where $I$ denotes the nuclear spin, $L,S,J$ the orbital, spin and total electronic angular momenta, and $F$ and $m_F$ the total atomic angular momentum and its orientation. $\alpha$ and $\beta$ label the nuclear and electronic configurations. In the nuclear transition, the nuclear and total angular momentum quantum numbers ($\alpha, I, F, m_F$) can change, while the purely electronic quantum numbers ($\beta,L,S,J$) remain constant. The nuclear transition frequency is independent of all mechanisms that produce level shifts depending only on the electronic quantum numbers ($\beta,L,S,J$), because these do not change and consequently the upper and the lower state of the transition are affected in the same way. This applies to the scalar part of the quadratic Stark effect, which typically is the dominant mechanism for the shift of electronic transition frequencies due to static electric fields, electromagnetic radiation, and collisions. The observed nuclear transition frequency is however shifted by the hyperfine Stark shift, which depends on $F$ and $m_F$, and has been studied in microwave atomic clocks. In the optical frequency range, a relative magnitude of typically $10^{-19}$ may be expected for the hyperfine Stark shift caused by the $\approx 10$ V/cm room temperature blackbody radiation field. In order to avoid the influence of the linear Zeeman effect, an electronic state can be chosen such that $F$ is an integer. In this case a Zeeman component $m_F=0 \rightarrow 0$ is available, that shows only a small quadratic Zeeman effect around zero magnetic field. Since this shift depends similarly on the electronic and the nuclear g-factor its magnitude will be comparable to those in other atomic clocks. Further field dependent shifts may arise from the tensor part of the quadratic Stark effect and from the quadrupole interaction between the atomic quadrupole moment and electric field gradients. Both these shifts can be expressed as a product of $J$-dependent and $F$-dependent terms and vanish if either $J<1$ or $F<1$. From these general considerations it can be seen that for every radiative nuclear transition, an electronic state can be selected which makes the hyperfine coupled nuclear transition frequency immune against the linear Zeeman effect and the quadratic Stark effect as well as the quadrupole shift. For electronic transitions, this combination of advantageous features can not be obtained. Since the selected electronic state has to be stable or at least long-lived, the choice could be made among the ground states of the differently charged positive ions of the element in question. In the case of a half integer nuclear spin (like in $^{229}$Th), the optimal electronic states are $^2S_{1/2}$ or $^2P_{1/2}$, and for an integer nuclear spin the states $^1S_0$ or $^3P_0$ fulfill all criteria. For a high precision nuclear clock, the case of trapped $^{229}$Th$^{3+}$-ions seems to be especially promising [@peik2] because its electronic level structure is suitable for laser cooling. The sensitive detection of excitation to the isomeric state will be possible using a double resonance scheme that probes the hyperfine structure of a transition in the electron shell. No electric dipole transitions originate from the electronic ground state of Th$^{3+}$ in the range of 1.8 – 15 eV so that resonant coupling between electronic and nuclear excitations is not expected to play an important role for the decay of the isomeric state. The $5f~^2F_{5/2}$ ground state of Th$^{3+}$ does not fulfill the condition $J<1$ for elimination of the tensor Stark effect and the quadrupole shift, but a metastable $7s~^2S_{1/2}$ state of lifetime $\approx 1$ s is also available. Alternatively, the method of quantum logic spectroscopy [@schmidt] with an auxiliary ion may be applied to other charge states of $^{229}$Th that can not be laser-cooled directly. [**3. A solid-state nuclear frequency standard**]{} A nuclear transition may also provide a resonance with very high resolution if the nuclei are embedded in a solid, as it is observed in Mö[ß]{}bauer spectroscopy. Thorium-229 opens the possibility to perform optical Mö[ß]{}bauer spectroscopy using a laser as a tuneable, coherent source of radiation. This may provide a compact and simple reference for an optical frequency standard with performance much superior to what is available in simple atomic systems like vapor cells [@peik2; @hudson]. The host crystal should be transparent at the nuclear resonance wavelength $\lambda_0$, a criterion that is fulfilled by a number of candidates like the fluorides of the alkaline earths. It would be only lightly doped with $^{229}$Th. If the broadening is dominantly homogeneous one nucleus per $\lambda_0^3$ may be used in order to avoid strong radiation trapping. Still, this would allow to handle $10^{11}$ nuclei in a cube of 1 mm dimension. With this number of nuclei direct fluorescence detection of the resonance radiation would be possible even if the resonant scattering rate is only of the order $10^{-4}$/s per nucleus. The uncertainty budget of such a solid-state nuclear clock will be quite different from that of a realization with trapped ions considered above. The crystal field shifts of the nuclear resonance frequency will be dominantly due to electric fields if a diamagnetic host is used. In insulators with high bandgap like fluorides rather high internal electric fields and field gradients will be found. The electron charge density at the position of the nucleus will lead to the isomer shift $\Delta f_{iso}=Ze^2\rho_0\langle r^2\rangle/(h\epsilon_0)$, where $Ze$ is the nuclear charge, $\rho_0$ the electron density at the nucleus and $\langle r^2\rangle$ the mean squared nuclear charge radius. The contribution of a 7s electron in thorium would shift the nuclear ground state by $\Delta f_{iso}\approx1$ GHz with respect to its energy in a bare nucleus. An electric field gradient will produce a quadrupole shift that may be of comparable magnitude: In the tetragonal crystal ThB$_4$, for example, the field gradient along the principal axis is about $5\times10^{21}$ V/m$^2$. Coupling to the ground state quadrupole moment of $^{229}$Th of $5\times 10^{-28}$ m$^2$ would produce a quadrupole shift of 0.6 GHz. The field gradient can be avoided in a crystal lattice of higher symmetry, like a cubic one. Both these shifts would be of less concern if they would be constant, which would only be the case if the positions of all charges in the lattice would be rigorously fixed. Thermal motion, however, will lead to a temperature-dependent broadening and shift of the line, where the line shape will depend on phonon frequencies and correlation times. Much information on these effects has been obtained in conventional ($\gamma$-ray) Mö[ß]{}bauer spectroscopy already. While the relativistic Doppler shift will lead to a temperature dependent relative frequency shift of about $10^{-15}/$K, the temperature dependence of the crystal field will critically depend on the choice of crystal host and may be significantly bigger. For a solid state nuclear clock of high accuracy (beyond $10^{-15}$) the temperature dependence may be eliminated if the crystal is cryogenically cooled to well below the Debye temperature, so that the influence of phonons is effectively frozen out. [**4. Conclusion**]{} Nuclear laser spectroscopy of $^{229}$Th seems to offer great potential for frequency metrology and promises to open a new field of research at the borderline between nuclear and atomic physics, shedding new light on familiar phenomena like nuclear radiative decay or hyperfine interactions. It may allow improved tests of fundamental physics, as it was recently shown that the resonance frequency would be the most sensitive probe in the search for temporal variations of the fundamental coupling constants [@flambaum1; @flambaum2]. [**Acknowledgments**]{} This work is supported by DFG through SFB 407 and the cluster of excellence QUEST. [99]{} L. A. Kroger and C. W. Reich, Nucl. Phys. A [**259**]{}, 29 (1976). R. G. Helmer and C. W. Reich, Phys. Rev. C [**49**]{}, 1845 (1994). K. Gulda et al., Nucl. Phys. A [**703**]{}, 45 (2002). V. Barci et al, Phys. Rev. C [**68**]{}, 034329 (2003). E. Ruchowska et al., Phys. Rev. C [**73**]{}, 044326 (2006). A. M. Dykhne and E. V. Tkalya, Pis’ma Zh. Eksp. Teor. Fiz. [**67**]{}, 233 (1998) \[JETP Lett. [**67**]{}, 251 (1998)\]. S. Matinyan, Phys. Rep. [**298**]{}, 199 (1998). E. V. Tkalya, Physics Uspekhi [**46**]{}, 315 (2003). G. M. Irwin and K. H. Kim, Phys. Rev. Lett. [**79**]{}, 990 (1997). D. S. Richardson et al., Phys. Rev. Lett. [**80**]{}, 3206 (1998). S. B. Utter et al., Phys. Rev. Lett. [**82**]{}, 505 (1999). R. W. Shaw, J. P. Young, S. P. Cooper and O. F. Webb, Phys. Rev. Lett. [**82**]{}, 1109 (1999). J. P. Young, R. W. Shaw and O. F. Webb, Inorg. Chem. [**38**]{}, 5192 (1999). E. Peik, J. Phys. D: Appl. Phys. [**40**]{}, 3330 (2007). Z. O. Guimaraes-Filho, O. Helene, Phys. Rev. C [**71**]{}, 044303 (2005). K. Zimmermann et al., to be published. B. R. Beck et al., Phys. Rev. Lett. [**98**]{}, 142501 (2007). S. Gerstenkorn et al. J. Physique (Paris) [**35**]{}, 483 (1974). E. Peik and Chr. Tamm, Europhys. Lett. [**61**]{}, 181 (2003). P. O. Schmidt, T. Rosenband, C. Langer, W. M. Itano, J. C. Bergquist and D. J. Wineland, Science [**309**]{}, 749 (2005). E. R. Hudson, A. C. Vutha, S. K. Lamoreaux, D. DeMille, Poster MO28, presented at the International Conference on Atomic Physics XXI, Storrs, Connecticut, USA (2008); http://www.phys.uconn.edu/icap2008/accepted/icap2008-posters1.pdf V. V. Flambaum, Phys. Rev. Lett. [**97**]{}, 092502 (2006). V. V. Flambaum, N. Auerbach and V. F. Dmitriev, arXiv:0807.3218; V. V. Flambaum and R. B. Wiringa, arXiv:0807.4943.

KEK-TH-1111\ [**Junji Hisano**]{}$^{(a)}$, [**Shigeki Matsumoto**]{}$^{(b)}$, [**Minoru Nagai**]{}$^{(a)}$, [**Osamu Saito**]{}$^{(a,b)}$, and [**Masato Senami**]{}$^{(a)}$ 0.15in [*$^{(a)}$[ICRR, University of Tokyo, Kashiwa 277-8582, Japan]{}\ $^{(b)}$[Theory Group, KEK, Oho 1-1 Tsukuba, Ibaraki 305-0801, Japan]{}\ *]{} Introduction ============ Precise measurements of cosmological parameters have achieved amazing progress in recent years. In particular, the observation of cosmic microwave background anisotropies by the Wilkinson Microwave Anisotropy Probe (WMAP) [@WMAP] confirmed that non-baryonic dark matter amounts to 20% of the energy of our universe. The existence of dark matter forces us to consider physics beyond the standard model (SM) for its constituent, because the SM has no candidate for the dark matter. Many models beyond the SM have been proposed for providing dark matter candidates. Models involving weakly interacting massive particles (WIMPs), such as supersymmetric models and universal extra dimension (UED) models [@UED; @UED2], have an advantage over other models, since the WIMPs can explain the observed abundance naturally in the thermal relic scenario [@UED2; @susydm; @UEDabundance]. Comparison of the predicted thermal abundance with observations is a powerful tool to constrain those models. It is now important for discussion of new physics signatures at collider experiments such as the LHC. In the thermal relic scenario, the annihilation cross sections of dark matter play a central role in evaluation of the abundance. Perturbative calculation of the cross sections is sufficient for it in usual cases. However, a non-perturbative effect on the cross sections has to be included, when dark matter particles are non-singlet under the SU(2)$_L$ interaction and much heavier than the weak gauge bosons. The weak interaction is not a short-distance force, but rather a long-distance one for non-relativistic two-bodies states of such heavy particles. The wave functions for the two-bodes states are modified from the plane waves by the interaction, and the annihilation cross sections are affected. The dark matter particles are non-relativistic at the freeze-out temperature. The attractive channels due to the weak force enhance the annihilation cross sections, compared with the perturbative ones. This effect is called the Sommerfeld enhancement, which was found in inelastic reactions between non-relativistic charged particles, historically. In this letter, we point out that the non-perturbative effect strongly alters the relic abundance of dark matter compared to that in perturbative calculation. We investigate the non-perturbative effect quantitatively in the wino-like neutralino dark matter as an example, and show that the abundance is reduced by 50% due to the effect. We also discuss non-perturbative effects on other dark matter candidates at the end of this letter. Sommerfeld enhancement in wino-like neutralino dark matter annihilation ======================================================================= Winos are the SU(2)$_L$ gauginos in the supersymmetric standard model, and they are an SU(2)$_L$ triplet. They are mixed with higgsinos, superpartners of the Higgs bosons, after the electroweak symmetry breaking. The neutral components in the mass eigenstates are called neutralino. The mixing with higgsinos is small for heavy winos, since it is suppressed by the electroweak scale. Thus, the wino-like neutralino ($\DM$) is highly degenerate with its charged SU(2)$_L$ partner ($\CP$) in mass when their masses are heavy enough. For the wino-like neutralino mass ($m$) of the order of 1 TeV, the mass difference between the neutralino and its SU(2)$_L$ partner is dominated by a radiative correction [@Hisano:2004ds]. It is calculated as $\delta m \simeq$ 0.17 GeV, which is used for numerical calculations in this letter. There are four processes related to the calculation of the wino-like neutralino relic abundance, and those are $\DM\DM$, $\CP\CPC$, $\DM\CP$, and $\CP\CP$ annihilation. We assume the CP conservation, and cross sections of $\DM\CPC$ and $\CPC\CPC$ annihilation are the same as those of $\DM\CP$ and $\CP\CP$, respectively. Furthermore, each process is decomposed into two ones with $S = 0$ and $S = 1$, where $S$ is the total spin of the two-bodies system at the initial state. In $\DM\DM$ and $\CP\CP$ annihilation, the $S = 1$ processes are forbidden at the $s$-wave annihilation. After all, processes we have to consider are $\DM\DM$, $\CP\CPC$, $\DM\CP$, $\CP\CP$ annihilation with $S = 0$, and those of $\CP\CPC$, $\DM\CP$ with $S = 1$. While gamma rays from the wino-like neutralino annihilation in the Galactic center [@Hisano:2004ds; @Hisano:2003ec] and anti-particles fluxes from that in Galactic Halo [@Hisano:2005ec] are evaluated including non-perturbative effects, the relic abundance of the neutralino have been calculated only within a perturbative method. However, as we will see later, the non-perturbative effect can significantly alter cross sections relevant to the abundance. Two-bodies effective Lagrangian, describing relative motion between two particles in the two-bodies system, is useful to evaluate the non-perturbative effect in the annihilation. The Lagrangian has a following form, $$\begin{aligned} {\cal L}_{\rm eff} = \int d^3 r~ \sum_{S_Z}\Phi^\dagger_{S_Z}(x,\vec{r}) \left[ i\partial_{x^0} + \frac{\nabla_x^2}{4m} + \frac{\nabla_r^2}{m} - V(r) + 2i \Gamma \delta(\vec{r}) \right] \Phi_{S_Z}(x,\vec{r})~, \label{twob}\end{aligned}$$ where $x$ is the center of mass coordinate for the two-bodies system, while the relative one is represented by $\vec{r}$. The index $S_Z$ is for the $z$-th component of the total spin $S$. The potential term $V(r)$ describes forces acting between two particles in the system. The absorptive part $\Gamma$ stands for the annihilation of the two-bodies system. For derivation of the effective Lagrangian and following evaluation of the annihilation cross section using it, see Ref. [@Hisano:2004ds]. In the $\CP\CP$ annihilation process with $S = 0$, the potential $V(r)$ and absorptive part $\Gamma$ in Eq. (\[twob\]) turn out to be $$\begin{aligned} V(r) = \frac{\alpha}{r} + \alpha_2 c_W^2 \frac{e^{-m_Z r}}{r}~, \quad \Gamma = \frac{\pi\alpha_2^2}{2m^2}~,\end{aligned}$$ where $\alpha$ is the fine structure constant, $\alpha_2 = g_2^2/4\pi$ is for the SU(2) gauge coupling constant, $m_Z$ is the $Z$ boson mass, and $c_W = \cos\theta_W$ is for the weak mixing angle $\theta_W$. The first term in the potential comes from the exchange of photons, while the second one is from the exchange of $Z$ bosons. In the calculation of $\Gamma$, we consider only the final states of SM particles, and neglect their masses, since they are light enough compared to the wino-like neutralino we are discussing. In both cases of the $\DM\CP$ annihilation processes with $S = 0$ and $1$, $V(r)$ is induced from the exchange of $W$ bosons, and both cases have the same form. On the other hand, the absorptive part $\Gamma$ is different each other. These are given by $$\begin{aligned} V(r) = - \alpha_2 \frac{e^{-m_W r}}{r}~, \quad \Gamma_{(S = 0)} = \frac{1}{2} \frac{\pi\alpha_2^2}{m^2}~, \quad \Gamma_{(S = 1)} = \frac{25}{24} \frac{\pi\alpha_2^2}{m^2}~,\end{aligned}$$ where $m_W$ is the $W$ boson mass. The potential and absorptive terms in the $\CP\CPC$ annihilation with $S = 1$ are $$\begin{aligned} V(r) = - \frac{\alpha}{r} - \alpha_2 c_W^2 \frac{e^{-m_Z r}}{r}~, \qquad \Gamma = \frac{25}{24} \frac{\pi \alpha_2^2}{m^2}~.\end{aligned}$$ The $\DM\DM$ two-bodies system is mixed with $\CP\CPC$ state with $S = 0$, in which mixing occurs through a $W$ boson exchange. Thus, the potential and absorptive terms are written by $2 \times 2$ matrices as $$\begin{aligned} {\bf V}(r) = \left( \begin{array}{cc} 2\delta m - \ds\frac{\alpha}{r} - \ds\alpha_2 c_W^2 \frac{e^{-m_Z r}}{r} & -\sqrt{2} \alpha_2 \ds\frac{e^{-m_W r}}{r} \\ -\sqrt{2} \alpha_2 \ds\frac{e^{-m_W r}}{r} & 0 \end{array} \right), ~ {\bf \Gamma} = \frac{\pi \alpha_2^2}{2 m^2} \left( \begin{array}{cc} 3 & \sqrt{2} \\ \sqrt{2} & 2 \end{array} \right).\end{aligned}$$ Off-diagonal elements describe the transition between $\CP\CPC$ and $\DM\DM$ systems. As seen in these potentials, all processes have attractive channels except for that of $\tilde \chi ^- \tilde \chi ^-$. The overlap between the wave functions of the incident particles are increased compared to the case without including the potentials, and it leads to enhancement of the annihilation cross sections. Once the two-bodies effective Lagrangian is obtained, annihilation cross sections including the non-perturbative effect can be calculated through the formula, $$\begin{aligned} \sigma v = c \Gamma |A|^2~, \qquad A \equiv \int d^3r~ e^{-i\vec{k}\cdot\vec{r}} \left( \frac{mv^2}{4} + \frac{\nabla^2}{m} \right) G(\vec{r},\vec{0})~, \label{formula}\end{aligned}$$ where $c = 2$ for an annihilation of identical particles, otherwise $c=1$. The relative velocity between incoming particles is denoted by $v$. The Green function $G$ satisfies the equation of motion of the effective Lagrangian, $$\begin{aligned} \left[ \frac{mv^2}{4} + \frac{\nabla^2}{m} - V(r) + 2i\Gamma\delta^3(\vec{r}) \right]G(\vec{r},\vec{r}') = \delta^3(\vec{r} - \vec{r}')~. \label{EOM}\end{aligned}$$ The boundary condition for the Green function is determined from following conditions. First, the Green function is analytic at any $\vec{r}$ and $\vec{r}'$ except the point $\vec{r} = \vec{r}'$. Second, only out-going waves survive at large $|\vec{r} - \vec{r}'|$. In $\DM\DM$ and $\CP\CPC$ annihilation processes with $S = 0$, the Green function has a $2\times2$ matrix form. For this case, it has been found in Ref. [@Hisano:2004ds; @Hisano:2003ec] that above formula is simply extended as $$\begin{aligned} \sigma_i v = c_i \sum_{j,j'} {\bf A}_{ij} {\bf \Gamma}_{jj'} {\bf A}^*_{ij'}~.\end{aligned}$$ The factor $|A|^2$ in Eq. (\[formula\]) is called the Sommerfeld enhancement factor, and it includes the non-perturbative effect due to the long-distance forces in $V(r)$. Note that if we neglect the non-perturbative effect, the enhancement factor becomes one and the annihilation cross section in Eq. (\[formula\]) coincides with the result in a usual perturbative method. Effect of Sommerfeld enhancement on dark matter abundance ========================================================= Now we evaluate the thermal relic abundance of the wino-like neutralino dark matter, including the non-perturbative effect. In the evaluation we have to include coannihilation processes in addition to the wino-like neutralino pair annihilation. We use the method developed in Ref. [@Griest:1990kh; @Gondolo:1990dk] for the calculation of the relic abundance including coannihilation effects. Under reasonable assumptions, the relic density obeys the following Boltzmann equation, $$\begin{aligned} \frac{dY}{dx} = -\frac{\langle\sigma_{\rm eff} v\rangle}{Hx} \left(1 - \frac{x}{3g_{*s}}\frac{dg_{*s}}{dx}\right) s\left(Y^2 - Y_{\rm eq}^2\right)~. \label{eq:Boltzmann}\end{aligned}$$ The yield of the dark matter, $Y$, is defined as $Y = n/s$, where $n$ is the sum of the number densities of $\DM$, $\CP$, and $\CPC$. The variable, $x = m/T$, is the scaled inverse temperature of the universe. The equilibrium abundance, $Y_{\rm eq}$, is given by $$\begin{aligned} Y_{\rm eq} = \frac{45}{2\pi^4} \left(\frac{\pi}{8}\right)^{1/2} \frac{g_{\rm eff}}{g_{*}} x^{3/2} e^{-x}~,\end{aligned}$$ where $g_{\rm eff}$ is the number of the effective degrees of freedom defined as $$\begin{aligned} g_{\rm eff}(x) = 2 + 4(1 + \delta m/m)^{3/2} e^{-x \delta m/m}~. \label{eq:geff}\end{aligned}$$ The entropy density $s$ and the Hubble parameter $H$ are given by $$\begin{aligned} s = \frac{2\pi^2}{45} g_{*s}\frac{m^3}{x^3}~, \qquad H = \left( \frac{g_*}{10} \right)^{1/2} \frac{\pi}{3 M_{\rm Pl}} \frac{m^2}{x^2} ~,\end{aligned}$$ where $M_{\rm Pl} = 2.4 \times 10^{18}$ GeV is the reduced Planck mass. The relativistic degrees of freedom of the thermal bath, $g_*$ and $g_{*s}$, should be treated as a function of the temperature for deriving the correct dark matter abundance in our calculation. The most important quantity to determine the abundance is the thermally averaged effective annihilation cross section $\langle\sigma_{\rm eff}v\rangle$ in Eq. (\[eq:Boltzmann\]), defined as $$\begin{aligned} \langle\sigma_{\rm eff}v\rangle &=& \sum_{i,j}\langle\sigma_{ij}v\rangle\frac{4}{g_{\rm eff}^2(x)} (1 + \Delta_i)^{3/2}(1 + \Delta_j)^{3/2} \exp[-x(\Delta_i + \Delta_j)]~, \nonumber \\ \langle\sigma_{ij}v\rangle &=& \left(\frac{m}{4\pi T}\right)^{3/2} \int 4\pi v^2dv~\left(\sigma_{ij}v\right) \exp\left(-\frac{mv^2}{4T}\right)~, \label{effective CS}\end{aligned}$$ where $i,j = \DM$, $\CP$ and $\CPC$, $\Delta_{\CP} = \Delta_{\CPC} = \delta m / m$, $\Delta_{\DM} = 0$, and $\sigma_{ij}$ is the annihilation cross section between $i$ and $j$. In Fig. \[CS\], the enhancement ratio of the averaged cross section, $\langle\sigma_{\rm eff}v\rangle$, to that in the perturbative calculation is shown as a function of $m$ with fixed $m/T = 20, ~200, ~2000$ (left figure) and as a function of $T$ with fixed $m = 2.8$ TeV (right figure). For comparison, the cross section in a perturbative calculation is also shown as a dotted line in the right figure. Note that the perturbative cross section is constant in time. The little drop at $x \sim 10^5$ is due to the decoupling of $\tilde \chi ^\pm$. In the calculation of the cross section, we used the running gauge coupling constant at the $m$ and $m_Z$ in the absorptive terms and the potentials, respectively. (-120,138)[$\langle \sigma_{\rm eff} \rangle / \langle \sigma_{\rm eff} \rangle_{\rm Tree}$]{} (-177,134)[Thermal averaged cross section, $\langle \sigma_{\rm eff} \rangle$]{} In these figures, large enhancement of the cross section is found due to the non-perturbative effect when $m$ is larger than $\sim 1$ TeV. A significant enhancement is shown at $m\sim$ 2.4 GeV. This originates in the bound state composed of $\DM\DM$ and $\CP\CPC$ pairs [@Hisano:2004ds; @Hisano:2003ec]. The enhancement by the non-perturbative effect is more significant for lower temperature. Since $\DM$ and $\CP$ are more non-relativistic for lower temperature, the long-range force acting between these particles strongly modifies their wave functions and alters the cross section significantly. (-115,136)[$Y / Y_{\rm Tree}$]{} (-165,136)[Thermal relic abundance, $\Omega_{\rm DM} h^2$]{} Since the averaged cross section depends on temperature in a non-trivial way as shown in Fig. \[CS\], we should integrate the Boltzmann equation numerically. After calculating the present value of the yield, $Y_0$, by the integration, we obtain the dark matter mass fraction in the current universe through the relation $\Omega_{DM} h^2 = m s_0 Y_0 h^2/\rho_c$, where $\rho_c$ is the critical density, $\rho_c = 1.05 \times 10^{-5} h^2 ~{\rm GeV cm}^{-3}$ ($h = 0.73^{+0.04}_{-0.03}$) [@PDG], and $s_0$ is the entropy density of the present universe. The result is shown in Fig. \[Omega\]. In the left figure, the ratio of the yield with the enhancement to one without the enhancement (perturbative result) is shown as a function of temperature. The mass of the wino-like neutralino is assumed $ m = 2.8$ TeV. The enhancement of the cross section at the departure from the equilibrium decreases the abundance by $20-30\%$, and it leads to quick deviation of the yield from the perturbative one after decoupling. Since the annihilation cross section is increased for lower temperature by the Sommerfeld enhancement, the sudden freeze-out phenomenon on the yield does not occur and the yield continues to be reduced by the non-perturbative effect even for $x > 100$ compared to the perturbative one. The resultant dark matter abundance is reduced by 50% compared to the perturbative result. In the right figure, the relic abundance of the dark matter in the present universe is shown as a function of $m$ in terms of $\Omega h^2$. The allowed regions by the WMAP at 1 and 2 $\sigma$ are also shown as shaded areas in this figure. We found that the mass in the wino-like neutralino dark matter consistent with the observation is shifted by 600 GeV due to the non-perturbative effect and the wino-like neutralino mass consistent with WMAP results turns out to be 2.7 TeV $\lesssim m \lesssim$ 3.0 TeV. Summary and discussion ====================== In this letter, we have pointed out the thermal relic abundance of dark matter, which is SU(2)$_L$ non-singlet and has a much larger mass than that of the weak gauge bosons, can be strongly reduced by the non-perturbative effect. We have investigated the non-perturbative effect on the relic abundance of wino-like neutralino as an example. Compared with the perturbative result, this effect reduces the abundance by about 50% and increases the mass of the wino-like neutralino dark matter consistent with the observation by about 600 GeV. As a result, the thermal relic abundance of the wino-like neutralino dark matter is consistent with observed abundances when 2.7 TeV $\lesssim m \lesssim$ 3.0 TeV. The non-perturbative effect can change relic abundances of other dark matter candidates with SU(2)$_L$ charge and heavy mass, such as higgsino-like neutralino. The non-perturbative effect on the thermal relic abundance of higgsino-like neutralino is expected to be roughly 10%, since winos are triplet under the SU(2)$_L$ gauge group, while higgsinos are SU(2)$_L$ doublet. Therefore, non-perturbative effect may change the abundance by O(10%) for other SU(2)$_L$ doublet candidates for the dark matter. A detailed analysis of this subject is studied elsewhere. The Sommerfeld enhancement occurs reasonably for particles with electric charge. In fact the non-perturbative effect through photon exchanges for charged particle annihilation can change the abundance by about 10%. Therefore, one may think that the relic abundance of dark matter will be changed in the stau coannihilation region or in the case that gravitino is the lightest supersymmetric particle and it is produced through decay of stau. However, the non-perturbative effect for stau does not change the dark matter abundance. In stau annihilation, $\sigma (\tilde \tau ^+ \tilde \tau ^-) \simeq \sigma (\tilde \tau ^+ \tilde \tau ^+)$ and the Sommerfeld enhancement is positive (negative) for the former (latter) process. Hence, the enhancement is almost canceled and the change of the abundance is 1% at most. Kaluza-Klein (KK) right-handed leptons ($E^{(1)}$) in UED models are highly degenerate with the lightest KK particle in mass and $\sigma(E^{(1)} \bar E ^{(1)}) > \sigma(E^{(1)} E ^{(1)})$ [@UED2; @Kakizaki:2005uy]. Hence, the enhancement could be expected to change the abundance of the KK dark matter. The change of the abundance is within 4% since $\sigma(E^{(1)} \bar E ^{(1)})$ contributes to the effective annihilation cross section by 40% at most. Finally, we comment on the non-perturbative effect of colored particles, such as gluino. The enhancement for colored particles are very effective [@Baer:1998pg]. However, it may be very complicated due to the existence of the QCD phase transition, which is discussed in Ref. [@Arvanitaki:2005fa]. Furthermore, colored particles can not be candidates for the dark matter and are not expected to be degenerate with a dark matter particle in mass due to large radiative corrections by the strong interaction. Hence, this subject is beyond the scope of this letter. Acknowledgments {#acknowledgments .unnumbered} =============== This work is supported in part by the Grant-in-Aid for Science Research, Ministry of Education, Science and Culture, Japan (No.1803422 and 15540255 for JH, 16081211 for SM and 18840011 for MS). Also, the work of MN is supported in part by JSPS. [99]{} C. L. Bennett [*et al.*]{}, Astrophys. J. Suppl.  [**148**]{}, 1 (2003); D. N. Spergel [*et al.*]{} \[WMAP Collaboration\], Astrophys. J. Suppl.  [**148**]{}, 175 (2003); D. N. Spergel [*et al.*]{}, arXiv:astro-ph/0603449. T. Appelquist, H. C. Cheng and B. A. Dobrescu, Phys. Rev. D [**64**]{}, 035002 (2001). G. Servant and T. M. P. Tait, Nucl. Phys. B [**650**]{}, 391 (2003); For reviews, see for instance, G. Jungman, M. Kamionkowski and K. Griest, Phys. Rept.  [**267**]{}, 195 (1996); L. Bergstrom, Rept. Prog. Phys.  [**63**]{}, 793 (2000); C. Munoz, Int. J. Mod. Phys. A [**19**]{}, 3093 (2004); G. Bertone, D. Hooper and J. Silk, Phys. Rept.  [**405**]{}, 279 (2005). M. Kakizaki, S. Matsumoto and M. Senami, Phys. Rev. D [**74**]{}, 023504 (2006). J. Hisano, S. Matsumoto, M. M. Nojiri and O. Saito, Phys. Rev. D [**71**]{}, 063528 (2005). J. Hisano, S. Matsumoto and M. M. Nojiri, Phys. Rev. Lett.  [**92**]{}, 031303 (2004). J. Hisano, S. Matsumoto, O. Saito and M. Senami, Phys. Rev. D [**73**]{}, 055004 (2006). K. Griest and D. Seckel, Phys. Rev. D [**43**]{}, 3191 (1991). P. Gondolo and G. Gelmini, Nucl. Phys. B [**360**]{}, 145 (1991). W. M. Yao [*et al.*]{} \[Particle Data Group\], J. Phys. G [**33**]{},1 (2006). M. Kakizaki, S. Matsumoto, Y. Sato and M. Senami, Nucl. Phys. B [**735**]{}, 84 (2006). H. Baer, K. m. Cheung and J. F. Gunion, Phys. Rev. D [**59**]{}, 075002 (1999). A. Arvanitaki, C. Davis, P. W. Graham, A. Pierce and J. G. Wacker, Phys. Rev. D [**72**]{}, 075011 (2005).

--- abstract: | A [*rack*]{} is a set equipped with a bijective, self-right-distributive binary operation, and a [*quandle*]{} is a rack which satisfies an idempotency condition. In this paper, we introduce a new definition of modules over a rack or quandle, and show that this definition includes the one studied by Etingof and Graña [@etingof/grana:orc] and the more general one given by Andruskiewitsch and Graña [@andr/grana:pointed-hopf]. We further show that this definition coincides with the appropriate specialisation of the definition developed by Beck [@beck:thesis], and hence that these objects form a suitable category of coefficient objects in which to develop homology and cohomology theories for racks and quandles. We then develop an Abelian extension theory for racks and quandles which contains the variants developed by Carter, Elhamdadi, Kamada and Saito [@carter/elhamdadi/saito:twisted; @carter/kamada/saito:diag] as special cases. address: | Mathematics Institute\ University of Warwick\ Coventry\ CV4 7AL\ United Kingdom author: - Nicholas Jackson title: Extensions of racks and quandles --- Introduction ============ A [*rack*]{} (or [*wrack*]{}) is a set $X$ equipped with a self-right-distributive binary operation (often written as exponentiation) satisfying the following two axioms: 1. For every $a,b \in X$ there is a unique $c \in X$ such that $c^b = a$. 2. For every $a,b,c \in X$, the [*rack identity*]{} holds: $$a^{bc} = a^{cb^c}$$ In the first of these axioms, the unique element $c$ is often denoted $a^{\overline{b}}$, although $\overline{b}$ should not itself be regarded as an element of the rack. Association of exponents should be understood to follow the usual conventions for exponential notation. In particular, the expressions $a^{bc}$ and $a^{cb^c}$ should be interpreted as $(a^b)^c$ and $(a^c)^{(b^c)}$ respectively. A rack which, in addition, satisfies the following idempotency criterion is said to be a [*quandle*]{}. 1. For every $a \in X$, $a^a = a$. There is an obvious notion of a [*homomorphism*]{} of racks: a function $f{\colon}X {\rightarrow}Y$ such that $f(a^b) = f(a)^{f(b)}$ for all $a,b \in X$. We may thus form the categories ${\mathsf{Rack}}$ and ${\mathsf{Quandle}}$. For any element $x \in X$ the map $\pi_x{\colon}a \mapsto a^x$ is a bijection. The subgroup of ${\operatorname{Sym}}X$ generated by $\{ \pi_x : x \in X \}$ is the [*operator group*]{} of $X$, denoted ${\operatorname{Op}}X$. This assignment is not functorial since there is not generally a well-defined group homomorphism ${\operatorname{Op}}f{\colon}{\operatorname{Op}}X {\rightarrow}{\operatorname{Op}}Y$ corresponding to an arbitrary rack homomorphism $f{\colon}X {\rightarrow}Y$. The group ${\operatorname{Op}}X$ acts on the rack $X$, and divides it into [*orbits*]{}. Two elements $x,y \in X$ are then said to be in the same orbit (denoted $x \sim y$ or $x \in [y]$) if there is a (not necessarily unique) word $w \in {\operatorname{Op}}X$ such that $y = x^w$. A rack with a single orbit is said to be [*transitive*]{}. The set of orbits of $X$ is denoted ${\operatorname{Orb}}X$. Given any group $G$, we may form the [*conjugation rack*]{} ${\operatorname{Conj}}G$ of $G$ by taking the underlying set of $G$ and defining the rack operation to be conjugation within the group, so $g^h := h^{-1}gh$ for all $g,h \in G$. This process determines a functor ${\operatorname{Conj}}{\colon}{\mathsf{Group}} {\rightarrow}{\mathsf{Rack}}$ which has a left adjoint, the [*associated group*]{} functor ${\operatorname{As}}{\colon}{\mathsf{Rack}} {\rightarrow}{\mathsf{Group}}$. For a given rack $X$, the associated group ${\operatorname{As}}X$ is the free group on the elements of $X$ modulo the relations $$a^b = b^{-1}ab$$ for all $a,b \in X$. Racks were first studied by Conway and Wraith [@conway/wraith:wracks] and later (under the name ‘automorphic sets’) by Brieskorn [@brieskorn:automorphic], while quandles were introduced by Joyce [@joyce:knot-quandle]. A detailed exposition may be found in the paper by Fenn and Rourke [@fenn/rourke:racks-links]. A [*trunk*]{} ${\mathsf{T}}$ is an object analogous to a category, and consists of a class of [*objects*]{} and, for each ordered pair $(A,B)$ of objects, a set ${\operatorname{Hom}}_{{\mathsf{T}}}(A,B)$ of [*morphisms*]{}. In addition, ${\mathsf{T}}$ has a number of [*preferred squares*]{} $$\bfig\square[A`B`C`D;f`g`h`k]\efig$$ of morphisms, a concept analogous to that of composition in a category. Morphism composition need not be associative, although it is in all the cases discussed in this paper, and particularly when the trunk in question is also a category. Given two arbitrary trunks ${\mathsf{S}}$ and ${\mathsf{T}}$, a [*trunk map*]{} or [*functor*]{} $F{\colon}{\mathsf{S}}{\rightarrow}{\mathsf{T}}$ is a map which assigns to every object $A$ of ${\mathsf{S}}$ an object $F(A)$ of ${\mathsf{T}}$, and to every morphism $f{\colon}A {\rightarrow}B$ of ${\mathsf{S}}$ a morphism $F(f){\colon}F(A) {\rightarrow}F(B)$ of ${\mathsf{T}}$ such that preferred squares are preserved: $$\bfig\square/>`>`>`>/<500,500>[F(A)`F(B)`F(C)`F(D);f_*`g_*`h_*`k_*]\efig$$ For any category ${\mathsf{C}}$ there is a well-defined trunk ${\operatorname{Trunk}({\mathsf{C}})}$ which has the same objects and morphisms as ${\mathsf{C}}$, and whose preferred squares are the commutative diagrams in ${\mathsf{C}}$. In particular, we will consider the case ${\operatorname{Trunk}({\mathsf{Ab}})}$, which we will denote ${\mathsf{Ab}}$ where there is no ambiguity. Trunks were first introduced and studied by Fenn, Rourke and Sanderson [@fenn/rourke/sanderson:trunks]. In this paper, we study extensions of racks and quandles in more generality than before, in the process describing a new, generalised notion of a module over a rack or quandle, which is shown to coincide with the general definition of a module devised by Beck [@beck:thesis]. Abelian groups ${\operatorname{Ext}}(X,{\mathcal{A}})$ and ${\operatorname{Ext}}_Q(X,{\mathcal{A}})$ are defined and shown to classify (respectively) Abelian rack and quandle extensions and to be generalisations of all known existing ${\operatorname{Ext}}$ groups for racks and quandles. This paper contains part of my doctoral thesis [@jackson:thesis]. I am grateful to my supervisor Colin Rourke, and to Alan Robinson, Ronald Brown, and Simona Paoli for many interesting discussions and much helpful advice over the past few years. I also thank the referees for their kind comments and helpful suggestions. Modules ======= Given a rack $X$ we define a trunk ${\mathsf{T}}(X)$ as follows: let ${\mathsf{T}}(X)$ have one object for each element $x \in X$, and for each ordered pair $(x,y)$ of elements of $X$, a morphism $\alpha_{x,y}{\colon}x {\rightarrow}x^y$ and a morphism $\beta_{y,x}{\colon}y {\rightarrow}y^x$ such that the squares $$\bfig\square/>`>`>`>/<1000,500>[x`x^y`x^z`x^{yz} = x^{zy^z}; \alpha_{x,y}`\alpha_{x,z}`\alpha_{x^y,z}`\alpha_{x^z,y^z}]\efig \qquad \bfig\square/>`>`>`>/<1000,500>[y`x^y`y^z`x^{yz} = x^{zy^z}; \beta_{y,x}`\alpha_{y,z}`\alpha_{x^y,z}`\beta_{y^z,x^z}]\efig$$ are preferred for all $x,y,z \in X$. Thus a trunk map $A{\colon}{\mathsf{T}}(X){\rightarrow}{\mathsf{Ab}}$, as defined in the previous section, determines Abelian groups $A_x$, and Abelian group homomorphisms $\phi_{x,y}{\colon}A_x {\rightarrow}A_{x^y}$ and $\psi_{y,x}{\colon}A_y {\rightarrow}A_{x^y}$, such that $$\begin{aligned} \phi_{x^y,z}\phi_{x,y} & = & \phi_{x^z,y^z}\phi_{x,z} \\ \mbox{and}\qquad \phi_{x^y,z}\psi_{y,x} & = & \psi_{y^z,x^z}\phi_{y,z}\end{aligned}$$ for all $x,y,z \in X$. It will occasionally be convenient to denote such a trunk map by a triple $(A,\phi,\psi)$. Rack modules ------------ Let $X$ be an arbitrary rack. Then a [*rack module*]{} over $X$ (or an [*$X$–module*]{}) is a trunk map ${\mathcal{A}} = (A,\phi,\psi){\colon}{\mathsf{T}}(X) {\rightarrow}{\mathsf{Ab}}$ such that each $\phi_{x,y}{\colon}A_x {\cong}A_{x^y}$ is an isomorphism, and $$\label{eqn:rmod} \psi_{z,x^y}(a) = \phi_{x^z,y^z}\psi_{z,x}(a) + \psi_{y^z,x^z}\psi_{z,y}(a)$$ for all $a \in A_z$ and $x,y,z \in X$. If $x,y$ lie in the same orbit of $X$ then this implies that $A_x {\cong}A_y$ (although the isomorphism is not necessarily unique). For racks with more than one orbit it follows that if $x \not\sim y$ then $A_x$ need not be isomorphic to $A_y$. Rack modules where the constituent groups are nevertheless all isomorphic are said to be [*homogeneous*]{}, and those where this is not the case are said to be [*heterogeneous*]{}. It is clear that modules over transitive racks must be homogeneous. An $X$–module ${\mathcal{A}}$ of the form $(A,{\operatorname{Id}},0)$ (so that $\phi_{x,y} = {\operatorname{Id}}{\colon}A_x {\rightarrow}A_{x^y}$ and $\psi_{y,x}$ is the zero map $A_y {\rightarrow}A_{x^y}$) is said to be [*trivial*]{}. \[exm:abgroup\] [Any Abelian group $A$ may be considered as a homogeneous trivial $X$–module ${\mathcal{A}}$, for any rack $X$, by setting $A_x = A, \phi_{x,y} = {\operatorname{Id}}_A$, and $\psi_{y,x} = 0_A$ for all $x,y \in X$.]{} \[exm:asx\] [Let $X$ be a rack, and let $A$ be an Abelian group equipped with an action of ${\operatorname{As}}X$. Then $A$ may be considered as a homogeneous $X$–module ${\mathcal{A}} = (A,\phi,\psi)$ by setting $A_x = A$, and defining $\phi_{x,y}(a) = a \cdot x$ and $\psi_{y,x}(a) = 0$ for all $a \in A$ and $x,y \in X$.]{} In particular, Etingof and Graña [@etingof/grana:orc] study a cohomology theory for racks, with ${\operatorname{As}}X$–modules as coefficient objects. \[exm:andr/grana\] In [@andr/grana:pointed-hopf], Andruskiewitsch and Graña define an [*$X$–module*]{} to be an Abelian group $A$ equipped with a family $\eta = \{\eta_{x,y} : x,y \in X\}$ of automorphisms of $A$ and another family $\tau = \{\tau_{x,y} : x,y \in X\}$ of endomorphisms of $A$ such that (after slight notational changes): $$\begin{aligned} \eta_{x^y,z}\eta_{x,y} & = & \eta_{x^z,y^z}\eta_{x,z} \\ \eta_{x^y,z}\tau_{y,x} & = & \tau_{y^z,x^z}\eta_{y,z} \\ \tau_{z,x^y} & = & \eta_{x^z,y^z}\tau_{z,x} + \tau_{y^z,x^z}\tau_{z,y}\end{aligned}$$ This may readily be seen to be a homogeneous $X$–module in the context of the current discussion. As a concrete example, let $X$ be $C_3=\{0,1,2\}$, the cyclic rack with three elements. This has rack structure given by $x^y = x+1 \pmod{3}$ for all $x,y \in X$. Let $A=\mathbb{Z}_5$ and define: $$\begin{aligned} &\eta_{x,y}{\colon}A {\rightarrow}A;\quad n \mapsto 2n\pmod{5}\\ &\tau_{y,x}{\colon}A {\rightarrow}A;\quad n \mapsto 4n\pmod{5}\end{aligned}$$ Then this satisfies Andruskiewitsch and Graña’s definition of a $C_3$–module, and (by setting $A_0=A_1=A_2=A=\mathbb{Z}_5$) is also a homogeneous $C_3$–module in the context of the current discussion. \[exm:alexander\] [Let $h = \{ h_i : i \in {\operatorname{Orb}}X \}$ be a family of Laurent polynomials in one variable $t$, one for each orbit of the rack $X$, and let $n = \{ n_i : i \in {\operatorname{Orb}}X \}$ be a set of positive integers, also one for each orbit. Then we may construct a (possibly heterogeneous) $X$–module ${\mathcal{A}} = (A,\phi,\psi)$ by setting $A_x = \mathbb{Z}_{n_{[x]}}[t,t^{-1}]/h_{[x]}(t)$, $\phi_{x,y}{\colon}a \mapsto ta$, and $\psi_{y,x}{\colon}b \mapsto (1-t)b$ for all $x,y \in X$, $a \in A_x$ and $b \in A_y$. The case where $A_x = \mathbb{Z}[t,t^{-1}]/h_{[x]}(t)$ for all $x$ in some orbit(s) of $X$ is also an $X$–module.]{} \[exm:dihedral\] [Let $n = \{ n_i : i \in {\operatorname{Orb}}X \}$ be a set of positive integers, one for each orbit of $X$. Then let ${\mathcal{D}} = (D,\phi,\psi)$ denote the (possibly heterogeneous) $X$–module where $D_x = \mathbb{Z}_{n_{[x]}}$, $\phi_{x,y}(a) = -a$, and $\psi_{y,x}(b) = 2b$ for all $x,y \in X$, $a \in A_x$ and $b \in A_y$. This module is isomorphic to the Alexander module where $h_i(t) = (1+t)$ for all $i \in {\operatorname{Orb}}X$. The case where $A_x = \mathbb{Z}$ for all $x$ in some orbit(s) of $X$, is also an $X$–module. The [*$n$th homogeneous dihedral $X$–module*]{} (where all the $n_i$ are equal to $n$) is denoted ${\mathcal{D}}_n$. The case where $D_x = \mathbb{Z}$ for all $x \in X$ is the [*infinite homogeneous dihedral $X$–module*]{} ${\mathcal{D}}_\infty$.]{} Given two $X$–modules ${\mathcal{A}} = (A,\phi,\psi)$ and ${\mathcal{B}} = (B,\chi,\omega)$, a [*homomorphism*]{} of $X$–modules, or an [*$X$–map*]{}, is a natural transformation $f{\colon}{\mathcal{A}} {\rightarrow}{\mathcal{B}}$ of trunk maps, that is, a collection $f = \{ f_x{\colon}A_x {\rightarrow}B_x : x \in X \}$ of Abelian group homomorphisms such that $$\begin{aligned} \phi_{x,y}f_x & = & f_{x^y}\phi_{x,y} \\ \mbox{and}\qquad\psi_{y,x}f_y & = & f_{x^y}\psi_{y,x}\end{aligned}$$ for all $x,y \in X$. We may thus form the category ${\mathsf{RMod}}_X$ whose objects are $X$–modules, and whose morphisms are $X$–maps. In his doctoral thesis [@beck:thesis], Beck gives a general definition of a ‘module’ in an arbitrary category. Given a category ${\mathsf{C}}$, and an object $X$ of ${\mathsf{C}}$, a [*Beck module*]{} over $X$ is an Abelian group object in the slice category ${\mathsf{C}}/X$. For any group $G$, the category ${\mathsf{Ab}}({\mathsf{Group}}/G)$, for example, is equivalent to the category of $G$–modules. Similar results hold for Lie algebras, associative algebras and commutative rings. The primary aim of this section is to demonstrate a categorical equivalence between the rack modules just defined, and the Beck modules in the category ${\mathsf{Rack}}$. For an arbitrary rack $X$ and an $X$–module ${\mathcal{A}} = (A,\phi,\psi)$, we define the [*semidirect product*]{} of ${\mathcal{A}}$ and $X$ to be the set $${\mathcal{A}} \rtimes X = \{ (a,x) : x \in X, a \in A_x \}$$ with rack operation given by $$(a,x)^{(b,y)} := \left(\phi_{x,y}(a) + \psi_{y,x}(b), x^y\right).$$ \[thm:semidirect-rack\] \[prp:semidirect-rack\] For any rack $X$ and $X$–module ${\mathcal{A}} = (A,\phi,\psi)$, the semidirect product ${\mathcal{A}} \rtimes X$ is a rack. For any three elements $(a,x), (b,y), (c,z) \in {\mathcal{A}} \rtimes X$, $$\begin{aligned} (a,x)^{(b,y)(c,z)}\! &= (\phi_{x,y}(a) + \psi_{y,x}(b),x^y)^{(c,z)} \\ &= (\phi_{x^y,z}\phi_{x,y}(a) + \phi_{x^y,z}\psi_{y,x}(b) + \psi_{z,x^y}(c),x^{yz}) \\ &= (\phi_{x^z,y^z}\phi_{x,z}(a) + \psi_{y^z,x^z}\phi_{y,z}(b) + \phi_{x^z,y^z}\psi_{z,x}(c) + \psi_{y^z,x^z}\psi_{z,y}(c), x^{zy^z}) \\ &= (\phi_{x,z}(a) + \psi_{z,x}(c),x^z)^{(\phi_{y,z}(b) + \psi_{z,y}(c),y^z)} \\ &= (a,x)^{(c,z)(b,y)^{(c,z)}}.\end{aligned}$$ Furthermore, for any two elements $(a,x), (b,y) \in {\mathcal{A}} \rtimes X$, there is a unique element $$(c,z) = (a,x)^{\overline{(b,y)}} = (\phi_{z,y}^{-1}(a - \psi_{y,z}(b)),x^{\overline{y}}) \in {\mathcal{A}} \rtimes X$$ such that $(c,z)^{(b,y)} = (a,x)$. Hence ${\mathcal{A}} \rtimes X$ satisfies the rack axioms. \[thm:rmod-beck\] For any rack $X$, the category ${\mathsf{RMod}}_X$ of $X$–modules is equivalent to the category ${\mathsf{Ab}}({\mathsf{Rack}}/X)$ of Abelian group objects over $X$. Given an $X$-module ${\mathcal{A}} = (A,\phi,\psi)$, let $T{\mathcal{A}}$ be the object $p{\colon}{\mathcal{A}} \rtimes X {\twoheadrightarrow}X$ in the slice category ${\mathsf{Rack}}/X$, where $p$ is defined as projection onto the second coordinate. Given an $X$–map $f{\colon}{\mathcal{A}}{\rightarrow}{\mathcal{B}}$, we obtain a slice morphism $Tf:{\mathcal{A}} \rtimes X {\rightarrow}{\mathcal{B}} \rtimes X$ defined by $T(f)(a,x) = (f_x(a),x)$ for all $a \in A_x$ and $x \in X$. This is functorial since, for any $X$–module homomorphism $g{\colon}{\mathcal{B}}{\rightarrow}{\mathcal{C}}$, $$\begin{aligned} T(fg)(a,x) &= ((fg)_x(a),x) \\ &= (f_xg_x(a),x) \\ &= T(f)(g_x(a),x) \\ &= T(f)T(g)(a,x)\end{aligned}$$ for all $a \in A_x$ and $x \in X$. We thus have a functor $T{\colon}{\mathsf{RMod}}_X{\rightarrow}{\mathsf{Rack}}/X$. Our aim is to show firstly that the image of $T$ is the subcategory ${\mathsf{Ab}}({\mathsf{Rack}}/X)$, and secondly that $T$ has a well-defined inverse. To show the first, that $T{\mathcal{A}}$ has a canonical structure as an Abelian group object, we must construct an appropriate section, and suitable multiplication and inverse morphisms. Let: $$\begin{aligned} & r{\colon}{\mathcal{A}} \rtimes X {\rightarrow}{\mathcal{A}} \rtimes X ; && (a,x) \mapsto (-a,x) \\ & m{\colon}({\mathcal{A}} \rtimes X)\times_X({\mathcal{A}} \rtimes X) {\rightarrow}{\mathcal{A}} \rtimes X ; && ((a_1,x),(a_2,x)) \mapsto (a_1+a_2,x) \\ & s{\colon}X {\rightarrow}{\mathcal{A}} \rtimes X ; && x \mapsto (0,x)\end{aligned}$$ The maps $r$ and $m$ both compose appropriately with the projection map $p$: $$\begin{aligned} &p(a,x) = x = p(-a,x) = p(r(a,x)) \\ &p(a_1,x) = p(a_2,x) = x = p(a_1+a_2,x) = p(m((a_1,x),(a_2,x))\end{aligned}$$ Furthermore, $ps = {\operatorname{Id}}_X$. Also $$\begin{aligned} m(m((a_1,x),(a_2,x)),(a_3,x)) &= m((a_1+a_2,x),(a_3,x)) \\ &= (a_1+a_2+a_3,x) \\ &= m((a_1,x),(a_2+a_3,x)) \\ &= m((a_1,x),m((a_2,x),(a_3,x))),\\ m(s(x),(a,x)) &= m((0,x),(a,x)) \\ &= (a,x) \\ &= m((a,x),(0,x)) \\ &= m((a,x),s(x)), \\ m((a_1,x),(a_2,x)) &= (a_1+a_2,x) \\ &= (a_2+a_1,x) \\ &= m((a_2,x),(a_1,x)),\end{aligned}$$ $$\begin{aligned} \mbox{and}\qquad m(r(a,x),(a,x)) &= m((-a,x),(a,x)) \\ &= (0,x) \\ &= m((a,x),(-a,x)) \\ &= m((a,x),r(a,x)),\end{aligned}$$ so $T{\mathcal{A}}$ is an Abelian group object in ${\mathsf{Rack}}/X$. Now, given an Abelian group object $p{\colon}R {\rightarrow}X$ in ${\mathsf{Rack}}/X$, with multiplication map $\mu$, inverse map $\nu$, and section $\sigma$, let $R_x$ be the preimage $p^{-1}(x)$ for each $x \in X$. Each of the $R_x$ has a canonical Abelian group structure defined in terms of the maps $\mu, \nu$, and $\sigma$: $\sigma(x)$ is the identity in $R_x$, and for any $u,v \in R_x$ let $u+v := \mu(u,v)$ and $-u := \nu(u)$. That the preimage $R_x$ is closed under addition and inversion follows immediately from the fact that $\mu$ and $\nu$ are rack homomorphisms over $X$. Next, we define maps $$\rho_{x,y}{\colon}R_x {\rightarrow}R_{x^y},\ \mbox{given by}\ u \mapsto u^{\sigma(y)},$$ for all $x,y \in X$ and $u \in R_x$. These are Abelian group homomorphisms, since $\rho_{x,y}\sigma(x) = \sigma(x)^{\sigma(y)} = \sigma(x^y)$ (which is the identity in $R_{x^y}$) and, for any $u_1,u_2 \in R_x$, $$\begin{aligned} \rho_{x,y}(u_1+u_2) &= \mu(u_1,u_2)^{\sigma(y)} \\ &= \mu(u_1,u_2)^{\mu(\sigma(y),\sigma(y))} \\ &= \mu(u_1^{\sigma(y)},u_2^{\sigma(y)}) \\ &= \rho_{x,y}(u_1) + \rho_{x,y}(u_2).\end{aligned}$$ It is also an isomorphism, since exponentiation by a fixed element of a rack is a bijection. Furthermore, for any $x,y,z \in X$ and any $u \in R_x$ $$\begin{aligned} \rho_{x^y,z}\rho_{x,y}(u) &= u^{\sigma(y)\sigma(z)} \\ &= u^{\sigma(z)\sigma(y)^{\sigma(z)}} \\ &= u^{\sigma(z)\sigma(y^z)} \\ &= \rho_{x^z,y^z}\rho_{x,z}(u).\end{aligned}$$ Now we define maps $$\lambda_{y,x}{\colon}R_y {\rightarrow}R_{x^y},\qquad \mbox{given by}\ v \mapsto \sigma(x)^v,$$ for all $x,y \in X$ and $v \in R_y$. These are also Abelian group homomorphisms since $$\lambda_{y,x}\sigma(y) = \sigma(x)^{\sigma(y)} = \sigma(x^y)$$ (which is the identity in $R_{x^y}$) and, for any $v_1,v_2 \in R_y$, $$\begin{aligned} \lambda_{y,x}(v_1+v_2) &= \sigma(x)^{\mu(v_1,v_2)} \\ &= \mu(\sigma(x),\sigma(x))^{\mu(v_1,v_2)} \\ &= \mu(\sigma(x)^{v_1},\sigma(x)^{v_2}) \\ &= \lambda_{y,x}(v_1) + \lambda_{y,x}(v_2).\end{aligned}$$ Also, for any $x,y,z \in X$, $v \in R_y$ and $w \in R_z$ $$\begin{aligned} \rho_{x^y,z}\lambda_{y,x}(v) &= \sigma(x)^{v\sigma(z)} \\ &= \sigma(x)^{\sigma(z)v^{\sigma(z)}} \\ &= \sigma(x^z)^{v^{\sigma(z)}} \\ &= \lambda_{y^z,x^z}\rho_{y,z}(v) \\ \mbox{and}\qquad\lambda_{z,x^y}(w) &= \sigma(x^y)^w \\ &= \sigma(x)^{\sigma(y)w} \\ &= \sigma(x)^{w\sigma(y)^w} \\ &= \mu(\sigma(x),\sigma(x))^{\mu(\sigma(z),w) \mu(\sigma(y),\sigma(y))^{\mu(w,\sigma(z))}} \\ &= \mu\left(\sigma(x)^{\sigma(z)\sigma(y)^w}, \sigma(x)^{w\sigma(y)^{\sigma(z)}}\right) \\ &= \sigma(x)^{\sigma(z)\sigma(y)^w} + \sigma(x)^{w\sigma(y)^{\sigma(z)}} \\ &= \sigma(x^z)^{\sigma(y)^w} + \sigma(x)^{w\sigma(y^z)} \\ &= \lambda_{y^z,x^z}\lambda_{z,y}(w) + \rho_{x^z,y^z}\lambda_{z,x}(w).\end{aligned}$$ Thus an Abelian group object $R {\rightarrow}X$ in ${\mathsf{Rack}}/X$ determines a unique rack module ${\mathcal{R}} = (R,\rho,\lambda)$ over $X$. For any two such Abelian group objects $p_1{\colon}R_1 {\rightarrow}X$ and $p_2{\colon}R_2 {\rightarrow}X$, together with a rack homomorphism $f_1{\colon}R_1 {\rightarrow}R_2$ over $X$, we may construct two $X$–modules ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$ as described above, and an $X$–map $g_1{\colon}{\mathcal{R}}_1{\rightarrow}{\mathcal{R}}_2$ by setting $(g_1)_x(u) = f_1(u)$ for all $u \in (R_1)_x$ and $x \in X$. It may be seen that $(g_1)_x:(R_1)_x {\rightarrow}(R_2)_x$ since $f_1$ is a rack homomorphism over $X$. It may also be seen that $g_1$ is a natural transformation of trunk maps ${\mathsf{T}}(X) {\rightarrow}{\mathsf{Ab}}$ since $$\begin{aligned} (g_1)_{x^y}((\rho_1)_{x,y}(u)) &= f_1((\rho_1)_{x,y}(u)) \\ &= f_1(u^{\sigma_1(y)}) \\ &= f_1(u)^{f_1\sigma_1(y)} \\ &= f_1(u)^{\sigma_2(y)} \\ &= (\rho_2)_{x,y}(g_1)_x(u) \\ \mbox{and}\qquad(g_1)_{x^y}(\lambda_1)_{y,x}(v) &= f_1((\lambda_1)_{y,x}(v)) \\ &= f_1(\sigma_1(x)^v) \\ &= f_1\sigma_1(x)^{f_1(v)} \\ &= \sigma_2(x)^{f_1(v)} \\ &= (\lambda_2)_{y,x}(g_1)_x(v)\end{aligned}$$ for all $u \in R_x, v \in R_y$, and $x,y \in X$. Given a third Abelian group object $p_3{\colon}R_3 {\rightarrow}X$ together with another slice morphism $f_2{\colon}R_2 {\rightarrow}R_3$, we may construct another $X$–module ${\mathcal{R}}_3$ and $X$–map $g_2{\colon}{\mathcal{R}}_2 {\rightarrow}{\mathcal{R}}_3$. From the composition $f_2f_1$ we may similarly construct a unique $X$–map $g{\colon}{\mathcal{R}}_1{\rightarrow}{\mathcal{R}}_3$. Then $$g_x(u) = (f_2f_1)(u) = (g_2)_x(f_1(u)) = (g_2)_x(g_1)_x(u).$$ Hence this construction determines a functor $S{\colon}{\mathsf{Ab}}({\mathsf{Rack}}/X){\rightarrow}{\mathsf{RMod}}_X$, which is the inverse of the functor $T{\colon}{\mathsf{RMod}}_X{\rightarrow}{\mathsf{Ab}}({\mathsf{Rack}}/X)$ described earlier. \[cor:rmod-abelian\] \[thm:rmod-abelian\] The category ${\mathsf{RMod}}_X$ is Abelian. The category ${\mathsf{RMod}}_X$ is additive, as for any $X$–modules ${\mathcal{A}}$ and ${\mathcal{B}}$, the set ${\operatorname{Hom}}_{{\mathsf{RMod}}_X}({\mathcal{A}},{\mathcal{B}})$ has an Abelian group structure given by $(f+g)_x(a) = f_x(a) + g_x(a)$ for all $f,g{\colon}{\mathcal{A}}{\rightarrow}{\mathcal{B}}$, all $x \in X$ and all $a \in A_x$. Furthermore, composition of $X$–maps distributes over this addition operation. The $X$–module with trivial orbit groups and structure homomorphisms is the zero object in ${\mathsf{RMod}}_X$, and for any two $X$–modules ${\mathcal{A}} = (A,\alpha,{\varepsilon})$ and ${\mathcal{B}} = (B,\beta,\zeta)$, the Cartesian product ${\mathcal{A}}\times{\mathcal{B}} = (A \times B, \alpha\times\beta, {\varepsilon}\times\zeta)$ is also an $X$–module. Given an $X$–map $f{\colon}{\mathcal{B}} = (B,\beta,\zeta) {\rightarrow}{\mathcal{C}} = (C,\gamma,\eta)$ let ${\mathcal{A}} = (A,\alpha,{\varepsilon})$ such that $A_x = \{a \in B_x : f_x(a) = 0\}$, with $\alpha_{x,y} = \beta_{x,y}|_{A_x}$ and ${\varepsilon}_{y,x} = \zeta_{y,x}|_{A_y}$. Then ${\mathcal{A}}$ is a submodule of ${\mathcal{B}}$ and the inclusion $\iota{\colon}{\mathcal{A}}\hookrightarrow{\mathcal{B}}$ is the (categorical) kernel of $f$. Now define ${\mathcal{D}} = (D,\delta,\xi)$ where $D_x = C_x/{\operatorname{im}}f_x$, and $\delta_{x,y} =\gamma_{x,y}+{\operatorname{im}}f_x$ and $\xi_{y,x} = \eta_{y,x} + {\operatorname{im}}f_y$. Then ${\mathcal{D}}$ is a quotient of ${\mathcal{C}}$ and the canonical projection map $\pi{\colon}{\mathcal{C}}{\rightarrow}{\mathcal{D}}$ is the (categorical) cokernel of $f$. Let $\mu{\colon}{\mathcal{H}}{\rightarrow}{\mathcal{K}}$ be an $X$–monomorphism. Then the inclusion $\iota{\colon}{\operatorname{im}}\mu{\rightarrow}{\mathcal{K}}$ is a kernel of the quotient map $\pi{\colon}{\mathcal{K}}{\rightarrow}{\mathcal{K}}/{\operatorname{im}}\mu$. Since $\mu$ is injective, $\mu'{\colon}{\mathcal{H}}{\cong}{\operatorname{im}}\mu$ where $\mu'_x(a) = \mu_x(a)$ for all $x \in X$ and $a \in H_x$. But since kernels are unique up to composition with an isomorphism, and since $\mu = \iota\mu'$, it follows that $\mu$ is the kernel of its cokernel, the canonical quotient map $\pi$. Let $\nu{\colon}{\mathcal{H}}{\rightarrow}{\mathcal{K}}$ be an $X$–epimorphism. Then the inclusion map $\iota{\colon}\ker\nu\hookrightarrow{\mathcal{H}}$ is a kernel of $\nu$. Given another $X$–map $\kappa{\colon}{\mathcal{H}}{\rightarrow}{\mathcal{L}}$ such that $\kappa\iota = 0$, then $\ker\nu\subseteq\ker\kappa$ so that $\nu(a) = \nu(b)$ implies that $\kappa(a)=\kappa(b)$. But since $\nu$ is surjective we can define an $X$–map $\theta{\colon}{\mathcal{K}}{\rightarrow}{\mathcal{L}}$ by $\theta_x\nu_x(a) = \kappa_x(a)$ for all $a \in H_x$ and $x \in X$. Then $\theta\nu = \kappa$ and so $\nu$ is a cokernel of $\iota$. So, every $X$–map has a kernel and a cokernel, every monic $X$–map is the kernel of its cokernel, and every epic $X$–map is the cokernel of its kernel, and hence ${\mathsf{RMod}}_X$ is an Abelian category. These results justify the use of the term ‘rack module’ to describe the objects under consideration, and show that ${\mathsf{RMod}}_X$ is an appropriate category in which to develop homology theories for racks. Papers currently in preparation will investigate new homology theories for racks, based on the derived functor approach of Cartan and Eilenberg [@cartan/eilenberg:homalg] and the cotriple construction of Barr and Beck [@barr/beck:standard-constructions]. We now introduce a notational convenience which may serve to simplify matters in future. Let $X$ be a rack, ${\mathcal{A}} = (A,\phi,\psi)$ an $X$–module, and $w = y_1 y_2 \ldots y_n$ a word in ${\operatorname{As}}X$. Then we may denote the composition $$\phi_{x^{y_1 \ldots y_{n-1}},y_n} \phi_{x^{y_1 \ldots y_{n-2}},y_{n-1}} \ldots \phi_{x,y_1}$$ by $\phi_{x,w} = \phi_{x,y_1 \ldots y_n}$. This shorthand is well-defined as the following lemma shows: \[lem:asword\] \[thm:asword\] If $y_1 \ldots y_n$ and $z_1 \ldots z_m$ are two different representative words for the same element $w \in {\operatorname{As}}X$, then the compositions $$\begin{aligned} &\phi_{x^{y_1 \ldots y_{n-1}},y_n} \phi_{x^{y_1 \ldots y_{n-2}},y_{n-1}} \ldots \phi_{x,y_1} \\ \mbox{and}\qquad &\phi_{x^{z_1 \ldots z_{m-1}},z_m} \phi_{x^{z_1 \ldots z_{m-2}},z_{m-1}} \ldots \phi_{x,z_1}\end{aligned}$$ are equal, for all $x \in X$. Furthermore, $\phi_{x,1} = {\operatorname{Id}}_{A_x}$, where $1$ denotes the identity in ${\operatorname{As}}X$. Let $T{\colon}{\mathsf{RMod}}_X{\rightarrow}{\mathsf{Ab}}({\mathsf{Rack}}/X)$ be the functor constructed in the proof of Theorem \[thm:rmod-beck\], and recall that $R_x = T({\mathcal{A}})_x$ has an Abelian group structure. For any $x,y \in X$, the homomorphism $T(\phi_{x,y}){\colon}R_x {\rightarrow}R_{x^y}$ maps $u \mapsto u^{\sigma(y)}$, where $\sigma$ is the section of $T{\mathcal{A}}$. Then for any $u \in R_x$ $$\begin{aligned} T(\phi_{x^{y_1 \ldots y_{n-1}},y_n} \phi_{x^{y_1 \ldots y_{n-2}},y_{n-1}} \ldots &\phi_{x,y_1})(u) \\ &= u^{\sigma(y_1)\ldots\sigma(y_n)} \\ &= u^{\sigma(y_1 \ldots y_n)} \\ &= u^{\sigma(z_1 \ldots z_m)} \\ &= u^{\sigma(z_1)\ldots\sigma(z_m)} \\ &= T(\phi_{x^{z_1 \ldots z_{m-1}},z_m} \phi_{x^{z_1 \ldots z_{m-2}},z_{m-1}} \ldots \phi_{x,z_1})(u)\end{aligned}$$ where the equality in the second and third lines follows from the functoriality of the associated group. The final statement follows from the observation $$T(\phi_{x,1})(u) = u^1 = u = T({\operatorname{Id}}_{A_x})(u).$$ Hence this notation is well-defined. Quandle modules --------------- We now study the specialisation of rack modules to the subcategory ${\mathsf{Quandle}}$. A [*quandle module*]{} is a rack module ${\mathcal{A}} = (A,\phi,\psi)$ which satisfies the additional criterion $$\label{eqn:qmod} \psi_{x,x}(a) + \phi_{x,x}(a) = a$$ for all $a \in A_x$ and $x \in X$. Where the context is clear, we may refer to such objects as [*$X$–modules*]{}. There is an obvious notion of a [*homomorphism*]{} (or, in the absence of ambiguity, an [*$X$–map*]{}) of quandle modules, and thus we may form the category ${\mathsf{QMod}}_X$ of quandle modules over $X$. Similarly to example \[exm:andr/grana\], Andruskiewitsch and Graña’s definition of quandle modules coincides with the definition of a homogeneous quandle module in the sense of the current discussion. Examples \[exm:abgroup\], \[exm:alexander\], and \[exm:dihedral\] of the previous subsection, are also quandle modules. Example \[exm:asx\] is not, but the variant obtained by setting $\psi_{y,x} = {\operatorname{Id}}_{A} - \phi_{x,y}$, for all $x,y \in X$, is. \[exm:andr/grana-qmod\] [For an arbitrary quandle $X$, Andruskiewitsch and Graña [@andr/grana:pointed-hopf] further define a [*quandle $X$–module*]{} to be a rack module (as in example \[exm:andr/grana\]) which satisfies the additional condition $$\eta_{x,x} + \tau_{x,x} = {\operatorname{Id}}_A$$ for all $x \in X$. This may be seen to be a homogeneous quandle $X$–module in the context of the current discussion.]{} Given a quandle $X$ and a quandle $X$–module ${\mathcal{A}}$, the semidirect product ${\mathcal{A}} \rtimes X$ has the same definition as before. \[thm:semidirect-quandle\] \[prp:semidirect-quandle\] If $X$ is a quandle and ${\mathcal{A}} = (A,\phi,\psi)$ a quandle module over $X$, the semidirect product ${\mathcal{A}} \rtimes X$ is a quandle. By proposition \[prp:semidirect-rack\], ${\mathcal{A}} \rtimes X$ is a rack, so we need only verify the quandle axiom. For any element $(a,x) \in {\mathcal{A}} \rtimes X$, $$(a,x)^{(a,x)} = (\phi_{x,x}(a) + \psi_{x,x}(a),x^x) = (a,x)$$ and so ${\mathcal{A}} \rtimes X$ is a quandle. These objects coincide with the Beck modules in the category ${\mathsf{Quandle}}$. \[thm:qmod-beck\] For any quandle $X$, there is an equivalence of categories $${\mathsf{QMod}}_X {\cong}{\mathsf{Ab}}({\mathsf{Quandle}}/X)$$ As in the proof of Theorem \[thm:rmod-beck\], we identify the quandle module ${\mathcal{A}} = (A,\phi,\psi)$ with ${\mathcal{A}} \rtimes X {\rightarrow}X$ in the slice category ${\mathsf{Quandle}}/X$. Proposition \[prp:semidirect-quandle\] ensures that this object is indeed a quandle over $X$, and hence we obtain a well-defined functor $T{\colon}{\mathsf{QMod}}_X{\rightarrow}{\mathsf{Ab}}({\mathsf{Quandle}}/X)$. Conversely, suppose that $R{\rightarrow}X$ is an Abelian group object in ${\mathsf{Quandle}}/X$, with multiplication map $\mu$, inverse map $\nu$, and section $\sigma$. As before, we may construct a rack module ${\mathcal{R}} = (R,\rho,\lambda)$ over $X$. It remains only to show that this module satisfies the additional criterion (\[eqn:qmod\]) for it to be a quandle module over $X$. But $$\begin{aligned} \lambda_{x,x}(a) + \rho_{x,x}(a) &= \mu(\sigma(x)^a,a^{\sigma(x)}) \\ &= \mu(\sigma(x),a)^{\mu(a,\sigma(x))} \\ &= \mu(\sigma(x),a)^{\mu(\sigma(x),a)} \\ &= \mu(\sigma(x)^{\sigma(x)},a^a) = a\end{aligned}$$ and so ${\mathcal{R}}$ is indeed a quandle $X$–module. The category ${\mathsf{QMod}}_X$ is Abelian. This proof is exactly the same as the proof of Theorem \[cor:rmod-abelian\]. Analogously to the previous subsection, we may conclude that our use of the term ‘quandle module’ is justified, and that the category ${\mathsf{QMod}}_X$ is a suitable environment in which to study the homology and cohomology of quandles. Abelian extensions ================== Having characterised suitable module categories, we may now study extensions of racks and quandles by these objects. Rack extensions have been studied before, in particular by Ryder [@ryder:thesis] under the name ‘expansions’; the constructs which she dubs ‘extensions’ are in some sense racks formed by disjoint unions, whereby the original rack becomes a subrack of the ‘extended’ rack. Ryder’s notion of rack expansions is somewhat more general than the extensions studied here, as she investigates arbitrary congruences (equivalently, rack epimorphisms onto a quotient rack) whereas we will only examine certain classes of such objects. Abelian extensions of racks --------------------------- An [*extension*]{} of a rack $X$ by an $X$–module ${\mathcal{A}} = (A,\phi,\psi)$ consists of a rack $E$ together with an epimorphism $f{\colon}E {\twoheadrightarrow}X$ inducing a partition $E = \bigcup_{x \in X} E_x$ (where $E_x$ is the preimage $f^{-1}(x)$), and for each $x \in X$ a left $A_x$–action on $E_x$ satisfying the following three conditions: 1. The $A_x$–action on $E_x$ is simply transitive, which is to say that for any $u,v \in E_x$ there is a unique $a \in A_x$ such that $a \cdot u = v$. 2. For any $u \in E_x$, $a \in A_x$, and $v \in E_y$, $(a \cdot u)^v = \phi_{x,y}(a) \cdot (u^v)$. 3. For any $u \in E_y$, $b \in A_y$, and $v \in E_y$, $u^{(b \cdot v)} = \psi_{y,x}(b) \cdot (u^v)$. Two extensions $f_1{\colon}E_1 {\twoheadrightarrow}X$ and $f_2{\colon}E_2 {\twoheadrightarrow}X$ by the same $X$–module ${\mathcal{A}}$ are [*equivalent*]{} if there exists a rack isomorphism (an [*equivalence*]{}) $\theta{\colon}E_1 {\rightarrow}E_2$ which respects the projection maps and the group actions: 1. $f_2\theta(u) = f_1(u)$ for all $u \in E_1$ 2. $\theta(a \cdot u) = \theta(a) \cdot u$ for all $u \in E_x$, $a \in A_x$ and $x \in X$. Let $f{\colon}E {\twoheadrightarrow}X$ be an extension of $X$ by ${\mathcal{A}}$. Then a [*section*]{} of $E$ is a function (not necessarily a rack homomorphism) $s{\colon}X {\rightarrow}E$ such that $fs = {\operatorname{Id}}_X$. Since the $A_x$ act simply transitively on the $E_x$, there is a unique $x \in X$ and a unique $a \in A_x$ such that a given element $u \in E_x$ can be written as $u = a \cdot s(x)$. Since $f$ is a homomorphism, it follows that $s(x)^{s(y)} \in E_{x^y}$ and so there is a unique $\sigma_{x,y} \in A_{x^y}$ such that $s(x)^{s(y)} = \sigma_{x,y} \cdot s(x^y) $. The set $\sigma = \{ \sigma_{x,y} : x,y \in X \}$ is the [*factor set*]{} of the extension $E$ [*relative to*]{} the section $s$, and may be regarded as an obstruction to $s$ being a rack homomorphism. It follows that, for all $x,y \in X$, $a \in A_x$, and $b \in A_y$ $$\begin{aligned} (a \cdot s(x))^{(b \cdot s(y))} &= \phi_{x,y}(a) \cdot s(x)^{(b \cdot s(y))} \\ &= (\psi_{y,x}(b) + \phi_{x,y}(a)) \cdot s(x)^{s(y)} \\ &= (\psi_{y,x}(b) + \phi_{x,y}(a) + \sigma_{x,y}) \cdot s(x^y) \end{aligned}$$ Thus the rack structure on $E$ is determined completely by the factor set $\sigma$. The next result gives necessary and sufficient conditions on factor sets of arbitrary rack extensions. \[thm:rack-ext1\] \[prp:rack-ext1\] Let $X$ be a rack, and ${\mathcal{A}} = (A,\phi,\psi)$ be an $X$–module. Let $\sigma = \{ \sigma_{x,y} \in A_{x^y} : x,y \in X \}$ be a collection of group elements. Let $E[{\mathcal{A}},\sigma]$ be the set $\{ (a,x) : a \in A_x, x \in X \}$ with rack operation $$(a,x)^{(b,y)} = (\phi_{x,y}(a) + \sigma_{x,y} + \psi_{y,x}(b), x^y)$$ for all $a \in A_x$, $b \in A_y$, and $x,y \in X$. Then $E[{\mathcal{A}},\sigma]$ is an extension of $X$ by ${\mathcal{A}}$ with factor set $\sigma$ if $$\label{eqn:rack-ext1} \sigma_{x^y,z} + \phi_{x^y,z}(\sigma_{x,y}) = \phi_{x^z,y^z}(\sigma_{x,z}) + \sigma_{x^z,y^z} + \psi_{y^z,x^z}(\sigma_{y,z})$$ for all $x,y,z \in X$. Conversely, if $E$ is an extension of $X$ by ${\mathcal{A}}$ with factor set $\sigma$ then (\[eqn:rack-ext1\]) holds, and $E$ is equivalent to $E[{\mathcal{A}},\sigma]$. To prove the first part, we require that $E[{\mathcal{A}},\sigma]$ satisfy the rack axioms. Given $(a,x), (b,y) \in E[{\mathcal{A}},\sigma]$, there is a unique $(c,z) \in E[{\mathcal{A}},\sigma]$ such that $(c,z)^{(b,y)} = (a,x)$, given by $$(c,z) = (\phi_{x,y}^{-1}(a - \sigma_{x,y} - \psi_{y,x}(b)),x^{\overline{y}})$$ Also, for any $(a,x), (b,y), (c,z) \in E[{\mathcal{A}},\sigma]$, $$\begin{aligned} (a,x)^{(b,y)(c,z)} &= (\phi_{x,y}(a) + \sigma_{x,y} + \psi_{y,x}(b),x^y)^{(c,z)} \\ &= (\phi_{x^y,z}\phi_{x,y}(a) + \phi_{x^y,z}(\sigma_{x,y}) + \phi_{x^y,z}\psi_{y,x}(b) + \sigma_{x^y,z} + \psi_{z,x^y}(c), x^{yz})\end{aligned}$$ and $$\begin{aligned} (a,x)^{(c,z)(b,y)^{(c,z)}} &= (\phi_{x,z}(a) + \sigma_{x,z} + \psi_{z,x}(c),x^z)^{(\phi_{y,z}(b) + \sigma_{y,z} + \psi_{z,y}(c),y^z)} \\ &= (\phi_{x^z,y^z}\phi_{x,z}(a) + \phi_{x^z,y^z}(\sigma_{x,z}) + \phi_{x^z,y^z}\psi_{z,x}(c) + \sigma_{x^z,y^z} \\ &\qquad+ \psi_{y^z,x^z}\phi_{y,z}(b) + \psi_{y^z,x^z}(\sigma_{y,z}) + \psi_{y^z,x^z}\psi_{z,y}(c), x^{zy^z})\end{aligned}$$ are equal if (\[eqn:rack-ext1\]) holds, and so $E[{\mathcal{A}},\sigma]$ is a rack. Now define $f{\colon}E[{\mathcal{A}},\sigma] {\twoheadrightarrow}X$ to be projection onto the second coordinate, and let $A_x$ act on $E[{\mathcal{A}},\sigma]_x = f^{-1}(x)$ by $a_1 \cdot (a_2,x) := (a_1 + a_2,x)$ for each $a_1,a_2 \in A_x$ and all $x \in X$. These actions are simply transitive and satisfy the requirements $$\begin{aligned} (a_1 \cdot (a_2,x))^{(b,y)} = (a_1+a_2, x)^{(b,y)} &= (\phi_{x,y}(a_1+a_2) + \sigma_{x,y} + \psi_{y,x}(b),x^y) \\ &= (\phi_{x,y}(a_1) + \phi_{x,y}(a_2) + \sigma_{x,y} + \psi_{y,x}(b),x^y) \\ &= \phi_{x,y}(a_1) \cdot (\phi_{x,y}(a_2) + \sigma_{x,y} + \psi_{y,x}(b),x^y) \\ &= \phi_{x,y}(a_1) \cdot (a_2,x)^{(b,y)}\\ \mbox{and}\qquad(a,x)^{b_1 \cdot (b_2,y)} &= (a,x)^{(b_1+b_2,y)} \\ &= (\phi_{x,y}(a) + \sigma_{x,y} + \psi_{y,x}(b_1+b_2),x^y) \\ &= (\phi_{x,y}(a) + \sigma_{x,y} + \psi_{y,x}(b_1) + \psi_{y,x}(b_2),x^y) \\ &= \psi_{y,x}(b_1) \cdot (\phi_{x,y}(a) + \sigma_{x,y} + \psi_{y,x}(b_2),x^y) \\ &= \psi_{y,x}(b_1) \cdot (a,x)^{(b_2,y)}\end{aligned}$$ so $E[{\mathcal{A}},\sigma]$ is an extension of $X$ by ${\mathcal{A}}$. Now define $s{\colon}X {\twoheadrightarrow}E[{\mathcal{A}},\sigma]$ by $s(x) = (0,x)$ for all $x \in X$. This is clearly a section of this extension. Also, $$s(x)^{s(y)} = (0,x)^{(0,y)} = (\sigma_{x,y},x^y) = \sigma_{x,y} \cdot s(x^y)$$ so $\sigma$ is the factor set of this extension relative to the section $s$. Conversely, let $f{\colon}E {\twoheadrightarrow}X$ be an extension of $X$ by a given $X$–module ${\mathcal{A}}$, with factor set $\sigma$ relative to some extension $s{\colon}X {\rightarrow}E$. By the simple transitivity of the $A_x$–action on the $E_x = f^{-1}(x)$, the map $\theta{\colon}(a,x) \mapsto a \cdot s(x)$ is an isomorphism $E[{\mathcal{A}},\sigma] {\cong}E$. Since $E$ is a rack, the earlier part of the proof shows that (\[eqn:rack-ext1\]) holds, and so $E[{\mathcal{A}},\sigma]$ is another extension of $X$ by ${\mathcal{A}}$. Furthermore, $\theta$ respects the projection maps onto $X$, and $$\theta(a_1 \cdot (a_2,x)) = \theta(a_1+a_2,x) = (a_1+a_2) \cdot s(x) = a_1 \cdot (a_2 \cdot s(x)) = a_1 \cdot \theta(a_2,x)$$ so $\theta$ is an equivalence of extensions. Andruskiewitsch and Graña [@andr/grana:pointed-hopf] introduce the notion of an extension by a [*dynamical cocycle*]{}. Given an arbitrary rack $X$ and a non-empty set $S$, we select a function $\alpha{\colon}X \times X {\rightarrow}{\operatorname{Hom}}_{\mathsf{Set}}(S \times S,S)$ (which determines, for each ordered pair $x,y \in X$, a function $\alpha_{x,y}{\colon}S \times S {\rightarrow}S$) satisfying the criteria 1. $\alpha_{x,y}(s,-)$ is a bijection on $S$ 2. $\alpha_{x^y,z}(s,\alpha_{x,y}(t,u)) = \alpha_{x^z,y^z}(\alpha_{x,z}(s,t),\alpha_{x,y}(s,u))$ for all $x,y,z \in X$ and $s,t,u \in S$. Then we may define a rack structure on the set $X \times S$ by defining $(x,s)^{(y,t)} = (x^y,\alpha_{x,y}(s,t))$. This rack, denoted $X \times_\alpha S$, is the extension of $X$ by $\alpha$. In the case where $S$ is an Abelian group, and $\alpha_{x,y}(s,t) = \phi_{x,y}(s) + \sigma_{x,y} + \psi_{y,s}(t)$ for some suitably-chosen Abelian group homomorphisms $\phi_{x,y},\psi_{y,x}{\colon}S {\rightarrow}S$, and family $\sigma = \{ \sigma_{x,y} \in S : x,y \in X \}$ of elements of $S$, then this is equivalent to the construction $E[{\mathcal{A}},\sigma]$ just discussed, for a homogeneous $X$–module ${\mathcal{A}} = (A,\phi,\psi)$. \[thm:rack-ext2\] \[prp:rack-ext2\] Let $\sigma$ and $\tau$ be factor sets corresponding to extensions of a rack $X$ by an $X$–module ${\mathcal{A}}$. Then the following are equivalent: 1. $E[{\mathcal{A}},\sigma]$ and $E[{\mathcal{A}},\tau]$ are equivalent extensions of $X$ by ${\mathcal{A}}$ 2. there exists a family $\upsilon = \{ \upsilon_x \in A_x : x \in X \}$ such that $$\label{eqn:rack-ext2} \tau_{x,y} = \sigma_{x,y} + \phi_{x,y}(\upsilon_x) + \psi_{y,x}(\upsilon_y) - \upsilon_{x^y}$$ for $x,y \in X$. 3. $\sigma$ and $\tau$ are factor sets of the same extension of $X$ by ${\mathcal{A}}$, relative to different sections. Let $\theta{\colon}E[{\mathcal{A}},\tau] {\cong}E[{\mathcal{A}},\sigma]$ be the hypothesised equivalence. Then it follows that $\theta(0,x) = (\upsilon_x,x)$ for some $\upsilon_x \in A_x$ and, furthermore, $$\theta(a,x) = \theta(a \cdot (0,x)) = a \cdot \theta(0,x) = a \cdot (\upsilon_x,x) = (a + \upsilon_x,x)$$ for all $a \in A_x$, since $\theta$ preserves the $A_x$–actions. Then $$\begin{aligned} &\theta\bigl((a,x)^{(b,y)}\bigr) = (\phi_{x,y}(a) + \psi_{y,x}(b) + \tau_{x,y} + \upsilon_{x^y}, x^y) \\ \mbox{and}\qquad&\theta(a,x)^{\theta(b,y)} = (a + \upsilon_x,x)^{(b+\upsilon_y,y)} = (\phi_{x,y}(a+\upsilon_x) + \psi_{y,x}(b+\upsilon_y) + \sigma_{x,y},x^y)\end{aligned}$$ which are equal since $\theta$ is a rack homomorphism, and so (\[eqn:rack-ext2\]) holds. This argument is reversible, showing the equivalence of the first two statements. Now, given such an equivalence $\theta$, define a section $s{\colon}X {\rightarrow}E[{\mathcal{A}},\tau]$ by $x \mapsto (\upsilon_x,x)$. Then the above argument also shows that $$\begin{aligned} &s(x)^{b \cdot s(y)} = (\upsilon_x,x)^{(\upsilon_y+b,y)} = (\sigma_{x,y} + \psi_{y,x}(b)) \cdot s(x^y) \\ \mbox{and}\qquad &(a \cdot s(x))^{s(y)} = (\upsilon_x+a,x)^{(\upsilon_y,y)} = (\sigma_{x,y} + \phi_{x,y}(a)) \cdot s(x^y)\end{aligned}$$ so $\sigma$ is the factor set of $E[{\mathcal{A}},\tau]$ relative to the section $s$. This property holds for any extension equivalent to $E[{\mathcal{A}},\tau]$. Conversely, if $\sigma$ and $\tau$ are factor sets of some extension $E$ of $X$ by ${\mathcal{A}}$ relative to different sections $s,t{\colon}X {\rightarrow}E$ then $s(x) = \upsilon_x \cdot t(x)$ for some $\upsilon_x \in A_x$, and so the first and third conditions are equivalent. The following corollary justifies the earlier assertion that the factor set is in some sense the obstruction to a section being a rack homomorphism. \[thm:rack-ext3\] \[cor:rack-ext3\] For an extension $f{\colon}E {\twoheadrightarrow}X$ by an $X$–module ${\mathcal{A}} = (A,\phi,\psi)$, the following statements are equivalent: 1. There exists a rack homomorphism $s{\colon}X {\rightarrow}E$ such that $fs = {\operatorname{Id}}_X$ 2. Relative to some section, the factor set of $E {\twoheadrightarrow}X$ is trivial 3. Relative to any section there exists, for the factor set $\sigma$ of $E {\twoheadrightarrow}X$, a family $\upsilon = \{\upsilon_x \in A_x : x \in X \}$ such that for all $x,y \in X$ $$\label{eqn:rack-ext3} \sigma_{x,y} = \phi_{x,y}(\upsilon_x) - \upsilon_{x^y} + \psi_{y,x}(\upsilon_y)$$ Extensions of this type are said to be [*split*]{}. We are now able to classify rack extensions: \[thm:rack-ext4\] Let $X$ be a rack and ${\mathcal{A}} = (A,\phi,\psi)$ an $X$–module. Then there is an Abelian group ${\operatorname{Ext}}(X,{\mathcal{A}})$ whose elements are in bijective correspondence with extensions of $X$ by ${\mathcal{A}}$. Let the set $Z(X,{\mathcal{A}})$ consist of extensions of $X$ by ${\mathcal{A}}$. As shown above, these are determined by factor sets $\sigma$ satisfying (\[eqn:rack-ext1\]). Defining an addition operation by $(\sigma+\tau)_{x,y} := \sigma_{x,y} + \tau_{x,y}$ gives this an Abelian group structure with the trivial factor set as identity. A routine calculation confirms that the set $B(X,{\mathcal{A}})$ of split extensions (equivalently, factor sets satisfying (\[eqn:rack-ext3\])) forms an Abelian subgroup of $Z(X,{\mathcal{A}})$, and so we may define ${\operatorname{Ext}}(X,{\mathcal{A}}) := Z(X,{\mathcal{A}})/B(X,{\mathcal{A}})$. In the case where ${\mathcal{A}}$ is a trivial homogeneous $X$–module (equivalently, an Abelian group $A$) the group ${\operatorname{Ext}}(X,{\mathcal{A}})$ coincides with $H^2(BX;A)$, the second cohomology group of the rack space of $X$ as defined by Fenn, Rourke and Sanderson [@fenn/rourke/sanderson:trunks]. Abelian extensions of quandles ------------------------------ We now turn our attention to the case where $X$ is a quandle. Extensions of $X$ by a quandle $X$–module ${\mathcal{A}}$ and their corresponding factor sets are defined in an analogous manner. \[thm:quandle-ext1\] \[prp:quandle-ext1\] Let $X$ be a quandle and ${\mathcal{A}} = (A,\phi,\psi)$ be a quandle module over $X$. Then extensions $f{\colon}E {\twoheadrightarrow}X$ such that $E$ is also a quandle are in bijective correspondence with factor sets $\sigma$ satisfying hypothesis (\[eqn:rack-ext1\]) of proposition \[thm:rack-ext1\] together with the additional criterion $$\label{eqn:quandle-ext1} \sigma_{x,x} = 0$$ for all $x \in X$. Following the reasoning of proposition \[thm:rack-ext1\], for $E$ to be a quandle is equivalent to the requirement that $$(a,x)^{(a,x)} = (\phi_{x,x}(a) + \sigma_{x,x} + \psi_{x,x}(a),x^x) = (a,x)$$ for all $x \in X$ and $a \in A_x$. Since ${\mathcal{A}}$ is a quandle module, this is equivalent to the requirement that (\[eqn:quandle-ext1\]) holds. We may now classify quandle extensions of $X$ by ${\mathcal{A}}$: \[thm:quandle-ext2\] For any quandle $X$ and quandle $X$–module ${\mathcal{A}}$, there is an Abelian group ${\operatorname{Ext}}_Q(X,{\mathcal{A}})$ whose elements are in bijective correspondence with quandle extensions of $X$ by ${\mathcal{A}}$. We proceed similarly to the proof of Theorem \[thm:rack-ext4\]. Let $Z_Q(X,{\mathcal{A}})$ be the subgroup of $Z(X,{\mathcal{A}})$ consisting of factor sets satisfying the criterion (\[eqn:quandle-ext1\]), and let $B_Q(X,{\mathcal{A}}) = B(X,{\mathcal{A}})$. Then we define ${\operatorname{Ext}}_Q(X,{\mathcal{A}}) = Z_Q(X,{\mathcal{A}})/B_Q(X,{\mathcal{A}})$. In the case where ${\mathcal{A}}$ is trivial homogeneous (and hence equivalent to an Abelian group $A$), extensions of $X$ by ${\mathcal{A}}$ correspond to Abelian quandle extensions, in the sense of Carter, Saito and Kamada [@carter/kamada/saito:diag] and so ${\operatorname{Ext}}_Q(X,{\mathcal{A}}) = H^2_Q(X;A)$. If the module ${\mathcal{A}}$ is a homogeneous Alexander module as defined in example \[exm:alexander\], then extensions of $X$ by ${\mathcal{A}}$ are exactly the twisted quandle extensions described by Carter, Saito and Elhamdadi [@carter/elhamdadi/saito:twisted], and so ${\operatorname{Ext}}_Q(X,{\mathcal{A}}) = H^2_{TQ}(X;A)$. [99]{} , [Matías Graña]{}, *From racks to pointed Hopf algebras*, Advances in Mathematics **178** (2003) 177–243 , [Jonathan Beck]{}, *Homology and standard constructions*, from: “Seminar on Triples and Categorical Homology Theory”, volume 80 of Lecture Notes in Mathematics, Springer–Verlag (1969) 245–335 , *Triples, algebras and cohomology*, PhD thesis, Columbia University (1967). Republished as: Reprints in Theory and Applications of Categories **2** (2003) 1–59 , *Automorphic sets and singularities*, Contemporary Mathematics **78** (1988) 45–115 , [Samuel Eilenberg]{}, *Homological Algebra*, Princeton University Press (1999) , [Mohamed Elhamdadi]{}, [Masahico Saito]{}, *Twisted quandle homology theory and cocycle knot invariants*, Algebraic and Geometric Topology **2** (2002) 95–135 , [Seiichi Kamada]{}, [Masahico Saito]{}, *Diagrammatic computations for quandles and cocycle knot invariants*, from: “Diagrammatic morphisms and applications (San Francisco, CA, 2000)”, Contemporary Mathematics **318** (2003) 51–74 , [Gavin Wraith]{}, unpublished correspondence (1959) , [Matías Graña]{}, *On rack cohomology*, Journal of Pure and Applied Algebra **177** (2003) 49–59 , [Colin Rourke]{}, *Racks and links in codimension 2*, Journal of Knot Theory and its Ramifications **1** (1992) 343–406 , [Colin Rourke]{}, [Brian Sanderson]{}, *Trunks and classifying spaces*, Applied Categorical Structures **3** (1995) 321–356 , *Homological algebra of racks and quandles*, PhD thesis, Mathematics Institute, University of Warwick (2004) , *A classifying invariant of knots: the knot quandle*, Journal of Pure and Applied Algebra **23** (1982) 37–65 , *The structure of racks*, PhD thesis, Mathematics Institute, University of Warwick (1993)

--- abstract: | Let $n$ and $p$ be non-negative integers with $n \geq p$, and $S$ be a linear subspace of the space of all $n$ by $p$ matrices with entries in a field $\K$. A classical theorem of Flanders states that $S$ contains a matrix with rank $p$ whenever ${\operatorname{codim}}S <n$. In this article, we prove the following related result: if ${\operatorname{codim}}S<n-1$, then, for any non-zero $n$ by $p$ matrix $N$ with rank less than $p$, there exists a line that is directed by $N$, has a common point with $S$ and contains only rank $p$ matrices. author: - 'Clément de Seguins Pazzis[^1] [^2]' title: Lines of full rank matrices in large subspaces --- *AMS Classification:* 15A03, 15A30. *Keywords:* Full rank, Matrices, Dimension, Flanders’s theorem. Introduction ============ Throughout the article, $\K$ denotes an arbitrary field. Let $n$ and $p$ be non-negative integers. We denote by ${\operatorname{M}}_{n,p}(\K)$ the space of all $n$ by $p$ matrices with entries in $\K$. In particular, we set ${\operatorname{M}}_n(\K):={\operatorname{M}}_{n,n}(\K)$ and we denote by ${\operatorname{GL}}_n(\K)$ its group of units. We denote by $E_{i,j}$ the matrix of ${\operatorname{M}}_{n,p}(\K)$ with zero entries everywhere except at the $(i,j)$-spot where the entry equals $1$. In a landmark article [@Flanders], Flanders proved the following classical result: Let $n,p,r$ be non-negative integers such that $n \geq p \geq r$. Let $S$ be a linear subspace of ${\operatorname{M}}_{n,p}(\K)$ in which every matrix has rank less than or equal to $r$. Then, $\dim S \leq nr$. The upper-bound $nr$ is optimal, as shown by the example of the space of all matrices with zero entries in the last $p-r$ columns. Before Flanders, Dieudonné [@Dieudonne] had already studied spaces of singular square matrices and obtained the special case $n=p$ and $r=n-1$ in the above theorem. Flanders actually had to assume that $\# \K>r$ due to his use of polynomials. This provision was lifted by Meshulam [@Meshulam] (for more recent proofs, see [@affpres; @dSPFlandersskew]). Here is a reformulation of Flanders’s theorem: if $n \geq p$, a linear subspace $S$ of ${\operatorname{M}}_{n,p}(\K)$ such that $\dim S>nr$ must contain a matrix with rank greater than $r$. In this work, we shall be concerned with not only finding one such matrix, but a whole line of matrices with large rank. Better, we want to control the direction of such a line. Before we formulate the problem, some basic considerations are necessary. Let $N \in {\operatorname{M}}_n(\K) {\smallsetminus}\{0\}$. If $N$ is invertible and $\K$ is algebraically closed, then every line directed by $N$ must contain a singular matrix: indeed, for all $A \in {\operatorname{M}}_n(\K)$, we can write $\forall \lambda \in \K, \; \det(A-\lambda N)=(-1)^n (\det N)\, p(\lambda)$ where $p$ denotes the characteristic polynomial of $N^{-1}A$, and $p$ must have a root. Conversely, every non-zero matrix with non-full rank directs a line of full rank matrices, as stated in the following lemma. \[fullspacelemma\] Let $n \geq p$ be non-negative integers and $N \in {\operatorname{M}}_{n,p}(\K)$ be such that ${\operatorname{rk}}N<p$. Then, there exists $A \in {\operatorname{M}}_{n,p}(\K)$ such that every matrix of $A+\K N$ has rank $p$. Set $r:={\operatorname{rk}}N$. Without loss of generality, we can assume that $$N=\begin{bmatrix} I_r & [0]_{r \times (p-r)} \\ [0]_{(n-r) \times r} & [0]_{(n-r) \times (p-r)} \end{bmatrix}.$$ If $n>p$, one checks that $A:=\underset{j=1}{\overset{p}{\sum}} E_{j+1,j}$ has the requested property.\ If $n=p$ one checks that the matrix $A:=E_{1,n}+\underset{j=1}{\overset{n-1}{\sum}} E_{j+1,j}$ has the requested property. Now, here is our problem for square matrices: given a linear subspace $S$ of ${\operatorname{M}}_n(\K)$ and a non-zero *singular* matrix $N \in S$, under what conditions on $\dim S$ can we guarantee that there exists $A \in S$ for which every matrix of $A+\K N$ is invertible? More generally, if $n \geq p$, and given a linear subspace $S$ of ${\operatorname{M}}_{n,p}(\K)$ and a non-zero matrix $N \in S$ with rank less than $p$, under what conditions on $\dim S$ can we guarantee that there exists $A \in S$ for which every matrix of $A+\K N$ has rank $p$? These questions are motivated by potential applications to the structure of spaces of bounded rank matrices over small finite fields. The following theorem, which is the main point of the present article, gives a full answer to them. \[rectangulartheorem\] Let $n \geq p \geq 2$ be integers. Let $S$ be a linear subspace of ${\operatorname{M}}_{n,p}(\K)$ with ${\operatorname{codim}}S\leq n-2$, and let $N \in {\operatorname{M}}_{n,p}(\K)$ be such that ${\operatorname{rk}}N<p$. Then, there exists $A \in S$ such that every matrix of $A+\K N$ has rank $p$. Here is a reformulation in terms of operator spaces: \[operatortheorem\] Let $U$ and $V$ be finite-dimensional vector spaces with $\dim U \leq \dim V$. Let $S$ be a linear subspace of $\calL(U,V)$ such that ${\operatorname{codim}}S \leq \dim V-2$, and $t \in \calL(U,V)$ be a non-injective operator. Then, there exists $a \in S$ such that every operator in $a+\K t$ is injective. Note, in the above theorems, that we do not require that the direction of the line be included in $S$! Let us immediately show that the upper-bound $n-2$ from Theorem \[rectangulartheorem\] is optimal. Consider the matrix $N:=\begin{bmatrix} I_{p-1} & [0]_{(p-1) \times 1} \\ [0]_{(n-p+1) \times (p-1)} & [0]_{(n-p+1) \times 1} \end{bmatrix}$, and the space $S$ of all matrices of the form $$\begin{bmatrix} ? & [?]_{1 \times (p-1)} \\ [0]_{(n-1)\times 1} & [?]_{(n-1) \times (p-1)} \end{bmatrix}.$$ Then, for all $A \in S$, some matrix in $A+\K N$ has zero as its first column, and hence not every matrix in $A+\K N$ has rank $p$. Yet, ${\operatorname{rk}}N<p$ and ${\operatorname{codim}}S=n-1$. Theorem \[rectangulartheorem\] will be proved in three steps. In the first step, we shall consider the case of square matrices with ${\operatorname{rk}}N=n-1$. The result actually deals with affine subspaces instead of just linear subspaces. \[penciltheorem\] Let $n$ be a non-negative integer. Let $N$ be a rank $n-1$ matrix of ${\operatorname{M}}_n(\K)$. Let $\calS$ be an affine subspace of ${\operatorname{M}}_n(\K)$ such that ${\operatorname{codim}}\calS \leq n-2$. Assume that at least one matrix of $\calS$ maps ${\operatorname{Ker}}N$ into ${\operatorname{Im}}N$. Then, there exists $A \in \calS$ such that every matrix of $A+\K N$ is invertible. Assume that $\K$ is algebraically closed. Then, the condition that some matrix of $\calS$ maps ${\operatorname{Ker}}N$ into ${\operatorname{Im}}N$ is unavoidable in Theorem \[penciltheorem\]. Consider indeed the matrix $N:=\begin{bmatrix} I_{n-1} & [0]_{(n-1) \times 1} \\ [0]_{1 \times (n-1)} & 0 \end{bmatrix}$ and the affine hyperplane $\calS$ of all matrices of ${\operatorname{M}}_n(\K)$ with entry $1$ at the $(n,n)$-spot. For all $A \in S$, the polynomial $\det(A+tN)$ reads $t^{n-1}+\underset{k=0}{\overset{n-2}{\sum}} b_k t^k$, and hence it is non-constant whenever $n\geq 2$, which yields that $A+\K N$ contains a singular matrix. If $\# \K>2$, the proof of Theorem \[penciltheorem\] will actually demonstrate that there exists a matrix $A \in \calS$ such that the (formal) polynomial $\det(A+tN)$ is constant and non-zero. As ${\operatorname{rk}}N=n-1$, this can be restated in terms of matrix pencils as saying that the matrix pencil $A+tN$ is equivalent to the pencil $I_n+t J$, where $J$ is the Jordan matrix $(\delta_{i,j-1})_{1 \leq i,j \leq n}$. If $\# \K=2$, this result fails for $n=3$: one considers the space $\calS$ of all matrices of the form $$\begin{bmatrix} ? & ? & a \\ ? & ? & ? \\ ? & a+1 & ? \end{bmatrix} \quad \text{with $a \in \K$},$$ and the matrix $$N:=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}.$$ One sees that $\calS$ has codimension $1$ in ${\operatorname{M}}_3(\K)$. Let $M=\begin{bmatrix} A & C \\ B & d \end{bmatrix}\in \calS$, with $A \in {\operatorname{M}}_2(\K)$, $B \in {\operatorname{M}}_{1,2}(\K)$, $C \in \K^2$ and $d \in \K$. We have $$\begin{aligned} \det(M+tN)& = d \det(A+tI_2)-B (A+t I_2)^\ad C \\ & =d \det(A+tI_2)+B (A^\ad+t I_2) C \\ & =d \det(A+tI_2)+t BC+B A^\ad C,\end{aligned}$$ where $A^\ad$ denotes the transpose of the matrix of cofactors of $A$. Assume that the polynomial $\det(M+tN)$ is constant. As $\det(A+t I_2)$ has degree $2$, we successively obtain $d=0$ and $BC=0$. From the definition of $\calS$, it follows that $B=0$ or $C=0$, and hence $\det(M+tN)=0$. Finally, by checking the proof of Theorem \[penciltheorem\], one can prove that, if $\# \K=2$, if ${\operatorname{codim}}\calS \leq n-3$ and some matrix of $\calS$ maps ${\operatorname{Ker}}N$ into ${\operatorname{Im}}N$, then $\det(A+tN)$ is constant and non-zero for some $A$ in $\calS$. We suspect that this result still holds, provided that $n>3$, under the weaker assumption that ${\operatorname{codim}}\calS \leq n-2$. In Section \[pencilproofsection\], Theorem \[penciltheorem\] will be proved by induction over $n$. In the next section, we shall extend it as follows, by considering an arbitrary singular matrix $N$. \[squaretheorem\] Let $n$ be a non-negative integer. Let $N$ be a singular matrix of ${\operatorname{M}}_n(\K)$. Let $\calS$ be an affine subspace of ${\operatorname{M}}_n(\K)$ such that ${\operatorname{codim}}\calS \leq n-2$. Assume that there exists $M \in \calS$ such that the operator $X \in {\operatorname{Ker}}N \mapsto \overline{MX} \in \K^n/{\operatorname{Im}}N$ is non-injective. Then, there exists $A \in \calS$ such that every matrix of $A+\K N$ is invertible. Again, this result will be proved by induction over $n$. In the last step, by far the easiest one, we shall derive Theorem \[rectangulartheorem\] from Theorem \[squaretheorem\] (see Section \[conclusionsection\]). The remaining open problem is the generalization of the above results to arbitrary ranks: given non-negative integers $n,p,r$ such that $n \geq p \geq r$, what is the smallest integer $d$ for which there exists a matrix $N \in {\operatorname{M}}_{n,p}(\K)$ with rank less than $r$ and a linear subspace $S$ of ${\operatorname{M}}_{n,p}(\K)$ with codimension $d$ that contains no element $A$ for which all the matrices of $A+\K N$ have rank greater than or equal to $r$? At the moment, we do not have a reasonable conjecture to suggest. Proof of Theorem \[penciltheorem\] {#pencilproofsection} ================================== The proof of Theorem \[penciltheorem\] will be performed by induction over $n$, using several steps. If $n\leq 1$ then the result is vacuous. If $n=2$, it is given by Lemma \[fullspacelemma\]. Assume now that $n \geq 3$. We use a *reductio ad absurdum*, by assuming that there is no matrix $A \in \calS$ such that every matrix of $A+\K N$ is invertible. Without loss of generality, we can assume that $$N=\begin{bmatrix} I_{n-1} & [0]_{(n-1) \times 1} \\ [0]_{1 \times (n-1)} & 0 \end{bmatrix}.$$ Then, we can split every matrix $M$ of ${\operatorname{span}}(\calS)$ up as $$M=\begin{bmatrix} A(M) & C(M) \\ L(M) & d(M) \end{bmatrix}$$ with $A(M) \in {\operatorname{M}}_{n-1}(\K)$, $L(M) \in {\operatorname{M}}_{1,n-1}(\K)$, $C(M) \in \K^{n-1}$ and $d(M) \in \K$. In $\calS$, we have the affine subspace $$\calV:=\bigl\{M \in \calS : \; d(M)=0\bigr\}$$ with codimension at most $1$ (it is non-empty because we have assumed that at least one matrix of $\calS$ maps ${\operatorname{Ker}}N$ into ${\operatorname{Im}}N$). We denote by $V$ the translation vector space of $\calV$. In $V$, we have two specific linear subspaces $$T:=\{M \in V : \; L(M)=0 \; \text{and}\; C(M)=0\}$$ and $$U:=\{M \in V : \; C(M)=0\}.$$ By the rank theorem, we have $$\label{ranktheorem} \dim A(T)+\dim L(U)+\dim C(\calV)=\dim \calV.$$ In particular, since $\dim \calV > n(n-1)$ and $\dim A(T) \leq (n-1)^2$ we find $$\label{dimC+dimL} \dim C(\calV)+\dim L(U)>n-1.$$ Given $X \in \K^{n-1} {\smallsetminus}\{0\}$, we denote by $A(T)_X$ the linear subspace of $A(T)$ consisting of the matrices with column space included in $\K X$. The bilinear form $$b : (Y,X) \in {\operatorname{M}}_{1,n-1}(\K) \times \K^{n-1} \mapsto YX$$ is non-degenerate on both sides, and in the rest of the proof we shall consider orthogonality with respect to it. Note in particular that yields $C(\calV) {\smallsetminus}L(U)^\bot \neq \emptyset$. Note that, for all $P \in {\operatorname{GL}}_{n-1}(\K)$, neither the previous assumptions nor the conclusion are affected in replacing $\calS$ with $Q \calS Q^{-1}$ where $Q:=P \oplus I_1$. In this transformation the spaces $L(U)$ and $C(\calV)$ are respectively replaced with $L(U)P^{-1}$ and $P C(\calV)$, whereas $b(YP^{-1},PX)=b(Y,X)$ for all $(Y,X) \in {\operatorname{M}}_{1,n-1}(\K) \times \K^{n-1}$. \[claim1\] For all $X \in C(\calV) {\smallsetminus}L(U)^\bot$, there exists $M \in \calV$ such that $C(M)=X$ and $L(M)C(M)=0$. Let $X \in C(\calV) {\smallsetminus}L(U)^\bot$. We can find $(M_1,M_0) \in \calV\times U$ such that $C(M_1)=X$ and $L(M_0)X \neq 0$. For all $\lambda \in \K$, we see that $C(M_1+\lambda M_0)=X$ and $$L(M_1+\lambda M_0)C(M_1+\lambda M_0)=L(M_1)X+\lambda L(M_0)X,$$ and hence for a well-chosen $\lambda$ we find $L(M_1+\lambda M_0)C(M_1+\lambda M_0)=0$. This proves our claim. \[claim2\] For all $X \in C(\calV) {\smallsetminus}L(U)^\bot$, one has $$\label{diminequality1} \dim C(\calV)+\dim A(T)_X \geq 2n-3.$$ We lose no generality in assuming that $X=\begin{bmatrix} 1 \\ [0]_{(n-2) \times 1} \end{bmatrix}$. Denote by $\calV'$ the affine subspace of $\calV$ consisting of the matrices $M \in \calV$ such that $C(M)=X$. Every matrix $M \in \calV'$ splits up as $$M=\begin{bmatrix} [?]_{1 \times (n-1)} & 1 \\ K(M) & [0]_{(n-1) \times 1} \end{bmatrix}$$ with $$K(M)=\begin{bmatrix} [?]_{(n-2) \times 1} & [?]_{(n-2) \times (n-2)} \\ ? & [?]_{1 \times (n-2)} \end{bmatrix}\in {\operatorname{M}}_{n-1}(\K).$$ Likewise, we write $$N=\begin{bmatrix} [?]_{1 \times (n-1)} & 0 \\ N' & [0]_{(n-1) \times 1} \end{bmatrix}$$ with $$N'=\begin{bmatrix} [0]_{(n-2) \times 1} & I_{n-2} \\ 0 & [0]_{1 \times (n-2)} \end{bmatrix}.$$ By Claim \[claim1\], there exists $M \in \calV$ such that $C(M)=X$ and $L(M)X=0$, and hence $K(M)$ maps ${\operatorname{Ker}}N'$ into ${\operatorname{Im}}N'$. Moreover, $N'$ has rank $n-2$. Thus, if ${\operatorname{codim}}K(\calV') \leq n-3$, then by induction we find a matrix $M \in \calV'$ such that $\det (K(M)+tN') \neq 0$ for all $t \in \K$; by developing the determinant along the last column, it would follow that $$\forall t \in \K, \; \det(M+tN)=(-1)^{n+1} \det(K(M)+tN')\in \K {\smallsetminus}\{0\}.$$ This would contradict our assumptions. Therefore, ${\operatorname{codim}}K(\calV') \geq n-2$. However, by the rank theorem, we see that $${\operatorname{codim}}K(\calV')= {\operatorname{codim}}\calV+\bigl(\dim C(\calV)-(n-1))+\bigl(\dim A(T)_X-(n-1)\bigr).$$ Thus, as our assumptions yield that ${\operatorname{codim}}\calV \leq n-1$, we obtain claimed inequality . It follows in particular that $$\label{minorCS} \dim C(\calV) \geq n-2.$$ One has $A(T)\subsetneq {\operatorname{M}}_{n-1}(\K)$. Assume on the contrary that $A(T)={\operatorname{M}}_{n-1}(\K)$. First, assume further that there exists $M \in \calV$ such that $L(M) \neq 0$, $C(M) \neq 0$ and $L(M)C(M)=0$. As $A(T)={\operatorname{M}}_{n-1}(\K)$, we can assume, without loss of generality, that $$L(M)=\begin{bmatrix} [0]_{1 \times (n-2)} & 1 \end{bmatrix}, \; C(M)=\begin{bmatrix} 1 \\ [0]_{(n-2) \times 1} \end{bmatrix}\; \text{and} \; A(M)=\begin{bmatrix} [0]_{1 \times (n-2)} & 0 \\ I_{n-2} & [0]_{(n-2) \times 1} \end{bmatrix}.$$ Then, it is easily checked that $\det(M+t N)=(-1)^{n+1}$, contradicting our basic assumptions on $\calV$. Therefore, $$\label{keyimp} \forall M \in \calV, \; L(M)C(M)=0 \Rightarrow (L(M)=0 \; \text{or}\; C(M)=0).$$ Choose $X \in C(\calV) {\smallsetminus}L(U)^\bot$. We know from Claim \[claim1\] that there exists $M_1 \in \calV$ such that $C(M_1)=X$ and $L(M_1)X=0$. Let $M_2 \in U$ be such that $L(M_2) \bot X$. Then, $C(M_1+M_2)=X$ and $L(M_1+M_2)=L(M_1)+L(M_2)$ is orthogonal to $X$. It follows from that $L(M_1+M_2)=0$ and $L(M_1)=0$, whence $L(M_2)=0$. Therefore $L(U)\cap \{X\}^\bot=\{0\}$, whence $\dim L(U)\leq 1$. By inequality , we deduce that $C(\calV)=\K^{n-1}$ and $\dim L(U)=1$. From there, we split the discussion into two (non-disjoint) cases. - **Case 1: $\# \K>2$.**\ Let $M \in \calV$ be such that $C(M)\not\in L(U)^\bot$. We can choose $M_0 \in U$ such that $L(M_0)C(M) \neq 0$. Then, for all $\lambda \in \K$, we have $C(M+\lambda M_0)=C(M)$ and $L(M+\lambda M_0) C(M+\lambda M_0)=L(M)C(M)+\lambda L(M_0)C(M)$; we can then choose $\lambda \in \K$ such that $L(M+\lambda M_0) C(M+\lambda M_0)=0$, leading, by , to $L(M+\lambda M_0)=0$, and hence $L(M)=L(-\lambda M_0) \in L(U)$. Hence, we have shown that $L(M) \in L(U)$ for all $M \in \calV$ such that $C(M)\not\in L(U)^\bot$. Yet, as $L(U)^\bot$ is a proper affine subspace of $\K^{n-1}$, its complementary subset in $\K^{n-1}$ generates the affine space $\K^{n-1}$ (remember that $\# \K>2$). Hence, $L(\calV) \subset L(U)$, leading to $\dim L(\calV) \leq 1$. Then, by applying the same line of reasoning to $\calS^T$, which satisfies the same assumptions, we would obtain $\dim C(\calV) \leq 1$, contradicting $C(\calV)=\K^{n-1}$ (remember that $n-1 \geq 2$). - **Case 2: $\K$ is finite.**\ Then, we use a different strategy. Since $\dim L(U)=1$ and ${\operatorname{codim}}\calS \leq n-2$, we find a matrix $M_1 \in \calS$ such that $d(M_1) \neq 0$. Since $C(\calV)=\K^{n-1}$, we also have $C(V)=\K^{n-1}$. Hence, we can choose $M'_1 \in V$ such that $C(M'_1)=-C(M_1)$. Hence, $M_2:=M_1+M'_1$ belongs to $\calS$ and satisfies $d(M_2) \neq 0$ and $C(M_2)=0$. As $n-1 \geq 2$ and $\K$ is a finite field, there exists a matrix $P \in {\operatorname{M}}_{n-1}(\K)$ with no eigenvalue: it suffices to take $P$ as the companion matrix of an irreducible polynomial over $\K$ with degree $n-1$. Since $A(T)={\operatorname{M}}_{n-1}(\K)$, we can add a well-chosen matrix of $T$ to $M_3$ so as to find a matrix $M_3 \in \calS$ such that $d(M_3) \neq 0$, $C(M_3)=0$ and $A(M_3)=P$. Then, $\det(M_3+t N)=d(M_3) \det(P+t I_{n-1}) \neq 0$ for all $t \in \K$, which contradicts our assumptions. In any case, we have found a contradiction, which yields $A(T) \subsetneq {\operatorname{M}}_{n-1}(\K)$. Combining the previous claim with identity and $\dim \calV>n(n-1)$ yields $$\dim C(\calV)+\dim L(U) > n.$$ In particular, $$\dim L(U)\geq 2.$$ \[claimCStotal\] One has $C(\calV)=\K^{n-1}$. Assume on the contrary that $C(\calV)\subsetneq \K^{n-1}$. Then, $\dim C(\calV)=n-2$ by inequality . We deduce from inequality that, for all $X \in C(\calV)$, the space $A(T)_X$ has dimension $n-1$, and hence it contains every matrix of ${\operatorname{M}}_{n-1}(\K)$ with column space $\K X$. As $A(T)\subsetneq {\operatorname{M}}_{n-1}(\K)$, we deduce that ${\operatorname{span}}(C(\calV)) \subsetneq \K^{n-1}$, whence $C(\calV)$ is a linear hyperplane of $\K^{n-1}$. Next, let $Y_0 \in C(\calV)^\bot$. We claim that $Y_0\, A(T) \subset \K Y_0$, that is $Y_0\, A(T) \bot C(\calV)$. Let $X \in C(\calV) {\smallsetminus}L(U)^\bot$. Let us prove that $Y_0 A(T) \bot X$. No generality is lost in assuming that $$X=\begin{bmatrix} 1 \\ [0]_{1 \times (n-2)} \end{bmatrix} \quad \text{and} \quad Y_0=\begin{bmatrix} [0]_{1 \times (n-2)} & 1 \end{bmatrix},$$ so that $C(\calV)=\K^{n-2} \times \{0\}$. As $\dim C(\calV)=n-2$ and ${\operatorname{codim}}A(T)>0$, inequality yields $\dim L(U) \geq 3$. Then, we can find $M \in \calV$ such that $C(M)=X$, $L(M)X=0$ and $L(M) \notin \K Y_0$: indeed, we know that we can find $M_1 \in \calV$ such that $C(M_1)=X$ and $L(M_1)X=0$ (see Claim \[claim1\]). Then, $L(U) \cap \{X\}^\bot$ has dimension at least $2$; we can choose $Z$ in $(L(U) \cap \{X\}^\bot) {\smallsetminus}\K Y_0$; then, we can choose $M_2 \in U$ such that $L(M_2)=Z$, and we check that one of the matrices $M_1$ or $M_1+M_2$ must fulfill our needs. Without further loss of generality, we can assume that $L(M)=\begin{bmatrix} 0 & 1 & [0]_{1 \times (n-3)} \end{bmatrix}$. Assume that there exists a matrix $J$ of $A(T)$ such that $Y_0 J$ is not orthogonal to $X$. Then, for some $a \in \K {\smallsetminus}\{0\}$, we have $$J=\begin{bmatrix} [?]_{(n-2) \times 1} & [?]_{(n-2) \times (n-2)} \\ a & [?]_{1 \times (n-2)} \end{bmatrix}.$$ Since $A(T)$ contains every matrix with column space $\K X'$, for all $X' \in \K^{n-2} \times \{0\}$, we deduce that there is a matrix $M'$ of $\calV$ such that $C(M')=X$, $L(M')=L(M)$ and $$A(M')=\begin{bmatrix} 0 & 0 & [0]_{1 \times (n-3)} \\ [0]_{(n-3) \times 1} & [0]_{(n-3) \times 1} & I_{n-3} \\ a & ? & [?]_{1 \times (n-3)} \end{bmatrix}$$ Then, one checks that $\det (M'+t N)=(-1)^{n-1} a$, which contradicts our assumptions. Hence, $Y_0\, A(T) \bot X$ for all $X \in C(\calV) {\smallsetminus}L(U)^\bot$. Since $\dim L(U) \geq 2$ and $\dim C(\calV)=n-2$, we find that $L(U)^\bot \cap C(\calV)$ is a proper linear subspace of $C(\calV)$, and we conclude that $Y_0\, A(T) \bot C(\calV)$, as claimed. Hence, $Y_0\, A(T) \subset \K Y_0$. In turn, this shows that ${\operatorname{codim}}A(T) \geq n-2$, and as ${\operatorname{codim}}C(\calV)=1$ we deduce that ${\operatorname{codim}}\calV \geq n$, contradicting our assumptions. \[claimcodimAT\] One has ${\operatorname{codim}}A(T)=1$. Assume that such is not the case. Let us consider the orthogonal $W$ of $A(T)$ for the non-degenerate symmetric bilinear form $(Z_1,Z_2) \mapsto {\operatorname{tr}}(Z_1Z_2)$ on ${\operatorname{M}}_{n-1}(\K)$. Then, $\dim W \geq 2$. The set $\widehat{W}:=\{Z \in W \mapsto ZX \mid X \in \K^{n-1}\}$ is a linear subspace of $\calL(W,\K^{n-1})$, and we claim that every operator in it has rank at most $1$. Assume that such is not the case. Then, we can find respective bases of $W$ and $\K^{n-1}$ in which one of the operators of $\widehat{W}$ is represented by $\begin{bmatrix} I_s & [0] \\ [0] & [0] \end{bmatrix}$ for some integer $s \geq 2$. By assigning to every $X \in \K^{n-1}$ the determinant of the upper-left $2$ by $2$ submatrix of the matrix representing $Z \mapsto ZX$ in the said bases, we define a non-zero quadratic form $q$ on $\K^{n-1}$ that vanishes at every vector $X \in \K^{n-1}$ such that $Z \in W \mapsto ZX$ has rank $1$. For all $X \in \K^{n-1} {\smallsetminus}L(U)^\bot$, we know that $\dim A(T)_X \geq n-2$ (see Claim \[claim2\]) and hence ${\operatorname{rk}}(Z \in W \mapsto ZX) \leq 1$. Therefore, $q$ vanishes at every vector of $\K^{n-1} {\smallsetminus}L(U)^\bot$. Yet, $L(U)^\bot$ has codimension at least $2$ in $\K^{n-1}$. Then, we deduce that $q=0$: if $\# \K>2$, this is easily obtained by choosing a non-zero linear form $\varphi$ on $\K^{n-1}$ that vanishes everywhere on $L(U)^\bot$, and by noting that the homogenous polynomial $x \mapsto q(x)\varphi(x)$ with degree $3$ vanishes everywhere on $\K^{n-1}$; if $\# \K=2$ the statement follows directly from Lemma 5.2 of [@dSPRC1]. This contradicts our assumptions. Thus, $\widehat{W}$ is a linear subspace of $\calL(W,\K^{n-1})$ in which every operator has rank at most $1$. As $\dim W>1$ and no vector of $W {\smallsetminus}\{0\}$ is annihilated by all the operators in $\widehat{W}$, the classification of vector spaces of rank $1$ operators shows that there exists a $1$-dimensional linear subspace $D$ of $\K^{n-1}$ that includes the range of every operator in $\widehat{W}$, which shows that ${\operatorname{Im}}Z \subset D$ for all $Z \in W$. Finally, as neither our assumptions nor our conclusion are modified in transposing both $N$ and $\calS$, we obtain that the above property holds for $W^T$ as well, yielding a linear hyperplane $H$ of $\K^{n-1}$ such that $H \subset {\operatorname{Ker}}Z$ for all $Z \in W$. However, the space of all matrices $M \in {\operatorname{M}}_{n-1}(\K)$ such that ${\operatorname{Im}}M \subset D$ and $H \subset {\operatorname{Ker}}M$ has dimension $1$, contradicting the assumption that $\dim W \geq 2$. Now, we are about to conclude. We know that $C(\calV)=\K^{n-1}$ and that $L(U)^\bot$ is a proper linear subspace of $\K^{n-1}$ (since $\dim L(U)>0$). If, for all $X \in C(\calV) {\smallsetminus}L(U)^\bot$, we had $\dim A(T)_X=n-1$, it would follow that $A(T)={\operatorname{M}}_{n-1}(\K)$, contradicting Claim \[claimcodimAT\]. Thus, we can find $X \in C(\calV) {\smallsetminus}L(U)^\bot$ such that $\dim A(T)_X<n-1$. As in the proof of Claim \[claimCStotal\] (see its second paragraph), since $\dim L(U) \geq 2$ we can find a matrix $M_1 \in \calV$ such that $C(M_1)=X$, $L(M_1)C(M_1)=0$ and $L(M_1) \neq 0$. Without loss of generality we can assume that $X=\begin{bmatrix} 1 \\ [0]_{(n-2) \times 1} \end{bmatrix}$ and $L(M_1)=\begin{bmatrix} [0]_{1 \times (n-2)} & 1 \end{bmatrix}$. Now, as ${\operatorname{codim}}A(T)=1$ and $\dim A(T)_X<n-1$, the rank theorem yields that for every $H \in {\operatorname{M}}_{n-2,n-1}(\K)$, there exists a matrix of $A(T)$ of the form $\begin{bmatrix} [?]_{1 \times (n-1)} \\ H \end{bmatrix}$. Thus, by adding a well-chosen matrix of $T$ to $M_1$, we reduce the situation to the one where $$M_1=\begin{bmatrix} [?]_{1 \times (n-2)} & ? & 1 \\ I_{n-2} & [0]_{(n-2) \times 1} & [0]_{(n-2) \times 1} \\ [0]_{1 \times (n-2)} & 1 & 0 \end{bmatrix}.$$ Then, one checks that $\det(M_1+t N)=(-1)^{n+1}$, which contradicts our initial assumptions. This final contradiction shows that $\calS$ contains a matrix $M$ such that $\forall t \in \K, \; \det(M+tN) \neq 0$. This completes the inductive proof. Proof of Theorem \[squaretheorem\] ================================== We shall prove Theorem \[squaretheorem\] by induction on $n$ and $r$. Without loss of generality, we can assume that $N=\begin{bmatrix} I_r & [0]_{r \times (n-r)} \\ [0]_{(n-r) \times r} & [0]_{(n-r) \times (n-r)} \end{bmatrix}$ where $r:={\operatorname{rk}}N$. If $\calS={\operatorname{M}}_n(\K)$ the result is known from Lemma \[fullspacelemma\]. In the rest of the proof, we assume that $\calS$ is a proper subspace of ${\operatorname{M}}_n(\K)$, and we denote by $S$ its translation vector space. In particular, the case $n\leq 2$ is settled, and we assume that $n \geq 3$. We perform a *reductio ad absurdum*, by assuming that $\calS$ does not contain a matrix $A$ of the required form. Theorem \[penciltheorem\] gives the case when $r=n-1$. In the rest of the proof, we assume that $r<n-1$. We write every matrix $M$ of ${\operatorname{M}}_n(\K)$ as $$M=\begin{bmatrix} A(M) & C(M) \\ B(M) & D(M) \end{bmatrix}$$ with $A(M) \in {\operatorname{M}}_r(\K)$, $B(M) \in {\operatorname{M}}_{n-r,r}(\K)$, $C(M) \in {\operatorname{M}}_{r,n-r}(\K)$ and $D(M) \in {\operatorname{M}}_{n-r}(\K)$. The assumptions tell us that there exists $M_1 \in \calS$ such that $D(M_1)$ has rank less than $n-r$. We distinguish between two cases. **Case 1: There exists a matrix $M_1 \in \calS$ such that $0<{\operatorname{rk}}D(M_1)<n-r$.**\ Set $s:={\operatorname{rk}}D(M_1)$. By conjugating $\calS$ with a matrix of the form $I_r \oplus P$ for some well-chosen $P \in {\operatorname{GL}}_{n-r}(\K)$, we see that no generality is lost in assuming that $D(M_1)=\begin{bmatrix} [0] & [0] \\ [0] & I_s \end{bmatrix}$. Then, by applying row operations of the form $L_i \leftarrow L_i+\lambda L_n$ with $i \in \lcro 1,r\rcro$ and $\lambda \in \K$ and column operations of the form $C_j \leftarrow C_j+\mu C_n$ with $j \in \lcro 1,r\rcro$ and $\mu \in \K$, no further generality is lost in assuming that the last row of $B(M_1)$ is zero and the last column of $C(M_1)$ is zero. Denote by $\calS'$ the affine subspace of $\calS$ consisting of the matrices with the same last row as $M_1$. Let us then write every matrix $M$ of $\calS'$ as $$M=\begin{bmatrix} K(M) & [?]_{(n-1) \times 1} \\ [0]_{1 \times (n-1)} & 1 \end{bmatrix} \quad \text{with $K(M) \in {\operatorname{M}}_{n-1}(\K)$.}$$ Then, with $N':=\begin{bmatrix} I_r & [0]_{r \times (n-r-1)} \\ [0]_{(n-1-r) \times r} & [0]_{(n-1-r) \times (n-1-r)} \end{bmatrix} \in {\operatorname{M}}_{n-1}(\K)$, we see that $K(M_1)$ is a matrix of $K(\calS')$ such that $X \in {\operatorname{Ker}}N' \mapsto \overline{K(M_1) X} \in \K^{n-1}/{\operatorname{Im}}N'$ has rank at most $n-2-r$ (as the first column of $D(M_1)$ is zero). If ${\operatorname{codim}}K(\calS')\leq n-3$, then by induction we find that $K(\calS')$ contains a matrix $A'$ such that every matrix of $A'+\K N'$ is invertible: writing $A'=K(A)$ for some $A \in \calS'$, we readily obtain that $\det(A+t N)=\det(A'+t N')$ for all $t$ in $\K$, which yields that $A+tN$ is invertible for all $t \in \K$. Hence, ${\operatorname{codim}}K(\calS')\geq n-2$, and as ${\operatorname{codim}}\calS \leq n-2$ we deduce from the rank theorem that $S$ contains $E_{1,n},E_{2,n},\dots,E_{n-1,n}$. Similarly, by considering the subspace of all matrices of $\calS$ with the same last column as $M_1$, we find that $S$ contains $E_{n,1},\dots,E_{n,n-1}$. Now, let $i \in \lcro 1,n-1\rcro$. Denote by $\calS_1$ the affine space deduced from $\calS$ by the row operation $L_i \leftarrow L_i-L_n$ (which leaves $N$ invariant). As $\calS$ contains $M_1+E_{i,n}$, we see that $\calS_1$ also contains $M_1$. Now, obviously $\calS_1$ satisfies all our assumptions with respect to $N$, and it follows from our first step that the translation vector space of $\calS_1$ contains $E_{n,1},\dots,E_{n,n-1}$. Hence, $S$ contains $E_{n,1}+E_{i,1},\dots,E_{n,n-1}+E_{i,n-1}$. As $S$ also contains $E_{n,1},\dots,E_{n,n-1}$, we deduce that it contains $E_{i,1},\dots,E_{i,n-1}$. Similarly, we obtain that, for all $j \in \lcro 1,n\rcro$, the space $S$ contains $E_{1,j},\dots,E_{n-1,j}$. Hence, $S$ contains $E_{i,j}$ for all $(i,j)\in \lcro 1,n\rcro^2 {\smallsetminus}\{(n,n)\}$. Then, the matrix $A:=E_{n,n}+E_{1,n-1}+\underset{i=1}{\overset{n-2}{\sum}} E_{i+1,i}$ belongs to $\calS$, and one checks that the polynomial $\det(A+tN)$ is constant and non-zero, whence every matrix of $A+\K N$ is invertible. This contradicts our assumptions. **Case 2: For every matrix $R$ of $D(\calS)$, either $R=0$ or $R$ is invertible.**\ Our assumptions then show that $D(\calS)$ contains $0$, and hence it is a linear subspace of ${\operatorname{M}}_{n-r}(\K)$. Every matrix of $D(\calS)$ with first row zero equals zero, and hence $\dim D(\calS) \leq n-r$. Now, denote by $\calT$ the affine subspace of $\calS$ consisting of its matrices $M$ such that $D(M)=0$. For $M \in \calT$, let us write $$C(M)=\begin{bmatrix} C_1(M) & \cdots & C_{n-r}(M) \end{bmatrix}.$$ If $C_1(\calT)=\{0\}$ then the rank theorem would yield ${\operatorname{codim}}\calS \geq r+(n-r)=n$, contradicting our assumptions. Thus, there exists $M_1 \in \calT$ such that $C_1(M_1) \neq 0$. Without loss of generality, we can assume that $C_1(M_1)=\begin{bmatrix} 1 \\ [0]_{(r-1) \times 1} \end{bmatrix}$. Denote by $\calT'$ the space of all matrices of $\calT$ with the same $(r+1)$-th column as $M_1$. For all $M \in {\operatorname{M}}_n(\K)$, we denote by $K(M)$ the submatrix of $M$ obtained by deleting the first row and the $(r+1)$-th column. Assume that ${\operatorname{codim}}K(\calT') \leq n-3$. Then, the induction hypothesis applies to $K(\calT')$ and to $K(N')$: indeed, every matrix of $K(\calT')$ maps ${\operatorname{Ker}}K(N)$ into ${\operatorname{Im}}K(N)$, and hence no such matrix induces an isomorphism from ${\operatorname{Ker}}K(N)$ to $\K^{n-1}/{\operatorname{Im}}K(N)$ (because $n-1>r$). Thus, we recover a matrix $M \in \calT'$ such that $K(M)+t K(N)$ is invertible for all $t$ in $\K$, and as $\det(M+t N)=(-1)^r \det(K(M)+t K(N))$ for all $t \in \K$, we see that $M+t N$ in invertible for all $t \in \K$. Hence, ${\operatorname{codim}}K(\calT) \geq n-2$. Yet, ${\operatorname{codim}}\calS \leq n-2$. By the rank theorem, it follows that $C_1(\calT)=\K^r$ and that $S$ contains $E_{1,1},\dots,E_{1,r},E_{1,r+2},\dots,E_{1,n}$. As $C_1(\calT)=\K^r$, we can apply the previous step to every non-zero vector of $\K^r$ rather than only to the first one of the standard basis. It follows that $S$ contains $E_{i,j}$ for all $j \in \lcro 1,n\rcro {\smallsetminus}\{r+1\}$ and all $i \in \lcro 1,r\rcro$. With the same method applied to $C_k$, for all $k \in \lcro r+1,n\rcro$, we obtain that $S$ contains $E_{i,j}$ for all $(i,j)\in \lcro 1,r\rcro \times \lcro 1,n-1\rcro$. Now, by applying the previous step to $\calS^T$ we obtain that $S$ contains $E_{i,j}$ for all $(i,j)\in \lcro 1,n\rcro \times \lcro 1,r\rcro$. Therefore, $\calT$ is the set of all $M \in {\operatorname{M}}_n(\K)$ such that $D(M)=0$. We are about to conclude. As $\dim D(\calS) \leq n-r$ and ${\operatorname{codim}}\calS \leq n-2$, we see that $(n-r)(n-r-1) \leq n-2$. Setting $s:=n-r$, we deduce that if $s >\frac{n}{2}$ then $\frac{n+1}{2}\,\frac{n-1}{2} \leq n-2$ (since $n>1$) which would lead to $n^2-4n+7 \leq 0$, that is $(n-2)^2+3 \leq 0$. Therefore $s \leq \frac{n}{2}$, that is $r \geq n-r$. It follows that the matrix $A:=\underset{i=1}{\overset{r}{\sum}} E_{i,n-r+i}+\underset{j=1}{\overset{n-r}{\sum}} E_{r+j,j}$ belongs to $\calT$, and one checks that the polynomial $\det(A+t N)$ is constant and non-zero, whence every matrix of $A+\K N$ is invertible. This completes our inductive proof of Theorem \[squaretheorem\]. Proof of Theorem \[rectangulartheorem\] {#conclusionsection} ======================================= We actually prove the “operator space" version of Theorem \[rectangulartheorem\], that is Theorem \[operatortheorem\]. Once more, we use an induction over $\dim V$, with $U$ fixed. Set $n:=\dim V$ and $p:=\dim U$. The case $\dim U=\dim V$ is known by the operator space reformulation of Theorem \[squaretheorem\]: in that case indeed the zero operator belongs to $S$ and does not induce an injective operator from ${\operatorname{Ker}}t$ to $V/{\operatorname{Im}}t$. In the remainder of the proof, we assume that $\dim V>\dim U$. Given a non-zero vector $y \in V$, we denote by $\pi_y : V \rightarrow V/\K y$ the canonical projection and we set $$S {\operatorname{mod}}y:=\{\pi_y \circ s \mid s \in S\},$$ which is a linear subspace of $\calL(U,V/\K y)$. We perform a *reductio ad absurdum*, by assuming that there is no operator $a \in S$ such that every operator of $a+\K t$ is injective. Let $y \in V {\smallsetminus}\{0\}$. Note that $\pi_y \circ t$ is non-injective. We claim that $S {\operatorname{mod}}y$ contains no operator $a$ such that every operator in $a+\K(\pi_y \circ t)$ is injective: indeed, if such an operator $a$ existed, then $a=\pi_y \circ a'$ for some $a' \in S$, and hence, for all $\lambda \in \K$, the operator $\pi_y \circ (a'+\lambda t)$ would be injective, which would show that $a'+\lambda t$ is injective. By induction, we deduce that ${\operatorname{codim}}(S {\operatorname{mod}}y) > (\dim V-1)-2$ and hence ${\operatorname{codim}}(S {\operatorname{mod}}y)\geq {\operatorname{codim}}S$. It follows from the rank theorem that $S$ contains every operator of $\calL(U,V)$ with range $\K y$.\ Varying $y$ shows that $S=\calL(U,V)$, and then Lemma \[fullspacelemma\] yields a contradiction. This completes the proof of Theorem \[rectangulartheorem\]. [1]{} J. Dieudonné, [Sur une généralisation du groupe orthogonal à quatre variables,]{} Arch. Math. [**1**]{} (1948) 282–287. H. Flanders, [On spaces of linear transformations with bounded rank,]{} J. Lond. Math. Soc. [**37**]{} (1962) 10–16. R. Meshulam, [On the maximal rank in a subspace of matrices,]{} Quart. J. Math. Oxford (2) [**36**]{} (1985) 225–229. C. de Seguins Pazzis, [Range-compatible homomorphisms on matrix spaces,]{} Linear Algebra Appl. [**484**]{} (2015) 237-289 C. de Seguins Pazzis, [The affine preservers of non-singular matrices,]{} Arch. Math. [**95**]{} (2010) 333–342. C. de Seguins Pazzis, [The Flanders theorem over division rings,]{} Preprint, arXiv: http://arxiv.org/abs/1504.01986 [^1]: Université de Versailles Saint-Quentin-en-Yvelines, Laboratoire de Mathématiques de Versailles, 45 avenue des Etats-Unis, 78035 Versailles cedex, France [^2]: e-mail address: [email protected]

--- abstract: 'This paper addresses the question of whether a “rigid molecule” (one which does not deform in an external field) used as the conducting channel in a standard three-terminal MOSFET configuration can offer any performance advantage relative to a standard silicon MOSFET. A self-consistent solution of coupled quantum transport and Poisson’s equations shows that even for extremely small channel lengths (about $1~nm$), a “well-tempered” molecular FET demands much the same electrostatic considerations as a “well-tempered” conventional MOSFET. In other words, we show that just as in a conventional MOSFET, the gate oxide thickness needs to be much smaller than the channel length (length of the molecule) for the gate control to be effective. Furthermore, we show that a rigid molecule with metallic source and drain contacts has a temperature independent subthreshold slope much larger than $60~mV/decade$, because the metal-induced gap states in the channel prevent it from turning off abruptly. However, this disadvantage can be overcome by using semiconductor contacts because of their band-limited nature.' author: - | Prashant Damle, Titash Rakshit, Magnus Paulsson and Supriyo Datta [^1] [^2] [^3] [^4] [^5]\ School of Electrical and Computer Engineering\ Purdue University\ West Lafayette, IN 47907 title: 'Current-voltage characteristics of molecular conductors: two versus three terminal' --- Molecular electronics, MOSFETs, electrostatic analysis, quantum transport, Non-equilibrium Green’s function (NEGF) formalism. Introduction {#sec:intro} ============ are promising candidates as future electronic devices because of their small size, chemical tunability and self-assembly features. Several experimental molecular devices have recently been demonstrated (for a review of the experimental work see [@reed_review]). These include two terminal devices where the conductance of a molecule coupled to two contacts shows interesting features such as a conductance gap [@reed_expt], asymmetry [@reichert_asymm_iv] and switching [@chen]. Molecular devices where a third terminal produces a negative differential resistance [@lang_fet], or suppresses the two terminal current [@emberly_fet] have been theoretically studied, but most of the work on modeling the current-voltage (IV) characteristics of molecular conductors has focused on two-terminal devices (see, for example, [@datta_expt; @emberly_two_pdt; @diVentra; @rdamle; @taylor_siesta; @palacios_fullerene] and references therein). The purpose of this paper is to analyze a three-terminal molecular device assuming that the molecule behaves essentially like a rigid solid. Unlike solids, molecules are capable of deforming in an external field and it may be possible to take advantage of such conformational effects to design transistors with superior characteristics. However, in this paper we do not consider this possibility and simply address the question of whether a “rigid molecule” used as the conducting channel in a standard three-terminal MOSFET configuration can offer any performance advantage relative to a standard silicon MOSFET. Although rigorous ab initio models are available in the literature [@diVentra; @rdamle; @taylor_siesta; @palacios_fullerene], they normally do not account for the three-terminal electrostatics that is central to the operation of transistors. For this reason we have used a simple model Hamiltonian whose parameters have been calibrated by comparing with ab initio models. We believe that a simple model Hamiltonian with rigorous electrostatics is preferable to an ab initio Hamiltonain with simplified electrostatics since the essential physics of a rigid molecular FET lies in its electrostatics. The role of electrostatic considerations in the design of conventional silicon MOSFETs (with channel lengths ranging from $10~nm$ and above) is well understood. For the gate to have good control of the channel conductivity, the gate insulator thickness has to be much smaller than the channel length. Also, for a given channel length and gate insulator thickness, a double gated structure is superior to a single gated one, simply by virtue of having two gates as opposed to one. If a molecule is used as the channel in a standard three-terminal MOSFET configuration, the effective channel length is very small - about $1~nm$. Would similar electrostatic considerations apply for such small channel lengths? In this paper we answer this question in the affirmative. Specifically we will show that: - [ The only advantage gained by using a molecular conductor for an FET channel is due to the reduced dielectric constant of the molecular environment. To get good gate control with a single gate the gate oxide thickness needs to be less than 10% of the channel (molecule) length, whereas in conventional MOSFETs the gate oxide thickness needs to be less than 3% of the channel length [@taur_ning]. With a double gated structure, the respective percentages are 60% and 20% [@zhibin_ballistic].]{} - [ Relatively poor subthreshold characteristics (a [*temperature independent*]{} subthreshold slope much larger than $60~mV/decade$) are obtained even with good gate control, if metallic contacts (like gold) are used, because the metal-induced gap states in the channel preclude it from turning off abruptly. Preliminary results with a molecule coupled to doped silicon source and drain contacts, however, show a temperature dependent subthreshold slope ($\sim k_BT/q$). We believe this is due to the band-limited nature of the silicon contacts, and we are currently investigating this effect.]{} Overall this study suggests that superior saturation and subthreshold characteristics in a molecular FET can only arise from novel physics beyond that included in our model. Further work on molecular transistors should try to capitalize on the additional degrees of freedom afforded by the “soft” nature of molecular conductors [@titash] - a feature that is not included in this study. Although there has been no experimental report of a moleculer FET to date [^6] , judging from the historical development of the conventional silicon MOSFET, it is reasonable to expect that a single gated structure would be easier to fabricate than a double gated one. With this in mind, in this paper we mainly focus on a single gated molecular FET geometry (see Fig. \[fig:scheme\]). Few key results with a double gated geometry will be shown wherever appropriate to emphasize the differences between the single and double gated structures. The paper is organized as follows: Section \[sec:theory\] contains a brief description of the theoretical formulation and the simulation procedure. Section \[sec:results\] presents the simulation results along with an explanation of the underlying physics. Section \[sec:conclusion\] summarizes this paper. Theory {#sec:theory} ====== A schematic figure of a molecule coupled to gold contacts (source and drain) is shown in Fig. \[fig:scheme\]a. As an example we use the Phenyl Dithiol (PDT) molecule which consists of a phenyl ring with thiol (-SH) end groups. A gate terminal modulates the conductance of the molecule. We use a simple model Hamiltonian $H$ to describe the molecule (Fig. \[fig:scheme\]b). The effect of the source and drain contacts is taken into account using self-energy functions $\Sigma_1$ and $\Sigma_2$ [@datta_book]. Scattering processes may be described using another self-energy matrix $\Sigma_p$. However, in this paper we focus on coherent or ballistic transport ($\Sigma_p=0$). The source and drain contacts are identified with their respective Fermi levels $\mu_1$ and $\mu_2$. Our simulation consists of iteratively solving a set of coupled equations (Fig. \[fig:scheme\]c) - the Non-Equilibrium Green’s Function (NEGF) formalism [@datta_book; @datta_tut] equations for the density matrix $\rho$ and the Poisson’s equation for the self-consistent potential $U_{SC}$. Given $H$, $U_{SC}$, $\Sigma_1$, $\Sigma_2$, $\mu_1$ and $\mu_2$ the NEGF formalism has clear prescriptions to obtain the density matrix $\rho$ from which the electron density and the current may be calculated. Once the electron density is calculated we solve the Poisson’s equation to obtain the self-consistent potential $U_{SC}$. The solution procedure thus consists of two iterative steps: - [ [**Step 1**]{}: calculate $\rho$ given $U_{SC}$ using NEGF]{} - [ [**Step 2**]{}: calculate $U_{SC}$ given $\rho$ using Poisson’s equation]{} The above two steps are repeated till neither $U_{SC}$ nor $\rho$ changes from iteration to iteration. It is worth noting that the self-consistent potential obtained by solving Poisson’s equation (Eq. \[eq:poisson\]) may be augmented by an appropriate exchange-correlation potential that accounts for many electron effects using schemes like Hartree-Fock (HF) or Density Functional Theory (DFT) [@szabo]. In this paper we do not consider the exchange-correlation effects. Step 1: To obtain $\rho$ from $U_{SC}$ -------------------------------------- The central issue in non-equilibrium statistical mechanics is to determine the density matrix $\rho$; once it is known, all quantities of interest (electron density, current etc.) can be calculated. A good introductory discussion of the concept of density matrix may be found in [@datta_tut]. To obtain the density matrix $\rho$ from the self-consistent potential $U_{SC}$ using the NEGF formalism, we need to know the Hamiltonian $H$, the contact self-energy matrices $\Sigma_{1,2}$ and the contact Fermi levels $\mu_{1,2}$. In this section we describe how we obtain these quantities, and then present a brief outline of the NEGF equations. [*Hamiltonian*]{}: We use a simple basis consisting of one $p_z$ (or $\pi$) orbital on each carbon and sulfur atom. It is well known that the PDT molecule has $\pi$ conjugation - a cloud of $\pi$ electrons above and below the plane of the molecule that dictate the transport properties of the molecule [@magnus_paper]. The on-site energies of our $p_z$ orbitals correspond to the energies of valence atomic $p_z$ orbitals of sulfur and carbon (apart from a constant shift of all levels which is allowed as it does not affect the wavefunctions). The carbon-carbon interaction energy is $2.5~eV$ which is widely used to describe carbon nanotubes [@saito_cnt_book]. The sulfur-carbon coupling of $1.5~eV$ is empirically determined to obtain a good fit to the ab initio energy levels obtained using the commercially available quantum chemistry software Gaussian ’98 [@gaussian] (Fig. \[fig:pi\]). Our model is very similar to the well established $p_z$ orbital based Hückel theory used by many quantum chemists. Although we use a simple model Hamiltonian to describe the molecule, we believe that the essential qualitative physics and chemistry of the molecule is captured. This is because both the molecular energy levels and the wavefunctions closely resemble those calculated from a much more sophisticated ab initio theory (Fig. \[fig:pi\]). [*Self-energy*]{}: Self-energy formally arises out of partitioning the molecule-contact system into a molecular subsystem and a contact subsystem. The contact self-energy $\Sigma$ is calculated knowing the contact surface Green’s function $g$ and the coupling between the molecule and contact $\tau$. For a molecule coupled to two contacts (source and drain) the molecular Green’s function at an energy $E$ is then written as [@datta_book] ($I$: identity matrix, $H$: molecular Hamiltonian, $U_{SC}$: self-consistent potential): $$G=[EI-H-U_{SC}-\Sigma_1-\Sigma_2]^{-1} \label{eq:G}$$ where the contact self-energy matrices are $$\Sigma_{1,2}=\tau_{1,2}g_{1,2}\tau_{1,2}^\dagger \label{eq:Sig}$$ We model the gold FCC (111) contacts using one $s$-type orbital on each gold atom. The coupling matrix element between neighboring $s$ orbitals is taken equal to $-4.3~eV$ - this gives correct surface density of states (DOS) of $0.07~/(eV-atom)$ for the gold (111) surface [@papaconst]. The site energy for each $s$ orbital is assumed to be $-8.74~eV$ in order to get the correct gold Fermi level of $\sim -5.1~eV$. The surface Green’s function $g$ is calculated using a recursive technique explained in detail in [@manoj_thesis]. The contact-molecule coupling $\tau$ is determined by the geometry of the contact-molecule bond. It is believed [@larsen] that when a thiol-terminated molecule like Phenyl Dithiol is brought close to a gold substrate, the sulfur bonds with three gold atoms arranged in an equilateral triangle. For a good contact extended Hückel theory predicts a coupling matrix element of about $2~eV$ between the sulfur $p_z$ orbital and the three gold $s$ orbitals. However to simulate the bad contacts typically observed in experiments [@reed_expt; @diVentra] we reduce the coupling by a factor of five (this factor is also treated as a parameter, and our results do not change qualitatively for a range of values of this parameter). Unlike the Hamiltonian, the self-energy matrices are non-Hermitian. The anti-Hermitian part of the self-energy, also known as the broadening function: $$\Gamma_{1,2}=i(\Sigma_{1,2}-\Sigma_{1,2}^\dagger) \label{eq:Gam}$$ is related to the lifetime of an electron in a molecular eigenstate. Thus upon coupling to contacts, the molecular density of states (Fig. \[fig:dos\]) looks like a set of broadened peaks. [*Where is the Fermi energy?*]{}: When a molecule is coupled to contacts there is some charge transfer between the molecule and the contacts, and the contact-molecule-contact system attains equilibrium with one Fermi level $E_f$. A good question to ask is where $E_f$ lies relative to the molecular energy levels. The answer is not clear yet, the position of $E_f$ seems to depend on what contact model one uses. A jellium model [@diVentra] for the contacts predicts that $E_f$ is closer to the LUMO level for PDT whereas an extended Hückel theory based model [@tian] predicts that $E_f$ is closer to the HOMO level (see Fig. \[fig:pi\] and the related caption for a description of HOMO and LUMO levels). Our ab initio model [@rdamle] seems to suggest that for gold contacts, $E_f$ ($\sim -5.1~eV$) lies a few hundred millivolts above the PDT HOMO. In this paper we will use $E_f=-5.1~eV$ and set the molecular HOMO level (obtained from the $\pi$ model) equal to the ab initio HOMO level ($\sim -5.4~eV$) (see Figs. \[fig:pi\], \[fig:dos\]). Once the location of the equilibrium Fermi energy $E_f$ is known, we can obtain the source and drain Fermi levels $\mu_1$ and $\mu_2$ under non-equilibrium conditions (non-zero $V_{DS}$): $\mu_1=E_f$ and $\mu_2=E_f-qV_{DS}$. [*NEGF equations*]{}: Given $H$, $\Sigma_{1,2}$, contact Fermi energies $\mu_{1,2}$ and the self-consistent potential $U_{SC}$, NEGF has clear prescriptions [@datta_book] to obtain the density matrix $\rho$. The density matrix can be expressed as an energy integral over the correlation function $-iG^<(E)$, which can be viewed as an energy-resolved density matrix: $$\rho = \int dE[-iG^<(E)/2\pi] \label{rho}$$ The correlation function is obtained from the Green’s function $G$ (eq. \[eq:G\]) and the broadening functions $\Gamma_{1,2}$ (eq. \[eq:Gam\]): $$-i{G}^< = G\left({f_1\Gamma_1 + f_2\Gamma_2}\right)G^\dagger$$ where $f_{1,2}(E)$ are the Fermi functions with electrochemical potentials $\mu_{1,2}$: $$f_{1,2}(E) = \left( 1 + \exp{\left[ {{E-\mu_{1,2}}\over{k_BT}}\right]}\right)^{-1}$$ The density matrix so obtained can be used to calculate the electron density $n{(\vec{r})}$ in real space using the eigenvectors of the Hamiltonian $\Psi_\alpha {(\vec{r})}$ expressed in real space: $$n{(\vec{r})}=\sum_{\alpha,\beta} \Psi_\alpha {(\vec{r})}\Psi_\beta ^* {(\vec{r})}\rho_{\alpha \beta} \label{eq:nofr}$$ The total number of electrons $N$ may be obtained from the density matrix as: $$N={\rm trace}(\rho) \label{eq:N}$$ The density matrix may also be used to obtain the terminal current [@datta_book]. For coherent transport, we can simplify the calculation of the current by using the transmission formalism where the transmission function [@datta_book]: $$T(E) = {\rm trace} \left[ \Gamma_1 G \Gamma_2 G^\dagger \right]$$ is used to calculate the terminal current $$I = (2q/h)\int_{-\infty}^\infty dE~T(E)~ \left(f_1(E)-f_2(E) \right)$$ Step 2: To obtain $U_{SC}$ from $\rho$ -------------------------------------- The Poisson’s equation relates the real space potential distribution $U {(\vec{r})}$ in a system to the charge density $n {(\vec{r})}$. We assume a nominal charge density $n_0 {(\vec{r})}$ obtained by solving the NEGF equations with $U {(\vec{r})}=0$ (at $V_{GS}=V_{DS}=0$). The Poisson’s equation is then solved for the [*change*]{} in the charge density ($n-n_0$) from the nominal value $n_0$ [^7] : $$\vec{\nabla}\cdot\left(\epsilon\vec{\nabla}U {(\vec{r})}\right) = -q^2(n {(\vec{r})}-n_0 {(\vec{r})}) \label{eq:poisson}$$ The Poisson (or Hartree) potential $U$ may be augmented by an appropriate exchange-correlation potential $U_{xc}$. In this paper, we do not take into account the exchange-correlation effects ($U_{xc}=0$). We have two schemes to solve the Poisson’s equation: (1) simple Capacitance Model and (2) rigorous numerical solution over a 2D grid in real space. [*Capacitance Model*]{}: We use a simplified picture of the molecule as a quantum dot with some nominal [*total*]{} charge $N_0$ (at $V_{GS}=V_{DS}=0$) and some average potential $U$ arising because of the [*change*]{} $N-N_0$ in this nominal charge due to the applied bias. Thus $U$, $N_0$ and $N$ are numbers and not matrices. The total charge $N$ can be calculated from the NEGF density matrix using Eq. \[eq:N\]. $U$ is the solution to the Poisson’s equation, and may be written as the sum of two terms: (1) A Laplace (or homogeneous) solution $U_L$ with zero charge on the molecule but with applied bias and (2) A particular (or inhomogeneous) solution $U_P$ with zero bias but with charge present on the molecule. Thus $U=U_L+U_P$. $U_L$ is easily written down in terms of the capacitative couplings $C_{MS}$, $C_{MD}$ and $C_{MG}$ of the molecule (Fig. \[fig:RC\]) with the source, drain and gate respectively: $$U_L=\beta (-qV_{GS}) + \frac{(1-\beta)}{2} (-qV_{DS}) \label{eq:UL}$$ where $$\beta=\frac{C_{MG}}{C_{MS}+C_{MD}+C_{MG}} \label{eq:beta}$$ is a parameter ($0 < \beta < 1$) and is a measure of how good the gate control is. Gate control is ideal when $C_{MG}$ is very large as compared to $C_{MS}$ and $C_{MD}$ [^8] . In this case, $\beta=1$ and the Laplace solution $U_L=-qV_{GS}$ is essentially tied to the gate. An estimate of gate control may be obtained from the numerical grid solution explained below by plotting $\beta$ as a function of gate oxide thickness (Fig. \[fig:beta\]). The particular solution $U_P$ may be written in terms of a charging energy $U_0$ as: $$U_P=U_0(N-N_0) \label{eq:UP}$$ The charging energy is treated as a parameter, and may be estimated as follows. The capacitance of a sphere of radius $R$ is $4\pi \epsilon R$. If we distribute a charge of one electron on this sphere, the potential of the sphere is $q/4\pi \epsilon R$. For $R=1~nm$ the value of this potential is about $1.4~eV$. Thus we use a charging energy $U_0 \sim 1~eV$. $U_0$ is the charging energy per electron per molecule and may also be estimated from the numerical grid solution by finding the average potential in the region occupied by the molecule and carrying one electronic charge distributed equally. This numerical procedure also yields $U_0 \sim 1~eV$ and is used to estimate the charging energy while comparing the capacitance model with the numerical grid solution (see Fig. \[fig:compare\] and the related caption). With the simple capacitance model just described, the Poisson’s solution $U$ is just a number. The self-consistent potential that adds to the $p_z$ Hamiltonian (see Eq. \[eq:G\]) is then calculated as $U_{SC}=UI$, where $I$ is the identity matrix of the same size as that of the Hamiltonian. [*Numerical solution*]{}: We use a 2D real space grid to solve the discretized Poisson’s equation for the geometry shown in Fig. \[fig:scheme\]a. The applied gate, source and drain voltages provide the boundary conditions. We use a dielectric constant of 3.9 for silicon dioxide and 2 for the self-assembled monolayer (SAM) [@sam_dielectric]. The correct procedure to obtain the real space charge density $n {(\vec{r})}$ (see Eq. \[eq:poisson\]) from the $p_z$ orbital space density matrix $\rho$ is to make use of Eq. \[eq:nofr\]. However, we simplify the calculation of $n {(\vec{r})}$ by observing that a carbon or sulfur $p_z$ orbital has a spread of about five to six Bohr radii (1 Bohr radius $a_B=0.529~\AA$). So for each atomic site $\alpha$ we distribute a charge equal to $\rho_{\alpha \alpha}$ equally in a cube with side $\sim~10a_B$ centered at site $\alpha$. The solution to Poisson’s equation yields the real space potential distribution. However, the self-consistent potential $U_{SC}$ that needs to be added to the Hamiltonian (Eq. \[eq:G\]) is in the $p_z$ orbital space. We assume that $U_{SC}$ is a diagonal matrix with each diagonal entry as the value of the real space solution $U$ at the appropriate atomic position. For example, the diagonal entry in $U_{SC}$ corresponding to the sulfur based $p_z$ orbital would be equal to $U(\vec{r}_S)$ where $\vec{r}_S$ is the position vector of the sulfur atom. Results {#sec:results} ======= The self-consistent procedure (Fig. \[fig:scheme\]c) is done with the two types of Poisson solutions discussed above. The simple capacitance model is fast while the 2D numerical solution is slow but more accurate. The capacitance model has two parameters, namely $\beta$ which is a measure of the gate control, and $U_0$ which is the charging energy. These parameters can be extracted using the 2D numerical solution. We will first present results with the capacitance model by assuming ideal gate control, or $\beta=1$. This ideal case is useful to explain the current saturation mechanism. We will then compare the results obtained from the capacitance model with those obtained from the numerical solution, and show that the two match reasonably well. Ideal gate control, on state ---------------------------- Fig. \[fig:ideal\_iv\] shows the molecular IV characteristic obtained by self-consistently solving the coupled NEGF - capacitance model Poisson’s equations. We contrast the IV for ideal gate control ($\beta=1$, Fig. \[fig:ideal\_iv\]a,b) with that for no gate control ($\beta=0$, Fig. \[fig:ideal\_iv\]c,d). For each case, we have shown the IV for positive as well as negative drain voltage. We observe the following: - [With ideal gate control the IV is asymmetric with respect to the drain bias. For positive drain bias, we see very little gate modulation of the current. For negative drain bias we see current saturation and good gate modulation - the IV looks like that of a MOSFET.]{} - [With no gate control the IV is symmetric with respect to the drain bias. There is no gate modulation.]{} These features of the IV characteristic may be understood as follows (Fig. \[fig:mechanism\]). Let us first consider the ideal gate case. Since the gate is held at a fixed potential [*with respect to the source*]{}, the molecular DOS does not shift relative to the source Fermi level $\mu_1$ as the drain bias is changed [^9] . For negative drain bias (Fig. \[fig:mechanism\]a), the drain Fermi level $\mu_2$ moves up (towards the LUMO) with respect to the molecular DOS. Since the drain current depends on the DOS lying between the source and drain Fermi levels, the current saturates for increasing negative drain bias because the tail of the DOS dies out as the drain Fermi level moves towards the LUMO. If the gate bias is now made more negative, the molecular levels shift up relative to the source Fermi level, thereby bringing in more DOS in the energy range between $\mu_1$ and $\mu_2$ (referred to as the $\mu_1$-$\mu_2$ window from now on) , and the current increases. Thus we get current saturation and gate modulation. For positive drain bias (Fig. \[fig:mechanism\]b), $\mu_2$ moves down (towards the HOMO) with respect to the molecular DOS. The current increases with positive drain bias because more and more DOS is coming inside the $\mu_1$-$\mu_2$ window as $\mu_2$ moves towards the HOMO peak. Once $\mu_2$ crosses the HOMO peak, the current levels off. This is the resonant tunneling mechanism. If the gate bias is now made more negative, no appreciable change is made in the DOS inside the $\mu_1$-$\mu_2$ window, and the maximum current remains almost independent of the gate bias. Now let us contrast this with the case where no gate is present. In this case, due to the applied drain bias $V_{DS}$, the molecular DOS floats up by roughly $-qV_{DS}/2$ with respect to the source Fermi level. For either negative (Fig. \[fig:mechanism\]c) or positive (Fig. \[fig:mechanism\]d) drain bias, the current flow mechanism is resonant tunneling. Since the equilibrium Fermi energy lies closer to the HOMO, for negative drain bias $\mu_1$ crosses HOMO while for positive drain bias $\mu_2$ crosses HOMO [@datta_expt; @toymodel]. No gate modulation is seen as expected, and the IV is symmetric with respect to $V_{DS}$. Ideal gate control - off state ------------------------------ Fig. \[fig:subth\] shows the log scale drain current as a function of gate bias at high drain bias. We note that despite assuming ideal gate control, the subthreshold slope of this molecular FET is about $300~mV/decade$ which is much worse than the ideal room temperature $k_BT/q=60~mV/decade$ that a good MOSFET can come close to achieving. It is also worth noting here that our simulation is done at low temperature - the subthreshold slope of the molecular FET is [*temperature independent*]{} and only depends on the molecular DOS as explained below. The poor subthreshold slope may be understood as follows. As the gate voltage is made more positive, the molecular DOS shifts down with respect to the $\mu_1$-$\mu_2$ window. The HOMO peak thus moves away from the $\mu_1$-$\mu_2$ window, and fewer states are available to carry the current. The rate at which the current decreases with increasing positive gate bias thus depends on the rate at which the tail of the DOS in the HOMO-LUMO gap dies away with increasing energy (Fig. \[fig:dos\]). Typically we find that the tail of the DOS dies away at the rate of several hundred milli electron-volts of energy per decade, and this slow fall in the DOS determines the subthreshold slope. The slow fall in the molecular DOS may be attributed to the metal-induced gap (MIG) states - the gold source and drain contacts have a sizeable DOS near the Fermi energy, and are separated only by a few angstroms [^10]. Since the molecule is assumed to be rigid, the molecular DOS has no temperature dependence and hence neither does the subthreshold slope. Thus the molecular FET with a rigid molecule acting as the channel is a very poor switching device even with ideal gate control. Estimate of Gate Control ------------------------ The 2D numerical Poisson’s solution may be used to estimate the gate control as follows. From Eq. \[eq:UL\] we see that $$\beta=\left. -\frac{1}{q} ~ \frac{\partial U_L}{\partial V_{GS}}\right | _{V_{DS}} \label{eq:beta_num}$$ Thus $\beta$ may be estimated from the numerical solution by slightly changing $V_{GS}$ (keeping $V_{DS}$ constant) and calculating how much the Laplace’s solution changes over the region occupied by the molecule. A plot of $\beta$ calculated using this method as a function of the gate oxide thickness $T_{ox}$ is shown in Fig. \[fig:beta\]. Knowing that the channel length (length of the PDT molecule) is about $1~nm$, It is evident from Fig. \[fig:beta\] that in order to get good gate control ($\beta > 0.8$) the gate oxide thickness ($T_{ox}$) needs to be about one tenth of the channel length ($L_{ch}$), or about $1~\AA$! Thus we need $L_{ch}/T_{ox} \sim 10$ to get a good molecular FET. In a well-designed conventional bulk MOSFET, $L_{ch}/T_{ox} \sim 40$ [@taur_ning]. This difference between a molecular FET and a conventional FET may be understood by noting that the dielectric constant of the molecular environment (=2) is about 6 times smaller than that of silicon (=11.7) [@lundstrom_private]. Fig. \[fig:beta\] also shows $\beta$ as a function of $T_{ox}$ calculated using the 2D numerical Laplace’s solution over a double gated molecular FET structure. In this case, we find that to get good gate control, we need $L_{ch}/T_{ox} \sim 1.6$. Thus a double gated structure is superior to a single gated one for a given $L_{ch}$ and $T_{ox}$, as is also expected for conventional silicon MOSFETs. The reason for this is simply that two gates can better control the channel than one. Comparison: Capacitance model vs. Numerical Poisson’s solution -------------------------------------------------------------- Fig. \[fig:compare\] compares the IV characteristic obtained by solving the self-consistent NEGF-Poisson’s equations with the numerical Poisson’s solution and the capacitance model. The parameters $\beta$ and $U_0$ for the capacitance model were extracted from the numerical solution. We see a reasonable agreement between the two solutions despite the simplifications made in the capacitance model (the capacitance model assumes a flat potential profile in the region occupied by the molecule, which may not be true, especially at high bias) . For $t_{ox}=1.5~nm$ (Fig. \[fig:compare\]a) there is very little gate modulation and no saturation as expected. In this case $\beta=0.28$ (Fig. \[fig:beta\]) and the IV resembles that shown in Fig. \[fig:ideal\_iv\]c more than the one in Fig. \[fig:ideal\_iv\]a. Also seen in Fig. \[fig:compare\] are the results for $t_{ox}=1~\AA$. For this case $\beta=0.82$ and we observe reasonable saturation and gate control. For realistic oxide thicknesses, however, we expect to observe an IV like the one shown in Fig. \[fig:compare\]a. We have also calculated the IV characteristics with a double gated geometry (not shown here), and as expected from Fig. \[fig:beta\], saturating IVs can be obtained for more realistic oxide thicknesses ($\sim ~ 7~\AA$). Conclusion {#sec:conclusion} ========== We have presented simulation results for a three terminal molecular device with a rigid molecule acting as the channel in a standard MOSFET configuration. The NEGF equations for quantum transport are self-consistently solved with the Poisson’s equation. We conclude the following: 1. [The current-voltage (IV) characteristics of molecular conductors are strongly influenced by the electrostatics, just like conventional semiconductors. With good gate control, the IV characteristics will saturate for one polarity of the drain bias and increase monotonically if the polarity is reversed. By contrast two-terminal symmetric molecules typically show symmetric IV characteristics.]{} 2. [The only advantage gained by using a molecular conductor for an FET channel is due to the reduced dielectric constant of the molecular environment. To get good gate control with a single gate the gate oxide thickness needs to be less than 10% of the channel (molecule) length, whereas in conventional MOSFETs the gate oxide thickness needs to be less than 3% of the channel length. With a double gated structure, the respective percentages are 60% and 20%.]{} 3. [Relatively poor subthreshold characteristics (a [*temperature independent*]{} subthreshold slope much larger than $60~mV/decade$) are obtained even with good gate control, if metallic contacts (like gold) are used, because the metal-induced gap states in the channel preclude it from turning off abruptly. Preliminary results with a molecule coupled to doped silicon source and drain contacts, however, show a temperature dependent subthreshold slope ($\sim k_BT/q$). We believe this is due to the band-limited nature of the silicon contacts, and we are currently investigating this effect.]{} 4. [Overall this study suggests that superior saturation and subthreshold characteristics in a molecular FET can only arise from novel physics beyond that included in our model. Further work on molecular transistors should try to exploit the additional degrees of freedom afforded by the “soft” (as opposed to rigid) nature of molecular conductors [@titash].]{} [*Acknowledgments*]{}: We would like to thank M. Samanta, A. Ghosh, R. Venugopal and M. Lundstrom for useful discussions. This work was supported by the NSF under grants number 9809520-ECS and 0085516-EEC and by the Semiconductor Technology Focus Center on Materials, Structures and Devices under contract number 1720012625. [^1]: Corresponding author: Prashant Damle [^2]: Telephone: (765) 494 3383 [^3]: Fax: (765) 494 2706 [^4]: email: [email protected] [^5]: ©2002 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. [^6]: The authors are aware of one experimental claim (J.H. Schön et al., Nature 413, page 713, 2001) reporting superior molecular FET with a single gated geometry. This claim, however, has been strongly questioned (see article by R.F. Service in Science 298, page 31, 2002). [^7]: The potential distribution corresponding to the nominal charge density (when no drain or gate bias is applied) is included in the calculation of the molecular Hamiltonian [@datta_expt]. [^8]: We have assumed that $C_{MS}=C_{MD}$ in eq. \[eq:UL\], which is reasonable because the center of the molecule is equidistant from the source and drain contacts in our model (see Fig. \[fig:scheme\]). In general, if the source (drain) is closer to the molecule, then $C_{MS}$ ($C_{MD}$) will be bigger [@datta_expt]. With $C_{MS}=C_{MD}$, the molecular Laplace potential is $V_{DS}/2$ in the absence of a gate ($\beta=0$), as is evident from eq. \[eq:UL\] (also see Fig. \[fig:mechanism\]c,d and the related caption). [^9]: This is true as long as the charging energy $U_0 \sim 1~eV$, which is typically the case. For high charging energies the particular solution $U_P$ can dominate the Laplace solution $U_L$ (see eqs. \[eq:UL\],\[eq:UP\] and related discussion), thereby reducing gate control. [^10]: For ballistic silicon MOSFETs, due to the band-limited nature of the doped silicon source/drain contacts, the MIG DOS is negligible. The subthreshold slope at a finite temperature is thus determined by the rate at which the difference in the source and drain [*Fermi function tails*]{} falls as a function of energy. This rate depends on the temperature, and the subthreshold slope is thus proportional to $k_BT/q$ ($\approx 60~mV$ at room temperature) for ballistic Si MOSFETs [@zhibin_ballistic]. Preliminary results for a molecular FET with doped silicon source and drain contacts do show a subthreshold slope proportional to $k_BT/q$; we are currently investigating this effect.

--- abstract: 'Semi-grant-free (SGF) transmission has recently received significant attention due to its capability to accommodate massive connectivity and reduce access delay by admitting grant-free users to channels which would otherwise be solely occupied by grant-based users. In this paper, a new SGF transmission scheme that exploits the flexibility in choosing the decoding order in non-orthogonal multiple access (NOMA) is proposed. Compared to existing SGF schemes, this new scheme can ensure that admitting the grant-free users is completely transparent to the grant-based users, i.e., the grant-based users’ quality-of-service experience is guaranteed to be the same as for orthogonal multiple access. In addition, compared to existing SGF schemes, the proposed SGF scheme can significantly improve the robustness of the grant-free users’ transmissions and effectively avoid outage probability error floors. To facilitate the performance evaluation of the proposed SGF transmission scheme, an exact expression for the outage probability is obtained and an asymptotic analysis is conducted to show that the achievable multi-user diversity gain is proportional to the number of participating grant-free users. Computer simulation results demonstrate the performance of the proposed SGF transmission scheme and verify the accuracy of the developed analytical results.' author: - 'Zhiguo Ding, , Robert Schober, , and H. Vincent Poor, [^1]' bibliography: - 'IEEEfull.bib' - 'trasfer.bib' title: ' [ A New QoS-Guarantee Strategy for NOMA Assisted Semi-Grant-Free Transmission ]{}' --- Introduction ============ The next generation Internet of Things (NGIoT) is envisioned to be an important use case for beyond 5G mobile networks [@ngiot]. The key challenge for supporting NGIoT is, given the scarce radio spectrum, how to support a massive number of devices, each of which might send a small number of packets only. For this emerging application, conventional grant-based transmission is not suitable, since the amount of signalling needed for handshaking could exceed the amount of data sent by the devices. This motivates the development of grant-free transmission schemes, which grant the devices access without lengthy handshaking protocols [@6933472]. Most existing grant-free schemes can be categorized into three groups. The first group applies random access protocols originally developed for computer networks [@randomaccess1], the second group relies on the excess spatial degrees of freedom offered by multiple-input multiple-output (MIMO) techniques [@8454392; @8734871], and the third group employs non-orthogonal multiple access (NOMA) which encourages spectrum sharing among the devices [@mojobabook; @8419284; @8703780; @7976275; @8533378]. It is noted that there are many works which use a combination of the three types of grant-free schemes and hence can potentially offer a significant performance improvement in terms of connectivity and transmission robustness [@8674774; @8719976; @jsacnoma10]. In this paper, we focus on a special case of NOMA based grant-free transmission, termed semi-grant-free (SGF) transmission [@8662677]. Unlike the aforementioned pure grant-free schemes, SGF transmission does not assume that a certain number of resource blocks, such as time slots or subcarriers, are reserved for contention among the grant-free users, since this assumption would put a strict cap on the number of grant-free users which can be served, particularly if the base station has a limited number of antennas and cannot use massive MIMO to improve connectivity. The key idea of SGF transmission is to opportunistically admit grant-free users to those resource blocks which would otherwise be solely occupied by grant-based users. An immediate advantage of SGF over conventional grant-free schemes is that the number of grant-free users is constrained not by the number of resource blocks reserved for grant-free transmission, but by the total number of resource blocks available in the system. Take an orthogonal frequency-division multiple access (OFDMA) system with $128$ subcarriers as an example. If only $8$ subcarriers are reserved for grant-free transmission, at most $8$ grant-free users can be served, but the use of SGF transmission can potentially provide service to $120$ additional grant-free users. In SGF transmission, a crucial task is how to guarantee a grant-based user’s quality of service (QoS) experience when admitting grant-free users to the same resource block. In [@8662677], two SGF transmission schemes, termed SGF Scheme I and Scheme II, were developed to realize this goal. In particular, SGF Scheme I requires the base station to decode the grant-based user’s signal first by treating the grant-free users’ signals as interference, and schedules grant-free users with weak channel conditions in order to limit the interference they cause to the grant-based user. Therefore, Scheme I is ideal for situations, where the grant-free users are cell-edge users, i.e., their connections to the base station are weaker than that of the grant-based user. SGF Scheme II schedules grant-free users with strong channel conditions, and requires the base station to decode the grant-free users’ signals first. Therefore, Scheme II is ideal for situations, where the grant-free users’ connections to the base station are strong. Two types of distributed contention control were applied in [@8662677] to reduce the system overhead and to control the number of admitted grant-free users [@Zhao2005s; @Bletsas06; @6334506]. In this paper, we consider the same grant-free communication scenario as in [@8662677], i.e., one grant-based user and $M$ grant-free users communicate in one resource block with the same base station. A new SGF transmission scheme is proposed which can be interpreted as an opportunistic combination of the two existing SGF schemes and offers the following three advantages: - Recall that SGF Scheme I decodes the grant-based user’s signal first by treating the grant-free users’ signals as interference. Hence, it is inevitable that the grant-based user’s QoS experience is negatively affected by the admission of the grant-free users into the channel. The new SGF scheme can strictly guarantee that admitting grant-free users is transparent to the grant-based user, and the grant-based user’s QoS experience is the same as when it occupies the channel along. - Recall that SGF Scheme II directly decodes the grant-free users’ signals by treating the grant-based user’s signal as interference, which means that interference always exists for the grant-free users. Hence, for SGF Scheme II, the data rates available for the grant-free users can be small. The new SGF scheme can realize interference-free transmission for the grant-free users, and hence can offer significantly improved achievable data rates for grant-free transmission. - For both existing SGF schemes, their outage probabilities exhibit error floors, when there is no transmit power control, e.g., the grant-free and grant-based users increase their transmit powers without coordination. Take Scheme I as an example, which decodes the grant-based user’s signal first and then decodes the grant-free users’ signals via successive interference cancellation (SIC). An outage probability error floor exists because increasing the grant-free users’ transmit powers might help the second stage of SIC but increases the outage probability in the first stage. A similar error floor exists for Scheme II. The new SGF scheme can effectively avoid these error floors and significantly improve transmission robustness, even without careful power control among the users. In order to facilitate the performance analysis, an exact expression for the outage probability achieved by the proposed SGF transmission scheme is developed based on order statistics. Because the outage probability achieved by the proposed SGF scheme can be a function of four random variables, including three dependent order statistics, the developed exact expression has an involved form and hence cannot provide much insight into the properties of SGF transmission. Therefore, two high SNR approximations are developed based on an asymptotic analysis of the derived exact expression. The asymptotic expressions demonstrate that the proposed SGF transmission scheme avoids an outage probability error floor and realizes a multi-user diversity gain of $M$. The remainder of the paper is organized as follows. In Section \[section II\], the existing SGF schemes are briefly introduced first, and then, the proposed new SGF scheme is described. In Section \[section III\], the outage performance achieved by the proposed SGF transmission scheme is analyzed, where an asymptotic analysis is also conducted to illustrate the multi-user diversity gain realized by the proposed scheme. Computer simulations are provided in Section \[section IV\], and Section \[section V\] concludes the paper. We collect the details of all proofs in the appendix. Existing and Newly Proposed SGF Schemes {#section II} ======================================= Consider an SGF communication scenario, where $M$ grant-free users compete with each other for admission to a resource block which would otherwise be solely occupied by a grant-based user for conventional orthogonal multiple access (OMA). Denote the grant-based user’s channel gain by $g$, and the grant-free users’ channel gains by $h_m$, $1\leq m \leq M$. We assume that the SGF system operates in quasi-static Rayleigh fading environments, i.e., all the channel gains are complex Gaussian distributed with zero mean and unit variance. Without loss of generality, we also assume that the grant-free users’ channel gains are ordered as follows: $$\begin{aligned} \label{channel order} |h_1|^2\leq \cdots\leq |h_M|^2.\end{aligned}$$ We note that this ordering assumption is to facilitate the performance analysis, and that this information is not available to any of the nodes in the system, including the base station. Prior to transmission, we assume that the grant-free users can overhear the information exchange between the grant-based user and the base station, and hence know the grant-based user’s channel state information (CSI) as well as the grant-based user’s transmit power, denoted by $P_0$. In addition, each grant-free user has acquired the knowledge of its own CSI, by exploiting the pilot signals broadcasted by the base station. Two Existing SGF Schemes ------------------------- SGF Scheme I in [@8662677] requires the base station to decode the grant-based user’s signal during the first stage of SIC. If grant-free user $m$ is admitted to the channel, the grant-based user’s achievable data rate is $\log\left(1+\frac{P_0|g|^2}{P_s|h_m|^2+1}\right)$, and grant-free user $m$’s data rate is $\log(1+P_s|h_m|^2)$ if the first stage of SIC is successful, where $P_s$ denotes the transmit power of the grant-free users, and the noise power is assumed to be normalized to one. In order to guarantee the grant-based user’s QoS requirement, the base station broadcasts a predefined threshold, denoted by $\tau_I$, and only the grant-free users whose channel gains fall below the threshold participate in contention. In this way, a user which has a strong channel and hence can cause strong interference to the grant-based user will not be granted access. The use of distributed contention control ensures that contention can be carried out in a distributed manner. Thereby, each grant-free user’s backoff period is proportional to its channel gain and therefore the user with the weakest channel gain will be granted access[^2]. Therefore, for SGF Scheme I, the admitted grant-free user’s data rate is given by $$\begin{aligned} R_{I} = \left\{\begin{array}{ll} \log(1+P_s|h_1|^2), &{\rm if} \text{ } |h_1|^2\leq \tau_I\quad \& \quad \log\left(1+\frac{P_0|g|^2}{P_s|h_1|^2+1}\right)>R_0\\ 0, &{\rm otherwise}\end{array}\right.,\end{aligned}$$ where $R_0$ denotes the grant-based user’s target data rate. SGF Scheme II in [@8662677] requires the base station to decode a grant-free user’s signal during the first stage of SIC. Similar to Scheme I, the base station broadcasts a threshold, denoted by $\tau_{II}$, and only the grant-free users whose channel gains are stronger than the threshold participate in contention. By using distributed contention control, each participating grant-free user sets its backoff time inversely propotional to its channel gain, which means that the grant-free user with the strongest channel condition is granted access, and its achievable data rate is given by $$\begin{aligned} R_{II} = \left\{\begin{array}{ll} \log\left(1+\frac{P_s|h_M|^2}{P_0|g|^2+1}\right), &{\rm if} \text{ } |h_M|^2\geq \tau_{II} \\ 0, &{\rm otherwise}\end{array}\right.. \end{aligned}$$ [*Remark 1:*]{} We note that, when $P_s\rightarrow \infty$ and $P_0\rightarrow \infty$, there is an error floor for the admitted grant-free user’s outage probability. Take SGF Scheme II as an example. $\log\left(1+\frac{P_s|h_M|^2}{P_0|g|^2+1}\right)$ becomes a constant, when $P_s\rightarrow \infty$ and $P_0\rightarrow \infty$, which means that there will be an error floor for the outage probability suffered by Scheme II. This error floor can be reduced, if $P_s$ is much larger than $P_0$. In other words, the existing SGF schemes require careful power control to guarantee the grant-free user’s target outage performance, which might not be possible in practice. Proposed SGF Scheme ------------------- In the proposed SGF scheme, prior to transmission, the base station broadcasts a threshold, denoted by $\tau(|g|^2)$, which needs to ensure the following inequality: $$\begin{aligned} \label{tau1} \log\left(1+\frac{P_0|g|^2}{\tau(|g|^2)+1}\right)\geq R_0.\end{aligned}$$ The proposed SGF scheme chooses $\tau(|g|^2)$ such that the above inequality constraint holds with equality: $$\begin{aligned} \tau(|g|^2) =\max\left\{0, \frac{P_0|g|^2}{2^{R_0}-1}-1\right\},\end{aligned}$$ where $\max (a,b)$ denotes the maximum of $a$ and $b$. Upon receiving this threshold, each grant-free user compares its channel gain with the threshold individually. Unlike the two existing schemes, the proposed SGF scheme allows all the grant-free users to participate in contention. Each user’s backoff time is determined by how its channel gain compares to $\tau(|g|^2)$, as shown in the following: - Group $1$ contains the users whose channel gains are above the threshold, i.e., $P_s|h_m|^2>\tau(|g|^2)$. If a user in Group $1$ is granted access, its signal has to be detected during the first stage of SIC. Otherwise, $P_s|h_m|^2>\tau(|g|^2)$ leads to $\log\left(1+\frac{P_0|g|^2}{P_s|h_m|^2+1}\right)<R_0$, which would mean that the grant-based user’s signal cannot be decoded correctly [^3]. Therefore, if a user in Group $1$ is granted access, its achievable data rate is $\log\left(1+\frac{P_s|h_m|^2}{P_0|g|^2+1}\right) $, and hence its backoff time is set to be inversely proportional to this achievable rate. - Group $2$ contains the users whose channel gains are below the threshold, i.e., $P_s|h_m|^2<\tau(|g|^2)$. For a user in Group $2$, its signal can be decoded in either one of the two SIC stages, without affecting the grant-based user’s QoS. In particular, if its signal is decoded in the first stage of SIC, its achievable data rate is $ \log\left(1+\frac{P_s|h_m|^2}{P_0|g|^2+1}\right) $. If its signal is decoded in the second stage of SIC, its achievable data rate is $\log(1+P_s|h_m|^2)$. We note that if a user from Group $2$ is granted access, it is guaranteed that the grant-based user’s signal can be successfully decoded in the first stage of SIC, since $P_s|h_m|^2<\tau(|g|^2)$ leads to $\log\left(1+\frac{P_0|g|^2}{P_s|h_m|^2+1}\right)>R_0$. In other words, $\log(1+P_s|h_m|^2)$ is always achievable for a user from Group $2$. Therefore, its backoff time is set to be inversely proportional to $ \log(1+P_s|h_m|^2) $, since $ \log(1+P_s|h_m|^2)\geq \log\left(1+\frac{P_s|h_m|^2}{P_0|g|^2+1}\right) $. By carrying out distributed contention control [@Zhao2005s; @Bletsas06; @6334506], either a user from Group 1 with the largest $\log\left(1+\frac{P_s|h_m|^2}{P_0|g|^2+1}\right) $ or a user from Group 2 with the largest $ \log(1+P_s|h_m|^2) $ is granted access in a distributed manner. [*Remark 2:*]{} The proposed SGF scheme can be viewed as a hybrid version of the two existing schemes. In particular, under the condition that admitting a grant-free user needs to be transparent to the grant-based user, the users in Group 1 can support SGF Scheme II, whereas the users in Group 2 can support either of the two schemes. The proposed scheme will select the grant-free user with the largest achievable data rate in an opportunistic manner . [*Remark 3:* ]{} We note that, among the $M$ grant-free users, only two users have the chance of being granted access, if the grant-free users’ channel gains are ordered as in . One is grant-free user $M$, if Group $1$ is not empty, since $\log\left(1+\frac{P_s|h_m|^2}{P_0|g|^2+1}\right) \leq \log\left(1+\frac{P_s|h_j|^2}{P_0|g|^2+1}\right) $ always holds for any $m\leq j$. The other one is the grant-free user which has the strongest channel gain in Group $2$, if Group $2$ is not empty, since $ \log(1+P_s|h_m|^2) \leq \log(1+P_s|h_j|^2) $ for $m\leq j$. If a grant-based protocol is used, i.e., global CSI is available at the base station, the base station can decide which user is to be admitted by simply comparing the two users’ data rates. The use of the proposed distributed contention control can ensure that the same goal is achieved without acquiring global CSI at the base station. Outage Performance Analysis {#section III} =========================== It is straightforward to show that the use of the proposed SGF scheme can strictly guarantee that admitting grant-free users is completely transparent to the grant-based user, and the grant-based user’s experience is the same as with OMA. Therefore, in this paper, we mainly focus on the outage performance of the admitted grant-free user, where we assume that all the grant-free users have the same target data rate, denoted by $R_s$. To characterize the outage event, denote the event that there are $m$ users in Group 2 by $E_m $, where $E_m$ can be explicitly defined as follows: $$\begin{aligned} E_m = \left\{ |h_{m}|^2<\frac{\tau(|g|^2)}{P_s}, |h_{m+1}|^2> \frac{\tau(|g|^2)}{P_s} \right\} ,\end{aligned}$$ for $1\leq m \leq M-1$. Furthermore, the two extreme cases with no user in Group 1 and Group 2 can be defined as $E_M=\left\{ |h_{M}|^2<\frac{\tau(|g|^2) }{P_s}\right\}$ and $E_0 = \left\{ |h_{1}|^2>\frac{\tau(|g|^2)}{P_s} \right\} $, respectively. The overall outage probability experienced by the admitted grant-free users is given by $$\begin{aligned} \nonumber {\rm P}_{out} = & \sum^{M-1}_{m=1}{\rm P}\left(E_m, \max\left\{ R_{k,I} ,1\leq k\leq m \right\}<R_s,\max\left\{R_{k,II} ,m< k\leq M \right\}<R_s \right)\\ &+ {\rm P}\left(E_M, \max\left\{ R_{k,I} ,1\leq k\leq M \right\}<R_s\right) \label{out} + {\rm P}\left(E_0, \max\left\{R_{k,II},1\leq k\leq M \right\}<R_s\right),\end{aligned}$$ where $R_{k,I} = \log\left(1+P_s|h_k|^2\right)$ and $R_{k,II}=\log\left(1+\frac{P_s|h_k|^2}{P_0|g|^2+1}\right)$. Because the grant-free users’ channel gains are ordered as in , the outage probability can be simplified as follows: $$\begin{aligned} \label{out overall} {\rm P}_{out} = & \sum^{M}_{m=1}{\rm P}\left(E_m, \max \left\{R_{m,I},R_{M,II} \right\}<R_s\right)+ {\rm P}\left(E_0, R_{M,II} <R_s\right). \end{aligned}$$ Define $\epsilon_0=2^{R_0}-1$, $\epsilon_s=2^{R_s}-1$, $\alpha_0=\frac{\epsilon_0}{P_0}$, and $\alpha_s=\frac{\epsilon_s}{P_s}$. We note that if $|g|^2<\alpha_0$, $$\begin{aligned} \label{fact1} \tau(|g|^2) =\max\left\{0, \frac{P_0|g|^2}{2^{R_0}-1}-1\right\}=0. \end{aligned}$$ By using , the outage probability can be rewritten as follows: $$\begin{aligned} \nonumber {\rm P}_{out} =&\sum^{M}_{m=1}\underset{Q_m}{\underbrace{{\rm P}\left(E_m,|g|^2>\alpha_0, \max \left\{R_{m,I}, R_{M,II}\right\} <R_s\right)}}\\ &+ \underset{Q_0}{\underbrace{{\rm P}\left(E_0,|g|^2>\alpha_0, R_{M,II} <R_s\right)}} + \underset{Q_{M+1}}{\underbrace{{\rm P}\left(R_{M,II}<R_s , |g|^2<\alpha_0\right)}}.\label{overall}\end{aligned}$$ The following theorem provides an exact expression for the outage probability achieved by the proposed SGF scheme. \[theorem1\] Assume that $\epsilon_s\epsilon_0<1$ and $M\geq 2$. The outage probability achieved by the proposed SGF transmission scheme can be expressed as follows: $$\begin{aligned} \nonumber {\rm P}_{out} =&\sum^{M-2}_{m=1} \bar{\eta}_m \sum^{M-m}_{l=0}{M-m \choose l}(-1)^l \sum^{m}_{p=0}{m \choose p}(-1)^p \tilde{\mu}_4 \phi(p,\tilde{\mu}_2) \\\nonumber &+\sum^{M-1}_{i=0}{M-1 \choose i} \frac{(-1)^i\tilde{\eta}_0}{M-1} \left( e^{\frac{ 1}{P_s} } \phi(i,\mu_7) - e^{-\alpha_s } \phi(i,\mu_8) \right) \\\nonumber & + \frac{ \tilde{\eta}_0}{M(M-1)} \sum^{M}_{l=0}(-1)^l {M \choose l}e^{ - l\alpha_s } e^{\frac{M-l}{P_s} } g_{\tilde{\mu}_{12}}(\alpha_0,\alpha_2) +\sum^{M}_{i=0}{M\choose i} (-1)^i e^{\frac{ i}{P_s}}g_{\frac{i }{\alpha_0P_s}}\left(\alpha_0,\alpha_1\right) \\ &+\left(1-e^{- \alpha_s} \right) ^M e^{- \alpha_1 } +\sum^{M}_{i=0}{M\choose i}(-1)^ie^{- i\alpha_s } \frac{1-e^{-\left(1+ i\alpha_sP_0 \right)\alpha_0 } }{1+ i\alpha_sP_0 }\label{overalxx},\end{aligned}$$ where $\bar{\eta}_m = \frac{M!}{m!(M-m)!}$, $\tilde{\eta}_0 = \frac{M!}{(M-2)!}$, $\tilde{\mu}_2 = l\alpha_sP_0+(M-m-l) \frac{\epsilon_0^{-1}P_0}{P_s} $, $\tilde{\mu}_4 = e^{-l\alpha_s+(M-m-l)\frac{ 1}{P_s} } $, $\mu_7=\frac{1}{P_s\alpha_0}$, $\mu_8=\alpha_sP_0$, $\tilde{\mu}_{12}= l\alpha_sP_0+ (M-l)\frac{\alpha_0^{-1}}{P_s} $, $\alpha_1 =(1+\epsilon_s)\alpha_0 $, $\alpha_2=\frac{\epsilon_0(\epsilon_s+1)}{(1-\epsilon_0\epsilon_s) P_0}$, $g_{\mu}(x_1, x_2) = \frac{e^{-(1+\mu) x_1}-e^{-(1+\mu) x_2}}{1+\mu}$, and $\phi(p,\mu) = e^{-p\alpha_s}g_{\mu}(\alpha_1,\alpha_2) +e^{\frac{ p}{P_s}} g_{\mu+\frac{ p}{P_s\alpha_0}}(\alpha_0,\alpha_1)$. See Appendix A. Following the steps in the proof for Theorem \[theorem1\], the outage probability for the case $M=1$ can be obtained straightforwardly as shown in the following corollary. \[corollary0\] Assume that $\epsilon_s\epsilon_0<1$ and $M=1$. The outage probability achieved by the proposed SGF transmission scheme can be expressed as follows: $$\begin{aligned} {\rm P}_{out} =& e^{\frac{ 1}{P_s}}g_{\frac{\alpha_0^{-1}}{P_s}}\left(\alpha_0, \alpha_2\right) - e^{ -\alpha_s} g_{\alpha_sP_0} \left(\alpha_0, \alpha_2\right) +\sum^{1}_{i=0}{1\choose i} (-1)^i e^{\frac{ i}{P_s}}g_{\frac{i }{\alpha_0P_s}}\left(\alpha_0,\alpha_1\right) +\left(1-e^{- \alpha_s} \right) e^{- \alpha_1 } .\end{aligned}$$ [*Remark 4:*]{} In this paper, we mainly focus on the case $\epsilon_s\epsilon_0<1$ because the error floor of ${\rm P}_{out}$ can be avoided in this case, i.e., the scenario with $\epsilon_s\epsilon_0<1$ is ideal for the application of the proposed SGF scheme. $\epsilon_s\epsilon_0<1$ means that $R_s$ needs to be small for a given $R_0$, which is a realistic assumption in practice since SGF is invoked to encourage spectrum sharing between a grant-based user and a grant-free user with a small target data rate. [*Remark 5:* ]{} We note that for the case $\epsilon_s\epsilon_0\geq 1$, the proposed SGF scheme still works and offers significant performance gains compared to the two existing SGF schemes, as shown in the simulation section. However, for $\epsilon_s\epsilon_0\geq 1$, the outage probability achieved by the proposed SGF scheme exhibits an error floor, similar to the existing schemes. More detailed discussions will be provided in Section \[section IV\]. [*Remark 6:*]{} The outage probability expression shown in Theorem \[theorem1\] is complicated, mainly due to the fact that $Q_m$ depends on the choice of $m$. For example, for $1\leq m \leq M-2$, $Q_m$ is a function of the four channel gains, $h_m$, $h_{m-1}$, $h_M$, and $g$, whereas $Q_{M-1}$ is a function of only $h_{M-1}$, $h_M$, and $g$. The fact that the $h_m$, $h_{m-1}$, and $h_M$ are dependent order statistics makes the expression even more involved. However, at high SNR, insightful approximations can be obtained as shown in the following theorem. \[theorem2\] Assuming that $\epsilon_s\epsilon_0<1$, $M\geq 2$ and $P_s=P_0\rightarrow \infty$, the outage probability ${\rm P}_{out}$ can be approximated at high SNR as follows: $$\begin{aligned} {\rm P}_{out} \approx& \sum_{m=1}^{M-2} \frac{\bar{\eta}_m}{P_s^{M+1}}\epsilon_s^{m}\sum^{M-m}_{i=0} {M-m \choose i} \left( \epsilon_s+1 \right)^{M-m-i} \left( \epsilon_s- \epsilon_0^{-1} \right)^{i} \epsilon_0^{i+1}\frac{\tilde{\alpha}_2^{i+1}-(1+\epsilon_s)^{i+1}}{i+1} \\\nonumber &+ \sum_{m=1}^{M-2} \frac{\tilde{\eta}_0 \left(1+\epsilon_s\right)\epsilon_s ^{M-1} \epsilon_0}{P_s^{M+1}(M-1)} \left( (\tilde{\alpha}_2 -1-\epsilon_s ) +\frac{\epsilon_s}{M} \right) \\\nonumber &+ \frac{\bar{\eta}_m}{P_s^{M+1} } \sum^{M-m}_{i=0} {M-m \choose i} \left( \epsilon_s+\epsilon_0 \epsilon_s \right)^{M-m-i} \left( \epsilon_s- \epsilon_0^{-1} \right)^{i}\epsilon_0^{i+1}\frac{\epsilon_s^{m+i+1}}{m+i+1} \\\nonumber &+ \frac{ \tilde{\eta}_0}{P_s^{M+1}M(M-1)}e^{\frac{M}{P_s} } \sum^{M}_{i=0}{M\choose i}\left(\epsilon_s + 1 \right)^{M-i} \left( \epsilon_s -\epsilon_0^{-1} \right)^i \epsilon_0^{i+1} \frac{\tilde{\alpha}_2^{i+1}-1}{i+1} \\\nonumber &+\frac{1}{P_s^{M+1}(M+1)\epsilon_0^M} \epsilon_0^{M+1} \epsilon_s^{M+1} + \frac{\epsilon_s ^M}{P_s^M} + \frac{\epsilon_s ^M(1+\epsilon_0)^{M+1}-1}{P_s^{M+1}(M+1)},\end{aligned}$$ where $\tilde{\alpha}_2=\frac{ (\epsilon_s+1)}{(1-\epsilon_0\epsilon_s) } $. See Appendix B. [*Remark 7:*]{} Following the same steps as in the proof of Theorem \[theorem2\], the outage probability for the case $M=1$ can be approximated as follows: $$\begin{aligned} {\rm P}_{out} \approx& \frac{1}{2P_s^{2}} \epsilon_0 \epsilon_s^{2} + \frac{\epsilon_s }{P_s} + \frac{1}{P_s^2}\left( 1 +\epsilon_s \right)\epsilon_0 (\tilde{\alpha}_2-1).\end{aligned}$$ By comparing the terms in Theorem \[theorem2\], one can find that there is one term proportional to $\frac{1}{P_s^{M}}$, and the other ones are proportional to $\frac{1}{P_s^{M+1}}$. Therefore, a further approximation can be straightforwardly obtained as shown in the following corollary. \[corollary1\] Assuming that $\epsilon_s\epsilon_0<1$ and $P_s=P_0\rightarrow \infty$, the outage probability ${\rm P}_{out}$ can be further approximated as follows: $$\begin{aligned} {\rm P}_{out} \approx \frac{\epsilon_s ^M}{P_s^M} . \end{aligned}$$ A diversity gain of $M$ is achievable for the proposed SGF transmission scheme. [*Remark 8:*]{} Recall that for the two existing SGF schemes, their outage probabilities suffer from error floors, when $P_s$ and $P_0$ go to infinity. Corollary \[corollary1\] demonstrates that not only can the proposed SGF transmission scheme avoid these error floors, but also can it ensure that the achievable diversity gain is proportional to the number of participating grant-free users, i.e., the more grant-free users there are, the better the outage performance. [*Remark 9:*]{} The main reason why the proposed SGF scheme avoids an error floor can be explained as follows. By using , an upper bound on the outage probability achieved by the proposed SGF scheme can be obtained as follows: $$\begin{aligned} {\rm P}_{out} = & \sum^{M}_{m=1}{\rm P}\left(E_m, \max \left\{R_{m,I},R_{M,II} \right\}<R_s\right)+ {\rm P}\left(E_0, R_{M,II} <R_s\right) \\\nonumber \leq & \sum^{M}_{m=1}{\rm P}\left( R_{m,I} <R_s\right)+ {\rm P}\left(E_0, R_{M,II} <R_s\right) \\\nonumber = & \sum^{M}_{m=1}{\rm P}\left( \log\left(1+P_s|h_m|^2\right)<R_s\right)+ {\rm P}\left(E_0, \log\left(1+\frac{P_s|h_M|^2}{P_0|g|^2+1}\right) <R_s\right),\end{aligned}$$ where the last step follows from the definitions of $R_{m,I}$ and $R_{M,II}$. By using the fact that the users are ordered as in , ${\rm P}_{out}$ can be further upper bounded as follows: $$\begin{aligned} \label{lax} {\rm P}_{out} \leq & M{\rm P}\left( \log\left(1+P_s|h_1|^2\right)<R_s\right)+ \underset{Q_u}{\underbrace{{\rm P}\left(E_0, \log\left(1+\frac{P_s|h_M|^2}{P_0|g|^2+1}\right) <R_s\right)}}.\end{aligned}$$ Recall that the outage probability for SGF Scheme II is ${\rm P}_{out}^{II}\triangleq {\rm P}\left(\log\left(1+\frac{P_s|h_M|^2}{P_0|g|^2+1}\right)<R_s\right)$, where an error floor exists since its signal-to-interference-plus-noise ratio (SINR) becomes a constant when $P_s$ and $P_0$ go to infinity. The probability $Q_u$ is quite similar to ${\rm P}_{out}^{II}$, but $Q_u$ does not exhibit an error floor, as explained in the following. By using the definition of $E_0$, $Q_u$ can be rewritten as follows: $$\begin{aligned} Q_u=&{\rm P}\left(|h_{1}|^2>\frac{\tau(|g|^2)}{P_s} , |h_M|^2 <\alpha_s(P_0|g|^2+1)\right) .\end{aligned}$$ Since $|h_{1}|^2\leq |h_{M}|^2$, the lower bound on $|h_{1}|^2$, $\frac{\tau(|g|^2)}{P_s}$, needs to be smaller than the upper bound on $|h_M|^2$, $\alpha_s(P_0|g|^2+1)$, which introduces an additional constraint $|g|^2< \alpha_2$, if $\epsilon_s\epsilon_0< 1$, as shown in - . This additional constraint $|g|^2< \alpha_2$ effectively removes the error floor since $$\begin{aligned} Q_u =&{\rm P}\left(|h_{1}|^2>\frac{\tau(|g|^2)}{P_s} , \log\left(1+\frac{P_s|h_M|^2}{P_0|g|^2+1}\right)<R_s,|g|^2<\alpha_2 \right) \\\nonumber \leq& {\rm P}\left(|g|^2<\alpha_2 \right) = 1-e^{-\alpha_2}\rightarrow 0,\end{aligned}$$ for $P_0\rightarrow \infty$. On the other hand, it is straightforward to show that the first term in , ${\rm P}\left( \log\left(1+P_s|h_1|^2\right)<R_s\right)$, also goes to zero at high SNR. Therefore, ${\rm P}_{out}$ does not have an error floor. Simulation Results {#section IV} ================== In this section, the performance of the proposed SGF transmission scheme is studied via computer simulations, where the accuracy of the developed analytical results is also evaluated. To facilitate performance evaluation, the two existing SGF schemes proposed in [@8662677] are used as benchmark schemes. We note that the proposed SGF scheme allows all the users to participate in contention. Therefore, for a fair comparison, we choose $\tau_{I}=\infty$ and $\tau_{II}=0$ for the two benchmarking schemes, which allow all grant-free users to participate in contention and hence yield the best performance for the two schemes. In Fig. \[fig 1\], the outage performance achieved by the proposed SGF transmission scheme is compared to those of the two existing schemes for different choices of $P_s$ and $P_0$. In particular, in Fig. \[fig1a\], we assume $P_s=\frac{P_0}{10}$, which is equivalent to the case where the grant-free users have weaker channel conditions than the grant-based user, if all the users use the same transmit power. Recall that SGF Scheme I first decodes the grant-based user’s signal by treating the grant-free user’s signal as interference. Therefore, the situation with $P_s=\frac{P_0}{10}$ is ideal for the application of SGF Scheme I, and Fig. \[fig1a\] confirms this conclusion since SGF Scheme I outperforms SGF Scheme II. We note that for SGF Scheme I, the outage probability for $M=1$ can be better than that for $M=5$, since a larger $M$ can reduce ${\rm P}\left(\log\left(1+\frac{P_0|g|^2}{P_s|h_1|^2+1}\right)<R_0\right)$ but may increase ${\rm P}\left(\log\left(1+P_s|h_1|^2\right)<R_s\right)$. In Fig. \[fig1b\], we focus on the situation, where $P_s\rightarrow \infty$ and $P_0$ is kept constant. This is equivalent to the case where the grant-free users have stronger channel conditions than the grant-based user, if all the users use the same transmit power. Therefore, this situation is ideal for the application of SGF Scheme II, and Fig. \[fig1b\] shows that SGF Scheme II indeed outperforms SGF Scheme I. For both considered scenarios, the proposed SGF scheme outperforms the two existing schemes, and can also effectively avoid error floors in both considered scenarios, as shown in the two figures. In Fig. \[fig 2\], we examine the accuracy of the developed analytical results for the outage probability. In Fig. \[fig2a\], the exact expressions for the outage probabilities shown in Theorem \[theorem1\] and Corollary \[corollary0\] are used, and the figure shows that the curves for the analytical results perfectly match the curves obtained by simulations, which demonstrates the accuracy of the result provided in Theorem \[theorem1\]. In Fig. \[fig2b\], the accuracy of the approximations developed in Theorem \[theorem2\] and Corollary \[corollary1\] is studied. As can be observed from the figure, both approximations are accurate at high SNR. We note that the approximation in Corollary \[corollary1\] becomes less accurate as $M$ increases. This is due to the fact that the approximation in Corollary \[corollary1\] disregards the terms, $Q_m$, $0\leq m \leq M-1$, and $Q_{M+1}$, and considers $Q_M$ only. When $M$ is small, such an approximation is accurate. But the gap between the approximation and the actual probability becomes noticeable when $M$ becomes large. The fact that the curves for the approximation in Corollary \[corollary1\] are below the other curves is also due to the same reason. In Fig. \[fig3\], the impact of different choices for the target data rate and the transmit power on the outage performance is studied. The figure shows that reducing the grant-free user’s target rate can affect the outage probability more significantly than reducing the grant-based user’s target rate. In addition, the figure shows that, for a fixed $P_0$, increasing $P_s$ can improve the grant-free user’s outage performance, i.e., a grant-free user can improve its performance by increasing its own transmit power. This is not true for SGF Scheme I since increasing $P_s$ deteriorates the probability ${\rm P}\left(\log\left(1+\frac{P_0|g|^2}{P_s|h_1|^2+1}\right)<R_0\right)$. In Fig. \[fig4\], the performance of the proposed SGF transmission scheme is evaluated under the condition that $\epsilon_s\epsilon_0\geq 1$. As discussed in Remarks 3 and 7, the condition $\epsilon_s\epsilon_0< 1$ is important to avoid error floors. If this condition does not hold, error floors appear, as shown in Fig. \[fig4\]. However, we note that the outage performance achieved by the proposed SGF transmission scheme is still significantly better than those of the two existing schemes. For example, for the case with $R_0=1.5$ bits per channel use (BPCU), and $R_s=1$ BPCU, the proposed scheme can achieve an outage probability of $1\times 10^{-4}$, whereas the outage probabilities achieved by the two existing schemes exceed $1\times 10^{-1}$. In Fig. \[fig5\], the ergodic data rate is used to evaluate the performance of the considered SGF schemes. As can be observed from the figure, the proposed SGF scheme outperforms both existing schemes, particularly at high SNR, which is consistent with the figures showing the outage probability. In addition, Fig. \[fig5\] shows that the slope of the curves for the proposed SGF scheme is larger than those of the two existing schemes, which demonstrates that the proposed scheme can effectively exploit multi-user diversity. An interesting observation is that, for high transmit powers, the curves for SGF Scheme II become flat, whereas the curves for the other two schemes do not. This is due to the fact that the data rate achieved by Scheme II is $ \log\left(1+\frac{P_s|h_M|^2}{P_0|g|^2+1}\right) $, which becomes a constant when both $P_s$ and $P_0$ approach infinity. On the other hand, once the grant-based user’s target data rate can be realized, the achievable data rates for SGF Scheme I and the proposed SGF scheme are of the form $\log(1+P_s|h_m|^2)$, which means that their ergodic data rates are not bounded when $P_s$ goes to infinity, as confirmed by the figure. The performance gain of the proposed scheme over Scheme I is due to the fact that, when the grant-based user’s signal cannot be decoded correctly in the first stage of SIC, the data rate of Scheme I becomes zero, but the proposed scheme can still offer a non-zero data rate by changing the SIC order. Fig. \[fig 6\] shows the grant-free users’ admission probabilities, i.e., which grant-free user is admitted to the resource block by the proposed SGF scheme, for different choices of $R_0$ and the users’ transmit powers. We first note that the admission probability for grant-free user $m$ is given by $$\begin{aligned} {\rm P}_m = & {\rm P}\left(E_m, R_{m,I}>R_{M,II} \right) ,\end{aligned}$$ for $1\leq m \leq M-1$, and $$\begin{aligned} \nonumber {\rm P}_M = &\sum^{M-1}_{n=1} {\rm P}\left(E_m, R_{m,I}<R_{M,II} \right) + {\rm P}\left(E_M \right)+ {\rm P}\left(E_0 \right).\end{aligned}$$ Fig. \[fig 6\] shows that at low SNR, grant-free user $M$, the user with the strongest channel gain, is preferred over the other users, as explained as follows. At low SNR, the threshold $\tau(|g|^2) \triangleq \max\left\{0, \frac{P_0|g|^2}{2^{R_0}-1}-1\right\} $ is very likely to be zero, which means that Group 2 is empty, i.e., $E_0$ happens. As a result, grant-free user $M$ is granted access. In addition, Fig. \[fig 6\] shows that at high SNR, the users’ admission probabilities become constant, and increasing $R_0$ increases the admission probabilities of the grant-free users whose channel gains are weak, which can be explained as follows. By assuming $P_s=P_0\rightarrow \infty$, ${\rm P}_m$, $1\leq m \leq M-1$, can be approximated as follows:[^4] $$\begin{aligned} \label{pmxx} {\rm P}_m = & {\rm P}\left(E_m, R_{m,I}>R_{M,II} \right) \rightarrow {\rm P}\left(E_m \right)\\\nonumber =& {\rm P}\left( |h_{m}|^2<\frac{\tau(|g|^2)}{P_s}, |h_{m+1}|^2> \frac{\tau(|g|^2)}{P_s} \right) \rightarrow {\rm P}\left( |h_{m}|^2< \frac{|g|^2}{2^{R_0}-1} , |h_{m+1}|^2> \frac{|g|^2}{2^{R_0}-1} \right),\end{aligned}$$ which is indeed a constant and not a function of $P_s$ or $P_0$. Since ${\rm P}_m$, $1\leq m \leq M-1$, are constant at high SNR and $\sum^{M}_{m=1}{\rm P}_m =1$, ${\rm P}_M $ must also be constant at high SNR, as confirmed by the figure. By increasing $R_0$, $ \frac{|g|^2}{2^{R_0}-1} $ is reduced, and indicates that ${\rm P}_m$ is increased for small $m$, i.e., the weak users’ admission probabilities are increased, as shown in Fig. \[fig 6\]. Conclusions {#section V} =========== In this paper, we have proposed a new NOMA assisted SGF transmission scheme. Compared to the two existing SGF schemes, this new scheme can ensure that admitting a grant-free user is completely transparent to the grant-based user, i.e., the grant-based user communicates with its base station as if it solely occupied the channel. In addition, the proposed SGF scheme significantly improves the reliability of the grant-free users’ transmissions compared to the existing SGF schemes. To facilitate the performance evaluation of the proposed SGF scheme, an exact expression for the outage probability was derived, where an asymptotic analysis was also carried out to show that the full multi-user diversity gain of $M$ is achievable. Computer simulation results were provided to demonstrate the performance of the proposed SGF scheme and to verify the accuracy of the developed analytical results. In this paper, Rayleigh fading is assumed for the users’ channel gains. An important direction for future research is to carry out a stochastic geometry analysis by taking the users’ path losses into consideration. In addition, we assumed that the admitted grant-free user uses only one time slot for transmission. In practice, the grant-free user may perform retransmission and occupy the channel for a few consecutive time slots. An interesting direction for future research is to develop hybrid automatic repeat request (HARQ) schemes for SGF transmission. Proof for Theorem \[theorem1\] ============================== The evaluation of probability $Q_m$ in depends on the value of $m$, as shown in the following subsections. Evaluation of $Q_m$, $1\leq m\leq M-2$ -------------------------------------- In this case, probability $Q_m$ involves three order statistics, $h_m$, $h_{m+1}$, and $h_M$, and can be expressed as follows: $$\begin{aligned} \nonumber {Q}_{m} =&{\rm P}\left(E_m,|g|^2>\frac{\epsilon_0}{P_0}, \log\left(1+P_s|h_m|^2\right)<R_s, \log\left(1+\frac{P_s|h_M|^2}{P_0|g|^2+1}\right) <R_s \right) \\ =&\underset{|g|^2>\frac{\epsilon_0}{P_0}} {\mathcal{E}}\left\{{\rm P}\left( |h_m|^2 <\xi, |h_{m+1}|^2 >\frac{P_0\epsilon_0^{-1}|g|^2-1}{P_s} , |h_M|^2<\frac{\epsilon_s(1+P_0|g|^2) }{P_s} \right)\right\}, \label{qmz1}\end{aligned}$$ where $\mathcal{E}\{\cdot\}$ denotes the expectation operation, and $\xi=\min\left\{\frac{\epsilon_s}{P_s}, \frac{ P_0\epsilon_0^{-1}|g|^2-1}{P_s}\right\}$. For the case $1\leq m\leq M-2$, $h_{m+1}$ and $h_M$ are different. As a result, there is a hidden constraint in that the lower bound on $h_{m+1}$ should be smaller than the upper bound on $h_M$, i.e., $\frac{\epsilon_s(1+P_0|g|^2) }{P_s}>\frac{P_0\epsilon_0^{-1}|g|^2-1}{P_s}$. We first note that whether $\frac{\epsilon_s(1+P_0|g|^2) }{P_s}>\frac{P_0\epsilon_0^{-1}|g|^2-1}{P_s}$ holds depends on the value of $g$ as shown in the following: $$\begin{aligned} \label{range} &\epsilon_s (1+P_0|g|^2)- \left(P_0\epsilon_0^{-1}|g|^2-1\right) \\\nonumber =&(\epsilon_s-\epsilon_0^{-1}) P_0|g|^2 +\epsilon_s+1 \left\{ \begin{array}{ll} <0, & {\rm if} \text{ } |g|^2 >\frac{\epsilon_0(\epsilon_s+1)}{(1-\epsilon_0\epsilon_s) P_0}\\ >0, & \text{otherwise} \end{array}\right.,\end{aligned}$$ where the assumption that $\epsilon_s\epsilon_0<1$ was used. Furthermore, we note that the following inequality always holds $$\begin{aligned} \label{range2} \frac{\epsilon_0(\epsilon_s+1)}{(1-\epsilon_0\epsilon_s) P_0}>\frac{\epsilon_0}{P_0},\end{aligned}$$ since $(\epsilon_s+1 )- (1-\epsilon_s\epsilon_0 ) = \epsilon_s+\epsilon_s\epsilon_0\geq 0$. Therefore, denoting the probability inside the expectation in by $S_m$, $Q_m$ can be expressed as follows: $$\begin{aligned} \label{range3} {Q}_{m} =&\underset{ \alpha_2>|g|^2>\alpha_0} {\mathcal{E}}\left\{ S_m\right\}+\underset{|g|^2>\alpha_1} {\mathcal{E}}\left\{ S_m\right\}\\\nonumber =&\underset{\alpha_2>|g|^2>\alpha_0} {\mathcal{E}}\left\{ S_m\right\},\end{aligned}$$ where the last step follows by using . For the case $1\leq m \leq M-2$, $S_m$ is a function of three order statistics, $|h_m|^2$, $|h_{m+1}|^2$, and $|h_{M}|^2$, whose joint probability density function (pdf) is given by [@David03] $$\begin{aligned} \label{3pdf} &f_{|h_{m}|^2,|h_{m+1}|^2, |h_{M}|^2}(x,y,z) \\\nonumber=& \eta_m e^{-x}\left(1-e^{-x}\right)^{m-1} e^{-y}\left(e^{-y}-e^{-z}\right)^{M-m-2} e^{-z}\\\nonumber =& \eta_m \sum^{M-m-2}_{i=0}{M-m-2 \choose i}(-1)^ie^{-x}\left(1-e^{-x}\right)^{m-1} e^{-y} e^{-(M-m-2-i)y}e^{-iz}\ e^{-z},\end{aligned}$$ where $x\leq y\leq z$ and $\eta_m = \frac{M!}{(m-1)!(M-m-2)!}$. For a fixed $g$ and by using the joint pdf shown in , $S_m$ can be expressed as follows: $$\begin{aligned} \nonumber S_m =& \eta_m \sum^{M-m-2}_{i=0}{M-m-2 \choose i}(-1)^i\int_0^{\xi}e^{-x}\left(1-e^{-x}\right)^{m-1}\\\nonumber &\times \int_{\frac{\alpha_0^{-1}|g|^2-1}{P_s} }^{\alpha_s(1+P_0|g|^2)} e^{-(M-m-1-i)y} \int^{\alpha_s(1+P_0|g|^2) }_{y} e^{-(i+1)z} dz dydx. \end{aligned}$$ With some algebraic manipulations, $S_m$ can be calculated as follows: $$\begin{aligned} S_m \nonumber = \eta_m \sum^{M-m-2}_{i=0}{M-m-2 \choose i} \sum^{m}_{p=0}{m \choose p}\frac{(-1)^{i+p}e^{-p\xi}}{m(i+1)} \\ \times \nonumber \left(\frac{\mu_3e^{-\mu_1 |g|^2}-\mu_5e^{-\mu_6 |g|^2} }{M-m} - \frac{\mu_4 e^{-\mu_2|g|^2}-\mu_5 e^{-\mu_{6}|g|^2}}{M-m-1-i}\right),\end{aligned}$$ where $\mu_1 = \frac{(M-m)\alpha_0^{-1}}{P_s}$, $\mu_2 = \left((i+1)\alpha_s+(M-m-1-i)\frac{\epsilon_0^{-1}}{P_s} \right)P_0$, $\mu_3=e^{\frac{M-m}{P_s} }$, $\mu_4 = e^{-(i+1)\alpha_s+(M-m-1-i)\frac{ 1}{P_s} } $, $\mu_5=e^{-(M-m)\alpha_s }$, and $\mu_6=(M-m)\alpha_sP_0$. Recall that $Q_m$ can be obtained by finding the expectation of $S_m$ for $\alpha_2>|g|^2>\alpha_0$, i.e., $ {Q}_{m} =\underset{\alpha_2>|g|^2>\alpha_0} {\mathcal{E}}\left\{ S_m\right\}$. We note that $S_m$ is a function $|g|^2$ via $\xi$. The complication is that $\xi$ can have two possible forms depending on the value of $|g|^2$ as shown in the following: $$\begin{aligned} \xi = &\left\{ \begin{array}{ll} \alpha_s, &\text{if } \epsilon_s<\alpha_0^{-1}|g|^2-1\\ \frac{\alpha_0^{-1}|g|^2-1}{P_s}, &\text{otherwise} \end{array}\right. \\\nonumber = &\left\{ \begin{array}{ll} \alpha_s, &\text{if } |g|^2> \alpha_1 \\ \frac{\alpha_0^{-1}|g|^2-1}{P_s}, &\text{otherwise} \end{array}\right.. \end{aligned}$$ It is important to note that $\alpha_0\leq \alpha_1\leq \alpha_2$ always holds since $$\begin{aligned} \frac{\epsilon_0}{P_0} \leq \frac{\epsilon_0(1+\epsilon_s)}{P_0} \leq \frac{\epsilon_0(1+\epsilon_s)}{P_0(1-\epsilon_0\epsilon_s)}. \end{aligned}$$ Therefore, a key step for evaluating $\underset{\alpha_2>|g|^2>\alpha_0} {\mathcal{E}}\left\{ S_m\right\}$ is to calculate the following general expectation: $$\begin{aligned} \nonumber &\underset{\alpha_2>|g|^2>\alpha_0} {\mathcal{E}}\left\{ e^{-p\xi}e^{-\mu|g|^2}\right\} \\\nonumber =&e^{-p\alpha_s}\int^{\alpha_2}_{\alpha_1}e^{-(1+\mu) x}dx +\int^{\alpha_1}_{\alpha_0}e^{-p\frac{\alpha_0^{-1} x-1}{P_s}}e^{-(1+\mu) x}dx \\ =&e^{-p\alpha_s}g_{\mu}(\alpha_1,\alpha_2) +e^{\frac{ p}{P_s}} g_{\mu+\frac{ p}{P_s\alpha_0}}(\alpha_0,\alpha_1)\label{define of g}. \end{aligned}$$ By using the result shown in , $ {Q}_{m} $ can be calculated as follows: $$\begin{aligned} \label{amx} {Q}_{m} = \eta_m \sum^{M-m-2}_{i=0}{M-m-2 \choose i}\frac{(-1)^i}{m(i+1)} \sum^{m}_{p=0}{m \choose p}(-1)^p \\\nonumber \times \left(\frac{\mu_3 \phi(p,\mu_1) -\mu_5 \phi(p,\mu_6) }{M-m} - \frac{\mu_4 \phi(p,\mu_2)-\mu_5 \phi(p,\mu_6) }{M-m-1-i}\right) .\end{aligned}$$ The form in is quite involved and cannot be directly used to obtain a high SNR approximation later. In the following, we will show that the expression of $Q_m$ can be simplified. In particular, $Q_m$ can be first rewritten as follows: $$\begin{aligned} {Q}_{m} =& \frac{ \eta_m}{m(M-m-1)} \sum^{M-m-2}_{i=0}{M-m-1 \choose i+1}(-1)^i \sum^{m}_{p=0}{m \choose p}(-1)^p \\\nonumber &\times \left(\frac{\mu_3 \phi(p,\mu_1) -\mu_5 \phi(p,\mu_6) }{M-m} - \frac{\mu_4 \phi(p,\mu_2)-\mu_5 \phi(p,\mu_6) }{M-m-1-i}\right) ,\end{aligned}$$ which is obtained by absorbing $i+1$ into the binomial coefficients ${M-m-1 \choose i+1}$. By letting $l=i+1$, $Q_m$ can be further rewritten as follows: $$\begin{aligned} {Q}_{m} =& \frac{ -\eta_m}{m(M-m-1)} \sum^{M-m-1}_{l=0}{M-m-1 \choose l}(-1)^l \sum^{m}_{p=0}{m \choose p}(-1)^p \\\nonumber &\times \left(\frac{\mu_3 \phi(p,\mu_1) -\mu_5 \phi(p,\mu_6) }{M-m} - \frac{\tilde{\mu}_4 \phi(p,\tilde{\mu}_2)-\mu_5 \phi(p,\mu_6) }{M-m-l}\right) .\end{aligned}$$ We note that the term $l=0$ can be added since $\frac{\mu_3 \phi(p,\mu_1) -\mu_5 \phi(p,\mu_6) }{M-m} - \frac{\tilde{\mu}_4 \phi(p,\tilde{\mu}_2)-\mu_5 \phi(p,\mu_6) }{M-m-l}=0$ for $l=0$. We further note the fact that $\mu_1$, $\mu_3$, $\mu_5$ and $\mu_6$ are not functions of $l$. Therefore, by using the fact that $ \sum^{n}_{l=0}(-1)^l {n \choose l}=0$, some terms in $Q_m$ can be eliminated. In particular, $Q_m$ can be simplified as follows: $$\begin{aligned} \nonumber {Q}_{m} =& \frac{ \eta_m}{m(M-m-1)} \sum^{M-m-1}_{l=0}{M-m-1 \choose l}(-1)^l \sum^{m}_{p=0}{m \choose p}(-1)^p \frac{\tilde{\mu}_4 \phi(p,\tilde{\mu}_2)-\mu_5 \phi(p,\mu_6) }{M-m-l}\\\nonumber \overset{(a)}{ =}& \bar{\eta}_m \sum^{M-m}_{l=0}{M-m \choose l}(-1)^l \sum^{m}_{p=0}{m \choose p}(-1)^p \left(\tilde{\mu}_4 \phi(p,\tilde{\mu}_2)-\mu_5 \phi(p,\mu_6) \right) \\ \label{am} \overset{(b)}{ =}& \bar{\eta}_m \sum^{M-m}_{l=0}{M-m \choose l}(-1)^l \sum^{m}_{p=0}{m \choose p}(-1)^p \tilde{\mu}_4 \phi(p,\tilde{\mu}_2) ,\end{aligned}$$ where step (a) follows by absorbing $M-m-l$ into the binomial coefficients, step (b) follows by using the fact that $ \sum^{n}_{l=0}(-1)^l {n \choose l}=0$. Again, we note that in step (a), the term $l=M-m$ can be added without changing the value of the summation since $\tilde{\mu}_4 \phi(p,\tilde{\mu}_2)-\mu_5 \phi(p,\mu_6) =0$ for $l=M-m$. Comparing to , we note that the expression for $Q_m$ has been simplified. Evaluation of $Q_{M-1}$ ----------------------- Recall that $ {Q}_{M-1} $ can be expressed as follows: $$\begin{aligned} \label{qm-1} {Q}_{M-1} &=\underset{|g|^2>\alpha_0} {\mathcal{E}}\left\{{\rm P}\left( |h_{M-1}|^2 <\xi,\right.\right. \\\nonumber &\left.\left. |h_{M}|^2 >\frac{\alpha_0^{-1}|g|^2-1}{P_s} , |h_M|^2<\alpha_s(1+P_0|g|^2) \right)\right\}.\end{aligned}$$ Denote the probability inside of the expectation in by $S_{M-1}$. Again, by applying , ${Q}_{M-1} $ can be expressed as follows: $$\begin{aligned} {Q}_{M-1} = &\underset{\alpha_2>|g|^2>\alpha_0} {\mathcal{E}}\left\{ S_{M-1}\right\}.\end{aligned}$$ Unlike $S_m$, $1\leq m \leq M-2$, $S_{M-1}$ becomes a function of two order statistics, $|h_{M-1}|^2$ and $|h_{M}|^2$, whose joint pdf is given by $$\begin{aligned} &f_{|h_{M-1}|^2, |h_{M}|^2}(x,y) = \tilde{\eta}_0 e^{-x} \left(1-e^{-x}\right)^{M-2} e^{-y},\end{aligned}$$ where $x\leq y$. By using this joint pdf, $S_{M-1}$ can be calculated as follows: $$\begin{aligned} \nonumber S_{M-1} = & \frac{\tilde{\eta}_0\left(1-e^{-\xi}\right)^{M-1} \left( e^{-\frac{\alpha_0^{-1}|g|^2-1}{P_s} } - e^{-\alpha_s(1+P_0|g|^2)} \right)}{M-1}\\\nonumber = & \sum^{M-1}_{i=0}{M-1 \choose i} (-1)^i \frac{\tilde{\eta}_0}{M-1} e^{-i\xi} \\&\times \left( e^{\frac{ 1}{P_s} } e^{-\mu_7 |g|^2} - e^{-\alpha_s } e^{-\mu_8|g|^2} \right). \end{aligned}$$ By applying , $ {Q}_{M-1} $ can be obtained as follows: $$\begin{aligned} \label{am-1} {Q}_{M-1} = & \sum^{M-1}_{i=0}{M-1 \choose i} (-1)^i \frac{\tilde{\eta}_0}{M-1} \\&\times \left( e^{\frac{ 1}{P_s} } \phi(i,\mu_7) - e^{-\alpha_s } \phi(i,\mu_8) \right).\end{aligned}$$ Evaluation of $Q_M$ ------------------- Unlike $Q_m$, $1\leq m \leq M-1$, $Q_M$ is a function of $h_M$ and $g$. In particular, recall that $Q_M$ can be expressed as follows: $$\begin{aligned} Q_M= &{\rm P}\left(\log\left(1+P_s|h_M|^2\right)<R_s,\right.\\\nonumber &\left.\log\left(1+\frac{P_s|h_M|^2}{P_0|g|^2+1}\right) <R_s, E_M,|g|^2>\alpha_0 \right)\\\nonumber = &{\rm P}\left(\log\left(1+P_s|h_M|^2\right)<R_s ,|h_{M}|^2<\tau(|g|^2) P_s^{-1}, |g|^2>\alpha_0 \right),\end{aligned}$$ where the last step follows from the fact that $P_s|h_M|^2\geq \frac{P_s|h_M|^2}{P_0|g|^2+1}$. Therefore, we can rewrite $Q_m$ as follows: $$\begin{aligned} \nonumber Q_M= &{\rm P}\left( |h_M|^2<\alpha_s ,|h_{M}|^2<\frac{\alpha_0^{-1}|g|^2 -1}{P_s},|g|^2>\alpha_0 \right) \\ = &{\rm P}\left( |h_{M}|^2<\frac{\alpha_0^{-1}|g|^2 -1}{P_s},\alpha_0 <|g|^2<\alpha_1 \right) +{\rm P}\left( |h_M|^2<\alpha_s ,|g|^2>\alpha_1 \right),\end{aligned}$$ where we use the fact that $\alpha_s <\frac{\alpha_0^{-1}|g|^2 -1}{P_s}$ is guaranteed if $|g|^2>\alpha_1$. By applying the fact that $h_M$ and $g$ are independent, $Q_M$ can be calculated as follows: $$\begin{aligned} Q_M= &\int^{\alpha_1 }_{\alpha_0 }\left(1-e^{-\frac{\alpha_0^{-1}x -1}{P_s}}\right)^M e^{-x}dx \\\nonumber &+\left(1-e^{- \alpha_s} \right) ^M e^{- \alpha_1 } .\end{aligned}$$ With some algebraic manipulations, $Q_M$ can be finally expressed as follows: $$\begin{aligned} Q_M = &\sum^{M}_{i=0}{M\choose i} (-1)^i e^{\frac{ i}{P_s}}g_{\frac{i }{\alpha_0P_s}}\left(\alpha_0,\alpha_1\right) \\\nonumber &+\left(1-e^{- \alpha_s} \right) ^M e^{- \alpha_1 } .\label{aM}\end{aligned}$$ Evaluation of $Q_0$ ------------------- $Q_0$ is surprisingly more complicated to analyze, compared to $Q_M$. Recall that $Q_0$ can be expressed as follows: $$\begin{aligned} Q_0=& {\rm P}\left(\log\left(1+\frac{P_s|h_M|^2}{P_0|g|^2+1}\right)<R_s , |h_{1}|^2>\frac{P_0\epsilon_0^{-1}|g|^2-1}{P_s} ,|g|^2>\frac{\epsilon_0}{P_0}\right) \\\nonumber =& {\rm P}\left( |h_M|^2<\alpha_s(P_0|g|^2+1) , |h_{1}|^2>\frac{\alpha_0^{-1}|g|^2-1}{P_s} ,|g|^2>\alpha_0\right).\end{aligned}$$ Again, by applying the fact that the lower bound on $|h_M|^2$ needs to be larger than the upper bound on $|h_1|^2$ as discussed in , the probability $Q_0$ can be expressed as follows: $$\begin{aligned} \nonumber Q_0=& \underset{\alpha_0<|g|^2<\alpha_2}{\mathcal{E}} \left\{{\rm P}\left( |h_M|^2<\alpha_s(P_0|g|^2+1) , |h_{1}|^2>\frac{\alpha_0^{-1}|g|^2-1}{P_s} \right)\right\}. \label{q00}\end{aligned}$$ Denote the probability inside the expectation in by $S_0$. $S_0$ is a function of two order statistics, $|h_1|^2$ and $|h_M|^2$, whose joint pdf is given by $$\begin{aligned} f_{|h_1|^2,|h_M|^2}(x,y)=& \tilde{\eta}_0 e^{-x}\left(e^{-x}-e^{-y}\right)^{M-2}e^{-y} \\\nonumber= & \tilde{\eta}_0 \sum^{M-2}_{i=0}(-1)^i {M-2 \choose i}e^{-(M-1-i)x} e^{-(i+1)y} ,\end{aligned}$$ for $x\leq y$. For a fixed $|g|^2$ and by applying the joint pdf, $S_0$ can be calculated as follows: $$\begin{aligned} S_0&= \tilde{\eta}_0 \sum^{M-2}_{i=0}(-1)^i {M-2 \choose i}\\\nonumber &\times \int^{\alpha_s(P_0|g|^2+1) }_{\frac{\alpha_0^{-1}|g|^2-1}{P_s} }e^{-(M-1-i)x}\int^{\alpha_s(P_0|g|^2+1) }_{x} e^{-(i+1)y} dydx. \end{aligned}$$ With some algebraic manipulations, $S_0$ can be expressed as follows: $$\begin{aligned} S_0&= \tilde{\eta}_0 \sum^{M-2}_{i=0}(-1)^i {M-2 \choose i} \\\nonumber &\times \left(\frac{ e^{\frac{ M}{P_s} }e^{-\mu_{10}|g|^2 } - e^{- M\alpha_s }e^{-\mu_{11}|g|^2 } }{M(i+1)}\right.\\\nonumber & -\frac{ \left( e^{\frac{M-1-i}{P_s}- (i+1)\alpha_s } e^{-\mu_{12}|g|^2} - e^{-M \alpha_s } e^{-\mu_{11}|g|^2 } \right) }{(i+1)(M-1-i)} ,\end{aligned}$$ where $\mu_{10}=\frac{M}{\alpha_0P_s}$ and $\mu_{11}= M\alpha_sP_0 $, and $\mu_{12}= (i+1)\alpha_sP_0+ (M-1-i)\frac{\alpha_0^{-1}}{P_s} $. By applying the integration result in , $Q_0$ can be expressed as follows: $$\begin{aligned} \nonumber Q_{0}&= \tilde{\eta}_0 \sum^{M-2}_{i=0}(-1)^i {M-2 \choose i}\\\nonumber &\times \left(\frac{ e^{\frac{ M}{P_s} }g_{\mu_{10}}(\alpha_0,\alpha_2) - e^{- M\alpha_s }g_{\mu_{11}}(\alpha_0,\alpha_2) }{M(i+1)}\right.\\ & \left.-\frac{ \left( e^{\frac{M-1-i}{P_s}- (i+1)\alpha_s } g_{\mu_{12}}(\alpha_0,\alpha_2) - e^{-M \alpha_s } g_{\mu_{11}}(\alpha_0,\alpha_2) \right) }{(i+1)(M-1-i)} \right).\label{a00}\end{aligned}$$ The expression in is quite involved, and cannot be used directly to obtain a high SNR approximation. In the following, we will show that can be simplified. First, the expression for $Q_0$ is modified as follows: $$\begin{aligned} Q_{0}=& \frac{\tilde{\eta}_0}{M-1} \sum^{M-2}_{i=0}(-1)^i {M-1 \choose i+1}\\\nonumber &\times \left(\frac{ e^{\frac{ M}{P_s} }g_{\mu_{10}}(\alpha_0,\alpha_2) - e^{- M\alpha_s }g_{\mu_{11}}(\alpha_0,\alpha_2) }{M}\right.\\\nonumber & \left.-\frac{ \left( e^{\frac{M-1-i}{P_s}- (i+1)\alpha_s } g_{\mu_{12}}(\alpha_0,\alpha_2) - e^{-M \alpha_s } g_{\mu_{11}}(\alpha_0,\alpha_2) \right) }{(M-1-i)} \right) ,\end{aligned}$$ which is obtained by absorbing $i+1$ into the binomial coefficients. The expression for $Q_0$ can be further modified as follows: $$\begin{aligned} \label{high x1} Q_{0}=& - \frac{\tilde{\eta}_0}{M-1} \sum^{M-1}_{l=0}(-1)^l {M-1 \choose l}\\\nonumber &\times \left(\frac{ e^{\frac{ M}{P_s} }g_{\mu_{10}}(\alpha_0,\alpha_2) - e^{- M\alpha_s }g_{\mu_{11}}(\alpha_0,\alpha_2) }{M }\right.\\\nonumber & \left.-e^{ - l\alpha_s }\frac{ \left( e^{\frac{M-l}{P_s} } g_{\tilde{\mu}_{12}}(\alpha_0,\alpha_2) - e^{-(M-l) \alpha_s } g_{\mu_{11}}(\alpha_0,\alpha_2) \right) }{(M-l)} \right)\end{aligned}$$ which is obtained by substituting $l=i+1$. We note that the term $l=0$ can be added without changing the value of the summation, since the terms in the second and third lines in cancel each other when $l=0$. By using the fact that $ \sum^{n}_{l=0}(-1)^l {n \choose l}=0$, $Q_0$ can be further simplified as follows: $$\begin{aligned} Q_{0} =& \frac{\tilde{\eta}_0}{M-1} \sum^{M-1}_{l=0}(-1)^l {M-1 \choose l}e^{ - l\alpha_s }\\\nonumber &\times \left( \frac{ e^{\frac{M-l}{P_s} } g_{\tilde{\mu}_{12}}(\alpha_0,\alpha_2) - e^{-(M-l) \alpha_s } g_{\mu_{11}}(\alpha_0,\alpha_2) }{(M-l)} \right) \\\nonumber =& \frac{ \tilde{\eta}_0}{M(M-1)} \sum^{M}_{l=0}(-1)^l {M \choose l}e^{ - l\alpha_s }\\\nonumber &\times \left( e^{\frac{M-l}{P_s} } g_{\tilde{\mu}_{12}}(\alpha_0,\alpha_2) - e^{-(M-l) \alpha_s } g_{\mu_{11}}(\alpha_0,\alpha_2) \right),\end{aligned}$$ where the last step is obtained by absorbing $(M-l)$ into the binomial coefficients. In addition, we also note that the term $l=M$ can be added since $ e^{\frac{M-l}{P_s} } g_{\tilde{\mu}_{12}}(\alpha_0,\alpha_2) - e^{-(M-l) \alpha_s } g_{\mu_{11}}(\alpha_0,\alpha_2) =0$ when $l=M$. Again, by using the fact that $ \sum^{n}_{l=0}(-1)^l {n \choose l}=0$, $Q_0$ can be further simplified as follows: $$\begin{aligned} \label{a0} Q_{0} =& \frac{ \tilde{\eta}_0}{M(M-1)} \sum^{M}_{l=0}(-1)^l {M \choose l}e^{ - l\alpha_s } e^{\frac{M-l}{P_s} } g_{\tilde{\mu}_{12}}(\alpha_0,\alpha_2) . \end{aligned}$$ $Q_{M+1}$ can be evaluated similarly to $Q_M$ since both are functions of $h_M$ and $g$, and it can be expressed as follows: $$\begin{aligned} \nonumber Q_{M+1}=&{\rm P}\left(\log\left(1+\frac{P_s|h_M|^2}{P_0|g|^2+1}\right)<R_s , |g|^2<\alpha_0\right) \\ =&\sum^{M}_{i=0}{M\choose i}(-1)^ie^{- i\alpha_s } \frac{1-e^{-\left(1+ i\alpha_sP_0 \right)\alpha_0 } }{1+ i\alpha_sP_0 }. \label{aM+1}\end{aligned}$$ Therefore, by combining , , , and , the overall outage probability is obtained as shown in the theorem and the proof is complete. Proof for Theorem \[theorem2\] ============================== As discussed in the proof for Theorem $\ref{theorem1}$, $Q_m$ depends on the value of $m$. Therefore, the high SNR approximations for different $Q_m$ will be discussed separately in the following subsections. High SNR Approximation for $Q_m$, $1\leq m \leq M-2$ ---------------------------------------------------- Among all the terms in , the expression for $Q_m$, $1\leq m\leq M-2$, is the most complicated one, as is evident from the proof of Theorem \[theorem1\]. First, recall $ {Q}_{m} $ can be expressed as follows: $$\begin{aligned} \label{qm simplified} {Q}_{m} =& \bar{\eta}_m \sum^{M-m}_{l=0}{M-m \choose l}(-1)^l \sum^{m}_{p=0}{m \choose p}(-1)^p \tilde{\mu}_4 \phi(p,\tilde{\mu}_2) \\\nonumber =& \bar{\eta}_m \sum^{M-m}_{l=0}{M-m \choose l}(-1)^l \sum^{m}_{p=0}{m \choose p}(-1)^p \\\nonumber &\times \left(\tilde{\mu}_4 e^{-p\alpha_s}g_{\tilde{\mu}_2}(\alpha_1,\alpha_2) +\tilde{\mu}_4 e^{\frac{ p}{P_s}} g_{\tilde{\mu}_2+\frac{ p}{P_s\alpha_0}}(\alpha_0,\alpha_1) \right) .\end{aligned}$$ In order to facilitate a high SNR approximation, $ {Q}_{m}$ is rewritten as follows: $$\begin{aligned} {Q}_{m} =& \bar{\eta}_m \int^{\alpha_2}_{\alpha_1} \sum^{M-m}_{l=0}{M-m \choose l}(-1)^l \sum^{m}_{p=0}{m \choose p}(-1)^p \tilde{\mu}_4 e^{-p\alpha_s}e^{-(1+\tilde{\mu}_2)x} dx \\\nonumber&+ \bar{\eta}_m \int^{\alpha_1}_{\alpha_0} \sum^{M-m}_{l=0}{M-m \choose l}(-1)^l \sum^{m}_{p=0}{m \choose p}(-1)^p \tilde{\mu}_4 e^{\frac{ p}{P_s}} e^{-(1+\tilde{\mu}_2+\frac{ p}{P_s\alpha_0})x}dx ,\end{aligned}$$ By applying the approximation, $e^{-x}\approx 1-x$ for $x\rightarrow 0$ and also using the definitions of $\tilde{\mu}_2$ and $\tilde{\mu}_4$, $Q_m$ can be approximated as follows: $$\begin{aligned} {Q}_{m} \approx& \bar{\eta}_m \int^{\alpha_2}_{\alpha_1} \sum^{M-m}_{l=0}{M-m \choose l}(-1)^l \sum^{m}_{p=0}{m \choose p}(-1)^p \\\nonumber &\times e^{-l\alpha_s-\frac{ l}{P_s} }e^{-p\alpha_s}e^{-( l\epsilon_s-l \epsilon_0^{-1})x} dx \\\nonumber&+ \bar{\eta}_m \int^{\alpha_1}_{\alpha_0} \sum^{M-m}_{l=0}{M-m \choose l}(-1)^l \sum^{m}_{p=0}{m \choose p}(-1)^p \\\nonumber &\times e^{-l\alpha_s-l\frac{ 1}{P_s} } e^{\frac{ p}{P_s}} e^{-( l\epsilon_s-l \epsilon_0^{-1}+\frac{ p}{P_s\alpha_0})x}dx.\end{aligned}$$ By using the fact that $ \sum^{n}_{l=0}(-1)^l {n \choose l}a^l=(1-a)^n$, the approximation of $Q_m$ can be further simplified as follows: $$\begin{aligned} {Q}_{m} \approx& \bar{\eta}_m\left(1-e^{-\alpha_s}\right)^{m} \int^{\alpha_2}_{\alpha_1} \sum^{M-m}_{l=0}{M-m \choose l}(-1)^l e^{-l\alpha_s-\frac{ l}{P_s} }e^{-( l\epsilon_s-l \epsilon_0^{-1})x} dx \\\nonumber&+ \bar{\eta}_m \int^{\alpha_1}_{\alpha_0} \left(1-e^{\frac{ 1}{P_s}-\frac{ x}{P_s\alpha_0}} \right)^m\sum^{M-m}_{l=0}{M-m \choose l}(-1)^l e^{-l\alpha_s-l\frac{ 1}{P_s} } e^{-( l\epsilon_s-l \epsilon_0^{-1})x}dx \\\nonumber =& \bar{\eta}_m\left(1-e^{-\alpha_s}\right)^{m} \int^{\alpha_2}_{\alpha_1} \left(1-e^{-(\alpha_s+\frac{ 1}{P_s} +( \epsilon_s- \epsilon_0^{-1})x)} \right)^{M-m} dx \\\nonumber&+ \bar{\eta}_m \int^{\alpha_1}_{\alpha_0}\left(1-e^{\frac{ 1}{P_s}-\frac{ x}{P_s\alpha_0}} \right)^m \left(1-e^{-(\alpha_s+\frac{ 1}{P_s} +( \epsilon_s- \epsilon_0^{-1} )x}\right)^{M-m}dx.\end{aligned}$$ A more simplified form of $ {Q}_{m} $ can be obtained by carrying out the following high SNR approximations: $$\begin{aligned} {Q}_{m} \approx& \bar{\eta}_m\left(1-e^{-\alpha_s}\right)^{m} \int^{\alpha_2}_{\alpha_1} \left( \alpha_s+\frac{ 1}{P_s} +( \epsilon_s- \epsilon_0^{-1})x \right)^{M-m} dx \\\nonumber&+ \frac{\bar{\eta}_m}{P_s^m\alpha_0^m} \int^{\alpha_1}_{\alpha_0}\left( x-\alpha_0 \right)^m \left( \alpha_s+\frac{ 1}{P_s} +( \epsilon_s- \epsilon_0^{-1} )x\right)^{M-m}dx ,\end{aligned}$$ With some algebraic manipulations, the high SNR approximation for $Q_m$ can be obtained as follows: $$\begin{aligned} \nonumber {Q}_{m} \approx & \frac{\bar{\eta}_m}{P_s^{M+1}}\epsilon_s^{m}\sum^{M-m}_{i=0} {M-m \choose i} \left( \epsilon_s+1 \right)^{M-m-i} \left( \epsilon_s- \epsilon_0^{-1} \right)^{i} \\\nonumber &\times \epsilon_0^{i+1}\frac{\tilde{\alpha}_2^{i+1}-(1+\epsilon_s)^{i+1}}{i+1} + \frac{\bar{\eta}_m}{P_s^{M+1}\epsilon_0^m} \sum^{M-m}_{i=0} {M-m \choose i} \\ \label{aam} &\times \left( \epsilon_s+\epsilon_0 \epsilon_s \right)^{M-m-i} \left( \epsilon_s- \epsilon_0^{-1} \right)^{i}\epsilon_0^{m+i+1}\frac{\epsilon_s^{m+i+1}}{m+i+1} .\end{aligned}$$ High SNR Approximation for $Q_0$ -------------------------------- To facilitate the asymptotic analysis, $Q_0$ can be rewritten as follows: $$\begin{aligned} \nonumber Q_{0} =& \frac{ \tilde{\eta}_0}{M(M-1)} \sum^{M}_{l=0}(-1)^l {M \choose l}e^{ - l\alpha_s } e^{\frac{M-l}{P_s} } g_{\tilde{\mu}_{12}}(\alpha_0,\alpha_2) \\\nonumber =& \frac{ \tilde{\eta}_0}{M(M-1)} \sum^{M}_{l=0}(-1)^l {M \choose l}e^{ - l\alpha_s } e^{\frac{M-l}{P_s} } \frac{e^{-(1+\tilde{\mu}_{12}) \alpha_0}-e^{-(1+\tilde{\mu}_{12}) \alpha_2}}{1+\tilde{\mu}_{12}} . \end{aligned}$$ To carry out high SNR approximations, $Q_{0} $ can be first rewritten as follows: $$\begin{aligned} Q_{0} =& \frac{ \tilde{\eta}_0}{M(M-1)} \sum^{M}_{l=0}(-1)^l {M \choose l}e^{ - l\alpha_s } e^{\frac{M-l}{P_s} } \int^{\alpha_2}_{\alpha_0} e^{-(1+\tilde{\mu}_{12})x}dx \\\nonumber=& \frac{ \tilde{\eta}_0}{M(M-1)}e^{\frac{M}{P_s} } \int^{\alpha_2}_{\alpha_0}e^{-x} e^{- M\epsilon_0^{-1} x}\sum^{M}_{l=0}(-1)^l {M \choose l} \\\nonumber &\times e^{ - l\left(\alpha_s + \frac{1}{P_s}+\epsilon_sx -\epsilon_0^{-1} x\right)} dx .\end{aligned}$$ By using the fact that $ \sum^{n}_{l=0}(-1)^l {n \choose l}a^l=(1-a)^n$, $Q_0$ can be expressed as follows: $$\begin{aligned} Q_{0} =& \frac{ \tilde{\eta}_0}{M(M-1)}e^{\frac{M}{P_s} } \int^{\alpha_2}_{\alpha_0} \left(1-e^{ - l\left(\alpha_s + \frac{1}{P_s}+\epsilon_sx -\epsilon_0^{-1} x\right)} \right)^M dx \\\nonumber\approx& \frac{ \tilde{\eta}_0}{M(M-1)}e^{\frac{M}{P_s} } \int^{\alpha_2}_{\alpha_0} \left( \alpha_s + \frac{1}{P_s}+\epsilon_sx -\epsilon_0^{-1} x \right)^M dx ,\end{aligned}$$ where the last step is obtained by applying the following approximation, $e^{-x}\approx 1-x$ for $x\rightarrow 0$. By applying the binomial expansion, $Q_0$ can be expressed as follows: $$\begin{aligned} Q_{0} \nonumber \approx& \frac{ \tilde{\eta}_0}{M(M-1)}e^{\frac{M}{P_s} } \sum^{M}_{i=0}{M\choose i}\left(\alpha_s + \frac{1}{P_s} \right)^{M-i} \left( \epsilon_s -\epsilon_0^{-1} \right)^i \int^{\alpha_2}_{\alpha_0}x^i dx \\\nonumber =& \frac{ \tilde{\eta}_0}{M(M-1)}e^{\frac{M}{P_s} } \sum^{M}_{i=0}{M\choose i}\left(\alpha_s + \frac{1}{P_s} \right)^{M-i} \\ \label{aa0} &\times \left( \epsilon_s -\epsilon_0^{-1} \right)^i \frac{\alpha_2^{i+1}-\alpha_0^{i+1}}{i+1}. \end{aligned}$$ High SNR Approximation for $Q_{M-1}$ ------------------------------------ First, we recall that $Q_{M-1}$ can be expressed as follows: $$\begin{aligned} {Q}_{M-1} = &\sum^{M-1}_{i=0}{M-1 \choose i} (-1)^i \frac{\tilde{\eta}_0}{M-1} \\\nonumber&\times \left( e^{\frac{ 1}{P_s} } e^{-i\alpha_s}g_{\mu_7}(\alpha_1,\alpha_2) +e^{\frac{ 1}{P_s} } e^{\frac{ i}{P_s}} g_{\mu_7+\frac{ i}{P_s\alpha_0}}(\alpha_0,\alpha_1)\right. \\\nonumber &\left. - e^{-\alpha_s } e^{-i\alpha_s}g_{\mu_8}(\alpha_1,\alpha_2) -e^{-\alpha_s } e^{\frac{ i}{P_s}} g_{\mu_8+\frac{ i}{P_s\alpha_0}}(\alpha_0,\alpha_1) \right).\end{aligned}$$ In order to obtain the high SNR approximation, we first express $Q_{M-1}$ as follows: $$\begin{aligned} {Q}_{M-1} = & \sum^{M-1}_{i=0}{M-1 \choose i} (-1)^i \frac{\tilde{\eta}_0}{M-1} \\&\times \left( \int^{\alpha_2}_{\alpha_1} e^{\frac{ 1}{P_s} } e^{-i\alpha_s}e^{-(1+ \mu_7)x} dx +\int^{\alpha_1}_{\alpha_0} e^{\frac{ 1}{P_s} } e^{\frac{ i}{P_s}} e^{-(1+ \mu_7+\frac{ i}{P_s\alpha_0})x}dx \right. \\\nonumber &\left. - \int^{\alpha_2}_{\alpha_1}e^{-\alpha_s } e^{-i\alpha_s}e^{-(1+ \mu_8)x} dx -\int^{\alpha_1}_{\alpha_0}e^{-\alpha_s } e^{\frac{ i}{P_s}} e^{-(1+ \mu_8+\frac{ i}{P_s\alpha_0})x}dx \right) .\end{aligned}$$ By using the fact that $ \sum^{n}_{l=0}(-1)^l {n \choose l}=0$, $ {Q}_{M-1} $ can be further expressed as follows: $$\begin{aligned} \nonumber & {Q}_{M-1} = \frac{\tilde{\eta}_0}{M-1} \left(e^{\frac{ 1}{P_s} } \left(1-e^{-\alpha_s}\right)^{M-1} \int^{\alpha_2}_{\alpha_1} e^{-(1+ \frac{1}{P_s\alpha_0})x} dx \right.\\\nonumber &\left. +\int^{\alpha_1}_{\alpha_0} e^{\frac{ 1}{P_s} } \left(1-e^{\frac{ 1}{P_s}-\frac{ 1}{P_s\alpha_0}x}\right)^{M-1} e^{-(1+ \frac{1}{P_s\alpha_0})x}dx \right. \\\nonumber &\left. -e^{-\alpha_s } \left(1-e^{-\alpha_s}\right)^{M-1} \int^{\alpha_2}_{\alpha_1}e^{-(1+ \alpha_sP_0)x} dx\right. \\ &\left. -\int^{\alpha_1}_{\alpha_0}e^{-\alpha_s } \left(1-e^{\frac{ 1}{P_s}-\frac{ 1}{P_s\alpha_0}x} \right)^{M-1}e^{-(1+ \alpha_sP_0)x}dx \right).\end{aligned}$$ Directly applying the approximation, $e^{-x}\approx 1-x$ for $x\rightarrow 0$, to the above equation results in a very complicated form. In order to facilitate the high SNR approximation, we rearrange the four terms in the above equation as follows: $$\begin{aligned} \nonumber {Q}_{M-1} = & \frac{\tilde{\eta}_0}{M-1} \left( \left(1-e^{-\alpha_s}\right)^{M-1}\right.\\\nonumber&\times \left( \int^{\alpha_2}_{\alpha_1} e^{\frac{ 1}{P_s}-(1+ \frac{1}{P_s\alpha_0})x} -e^{-\alpha_s -(1+ \alpha_sP_0)x} dx\right) \\\nonumber &\left. +\int^{\alpha_1}_{\alpha_0} \left(1-e^{\frac{ 1}{P_s}-\frac{ 1}{P_s\alpha_0}x}\right)^{M-1} \right. \\ \nonumber &\left. \times\left(e^{\frac{ 1}{P_s}-(1+ \frac{1}{P_s\alpha_0})x} - e^{-\alpha_s-(1+ \alpha_sP_0)x}\right)dx \right) .\end{aligned}$$ By applying the approximation, $e^{-x_1}-e^{-x_2}\approx x_2-x_1$ for $x_1\rightarrow 0$ and $x_2\rightarrow 0$, $ {Q}_{M-1} $ can be approximated as follows: $$\begin{aligned} {Q}_{M-1} \label{qm-dd} \approx& \frac{\tilde{\eta}_0}{M-1} \left( \left(1-e^{-\alpha_s}\right)^{M-1}\right.\\\nonumber&\times \left( \int^{\alpha_2}_{\alpha_1} \left(\frac{ 1}{P_s}- \frac{1}{P_s\alpha_0}x +\alpha_s + \alpha_sP_0x\right) dx\right) \\\nonumber &\left. +\int^{\alpha_1}_{\alpha_0} \left(1-e^{\frac{ 1}{P_s}-\frac{ 1}{P_s\alpha_0}x}\right)^{M-1} \right. \\ \nonumber &\left. \times\left(\frac{ 1}{P_s}- \frac{1}{P_s\alpha_0}x+\alpha_s+ \alpha_sP_0x\right)dx \right).\end{aligned}$$ The approximation shown in can be further approximated as follows: $$\begin{aligned} \nonumber {Q}_{M-1} \approx& \frac{\tilde{\eta}_0}{M-1} \left( \left(1-e^{-\alpha_s}\right)^{M-1} \left( \int^{\alpha_2}_{\alpha_1} \left(\frac{ 1}{P_s} +\alpha_s \right) dx\right) \right. \\\nonumber &\left. +\int^{\alpha_1}_{\alpha_0} \left(1-e^{\frac{ 1}{P_s}-\frac{ 1}{P_s\alpha_0}x}\right)^{M-1} \times\left(\frac{ 1}{P_s} +\alpha_s\right)dx \right)\\ \label{aaM-1} =& \frac{\tilde{\eta}_0 \left(1+\epsilon_s\right)\epsilon_s ^{M-1} \epsilon_0}{P_s^{M+1}(M-1)} \left( (\tilde{\alpha}_2 -1-\epsilon_s ) +\frac{\epsilon_s}{M} \right).\end{aligned}$$ Following similar steps as for the approximation of $Q_m$, $1\leq m \leq M-1$, $Q_M$ can be approximated as follows: $$\begin{aligned} \label{aaM} Q_M \approx &\frac{1}{(M+1)\epsilon_0^M} \alpha_0^{M+1} \epsilon_s^{M+1} + \alpha_s ^M ,\end{aligned}$$ and $Q_{M+1}$ can be approximated as follows: $$\begin{aligned} \label{aaM+1} Q_{M+1}& \alpha_s ^M \frac{(1+\epsilon_0)^{M+1}-1}{P_0(M+1)}. \end{aligned}$$ By combining , ,, and , the high SNR approximation for ${\rm P}_{out}$ can be obtained and the proof for Theorem \[theorem2\] is complete. [^1]: Z. Ding and H. V. Poor are with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA. Z. Ding is also with the School of Electrical and Electronic Engineering, the University of Manchester, Manchester, UK (email: <[email protected]>, <[email protected]>). R. Schober is with the Institute for Digital Communications, Friedrich-Alexander-University Erlangen-Nurnberg (FAU), Germany (email: <[email protected]>). [^2]: In this paper, we focus on the case, where a single grant-free user is admitted to the channel. However, as discussed in [@8662677] and [@6334506], more than one user can be granted access via distributed contention, which is beyond the scope of this paper. [^3]: In this paper, the grant-free users are assumed to use the same fixed transmit power, $P_s$. The use of distributed power control can further improve the performance of SGF transmission without increasing system overhead, but is beyond the scope of this paper. [^4]: Obtaining an exact expression for ${\rm P}_m$ is not a trivial task since ${\rm P}_m$ is a function of four random variables, $|g|^2$, $|h_m|^2$, $|h_{m+1}|^2$ and $|h_M|^2$. The fact that $|h_m|^2$, $|h_{m+1}|^2$ and $|h_M|^2$ are not independent makes it more difficult to analyze ${\rm P}_m$, which is an important direction for future research.

--- abstract: '[Dans cet article je donnerai une nouvelle démonstration courte et directe pour le Théorème des Nombres Premiers. C’est vrai que ce théorème a été complétement démontré au début du 20ème siecle mais la démonstration était basé sur des résultats élémentaires (théorème de **Chebyshev**) et aussi analytiques compliqués (théorème de **Ikehrara**), mais ici j’ai pas utilisé le théorème de Chebyshev ainsi que j’ai remplacé et j’ai généralisé le théorème de **Ikehara** grâce à la notion des fonctions à variation bornée qui est ancienne mais récent dans la théorie analytique des nombres.]{}' author: - | \ `[email protected]`\ `[email protected]` title: | <span style="font-variant:small-caps;">Tauberian Theorem of Laplace Transformation\ And\ Application of Prime Number Theorem</span> --- Préliminaires ============= Les fonctions à variation bornée -------------------------------- Les fonctions à variation bornée joue un rôle très important dans la théorie de l’intégration au sens de **Stieltjes**, ici on va s’interésser à les fonctions à variation bornée sur $\R^+$ à valeurs complexes. Soit $x$ un réel positif et soit $(x_k)_{k=0,\cdots, n}$ une suite finie et strictement croissante des réels de l’intervalle $[0,x]$ tels que $0=x_0 < x_1<x_2< \cdots < x_n = x$ est une subdivision de l’intervalle $[0,x]$, on note $\Sigma$ pour cette subdivion et $\mathcal{S}([0,x])$ l’ensemble de toutes les subdivions possibles de $[0,x]$. *La fonction variation totale* d’une fonction complexe définie sur $\R^+$, notée $T_f$, est la fonction définie par [$$T_f(x) := \sup_{\Sigma \in \mathcal{S}([0,x])} \sum_{k=1}^n|f(x_k) - f(x_{k-1})|\label{e1}$$]{} Il est bien clair que la fonction $T_f$ est une fonction croissante sur $\R^+$, par conséquent si $T_f$ est majorée sur $\R^+$ alors on dira que $f$ est à *variation bornée* sur $\R^+$ et on note $$V(f) = \lim_{x\to + \infty} T_f(x) \in \R^+$$ pour *la variation totale* de la fonction $f$. - Toute fonction $g$ de classe $\mathcal{C}^1$ sur $\R^+$ à valeurs complexes telle que $g' \in L^1(\R^+)$ est à variation bornée, en effet, pour une subdivision $\Sigma: 0=x_0 < x_1<\cdots < x_n=x$ et puisque $g$ est continue sur chaque intervalle $[x_{i-1},x_i]$ (pour $i=1,\cdots n$) et dérivable sur leurs interieurs topologique alors d’après le théorème des accroissements finis il existe des $c_i$ dans $]x_{i-1},x_i[$ tels que $$|g(x_i) - g(x_{i-1})|=|g'(c_i)||x_{i} - x_{i-1}|$$ D’où $$T_g(x) = \sup_{\Sigma \in \mathcal{S}([0,x])} \sum_{k=1}^n|g'(c_i)||x_{i} - x_{i-1}|$$ Or cette somme est une somme de **Darboux** ce qu’on peut déduire grâce à l’intégrale de **Riemann** que $$T_g(x) = \int_0^x|g'(t)| dt$$ Donc $$V(g) = \int_{0}^{+\infty}|g'(t)| dt$$ qui est finie puisque $g' \in L^1(\R^+)$, alors $g$ est à variation bornée sur $\R^+$. - toute fonction à variation bornée sur $\R^+$ est bornée sur $\R^+$, en effet, soit $f$ une fonction à variation bornée sur $\R^+$ alors pour un $x\geq 0$ $$\begin{aligned} |f(x) - f(0)| &= \left|\sum_{k=1}^n f(x_i) - f(x_{i-1})\right| \\ &\leq \sum_{k=1}^n|f(x_i) - f(x_{i-1})| \\ &\leq T_f(x) \\ &\leq V(f) <+\infty \end{aligned}$$ Alors $f$ est bornée sur $\R^+$. \[prop1\] On dit qu’une fonction $f$ définie de $\R^+$ à valeurs complexes admet une limite à gauche en $x \in \R^+$, notée $f(x^-)$ si à tout $\eps >0$ on peut associer un $0 \leq \delta < x$ tel que $$a < t < x \Longrightarrow |f(t) - f(x^-)| < \eps$$ Et en plus si $f(x^-) = f(x)$ on dit que $f$ est continue à gauche en $x$. On note pour $\vb$ la classe des fonctions, définies de $\R^+$ à valeurs complexes, à variation bornée, continues à gauche en tout point de $\R^+$ et qui s’annullent en $0$. Intégrale de Lebesgue-Stieltjes ------------------------------- Le théorème 8.14 page 156 du livre \[Rud\] a établi le lien entre la théorie de la mesure et la théorie des fonctions à variation bornée. Donc d’après le même théorème, soit $f \in \vb$ alors il existe une unique mesure complexe de Borel $\mu_f$ telle que [$$f(x) = \mu_f([0,x[), \qquad \forall x\geq 0 \label{e2}$$]{} Et en plus pour tout $x \in \R^+$ on a [$$T_f(x)=|\mu_f|([0,x[)$$]{} Où $|\mu_f|$ est une mesure positive de Borel, dite *la variation totale de la mesure complexe $\mu_f$*, qui est *finie* d’après le théorème 6.4 page 114 de \[2\]. - On peut facilement montrer que $|\mu_f|$ est finie autant que $f\in \vb$, en effet, soit $f\in \vb$ alors $$\begin{aligned} |\mu_f|(\R^+) &= \lim_{x\to +\infty} |\mu_f|([0,x[) \\ &= \lim_{x\to +\infty}T_f(x) \\&= V(f) <+ \infty\end{aligned}$$ - D’une autre part, si $f$ est à valeurs dans $\R$ alors $\mu_f$ est dite *une mesure signée* alors de la même manière on démontre que cette mesure est finie. - Soit $f\in \vb$, si $y>x$ alors $$\begin{aligned} f(y) - f(x) &= \mu_f([0,y[) - \mu_f([0,x[)\\ &= \mu_f([x,y[) \end{aligned}$$ Donc $$\mu_f(\{x\}) = f(x^+) - f(x)$$ D’où $f$ est continue en $x$ si et seulement si $$\mu_f(\{x\}) = 0$$ Le théorème de **Radon-Nikodym**, voir le théorème 6.12 page 120 de \[2\], assure que pour toute mesure complexe $\mu$ il existe une fonction mesurable complexe $h$ de module égal à $1$ telle que $$d\mu = h d|\mu| .$$ Ainsi, on déduit que pour toute fonction $g:\R^+ \longrightarrow \C$ mesurable et bornée sur $\R^+$ on a $g \in L_{\mu_f}^1(\R^+)$ où $f\in \vb$. En effet: $$\begin{aligned} \left| \int_{\R^+} g d\mu_f \right| &\leq \int_{\R^+} |g| d|\mu_f| \\ &\leq \|g\|_{\infty} |\mu_f|(\R^+)\\ &< +\infty \end{aligned}$$ Où $$\|g\|_{\infty} = \sup_{x \in \R^+}|g(x)|.$$ Maintenant, d’après le théorème 6.1.4 du livre \[1\] on constate que pour $f \in \vb$ on a[$$\int_0^x df(t) = \mu_f([0,x[), \qquad x\geq 0 \label{e4}$$]{} Soient donc $f\in \vb$ et $g:\R^+ \longrightarrow \C$ une fonction de classe $\mathcal{C}^1$ sur $\R^+$ telle que $g' \in L^1(\R^+)$, alors d’après le théorème 6.2.2 (grâce au résultat \[e4\]) du même livre on démontre que [$$\int_0^{+\infty} g(t) df(t) = \mu_{fg}(\R^+) - \int_0^{+\infty} f(t)g'(t) dt \label{e5}$$]{} la mesure complexe $\mu_{fg}$ a bien un sens, en effet d’après les propriétés \[prop1\] on démontre que $g$ est à variation bornée or le produit de deux éléments de $\vb$ est un élément de $\vb$ alors $fg \in \vb$ (car $fg$ est à variation bornée et continue à gauche à chaque point de $\R^+$ et $(fg)(0) =0$), et en plus $$\mu_{fg}([0,x[) = f(x)g(x), \qquad \forall x\in \R^+$$ Et $$|\mu_{fg}(\R^+)| \leq |\mu_{fg}|(\R^+) = \lim_{x\to + \infty} T_{fg}(x)< + \infty$$ Théorème Tauberien de la transformation de Laplace complexe =========================================================== Dans tout ce qui suit $s = \sigma + it$ où $\sigma , t \in \R$ est un nombre complexe et $\rho$ est une fonction de la classe $\vb^* := \{f \in \vb , \quad \ \Im(f) = 0 \}$. Ainsi $\C_*^+$ est l’ensemble des nombres complexes de partie réelle strictement positive. On définit la transformation de Laplace-Stieltjes de la fonction $\rho$ par $$\L_{\rho}^*(s) = \int_0^{+\infty} e^{-sx}d\rho(x), \qquad \sigma > 0$$ il est bien clair d’après ce qui précéde, puisque $x\mapsto e^{-sx}$ est continue et bornée sur $\R^+$ pour tout $\sigma > 0$, que la fonction $\L_{\rho}^*$ est bien définie. ****** On pose pour tout $(x,s) \in \R^+ \times \C_+^*$ $$\phi(x,s) = e^{-sx}$$ Alors on a - Pour tout $x\geq 0$ la fonction $s\mapsto \phi(x,s)$ est continue en $0^+$. - Pour tout $\sigma > 0$ la fonction $x \mapsto \phi(x,s)$ est continue donc mésurable sur $\R^+$. - Pour tout $\sigma > 0$ et pour $d\rho$-presque tout $x \in \R^+$ on a $$|\phi(x,s)|\leq 1$$ Où $1 \in L_{d\rho}^1(\R^+)$ car $$\int_{\R^+} d\rho(x) = \mu_{\rho}(\R^+) <+ \infty$$ Alors la fonction $x \mapsto \phi(x,s)$ est $d\rho$-intégrable sur $\R^+$ et la fonction $\L_{\rho}^*$ est est continue en $0^+$. Donc $$\begin{aligned} \lim_{s \to 0^+} \L_{\rho}^*(s) &= \L_{\rho}^*(0) \\ &= \int_{\R^+}d\rho (x) \\ &= \lim_{x\to + \infty} \mu_{\rho}([0,x[) \qquad \text{d'après \ref{e4}} \\ &= \lim_{+ \infty} \rho \qquad \qquad \qquad \quad \! \! \text{d'après \ref{e2}} \end{aligned}$$ $\blacksquare$ Maintenant on définit la transformation de Laplace complexe d’une fonction $\rho \in \vb^*$ par $$\L_{\rho}(s) = \int_{0}^{+ \infty} \rho(x)e^{-sx}dx, \qquad \sigma > 0 .$$ La fonction $\L_{\rho}$ est bien définie, en effet puisque $\rho \in \vb^*$ alors $\rho$ est bornée sur $\R^+$ en plus $$\left|\int_0^{+ \infty} \rho(x)e^{-sx}dx \right| \leq \| \rho\|_{\infty} \int_0^{+ \infty} e^{- \sigma x} dx = \frac{\|\rho\|_{\infty}}{\sigma} < +\infty$$ ****** Soit $s\in \C_+^*$, d’après l’équation \[e5\] on a $$\L_{\rho}^*(s) = \mu_{\rho e^{-s \cdot}}(\R^+) + s \int_0^{+\infty} \rho(x) e^{-sx} dx$$ Or $$|\mu_{\rho e^{-s \cdot}}(\R^+)| = |\mu_{\rho}(\R^+)|\lim_{x \to + \infty} e^{-\sigma x} = 0$$ Donc $$\L_{\rho}^*(s) = s \L_{\rho}(s)$$ Passons à la limite $s \to 0^+$ on a d’après le <span style="font-variant:small-caps;">Lemme</span> \[L1\] $$\lim_{x\to + \infty} \rho(x) =Res(\L_{\rho},0)$$ $\blacksquare$ D’une manière générale, soit $\alpha$ un réel positif alors il est clair, d’après ce qui précéde, que pour tout $\rho \in \vb^*$ on a $\varrho(x) = \rho(x) e^{-\alpha x}$ est un élément de $\vb^*$. Ainsi on déduit le résultat suivant: ****** Soit $\sigma > \alpha$, alors $$\L_{\rho}(s) = \int_0^{+\infty} \rho(x)e^{-sx}dx = \int_{0}^{+\infty} \rho(x) e^{-\alpha x} e^{-(s-\alpha)x}dx$$ On pose $$\varrho(x) = \rho(x)e^{- \alpha x}, \qquad \forall x \geq 0 .$$ Alors $$\L_{\rho}(s) = \int_0^{+ \infty} \varrho(x)e^{-(s-\alpha)x}dx = \L_{\varrho}(s - \alpha)$$ Donc $$(s-\alpha) \L_{\rho}(s) = (s-\alpha)\L_{\varrho}(s- \alpha) = z \L_{\varrho}(z)$$ D’où quand $s \to \alpha$ on aura $z \to 0$ et d’après le <span style="font-variant:small-caps;">Théorème</span> \[T1\] on a $$\lim_{x \to + \infty} \varrho(x) = Res(\L_{\varrho}(z),z=0) = Res(\L_{\rho},\alpha)$$ Alors $$\rho(x) \underset{x \to + \infty}{\sim} Res(\L_{\rho},\alpha) e^{\alpha x} .$$ $\blacksquare$ Théorème des Nombres Premiers (nouvelle démonstration) ====================================================== Soit $\chi: \N^* \longrightarrow \R^+$ une fonction arithmétique positive, on pose pour tout $x \in (1,+\infty)$ $$f(x) = \sum_{1 \leq n < x}\chi(n) \qquad \text{et} \qquad f(1) =0 .$$ Il est clair que la fonction $f$ est croissante sur $[1,+\infty)$ et continue à gauche en tout point de $[1,+\infty)$. Ainsi, les points de discontinuité de $f$ sont des éléments de $\N^*$. Si $f$ est continue en $x\in \N^*$ alors on aura $$f(x^+) = f(x)$$ Donc $$\begin{aligned} 0 &= f(x^+) - f(x) \\ &= \sum_{x \leq n < x^+}\chi(n) \\ &= \chi(x) \end{aligned}$$ Alors [$$f \ \text{est \ continue \ en } \ x \in \N^* \Longleftrightarrow \chi(x) = 0 \label{R1}$$]{} Soit maintenant $(a_k)_{k \in \N}$ une suite croissante des points de discontinuité de la fonction $f$ sur $[1,+\infty)$ alors $f$ est constante sur chaque intervalle $I_k=(a_{k-1},a_{k}]$ (où $k \in \N^*$). En effet, soit $k\in \N^*$ s’il existe $n \in I_k$ tel que $f$ est continue en $n$ alors d’après \[R1\] $\chi(n) = 0$ ainsi et d’une manière générale soit $(\beta_i)_{i \in \N^*}$ une suite strictement croissante des entiers de $\overset{\circ}{I_k}$ (l’interieur de $I_k$), alors $f$ est continue en chaque $\beta_i$ d’où $\chi(\beta_i)=0$ pour tout $i=1,2,\cdots$ et en conséquent pour tout $x \in (a_{k-1},a_k]$ on a $f(x) = f(a_{k-1}^+)$ ($k \in \N^*$). Soient $\alpha >1$ un réel et $\rho$ la fonction définie sur $\R^+$ par $$\rho(x) = f\left( e^{x} \right) e^{-\alpha x} .$$ Soit $k$ un entier strictement positif on note $ (\lambda_k)_{k \in \N}$ pour la suite croissante des points de discontinuité de la fonction $\rho$ sur $\R^+$ ($\lambda_k = \log a_k \in \log\N^*$). Alors la fonction $\rho$ est décroissante sur chaque intervalle $J_k = (\lambda_{k-1},\lambda_k]$, en effet: soient $x,y \in J_k$ tels que $x>y$, donc puisque $f$ est constante ($\equiv c_k$) sur $J_k$ alors $\rho(x) - \rho(y) = c_k \left(e^{-\alpha x} - e^{-\alpha y}\right) < 0$ d’où $\rho$ est strictement décroissante sur $J_k$ pour tout $k \in \N^*$. D’une autre part, pour tout $k \in \N^*$ $$\rho(\lambda_k^+) > \rho(\lambda_k).$$ En effet, puisque la fonction $x \mapsto e^{- \alpha x}$ est continue sur $\R^+$ alors $e^{-\alpha \lambda_k^+} = e^{- \alpha \lambda_k}$ donc $$\rho(\lambda_k^+)-\rho(\lambda_k) = (f(a_k^+) - f(a_k))e^{- \alpha \lambda_k}$$ et puisque $f$ est discontinue en $a_k$ et croissante sur $[1,+\infty)$ alors $f(a_k^+) > f(a_k)$. D’où $$\rho(\lambda_k^+) > \rho(\lambda_k).$$ ****** Soit $x \in \R^+$, on pose $0=x_0 < x_1 < \cdots < x_n = x$ une subdivision de l’intervalle $[0,x]$ et on note pour $m$ le plus grand entier naturel non nul tel que $\lambda_{m-1} < x \leq \lambda_m$ où les $(\lambda_k)_{k \in \N}$ sont les points de discontinuité de la fonction $\rho$ définis précédamment, alors $$\sum_{i=1}^n |\rho(x_i) - \rho(x_{i-1})| = \sum_{k=0}^m\sum_{\underset{x_i \in J_k}{i=1}}^n|\rho(x_i) - \rho(x_{i-1})|$$ où $(J_k)_{k \in \N^*}$ sont les intervalles $(\lambda_{k-1},\lambda_k]$ et $J_0=[0,\lambda_0]$, et on note bien que $\displaystyle \cup_{k=0}^mJ_k =[0, \lambda_m] $ donc puisque $\rho$ est strictement décroissante sur chaque $J_k$ alors $$\begin{aligned} \sum_{k=0}^m\sum_{\underset{x_i \in J_k}{i=1}}^n|\rho(x_i) - \rho(x_{i-1})| & \leq \rho(0) - \rho(\lambda_0) + \sum_{k=1}^m\left(\rho(\lambda_{k-1}^+) - \rho(\lambda_k)\right) - (\rho(x) - \rho(\lambda_m)) \\ &=-\rho(\lambda_0)+\rho(\lambda_0^+)-\rho(\lambda_1)+\rho(\lambda_1^+)+ \cdots -\rho(\lambda_m) - \rho(x) + \rho(\lambda_m)\\ &= - \rho(x) + \sum_{k=0}^{m-1}\left(\rho(\lambda_k^+) -\rho(\lambda_k)\right) \\ &= - \rho(x) + \sum_{k=0}^{m-1} \frac{f(a_k^+) - f(a_k)}{a_k^{\alpha}} \\ &= -\rho(x) + \sum_{k=0}^{m-1}\frac{\chi(a_k)}{a_k^{\alpha}}\end{aligned}$$ Où les $(a_k)_{k \in \N}$ sont les points de discontinuité de la fonction $f$ et qui sont des éléments de $\N^*$. Donc $$\sum_{k=0}^{m-1}\frac{\chi(a_k)}{a_k^{\alpha}} \leq \sum_{1\leq \ell < e^x} \frac{\chi(\ell)}{\ell^{\alpha}}$$ D’où $$\sum_{i=1}^n |\rho(x_i) - \rho(x_{i-1})| \leq - \rho(x) + \sum_{1\leq \ell < e^x} \frac{\chi(\ell)}{\ell^{\alpha}}.$$ Alors $$T_{\rho}(x) \leq - \rho(x) + \sum_{1 \leq \ell <e^x}\frac{\chi(\ell)}{\ell^{\alpha}} .$$ Or puisque $\rho$ est une fonction positive alors $$T_{\rho}(x) \leq \sum_{1 \leq \ell < e^x} \frac{\chi(\ell)}{\ell^{\alpha}} .$$ Donc $$\text{la série} \ \sum_{n \geq 1} \frac{\chi(n)}{n^{\alpha}} \ \text{converge} \ \Longrightarrow \rho \in \vb^* .$$ $\blacksquare$ Sans perte de généralité le résultat est vrai pour toute fonction arithmétique $\chi:\N^* \longrightarrow \R$ croissante. Dans ce cas, le <span style="font-variant:small-caps;">Lemme</span> \[L1\] peut être reformulé: $$\text{la série} \ \sum_{n \geq 1} \frac{\chi(n)}{n^{\alpha}} \ \text{est absolument convergente} \Longrightarrow \rho \in \vb^*.$$ où $\alpha > 1$. Maintenant on arrive au résultat le plus important dans cette section: ****** Soit $\alpha > 1$ un réel tel que la série du terme générale $\frac{\chi(n)}{n^{\alpha}}$ est convergente alors, d’après le <span style="font-variant:small-caps;">Lemme</span> \[L1\], la fonction $\rho(x) = f(e^x)e^{-\alpha x}$ est un élément de $\vb^*$. Or d’après le <span style="font-variant:small-caps;">Corollaire</span> \[C1\] on déduit que $$\rho(x) \underset{x \to + \infty}{\sim} Res(\L_{\rho},\beta) e^{\beta x} .$$ Ce qui est $$f(e^x) \underset{x \to + \infty}{\sim}Res(\L_{\rho},\beta) e^{(\alpha + \beta)x} .$$ D’où $$f(x) \underset{x \to + \infty}{\sim} Res(\L_{\rho},\beta) x^{\alpha + \beta} .$$ Ce qu’il fallait démontrer. $\blacksquare$ On rappelle que la fonction $\Lambda$ de **Von Mangoldt** est une fonction arithmétique définie sur $\N^*$ par $$\Lambda(n) := \begin{cases} \log p \quad \text{si} \ n=p^k , \quad k \in \N^*,p\in \P \\ \\ \quad 0 \qquad \text{sinon} \end{cases}.$$ La fonction définie pour tout $x \in [1,+ \infty)$ tel que $x \neq p^k$ où $k\in \N^*$ et $p\in \P$ par $ \displaystyle \psi(x) = \sum_{n < x} \Lambda(n)$ est dite la fonction de **Chebyshev**, ainsi pour démontrer le théorème des nombres premiers il faut et il suffit de démontrer que $$\psi(x) \underset{x\to + \infty}{\sim} x .$$ Il existe une forte relation entre la fonction $\zeta$ de **Riemann** et la fonction $\psi$, en effet $$-\frac{\zeta'(s)}{\zeta(s)} = \sum_{n \geq 1} \frac{\Lambda(n)}{n^s} = s \int_0^{+ \infty} \psi(e^x)e^{-sx}dx, \qquad \forall \sigma > 1.$$ On rappelle aussi que la fonction $\zeta$ est holomorphe sur $\{\sigma \geq 1\}$ sauf au $s=1$ qui est le seul pôle simple de la fonction $\zeta$, ainsi d’après **Hadamard** et **De La Vallée Poussin** la fonction $\zeta$ ne s’annulle en aucun point du demi-plan $\{\sigma \geq 1 \}$. Alors on déduit que la fonction $-\frac{\zeta'}{\zeta}$ est holomorphe sur $\{\sigma \geq 1 \}$ sauf au point $s=1$ qui est le seul pôle simple de résidu égal à $1$. Et on a le **Théorème des Nombres Premiers**: ****** Soit $\alpha > 1$ un réel donné, on pose $$\rho(x) = \psi(e^x)e^{-\alpha x}, \qquad \forall x \in \R^+.$$ Alors puisque la fonction $- \frac{\zeta'}{\zeta}$ est holomorphe sur $\{\sigma > 1\}$ alors la série su terme général $\frac{\Lambda(n)}{n^{\alpha}}$ est convergente pour tout $\alpha > 1$. D’une autre part, soit $\sigma > 1$ alors $$\begin{aligned} \L_{\rho}(s) &= \int_0^{+ \infty}\rho(x)e^{-sx}dx \\ &= \int_0^{+\infty} \psi(e^x)e^{-\alpha x} e^{-sx}dx \\ &= \int_0^{+ \infty} \psi(e^x)e^{-(s+\alpha)x}dx \\&= - \frac{\zeta'(s+\alpha)}{(s+\alpha)\zeta(s+\alpha)} \end{aligned}$$ Alors puisque la fonction $s\mapsto - \frac{\zeta'(s+\alpha)}{(s+\alpha)\zeta(s+\alpha)}$ est holomorphe sur $\{\sigma \geq 1-\alpha\}$ sauf au point $s=1-\alpha$ qui est le seul pôle simple de cette fonction où $$Res\left(- \frac{\zeta'(s+\alpha)}{(s+\alpha)\zeta(s+\alpha)},1-\alpha \right)=1$$ Alors d’après le <span style="font-variant:small-caps;">Théorème</span> \[T2\] on a $$\psi(x) \underset{x \to + \infty}{\sim}x^{\alpha + 1 - \alpha}$$ C’est à dire $$\psi(x) \underset{x \to + \infty}{\sim} x .$$ Ce qu’il fallait démontrer. $\blacksquare$ [plain]{} M.Carter et B. Van Brunt, *The Lebesgue-Stieltjes Integral* a practical introduction, Springer (2000). E.C. Titchmarsh, *The Theory of The Riemann Zeta-Function* 2nd ed, revised by D. R. Heath-Brown, Oxford University Press (1986). Walter Rudin, *Analyse réelle et complexe*, Troisième tirage MASSON Paris New York Barcelone Milan 1980.

--- abstract: | The notion of the geometrical $\Z/2 \oplus \Z/2$–control of self-intersection of a skew-framed immersion and the notion of the $\Z/2 \oplus \Z/4$-structure (the cyclic structure) on the self-intersection manifold of a $\D_4$-framed immersion are introduced. It is shown that a skew-framed immersion $f:M^{\frac{3n+q}{4}} \looparrowright \R^n$, $0 < q <<n$ (in the $\frac{3n}{4}+\varepsilon$-range) admits a geometrical $\Z/2 \oplus \Z/2$–control if the characteristic class of the skew-framing of this immersion admits a retraction of the order $q$, i.e. there exists a mapping $\kappa_0: M^{\frac{3n+q}{4}} \to \RP^{\frac{3(n-q)}{4}}$, such that this composition $I \circ \kappa_0: M^{\frac{3n+q}{4}} \to \RP^{\frac{3(n-q)}{4}} \to \RP^{\infty}$ is the characteristic class of the skew-framing of $f$. Using the notion of $\Z/2 \oplus \Z/2$-control we prove that for a sufficiently great $n$, $n=2^l-2$, an arbitrary immersed $\D_4$-framed manifold admits in the regular cobordism class (modulo odd torsion) an immersion with a $\Z/2 \oplus \Z/4$-structure. In the last section we present an approach toward the Kervaire Invariant One Problem. author: - 'P.M.Akhmet’ev [^1]' title: 'Geometric approach towards stable homotopy groups of spheres. The Kervaire invariant' --- \[section\] \[theorem\][Лемма]{} \[theorem\][Предложение]{} \[theorem\][Следствие]{} \[theorem\][Гипотеза]{} \[theorem\][Проблема]{} \[theorem\][Определение]{} \[theorem\][Замечание]{} Ø[[**O**]{}]{} Self-intersection of immersions and Kervaire Invariant ====================================================== The Kervaire Invariant One Problem is an open problem in Algebraic topology, for algebraic approach see \[B-J-M\], \[C-J-M\]. We will consider a geometrical approach; this approach is based on results by P.J.Eccles, see \[E1\]. For a geometrical approach see also \[C1\],\[C2\]. Let $f: M^{n-1} \looparrowright \R^n$, $n= 2^l -2$, $l>1$, be a smooth (generic) immersion of codimension 1. Let us denote by $g: N^{n-2} \looparrowright \R^n$ the immersion of self-intersection manifold. ### Definition 1 {#definition-1 .unnumbered} The Kervaire invariant of $f$ is defined as $$\Theta(f) = <w_2^{\frac{n-2}{2}}; [N^{n-2}] >,$$ where $w_2 = w_2(N^{n-2})$ is the normal Stiefel-Whitney of $N^{n-2}$. $$$$ The Kervaire invariant is an invariant of the regular cobordism class of the immersion $f$. Moreover, the Kervaire invariant is a well-defined homomorphism $$\Theta: Imm^{sf}(n-1,1) \to \Z/2. \eqno(1)$$ The normal bundle $\nu(g)$ of the immersion $g: N^{n-2} \looparrowright \R^n$ is a 2-dimensional bundle over $N^{n-2}$ equipped with a $\D_4$–framing. The classifying mapping $\eta: N^{n-2} \to K(\D_4,1)$ of this bundle is well-defined. The $\D_4$-structure of the normal bundle or the $\D_4$–framing is the prescribed reduction of the structure group of the normal bundle of the immersion $g$ to the group $\D_4$ corresponding to the mapping $\eta$. The pair $(g,\eta)$ represents an element in the cobordism group $Imm^{\D_4}(n-2,2)$. The homomorphism $$\delta: Imm^{sf}(n-1,1) \to Imm^{\D_4}(n-2,2) \eqno(2)$$ is well-defined. Let us recall that the cobordism group $Imm^{sf}(n-k,k)$ generalizes the group $Imm^{sf}(n-1,1)$. This group is defined as the cobordism group of triples $(f,\Xi,\kappa)$, where $f: M^{n-k} \looparrowright \R^n$ is an immersion with the prescribed isomorphism $\Xi: \nu(g) \cong k \kappa$, called a skew-framing, $\nu(f)$ is the normal bundle of $f$, $\kappa$ is the given line bundle over $M^{m-k}$ with the characteristic class $w_1(\kappa) \in H^1(M^{m-k};\Z/2)$. The cobordism relation of triples is standard. The generalization of the group $Imm^{\D_4}(n-2,2)$ is following. Let us define the cobordism groups $Imm^{\D_4}(n-2k,2k)$. This group $Imm^{\D_4}(n-2k,2k)$ is represented by triples $(g,\Xi,\eta)$, where $g: N^{n-2k} \looparrowright \R^n$ is an immersion, $\Xi$ is a dihedral $k$-framing, i.e. the prescribed isomorphism $\Xi: \nu_g \cong k \eta$, where $\eta$ is a 2-dimensional bundle over $N^{n-2k}$. The characteristic mapping of the bundle $\eta$ is denoted also by $\eta: N^{n-2k} \to K(\D_4,1)$. The mapping $\eta$ is the characteristic mapping for the bundle $\nu_g$, because $\nu_g \cong k \eta$. Obviously, the Kervaire homomorphism (1) is defined as the composition of the homomorphism (2) with a homomorphism $$\Theta_{\D_4} : Imm^{\D_4}(n-2,2) \to \Z/2. \eqno(3)$$ The homomorphism (3) is called the Kervaire invariant for $\D_4$-framed immersed manifolds. The Kervaire homomorphisms are defined in a more general situation by a straightforward generalization of the homomorphisms (1) and (3): $$\Theta^k: Imm^{sf}(n-k,k) \to \Z/2, \eqno(4a)$$ $$\Theta^k_{\D_4} : Imm^{\D_4}(n-2k,2k) \to \Z/2, \eqno(4b)$$ (for $k=1$ the new homomorphism coincides with the homomorphism (3) defined above) and the following diagram $$\begin{array}{ccccc} Imm^{sf}(n-1,1) & \stackrel {\delta}{\longrightarrow} & Imm^{\D_4}(n-2,2) & \stackrel{\Theta_{\D_4}}{\longrightarrow} & \Z/2 \\ \downarrow J^k & & \downarrow J^k_{\D_4} & & \vert \vert \\ Imm^{sf}(n-k,k) & \stackrel{\delta^k}{\longrightarrow} & Imm^{\D_4}(n-2k,2k) & \stackrel{\Theta_{\D_4}^k}{\longrightarrow} & \Z/2 \\ \end{array} \eqno(5)$$ is commutative. The homomorphism $J^k$ ($J^k_{\D_4}$) is determined by the regular cobordism class of the restriction of the given immersion $f$ ($g$) to the submanifold in $M^{n-1}$ ($N^{n-2}$) dual to $w_1(\kappa)^{k-1} \in H^{k-1}(M^{n-1};\Z/2)$ ($w_2(\eta)^{k-1} \in H^{2k-2}(N^{n-2};\Z/2)$). Let $(g,\Xi,\eta)$ be a $\D_4$-framed (generic) immersion in the codimension $2k$. Let $h: L^{n-4k} \looparrowright \R^n$ be the immersion of the self-intersection (double points) manifold of $g$. The normal bundle $\nu_h$ of the immersion $h$ is decomposed into a direct sum of $k$ isomorphic copies of a 4-dimensional bundle $\zeta$ with the structure group $\Z/2 \int \D_4$. This decomposition is given by the isomorphism $\Psi: \nu_h \cong k \zeta$. The bundle $\nu_h$ itself is classified by the mapping $\zeta: L^{n-4k} \to K(\Z/2 \int \D_4,1)$. All the triples $(h,\zeta,\Psi)$ described above (we do not assume that a triple is realized as the double point manifold for a $\D_4$-framed immersion) up to the standard cobordism relation form the cobordism group $Imm^{\Z/2 \int \D_4}(n-4k,4k)$. The self-intersection of an arbitrary $\D_4$-framed immersion is a $\Z/2 \int \D_4$-framed immersed manifold and the cobordism class of this manifold well-defines the natural homomorphism $$\delta_{\D_4}^k : Imm^{\D_4}(n-2k,2k) \to Imm^{\Z/2 \int \D_4}(n-4k,4k). \eqno(6)$$ The subgroup $\D_4 \oplus \D_4 \subset \Z/2 \int \D_4$ of index 2 induces the double cover $\bar L^{n-4k} \to L^{n-4k}$. This double cover corresponds with the canonical double cover over the double point manifold. Let $\bar \zeta: \bar L^{n-4k} \to K(\D_4,1)$ be the classifying mapping induced by the projection homomorphism $\D_4 \oplus \D_4 \to \D_4$ to the first factor. Let $\bar \zeta \to L^{n-4k}$ be the 2-dimensional $\D_4$–bundle defined as the pull-back of the universal 2-dimensional bundle with respect to the classifying mapping $\bar \zeta$. ### Definition 2 {#definition-2 .unnumbered} The Kervaire invariant $\Theta_{\Z/2 \int \D_4}^k: Imm^{\Z/2 \int \D_4}(n-4k,4k) \to \Z/2$ for a $\Z/2 \int \D_4$-framed immersion $(h,\Psi,\zeta)$ is defined by the following formula: $$\Theta_{\Z/2 \int \D_4}^k(h,\Psi,\zeta) = <w_2(\bar \eta)^{\frac{n-4k}{2}};[L^{n-4k}]>.$$ $$$$ This new invariant is a homomorphism $\Theta_{\Z/2 \int \D_4}^k: Imm^{\Z/2 \int \D_4}(n,n-4k) \to \Z/2$ included into the following commutative diagram: $$\begin{array}{ccc} Imm^{\D_4}(n-2k,2k) & \stackrel{\Theta_{\D_4}}{\longrightarrow} & \Z/2 \\ \downarrow \delta_{\D_4}^k & & \vert \vert \\ Imm^{\Z/2 \int \D_4}(n-4k,4k) & \stackrel{\Theta_{\Z/2 \int \D_4}^k}{\longrightarrow} & \Z/2. \\ \end{array} \eqno(7)$$ Let us formulate the first main results of the paper. In section 2 the notion of $\Z/2 \oplus \Z/2$-control ($\I_b$–control) on self-intersection of a skew-framed immersion is considered. Theorem 1 (for the proof see section 3) shows that under a natural restriction of dimensions the property of $\I_b$-control holds for an immersion in the regular cobordism class modulo odd torsion. In section 4 we formulate a notion of $\Z/2 \oplus \Z/4$–structure (or an $\I_4$–structure, or a cyclic structure) of a $\D_4$-framed immersion. In section 5 we prove Theorem 2. We prove under a natural restriction of dimension that an arbitrary $\D_4$-framed $\I_b$-controlled immersion admits in the regular homotopy class an immersion with a cyclic structure. For such an immersion Kervaire invariant is expressed in terms of $\Z/2 \oplus \Z/4$–characteristic numbers of the self-intersection manifold. The proof (based on the two theorems from \[A2\] (in Russian)) of the Kewrvaire Invariant One Problem is in section 6. The author is grateful to Prof. M.Mahowald (2005) and Prof. R.Cohen (2007) for discussions, to Prof. Peter Landweber for the help with the English translation, and to Prof. A.A.Voronov for the invitation to Minnesota University in (2005). This paper was started in 1998 at the Postnikov Seminar. This paper is dedicated to the memory of Prof. Yu.P.Soloviev. Geometric Control of self-intersection manifolds of skew-framed immersions ========================================================================== In this and the remining sections of the paper by $Imm^{sf}(n-k,k)$, $Imm^{\D_4}(n-2k,2k)$, $Imm^{\Z/2 \int \D_4}(n-4k,4k)$, etc., we will denote not the cobordism groups themselves, but the 2-components of these groups. In case the first argument (the dimension of the immersed manifold) is strictly positive, all the groups are finite 2-group. Let us recall that the dihedral group $\D_4$ is given by the representation (in terms of generators and relations) $\{a,b\vert a^4 = b^2 = e, [a,b]=a^2\}$. This group is a subgroup of the group $O(2)$ of isometries of the plane with the base $\{f_1,f_2\}$ that keeps the pair of lines generated by the vectors of the base. The element $a$ corresponds to the rotation of the plane through the angle $\frac{\pi}{2}$. The element $b$ corresponds to the reflection of the plane with the axis given by the vector $f_1 + f_2$. Let $\I_b(\Z/2 \oplus \Z/2)= \I_b \subset \D_4$ be the subgroup generated by the elements $\{a^2,b\}$. This is an elementary $2$-group of rank 2 with two generators. These are the transformations of the plane that preserve each line $l_1$, $l_2$ generated by the vectors $f_1+f_2$, $f_1-f_2$ correspondingly. The cohomology group $H^1(K(\I_b,1);\Z/2)$ is the elementary 2-group with two generators. The first (second) generator of this group detects the reflection of the line $l_2$ (of the line $l_1$) correspondingly. The generators of the cohomology group will be denoted by $\tau_1$, $\tau_2$ correspondingly. ### Definition 3 {#definition-3 .unnumbered} We shall say that a skew-framed immersion $(f,\Xi)$, $f: M^{n-k} \looparrowright \R^n$ has self-intersection of type $\I_b$, if the double-points manifold $N^{n-2k}$ of $f$ is a $\D_4$-framed manifold that admits a reduction of the structure group $\D_4$ of the normal bundle to the subgroup $\I_b \subset \D_4$. $$$$ Let us formulate the following conjecture. ### Conjecture {#conjecture .unnumbered} For an arbitrary $q > 0$, $q=2 (mod 4)$, there exists a positive integer $l_0=l_0(q)$, such that for an arbitrary $n = 2^l-2$, $l>l_0$ an arbitrary element $a \in Imm^{sf}(\frac{3n+q}{4},\frac{n-q}{4})$ is stably regular cobordant to a stably skew-framed immersion with $\I_b$-type of self-intersection (for the definition of stable framing see \[E2\], of stable skew-framing see \[A1\]). $$$$ Let us formulate and prove a weaker result toward the Conjecture. We start with the following definition. Let $\omega: \Z/2 \int \D_4 \to \Z/2$ be the epimorphism defined as the composition $\Z/2 \int \D_4 \subset \Z/2 \int \Sigma_4 \to \Sigma_4 \to \Z/2$, where $\Sigma_4 \to \Z/2$ is the parity of a permutation. Let $\omega^{!}: Imm^{\Z/2 \int \D_4}(n-4k,4k) \to Imm^{Ker \omega}(n-4k,4k)$ be the transfer homomorphism with respect to the kernel of the epimorphism $\omega$. Let $P$ be a polyhedron with $dim(P) < 2k-1$, $Q \subset P$ be a subpolyhedron with $dim(Q)=dim(P)-1$, and let $P \subset \R^n$ be an embedding. Let us denote by $U_P$ the regular neighborhood of $P \subset \R^n$ of the radius $r_P$ and by $U'_Q$ the regular neighborhood of $Q \subset \R^n$ of the radius $r_Q$, $r_Q > r_P$. Let us denote $U_Q = U_P \cap U'_Q$. The boundary $\partial U_P$ of the neighborhood $U_P$ is a codimension one submanifold in $\R^n$. This manifold $\partial U_P$ is a union of the two manifolds with boundaries $V_Q \cup_{\partial} V_P$, $V_Q = U_Q \cap \partial U_P$, $V_P = \partial U_P \setminus U_Q$ along the common boundary $\partial V_Q = \partial V_P$. Let us assume that the two cohomology classes $\tau_{Q,1} \in H^1(Q;\Z/2)$, $\tau_{Q,2} \in H^1(Q;\Z/2)$ are given. The projection $U_Q \to Q$ of the neighborhood on the central submanifold determines the cohomology classes $\tau_{U_Q,1}, \tau_{U_Q,2} \in H^1(U_Q;\Z/2)$ as the inverse images of the classes $\tau_{Q,1}, \tau_{Q,2}$ correspondingly. Let $(g,\Xi_N,\eta)$, $dim(N)=n-2k$ be a $\D_4$–framed generic immersion, $n-4k > 0$, and $g(N^{n-2k}) \cap \partial U_P$ be an immersed submanifold in $U_Q \subset \partial U_P$. Let us denote $g(N^{n-2k}) \setminus (g(N^{n-2k}) \cap (U_P))$ by $N^{n-2k}_{int}$, and the complement $N^{n-2k} \setminus N^{n-2k}_{int}$ by $N^{n-2k}_{ext}$. The manifolds $N^{n-2k}_{ext}$, $N^{n-2k}_{int}$ are submanifolds in $N^{n-2k}$ of codimension 0 with the common boundary, this boundary is denoted by $N_Q^{n-2k-1}$. The self-intersection manifold of $g$ is denoted by $L^{n-4k}$. By the dimensional reason ($n-4k=q<<n$) $L^{n-4k}$ is a submanifold in $\R^n$, parameterized by an embedding $h$, equipped by the $\Z/2 \int \D_4$-framing of the normal bundle denoted by $(\Psi, \zeta)$. The triple $(h,\Psi,\zeta)$ determines an element in the cobordism group $Imm^{\Z/2 \int \D_4}(n-4k,4k)$. ### Definition 4 {#definition-4 .unnumbered} We say that the $\D_4$–framed immersion $g$ is an $\I_b$–controlled immersion if the following conditions hold: –1. The structure group of the $\D_4$–framing $\Xi_N$ restricted to the submanifold (with boundary) $g(N^{n-2k}_{ext})$ is reduced to the subgroup $\I_b \subset \D_4$ and the cohomology classes $\tau_{U_Q,1}, \tau_{U_Q,2} \in H^1(U_Q;\Z/2)$ are mapped to the generators $\tau_1, \tau_2 \in H^1(N_Q^{n-2k-1};\Z/2)$ of the cohomology of the structure group of this $\I_b$-framing by the immersion $g \vert_{N^{n-2k-1}_Q} : N^{n-2k-1}_Q \looparrowright \partial(U_Q) \subset U_Q$. –2. The restriction of the immersion $g$ to the submanifold $N_Q^{n-2k-1} \subset N^{n-2k}$ is an embedding $g \vert_{N_Q^{n-2k-1}} : N_Q^{n-2k-1} \subset \partial U_Q$, and the decomposition $L^{n-4k} =L^{n-4k}_{int} \cup L^{n-4k}_{ext} \subset (U_P \cup \R^n \setminus U_P)$ of the self-intersection manifold of $g$ into two (probably, non-connected) $\Z/2 \int \D_4$-framed components is well-defined. The manifold $L^{n-4k}_{int}$ is a submanifold in $U_P$ and the triple $(L^{n-4k}_{int},\Psi_{int},\zeta_{int})$ represents an element in $Imm^{Ker \omega}(n-4k,4k)$ in the image of the homomorphism $\omega^!: Imm^{\Z/2 \int \D_4}(n-4k,4k) \to Imm^{Ker \omega}(n-4k,4k)$. $$$$ ### Definition 5 {#definition-5 .unnumbered} Let $(f,\Xi_M,\kappa) \in Imm^{sf}(n-k,k)$ be an arbitrary element, where $f: M^{n-k} \looparrowright \R^n$ is an immersion of codimension $k$ with the characteristic class $\kappa \in H^1(M^{n-k};\Z/2)$ of the skew-framing $\Xi_M$. We say that the pair $(M^{n-k},\kappa)$ admits a retraction of order $q$, if the mapping $\kappa : M^{n-k} \to \RP^{\infty}$ is represented by the composition $\kappa = I \circ \bar \kappa : M^{n-k} \to \RP^{n-k-q-1} \subset \RP^{\infty}$. The element $[(f, \Xi_M, \kappa)]$ admits a retraction of order $q$, if in the cobordism class of this skew-framed immersion there exists a triple $(M'^{n-k}, \Xi_{M'}, \kappa')$ that admits a retraction of order $q$. $$$$ ### Theorem 1 {#theorem-1 .unnumbered} Let $q = q(l)$ be a positive integer, $q=2(mod 4)$. Let us assume that an element $\alpha \in Imm^{sf}(\frac{3n+q}{4},\frac{n-q}{4})$ admits a retraction of the order $q$ and $3n-12k-4>0$. Then the element $\delta(\alpha) \in Imm^{\D_4}(n-2k,2k)$, $k=\frac{n-q}{4}$, is represented by a $\D_4$-framed immersion $[(g,\Psi_N,\eta)]$ with $\I_b$-control. $$$$ Proof of Theorem 1 ================== Let us denote $n-k-q-1 = 3k-1$ by $s$. Let $d: \RP^{s} \to \R^n$ be a generic mapping. We denote the self-intersection points of $d$ (in the target space) by $\Delta(d)$ and the singular points of $d$ by $\Sigma(d)$. Let us recall a classification of singular points of generic mappings $\RP^{s} \to \R^n$ in the case $4s < 3n$, for details see \[Sz\]. In this range generic mappings have no quadruple points. The singular values (in the target space) are of the following two types: – a closed manifold $\Sigma^{1,1,0}$; – a singular manifold $\Sigma^{1,0}$ (with singularities of the type $\Sigma^{1,1,0}$). The multiple points are of the multiplicities 2 and 3. The set of triple points form a manifold with boundary and with corners on the boundary. These “corner” singular points on the boundary of the triple points manifold coincide with the manifold $\Sigma^{1,1,0}$. The regular part of boundary of triple points is a submanifold in $\Sigma^{1,0}$. The double self-intersection points form a singular submanifold in $\R^n$ with the boundary $\Sigma^{1,0}$. This submanifold is not generic. After an arbitrary small alteration the double points manifold becomes a submanifold in $\R^n$ with boundary and with corners on the boundary of the type $\Sigma^{1,1,0}$. Let $U_{\Sigma}$ be a small regular neighborhood of the radius $\varepsilon_1$ of the singular submanifold $\Sigma^{1,0}$. Let $U_{\Delta}$ be a small regular neighborhood of the same radius of the submanifold $\Delta(d)$ (this submanifold is immersed with singularities on the boundary). The inclusion $U_{\Sigma} \subset U_{\Delta}$ is well-defined. Let us consider a regular submanifold in $\Delta$ obtained by excising a small regular neighborhood of the boundary. This immersed manifold with boundary will be denoted by $\Delta^{reg}$. The (immersed) boundary $\partial \Delta^{reg}$ will be denoted by $\Sigma^{reg}$. We will consider the pair of regular neighborhoods $U^{reg}_{\Sigma} \subset U^{reg}_{\Delta}$ of the pair $\Sigma^{reg} \subset \Delta^{reg}$ of the radius $\varepsilon_2$, $\varepsilon_2 << \varepsilon_1$. Because $2 dim(\Delta^{reg})<n$, after a small perturbation the manifold $\Delta^{reg}$ is a submanifold in $U^{reg}_{\Delta}$. Let $(f_0,\Xi_0,\kappa)$, $f_0: M^{n-k} \looparrowright \R^n$, $n-k=\frac{3n + q}{4}$ be a skew-framed immersion in the cobordism class $\alpha$. We will construct an immersion $f: M^{n-k} \looparrowright \R^n$ in the regular homotopy class of $f_0$ by the following construction. Let $\kappa_0: M^{n-k} \to \RP^{s}$ be a retraction of order $q$. Let $f: M \looparrowright \R^n$ be an immersion in the regular homotopy class of $f_0$ under the condition $dist(d \circ \kappa_0, f_0) < \varepsilon_3$. The caliber $\varepsilon_3$ of the approximation is given by the following inequality: $\varepsilon_3 << \varepsilon_2$. Let $g_1: N^{n-2k} \looparrowright \R^n$ be the immersion, parameterizing the double points of $f$. The immersion $g_1$ is not generic. After a small perturbation of the immersion $g_1$ with the caliber $\varepsilon_3$ we obtain a generic immersion $g_2: N^{n-2k} \looparrowright \R^n$. The immersed submanifold $g_2(N^{n-2k})$ is divided into two submanifolds $g_2(N^{n-2k}_{int})$, $g_2(N^{n-2k}_{ext})$ with the common boundary $g_2(\partial N^{n-2k}_{int}) = g_2(\partial N^{n-2k}_{ext})$ denoted by $g_2(N_Q^{n-2k-1})$. The manifold $g_2(N^{n-2k}_{int})$ is defined as the intersection of the immersed submanifold $g_2(N^{n-2k})$ with the neighborhood $U^{reg}_{\Delta}$. The manifold $g_2(N^{n-2k}_{ext})$ is defined as the intersection of the immersed submanifold $g_2(N^{n-2k})$ with the complement $\R^n \setminus (U^{reg}_{\Delta})$. We will assume that $g_2$ is regular along $\partial U^{reg}_{\Delta}$. Then $g_2(N^{n-2k}_Q)$ is an immersed submanifold in $\partial U^{reg}_{\Delta}$. By construction the structure group $\D_4$ of the normal bundle of the immersed manifold $g_2(N^{n-2k}_{ext})$ admits a reduction to the subgroup $\I_b \subset \D_4$. Let us denote by $L^{n-4k}$ the self-intersection manifold of the immersion $g_2$. This manifold is embedded into $\R^n$ by $h: L^{n-4k} \subset \R^n$. The normal bundle of this embedding $h$ is equipped with a $\Z/2 \int \D_4$-framing denoted by $\Psi_L$ and the characteristic class of this framing is denoted by $\zeta_L$. By the analogous construction the manifold $L^{n-4k}$ is decomposed as the union of the two manifolds over a common boundary, denoted by $\Lambda$: $L^{n-4k}=L^{n-4k}_{ext} \cup_{\Lambda} L^{n-4k}_{int}$. The manifold (with boundary) $L^{n-4k}_{int}$ is embedded by $h$ into $U^{reg}_{\Delta}$, the manifold $L^{n-4k}_{ext}$ (with the same boundary) is embedded in the complement $\R^n \setminus U^{reg}_{\Delta}$. The common boundary $\Lambda$ is embedded into $\partial U^{reg}_{\Delta}$. The manifold $L^{n-4k}$ is a $\Z/2 \int \D_4$-framed submanifold in $\R^n$. Let us describe the reduction of the structure group of this manifold to a corresponding subgroup in $\Z/2 \int \D_4$. We will describe the subgroups $\I_{2,j}(\Z/2 \oplus \D_4) \subset \Z/2 \int \D_4$, $j=x,y,z$. We will describe the transformations of $\R^4$ in the standard base $(f_1,f_2,f_3,f_4)$ determined by generators of the groups. Let us consider the subgroup $\I_{2,x}$. The generator $c_x$ (a generator will be equipped with the index corresponding to the subgroup) defines the transformation of the space by the following formula: $c_x(f_1)=f_3$, $c_x(f_3)=f_1$, $c_x(f_2)=f_4$, $c_x(f_4)=f_2$. For the generator $a_x$ (of the order 4) the transformation is the following: $a_x(f_1)=f_2$, $a_x(f_2)=-f_1$, $a_x(f_3)=f_4$, $a_x(f_4)=-f_3$. The generator $b_x$ (of order 2) defines the transformation of the space by the following formula: $b_x(f_1)=f_2$, $b_x(f_2)=f_1$, $b_x(f_3)=f_4$, $b_x(f_4)=f_3$. From this formula the subgroup $\D_4 \subset \D_4 \oplus \Z/2$ is represented by transformations that preserve the subspaces $(f_1,f_2)$, $(f_3,f_4)$. The generator of the cyclic subgroup $\Z/2 \subset \D_4 \oplus \Z/2$ permutes these planes. The subgroups $\I_{2,y}$ and $\I_{2,x}$ are conjugated by the automorphism $OP: \Z/2 \int \D_4 \to \Z/2 \int \D_4$ given in the standard base by the following formula: $f_1 \mapsto f_1$, $f_2 \mapsto f_3$, $f_3 \mapsto f_2$, $f_4 \mapsto f_4$. Therefore the generator $c_y \in \I_{2,y}$ is determined by the following transformation: $c_y(f_1)=f_2$, $c_y(f_2)=f_1$, $c_y(f_3)=f_4$, $c_y(f_4)=f_3$. The generator $a_y$ (of the order 4) is given by $a_y(f_1)=f_3$, $a_y(f_3)=-f_1$, $a_y(f_2)=f_4$, $a_y(f_4)=-f_2$. The generator $b_y$ (of the order 2) is given by $b_y(f_1)=f_3$, $b_y(f_3)=f_1$, $b_y(f_2)=f_4$, $b_y(f_4)=f_2$. Let us describe the subgroup $\I_{2,z}$. In this case the generator $c_z$ defines the transformation of the space by the following formula: $c_z(f_i)=-f_i$, $i=1,2,3,4$. For the generator $a_z$ (of order 4) the transformation is the following: $a_z(f_1)=f_2$, $a_z(f_2)=f_3$, $a_z(f_3)=f_4$, $a_z(f_4)=f_1$. The generator $b_x$ (of the order 2) defines the transformation of the space by the following formula: $b_z(f_1)=f_2$, $b_z(f_2)=f_1$, $b_z(f_3)=f_4$, $b_z(f_4)=f_3$. Obviously, the restriction of the epimorphism $\omega: \Z/2 \int \D_4 \to \Z/2$ to the subgroups $\I_{2,x}, \I_{2,y} \subset \Z/2 \int \D_4$ is trivial and the restriction of this homomorphism to the subgroup $\I_{2,z}$ is non-trivial. The subgroup $\I_3 \subset \I_{2,x}$ is defined as the subgroup with the generators $c_x, b_x, a_x^2$. This is an index 2 subgroup isomorphic to the group $\Z/2^3$. The image of this subgroup in $\Z/2 \int \D_4$ coincides with the intersection of arbitrary pair of subgroups $\I_{2,x}$, $\I_{2,y}$, $\I_{2,z}$. The subgroup $\I_3 \subset \I_{2,y}$ is generated by $c_y, b_y, a^2_y$. Moreover, one has $c_y=b_x$, $b_y=c_x$, $a^2_y=a^2_x$. It is easy to check that the following relations hold: $c_z=a^2_x$, $a^2_z=c_x=b_y$, $b_z=b_x=c_y$. Therefore $Ker(\omega \vert_{\I_2,z})$ coincides with the subgroup $\I_3 \subset \I_{2,z}$. The subgroups $\I_{2,x}, \I_{2,y}, \I_{2,z}, \I_3$ in $\Z/2 \int \D_4$ are well-defined. There is a natural projection $\pi_b: \I_3 \to \I_b$. We will also consider the subgroup $\I_{2,x \downarrow} \subset \Z/2 \int \D_4$ from geometrical considerations. This subgroup is a quadratic extension of the subgroup $\I_{2,x}$ such that $\I_{2,x}=Ker \omega \vert_{\I_{2,x \downarrow}} \subset \I_{2,x\downarrow}$. An algebraic definition of this group will not be required. In the following lemma we will describe the structure group of the framing of the triad $(L^{n-4k}_{int} \cup_{\Lambda} L^{n-4k}_{ext})$. The framings of the spaces of the triad will be denoted by $(\Psi_{\int} \cup_{\Psi_\Lambda} \cup \Psi_{ext}, \zeta_{int} \cup_{\zeta_\Lambda} \cup \zeta_{ext})$. ### Lemma 1 {#lemma-1 .unnumbered} There exists a generic regular deformation $g_1 \to g_2$ of the caliber $3 \varepsilon_3$ such that the immersed manifold $g_2(N^{n-2k}_{ext})$ admits a reduction of the structure group of the $\D_4$-framing to the subgroup $\I_b \subset \D_4$. The manifold $L^{n-4k}_{int}$ is divided into the disjoint union of the two manifolds (with boundaries) denoted by $(L^{n-4k}_{int,x \downarrow}, \Lambda_{x \downarrow})$, $(L^{n-4k}_{int,y}, \Lambda_{y})$. 1\. The structure group of the framing $(\Psi_{int,x \downarrow}, \Psi_{\Lambda_{x \downarrow}})$ for the submanifold (with boundary) $(L^{n-4k}_{int,x \downarrow},\Lambda_{x \downarrow})$ is reduced to the subgroups $(\I_{2,x\downarrow},\I_{2,z})$. (In particular, the 2-sheeted cover over $L^{n-4k}_{int,x \downarrow}$, classified by $\omega$ (denoted by $\tilde L^{n-4k}_{int,x} \to L^{n-4k}_{int,x \downarrow}$) is, generally speaking, a non-trivial cover.) 2\. The structure group of the framing $(\Psi_{int,y}, \Psi_{\Lambda})$ for the submanifold (with boundary) $(L^{n-4k}_{int,y},\Lambda_y)$ is reduced to the subgroup $(\I_{2,y},\I_3)$. (In particular, the 2-sheeted cover $\tilde L^{n-4k}_{int,y} \to L^{n-4k}_{int,y}$ classified by $\omega$, is the trivial cover.) Moreover, the double covering $\tilde L^{n-4k}_{x}$ over the component $L^{n-4k}_{x \downarrow}$ is naturally diffeomorphic to $\tilde L^{n-4k}_y$ and this diffeomorphism agrees with the restriction of the automorphism $OP: \Z/2 \int \D_4 \to \Z/2 \int \D_4$ on the subgroup $\I_{2,x}$, $OP(\I_{2,x})=\I_{2,y}$. 3\. The structure group of the framing $(\Psi_{ext}, \zeta_{ext})$ for the submanifold (with boundary) $h(L^{n-4k}_{ext}, \Lambda^{n-4k}) \subset (\R^n \setminus U^{reg}_{\Delta}, \partial(U^{reg}_{\Delta}))$ is reduced to the subgroup $\I_{2,z}$. (In particular, the 2-sheeted cover $\tilde L^{n-4k}_{ext} \to L^{n-4k}_{ext}$ classified by $\omega$, is, generally speaking, a nontrivial cover.) $$$$ ### Proof of Lemma 1 {#proof-of-lemma-1 .unnumbered} Components of the self-intersection manifold $g_1(N^{n-2k}) \setminus (g_1(N^{n-2k}) \cap U_{\Sigma})$ (this manifold is formed by double points $x \in g_1(N^{n-2k}), x \notin U_{\Sigma}$ with inverse images $\bar x_1, \bar x_2 \in M^{n-k}$) are classified by the following two types. Type 1. The points $\kappa(\bar x_1)$, $\kappa(\bar x_2)$ in $\RP^s$ are $\varepsilon_2$-close. Type 2. The distances between the points $\kappa(\bar x_1)$, $\kappa(\bar x_2)$ in $\RP^s$ are greater then the caliber $\varepsilon_2$ of the regular approximation. Points of this type belong to the regular neighborhood $U_{\Delta}$ (of the radius $\varepsilon_1$). Let us classify components of the triple self-intersection manifold $\Delta_3(f)$ of the immersion $f$. The a priori classification of components is the following. A point $x \in \Delta_3(f)$ has inverse images $\bar x_1, \bar x_2, \bar x_3$ in $M^{n-k}$. Type 1. The images $\kappa(\bar x_1), \kappa(\bar x_2), \kappa(\bar x_3)$ are $\varepsilon_2$-close in $\RP^s$. Type 2. The images $\kappa(\bar x_1), \kappa(\bar x_2)$ are $\varepsilon_2$-close in $\RP^s$ and the distance between the images $\kappa(\bar x_3)$ and $\kappa(\bar x_1)$ (or $\kappa(\bar x_2)$) are greater than the caliber $\varepsilon_2$ of the approximation. Type 3. The pairwise distances between the points $\kappa(\bar x_1), \kappa(\bar x_2), \kappa(\bar x_3)$ greater than the caliber $\varepsilon_2$ of the approximation. By a general position argument the component of the type 3 does not intersect $d(\RP^s)$. Therefore the immersion $f$ can be deformed by a small $\varepsilon_2$-small regular homotopy inside the $\varepsilon_3$-regular neighborhood of the regular part of $d(\RP^s)$ such that after this regular homotopy $\Delta_3(f)$ is contained in the complement of $U^{reg}_{\Delta}$. The codimension of the submanifold $\bar \Delta_2(d) \subset \RP^s$ is equal to $n-3k+1=q+k+1$ and greater then $dim(\Delta_3(f)) = n-3k$. By analogical arguments the component of triple points of the type 1 is outside $U^{reg}_{\Delta}$. Let us classify components of the quadruple self-intersection manifold $\Delta_4(f)$ of the immersion $f$. A point $x \in \Delta_4(f)$ has inverse images $\bar x_1, \bar x_2, \bar x_3, \bar x_4$ in $M^{n-k}$. The a priori classification is the following. Type 1. The images $\kappa(\bar x_1), \kappa(\bar x_2)$ are $\varepsilon_2$-close in $\RP^s$ and the pairwise distances between the images $\kappa(\bar x_1)$ (or $\kappa(\bar x_2)$), $\kappa(\bar x_3)$ and $\kappa(\bar x_4)$) are greater than the caliber $\varepsilon_2$ of the approximation. Type 2. The two pairs $(\kappa(\bar x_1), \kappa(\bar x_2))$ and $(\kappa(\bar x_3), \kappa(\bar x_4))$ of the images are $\varepsilon_2$-close in $\RP^s$ and the distance between the images $\kappa(\bar x_1)$ (or $\kappa(\bar x_2)$) and $\kappa(\bar x_3)$ (or $\kappa(\bar x_4)$) are greater than the calibre $\varepsilon_2$ of the approximation. (The described component is the complement of the regular $\varepsilon_2$ neighborhood of the triple points manifold of $d(\RP^s)$.) Type 3. Images $\kappa(\bar x_1), \kappa(\bar x_2)$ and $\kappa(\bar x_3)$ on $\RP^s$ are pairwise $\varepsilon_2$-close in $\RP^s$ and the distance between the images $\kappa(\bar x_1)$ (or $\kappa(\bar x_2)$, or $\kappa(\bar x_3)$) and $\kappa(\bar x_4)$ is greater than the caliber $\varepsilon_2$ of the approximation. Type 4. All the images $\kappa(\bar x_1), \kappa(\bar x_2)$, $\kappa(\bar x_3)$ and $\kappa(\bar x_4)$ are pairwise $\varepsilon_2$-close in $\RP^s$. Let us prove that there exists a generic $f$ such that the components of the type 1 and the type 3 are empty. For the component of the type 3 the proof is analogous to the proof for the component of the type 1. Let us prove that there exists a generic deformation $g_1 \to g_2$ with the caliber $3\varepsilon_3$ such that after this deformation in the neighborhood $U_{\Delta}^{reg}$ there are no self-intersection points of $g_2$ obtained by a generic resolution of triple points of $f$ of the types 1 and 2. Let us start with the proof for triple points of the type 1. For a generic small alteration of the immersion $g_2$ inside $U^{reg}_{\Delta}$ the points of the type 1 of the triple points manifold $\Delta_3(f)$ are perturbed into a component of the self-intersection points on $L^{n-4k}$. This component is classified by the following two subtypes: – Subtype [**a**]{}. Preimages of a point are $(\bar x_2, \bar x_1), (\bar x_2, \bar x'_1)$. —Subtype [**b**]{}. Preimages of a point are $(\bar x_1, \bar x'_1),(\bar x_1, \bar x_2)$. In the formula above the points with the common indeces have $\varepsilon_3$-close projections on the corresponding sheet of $d(\RP^s)$. The two points in a pair form a point on $N^{n-2k}$ and a couple of pairs forms a point on the component of $L^{n-4k}$. Let us prove that there exists a $2\varepsilon_3$-small regular deformation $g_1 \to g_2$, such that the component of $h(L^{n-4k}) \cap U^{reg}_{\Delta}$ of the subtype [**a**]{} is empty. Let $K^{s-k}$ be the intersection manifold of $f(M^{n-k})$ with $d(\RP^s)$ (this manifold is immersed into the regular part in $\RP^s$). By a general position argument, because $2s < n - 2k$, a generic perturbation $r \to r'$ of the immersion $r: K^{s-k} \looparrowright \RP^s \to \R^n$ is an embedding. Therefore there exists a $2\varepsilon_2$-small deformation of immersed manifold $r(K^{s-k}) \to r'(K^{s-k})$ in $\R^n$, such that the regular $\varepsilon_2$-neighborhood of the submanifold $r'(K^{s-k})$ has no self-intersection. The deformation of the immersed manifolds $r(K^{s-k}) \to r'(K^{s-k})$ is extended to the deformation of $g_1(N^{n-2k})$ in the regular neighborhoods of the constructed one-parameter family of immersed manifolds. After the described regular deformation the immersed manifold $g_2(N^{n-2k})$ has no self-intersection components of the subtype [**a**]{}. The case of the self-intersection of the subtype [**b**]{} is analogous. Let us describe a generic deformation $g_1 \to g_2$ with the support in $U^{reg}_{\Delta}$ that resolves self-intersection corresponding to quadruple points of $f$ of the type 2. This deformation could be arbitrarily small. After this deformation the component $\Delta_4(f)$ of the type 2 is resolved into two components of $L^{n-4k}$ of different subtypes. These two components will be denoted by $L^{n-4k}_x$, $L^{n-4k}_y$. The immersed submanifold $g_2(N^{n-2k}) \cap U^{reg}_{\Delta}$ is divided into two components. The first component is formed by pairs of points $(\bar x, \bar x')$ with the $3\varepsilon_3$-close images $(\kappa(\bar x), \kappa(\bar x')$ on $\RP^s$. This component is denoted by $g_2(N^{n-2k}_x)$. The last component of $g_2(N^{n-2k}) \cap U^{reg}_{\Delta}$ is denoted by $g_2(N^{n-2k}_y)$. This component is formed by pairs of points $(\bar x, \bar x')$ with the projections $(\kappa(\bar x), \kappa(\bar x'))$ on different sheets of $\RP^s$. The component $L^{n-4k}_{x\downarrow}$ is defined by pairs $(\bar x_1, \bar x'_1),(\bar x_2, \bar x'_2)$. The component $L^{n-4k}_{y}$ is defined by pairs $(\bar x_1, \bar x_2),(\bar x'_1, \bar x'_2)$. A common index of points in the pair means that the images of the points are $\varepsilon_3$-close on $\RP^s$. Each pair determines a point on $N^{n-2k}$ with the same image of $g_2$. It is easy to see that the component $L^{n-4k}_{x\downarrow}$ is the self-intersection of $g_2(N^{n-2k}_x)$ and the component $L^{n-4k}_{y}$ is the self-intersection of $g_2(N^{n-2k}_y)$. It is easy to see that the structure groups of the components agree with the corresponding subgroup described in the lemma. The component $L^{n-4k}_{x \downarrow}$ admits a reduction of the structure group to the subgroup $\I_{2,x \downarrow} \subset \Z/2 \int \D_4$. The component $L^{n-4k}_{y}$ admits a reduction of the structure group to the subgroup $\I_{2,y}$. Moreover, it is easy to see that the covering $\tilde L^{n-4k}_{x \downarrow}$ over $L^{n-4k}_x$ induced by the epimorphism $\omega: \Z/2 \int \D_4 \to \Z/2$ with the kernel $\I_{2,x} \subset \Z/2 \int \D_4$ is naturally diffeomorphic to $L^{n-4k}_y$. Also it is easy to see that this diffeomorphism agrees with the transformation $OP$ of the structure groups of the framing over the components. The last component of $L^{n-4k}$ is immersed in the $\varepsilon_2$-neighborhood of $d(\RP^s)$ outside of $U^{reg}_{\Delta}$ and will be denoted by $L^{n-4k}_z$. The structure group of the framing of this component is $\I_{2,z}$. Lemma 1 is proved. ### The last part of the proof of the Theorem 1 {#the-last-part-of-the-proof-of-the-theorem-1 .unnumbered} Let us construct a pair of polyhedra $(P',Q') \subset \R^n$, $dim(P') =2s-n=n-2k-q-2$, $dim(Q') = dim(P')-1$. Obviously, $dim(P')< 2k -1$. Take a generic mapping $d': \RP^s \to \R^n$. Let us consider the submanifold with boundary $(\Delta'^{reg}, \partial \Delta'^{reg}) \subset \R^n$ (see the denotation in Lemma 1). Let $\eta_{\Delta'^{reg}}: (\Delta'^{reg},\partial \Delta'^{reg}) \to (K(\D_4,1),K(\I_b,1))$ be the classifying mapping for the double point self-intersection manifold of $d$. By a standard argument we may modify the mapping $d$ into $d'$ such that the mapping $\eta_{\Delta^{reg}}$ is a homotopy equivalence of pairs up to the dimension $q+1$. After this modification $d' \to d$ we define $(P,Q)=(\Delta^{reg},\partial \Delta^{reg}) \subset \R^n$ and the mapping $\eta_{\Delta^{reg}}$ is a $(q+1)$-homotopy equivalence. The subpolyhedron $Q$ is equipped with two cohomology classes $\kappa_{Q,1}, \kappa_{Q,2} \in H^1(Q;\Z/2)$. Because $\Sigma$ is a submanifold in $\RP^s$, the restriction of the characteristic class $\kappa \in H^1(\RP^s;\Z/2)$ to $H^1(\Sigma;\Z/2)$ is well-defined. The inclusion $i_Q: Q \subset U_{\Sigma}$ determines the cohomology class $(i_Q)^{\ast}(\kappa) \in H^1(Q;\Z/2)$. The cohomology class $\kappa_{Q,1}$ is defined as the characteristic class of the canonical double points covering over $\Sigma$. The class $\kappa_{Q,2}$ is defined by the formula $\kappa_{Q,2} = (i_Q)^{\ast}(\kappa) + \kappa_{Q,1}$. The immersed manifold (with boundary) $(N^{n-2k} \cap U_{\Sigma}) \looparrowright U_{\Sigma}$ is equipped with an $\I_b$-framing. Obviously the classes $\kappa_{Q,1}, \kappa_{Q,2} \in H^1(U_{\Sigma};\Z/2)=H^1(Q;\Z/2)$ restricted to $H^1(g_2(N^{n-2k}_{ext});\Z/2)$ ( recall that $g_2(N^{n-2k}_{ext})=g_2(N^{n-2k}) \cap (\R^n \setminus U_{\Delta})$) agree with the two generated cohomology classes $\rho_1, \rho_2$ of the $\I_b$-framing correspondingly. Let us define the immersion $g: N^{n-2k} \looparrowright \R^n$ with $\I_b$-control over $(P,Q)$. Let us start with the immersion $g_2: N^{n-2k} \looparrowright \R^n$ constructed in the lemma. By a $2\varepsilon_2$–small generic regular deformation we may deform the immersion $g_2$ into $g_3$, such that this deformation pushes the component $g_2(N^{n-2k}_x)$ out of $U^{reg}_{\Delta}$. Therefore the component $L^{n-4k}_{x \downarrow} \subset L^{n-4k}$ of the self-intersection of $g_2$ is also deformed out of $U^{reg}_{\Delta}$. The immersed manifold (with boundary) $g_3(N^{n-2k}) \cap (\R^n \setminus U^{reg}_{\Delta})$ is equipped with an $\I_b$-framing of the normal bundle. Obviously, the classes $\kappa_{Q,1}, \kappa_{Q,2} \in H^1(U_{\Sigma};\Z/2)=H^1(Q;\Z/2)$, restricted to $H^1(g_2(N^{n-2k}) \cap U_{\Delta};\Z/2)$, agree with the two generated cohomological classes of the $\I_b$-framing. The immersed manifold $g_3(N^{n-2k}) \cap U^{reg}_{\Delta}$ coincides with $g_2(N^{n-2k}_{y})$ and has the general structure group of the framing. This immersed manifold has the self-intersection manifold (with boundary) $h(L^{n-4k}) \cap U^{reg}_{\Delta}$ with the reduction of the structure group to the pair of the subgroups $(\I_{2,y},\I_3)$. Let us prove that the immersed manifold (with boundary) $h(L^{n-4k}) \cap U^{reg}_{\Delta}$ is $\Z/2 \int \D_4$-framed cobordant (relative to the boundary) to a $\Z/2 \int \D_4$-framed manifold decomposed into the disjoint union of a closed $\Z/2 \int \D_4$-framed manifold that is the image of the transfer homomorphism $\omega^!$ and a relative $\I_3$-framed manifold. Take a $\Z/2 \int \D_4$-framed manifold $(\tilde L^{n-4k}, \tilde \Psi, \tilde \zeta)$ that is defined as the image of $\Z/2 \int \D_4$-framed manifold $(L^{n-4k}, \Psi, \zeta)$ by the transfer homomorphism (a double covering) with respect to the cohomology class $\omega \in H^1(\Z/2 \int \D_4;\Z/2)$. Recall that the manifold $\tilde L^{n-4k}$ is obtained by gluing the manifold $\tilde L^{n-4k}_{x} \cup \tilde L^{n-4k}_y$ with the manifold $\tilde L^{n-4k}_z$ along the common boundary $\tilde \Lambda^{n-4k-1}$. Note that the group of the framing of the last manifold $\tilde \Lambda^{n-4k-1}_z$ is the subgroup $\I_3 \subset \Z/2 \int \D_4$. Let $OP\alpha$ be the $\Z/2 \int \D_4$–framed immersion obtained from an arbitrary $\Z/2 \int \D_4$-framed immersion $\alpha$ by changing the structure group of the framing by the transformation $OP$. The $\Z/2 \int \D_4$-framed manifold (with boundary) $(\tilde L^{n-4k}_y, \tilde \Psi_y, \tilde \zeta_y)$ coincides with the two disjoint copies of $\Z/2 \int \D_4$-framed manifold (with boundary) $OP(\tilde L^{n-4k}_y, \tilde \Psi_y, \tilde \zeta_y)$. Let us put $\alpha_1=-OP(\tilde L^{n-4k}, \tilde \Psi, \tilde \zeta)$. Let us define the sequence of $\Z/2 \int \D_4$-framed immersions $\alpha_2 = -2 OP\alpha_1$, $\alpha_3 = -2 OP\alpha_2$, $\dots$, $\alpha_j = -2 OP\alpha_{j-1}$. Obviously, the $\D/4 \int \Z/2$-framed immersion $\alpha_1 + \alpha_2 = \alpha_1 + 2OP\alpha_1^{-1}$ is represented by 3 copies of the manifold $\tilde L^{n-4k}$. The second and the third copies are obtained from the first copy by the mirror image and the changing of structure group of the framing. The manifold $-OP[\tilde L^{n-4k}] \cup 2[\tilde L^{n-4k}]$ contains, in particular, a copy of $-OP[\tilde L^{n-4k}_x]$ inside the first component and the union $[\tilde L^{n-4k}_y \cup L^{n-4k}_y]$ of the mirror two copies of $-OP[\tilde L^{n-4k}_x]$ in the second and the third component. Therefore the manifold $-OP[\tilde L^{n-4k}] \cup 2[ \tilde L^{n-4k}]$ is $\Z/2 \int \D_4$-framed cobordant to a $\Z/2 \int \D_4$-framed manifold, obtained by gluing the union of a copy of $-OP[\tilde L^{n-4k}_x]$ and 4 copies of $\tilde L^{n-4k}_y$ by a $\I_3$-framing manifold along the boundary. This cobordism is relative with respect to the submanifold $-OP[\tilde L^{n-4k}_z] \cup 2[L^{n-4k}_z] \subset -OP[L^{n-4k}] \cup 2L^{n-4k}$. By an analogous argument it is easy to prove that the element $\aleph= \sum_{j=1}^{j_0} \alpha_j$ is $\Z/2 \int \D_4$-framed cobordant to the manifold obtained by gluing the union $-OP[\tilde L^{n-4k}_x] \cup 2^j(-OP)^{j-1}[\tilde L^{n-4k}_y]$ by an $\I_3$-manifold along the boundary. Moreover, this cobordism is relative with respect to all copies of $\tilde L^{n-4k}_z$ (with various orientations). If $j_0$ is great enough, the manifold (with $\I_3$-framed boundary) $2^j(-OP)^{j_0-1}[\tilde L^{n-4k}_y]$ is cobordant relative to the boundary to an $\I_3$-framed manifold. Therefore the manifold $L^{n-4k}_y$ is $\Z/2 \int \D_4$-framed cobordant relative to the boundary to the union of an $\I_3$-framed manifold with the same boundary and a closed manifold that is the double cover with respect to $\omega$ over a $\Z/2 \int \D_4$-framed manifold. This cobordism is realized as a cobordism of the self-intersection of a $\D_4$-framed immersion with support inside $U^{reg}_{\Delta}$. This cobordism joins the immersion $g_3$ with a $\D_4$–framed immersion $g_4$. After an additional deformation of $g_4$ inside a larger neighborhood of $\Delta^{reg}$ the relative $\I_b$-submanifold of the self-intersection manifold of $g_4$ is deformed outside of $U^{reg}_{\Delta}$. The $\D_4$-framed immersion obtained as the result of this cobordism admits an $\I_b$-control. The Theorem 1 is proved. An $\I_4$-structure (a cyclic structure) of a $\D_4$-framed immersion ====================================================================== Let us describe the subgroup $\I_4 \subset \Z/2 \int \D_4$. This subgroup is isomorphic to the group $\Z/2 \oplus \Z/4$. Let us recall that the group $ \Z/2 \int \D_4$ is the transformation group of $\R^4$ that permutes the $4$-tuple of the coordinate lines and two planes $(f_1, f_2)$, $(f_3, f_4)$ spanned by the vectors of the standard base $(f_1, f_2, f_3, f_4)$ (the planes can remin fixed or be permuted by a transformation). Let us denote the generators of $\Z/2 \oplus \Z/4$ by $l$, $r$ correspondingly. Let us describe the transformations of $\R^4$ given by each generator. Consider a new base $(e_1, e_2, e_3, e_4)$, given by $e_1=f_1+f_2$, $e_2=f_1-f_2$, $e_3=f_3+f_4$, $e_4=f_3-f_4$. The generator $r$ of order 4 is represented by the rotation in the plane $(e_2,e_4)$ through the angle $\frac{\pi}{2}$ and the reflection in the plane $(e_1,e_3)$ with respect to the line $e_1+e_3$. The generator $l$ of order 2 is represented by the central symmetry in the plane $(e_1, e_3)$. Obviously, the described representation of $\I_4$ admits invariant (1,1,2)-dimensional subspaces. We will denote subspaces by $\lambda_1, \lambda_2, \tau$. The lines $\lambda_1, \lambda_2$ are generated by the vectors $e_1+e_3$, $e_1-e_3$ correspondingly. The subspace $\tau$ is generated by the vectors $e_2, e_4$. The generator $r$ acts by the reflection in $\lambda_2$ and by the rotation in $\tau$ throught the angle $\frac{\pi}{2}$. The generator $l$ acts by reflections in the subspaces $\lambda_1$, $\lambda_2$. In particular, if the structure group $\Z/2 \int \D_4$ of a 4-dimensional bundle $\zeta: E(\zeta) \to L$ admits a reduction to the subgroup $\I_4$, then the bundle is decomposed into the direct sum $\zeta = \lambda_1 \oplus \lambda_2 \oplus \tau$ of $1,1,2$–dimensional subbundles. ### Definition 6 {#definition-6 .unnumbered} Let $(g: N^{n-2k} \looparrowright \R^n, \Xi_N, \eta)$ be an arbitrary $\D_4$-framed immersion. We shall say that this immersion is an $\I_b$–immersion (or a cyclic immersion), if the structure group $\Z/2 \int \D_4$ of the normal bundle over the double points manifold $L^{n-4k}$ of this immersion admits a reduction to the subgroup $\I_4 \subset \Z/2 \int \D_4$. In this definition we assume that the pairs $(f_1, f_2)$, $(f_3,f_4)$ are the vectors of the framing for the two sheets of the self-intersection manifold at a point in the double point manifold $L^{n-4k}$. $$$$ In particular, for a cyclic $\Z/2 \int \D_4$-framed immersion there exists the mappings $\kappa_a: L^{n-4k} \to K(\Z/2,1)$, $\mu_a: L^{n-4k} \to K(\Z/4,1)$ such that the characteristic mapping $\zeta: L^{n-4k} \to K(\Z/2 \int \D/4,1)$ of the $\Z/2 \int \D_4$-framing of the normal bundle over $L^{n-4k}$ is reduced to a mapping with the target $K(\I_b,1)$ such that the following equation holds: $$\zeta = i(\kappa_a \oplus \mu_a),$$ where $i: \Z/2 \oplus \Z/4 \to \I_4$ is the prescribed isomorphism. The following Proposition is proved by a straightforward calculation. ### Proposition 2 {#proposition-2 .unnumbered} Let $(g,\Psi_N, \eta)$ be a $\D_4$–framed immersion, that is a cyclic immersion. Then the Kervaire invariant, appearing as the top line of the diagram (7), can be calculated by following formula: $$\Theta_a=<\kappa_a^{\frac{n-4k}{2}}\mu_a^{\ast}(\tau)^{\frac{n-4k-2}{4}} \mu_a^{\ast}(\rho);[L]>, \eqno(8)$$ where $\tau \in H^2(\Z/4;\Z/2)$, $\rho \in H^1(\Z/4;\Z/2)$ are the generators. $$$$ ### Proof of Proposition 2 {#proof-of-proposition-2 .unnumbered} Let us consider the subgroup of index 2, $\I_b \subset \I_4$. This subgroup is the kernel of the epimorphism $\chi': \I_4 \to \Z/2$, that is the restriction of the characteristic class $\chi: \Z/2 \int \D_4 \to \Z/2$ of the canonical double cover $\bar L \to L$ to the subgroup $\I_4 \subset \Z/2 \int \D_4$. Obviously, the characteristic number (8) is calculated by the formula $$\Theta_a=<\hat \kappa_a^{\frac{n-4k}{2}} \hat \rho_a^{\frac{n-4k}{2}};\bar L>, \eqno(9)$$ where the characteristic class $\hat \kappa_a \in H^1(\bar L;\Z/2)$ is induced from the class $\kappa_a \in H^1(L;\Z/2)$ by the canonical cover $\bar L \to L$, and the class $\hat \rho_a \in H^1(\bar L;\Z/2)$ is obtained by the transfer of the class $\rho \in H^1(L;\Z/4)$. Note that $\hat \kappa_a = \tau_1$, $\hat \rho_a = \tau_2$, where $\tau_1$, $\tau_2$ are the two generating $\I_b$–characteristic classes. Therefore $\hat \kappa_a \hat \rho_a = \tau_1 \tau_2 = w_2(\eta)$, where $\eta$ is the two-dimensional bundle that determines the $\D_4$–framing (over the submanifold $\bar L^{n-4k} \subset N^{n-2k}$ this framing admits a reduction to an $\I_b$-framing) of the normal bundle for the immersion $g$ of $N^{n-2k}$ into $\R^n$. Therefore the characteristic number, given by the formula (8) in the case when the $\Z/2 \int \D_4$ framing over $L^{n-4k}$ is reduced to an $\I_4$-framing, coincides with the characteristic number, given by the formula (9). Proposition 2 is proved. ### Definition 7 {#definition-7 .unnumbered} We shall say that a $\D_4$-framed immersion $(g,\Xi_N,\eta)$ admits a $\I_4$–structure (a cyclic structure), if for the double points manifold $L^{n-4k}$ of $g$ there exist mappings $\kappa_a: L^{n-4k} \to K(\Z/2,1)$, $\mu_a: L^{n-4k} \to K(\Z/4,1)$ such that the characteristic number (8) coincides with Kervaire invariant, see Definition 2. $$$$ ### Theorem 2 {#theorem-2 .unnumbered} Let $(g, \Psi, \eta)$ be a $\D_4$-framed immersion, $g: N^{n-2k} \looparrowright \R^n$, that represents a regular cobordism class in the image of the homomorphism $\delta: Imm^{sf}(n-k,k) \to Imm^{\D_4}(n-2k,2k)$, $n-4k=62$, $n=2^l-2$, $l \ge 13$, and assume the conditions of the Theorem 1 hold, i.e. the residue class $\delta^{-1}(Imm^{sf}(n-k,k)$ (this class is defined modulo odd torsion) contains a skew-framed immersion that admits a retraction of order $62$. Then in the $\D_4$-framed cobordism class $[(g, \Psi, \eta)] = \delta[(f, \Xi, \kappa)] \in Imm^{\D_4}(n-2k,2k)$ there exists a $\D_4$-framed immersion that admits an $\I_4$–structure (a cyclic structure). $$$$ Proof of Theorem 2 ================== Let us formulate the Geometrical Control Principle for $\I_b$–controlled immersions. Let us take an $\I_b$–controlled immersion (see Definition 4) $(g,\Xi_N,\eta;(P,Q),\kappa_{Q,1}, \kappa_{Q,2})$, where $g:N \looparrowright \R^n$ is a $\D_4$-framed immersion, equipped with a control mapping over a polyhedron $i_P: P \subset \R^n$, $dim(P)=2k-1$; $Q \subset P$ $dim(Q)=dim(P)-1$. The characteristic classes $\kappa_{Q,i} \in H^1(Q;\Z/2)$, $i=1,2$ coincide with characteristic classes $\kappa_{i,N_Q} \in N_Q^{n-2k-1}$ by means of the mapping $\partial N^{n-2k}_{int} = N^{n-2k}_Q \to Q$, where $N^{n-2k}_{int} \subset N^{n-2k}$, $N^{n-2k}_{int}=g^{-1}(U_P)$, $U_P \subset \R^n$. ### Proposition 3. Geometrical Control Principle for $\I_b$–controlled immersions {#proposition-3.-geometrical-control-principle-for-i_bcontrolled-immersions .unnumbered} Let $j_P : P \subset \R^n$ be an arbitrary embedding; such an embedding is unique up to isotopy by a dimensional reason, because $2dim(P)+1=4k-1<n$. Let $g_1: N^{n-2k} \to \R^n$ be an arbitrary mapping, such that the restriction $g_1 \vert_{N_{int}}: (N^{n-2k}_{int}, N^{n-2k-1}_Q) \looparrowright (U_P, \partial U_P)$ is an immersion (the restriction $g \vert_{N^{n-2k-1}_Q}$ is an embedding) that corresponds to the immersion $g \vert_{N^{n-2k}_{int}}: (N^{n-2k}_{int},N^{n-2k-1}_Q) \looparrowright (U_P, \partial U_P)$ by means of the standard diffeomorphism of the regular neighborhoods $U_{i_P}=U_{j_P}$ of subpolyhedra $i(P)$ and $j(P)$. (For a dimension reason there is a standard diffeomorphism of $U_{i_P}$ and $U_{j_P}$ up to an isotopy.) Then for an arbitrary $\varepsilon >0$ there exists an immersion $g_{\varepsilon}: N^{n-2k} \looparrowright \R^n$ such that $dist_{C^0}(g_1,g_{\varepsilon})<\varepsilon$ and such that $g_{\varepsilon}$ is regular homotopy to an immersion $g$ and the restrictions $g_{\varepsilon} \vert _{N^{n-2k}_{int}}$ and $g_1 \vert_{N^{n-2k}_{int}}$ coincide. $$$$ We start the proof of Theorem 2 with the following construction. Let us consider the manifold $Z=S^{\frac{n}{2}+64}/i \times \RP^{\frac{n}{2}+64}$. This manifold is the direct product of the standard lens space $(mod 4)$ and the projective space. The cover $p_Z: \hat Z \to Z$ over this manifold with the covering space $\hat Z = \RP^{\frac{n}{2}+64} \times \RP^{\frac{n}{2}+64}$ is well-defined. Let us consider in the manifold $Z$ a family of submanifolds $X_i$, $i=0, \dots, \frac{n+2}{64}$ of the codimension $\frac{n+2}{2}$, defined by the formulas $X_0 = S^{\frac{n}{2}+64}/i \times \RP^{63}$, $X_1= S^{\frac{n}{2}+32}/i \times \RP^{95}, \dots$, $X_j = S^{\frac{n}{2} - 32(j-2)-1}/i \times \RP^{32(j+2)-1}, \dots$, $X_{\frac{n+2}{64}} = S^{63}/i \times \RP^{\frac{n}{2}+64}$. The embedding of the corresponding manifold in $Z$ is defined by the Cartesian product of the two standard embeddings. The union of the submanifolds $\{X_i\}$ is a stratified submanifold (with singularities) $X \subset Z$ of the dimension $\frac{n}{2}+127$, the codimension of maximal singular strata in $X$ is equal to $64$. The covering $p_X: \hat X \to X$, induced from the covering $p_Z: \hat Z \to Z$ by the inclusion $X \subset Z$, is well-defined. The covering space $\hat X$ is a stratified manifold (with singularities) and decomposes into the union of the submanifolds $\hat X_0 = \RP^{\frac{n}{2}+64} \times \RP^{63}, \dots, \hat X_j = \RP^{\frac{n}{2} - 32(j-2)} \times \RP^{32(j+2) -1}, \dots, \hat X_{\frac{n+2}{64}}=\RP^{63} \times \RP^{\frac{n}{2}+64}$. Each manifold $\hat X_i$ of the family is the $2$-sheeted covering space over the manifold $X_i$ over the first coordinate. Let us define $d_1(j)= \frac{n}{2} - 32(j-2)$, $d_2(j)=32(j+2)-1$. Then the formula for $X_i$ is the following: $X_j=\RP^{d_1(j)} \times \RP^{d_2(j)}$. The cohomology classes $\rho_{X,1} \in H^1(X;\Z/4)$, $\kappa_{X,2}\in H^1(X;\Z/2)$ are well-defined. These classes are induced from the generators of the groups $H^1(Z;\Z/4)$, $H^1(Z;\Z/2)$. Analogously, the cohomology classes $\kappa_{\hat X,i}\in H^1(\hat X;\Z/4)$, $i=1,2$ are well-defined. The cohomology class $\kappa_{\hat X,1}$ is induced from the class $\rho_{X,1} \in H^1(X;\Z/4)$ my means of the transfer homomorphim, and $\kappa_{\hat X,2} = (p_X)^{\ast}(\kappa_{X,2})$. Let us define for an arbitrary $j= 0, \dots, (\frac{n+2}{64})$ the space $J_j$ and the mapping $\varphi_j: X_j \to J_j$. We denote by $Y_1(k)$ the space $S^{31}/i \ast \dots \ast S^{31}/i$ of the join of $k$ copies, $k=1, \dots ,(\frac{n+2}{64}+1)$, of the standard lens space $S^{31}/i$. Let us denote by $Y_2(k)$, $k=2, \dots, (\frac{n+2}{64}+2)$, $Y_2(k) = \RP^{31} \ast \dots \ast \RP^{31}$ the joins of the $k$ copies of the standard projective space $\RP^{31}$. Let us define $J_j =Y_1(\frac{n+2}{64}-j+2)) \times Y_2(j+2)$ $Q = Y_1(\frac{n+2}{64}+2) \times Y_2(\frac{n+2}{64}+2)$. For a given $j$ the natural inclusions $J_j \subset Q$ are well-defined. Let us denote the union of the considered inclusions by $J$. The mapping $\varphi_j: X_j \to J_j$ is well-defined as the Cartesian product of the two following mappings. On the first coordinate the mapping is defined as the composition of the standard 2-sheeted covering $\RP^{d_1(j)} \to S^{\frac{n}{2}-64(j-1)}/i$ and the natural projection $S^{d_1(j)}/i \to Y_1(d_1(j))$. On the second coordinate the mapping is defined by the natural projection $\RP^{d_2(j)} \to Y_2(j+1)$. The family of mappings $\varphi_j$ determines the mapping $\varphi: \hat X \to J$, because the restrictions of any two mappings to the common subspace in the origin coincide. For $n+2 \ge 2^{13}$ the space $J$ embeddable into the Euclidean $n$-space by an embedding $i_J: J \subset \R^n$. Each space $Y_1(k)$, $Y_2(k)$ in the family is embeddable into the Euclidean $(2^6k -1-k)$–space. Therefore for an arbitrary $j$ the space $J_j$ is embaddable into the Euclidean space of dimension $n+126-\frac{n+2}{64}$. In particular, if $n+2 \ge 2^{13}$ the space $J_j$ is embeddable into $\R^n$. The image of an arbitrary intersection of the two embeddings in the family belongs to the standard coordinate subspace. Therefore the required embedding $i_J$ is defined by the gluing of embeddings in the family. Let us describe the mapping $\hat h: \hat X \to \R^n$. By $\varepsilon$ we denote the radius of a (stratified) regular neighborhood of the subpolyhedron $i_J(J) \subset \R^n$. Let us consider a small positive $\varepsilon_1$, $\varepsilon_1 << \varepsilon$, (this constant will be defined below in the proof of Lemma 4) and let us consider a generic $PL$ $\varepsilon_1$–deformation of the mapping $i_J \circ \varphi: \hat X \to J \subset \R^n$. The result of the deformation is denoted by $\hat h: \hat X \to \R^n$. Let us define the positive integer $k$ from the equation $n-4k=62$. In the prescribed regular homotopy class of an $\I_b$-controlled immersion $f: N^{n-2k} \looparrowright \R^n$ we will construct another $\I_b$–controlled immersion $g: N^{n-2k} \looparrowright \R^n$ that admits a $\I_b$–structure. Let the immersion $f$ be controlled over the embedded subpolyhedron $\psi_P: P \subset \R^n$. Let $\psi_Q: Q \to \hat X$ be a generic mapping such that $\kappa_{Q,i} = \psi_Q \circ \kappa_{\hat X,i}$, $i=1,2$. By the previous definition the manifolds $N^{n-2k}_{int}$, $N^{n-2k}_{ext}$ with the common boundary $N^{n-2k-1}_Q$, $N^{n-2k} = N^{n-2k}_{int} \cup_{N^{n-2k-1}_Q} N^{n-2k}_{ext}$ are well-defined. Let $\eta: N^{n-2k}_{ext} \to K(\I_b,1) \subset K(\D_4,1)$ be the characteristic mapping of the framing $\Xi_N$, restricted to $N^{n-2k}_{ext} \subset N^{n-2k}$. The restriction of this mapping to the boundary $\partial N^{n-2k}_{ext}= N^{n-2k-1}_Q$ is given by the composition $\partial N^{n-2k-1}_Q \to Q \to K(\I_b,1) \subset K(\D_4,1)$. The target space for the mapping $\eta$ is the subspace $K(\I_b,1) \subset K(\D_4,1)$. This mapping is determined by the cohomology classes $\kappa_{N^{n-2k}_{ext},s} \in H^1(N^{n-2k}_{ext},Q;\Z/2)$, $s=1,2$. Let us define the mapping $\lambda: N^{n-2k}_{ext} \to \hat X$ by the following conditions. This mapping transforms the cohomology classes $\kappa_{\hat X,i}$ into the classes $\kappa_i \in H^1(N^{n-2k}_{ext};\Z/2)$ and also the restriction $\lambda \vert_{N^{n-2k-1}_Q}$ coincides with the composition of the projection $N^{n-2k-1}_{Q} \to Q$ and the mapping $\psi_Q: Q \to \hat X$. The boundary conditions for the mapping $\psi_Q$ are $\kappa_{Q,i} = \psi_Q \circ \kappa_{\hat X,i}$, $i=1,2$. The submanifold with singularities $\hat X \subset \hat Z$ contains the skeleton of the space $\hat Z$ of the dimension $\frac{n}{2}+62$. Because $n-2k=\frac{n}{2}+31$, the mapping $\lambda$ is well-defined. Let us denote the composition $\hat h \circ \lambda: N^{n-2k}_{ext} \to \hat X \to \R^n$ by $g_1$. Let us denote the mapping $ \hat h \circ \psi_Q: Q \to \hat X \to \R^n$ by $ \varphi_Q$. One can assume that the mapping $\varphi_Q$ is an embedding. Moreover, without loss of generality one may assume that this embedding is extended to a generic embedding $\varphi_P: P \subset \R^n$ such that the embedded polyhedron $\varphi_P: P \subset \R^n$ does not intersect $g_1(N^{n-2k}_{ext})$. Let us denote by $U_{\varphi}(P)$ a regular neighborhood of the subpolyhedron $\varphi_P(P) \subset \R^n$ (we may assume that the radius of this neighborhood is equal to $\varepsilon$). Up to an isotopy a regular neighborhood $U_{\varphi}(P)$ is well-defined, in particular, this neighborhood does not depend on the choice of a regular embedding of $P$, moreover $U_{\varphi}(P)$ and $U(P)$ are diffeomorphic. Without loss of generality after an additional small deformation we may assume that the restriction $g_1 \vert_{N^{n-2k}_{int}}$ is a regular immersion $g_1: N^{n-2k}_{int} \subset \R^n$ with the image inside $U_{\varphi}(P)$. In particular, the restriction of $g_1$ to the boundary $N^{n-2k-1}_Q = \partial(N^{n-2k}_{int})$ is a regular embedding $N^{n-2k-1}_Q \subset \partial U(P)$. The immersion $g_1 \vert_{N_{int}}$ is conjugated to the immersion $f \vert_{N_{int}}$ by means of a diffeomorphism of $U_{\varphi}(P)$ with $U(P)$. By Proposition 3, for an arbitrary $\varepsilon_2 >0$, $\varepsilon_2 << \varepsilon_1 << \varepsilon$, there exists an immersion $g : N^{n-2k} \looparrowright \R^n$ in the regular homotopy class of $f$, such that $g$ coincides with $g'$ (and with $g_1$) on $N^{n-2k}_{int}$ and, moreover, $dist(g,g_1)< \varepsilon_2$. Let us consider the self-intersection manifold $L^{n-4k}$ of the immersion $g$. This manifold is a submanifold in $\R^n$. Let us construct the mappings $\kappa_a: L^{n-4k} \to K(\Z/2,1)$, $\mu_a: L^{n-4k} \to K(\Z/4,1)$. Then we check the conditions (8) and (9). The manifold $L^{n-4k}$ is naturally divided into two components. The first component $L^{n-4k}_{int}$ is inside $U_{\varphi_P}(P)$. The last component (we will denote this component again by $L^{n-4k}$) consists of the last self-intersection points. This component is outside the $\varepsilon$–neighborhood of the submanifold with singularities $h(X)$. The mappings $\kappa_a$, $\mu_a$ over $L^{n-4k}_{int}$ are defined as the trivial mappings. Let us define the mappings $\kappa_a$, $\mu_a$ on $L^{n-4k}$. Let us consider the mapping $\varphi: \hat X \to J$ and the singular set (polyhedron) $\Sigma$ of this mapping. This is the subpolyhedron $ \Sigma \subset \{ \hat X^{(2)}= \hat X \times \hat X \setminus \Delta_{\hat X} / T'\}$, where $T': \hat X^({2}) \to \hat X^{(2)}$– is the involution of coordinates in the delated product $\hat X^{(2)}$ of the space $\hat X$. The subpolyhedron (it is convenient to view this polyhedron as a manifold with singularities) $\Sigma$ is naturally decomposed into the union of the subpolyhedra $\Sigma(j)$, $j=0, \dots, \frac{n+2}{128}$. The subpolyhedron $\Sigma(j)$ is the singular set of the mapping $\varphi(j): \RP^{d_1(j)} \times \RP^{d_2(j)} \to S^{d_1(j)}/i \times \RP^{d_2(j)} \to J_j$. This subpolyhedron consists of the singular points of the mapping $\varphi$ in the inverse image $(\varphi)^{-1}(J_j)= \RP^{d_1(j)} \times \RP^{d_2(j)}$ of the subspace $J_j \subset J$. Let us consider the subspace $\Sigma^{reg} \subset \Sigma$, consisting of points on strata of length 0 (regular strata) and of length 1 (singular strata of the codimension 32) after the regular $\varepsilon_2$ –neighborhoods ($\varepsilon_2 << \varepsilon_1$) of the diagonal $\Delta^{diag}$ and the antidiagonal $\Delta^{antidiag}$ of $\Sigma^{reg}$ are cut out. The manifold with singularities $\Sigma^{reg}$ admits a natural compactification (closure) in the neighborhood of $\Delta^{diag}$ and $\Delta^{antidiag}$; the result of the compactification will be denoted by $K_{reg}$. The space $RK$, called the space of resolution of singularities, equipped with the natural projection $RK \to K_{reg}$ is defined by the analogous construction; see the short English translation of \[A1\], Lemma 7. The cohomology classes $\rho_{RK,1} \in H^1(RK;\Z/4)$, $\kappa_{RK,2} \in H^1(RK;\Z/2)$ are well-defined. The cohomology classes $\kappa_{K_{reg},1} \in H^1(RK;\Z/2)$, $\kappa_{RK,1} \in H^1(RK;\Z/2)$ are the images of the class $\kappa_{\Sigma,1} \in H^1(\Sigma;\Z/2)$ with respect to the inclusion $K_{reg} \subset \Sigma$ and the projection $RK \to K_{reg}$. The class classifies the transposition of the two non-ordered preimages of a point in the singular set. Let us consider the restrictions of the classes $\kappa_{K_{reg},1}, \kappa_{RK,1}, \kappa_{\Sigma,1}$ to neighborhoods of the diagonal and the antidiagonal. The natural projection $\Delta^{diag} \to \hat X$ is well-defined. The restrictions of the classes $\rho_1$ and $\kappa_2$ to neighborhoods of the diagonal coincide with the restrictions of the classes $\rho_{\hat X,1} \in H^1(\hat X;\Z/4)$, $\kappa_{\hat X,2} \in H^1(\hat X;\Z/2)$. (These classes $\rho_{\hat X,1}, \kappa_{\hat X,2}$ are extended to neighborhoods of the diagonal). Let us recall that the mapping $\hat h: \hat X \to \R^n$ is defined as the result of an $\varepsilon_1$–small regular deformation of the mapping $\hat X \to X \stackrel{h}{\longrightarrow} \R^n$. The singular set of the mapping $\hat h$ will be denoted by $\Sigma_{\hat h}$. This is a $128$–dimensional polyhedron, or a manifold with singularities in the codimensions $32,64,96,128$. Moreover, the inclusion $\Sigma_{\hat h} \subset \hat X^{(2)}$ is well-defined. The image of this inclusion is in the regular $\varepsilon_1$–small neighborhood of the singular polyhedron $\Sigma \subset X^{(2)}$. Let us denote by $\Sigma_{\hat h}^{reg}$ the part of the singular set after cutting out the regular $\varepsilon_1$–neighborhood of the points in singular strata of length at least 2 (of the codimension 64) and self-intersection points of all singular strata (these strata are also of the codimension 64). The boundary $\partial \Sigma_{\hat h}$ is a submanifold with singularities in $\hat X$ and therefore. by a general position argument, we may also assume that the boundary $\partial \Sigma_{\hat h}^{reg}$ is a regular submanifold with singularities in $\hat X$. Additionally, by general position arguments, the intersection of the image $Im(\lambda (N^{n-2k}_{ext}))$ inside the singular set $\Sigma_{\hat h}$ (this is a polyhedron of the dimension $62$) on $X$ are outside (with respect to the caliber $\varepsilon$) of the projection of the singular submanifold with singularities (this singular part is of the codimension $64$) in the complement of the regular submanifold with singularities $\Sigma_{\hat h}^{reg} \subset \Sigma_{\hat h}$. Therefore the image $Im(\lambda (N^{n-2k}_{ext}))$ is inside the regular part $\Sigma_{\hat h}^{reg} \subset \Sigma_{\hat h}$. Let us denote by $L^{62}_{cycl} \subset L^{62}$ the submanifold (with boundary) given by the formula $L^{62}_{cycl}= L^{62} \cap U_{\Sigma^{reg}}$. The mappings $\kappa_a$, $\rho_a$ are extendable from $U_{\Sigma^{reg}}$ to $L^{62}_{cycl} \subset L^{62}$. Let us prove that these mappings are extendable to mappings $\kappa_a: L^{62} \to K(\Z/2,1)$, $\rho_a: L^{62} \to K(\Z/4,1)$. The complement of thе submanifold $L^{62}_{cycl} \subset L^{62}$ is denoted by $L^{62}_{\I_3} = L^{62} \setminus L^{62}_{cycl}$. The submanifold $L^{62}_{\I_3}$ is a submanifold in the regular $\varepsilon$–neighborhood of $h(X) \subset \R^n$. Obviously, the structure group of the $\Z/2 \int \D_4$–framing of the normal bundle of the manifold (with boundary) $L^{62}_{\I_3}$ is reduced to the subgroup $\I_3 \subset \Z/2 \int \D_4$. Let us consider the mapping of pairs $\mu_a \times \kappa_a: (L^{62}_{cycl}, \partial L^{62}_{cycl}) \to (K(\Z/4,1) \times K(\Z/2,1),K(\Z/2,1) \times K(\Z/2,1))$. Let us consider the natural projection $\pi_b: \I_3 \to \I_b$. The extension of the mapping $\mu_a \times \kappa_a$ to the required mapping $L^{62} \to K(\Z/4,1) \times K(\Z/2,1)$ is given by the composition $L^{62}_{\I_3} \to K(\I_3,1) \stackrel{\pi_{b,\ast}}{\longrightarrow} K(\I_b,1) \subset K(\Z/4,1) \times K(\Z/2,1)$, where $\kappa_1 \in K(\I_b;\Z/2)$ determines the inclusion $K(\I_b,1) \subset K(\Z/2,1) \subset K(\Z/4,1)$. Let us formulate the results in the following lemma. ### Lemma 4 {#lemma-4 .unnumbered} –1. Let $n \ge 2^{13}-2$ and $k$, $n-4k=62$ satisfy the conditions of Theorem 1 (in particular, an arbitrary element in the group $Imm^{sf}(n-k,k)$ admits a retraction of the order $62$. Then for arbitrarily small positive numbers $\varepsilon_1$, $\varepsilon_2$, $\varepsilon_1 >> \varepsilon_2$ (the numbers $\varepsilon_1$, $\varepsilon_2$ are the calibers of the regular deformations in the construction of the $PL$–mapping $\hat h :\hat X \to \R^n$ and of the immersion $g: N^{n-2k} \looparrowright \R^n$ correspondingly) there exists the mapping $m_a =(\kappa_a \times \mu_a): \Sigma_{h}^{reg} \to K(\Z/4,1) \times K(\Z/2,1)$ under the following condition. The restriction $m_a \vert_{\partial \Sigma_h^{reg}}$ (by $\partial \Sigma_h^{reg}$ is denoted the part of the singular polyhedron consisting of points on the diagonal) has the target $K(\Z/2,1) \times K(\Z/2,1) \subset K(\Z/4,1) \times K(\Z/2,1)$ and is determined by the cohomological classes $\kappa_{\hat X,1}, \kappa_{\hat X,2}$. –2. The mappings $\kappa_a$, $\mu_a$ induces a mapping $(\mu_a \times \kappa_a): L^{62} \to K(\Z/4,1) \times K(\Z/2,1)$ on the self-intersection manifold of the immersion $g$. $$$$ Let us prove that the mapping $(\mu_a \times \kappa_a)$ constructed in Lemma 4 determines a $\Z/2 \oplus \Z/4$–structure for the $\D_4$–framed immersion $g$. We have to prove the equation (9). Let us recall that the component $L^{62}_{int}$ of the self-intersection manifold of the immersion $g$ is a $\Z/2 \int \D_4$–framed manifold with trivial Kervaire invariant: the corresponding element in the group $Imm^{\Z/2 \int \D_4}(62, n-62)$ is in the image of the transfer homomorphism. Therefore it is sufficient to prove the equation $$<m_a^{\ast}(\rho \tau^{15} t^{31});[L^{62}]>=\Theta,$$ or, equivalently, the equation $$<(\hat \rho_a^{31} \hat \kappa_a^{31});[\hat L^{62}]>=\Theta, \eqno(10)$$ where $\hat L \to L$ is the canonical cover over the self-intersection manifold, $\hat L \subset N^{n-2k}_{ext}$ is the canonical inclusion. By Herbert’s theorem (see \[A1\] for the analogous construction) we may calculate the right side of the equation by the formula $$<\eta^{\ast}(w_2(\I_b))^{\frac{n-2k}{2}};[N^{n-2k}_{ext}/\sim]>. \eqno(11)$$ In this formula by $N^{n-2k}_{ext}/\sim$ is denoted the quotient of the boundary $\partial N^{n-2k}_{ext}=N^{n-2k-1}_Q$ that is contracted onto the polyhedron $Q$ with the loss of the dimension. Note that the mapping $m_a \vert_{N^{n-2k-1}_Q}$ is obtained by the composition of the mapping $p_Q: N^{n-2k-1} \to Q$ with a loss of dimension with the mapping $Q \to K(\I_b,1)$, the last mapping is determined by the cohomology classes $\kappa_{i,Q} \in H^1(Q;\Z/2)$, $i=1,2$. Therefore, $m_{a\ast}([N^{n-2k}_{ext}/\sim]) \in H_{n-2k}(\I_b;\Z/2)$ is a permanent cycle and the integration over the cycle $[N^{n-2k}_{ext}/\sim]$ of the inverse image of the universal cohomology class in (11) is well-defined. It is convenient to consider the characteristic number $\Theta_a$ as the value of a homomorphism $H_{n-2k}(X;\Z/2) \to \Z/2$ on the cycle $\lambda_{\ast}[N^{n-2k}_{ext}/\sim] \in H_{n-2k}(X;\Z/2)$. This homomorphism is the result of the calculation of the characteristic class $w_2(\I_b) \in H^2(K(\I_b,1);\Z/2)$ on the prescribed cycle, i.e. on the image of the fundamental cycle $[N^{n-2k}_{ext}/\sim]$ with respect to the mapping $N^{n-2k}_{ext}/\sim \to \hat X \to K(\I_b,1)$. The cycle $\lambda_{\ast}[N^{n-2k}_{ext}/\sim] \in H_{n-2k}(X;\Z/2)$ is the modulo 2 reduction of an integral homology class. Therefore this cycle is given by a sum of fundamental classes of the product of the two odd-dimensional projective spaces, the sum of the dimensions of this spaces being equal to $n-2k$. Let us consider an arbitrary submanifold $S^{k_1}/i \times \RP^{k_2} \subset X$, $k_1 + k_2 = \frac{n}{2}+31$, $k_1, k_2$ being odd. Let us consider the cover $\RP^{k_1} \times \RP^{k_2} \to S^{k_1}/i \times \RP^{k_2}$ and the composition $\RP^{k_1} \times \RP^{k_2} \subset \hat X \stackrel{\hat h}{\looparrowright} \R^n$ after an $\varepsilon_1$–small generic perturbation. Let us denote this mapping by $s_{k_1,k_2}$. The self-intersection manifold of the generic mapping $s_{k_1,k_2}: \RP^{k_1} \times \RP^{k_2} \to \R^n$ is a manifold with boundary denoted by $\Lambda^{62}_{k_1,k_2}$. The mapping $$\mu_a \times \kappa_a : (\Lambda^{62}_{k_1,k_2}, \partial N^{n-2k}_{k_1,k_2}) \to (K(\Z/4,1)\times K(\Z/2,1), K(\Z/2,1) \times K(\Z/2,1))$$ is well-defined. The $61$-dimensional homology fundamental class $[\partial \Lambda]$ is integral, therefore the image of this fundamental class $(\mu_a \times \kappa_a)_{\ast}([\partial \Lambda^{62}_{k_1,k_2}]) \in H_{61}(K(\Z/4,1) \times K(\Z/2,1);\Z/2)$ is trivial for a dimensional reason. Therefore the homology class $$(\mu_a \times \kappa_a)_{\ast}([\Lambda^{62}_{k_1,k_2}, \partial \Lambda^{62}_{k_1,k_2}]) \in$$ $$H_{62}(K(\Z/4,1)\times K(\Z/2,1),K(\Z/2,1) \times K(\Z/2,1);\Z/2)$$ is well-defined. Let us consider the (permanent) homology class $$(\mu_a \times \kappa_a)^{!}_{\ast}([\bar \Lambda^{62}_{k_1,k_2}]) \in H_{62}(K(\Z/2,1)\times K(\Z/2,1);\Z/2), \eqno(12)$$ defined from the relative class above by the transfer homomorphism. To prove (10) it is sufficient to prove that the class (12) coincides with the characteristic class $$p_{\ast, b} \circ \hat \eta_{\ast}([\hat \Lambda]) \in H_{62}(K(\I_b,1);\Z/2)$$ under the following isomorphism of the target group $\I_b = \Z/2 \oplus \Z/2$. By this isomorphism the prescribed generators in $H^1(\Z/2 \oplus \Z/2;\Z/2)$ are identified with the cohomology classes $\tau_{1}, \tau_2 \in H^1(K(\I_b,1);\Z/2)$ (compare with Lemma 8 in \[A1\]). Theorem 2 is proved. Kervaire Invariant One Problem ============================== In this section we will prove the following theorem. ### Main Theorem {#main-theorem .unnumbered} There exists an integer $l_0$ such that for an arbitrary integer $l \ge l_0$, $n=2^l-2$ the Kervaire invariant given by the formula (1) is trivial. $$$$ ### Proof of Main Theorem {#proof-of-main-theorem .unnumbered} Take the integer $k$ from the equation $n-4k=62$. Consider the diagram (5). By the Retraction Theorem \[A2\], Section 8 there exists an integer $l_0$ such that for an arbitrary integer $l \ge l_0$ an arbitrary element $[(f,\Xi,\kappa)]$ in the 2-component of the cobordism group $Imm^{sf}(\frac{3n+q}{4},\frac{n-q}{4})$ admits a retraction of order $62$. By Theorem 2 in the cobordism class $\delta[(f,\Xi,\kappa)]$ there exists a $\D_4$-framed immersion $(g,\Psi,\eta)$ with an $\I_4$-structure. Take the self-intersection manifold $L^{62}$ of $g$ and let $L_0^{10} \subset L^{62}$ be the submanifold dual to the cohomology class $\kappa_a^{28}\mu_a^{\ast}(\tau)^{12} \in H^{52}(L^{62};\Z/2)$. By a straightforward calculation the restriction of the normal bundle of $L^{62}$ to the submanifold $L_0^{10} \subset L^{62}$ is trivial and the normal bundle of $L_0^{10}$ is the Whitney sum $12 \kappa_a \oplus 12 \mu_a$, where $\kappa_a$ is the line $\Z/2$-bundle, $\mu_a$ is the plane $\Z/4$-bundle with the characteristic classes $\kappa_a$, $\mu_a^{ast}(\tau)$ described in the formula (8). By Lemma 6.1 (in the proof of this lemma we have to assume that the normal bundle of the manifold $L^{10}_0$ is as above) and by Lemma 7.1 \[A2\] the characteristic class (8) is trivial. The Main Theorem is proved. $$$$ $$$$ Moscow Region, Troitsk, 142190, IZMIRAN. [email protected] $$$$ [99]{} P.M.Akhmet’ev, [*Geometric approach towards stable homotopy groups of spheres. The Steenrod-Hopf invariants*]{}, a talk at the M.M.Postnikov Memorial Conference (2007)– “Algebraic Topology: Old and New” and at the Yu.P.Soloviev Memorial Conference (2005) “Topology, analysis and applications to mathematical physics” arXiv:??. The complete version (in Russian) arXiv:0710.5779. P.M.Akhmetiev, [*Geometric approach towards stable homotopy groups of spheres. The Kervaire invariants*]{} (in Russian) arXiv:0710.5853. M.G. Barratt, J.D.S. Jones and M.E. Mahowald, [*The Kervaire invariant one problem*]{}, Contemporary Mathematics Vol 19, (1983) 9-22. Carter, J.S., [*Surgery on codimension one immersions in $\R^{n+1}$: removing $n$-tuple points.*]{} Trans. Amer. Math. Soc. 298 (1986), N1, pp 83-101. Carter, J.S., [*On generalizing Boy’s surface: constructing a generator of the third stable stem.*]{} Trans. Amer. Math. Soc. 298 (1986), N1, pp 103-122. R.L.Cohen, J.D.S.Jones and M.E.Mahowald [*The Kervaire invariant of immersions*]{} Invent. math. 79, 95-123 (1985). P.J. Eccles, [*Codimension One Immersions and the Kervaire Invariant One Problem*]{}, Math. Proc. Cambridge Phil. Soc., vol.90 (1981) 483-493. P.J. Eccles, [*Representing framed bordism classes by manifolds embedded in low codimension*]{},  LNM, Springer,  N657 (1978), 150-155. A. Szucs, [*Topology of $\Sigma\sp {1,1}$-singular maps*]{}, Math. Proc. Cambridge Philos. Soc. 121 (1997), no. 3, 465–477. [^1]: This work was supported in part by the London Royal Society (1998-2000), RFBR 08-01-00663, INTAS 05-1000008-7805.

--- author: - | Shay Elmalem, Raja Giryes and Emanuel Marom\ Tel Aviv University\ bibliography: - 'egbib.bib' title: Motion Deblurring using Spatiotemporal Phase Aperture Coding --- Introduction ============ Related work {#prevWork} ============ Spatiotemporal aperture coding {#mask} ============================== The color-coded motion deblurring network {#CNN} ========================================= Experiments {#exp} =========== Conclusion {#sum} ========== PSF spectral analysis {#psf_spect} ===================== CNN structure and details {#cnn_det} ========================= Test-set results {#testSet} ================ Quantitative comparison statistics {#stat} ================================== Experimental setup description {#expSetup} ============================== Additional experimental results {#expRes} ===============================

--- author: - | [C.G.Bao$^1$, Y.X.Liu$^2$]{}\ [$^1$ Department of Physics,Zhongshan University, Guangzhou,510275,P.R.China.]{}\ [$^2$ Department of Physics, Beijing University, Beijing, 100871, P.R.China]{} --- 148 true mm 225 true mm tcilatex ABSTRACT: The inherent nodal structures of the wavefunctions of 6-nucleon systems are investigated. A group of six low-lying states (including the ground states) dominated by total orbital angular momentum L=0 components are found, the quantum numbers of each of these states are deduced. In particular, the spatial symmetries of these six states are found to be mainly the {4,2} and {2,2,2}. PACS: 21.45.+v, 02.20.-a, 27.20.+n As a few-body system the 6-body system has been scarcely investigated theoretically due to the complexity arising from the 15 spatial degrees of freedom. The existing related literatures concern mainly the ground states and a few resonances \[1-5\]. The study of the character of the excited states is very scarce. On the other hand, the particles of 6-body systems are neither too few nor too many. The study of them is attractive because it may lead to an understanding of the connection between the few-body theory and the modal theories for nuclei. Before solving the 6-body Schrödinger equation precisely, if we can have some qualitative understanding of the spectrum, it would be very helpful. This understanding , together with the results from calculations and experiments, will lead to a complete comprehension of the physics underlying the spectrum. In \[6\] the qualitative feature of 4-nucleon systems has been studied based on symmetry. In this paper we shall generalize the idea of \[6\] to extract qualitative character of the low-lying states of 6-nucleon systems. There are two noticeable findings in \[6\]. (i) The ground state is dominated by total orbital angular momentum L=0 component, while all the resonances below the 2n+2p threshold are dominated by L=1 components, there is a very large gap lying between them. Experimentally, this gap is about 20 MeV. This fact implies that the collective rotation is difficult to be excited. (ii) The internal wavefunctions (the wavefunction relative to a body-frame) of all the states below the 2n+2p threshold do not contain nodal surfaces. This fact implies that the excitation of internal oscillation takes a very large energy. Therefore, ti would be reasonable to assume that the L=0 nodeless component will be also important in the low-lying spectrum of the 6-nucleon systems. It was found in \[7,8\] that a specific kind of nodal surfaces may be imposed on the wavefunctions by symmetry. Let $\Psi $ be an eigenstate. Let $A$ denotes a geometric configuration. In some cases $A$ may be invariant to specific combined operations $O_i$ ( i=1 to m). For example, when $A$ is a regular octahedron (OCTA) for a 6-body system, then $A$ is invariant to a rotation about a 4-fold axis of the OCTA by 90$^{\circ }$ together with a cyclic permutation of four particles. In this case we have $$\stackrel{\wedge }{O_i}\Psi (A)=\Psi (O_iA)=\Psi (A)\hspace{1.0in}(1)$$ Owing to the inherent transformation property of $\Psi $ (the property with respect to rotation, inversion, and permutation), (1) always can be written in a matrix form (as we shall see) and appears as a set of homogeneous linear algebra equations. They impose a very strong constraint on $\Psi $ so that $\Psi $ may be zero at $A$. This is the origin of this specific kind of nodal surfaces, they are called the inherent nodal surfaces (INS). The INS appear always at geometric configurations with certain geometric symmetry. For a 6-body system the OCTA is the configuration with the strongest geometric symmetry. Let us assume that the six particles form an OCTA. Let k’ be a 4-fold axis of the OCTA, and let the particles 1,2,3, and 4 form a square surrounding k’. Let $R_\delta ^{k^{\prime }}$ denote a rotation about k’ by the angle $\delta $ (in degree), let $p(1432)$ denotes a cyclic permutation. Evidently, the OCTA is invariant to $$O_1=p(1432)R_{-90}^{k^{\prime }}\hspace{1.0in}(2)$$ Let $p_{ij}$ denotes an interchange of the locations of particles i and j, $% P $ denotes a space inversion. The OCTA is also invariant to $$O_2=p_{13}p_{24}p_{56}P\hspace{1.0in}(3)$$ Let i’ be an axis vertical to k’ and parallel to an edge of the above square; say, parallel to $\stackrel{\rightarrow }{r_{12}}$. Then the OCTAis also invariant to $$O_3=p_{14}p_{23}p_{56}R_{180}^{i^{\prime }}.\hspace{1.0in}(4)$$ Let $OO^{\prime }$ be a 3-fold axis of the OCTA, where $O$ denotes the center of mass. Let particles 2,5, and 3 form a regular triangle surroundung the $OO^{\prime }$; 1,4, and 6 form another triangle. Then the OCTA is also invariant to $$O_3=p(253)p(146)R_{-120}^{oo^{\prime }}\hspace{1.0in}(5)$$ Besides, the OCTA is also invariant to some other operators, e.g., the $% p(152)p(364)R_{-120}^{oo"}$ (where $OO"$ is another 3-fold axis). However, since the rotations about two different 3-fold axes are equivalent, one can prove that this additional operator does not introduce new constraints, and the operators $O_1$ to $O_4$ are sufficient to specify the constraints arising from symmetry. Let an eigenstate of a 6-nucleon system with a given total angular momentum J, parity $\Pi $, and total isospin T be written as $$\Psi =\sum_{L,S}\Psi _{LS}\hspace{1.0in}(6)$$ where S is the total spin, $$\Psi _{LS}=\sum_{\lambda i}F_{LSM}^{\lambda i}\chi _S^{\stackrel{\symbol{126}% }{\lambda }i}\hspace{1.0in}(7)$$ Where $M$ is the Z-component of L, $F_{LSM}^{\lambda i}$ is a function of the spatial coordinates, which is the i$^{th}$ basis function of the $% \lambda -$representation of the S$_6$ permutation group. The $\chi _S^{% \stackrel{\sim }{\lambda }i}$ is a basis function in the spin-isospin space with a given S and T and belonging to the $\stackrel{\sim }{\lambda }-$representation, the conjugate of $\lambda .$ In (7) the allowed $\lambda $ are listed in Table 1, they depend on S and T \[9\]. S T $\lambda $ --- --- --------------------------------------------------------------------- 0 0 {1$^6$}, {2,2,1,1},{3,3},{4,1,1} 1 0 {2,1$^4$}, {3,1$^3$}, {2,2,2}, {3,2,1}, {4,2} 2 0 {2,2,1,1}, {3,2,1} 3 0 {2,2,2} 0 1 {2,1$^4$}, {3,1$^3$}, {2,2,2}, {3,2,1}, {4,2} 1 1 {1$^6$}, {2,1$^4$}, 2{2,2,1,1}, {3,1$^3$}, 2{3,2,1}, {3,3}, {4,1,1} 2 1 {2,1$^4$}, {2,2,1,1}, {3,1$^3$}, {2,2,2}, {3,2,1} 3 1 {2,2,1,1} Tab.1, The allowed representation $\lambda $ in (7) From k’ and i’ defined before one can introduce a body frame i’-j’-k’. In the body-frame the $F_{LSM}^{\lambda i}$ can be expanded $$F_{LSM}^{\lambda i}(123456)=\sum_Q D_{QM}^L(-\gamma ,-\beta ,-\alpha )F_{LSQ}^{\lambda i}(1^{\prime }2^{\prime }3^{\prime }4^{\prime }5^{\prime }6^{\prime })\hspace{1.0in}(8)$$ Where $\alpha \beta \gamma $ are the Euler angles to specify the collective rotation, $D_{QM}^L$ is the well known Wigner function, Q are the projection of L along the k’-axis. The (123456) and (1’2’3’4’5’6’) specifies that the coordinates are relative to a fixed frame and the body-frame, respectively. Since the $F_{LSQ}^{\lambda i}$ span a representation of the rotation group, space inversion group, and permutation group, the invariance of the OCTA to the operations $O_1$ to $O_4$ leads to four sets of equations. For example, from $$\hat O_1F_{LSQ}^{\lambda i}(A)=F_{LSQ}^{\lambda i}(O_1A)=F_{LSQ}^{\lambda i}(A)\hspace{1.0in} (9)$$ where $F_{LSQ}^{\lambda i}(A)$ denotes that the coordinates in $% F_{LSQ}^{\lambda i}$ are given at an OCTA, for all $Q$ with $|Q|\leq L$ we have $$\sum_{i^{\prime }}[g_{ii^{\prime }}^\lambda (p(1234))e^{-i\frac \pi 2Q}-\delta _{ii^{\prime }}]F_{LSQ}^{\lambda i^{\prime }}(A)=0\hspace{1.0in}% (10)$$ where $g_{ii^{\prime }}^\lambda $ are the matrix elements belonging to the representation $\lambda $, which are known from the textbooks of group theory (e.g., refer to \[10\]). From $\hat O_2$ and $\hat O_4,$ we have $$\sum_{i^{\prime }}[g_{ii^{\prime }}^\lambda (p_{13}p_{24}p_{56})\Pi -\delta _{ii^{\prime }}]F_{LSQ}^{\lambda i^{\prime }}(A)=0\hspace{1.0in}(11)$$ and $$\sum_{Q^{\prime }i^{\prime }}[(-1)^Lg_{ii^{\prime }}^\lambda (p_{14}p_{23}p_{56})\delta _{\stackrel{-}{Q}Q^{\prime }}-\delta _{ii^{\prime }}\delta _{QQ^{\prime }}]F_{LSQ^{\prime }}^{\lambda i^{\prime }}(A)=0\ % \hspace{1.0in}(12)$$ where $\stackrel{-}{Q}=-Q.$ It is noted that $$R_{-120}^{oo^{\prime }}=R_\theta ^{j^{\prime }}R_{-120}^{k^{\prime }}R_{% \stackrel{-}{\theta }}^{j^{\prime }}\hspace{1.0in}(13)$$ where $\theta =\arccos (\sqrt{\frac 13})$. Thus from $\hat O_3$ we have $$\sum_{Q^{\prime }i^{\prime }}[g_{ii^{\prime }}^\lambda [p(235)p(164)]\sum_{Q^{\prime \prime }}D_{QQ"}^L(0,\theta ,0)e^{-i\frac{2\pi }3Q"}D_{Q^{\prime }Q"}^L(0,\theta ,0)-\delta _{ii^{\prime }}\delta _{QQ^{\prime }}]F_{LSQ^{\prime }}^{\lambda i^{\prime }}(A)=0\hspace{1.0in}% (14)$$ Eq.(10), (11), (12), and (14) are the equations that the $F_{LSQ}^{\lambda i}(A)$ have to fulfilled. In some cases there is one or more than one nonzero solution(s) (i.e., not all the $F_{LSQ}^{\lambda i}(A)$ are zero) to all these equations . But in some other cases, there are no nonzero solutions. In the latter case, the $\Psi _{LS}$ has to be zero at the OCTA configurations disregarding their size and orientation. Accordingly, an INS emerges and the OCTA is not accessible. Evidently, the above equations depend on and only on L, $\Pi $, and $\lambda .$ Therefore the existence of the INS does not at all depend on dynamics (e.g., not on the interaction, mass, etc.). Since the search of nonzero solutions of linear equations is trivial, we shall neglect the details but give directly the results of the L=0 components in the second and fourth columns of Tab.2 0$^{+}$ 0$^{+}$ 0$^{-}$ 0$^{-}$ ------------ --------- --------- --------- --------- $\lambda $ OCTA C-PENTA OCTA C-PENTA {6} 1 1 0 0 {5,1} 0 1 0 0 {4,2} 1 1 0 0 {3,3} 0 1 0 0 {2,2,2} 1 1 1 0 {2,2,1,1} 0 1 0 0 {2,1$^4$} 0 1 0 0 {1$^6$} 0 1 0 0 {3,2,1} 0 2 0 0 {4,1,1} 0 0 0 0 {3,1$^3$} 0 0 1 0 Tab.2, The accessibility of the OCTA (regular octahedron) and the C-PENTA (regular centered-pentagon) to the L$^\Pi =0^{+}$ and $0^{-}$ wavefunctions with different spatial permutation symmetry $\lambda $. Where the figures in the blocks are the numbers of independent nonzero solutions. The figure 0 implies that nonzero solutions do not exist. The INS existing at the OCTA may even extend beyond the OCTA. For example, when the shape in Fig.1a is prolonged along k’, then the shape is called a prolonged-octahedron. This shape (denoted by $B$ ) is invariant to $O_1,O_2,$ and $O_4$, but not to $O_3$. Hence, the $F_{LSQ^{\prime }}^{\lambda i^{\prime }}(B)$ should fulfill only (10) to (12), but not (14). When nonzero common solutions of (10), (11), (12), and (14) do not exist, while nonzero solutions of only (10) to (12) also do not exist, the INS extends from the OCTA to the prolonged-octahedrons. An OCTA has many ways to deform;e.g., instead of a square, the particles 1,2,3, and 4 form a rectangle or form a diamond, etc.. Hence, the INS at the OCTA has many possibilities to extend. How it extend is determined by the (L$\Pi \lambda$) of the wavefunction. Thus, in the coordinate space, the OCTA is a source where the INS may emerge and extend to the neighborhood surrounding the OCTA. This fact implies that specific inherent nodal structure exists. The details of the inherent nodal structure will not be concerned in this paper. However, it is emphasized that for a wavefunction, if the OCTA is accessible, all the shapes in the neighborhood of the OCTA are also accessible, therefore this wavefunction is inherent nodeless in this domain. Another shape with also a stronger geometric symmetry is a regular centered-pentagons(C-PENTA, the particle 6 is assumed to be located at the center of mass O). Let k’ be the 5-fold axis. The C-PENTA is invariant to (i) a rotation about k’ by $\frac{2\pi }5$ together with a cyclic permutation of the five particles of the pentagon , (ii) a rotation about k’ by $\pi $ together with a space inversion, (iii) a rotation about i’ by $\pi $ together with $p_{14}p_{23}$ (here i’ is the axis vertical to k’ and connecting O and particle 5). These invariances will lead to constraints embodied by sets of homogeneous equations, and therefore the accessibility of the C-PENTA can be identified as also given in Tab.2. In addition to the OCTA, the C-PENTA is another source where the INS may emerge and extend to its neighborhood; e.g., extend to the pentagon-pyramid as shown in Fig.1b with h$\neq $0. There are also other sources. For example, the one at the regular hexagons. However, among the 15 bonds, 12 can be optimized at an OCTA, 10 at a pentagon-pyramid, but only 6 at a hexagon. Therefore in the neighborhood of the hexagon (and also other regular shapes) the total potential energy is considerably higher. Since the wavefunctions of the low-lying states are mainly distributed in the domain with a relatively lower potential energy, we shall concentrate only in the domains surrounding the OCTA and the C-PENTA. When (L$\Pi \lambda $) =(0+{6}), (0+{4,2}), or (0+{2,2,2}), the wavefunction can access both the OCTA and the C-PENTA (refer to Table 2). These and only these wavefunctions are inherent-nodeless in the two most important domains, and they should be the dominant components for the low-lying states. All the other L=0 components must contain at least an INS resulting in a great increase in energy. From Tab.1 it is clear that the (0+{6}) component is not allowed, while the (0+{4,2}) component can be contained in \[S,T\]=\[1,0\] and \[0,1\] states, and the (0+{2,2,2}) component can be contained in \[S,T\]=\[1,0\], \[3,0\], \[0,1\], and \[2,1\] states. When \[S,T\]=\[1,0\] , the $\lambda $ can be {4,2} or {2,2,2}, therefore two J$% ^\Pi =1^{+}$ partner-states with their spatial wavefunctions orthogonal to each other exist, each of them is a specific mixture of {4,2} and {2,2,2}. Similarly, two partner-states with \[S,T\]=\[0,1\] and J$^\Pi =0^{+}$ exist also. When \[S,T\]=\[3,0\] or \[2,1\], the $\lambda $ has only one choice, therefore in each case only one state exists. Thus we can predict that there are totally six low-lying states dominated by L=0 components without nodal surfaces as listed in Tab.3, where the L,S, and $\lambda $ are only the quantum numbers of the dominant component. $\lambda $ E -- --- --- --- -- ------------------- ------ {4,2} and {2,2,2} 0 {4.2} and {2,2,2} 5.65 {2,2,2} 2.19 0 2 + 4.31 {4,2} and {2,2,2} 3.56 {4,2} and {2,2,2} {2,2,2} 5.37 Tab.3, Prediction of the quantum numbers of low-lying states (dominated by L=0 components) of the 6-nucleon systems based on symmetry. The last column is the energies (in MeV) of the states of $^6$Li taken from \[11\]. It is expected that these low-lying states should be split by the nuclear force. Owing to the interference of the {4,2} and {2,2,2} components, there would be an larger energy gap lying between the two partner-states of each pair. Ajzenberg-selove has made an analysis on $^6$Li based on experimental data \[11\], the results are listed in Tab.3. Although our analysis is based simply on symmetry, but the results of the two analyses are close. For the T=0 states, there are two J$^\Pi =$ 1$^{+}$ states (\[S,T\]=\[1,0\]) in \[11\] with a split, they are just the expected partners. The split is so large (5.65 MeV) that the lower one becomes the ground state while the higher one becomes the highest state of this group. There is a T=0 state in \[11\] at 2.19 MeV with exactly the predicted quantum numbers J$^\Pi =3^{+}$. Nonetheless, there is a T=0 state in \[11\] at 4.31 MeV with J$^\Pi $ = 2$^{+}$, which do not appear in our analysis. May be this state is dominated by L=1 component, may be there is another origin to be clarified. For the T=1 states, one of the expected partners with J$^\Pi =0^{+}$ (\[S,T\]=\[0,1\]) was found in \[11\] at 3.56 MeV . However, the other partner ( it would be considerably higher) has not yet been identified in \[11\], this is an open problem. Nonetheless, if this state exists, the structure of its spatial wavefunction would be similar to the T=0 state at 5.65 MeV . The third expected T=1 state was found in \[11\] at 5.37MeV with exactly the predicted J$^\Pi =2^{+}$. In summary we have explained the origin of the quantum numbers of the low-lying states of 6-nucleon systems. The explanation is very different from that based on the shell model \[12,13\]. For example, according to our analysis, the J$^\Pi =3^{+}$ state at 2.19 MeV has S=3 and L=0. On the contrary, in the shell model the four nucleons in the 1s orbit must have their total spin zero and total isospin zero; therefore this state should have S $\leq 1$ and L $\geq 2.$ However, it is noted that the 2$_1^{+}$ state (having S=0 and L=2) of the $^{12}C$ lies at 4.44 MeV \[14\]. Since the $% ^6$Li is considerably lighter and smaller than the $^{12}$C, the L=2 state of $^6$Li should be much higher than 4.44 MeV due to having a much smaller moment of inertia. Therefore the 3$^{+}$ state at 2.19 MeV is difficult to be explained as a L $\geq $ 2 state. In particular, it is found that the {2,2,2} component is important; however this component is suppressed by the shell model. Thus, our analysis raises a challenge to the shell model in the case that the number of nucleons is not large enough. Evidently, much work should be done to clarify the physics underlying these systems. It has been shown that sources of INS may exist in the quantum states. Nonetheless, there are essentially inherent-nodeless components of wavefunctions (each with a specific set of (L$\Pi \lambda $)). They are the most important building blocks to constitute the low-lying states. The identification of these particularly favorable components is a key to understand the low-lying spectrum. The idea of this paper can be generalized to investigate different kinds of systems, thereby we can understand them in an unified way. ACKNOWLEDGEMENT: This work is supported by the NNSF of the PRC, and by a fund from the National Educational Committee of the PRC. REFERENCES 1, B.S.Pudliner, V.R. Pandharipande, J.Carlson, and R.B.Wiringa, Phys. Rev. Lett. 74, 4396, (1995) 2, B.S.Pudliner, V.R. Pandharipande, J.Carlson, S.C.Pieper, and R.B.Wiringa, Phys. Rev. C56, 1720 (1997) 3, K.Varga, Y.Suzuki, Phys. Rev. C52, 2885, (1995) 4, A.Csótó, Phys. Rev. C49, 3035, (1994) 5, Y. Fujiwara and Y.C.Tang, Phys. Rev.C43, 96, 1991; Few-Body Systems 12, 21, (1992.) 6, C.G.Bao, Conference Handbook of XVth International Conference on Few-Body Problems in Physics, edited by L.P.Kok, J.C.S.Bacelar, and A.E.L.Dieperink, Gröningen, p.496, 1997: nucl-th/9805001(preprint). 7, C.G.Bao, Few-Body Systems, 13, 41, (1992); Phys. Rev. A47, 1752 , (1993); Phys. Rev. A50, 2182, (1994); Chinese Phys. Lett. 14, 20, (1997); Phys. Rev. Lett., 79, 3475,(1997.) 8, W.Y.Ruan, and C.G.Bao, Few-Body Systems, 14, 25, (1993) 9, C.Itzykson and M.Nauenberg, Rev. Mod. Phys. 38, 95, (1966) 10, J.Q.Chen, ”Group Representation Theory for Physicists”, World Scientific, Singapore ,1989 11, F.Ajzenberg-Selove, Nucl.Phys. A490, 1, 1(988) 12, M.G.Mayer and J.H.D.Jensen, ”Elementary Theory of Nuclear Shell Structure”, Willey, New York, 1955 13, A. deShalit and I.Talmi, ”Nuclear Shell Theory”, Academic, New York, 1963 14, Y.Fujiwara, H.Horiuchi, K.Ikeda, M.Kamimura, K.Kato, Y.Suzuki, and E.Uegaki, Prog. Theor. Phys., Supplement 68, 29, (1980)

--- abstract: 'Scintillating CaWO$_4$ single crystals are a promising multi-element target for rare-event searches and are currently used in the direct Dark Matter experiment CRESST (Cryogenic Rare Event Search with Superconducting Thermometers). The relative light output of different particle interactions in CaWO$_4$ is quantified by Quenching Factors (QFs). These are essential for an active background discrimination and the identification of a possible signal induced by weakly interacting massive particles (WIMPs). We present the first precise measurements of the QFs of O, Ca and W at mK temperatures by irradiating a cryogenic detector with a fast neutron beam. A clear energy dependence of the QFs and a variation between different CaWO$_4$ single crystals were observed for the first time. For typical CRESST detectors the QFs in the region-of-interest (10-40keV) are $QF_O^{ROI}=(11.2{\pm}0.5)$%, $QF_{Ca}^{ROI}=(5.94{\pm}0.49)$% and $QF_W^{ROI}=(1.72{\pm}0.21)$%. The latest CRESST data (run32) is reanalyzed using these fundamentally new results on light quenching in CaWO$_4$ having moderate influence on the WIMP analysis. Their relevance for future CRESST runs and for the clarification of previously published results of direct Dark Matter experiments is emphasized.' author: - 'R. Strauss' - 'G. Angloher' - 'A. Bento' - 'C. Bucci' - 'L. Canonica' - 'A. Erb' - 'F.v.Feilitzsch' - 'P.Gorla' - 'A. Gütlein' - 'D. Hauff' - 'J. Jochum' - 'H. Kraus' - 'J.-C. Lanfranchi' - 'J. Loebell' - 'A. Münster' - 'F.Petricca' - 'W. Potzel' - 'F. Pröbst' - 'F. Reindl' - 'S. Roth' - 'K. Rottler' - 'C. Sailer' - 'K. Schäffner' - 'J.Schieck' - 'S. Scholl' - 'S. Schönert' - 'W. Seidel' - 'M.v.Sivers' - 'L. Stodolsky' - 'C. Strandhagen' - 'A.Tanzke' - 'M. Uffinger' - 'A. Ulrich' - 'I. Usherov' - 'S. Wawoczny' - 'M. Willers' - 'M. Wüstrich' - 'A. Zöller' - 'W. Carli' - 'C. Ciemniak' - 'H. Hagn' - 'D. Hellgartner' bibliography: - 'quenching\_final.bib' title: 'Precision Measurements of Light Quenching in CaWO$_4$ Crystals at mK Temperatures' --- Rare-event searches for Dark Matter (DM) in the form of weakly interacting massive particles (WIMPs) [@Bertone:2004pz; @Jungman:1995df] have reached impressive sensitivities during the last decade [@Cushman:2013zza]. Well motivated WIMP candidates with masses $m_\chi$ between a few GeV/$c^2$ and a few TeV/$c^2$ might be detectable via nuclear recoils of few keV in terrestrial experiments [@Lewin:1995rx]. While the DAMA/LIBRA [@Bernabei:2010mq], and recently the CoGeNT [@Aalseth:2010vx], CRESST [@Angloher:2012vn], and the CDMS(Si) [@PhysRevLett.111.251301] experiments observed excess signals that might be interpreted as induced by DM particles with $m_\chi{\sim10}$GeV/$c^2$ at WIMP-nucleon cross-sections of ${\sim}10^{-4}$pb, this scenario is ruled out by the LUX [@Akerib:2013tjd] and XENON100 [@Aprile:2012nq] experiments, and almost excluded by the CDMS(Ge) [@Ahmed:2009zw; @Ahmed:2010wy] and EDELWEISS [@Armengaud:2011cy; @Armengaud:2012pfa] experiments. It is strongly disfavoured by accelerator constraints [@ATLAS:2012ky; @Chatrchyan:2012me] and in mild tension with an extended analysis [@PhysRevD.85.021301] of published CRESST data [@Angloher2009270].\ The CRESST experiment [@Angloher:2012vn] employs scintillating CaWO$_4$ crystals [@edison; @PhysRevB.75.184308] as a multi-element target material. The key feature of a CRESST detector module is the simultaneous measurement of the recoil energy $E_r$ by a particle interaction in the crystal (operated as cryogenic calorimeter at mK temperatures [@Probst:1995fk]) and the corresponding scintillation-light energy $E_l$ by a separate cryogenic light absorber. Since the relative light yield $LY{=}E_l/E_r$ is reduced for highly ionizing particles compared to electron recoils (commonly referred to as quenching) nuclear-recoil events can be discriminated from e$^-$/$\gamma$ and $\alpha$ backgrounds. The phenomenological Birks model [@birks1964theory] predicts this quenching effect to be stronger the higher the mass number $A$ of the recoiling ion, which allows to distinguish, in general, between O ($A{\approx}16$), Ca ($A{\approx}40$) and W ($A{\approx}184$) recoils. The expected WIMP-recoil spectrum - assuming coherent scattering - is completely dominated by W-scatters for $m_\chi\,\gtrsim 20$GeV/c$^2$. However, the light targets O and Ca make CRESST detectors particularly sensitive to low-mass WIMPs of $1$GeV$\,\lesssim\, m_\chi\,\lesssim$20GeV. Furthermore, the knowledge of the recoil composition of O, Ca and W allows a test of the assumed $A^2$-dependence of the spin-independent WIMP-nucleon cross-section [@Jungman:1995df]. In addition, background neutrons, which are mainly visible as O-scatters (from kinematics [@scholl_paper]), can be discriminated statistically.\ The mean LY of e$^-$/$\gamma$ events ($LY_{\gamma}$) is energy dependent and phenomenologically parametrized as ${LY_{\gamma}(E_r){=}(p_0+p_1E_r)(1-p_2\exp(-E_r/p_3)}$ [@strauss_PhD]. By convention, $LY_{\gamma}$(122keV) is normalized to unity. The parameters $p_{0}$, $p_1$, $p_2$ and $p_3$ are derived from a maximum-likelihood (ML) fit for every detector module individually. For the module used in this work the fit yields: $p_0\,{=}\,1.07$, $p_1\,{=}\,{-}1.40\cdot 10^{-5}$keV$^{-1}$, $p_2\,{=}\,6.94\cdot 10^{-2}$ and $p_3\,{=}\,147$keV (errors are negligible for the following analysis). The exponential decrease towards lower recoil energies (quantified by $p_2$ and $p_3$) accounts for the scintillator non-proportionality [@Lang:2009uh]. The Quenching Factor (QF) of a nucleus $x$ - in general energy dependent - is defined as $QF_x(E_r)=LY_x(E_r)/LY_{\gamma,corr}(E_r)$ where $LY_x$ is the mean LY of a nuclear recoil x. For normalization, the LY of e$^{-}$/$\gamma$ events corrected for the scintillator non-proportionality (which is not observed for nuclear recoils) is used by convention: $LY_{\gamma,corr}=p_0+p_1E_r$. For typical CRESST detector modules, the uncertainties in energy and LY are well described by gaussians [@Angloher:2012vn] consistent with photon-counting statistics in the energy range considered in this work.\ Since the resolution of light-detectors operated in the CRESST setup at present is not sufficient to disentangle O, Ca and W recoils unambiguously, dedicated experiments to measure the QFs of CaWO$_4$ are necessary. Earlier attempts yield inconclusive results, in particular for the value of $QF_W$ [@Jagemann:2006sx; @Ninkovic:2006xy; @Bavykina:2007ze].\ At the accelerator of the Maier-Leibnitz-Laboratorium (MLL) in Garching a dedicated neutron-scattering facility for precision measurements of QFs at mK temperatures was set up (see FIG.\[fig:setupMLL\]). A pulsed $^{11}$B beam of ${\sim}65$MeV in bunches of 2-3ns (FWHM) produces monoenergetic neutrons of ${\sim}11$MeV via the nuclear reaction p($^{11}$B,n)$^{11}$C in a pressurized H$_2$ target [@Jagemann2005245]. These neutrons are irradiated onto a CRESST-like detector module consisting of a ${\sim}$10g cylindrical CaWO$_4$ single crystal (20mm in diameter, 5mm in height) and a separated Si light absorber (20mm in diameter, 500$\mu$m thick) [@Strauss:2012fk]. Both are operated as cryogenic detectors in a dilution refrigerator at ${\sim}20\,$mK [@Lanfranchi20091405]. Undergoing elastic (single) nuclear scattering in CaWO$_4$ the neutrons are tagged at a fixed scattering angle $\Theta$ in an array of 40 liquid-scintillator (EJ301) detectors which allow fast timing (${\sim}2$ns) and n/$\gamma$ discrimination. Depending upon which of the three nuclei is hit a distinct amount of energy is deposited by the neutron in the crystal. Triple-coincidences between (1) a $^{11}$B pulse on the H$_2$ target, (2) a neutron pulse in a liquid-scintillator detector and (3) a nuclear-recoil event in the CaWO$_4$ crystal can be extracted from the data set. A neutron time-of-flight (TOF) measurement between neutron production and detection combined with a precise phononic measurement of the energy deposition in the crystal (resolution ${\sim}1$keV (FWHM)) allows an identification of the recoiling nucleus. To derive the individual QF the corresponding scintillation-light output is measured simultaneously by the light detector. Since the onset uncertainty of cryodetector pulses is large (${\sim}5\,\mu$s) compared to typical neutron TOFs (${\sim}$50ns) an offline coincidence analysis has to be performed [@strauss_PhD].\ ![Schematic experimental setup of the neutron-scattering facility. Neutrons produced by the accelerator are scattered off a CRESST-like detector module (operated at 20mK) and tagged in liquid-scintillator neutron detectors at a fixed scattering angle $\Theta$.[]{data-label="fig:setupMLL"}](facility.pdf){width="38.00000%"} The experiment was optimized for the measurement of $QF_W$ [@strauss_PhD; @strauss_ltd15]. To enhance the number of W-scatters a scattering angle of ${\Theta\,{=}\,80^\circ}$ was chosen due to scattering kinematics [@Jagemann:2006sx]. For this specific angle, the expected recoil energy of triple-coincident events is ${\sim}\,100$keV for W, ${\sim}\,450$keV for Ca, and ${\sim}\,1.1$MeV for O . In ${\sim}\,3$ weeks of beam time a total of ${\sim}\,10^8$ cryodetector pulses were recorded. FIG.\[fig:timing\] shows the time difference $\Delta t$ between neutron events with the correct TOF identified in one of the liquid-scintillator detectors and the closest W-recoil (in time) in the CaWO$_4$ crystal ($E_r=100{\pm}20$keV). A gaussian peak of triple-coincidences on W (dashed red line) at $\Delta t\,{\approx}\,0.016$ms and a width of $\sigma_t\,{\approx}\,4.8\,\mu$s (onset resolution of the cryodetector) is observed above a background due to accidental coincidences uniformly distributed in time (shaded area). Within the $2\sigma$-bounds of the peak 158 W-scatters are identified with a signal-to-background (S/B) ratio of ${\sim}\,7:1$.\ ![Histogram of the time difference $\Delta t$ between neutron events with the correct TOF and the closest W-recoil in the CaWO$_4$ crystal ($E_r\,{=}\,100\,{\pm}\,20$keV). A fit to the distribution (solid black line) including a constant for the accidental background (shaded area) and a gaussian for the triple-coincidences on W (dashed red line) is shown. 158 W-scatters are identified with a signal-to-background ratio of ${\sim}\,7:1$. []{data-label="fig:timing"}](time_final.eps){width="48.00000%"} The mean LY of the extracted W-scatters is found at a lower value compared to the mean LY of all nuclear recoils, i.e., the (overlapping) contributions of O, Ca and W if no coincidence measurement is involved. The accidental coincidences have a LY-distribution equal to that which is modelled by a probability-density function (background-pdf) [@strauss_PhD]. A simultaneous maximum-likelihood (ML) fit is performed including (1) the timing distribution which fixes the S/B ratio and the number of identified W-events, and (2) the LY distribution described by a gaussian (W-events) and the background-pdf. The final results are $LY_W\,{=}\,0.0208\,{\pm}\,0.0024$ and $QF_W\,{=}\,(1.96\,{\pm}\,0.22)$%, correspondingly (errors are dominated by statistics). FIG.\[fig:LY\] shows the LY histogram of the identified events and the fit by the gaussian (dashed red line) and the background-pdf (shaded area).\ ![LY histogram of the 158 events identified as triple-coincidences on W. A fit to the distribution (solid black line) is shown which includes a gaussian (dashed red line) accounting for W-scatters and the background-pdf (shaded area) describing accidental coincidences. The simultaneous ML fit including the timing distribution yields $QF_W\,{=}\,(1.96\,{\pm}\,0.22)$%. []{data-label="fig:LY"}](LY_final.eps){width="48.00000%"} For the measurement of $QF_{Ca}$ and $QF_O$ no coincidence signals are necessary, instead, an analysis of the nuclear-recoil data alone is sufficient. Commonly CRESST data is displayed in the energy-LY plane [@Angloher:2012vn] giving rise to nearly horizontal bands which correspond to different types of particle interactions ($LY\,{\approx}\,1$ for electron and $LY\,{\lesssim}\,0.2$ for nuclear recoils). The nuclear-recoil bands of the data recorded during ${\sim}1$ week of beam time (${\sim}\,5\cdot 10^5$ pulses) are shown in FIG.\[fig:energyDependence\] bottom (grey dots). From kinematics using ${\sim}11$MeV neutrons as probes the O-recoil band extends up to ${\sim}2.4$MeV while the Ca- and W-bands extend up to ${\sim}1.05$MeV and ${\sim}240$keV, respectively [@Jagemann2005245]. Despite the strong overlap of the 3 nuclear-recoil bands the contributions of O and Ca fitted by two gaussians can be disentangled at $E_r\,{\gtrsim}\,350$keV (see FIG.\[fig:energyDependence\] top) due to high statistics and a good light-detector resolution. In FIG.\[fig:qf\_results\] the results for $QF_O$ and $QF_{Ca}$ (red error bars) derived by these independent one-dimensional (1-dim) fits are shown for selected recoil-energy slices of 20keV in width. All parameters in the fit are left free except for the LY-resolutions which are fixed by a ML fit of the electron-recoil band [@strauss_PhD]. While $QF_O$ clearly rises towards lower recoil energies, this effect is less pronounced for $QF_{Ca}$.\ ![ Top/Middle: LY histograms of two energy slices at 350 and 40keV (of 20keV in width) fitted by gaussians. Bottom: Neutron-induced nuclear-recoil events of O, Ca, and W plotted in the LY-energy plane (grey dots). The corresponding 1$\sigma$ acceptance bounds as derived from the correlated ML fit are indicated (see text). []{data-label="fig:energyDependence"}](canLy300.eps "fig:"){width="48.00000%"} ![ Top/Middle: LY histograms of two energy slices at 350 and 40keV (of 20keV in width) fitted by gaussians. Bottom: Neutron-induced nuclear-recoil events of O, Ca, and W plotted in the LY-energy plane (grey dots). The corresponding 1$\sigma$ acceptance bounds as derived from the correlated ML fit are indicated (see text). []{data-label="fig:energyDependence"}](canLy40.eps "fig:"){width="48.00000%"} ![ Top/Middle: LY histograms of two energy slices at 350 and 40keV (of 20keV in width) fitted by gaussians. Bottom: Neutron-induced nuclear-recoil events of O, Ca, and W plotted in the LY-energy plane (grey dots). The corresponding 1$\sigma$ acceptance bounds as derived from the correlated ML fit are indicated (see text). []{data-label="fig:energyDependence"}](LY_allNR_v2.eps "fig:"){width="48.60000%"} Below ${\sim}350$keV, due to the strong overlap of the nuclear-recoil bands, this simple approach fails. Instead, a correlated ML fit was performed based on the following assumptions: (1) for the mean LY of O- and Ca-scatters the phenomenological parametrization $LY_x(E_r)\,{=}\,LY_x^\infty\left(1+f_x\cdot \exp{(-E_r/\lambda_x)}\right)$ is proposed with the free parameters $LY_x^\infty$ (LY at $E_r\,{=}\,\infty$), $f_x$ (fraction of energy-dependent component) and $\lambda_x$ (exponential decay with energy), and (2) the mean LY of W-scatters is approximated to be constant in the relevant energy range (up to $\sim$240keV) at the value precisely measured with the triple-coincidence technique ($LY_W\,{=}\,0.0208\,{\pm}\,0.0024$). These assumptions are supported by the result of the 1-dim fits (see FIG.\[fig:qf\_results\]), by Birks’ model [@birks1964theory], and by a recent work [@sabine_phD] which predict the strength of the energy-dependence to decrease with $A$. The nuclear-recoil bands are cut into energy intervals of 10keV (20keV to 1MeV), of 20keV (1MeV to 1.4MeV) and 50keV (above 1.4MeV) and fitted with up to 3 gaussians depending on the recoil energy (e.g., shown in FIG.\[fig:energyDependence\] middle for $E_r{=}40$keV). Except for the assumptions mentioned above and the LY-resolution all parameters are left free in the fit. The fit converges over the entire energy range (20-1800keV). In TABLE\[tab:energydependence\] the results for $LY_x^\infty$, $f_x$ and $\lambda_x$ are presented which correspond, e.g. at 40keV, to $QF_O{=}(12.6{\pm}0.5)$%, $QF_{Ca}{=}(6.73{\pm}0.43)$% (here, the errors are dominated by systematics) and $QF_W{=}(1.96{\pm}0.22)$% at $1\sigma$ C.L. The 1-$\sigma$ acceptance bounds of O, Ca and W recoils as obtained in the correlated ML fit are shown in FIG.\[fig:energyDependence\] bottom. The final results for $QF_O$, $QF_{Ca}$ and $QF_W$ are presented in and are found to be in perfect agreement with the outcome of the 1-dim fits (red error bars). These are the first experimental results which clearly show a rise of the QFs of O (Ca) of ${\sim}28$% (${\sim}6$%) towards the ROI (10-40keV) compared to that at a recoil energy of 500keV.\ --------- --------------------- ------------------- ---------------- nucleus $LY_x^\infty$ $f_x$ $\lambda_x$ O $0.07908\pm0.00002$ $0.7088\pm0.0008$ $567.1\pm0.9$ Ca $0.05949\pm0.00078$ $0.1887\pm0.0022$ $801.3\pm18.8$ --------- --------------------- ------------------- ---------------- : \[tab:energydependence\]Results for the free parameters $LY_x^\infty$, $f_x$ and $\lambda_x$ of the ML analysis. The statistical errors are given at $1\sigma$ C.L. ![Results of the correlated ML analysis for $QF_O$, $QF_{Ca}$ and $QF_W$ (solid lines). The shaded areas indicate the 1$\sigma$ and 2$\sigma$ bounds. For the first time a clear energy dependence of $QF_O$ and $QF_{Ca}$ is observed. These results are in agreement with that of the 1-dim fits of discrete energy intervals (see text) shown as red error bars. $QF_W$ is fixed (in the correlated fit) at the value measured by the triple-coincidence technique. []{data-label="fig:qf_results"}](bands_v7.eps){width="48.00000%"} In previous works, the QFs of CaWO$_4$ were assumed to be constant over the entire energy range [@Angloher:2012vn]. A statistical analysis shows that this simple model is clearly disfavoured. Employing a likelihood-ratio test in combination with Monte-Carlo simulations gives a p-value of $p<10^{-5}$ for the data presented here to be consistent with constant QFs. Furthermore, the derived energy spectra of the individual recoiling nuclei agree with the expectation from incident 11MeV neutrons while the constant QF approach provides non-physical results.\ In the present paper, using the 8 detector modules operated in the last CRESST measurement campaign (run32) an additional aspect was investigated: the variation of the quenching behaviour among *different* CaWO$_4$ crystals [@strauss_PhD]. Nuclear recoils acquired during neutron-calibration campaigns of CRESST run32 are completely dominated by O-scatters at $E_r\gtrsim 150$keV (from kinematics) [@Angloher:2012vn]. Despite low statistics (a factor of ${\sim}100$ less compared to the measurement presented here) in the available data, the mean LY of O-events can be determined by a gaussian fit with a precision of $\mathcal{O}$(1%) for every module. In this way, the mean QF of O between 150 and 200keV was determined individually for the 8 detector modules operated in run32 ($\overline{QF_O^\ast}$) and for the reference detector operated at the neutron-scattering facility ($\overline{QF_O}$). Different values of $\overline{QF_O^\ast}$ are observed for the CRESST detector crystals (variation by ${\sim}11$%) and for the reference crystal (${\sim}12$% higher than the mean of $\overline{QF_O^\ast}$). This variation appears to be correlated with the crystal’s optical quality. The QF - which is a relative quantity - is found to be lower if a crystal has a smaller defect density and thus a higher absolute light output, i.e., the LY of nuclear recoils is less affected by an increased defect density. This is in agreement with the prediction described in a recent work [@sabine_phD]. In the present paper, a simple model to account for this variation is proposed: For every detector module which is to be calibrated a scaling factor $\epsilon$ is introduced, $\epsilon\,{=}\,\overline{QF_O^\ast}/\overline{QF_O}$. Then, within this model the QFs of the nucleus $x$ can be calculated for every module by $QF_{x}^\ast(E_r)\,{=}\,\epsilon\cdot QF_{x}(E_r)$ where $QF_{x}$ is the value precisely measured within this work. The nuclear-recoil behaviour of CRESST modules is well described by energy-dependent QFs. In the QFs, averaged over the ROI (10-40keV), and the scaling factor $\epsilon$ are listed for two selected detector modules (Rita and Daisy, with the lowest and highest absolute light output, respectively) and the mean of all 8 detector modules of run32 (Ø), $QF_O^{ROI}\,{=}\,(11.2{\pm}0.5)$%, $QF_{Ca}^{ROI}\,{=}\,(5.94{\pm}0.49)$% and $QF_W^{ROI}\,{=}\,(1.72{\pm}0.21)$%.\ ------- ------------------- -------------------------- ----------------------------- -------------------------- $\epsilon$ $QF_O^\mathrm{ROI}$\[%\] $QF_{Ca}^\mathrm{ROI}$\[%\] $QF_W^\mathrm{ROI}$\[%\] Rita $0.844{\pm}0.006$ $10.8{\pm}0.5$ $5.70{\pm}0.44$ $1.65{\pm}0.19$ Daisy $0.939{\pm}0.021$ $12.0{\pm}0.7$ $6.33{\pm}0.58$ $1.84{\pm}0.24$ Ø $0.880{\pm}0.011$ $11.2{\pm}0.5$ $5.94{\pm}0.49$ $1.72{\pm}0.21$ ------- ------------------- -------------------------- ----------------------------- -------------------------- : \[tab:CRESST\] QF results averaged over the ROI (10-40keV) and adjusted by the scaling factor $\epsilon$ for the modules Rita and Daisy, and the mean (Ø) of all run32 detectors ($1\sigma$ errors). We now turn to the effect of energy-dependent quenching since constant QFs as assumed in earlier CRESST publications do not sufficiently describe the behaviour of the nuclear-recoil bands. The value of $QF_O$ in the ROI was underestimated by ${\sim}8$% while the room-temperature measurements overestimated the values of $QF_{Ca}$ and $QF_W$ by ${\sim}7$% and ${\sim}130$%, respectively [@Angloher:2012vn]. Therefore, the parameter space of accepted nuclear recoils is larger than assumed in earlier publications (by ${\sim}46$%) requiring a re-analysis of the published CRESST data.\ During the latest measuring campaign (run32) a statistically significant signal ($4.2\sigma$) above known backgrounds was observed. If interpreted as induced by DM particles two WIMP solutions were found [@Angloher:2012vn], e.g. at a mass of $m_\chi=11.6\,$GeV/c$^2$ with a WIMP-nucleon cross section of $\sigma_\chi=3.7\cdot10^{-5}$pb. The dedicated ML analysis was repeated using the new QF values ($\O$ in TABLE\[tab:CRESST\]) yielding $m_\chi=12.0$GeV/c$^2$ and $\sigma_\chi=3.2\cdot10^{-5}$pb at 3.9$\sigma$. Beside this moderate change of the WIMP parameters also the background composition ($e^-$, $\gamma$, neutrons, $\alpha$’s and $^{206}$Pb) is influenced. This is mainly due to the significantly lower value of $QF_W$ which increases the leakage of $^{206}$Pb recoils into the ROI (by ${\sim}$18%). The other WIMP solution is influenced similarly: $m_\chi$ changes from 25.3 to 25.5GeV/c$^2$, $\sigma_\chi$ from $1.6\cdot10^{-6}$ to $1.5\cdot10^{-6}$pb and the significance drops slightly from 4.7 to 4.3$\sigma$.\ In conclusion, the first precise measurement of $QF_W$ at mK temperatures and under conditions comparable to that of the CRESST experiment was obtained at the neutron-scattering facility in Garching by an extensive triple-coincidence technique. Furthermore, the QFs of all three nuclei in CaWO$_4$ were precisely determined by a dedicated maximum-likelihood analysis over the entire energy range (${\sim}20{-}1800$keV). The observed energy dependence of the QFs, which is more pronounced for lighter nuclei, has significant influence on the determination of the ROI for DM search with CRESST. The observed variation of the QFs between different CaWO$_4$ crystals is related to the optical quality and can be adapted to every individual crystal by the simple model proposed above. The updated values of the QFs are highly relevant to disentangle the recoil composition (O, Ca and W) of a possible DM signal and, therefore, to determine the WIMP parameters. Since the separation between the O and W recoil bands is higher by ${\sim}46$% compared to earlier assumptions, background neutrons which are mainly visible as O-scatters [@scholl_paper] can be discriminated more efficiently from possible WIMP-induced events. A reanalysis of the run32 data shows a moderate influence of the new QF values on the WIMP parameters.\ The results obtained here are of importance for the current CRESST run (run33) and upcoming measuring campaigns. Providing a highly improved background level run33 has the potential to clarify the origin of the observed excess signal and to set competitive limits for the spin-independent WIMP-nucleon cross section in the near future. For the planned multi-material DM experiment EURECA (European Underground Rare Event Calorimeter Array) [@Kraus:2011zz] the neutron-scattering facility will be an important tool to investigate the light quenching of alternative target materials in the future.\ This research was supported by the DFG cluster of excellence: “Origin and Structure of the Universe”, the DFG “Transregio 27: Neutrinos and Beyond”, the “Helmholtz Alliance for Astroparticle Phyiscs”, the “Maier-Leibnitz-Laboratorium” (Garching) and by the BMBF: Project 05A11WOC EURECA-XENON.

--- author: - 'Z. Shah' - 'N. Mankuzhiyil' - 'A. Sinha' - 'R. Misra' - 'S. Sahayanathan' - 'N. Iqbal' bibliography: - 'ms89.bib' title: 'Log-normal flux distribution of bright *Fermi* blazars' --- Introduction ============ Blazars are subclass of Active Galactic Nuclei (AGNs) with their relativistic jets pointing towards line of sight of the observer [@Blandford1979]. Even though the mechanism behind the formation of relativistic jets is not fully understood yet, it is most likely related to the focusing properties of the fully ionized, rotating accretion disk [@Blandford1977]. Blazars include BL Lac objects and Flat-Spectrum Radio Quasars (FSRQs), where the significant difference between the two classes being their optical emission/absorption lines, which are strong for FSRQs, while weak or absent for BL Lacs [@Urry1995]. The spectral energy distribution (SED) of blazars consists of two broad emission components, where the low energy component peaks at optical to X-ray band, while the high energy component peaks at MeV to TeV band. BLLac objects are further subdivided based on the peak frequency ($\nu_s$) of their low energy component namely, high energy peaked BLLac (HBL; $\nu_s > 10^{15.3}$ Hz), intermediate energy peaked BLLac (IBL; $10^{14}<\nu_s\leq10^{15.3}$ Hz), and low energy peaked BLLac (LBL; $\rm \nu_s\leq10^{14}$ Hz) [@Fan2016] . In case of FSRQs, $\nu_s$ usually falls at relatively lower frequencies ($\lesssim10^{14}$ Hz). The low energy component of the blazar SED is commonly attributed to the synchrotron emission due to the interaction of relativistic electrons in the jet magnetic field; whereas the high energy component is explained as inverse Compton (IC) scattering process. If the target low energy photons for the IC process is the synchrotron photon itself then the IC mechanism is called Synchrotron Self Compton (SSC; @Marscher1985 [@Band1985]). On the other hand, if the photon origin is external to the jet, e.g. broad line region (BLR), obscuring torus, Cosmic Microwave Background (CMB) etc., then the process is called external Compton (EC) mechanism [@Dermer1992; @Sikora1994; @Shah2017]). Alternate to this leptonic interpretation of the high energy emission, hadronic models involving nuclear cascades were also put forth and are successful in explaining many observed features of blazars [@Mannheim1992; @Bottcher2007]. One of the distinct property of blazars is their rapid flux and spectral variability across the entire electromagnetic spectrum on time scales ranging from minutes to years. Though the cause of variability is still not well understood, plausible clues can be obtained by studying the long term flux distribution of blazars. Such studies have been performed in detail at X-ray energies for Seyfert galaxies and X-ray binaries, where the emission at these energies is dominated by the accretion disk or its corona. The X-ray flux of Seyfert1 IRAS13224-3809 using ASCA observations in different epochs, exhibit a log-normal distribution [@Gaskell2004]. In another study, [@Uttley2005] found that the X-ray flux of Seyfert1 NGC4051 also shows a log-normal distribution, which was comparable to the X-ray flux of the black-hole X-ray binary CygX-1. Linear relationship between the optical flux and the corresponding variation were noticed in Seyfert1 NGC4151 [@Lyutyi1987], which in turn is an indication of log-normality of flux distribution. A similar relationship was also noticed in X-ray band in both Seyfert1 Mrk766 [@Vaughan2003a], and Seyfert2 MCG6-30-15 [@Vaughan2003b]. The log-normality of flux distribution in a blazar was first detected in BL Lacertae, from the RXTE observations [@Giebels2009]. This result is particularly interesting since for blazars, X-ray emission originate from jets rather than the accretion disk or its environment. Hence, this result may hint the plausible disk-jet connection in blazars, which is still not clearly understood. The log-normality was later observed in many blazars at different energies. For instance, such behavior was inferred in Mrk421 and Mrk501 at Very High Energy (VHE $>$100GeV) band, though the data was noncontinuous [@Tluczykont2010]. Similarly, the 4-year flux distribution of blazars given in the third Fermi-LAT catalog of AGNs (3LAC; [@fermi3agn]), showed a log-normal behaviour. While quantifying the flux variability in Mrk421, [@Sinha2016] also noticed a log-normal flux distribution (more than normal) trend, through out the frequencies from radio to VHE. On the contrary, a detailed multi-wavelength study of FSRQ PKS1510-089, [@Kushwaha2016] found that the flux distribution follow two distinctive log-normal profiles in both optical and $\gamma$-rays, while X-ray flux distribution follow a single log-normal distribution. Interestingly, the $\gamma$-ray flux distribution of the same source, obtained from a near continuous data during August2008-October2015, was well fitted by a log-normal distribution and similar was the case of HBL Mrk421 and FSRQs B21520+31. On the other hand, the $\gamma$-ray flux distribution of FRI radio galaxy NGC1275 was not able to be represented by a log-normal or normal function, even though the rms increases linearly with flux [@Kushwaha2017]. In this work, we aim to study the flux distribution properties of the brightest Fermi blazars using the data collected in more than 8 years. We also investigate the associated spectral properties of these brightest blazars. Further, we examine the above properties in order to associate the unclassified blazar types (BCUs) with the known blazar classes. We select bright blazars from the 3LAC, and analyze the data (described in Sect.2). In order to overcome the effect of short-term flux variations, which are most likely associated with the change in the emission region geometry, we consider the flux in monthly bins for our study. After analyzing the features of the flux distribution, and verifying the log-normality (Sect.3), we study the association of flux distribution with spectral properties (Sect.4). The results and possible implications are discussed in Sect.5. *Fermi*-LAT analysis ==================== The Large Area Telescope (LAT) on board *Fermi* satellite is a pair conversion detector [@Atwood2009] with an effective area $\sim 8000 \,\rm cm^2/GeV$ photon, and field of view $\sim 2.4 $sr, in the energy range from 20MeV to more than 300GeV, which scans entire sky in every 3 hours. We made a primary selection of 25 FSRQs and 25 BLLacs from the four year NASA’S Fermi 3LAC interactive table [^1]. The selection was based on the criteria such that the chosen FSRQs and BLLacs should have monthly averaged photon flux $\rm > 6.5\times 10^{-9}\,photons\,cm^{-2} s^{-1}$ and $\rm > 5.5\times 10^{-9}\,photons\,cm^{-2} s^{-1}$ respectively, and the number of upper limits (i.e, non detections) should be less than or equal to 4. We have then downloaded the first 8.4 years of data ( from 2008 August to 2016 December) for the selected sources. The data were analyzed in the energy range from 100MeV to 500GeV, in a region of interest (ROI) of $10^o$ centering the nominal source positions. The analysis was carried out using the maximum likelihood method (*gtlike*) and standard *Fermi* [SCIENCE TOOLS]{} (version v9r12) with the instrument response function [‘$\rm P8R2\_SOURCE\_V6$’]{}, Galactic diffuse model [‘$\rm gll\_iem\_v06.fit$’]{} and isotropic background model [‘$\rm iso\_p8R2\_SOURCE\_V6\_v06.txt$’]{}. Events which were contaminated by the bright Earths limb were excluded using zenith angle cut of $90^o$. Further the time bins with $TS < 9$ were excluded, which correspond to a detection significance of $\sqrt{TS}\approx 3\sigma$ We estimated monthly photon flux, energy flux, and spectral index for all the sources using the maximum likelihood analysis. Flux distribution ================= The monthly average $\gamma$-ray flux obtained in the analysis of $\sim$ 100 months of data were distributed to a histogram of fluxes, for each source. An adaptive binning was used for each source to ensure the bin width is larger than the average error of the flux within a bin. Apart from the flux that corresponds to a TS value $TS\leq9$, we have also excluded the flux with larger uncertainty, such that F/$\delta$F$<$2. In order to avoid the bias due to a possible lack of lower luminosity flux states, we restrict our focus only on the blazars, for which the total excluded flux points (after the cuts mentioned above) are less than 10$\%$. After this cut, 38 (out of 50) blazars survived, which include 19 BLLacs and 19 FSRQs. We fit all 38 flux histograms in log-scale, with functions $$\rm{L(x)}=\frac{1}{\sqrt{2\pi}\sigma}\exp^{\frac{-(x-\mu)^2}{2\sigma^2}} \,\,\,\, [\rm{log-normal\,distribution}]$$ and $$\rm{G(X)}=\frac{1}{\sqrt{2\pi}\sigma}\exp^{\frac{(10^x-\mu)^2}{2\sigma^2}}10^x\,log_e(10)\,\,\,\, [\rm{normal\,distribution}]$$ where $\sigma$ and $\mu$ are the standard deviation and mean of the distribution, respectively. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Flux distribution of bright blazars in $\gamma$-ray band. The blue and red lines correspond to log-normal and normal fit respectively.[]{data-label="fig:blazar"}](ms89fig1_1.eps "fig:"){width="85.00000%"} ![Flux distribution of bright blazars in $\gamma$-ray band. The blue and red lines correspond to log-normal and normal fit respectively.[]{data-label="fig:blazar"}](ms89fig1_2.eps "fig:"){width="85.00000%"} ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------- ![(Continued)[]{data-label="fig:blazar"}](ms89fig1_3.eps "fig:"){width="85.00000%"} ![(Continued)[]{data-label="fig:blazar"}](ms89fig1_4.eps "fig:"){width="85.00000%"} ------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------- ![(Continued)[]{data-label="fig:blazar"}](ms89fig1_5.eps "fig:"){width="85.00000%"} ![(Continued)[]{data-label="fig:blazar"}](ms89fig1_6.eps "fig:"){width="85.00000%"} ------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------- ![(Continued)[]{data-label="fig:blazar"}](ms89fig1_7.eps "fig:"){width="85.00000%"} ------------------------------------------------------------------------------------- The flux histograms of the blazars are plotted in log-scale in Fig.\[fig:blazar\]. The normal and log-normal fits are shown as red and blue lines respectively. Fig.\[fig:skw\_skw\] (left panel) shows the comparison between the reduced $\chi^2$ obtained from normal and log-normal distributions. The fit parameters together with the computed skewness for both distributions are shown in Table\[table:blac\]. The flux distributions are found to be significantly skewed, whereas the skewness of the log of flux distribution is consistent with zero, thus suggesting log-normal trend (Fig.\[fig:skw\_skw\], right panel). Apart from calculating the reduced $\chi^2$, we have also performed Anderson-Darling (AD) test, in order to verify the normality/log-normality of the fits. The reduced $\chi^2$ of the fit, together with the AD test statistics and rejection/null hypothesis probability (p-value) are also shown in the Table\[table:blac\] . We note that the reduced $\chi^2$ for the normal flux distribution of some of the blazars fall in a reasonable range. However, the p-value estimated from the AD test rejects the normal distribution ($\rm p < 0.05$) for all sources, except for the FSRQ J0957.6+5523 (with a high reduced $\chi^2$ in this case). On the other hand, the AD test p-value, and the reduced $\chi^2$ of the flux distribution of J0957.6+5523 do not reject the log-normal distribution of flux either. It is interesting to note that the $\chi^2$ and AD tests do not reject the log-normality of the flux distribution of most of the blazars. Nevertheless, both tests reject the log-normality of the flux distribution of FSRQs J2329.3-4955, J1504.4+1029 and J1625.7-2527. Even though the AD tests marginally (eg: J1427.9-4206), and completely (J1512.8-0906) reject the log-normal fits of a few blazar flux distributions, the corresponding $\chi^2$ values are reasonable enough not to reject the log-normality. The standard deviation obtained from the log-normal fits, which is a measure of flux variability, is comparatively high for FSRQs. In the case of BL Lacs, HBLs show a lower variability, while the variability of LBLs are similar to that of FSRQs. The variability of IBLs roughly fall in between HBLs and LBLs. The mean values of the standard deviation obtained from the log-normal fit of the considered FSRQs, HBLs, IBLs and LBLs are 0.41$\pm$0.11, 0.21$\pm$0.04, 0.27$\pm$0.03, and 0.37$\pm$0.10 respectively. We have also noticed that the standard deviation of the flux distribution of J0957.6+5523 is significantly smaller compared to other FSRQs, and could be treated as a *steady FSRQ*. [@ l c c c c c c c c c c c c]{} & & & & & FSRQ & & & & & &\ Blazar & &&\ Name & width & centroid & skewness($\rm \kappa$) & $\chi^2/dof$ & AD (prob) && width$^*$ & centroid$^*$ & skewness$^*$ & $\chi^2/dof$ & AD (prob)\ J0457.0-2324 & 0.26$\pm$0.03 & -3.96$\pm$0.04 & -0.52$\pm$0.40 & 2.21 & 0.64 (0.09) && 8.33$\pm$1.40 & 11.1$\pm$1.60 & 0.71$\pm$0.40 & 1.12 & 1.50 (6.7e-04)\ J0730.2-1141 & 0.30$\pm$0.02 & -4.17$\pm$0.02 & -0.08$\pm$0.39 & 0.42 & 0.37 (0.41) && 6.39$\pm$1.17 & 6.71$\pm$2.01 & 1.40$\pm$0.39 & 2.88 & 3.32 (2.3e-08)\ J0957.6+5523 & 0.12$\pm$0.01 & -4.24$\pm$0.01 & -0.14$\pm$0.39 & 0.53 & 0.22 (0.84) && 1.54$\pm$0.21 & 5.85$\pm$0.28 & 0.62$\pm$0.39 & 2.89 & 0.69 (0.07122)\ J1127.0-1857 & 0.45$\pm$0.04 & -4.49$\pm$0.06 & 0.33$\pm$0.39 & 0.67 & 0.72 (0.06) && 6.48$\pm$2.04 & 3.17$\pm$3.18 & 1.83$\pm$0.39 & 7.38 & 6.40 (7.7e-16)\ J1224.9+2122 & 0.57$\pm$0.02 & -4.17$\pm$0.03 & 0.17$\pm$0.39 & 0.16 & 0.33 (0.51) && 23.8$\pm$8.34 & 8.26$\pm$17.6 & 3.41$\pm$0.39 & 11.7 & 10.97 ($<$2.2e-16)\ J1246.7-2547 & 0.30$\pm$0.03 & -4.45$\pm$0.04 & 0.31$\pm$0.39 & 0.93 & 0.72 (0.06) && 3.79$\pm$1.08 & 3.25$\pm$1.50 & 1.34$\pm$0.39 & 5.21 & 5.44 (1.6e-13)\ J1427.9-4206 & 0.33$\pm$0.02 & -3.75$\pm$0.03 & -0.88$\pm$0.40 & 0.75 & 1.02 (0.01) && 15.2$\pm$3.28 & 19.6$\pm$5.39 & 1.78$\pm$0.40 & 1.73 & 4.61 (1.6e-11)\ J1512.8-0906 & 0.37$\pm$0.04 & -3.73$\pm$0.05 & -0.80$\pm$0.38 & 0.71 & 2.44 (3.2e-06) && 15.7$\pm$4.52 & 17.6$\pm$6.85 & 2.27$\pm$0.38 & 5.26 & 10.36 ($<$2.2e-16)\ J0237.9+2848 & 0.33$\pm$0.02 & -4.43$\pm$0.02 & 0.59$\pm$0.39 & 0.36 & 0.71 (0.06) && 3.04$\pm$0.77 & 3.78$\pm$1.22 & 3.39$\pm$0.39 & 3.11 & 5.44 (1.6e-13)\ J2254.0+1608 & 0.60$\pm$0.10 & -3.37$\pm$0.12 & -0.57$\pm$0.60 & 1.44 & 0.83 (0.03) && 78.7$\pm$47.1 & 66.4$\pm$96.9 & 2.42$\pm$0.60 & 2.92 & 4.65 (9.7e-12)\ J1522.1+3144 & 0.27$\pm$0.02 & -4.07$\pm$0.02 & -1.02$\pm$0.39 & 0.84 & 0.62 (0.10) && 5.58$\pm$0.65 & 9.13$\pm$0.96 & 1.29$\pm$0.39 & 0.90 & 2.11 (2.1e-05)\ J1635.2+3809 & 0.38$\pm$0.03 & -4.14$\pm$0.03 & 0.27$\pm$0.39 & 0.72 & 0.76 (0.05) && 11.1$\pm$3.45 & 6.49$\pm$5.65 & 1.71$\pm$0.39 & 7.54 & 6.78 ($<$2.2e-16)\ J2329.3-4955 & 0.54$\pm$0.10 & -3.97$\pm$0.14 & -0.24$\pm$0.39 & 3.88 & 1.09 (0.01) && 21.7$\pm$6.67 & 15.6$\pm$12.9 & 1.90$\pm$0.39 & 6.33 & 4.16 (2.0e-10)\ J2345.2-1554 & 0.51$\pm$0.03 & -4.10$\pm$0.04 & -0.29$\pm$0.40 & 0.56 & 0.22 (0.84) && 14.5$\pm$5.63 & 10.7$\pm$9.85 & 2.09$\pm$0.40 & 5.70 & 6.66 ($<$2.2e-16)\ J0808.2-0751 & 0.45$\pm$0.05 & -4.53$\pm$0.06 & 0.51$\pm$0.41 & 0.77 & 0.65 (0.09) && 2.72$\pm$0.78 & 2.71$\pm$1.25 & 2.75$\pm$0.41 & 4.42 & 8.46 ($<$2.2e-16)\ J1229.1+0202 & 0.34$\pm$0.04 & -4.28$\pm$0.04 & 0.57$\pm$0.38 & 1.48 & 0.70 (0.06) && 3.89$\pm$1.25 & 5.29$\pm$1.93 & 3.50$\pm$0.38 & 3.12 & 12.89 ($<$2.2e-16)\ J1256.1-0547 & 0.34$\pm$0.02 & -3.94$\pm$0.02 & 0.29$\pm$0.37 & 0.38 & 0.27 (0.68) && 9.92$\pm$2.83 & 11.9$\pm$4.16 & 3.86$\pm$0.37 & 3.01 & 8.63 ($<$2.2e-16)\ J1504.4+1029 & 0.63$\pm$0.10 & -4.38$\pm$0.15 & 0.14$\pm$0.38 & 2.22 & 0.98 (0.01) && 17.9$\pm$7.48 & 5.29$\pm$11.4 & 1.84$\pm$0.38 & 11.7 & 8.54 ($<$2.2e-16)\ J1625.7-2527 & 0.30$\pm$0.04 & -4.28$\pm$0.05 & 0.47$\pm$0.45 & 2.12 & 1.11 (0.01) && 5.96$\pm$1.88 & 4.64$\pm$2.98 & 1.94$\pm$0.45 & 5.71 & 6.87 ($<$2.2e-16)\ & & & & & BL Lac & & & & & &\ J0222.6+4301 & 0.23$\pm$0.02 & -4.04$\pm$0.02 & 0.45$\pm$0.38 & 0.50 & 0.44 (0.28) && 5.23$\pm$1.07 & 8.88$\pm$1.42 & 2.35$\pm$0.38 & 2.30 & 3.55 (6.1e-09)\ J0238.6+1636 & 0.56$\pm$0.06 & -4.42$\pm$0.08 & 0.30$\pm$0.42 & 0.67 & 0.85 (0.03) && 5.19$\pm$2.04 & 3.57$\pm$2.97 & 2.98$\pm$0.42 & 6.97 & 7.70 ($<$2.2e-16)\ J0428.6-3756 & 0.33$\pm$0.03 & -3.95$\pm$0.04 & -0.90$\pm$0.38 & 1.84 & 11.45 (0.01) && 9.75$\pm$2.14 & 11.5$\pm$2.83 & 1.03$\pm$0.38 & 1.47 & 2.15 (1.7e-05)\ J0538.8-4405 & 0.45$\pm$0.06 & -3.48$\pm$0.08 & 0.42$\pm$0.38 & 1.30 & 1.02 (0.01) && 73.1$\pm$27.3 & 30.5$\pm$40.2 & 2.25$\pm$0.38 & 10.5 & 8.04 ($<$2.2e-16)\ J0721.9+7120 & 0.31$\pm$0.02 & -3.96$\pm$0.03 & -0.37$\pm$0.38 & 1.05 & 0.66 (0.08) && 9.66$\pm$1.75 & 11.1$\pm$2.56 & 1.03$\pm$0.38 & 1.94 & 2.25 (9.7e-06)\ J1104.4+3812 & 0.21$\pm$0.02 & -3.39$\pm$0.02 & -0.42$\pm$0.38 & 0.91 & 0.48 (0.22) && 23.0$\pm$1.89 & 45.1$\pm$3.15 & 1.67$\pm$0.38 & 0.92 & 2.67 (8.9e-07)\ J1427.0+2347 & 0.14$\pm$0.01 & -4.00$\pm$0.01 & -0.02$\pm$0.38 & 0.20 & 0.34 (0.50) && 3.74$\pm$0.56 & 9.92$\pm$0.68 & 0.61$\pm$0.38 & 1.78 & 1.13 (0.01)\ J1555.7+1111 & 0.17$\pm$0.01 & -3.84$\pm$0.01 & -0.04$\pm$0.38 & 0.22 & 0.5124 (0.19) && 6.88$\pm$0.78 & 14.6$\pm$0.97 & 1.00$\pm$0.38 & 1.04 & 2.12 (2.1e-05)\ J2158.8-3013 & 0.18$\pm$0.01 & -3.38$\pm$0.02 & 0.25$\pm$0.38 & 0.75 & 0.1918 (0.89) && 19.1$\pm$2.52 & 42.2$\pm$2.75 & 2.17$\pm$0.38 & 0.86 & 2.68 (8.4e-07)\ J0112.1+2245 & 0.31$\pm$0.01 & -4.42$\pm$0.01 & -0.05$\pm$0.38 & 0.13 & 0.14 (0.97) && 4.06$\pm$0.76 & 3.50$\pm$1.31 & 1.70$\pm$0.38 & 3.19 & 4.21 (9.2e-11)\ J0449.4-4350 & 0.23$\pm$0.02 & -4.02$\pm$0.03 & -0.44$\pm$0.38 & 1.57 & 0.51 (0.20) && 6.35$\pm$0.80 & 9.55$\pm$0.93 & 0.75$\pm$0.38 & 0.89 & 1.11 (0.01)\ J0509.4+0541 & 0.25$\pm$0.02 & -4.36$\pm$0.02 & 0.14$\pm$0.39 & 0.69 & 0.19 (0.89) && 3.63$\pm$0.95 & 3.65$\pm$1.27 & 1.53$\pm$0.39 & 4.44 & 3.02 (1.2e-07)\ J0818.2+4223 & 0.20$\pm$0.01 & -4.52$\pm$0.02 & -0.12$\pm$0.38 & 0.95 & 0.26 (0.72) && 1.61$\pm$0.16 & 3.02$\pm$0.18 & 0.91$\pm$0.38 & 0.72 & 2.05 (3.0e-05)\ J1015.0+4925 & 0.22$\pm$0.01 & -4.09$\pm$0.01 & 0.26$\pm$0.38 & 0.34 & 0.31 (0.56) && 4.45$\pm$0.66 & 8.11$\pm$0.97 & 2.80$\pm$0.38 & 1.36 & 3.94 (7.1e-10)\ J1058.5+0133 & 0.25$\pm$0.03 & -4.43$\pm$0.04 & -0.31$\pm$0.39 & 1.24 & 0.45 (0.27) && 2.70$\pm$0.43 & 3.83$\pm$0.41 & 0.82$\pm$0.39 & 1.09 & 2.67 (8.6e-07)\ J1653.9+3945 & 0.28$\pm$0.02 & -2.88$\pm$0.03 & -0.48$\pm$0.38 & 0.98 & 0.63 (0.10) && 9.33$\pm$1.22 & 13.6$\pm$1.58 & 0.86$\pm$0.38 & 1.06 & 1.30 (2.1e-3)\ J2202.7+4217 & 0.35$\pm$0.03 & -4.02$\pm$0.04 & -0.14$\pm$0.38 & 0.81 & 0.74 (0.05) && 11.6$\pm$3.02 & 9.69$\pm$4.56 & 1.09$\pm$0.38 & 4.85 & 2.82 (3.8e-07)\ J2236.5-1432 & 0.41$\pm$0.03 & -4.51$\pm$0.04 & 0.54$\pm$0.42 & 0.68 & 0.63 (0.10) && 7.91$\pm$2.48 & 2.23$\pm$4.92 & 3.63$\pm$0.42 & 8.65 & 12.24 ($<$2.2e-16)\ J0521.7+2113 & 0.28$\pm$0.02 & -4.05$\pm$0.02 & 0.45$\pm$0.46 & 0.34 & 0.38 (0.39) && 6.06$\pm$1.26 & 8.30$\pm$1.66 & 2.41$\pm$0.46 & 1.54 & 3.69 (2.6e-09)\ \ \[table:blac\] The case of Uncertain type blazars ---------------------------------- More than 500 sources were classified as blazars of uncertain type (BCU) in 3LAC. Even though these sources are associated with extra-galactic counterparts, and show some of the blazar characteristics, they lack reliable classification based on spectral information. In order to investigate the flux distribution properties of such sources, we analyzed the long term data (in the same time period as of the other bright blazars) of three brightest BCUs (namely, J0522.9-3628, J0532.0-4827, and J1328.9-5608). These sources are comparatively less brighter than the known classified blazars that we considered. After analyzing the *Fermi*-LAT data of three bright BCUs with the standard *Fermi* [SCIENCE TOOL]{} and using the quality cuts i.e. the flux after the cuts of $F/\delta F<2$ and $TS\leq9$, it was not possible to match with the acceptance criteria of 90%. Therefore, we modified our acceptance criteria from 90$\%$ to 60$\%$ of total flux. Since the flux states with TS$<$9 belong to low flux states (quiescent states) or less variable states, the difference in the acceptance criteria will not significantly bias our final results. The flux distribution of all three sources suggest log-normal distributions. The reduced $\chi^2$ of the log-normal flux distribution for the sources J0522.9-3628, J0532.0-4827, and J1328.9-5608 are found to be 0.29, 1.18 and 0.73 respectively (instead of 5.67, 3.09, 5.79 in the case of normal distribution). The AD test statistic p-values of the logarithmic flux distribution for the sources (in the same order as above) are 0.78, 0.80 and 0.17 (instead of $6.72\times10^{-15}$, $2.86\times10^{-16}$ and $2.37\times10^{-11}$ for normal distribution), which also propose log-normal distributions of flux over normal distributions. The obtained standard deviation from the log-normal fit of J0522.9-3628 and J1328.9-5608 are 0.29 and 0.34 respectively, while it is high (0.47, which is similar to FSRQs and LBLs) for the source J0532.0-4827. ![Flux distribution of bright blazars in $\gamma$-ray band. The pointers are same as in the Fig.\[fig:blazar\][]{data-label="fig:ucb"}](ms89fig3.eps "fig:"){width="100.00000%"}\ Association of flux distribution with spectral index ==================================================== In order to investigate the association of flux variability with spectral indices, we estimated the monthly spectra of the blazars. All the considered blazars were well described by either simple power law or log parabola models. In order to compare the spectral indices of the spectra described by both models, it would be meaningful to calculate the spectral indices at a specific energy, which was chosen to be 1 GeV (denoted by $\alpha_{1\rm{GeV}}$). The spectral index at energy E is defined by $$\alpha_E=\alpha+2\beta\log(E/E_p)$$ where $\alpha$ is spectral index at pivot Energy $\rm E_p$, and $\beta$ is the measure of spectral curvature. We have found that the average spectral indices for all FSRQs is 2.28$\pm$0.03. However, the FSRQ J2254.0+1608 (a.k.a. 3C 454.3, which is the brightest blazar in $\gamma-ray$ band) shows a harder spectrum of index 1.72$\pm$0.06. It has to be also noted that the spectral index of the *steady FSRQ* J0957.6+5523 is also smaller, which is 1.91$\pm$0.05. The mean spectral indices of all other FSRQs fall in the range of 2.08-2.75. However, the mean spectral indices of BLLacs fall in the range of 1.61-2.49, with a mean value of 2.01$\pm$0.11. The mean spectral indices of BCUs J0522.9-3628, J0532.0-4827 and J1328.9-5608 are 2.70, 2.64, and 2.74 respectively. These values fall in the range of spectral indices of other bright FSRQs, LBLs or IBLs, but comparatively higher than the spectral indices of HBLs. The flux variability has been plotted against the spectral index in Fig\[fig:lum\_sig\]. The HBLs (red square) fall in the left-low corner of the diagram, while FSRQs (pink circle) show a wider distribution. The IBLs (green triangle) and LBLs (gray dumbbell) fall in similar range of spectral index, though the variability distribution is wider in the case of former. The boxes represent the two-standard deviation uncertainty in $\alpha_{1\,GeV}$ and $\sigma$ from their respective mean value. The bright BCUs that were analyzed fall in the band of FSRQs. We also notice a slight correlation between the spectral index and standard deviation of the flux. The $\alpha_{1\,GeV}$ and $\sigma$ of all blazars can be roughly fitted by a straight line of slope 0.24$\pm$0.04, which is indicated by a blue dotted line in the figure\[fig:lum\_sig\]. The brightest BCUs show similar features as in the case of other blazars with respect to spectral index and flux distribution. All three BCUs (indicated as purple tilted square in fig.\[fig:lum\_sig\]) fall beyond the 2-standard deviation uncertainty region (dotted box) of BLLac sources. However, they are placed within the 2-standard deviation uncertainty region of FSRQ. ![The standard deviation ($\sigma$) of the blazar flux distribution vs their corresponding average spectral index at 1GeV ($\alpha_{1\,\rm{GeV}}$). Each blazar class has been marked with different pointers. HBL: red filled square, IBL: green triangle, LBL: gray dumbbell, FSRQ: pink circle, BCU: purple tilted square. The FSRQs which reject the log-normal flux distribution has been marked as pink open circle. The *steady FSRQ* has been shown separately black filled circle. The rectangle boxes correspond to two-standard-deviation uncertainty from the mean of $\sigma$ and $\alpha_{1\,\rm{GeV}}$ for each blazar class. The color of the box is chosen to be same as that of pointers for each blazar class. The blue-violet dotted box corresponds to two-standard-deviation uncertainty from the mean of $\sigma$ and $\alpha_{1\,\rm{GeV}}$ for all BLLac sources[]{data-label="fig:lum_sig"}](ms89fig4.eps) Additionally, we have also compared the monthly spectral indices of each source with their corresponding luminosity. It is interesting to note that every source shows *harder when brighter* phenomena. The monthly spectral indices and corresponding luminosity can be well fitted by a straight line. The average indices are found to be -0.09+0.01 for FSRQs, while it is -0.11$\pm$0.01, -0.13$\pm$0.01, -0.13$\pm$0.01 for HBL, IBL, and LBL respectively, suggesting the *harder when brighter* phenomena among the different bright blazar classes are not significantly different. Discussion ========== After a detailed study on the $\gamma$-ray flux of 38 brightest blazars, we found that the flux distributions predominantly suggest log-normal distribution rather than a normal distribution. We verified the log-normality (over normality) using both reduced $\chi^2$ and AD-test. The log-normality was rejected only in the case of three (in a sample of 38) blazars. However, the normal distribution was rejected for all blazars (except J0957.6+5523, though the reduced $\chi^2$ was high). The flux distribution of the three brightest BCUs follow log-normal distribution. From the obtained spectral index and the flux standard deviation parameters, they fall beyond the 2-standard deviation uncertainty limits of HBL, IBL and LBL. Though it can not be asserted, we are tempted to associate these sources with FSRQs. The log-normal distribution of the observed flux indicates the perturbation associated with the emission process to be of multiplicative nature rather than additive [@Lyubarskii1997; @Arevalo2006]. Flux variation in blazars can be attributed to the complex interplay between the intrinsic and source parameters. A simple scenario is to associate the flux variation with the fluctuation in the emitting electron number density or the magnetic field. However, the linear dependence of the these quantities with the differential flux suggests, this will cause a normal flux distribution contrary to the observations. Alternatively, the particle acceleration and the diffusion processes can modify the shape of the emitting electron distribution [@Kirk1998] and hence can be accounted for various flux distributions, including a log-normal one. The flux variation can also be associated with the change in the emission region geometry. Even though the change in volume associated with this can only produce normal flux distribution, inclusion of light travel time effects can significantly modify the same [@Chiaberge1999]. However, the timescales associated with these processes are too short and hence will not reflect the log-normal distribution obtained in our study, where we used monthly averaged fluxes. A log-normal flux distribution can directly hint the linkage of blazar jet with the accretion phenomena since the latter is well proven to produce such distribution through the study of galactic X-ray binaries (XRBs)[@Uttley2001]. The fluctuations in the disk at different radii are known to be produced independently by viscosity fluctuation on local viscous time scales, which modulates the mass accretion rate at larger distances from black-hole. The accretion rate variations then propagate to small radii through accretion flow and concoction of variations at different radii results in multiplicative emission. This model was put forward by [@Lyubarskii1997] for explanation of observed X-ray variability time-scales in XRBs. Also, for non beamed accreting objects the variability timescales are found to be proportional to $\rm M/\dot{m}$, where M is mass of black hole and $\rm \dot{m}$ is accretion rate [@Kording2007]. [@McHardy2008] had found that same relation surprisingly holds even for beamed jet emission from blazars e.g, 3C273, which should have otherwise shorter observed variability timescale due to relativistic time dilation than the timescale predicted using the black hole mass and accretion rate. Consequently, this lead to inference that source of variations in blazars lie out-side the jet i.e, in the accretion disks which then modulates the jet emission. A detailed study of month scale averaged flux distribution of blazars can hence be a key to understand disk-jet connection. On the contrary to the interpretations above, a log-normal flux distribution can also arise from additive processes under specific conditions. For example, if the blazar jet is assumed to be a large collection of mini-jets, then the logarithm of composite flux will show a normal distribution [@Biteau2012]. We note that the AD statistics does not reject the normality of flux distribution of J0957.6+5523. Moreover, the standard deviation obtained from the flux distribution of this source exhibits a significant difference from that of other blazars. 3LAC labeled this source as an FSRQ, based on the presence of broad optical emission lines, large redshift and high $\gamma$-ray luminosity of the order of $\rm \approx 10^{47} erg\,s^{-1}$. However, the integrated spectrum and morphological properties obtained from the VLBA observations question the FSRQ classification of the source, and suggest it as one of the weakest Compact Symmetric Object [@Rossetti2005]. Moreover, the brightness temperature of this source was found to be significantly lower ($2\times10^{8}$K at 5GHz, @Taylor2007), than that of other $\gamma$-ray blazars [@McConville2011]. These studies, together with our results, suggest that more multi-wavelength studies are required before associating this source to an FSRQ. ![Double log-normal fit to the flux distribution of the blazars J1625.7-2527, J1504.4+1029, and J2329.3-4955, which reject log-normal distribution. Since the statistics is not significant enough, these fits should be taken only as indicative.[]{data-label="fig:double"}](ms89fig5_1.eps "fig:"){width="\linewidth"} ![Double log-normal fit to the flux distribution of the blazars J1625.7-2527, J1504.4+1029, and J2329.3-4955, which reject log-normal distribution. Since the statistics is not significant enough, these fits should be taken only as indicative.[]{data-label="fig:double"}](ms89fig5_2.eps "fig:"){width="\linewidth"} ![Double log-normal fit to the flux distribution of the blazars J1625.7-2527, J1504.4+1029, and J2329.3-4955, which reject log-normal distribution. Since the statistics is not significant enough, these fits should be taken only as indicative.[]{data-label="fig:double"}](ms89fig5_3.eps "fig:"){width="\linewidth"} We have also investigated the possibility of double log-normal distribution of fluxes for the sources that reject log-normal distribution both in $\chi^2$ and AD test (J1625.7-2527, J1504.4+1029, and J2329.3-4955). The two distinct log-normal profiles may indicate different flux states correspond to a low and high states of the source [@Kushwaha2016]. It is interesting to note that the flux distribution of all three sources exhibit hints of double log-normal distribution (Fig.\[fig:double\]). However, the statistics of this distribution is not significant enough, hence these result should be taken only as an indicative. Another implication of our study is on the averaging of long term flux. We recommend the usage of the average of flux in log scale, rather than estimating average flux in a linear scale, especially for the highly variable sources. We show the difference of averaged flux in both linear and log-scale in Fig \[fig:flux\_vs\_flux\]. For example, in the case of FSRQs, the average value of Flux/Log$_{10}$(Flux) falls around $\sim$1.7, while the maximum value (in the case of 3C454.3) goes up-to 2.8. These values imply that averaging flux over a linear scale will significantly overestimate the same, which would in turn gives rise to inaccurate SED non-thermal emission model parameters. Conclusion ========== We studied in detail the flux distribution properties of 38 brightest $\gamma$-ray blazars using *Fermi*-LAT data of more than 8 years. The flux distribution suggest log-normal distribution, for 35 blazars, indicating a multiplicative perturbation associated with the emission process. Similar features were obtained also in the case of BCUs. On the other hand, the flux distributions of three FSRQs – J2329.3-4955, J1504.4+1029, and J1625.7-2527 – reject both log-normal and normal distribution. This could be due to two or more independent flux states associated with the source, however, more statistics is required to study these effects in detail. It would be also interesting to perform an elaborate study with better statistics for more blazars in $\gamma$-rays, and compare the properties with that of their X-ray counterparts. Acknowledgement =============== ZS, SS and NI are thankful to Indian Space Research Organization program (ISRO-RESPOND) for the financial support under grant no. ISRO/RES/2/396. [^1]: https://fermi.gsfc.nasa.gov/ssc/data/access/lat/4yr$_-$catalog/3FGL-table/

--- abstract: 'G 207-9 and LP 133-144 are two rarely observed ZZ Ceti stars located in the middle and close to the blue edge of the ZZ Ceti instability domain, respectively. We aimed to observe them at least during one observing season at Konkoly Observatory with the purpose of extending the list of known pulsation modes for asteroseismic investigations and detect any significant changes in their pulsational behaviour. We determined five and three new normal modes of G 207-9 and LP 133-144, respectively. In LP 133-144, our frequency analysis also revealed that at least at three modes there are actually triplets with frequency separations of $\sim4\mu$Hz. The rotational period of LP 133-144 based on the triplets is $\simeq42$h. The preliminary asteroseismic fits of G 207-9 predict $T_{\rmn{eff}}=12\,000$ or $12\,400$K and $M_*=0.855-0.870\,M_{\sun}$ values for the effective temperature and mass of the star, depending on the assumptions on the spherical degree ($l$) values of the modes. These results are in agreement with the spectroscopic determinations. In the case of LP 133-144, the best-fitting models prefer $T_{\rmn{eff}}=11\,800$K in effective temperature and $M_*\geq0.71\,M_{\sun}$ stellar masses, which are more than $0.1\,M_{\sun}$ larger than the spectroscopic value.' author: - | Zs. Bognár,$^{1}$[^1] M. Paparó,$^{1}$ L. Molnár,$^{1}$ P. I. Pápics,$^{2}$ E. Plachy,$^{1}$ E. Verebélyi,$^{1}$ and Á. Sódor$^{1}$\ $^{1}$Konkoly Observatory, MTA Research Centre for Astronomy and Earth Sciences, Konkoly Thege Miklós út 15-17, H–1121 Budapest\ $^{2}$Instituut voor Sterrenkunde, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium bibliography: - 'g207\_lp133\_mnras.bib' date: 'Accepted XXX. Received YYY; in original form ZZZ' title: 'G 207-9 and LP 133-144: light curve analysis and asteroseismology of two ZZ Ceti stars ' --- \[firstpage\] techniques: photometric – stars: individual: G 207-9, LP 133-144 – stars: interiors – stars: oscillations – white dwarfs Introduction ============ ZZ Ceti (or DAV) stars constitute the most populated group of pulsating white dwarfs. Their light variations are the results of local changes in their surface temperatures due to the excitation of nonradial $g$-mode pulsations in their non-degenerate envelope. This envelope consists of an inner helium and an outer hydrogen layer, therefore, the hydrogen Balmer-lines dominate the spectra of ZZ Ceti stars. The pulsations are driven by the so-called ‘convective driving’ mechanism [@1991MNRAS.251..673B; @1999ApJ...511..904G], as the driving region is associated with the base of the envelope convection zone. Otherwise, pulsating white dwarfs are just like their non-pulsating counterparts, and the information we gain on white dwarf structures by asteroseismic investigations can be essential to understand white dwarfs as a whole group. ZZ Ceti stars are short-period and low-amplitude pulsators with $11\,000-13\,000$K effective temperatures and modes excited typically in the $100-1400$s period range with $\sim$mmag amplitudes. However, within this period and amplitude range the stars exhibit a large variety of pulsational behaviour, from the star showing one rotational triplet only (G 226-29, @1995ApJ...447..874K) to the ‘rich’ DA pulsators with more than a dozen normal modes known (see e.g. @2009AIPC.1170..621B). Temporal variations in their pulsational behaviour are also well documented in white dwarfs, e.g. the case of GD 154 showing once a strongly non-sinusoidal light curve with one dominant mode and a series of its harmonic and near-subharmonic ($\sim n/2f_i$) peaks in its Fourier transform, or just behaving as a simple multiperiodic pulsator another time [@1978ApJ...220..614R; @2013MNRAS.432..598P]. For comprehensive reviews of the observational and theoretical aspects of pulsating white dwarf studies, see the papers of and . We also refer to the work of @2013ApJ...762...57V, in which the authors successfully reconstructed the boundaries of the empirical ZZ Ceti instability strip applying theoretical calculations, including its extension to lower effective temperatures and surface gravities, that is, further to the domain of the extremely low-mass DA pulsators. White dwarf observations with the *Kepler* space telescope revealed another new feature in ZZ Ceti stars, namely recurring increases in the stellar flux (‘outbursts’) in two cool DAVs being close to the red edge of the instability strip [@2015ApJ...809...14B; @2015ApJ...810L...5H]. G 207-9 and LP 133-144 were observed as part of our project aiming at least one-season-long local photometric time series measurements of white dwarf pulsators. Our purposes are to examine the short-term variability of pulsation modes in amplitude and phase, and to obtain precise periods for asteroseismic investigations. We have already published our findings on two cool ZZ Ceti stars (KUV 02464+3239, @2009MNRAS.399.1954B and GD 154, @2013MNRAS.432..598P), one ZZ Ceti located in the middle of the instability strip (GD 244; @2015ASPC..493..245B), and the DBV type KUV 05134+2605 . With the observations of G 207-9 and LP 133-144, we extended our scope of investigations to higher effective temperatures in the DAV instability domain. Observations and data reduction =============================== We collected photometric data both on G 207-9 ($B=14.8$mag, $\alpha_{2000}=18^{\mathrm h}57^{\mathrm m}30^{\mathrm s}$, $\delta_{2000}=+33^{\mathrm d}57^{\mathrm m}25^{\mathrm s}$) and LP 133-144 ($B=15.5$mag, $\alpha_{2000}=13^{\mathrm h}51^{\mathrm m}20^{\mathrm s}$, $\delta_{2000}=+54^{\mathrm d}57^{\mathrm m}43^{\mathrm s}$) in the 2007 observing season. We used the 1-m Ritchey-Chrétien-Coudé telescope at Piszkéstető mountain station of Konkoly Observatory. The detector was a Princeton Instruments VersArray:1300B back-illuminated CCD camera. The measurements were made in white light and with 10 or 30s integration times, depending on the weather conditions. We observed G 207-9 and LP 133-144 on 24 and 28 nights, respectively. Tables \[table:logg207\] and \[table:loglp\] show the journals of observations. Altogether, 85 and 137h of photometric data were collected on G 207-9 and LP 133-144, respectively. ----- --------- --------------- ------- -------- -------- Run UT date Start time Exp. Points Length no. (2007) (BJD-2450000) (s) (h) 01 Mar 26 4185.540 30 279 2.57 02 Mar 27 4186.533 30 280 2.70 03 Apr 02 4192.527 10 743 2.70 04 Apr 03 4193.540 30 208 1.88 05 Jun 15 4267.351 10 1001 4.46 06 Jun 16 4268.374 30 427 4.10 07 Jun 17 4269.396 10 1028 3.68 08 Jun 18 4270.341 30 500 4.75 09 Jun 19 4271.345 10 1173 4.89 10 Jun 20 4272.454 30 198 2.23 11 Jul 06 4288.389 30 184 2.42 12 Jul 07 4289.347 10 1417 5.18 13 Jul 08 4290.337 10 1449 5.60 14 Jul 09 4291.351 30 34 0.30 15 Jul 10 4292.434 30 227 2.27 16 Jul 26 4308.434 30 349 3.12 17 Jul 27 4309.337 10 1071 3.69 18 Jul 30 4312.452 30 134 1.42 19 Jul 31 4313.312 30 492 5.03 20 Aug 01 4314.460 10 885 3.05 21 Aug 10 4323.327 30 46 0.40 22 Aug 13 4326.323 30 721 6.43 23 Aug 14 4327.341 10 1646 5.74 24 Aug 15 4328.315 10 1896 6.53 85.16 ----- --------- --------------- ------- -------- -------- : Journal of observations of G 207-9. ‘Exp.’ is the exposure time used.[]{data-label="table:logg207"} ----- --------- --------------- -------- -------- -------- Run UT date Start time Exp. Points Length no. (2007) (BJD-2450000) (s) (h) 01 Jan 15 4115.614 30 299 2.81 02 Jan 17 4117.622 30 154 1.54 03 Jan 26 4126.615 30 233 2.19 04 Jan 28 4128.544 30 441 4.17 05 Jan 30 4130.528 30 493 4.58 06 Feb 17 4148.551 30 375 3.53 07 Mar 15 4175.283 30 725 9.27 08 Mar 16 4176.279 30 960 9.29 09 Mar 22 4182.356 30 562 6.04 10 Mar 24 4184.496 30 410 3.76 11 Mar 25 4185.398 30 338 3.12 12 Mar 26 4186.282 30 636 5.88 13 Mar 27 4187.273 30 718 7.94 14 Mar 30 4190.307 30 862 8.05 15 Mar 31 4191.357 30 325 2.98 16 Apr 01 4192.289 30 596 5.47 17 Apr 03 4194.276 30 202 1.92 18 Apr 12 4203.388 10 1483 5.58 19 Apr 13 4204.337 10 1808 6.80 20 Apr 14 4205.304 10 1946 7.66 21 Apr 15 4206.309 10 1608 7.27 22 Apr 16 4207.296 10 1422 5.51 23 Apr 17 4208.378 10 1439 5.26 24 May 10 4231.316 10 1026 4.08 25 May 12 4233.358 30 259 2.33 26 May 13 4234.315 10 1192 4.32 27 May 14 4235.372 10 763 2.82 28 May 16 4237.365 30 342 3.12 137.27 ----- --------- --------------- -------- -------- -------- : Journal of observations of LP 133-144. ‘Exp.’ is the exposure time used.[]{data-label="table:loglp"} We reduced the raw data frames following the standard procedure: we applied bias, dark and flat corrections on the frames using <span style="font-variant:small-caps;">iraf</span>[^2] routines, and performed aperture photometry of the variable and comparison stars with the <span style="font-variant:small-caps;">iraf daophot</span> package. We converted the observational times of every data point to barycentric Julian dates in barycentric dynamical time ($\mathrm{BJD_{TDB}}$) using the applet of @2010PASP..122..935E[^3]. We then checked the comparison star candidates for variability and instrumental effects. We selected three stars in the field of G 207-9 and two stars in the field of LP 133-144 and used the averages of these reference stars as comparisons for the differential photometry of the two pulsators. The panels of Fig. \[fig:ccd\] show the variable and the comparison stars in the CCD fields. We applied low-order polynomial fits to the light curves to correct for the instrumental trends and for the atmospheric extinction. This method did not affect the pulsation frequency domains. Figure \[fig:lcshort\] shows two illustrative light curve segments of G 207-9 and LP 133-144. All the light curves obtained for both pulsators are presented in Appendix \[app:g207\] and in Appendix \[app:lp133\]. Frequency analyses of the light curves ====================================== We determined the frequency content of the datasets on daily, weekly or monthly, and yearly time bases. We analysed the daily observations with custom developed software tools, as the command-line light curve fitting program <span style="font-variant:small-caps;">LCfit</span> [@2012KOTN...15....1S]. <span style="font-variant:small-caps;">LCfit</span> has linear (amplitudes and phases) and nonlinear (amplitudes, phases and frequencies) least-squares fitting options, utilizing an implementation of the Levenberg-Marquardt least-squares fitting algorithm. The program can handle unequally spaced and gapped datasets. <span style="font-variant:small-caps;">LCfit</span> is scriptable easily, which made the analysis of the relatively large number of nightly datasets very effective. We performed the standard Fourier analyses of the weekly or monthly data subsets and the whole light curves with the photometry modules of the Frequency Analysis and Mode Identification for Asteroseismology (<span style="font-variant:small-caps;">famias</span>) software package [@2008CoAst.155...17Z]. Following the traditional way, we accepted a frequency peak as significant if its amplitude reached the 4 signal-to-noise ratio (S/N). The noise level was calculated as the average amplitude in a $\pm1200\,\mu$Hz interval around the given frequency. {width="16cm"} {width="6cm"} {width="6cm"} G 207-9 ------- ### Previous observations G 207-9 was announced as the 8th known member of pulsating white dwarf stars in 1976 [@1976ApJ...207L..37R]. Four high ($\mathrm{F}_1$–$\mathrm{F}_4$) and one low ($\mathrm{F}_5$) amplitude peaks were detected at $\mathrm{F}_1=1354$, $\mathrm{F}_2=1794$, $\mathrm{F}_3=3145$, $\mathrm{F}_4=3425$ and $\mathrm{F}_5=3860\,\mu$Hz. Even though G 207-9 is a relatively bright target, and has been known as a pulsator for decades, no other time series photometric observations and frequency analysis have been published on this star up to now. ### Konkoly observations {#sect:g207freq} The analyses of the daily datasets revealed one dominant and four low amplitude peaks in the FTs. The dominant frequency of all nights’ observations in 2007 was at $3426\,\mu$Hz. This was the 3rd highest amplitude mode in 1975 ($\mathrm{F}_4=3425\,\mu$Hz). Two lower amplitude frequencies at $1672$ and $5098\,\mu$Hz reached the 4S/N detection limit in 12 and 14 of the daily datasets, respectively. Two additional low-amplitude frequencies were detected at $1608$ and $7725\,\mu$Hz in 3 and 4 cases, respectively. These frequencies are medians of the daily values. The $1608$ and $1672\,\mu$Hz frequencies could be determined separately only in the last three nights’ datasets. We present the FT of one night’s dataset (the second longest run) in the first panel of Fig. \[fig:g207FTa\]. Considering the consecutive nights of observations, five weekly time base datasets can be formed: Week 1 (JD2454185–193), Week 2 (JD2454267–272), Week 3 (JD2454288–292), Week 4 (JD2454308–314) and Week 5 (JD2454323–328). The Fourier analyses of these data verified the five frequencies found by the daily observations. In three cases the first harmonic of the dominant frequency was also detected. Additionally, the analysis of the Week 2 and Week 5 data suggested that the peaks at $\sim1672$ and $\sim5098\,\mu$Hz may be actually doublets or triplets and not singlet frequencies. The separations of the frequency components were found to be between $\sim2$ (close to the resolution limit) and $14\,\mu$Hz, but we mark these findings uncertain because of the effect of the 1d$^{-1}$ aliasing. The 2nd–6th panels of Fig. \[fig:g207FTa\] shows the FTs of the weekly datasets. We found only slight amplitude variations from one week to another. The amplitude of the dominant frequency varied between 8.6 and 10.5mmag. ![G 207-9: amplitude spectra of one night’s observation (*top panel*) and the weekly datasets (*lower panels*).[]{data-label="fig:g207FTa"}](g207FTa.eps){width="\columnwidth"} The standard pre-whitening of the whole dataset resulted 26 frequencies above the 4S/N limit. Most of them are clustering around the frequencies already known by the analyses of the daily and weekly datasets. Generally, amplitude and (or) phase variations during the observations can be responsible for the emergence of such closely spaced peaks. In such cases, these features are just artefacts in the FT, as we fit the light curve with fixed amplitudes and frequencies during the standard pre-whitening process. Another possibility is that some of the closely spaced peaks are rotationally split frequencies. We can resolve such frequencies if the time base of the observations is long enough. The Rayleigh frequency resolution ($1/\Delta T$) of the whole dataset is $0.08\,\mu$Hz. We also have to consider the 1d$^{-1}$ alias problem of single-site observations, which results uncertainties in the frequency determination. ---------- -------------------- ------- ------ ------ ------- $f_1$ $3426.303\pm0.001$ 291.9 10.1 111.5 $f_2$ $1678.633\pm0.003$ 595.7 2.0 15.4 $f_1^-?$ $3414.639\pm0.004$ 292.9 11.7 1.6 18.0 $f_3$ $5098.861\pm0.003$ 196.1 1.2 12.5 $f_2^-?$ $1667.328\pm0.005$ 599.8 11.3 1.1 8.6 $f_4$ $1603.071\pm0.004$ 623.8 1.1 8.1 $f_1^+?$ $3437.384\pm0.005$ 290.9 11.1 1.0 11.6 $f_5$ $7726.540\pm0.003$ 129.4 1.0 13.0 $f_4^-?$ $1595.481\pm0.004$ 626.8 7.6 0.8 6.1 $2f_1$ $6852.604\pm0.005$ 145.9 0.6 8.5 $f_6$ $3146.670\pm0.007$ 317.8 0.5 5.0 $f_7$ $3276.485\pm0.008$ 305.2 0.4 4.6 ---------- -------------------- ------- ------ ------ ------- : G 207-9: frequency content of the 2007 dataset. The errors were calculated by Monte Carlo simulations. $\delta f$ denotes the frequency differences of the closely spaced frequencies to $f_1$, $f_2$ or $f_4$.[]{data-label="table:g207freq"} {width="17.5cm"} ![G 207-9: comparison of the frequencies obtained in 1975 (*red dashed lines*) and in 2007 (*black solid lines*). The amplitudes of the 1975 observations are from the paper of @2006ApJ...640..956M.[]{data-label="fig:g207oldnew"}](g207oldnewfrek.eps){width="\columnwidth"} We checked the frequency content of the whole dataset by averaging three consecutive data points of the $10\,$s measurements as a test. That is, we created a new, more homogeneous dataset mimicking $30\,$s exposure times. We then compared the frequency solutions of this $30\,$s dataset with the frequencies of the original mixed $10$–$30\,$s data. Finally, we accepted as the frequencies characterizing the whole light curve the frequencies that could be determined in both datasets, that is, without 1d$^{-1}$ differences. This resulted a reduced frequency list of 12 frequencies. We list them in Table \[table:g207freq\]. There are still several closely spaced frequencies around three of the main frequencies ($f_1$, $f_2$ and $f_4$) remained with separations between $7.6$ and $11.7\,\mu$Hz. In the case of $f_1$ and $f_2$, these separations are close to $11.574\,\mu$Hz (1d$^{-1}$). It is possible that at least some of these frequency components are results of rotational splitting, but considering the uncertainties mentioned above, we do not accept them as rotationally split frequencies. In such cases when the frequency separations of the rotationally split components are around 1d$^{-1}$, multi-site or space-based observations are needed for reliable determination of the star’s rotational rate. Summing it up: besides the five frequencies ($f_1$–$f_5$) known also by the analyses of shorter (daily and weekly) data segments, we could detect two additional independent frequencies ($f_6$ and $f_7$) in the whole dataset. Frequency $f_6$ at $3146.7\,\mu$Hz was also detected in 1975 ($\mathrm{F}_3=3145\,\mu$Hz). Moreover, this was one of the dominant peaks at that time. Frequency $f_7$ at $3276.5\,\mu$Hz is a newly detected one. Note that the frequency $f_7$ is close to $f_2+f_4=3281.7\,\mu$Hz, however, the difference is $5.2\,\mu$Hz, which seems too large to claim that $f_7$ is the linear combination of these peaks considering the errors. Thus, we consider $f_7$ as an independent mode. Fig. \[fig:g207prewh\] shows the FT of the whole dataset and the frequency domains of $f_1-f_7$ on separate panels. Comparing the frequency content of the 1975 and 2007 observations, we can conclude that three of the five frequencies found in the 1975 dataset did not appear in 2007 ($\mathrm{F}_1$, $\mathrm{F}_2$ and $\mathrm{F}_5$), while two stayed at an observable level ($\mathrm{F}_3=f_6$ and $\mathrm{F}_4=f_1$). Figure \[fig:g207oldnew\] summarizes the frequencies of the two epochs. It seems that even though there were no large amplitude variations during our five-months observing season in 2007, on the time scale of years or decades, remarkable changes can happen in the pulsation of G 207-9: new frequencies can be excited to a significant level, while other modes can disappear. LP 133-144 ---------- ### Previous observations The variability of LP 133-144 was discovered in 2003 [@2004ApJ...600..404B]. Four pulsation frequencies were determined at that time, including two closely spaced peaks: $\mathrm{F}_1=3055.1$, $\mathrm{F}_2=3258.4$, $\mathrm{F}_3=3284.1$ and $\mathrm{F}_4=4780.6\,\mu$Hz. Similarly to the case of G 207-9, no further results of time series photometric observations have been published up to now. ### Konkoly observations {#sect:lp133freq} We found four recurring frequencies in the daily datasets at 3055, 3270, 3695 and 4780$\mu$Hz (median values). Their amplitudes varied from night to night, but the 4780$\mu$Hz peak was the dominant in almost all cases. One additional peak exceeded the 4S/N limit at 5573$\mu$Hz, but on one night only. We created four monthly datasets and analysed them independently. These are Month 1 (JD2454115–130), Month 2 (JD2454175–194), Month 3 (JD2454203–208) and Month 4 (JD2454231–237). The analyses of the monthly data revealed that at the 3270, 3695 and 4780$\mu$Hz frequencies there are actually doublets or triplets with 2.6–4.7$\mu$Hz frequency separations. This explains the different amplitudes in the daily FTs. The 3055$\mu$Hz frequency was found to be a singlet. In Month 3, the linear combination of the largest amplitude components of the 3270 and 4780$\mu$Hz multiplets also could be detected. The 5573$\mu$Hz frequency was significant in Month 2. The panels of Fig. \[fig:lp133FTa\] show the FT of one daily dataset and the monthly data. As in the case of G 207-9, there were no remarkable amplitude variations from one month to another. ![LP 133-144: amplitude spectra of one night’s observation (*top panel*) and the monthly datasets (*lower panels*).[]{data-label="fig:lp133FTa"}](lp133FTa.eps){width="\columnwidth"} The analysis of the whole 2007 dataset resulted in the detection of 19 significant frequencies in the $\sim2300-8000\,\mu$Hz frequency region. We also performed the test analysis utilizing the averaged 30s dataset, which confirmed the presence of the 14 largest amplitude frequencies (the other five peaks remained slightly under the significance level). Thus we accepted them as the frequencies characterizing the pulsation of LP 133-144 and list them in Table \[table:lp133freq\]. The Rayleigh frequency resolution of the whole dataset is $0.09\,\mu$Hz. --------- -------------------- ------- ----- ------ ------- $f_1$ 4780.555$\pm$0.001 209.2 10.9 100.9 $f_2$ 3269.302$\pm$0.001 305.9 3.9 35.4 $f_3$ 3695.083$\pm$0.002 270.6 3.5 31.2 $f_3^-$ 3691.627$\pm$0.002 270.9 3.5 3.4 30.5 $f_2^+$ 3272.475$\pm$0.002 305.6 3.2 3.0 26.7 $f_4$ 3055.125$\pm$0.002 327.3 2.8 25.1 $f_3^+$ 3698.551$\pm$0.003 270.4 3.5 2.0 18.4 $f_2^-$ 3266.125$\pm$0.005 306.2 3.2 1.2 10.4 $f_1^+$ 4784.696$\pm$0.005 209.0 4.1 1.1 10.6 $f_1^-$ 4776.400$\pm$0.007 209.4 4.2 1.0 8.7 $f_5$ 7116.986$\pm$0.010 140.5 0.6 5.9 $f_6^+$ 5574.381$\pm$0.009 179.4 4.8 0.6 6.0 $2f_1$ 9561.115$\pm$0.011 104.6 0.5 5.5 $f_6^-$ 5564.876$\pm$0.013 179.7 4.7 0.5 5.1 ($f_6$) 5569.618$\pm$0.020 179.5 0.4 4.1 --------- -------------------- ------- ----- ------ ------- : LP 133-144: frequency content of the 2007 dataset. The errors were calculated by Monte Carlo simulations. $\delta f$ denotes the frequency differences of the closely spaced frequencies to $f_1$, $f_2$ $f_3$ or $f_6$. We discuss the case of $f_6$ in the text. The signal-to-noise ratios refer to the original 10-30s dataset.[]{data-label="table:lp133freq"} The first eleven peaks in Table \[table:lp133freq\] are three triplets with frequency separations of $4.1-4.2\,\mu$Hz ($f_1$), $3.2\,\mu$Hz ($f_2$) or $3.5\,\mu$Hz ($f_3$), and two singlet frequencies ($f_4$ and $f_5$). In the case of $f_6$, three peaks can be determined in the original 10-30s dataset with frequency separations of $4.7-4.8\,\mu$Hz. However, the low amplitude central peak of this triplet at $f_6=5569.6\,\mu$Hz do not reach the 4S/N significance limit in the test 30s data. Still, to make the discussion of the triplet structures clear, we added $f_6$ to the list of Table \[table:lp133freq\] in parentheses. Besides these, the first harmonic of $f_1$ also appeared. Fig. \[fig:lp133prewh\] shows the FT of the whole dataset, the consecutive pre-whitening steps at the multiplet frequencies and at the frequency domains of $f_4$, $f_5$ and $2f_1$. We plot the frequencies of @2004ApJ...600..404B and the frequencies found in the 2007 Konkoly observations together in Fig. \[fig:lp133oldnew\]. Assuming that the closely spaced peaks at $\mathrm{F}_2$ and $\mathrm{F}_3$ are results of the not properly resolved components of the $f_2$ triplet, we found, with similar amplitudes, all the frequencies observed in 2003. Besides these, we detected three new frequencies: a relatively large amplitude mode at $f_3$, and two additional low-amplitude modes at $f_5$ and $f_6$. That is, we doubled the number of modes can be used for the asteroseismic fits. The schematic plot of the triplets can be seen in Fig. \[fig:lp133triplet\]. It is clearly visible that the frequency separations of the components are larger at higher frequencies. We discuss the rotation of LP 133-144 based on the investigation of these triplets in Sect. \[sect:lp133rot\]. {width="17.5cm"} ![LP 133-144: comparison of the frequencies obtained in 2003 (*red dashed lines*) and in 2007 (*black solid lines*).[]{data-label="fig:lp133oldnew"}](lp133oldnewfrek.eps){width="\columnwidth"} ![LP 133-144: schematic plot of the triplets found at different frequency domains. The frequency errors are comparable to the width of the lines.[]{data-label="fig:lp133triplet"}](lp133triplet.eps){width="\columnwidth"} Asteroseismology ================ We built our model grid for the asteroseismic investigations of our targets utilizing the White Dwarf Evolution Code (<span style="font-variant:small-caps;">wdec</span>; @1974PhDT........56L [@1969ApJ...156.1021K; @1975ApJ...200..306L; @1991PhDT.........XX; @1986PhDT.........2K; @1990PhDT.........5W; @1993PhDT.........4B; @1998PhDT........21M; @2008ApJ...675.1512B]). The <span style="font-variant:small-caps;">wdec</span> evolves a hot polytrope model ($\sim10^5$K) down to the requested temperature, and provides an equilibrium, thermally relaxed solution to the stellar structure equations. Then we are able to calculate the set of possible zonal ($m=0$) pulsation modes according to the adiabatic equations of non-radial stellar oscillations [@1989nos..book.....U]. We utilized the integrated evolution/pulsation form of the <span style="font-variant:small-caps;">wdec</span> code created by @2001PhDT.........1M to derive the pulsation periods for the models with the given stellar parameters. More details on the physics applied in the <span style="font-variant:small-caps;">wdec</span> can be found with references in @2008ApJ...675.1512B and in our previous papers on two ZZ Ceti stars [@2009MNRAS.399.1954B; @2013MNRAS.432..598P]. Considering the limited visibility of high spherical degree ($l$) modes due to geometric cancellation effects, we calculated the periods of dipole ($l=1$) and quadrupole ($l=2$) modes for the model stars only. The goodness of the fit between the observed ($P_i^{\rmn{obs}}$) and calculated ($P_i^{\rmn{calc}}$) periods was characterized by the root mean square ($\sigma_\mathrm{{rms}}$) value calculated for every model with the <span style="font-variant:small-caps;">fitper</span> program of @2007PhDT........13K: $$\sigma_\mathrm{{rms}} = \sqrt{\frac{\sum_{i=1}^{N} (P_i^{\rmn{calc}} - P_i^{\rmn{obs}})^2}{N}} \label{equ1} %\]$$ where *N* is the number of observed periods. We varied five main stellar parameters to build our model grid: the effective temperature ($T_{\rmn{eff}}$), the stellar mass ($M_*$), the mass of the hydrogen layer ($M_\rmn{H}$), the central oxygen abundance ($X_\rmn{O}$) and the fractional mass point where the oxygen abundance starts dropping ($X_{\rmn{fm}}$). We fixed the mass of the helium layer ($M_\rmn{He}$) at $10^{-2}\,M_*$. The grid covers the parameter range $11\,400-12\,800$K in $T_{\rmn{eff}}$ (the middle and hot part of the ZZ Ceti instability strip), $0.500-0.900\,M_{\sun}$ in stellar mass, $10^{-4}-10^{-8}\,M_*$ in $M_\rmn{H}$, $0.3-0.9$ in $X_\rmn{O}$ and $0.1-0.7$ in $X_{\rmn{fm}}$. We used step sizes of $200$K ($T_{\rmn{eff}}$), $0.005\,M_{\sun}$ ($M_*$), $0.2$dex (log$M_\rmn{H}$) and 0.1 ($X_\rmn{O}$ and $X_{\rmn{fm}}$). Period lists ------------ In the case of G 207-9, we could detect seven linearly independent pulsation frequencies by the 2007 Konkoly dataset ($f_1-f_7$; see Table \[table:g207freq\]). The question is, if we could add more frequencies to this list by the 1975 observations of @1976ApJ...207L..37R. As we mentioned already in Sect. \[sect:g207freq\], two of the frequencies detected in 1975 were also found in the Konkoly data ($\mathrm{F}_3=f_6$ and $\mathrm{F}_4=f_1$). The status of the remaining three 1975 frequencies is questionable. Assuming at least a couple of $\mu$Hz errors for the 1975 frequencies, $\mathrm{F}_1=\mathrm{F}_3-\mathrm{F}_2$ (or $\mathrm{F}_2=\mathrm{F}_3-\mathrm{F}_1$, or $\mathrm{F}_3=\mathrm{F}_1+\mathrm{F}_2$), thus, these three frequencies do not seem to be linearly independent. The fact that $\mathrm{F}_2$ and $\mathrm{F}_3$ are the two dominant peaks in the FT of @1976ApJ...207L..37R suggests that $\mathrm{F}_2$ and $\mathrm{F}_3$ might be the parent modes and $\mathrm{F}_1$ is a combination peak. Furthermore, @1976ApJ...207L..37R pointed out that $\mathrm{F}_5-\mathrm{F}_4\approx\mathrm{F}_2-\mathrm{F}_1$, thus, further combinations are possible. We also note that $f_5$ of the Konkoly dataset is almost at twice the value of $\mathrm{F}_5$ ($\delta f=6.5\mu$Hz), however, there is no sign of any pulsation frequency at $0.5f_5$ in the 2007 data. We used two sets of observed periods to fit the calculated ones. One set consists of the seven periods of $f_1-f_7$ observed in 2007, while we complemented this list with the period of $\mathrm{F}_2$ detected in 1975 to create another set. We selected $\mathrm{F}_2$ because it was the second largest amplitude peak in 1975, which makes it a good candidate for an additional normal mode. In LP 133-144, we found all the previously observed frequencies in our 2007 dataset, as we show in Sect. \[sect:lp133freq\]. Thus, we cannot add more frequencies to our findings, and performed the model fits with six periods. We summarized the periods utilized for modelling in Table \[table:periods\] for both stars. ----------------- -------- ------- -------- Period Period (s) (s) $f_1$ 291.9 $f_1$ 209.2 $f_2$ 595.7 $f_2$ 305.9 $f_3$ 196.1 $f_3$ 270.6 $f_4$ 623.8 $f_4$ 327.3 $f_5$ 129.4 $f_5$ 140.5 $f_6$ 317.8 $f_6$ 179.5 $f_7$ 305.2 +$\mathrm{F}_2$ 557.4 ----------------- -------- ------- -------- : G 207-9 and LP 133-144: periods utilized for the model fits.[]{data-label="table:periods"} Best-fitting models for G 207-9 ------------------------------- We determined the best-matching models considering several cases: at first, we let all modes to be either $l=1$ or $l=2$. Then we assumed that the dominant peak is an $l=1$, considering the better visibility of $l=1$ modes over $l=2$ ones. At last, we searched for the best-fitting models assuming that at least four of the modes is $l=1$, including the dominant frequency. We obtained the same model as the best-fitting asteroseismic solution both for the seven- and eight-period fits. It has $T_{\rmn{eff}}=12\,000$K, $M_*=0.870\,M_{\sun}$ and $M_\rmn{H}=10^{-4}\,M_*$. This model has the lowest $\sigma_\mathrm{{rms}}$ ($1.04-1.06$s) both if we do not apply any restrictions on the $l$ values of the modes, and as it gives $l=1$ solution to the dominant frequency, this model is also the best-fit if we assume that the $291.9$s mode is $l=1$. Note that in this model solution only this mode is an $l=1$, all the other six or seven modes are $l=2$. In the case of four expected $l=1$ modes and seven periods, the best-matching model has the same effective temperature ($T_{\rmn{eff}}=12\,000$K), a bit lower stellar mass ($M_*=0.865\,M_{\sun}$), and thinner hydrogen layer ($M_\rmn{H}=10^{-6}\,M_*$). Assuming four $l=1$ modes and eight periods, the best-matching model has $T_{\rmn{eff}}=12\,400$K, $M_*=0.855\,M_{\sun}$ and $M_\rmn{H}=10^{-4.6}\,M_*$. The second best-fit model is the same as for four $l=1$ modes and seven periods. We denoted with open circles these two latter models in Fig. \[fig:grids\] (left panel) on the $T_{\rmn{eff}}-M_*$ plane, together with the spectroscopic solution. Both in the case of G 207-9 and LP 133-144, we utilized the $T_{\rmn{eff}}$ and surface gravity ($\mathrm{log}\,g$) values provided by @2011ApJ...743..138G, and then corrected them according to the results of @2013AA...559A.104T based on radiation-hydrodynamics 3D simulations of convective DA stellar atmospheres. We accepted the resulting values as the best estimates for these atmospheric parameters. We converted the surface gravities to stellar masses utilizing the theoretical masses determined for DA stars by @1996ApJ...468..350B. Considering the mass of the hydrogen layer (see the left panel of Fig. \[fig:grids2\]), we found that most of the models up to $\sigma_\mathrm{{rms}}=3.0$s are in the $M_\rmn{H}=10^{-4}-10^{-6}\,M_*$ range, while about a dozen models predict thinner hydrogen layer down to $10^{-8}\,M_*$. The best-fitting models favour the $M_\rmn{H}=10^{-4.6}\,M_*$ value. We summarize the results of the spectroscopic atmospheric parameter determinations, the former modelling results based on the 1975 frequency list, the main stellar parameters of the models mentioned above and the calculated periods fitted with our observed ones in Table \[table:g207params\]. We also list the $\sigma_\mathrm{{rms}}$ values of the models. The $T_{\rmn{eff}}=12\,000$K solutions are in agreement with the spectroscopic value. The $T_{\rmn{eff}}=12\,400$K model seems somewhat too hot comparing to the $\sim12\,100$K spectroscopic temperature, but considering that the uncertainties of both values are estimated to be around $200$K, this model still not contradicts to the observations. The $0.855-0.870\,M_*$ stellar masses are also close to the value derived by spectroscopy, considering its uncertainty. Summing it up, we can find models with stellar parameters and periods close to the observed values even if we assume that at least half of the modes is $l=1$, including the dominant mode. {width="17.5cm"} {width="17.5cm"} [llrrrr]{} $T_{\rmn{eff}}$ (K) & & & & & Reference\ \ 12078$\pm$200 & 0.84$\pm$0.03 & & & & @2011ApJ...743..138G\ & & & & & @2013AA...559A.104T\ \ 12000 & 0.815 & 2.0 & 8.5 & 259.0, 292.0, 317.3, 557.3, 740.7, 787.5$^\star$ & @2009MNRAS.396.1709C\ 11700 & 0.530 & 3.5 & 6.5 & 259.0, 292.0, 317.3, 557.3, 740.7, 787.5$^\star$ & @2009MNRAS.396.1709C\ 12030 & 0.837 & 2.5 & 6–7 & 259.1 (1,4), 292.0 (2,10), 318.0 (1,5),& @2012MNRAS.420.1462R [@2013ApJ...779...58R]\ & & & & 557.4 (1,12), 740.4 (1,17) &\ & & & & &\ & & &\ & & & & &\ 12000 (1.06s) & 0.870 & 2.0 & 4.0 & 291.0 (1,7), 595.5 (2,32), 195.8 (2,9), &\ & & & & 625.6 (2,34), 129.0 (2,5), 319.7 (2,16), &\ & & & & 305.4 (2,13), 558.6 (2,28) &\ 12000 (1.61s) & 0.865 & 2.0 & 6.0 & 290.6 (1,5), 594.5 (1,14), 193.1 (2,6), &\ & & & & 623.9 (1,15), 130.4 (2,3), 316.6 (1,6), &\ & & & & 306.2 (2,12), 555.1 (2,24) &\ 12400 (1.50s) & 0.855 & 2.0 & 4.6 & 290.5 (1,6), 594.7 (1,16), 194.0 (1,3), &\ & & & & 624.7 (1,17), 132.3 (2,4), 318.7 (2,14), &\ & & & & 304.6 (2,13), 557.0 (2,27) &\ \ \ Best-fitting models for LP 133-144 ---------------------------------- The model with the lowest $\sigma_\mathrm{{rms}}$ ($0.46$s) has $T_{\rmn{eff}}=11\,800$K, $M_*=0.710\,M_{\sun}$ and $M_\rmn{H}=10^{-4.0}\,M_*$ if we do not apply any restrictions on the $l$ values of modes. Generally, the best-matching models have masses around $0.7\,M_{\sun}$, which are at least $0.1\,M_{\sun}$ larger than the spectroscopic value. These models provide $3-4$ $l=1$ solutions to the observed modes. We searched for the best-matching models in a second run, assuming that the three largest amplitude modes showing triplet structures at 209.2, 305.9 and 270.6s are all $l=1$ modes. The best-matching model has the same effective temperature ($T_{\rmn{eff}}=11\,800$K), slightly larger mass ($M_*=0.725\,M_{\sun}$) and much thinner hydrogen layer ($M_\rmn{H}=10^{-8.0}\,M_*$) than the previously selected model. The mass still seems too large comparing to the spectroscopic value, but it gives $l=1$ solutions for all the four modes with triplet frequencies, including the mode at 179.5s. These modes are consecutive radial overtones with $k=1-4$. We denoted this model with an open circle on the middle and right panels of Fig. \[fig:grids\]. The hydrogen layer masses versus the $\sigma_\mathrm{{rms}}$ values of these models are plotted in the right panel of Fig. \[fig:grids2\]. This figure also shows that the best-fitting models have thin hydrogen layer with $M_\rmn{H}=10^{-8.0}\,M_*$. Otherwise, two families of model solutions outlines: one with $M_\rmn{H}=10^{-4.0}-10^{-6.0}\,M_*$ and one with thinner, $M_\rmn{H}=10^{-7.6}-10^{-8.0}\,M_*$ hydrogen layers. If we restrict our period fitting to the models with effective temperatures and masses being in the range determined by spectroscopy, the best-matching model has $T_{\rmn{eff}}=12\,000$K, $M_*=0.605\,M_{\sun}$ and $M_\rmn{H}=10^{-4.2}\,M_*$. However, the 179.5s mode is $l=2$ in this case, while all the other frequencies are consecutive radial order $l=1$ modes. At last, we searched for models in this restricted parameter space and assuming that all the four frequencies showing triplets are $l=1$. Our finding with the lowest $\sigma_\mathrm{{rms}}$ has $T_{\rmn{eff}}=12\,000$K, $M_*=0.585\,M_{\sun}$ and $M_\rmn{H}=10^{-5.0}\,M_*$, however, its $\sigma_\mathrm{{rms}}$ is relatively large ($6.8$s), which means that there are major differences between the observed and calculated periods. Table \[table:lp133params\] lists the stellar parameters and theoretical periods of the models mentioned above. For completeness, we included this last model solution, too. We concluded, that our models predict at least $0.1\,M_{\sun}$ larger stellar mass for LP 133-144 than the spectroscopic value. Nevertheless, it is possible to find models with lower stellar masses, but in these cases not all the modes with triplet frequency structures has $l=1$ solutions and (or) the corresponding $\sigma_\mathrm{{rms}}$ values are larger than for the larger mass models. Considering the effective temperatures, the $T_{\rmn{eff}}=12\,000$K solutions are in agreement with the spectroscopic determination ($\sim12\,150$K) within its margin of error. As in the case of G 207-9, taking into account that the uncertainties for the grid parameters are of the order of the step sizes in the grid, the $T_{\rmn{eff}}=11\,800$K findings are still acceptable. [llrrrr]{} $T_{\rmn{eff}}$ (K) & & & & & Reference\ \ 12152$\pm$200 & 0.59$\pm$0.03 & & & & @2011ApJ...743..138G\ & & & & & @2013AA...559A.104T\ \ 11700 & 0.520 & 2.0 & 5.0 & 209.2 (1,2), 305.7 (2,7), 327.3 (2,8) & @2009MNRAS.396.1709C\ 12210 & 0.609 & 1.6 & $\sim6$ & 209.2 (1,2), 305.7 (2,8), 327.3 (2,9) & @2012MNRAS.420.1462R\ & & & & &\ & & &\ & & & & &\ 11800 (0.46s) & 0.710 & 2.0 & 4.0 & 208.8 (1,3), 305.6 (2,11), 270.1 (1,5), &\ & & & & 327.2 (1,6), 140.6 (2,4), 180.4 (1,2) &\ 11800 (1.46s) & 0.725 & 2.0 & 8.0 & 209.5 (1,2), 304.5 (1,4), 268.8 (1,3), &\ & & & & 328.3 (2,9), 138.5 (2,2), 181.0 (1,1) &\ 12000 (2.89s) & 0.605 & 2.0 & 4.2 & 204.5 (1,2), 307.9 (1,4), 271.5 (1,3), &\ & & & & 326.2 (1,5), 138.4 (1,1), 183.7 (2,4) &\ 12000 (6.83s) & 0.585 & 2.0 & 5.0 & 215.3 (1,2), 311.6 (1,4), 273.3 (1,3), &\ & & & & 326.6 (2,9), 126.6 (2,2), 176.4 (1,1) &\ ### Stellar rotation {#sect:lp133rot} A plausible explanation for the observed triplet structures is that these are rotationally split frequency components of $l=1$ modes. We used this assumption previously in searching for model solutions for our observed periods. Knowing the frequency differences of the triplet components ($\delta f$), we can estimate the rotation period of the pulsator. In the case of slow rotation, the frequency differences of the $m=-1,0,1$ rotationally split components can be calculated (to first order) by the following relation: $$\label{eq:rot} \delta f_{k,\ell,m} = \delta m (1-C_{k,\ell}) \Omega,$$ where the coefficient for high-overtone ($k\gg\ell$) $g$-modes and $\Omega$ is the (uniform) rotation frequency. In the case of LP 133-144, the presumed $l=1$ modes are low radial-order frequencies ($k=1-6$), but the $C_{k,\ell}$ values of the fitted modes can be derived by the asteroseismic models. We used the average of the frequency separations within a triplet and calculated the stellar rotation rate separately for $f_1$, $f_2$, $f_3$ and $f_6$ (see e.g. @2015MNRAS.451.1701H). We utilized the $T_{\rmn{eff}}=11\,800$K, $M_*=0.725\,M_{\sun}$ model. The resulting rotation periods are: $P_{f_1} = 1.83$d ($\overline{\delta f_1}=4.15\,\mu$Hz, $C_{k,\ell}=0.345$), $P_{f_2} = 1.82$d ($\overline{\delta f_2}=3.2\,\mu$Hz, $C_{k,\ell}=0.497$), $P_{f_3} = 1.69$d ($\overline{\delta f_3}=3.5\,\mu$Hz, $C_{k,\ell}=0.489$) and $P_{f_6} = 1.60$d ($\overline{\delta f_6}=4.75\,\mu$Hz, $C_{k,\ell}=0.343$). The average rotation period thus $1.74\pm0.11$d ($\sim42$h). This fits perfectly in the known rotation rates of the order of hours to days of ZZ Ceti stars (cf. Table 4 in @2008PASP..120.1043F). Note that the rotation periods calculated by the different multiplet structures are strongly depend on the actual values of observed frequency spacings and also on the $C_{k,\ell}$ values, which vary from model to model. Thus the different rotation periods calculated for the different modes does not of necessarily mean that e.g. in this case we detected differential rotation of the star, but we can provide a reasonable estimation on the global rotation period of LP 133-144. Summary and Conclusions ======================= We have presented the results of the one-season-long photometric observations of the ZZ Ceti stars G 207-9 and LP 133-144. These rarely observed pulsators are located in the middle and in the hot part of the instability strip, respectively. G 207-9 was found to be a massive object previously by spectroscopic observations, comparing its predicted $M_*>0.8\,M_{\sun}$ mass to the average $\sim0.6\,M_{\sun}$ value of DA stars (see e.g. @2013ApJS..204....5K). In contrary, the mass of LP 133-144 was expected to be around this average value. With our observations performed at Konkoly Observatory, we extended the number of known pulsation frequencies in both stars. We found seven linearly independent modes in G 207-9, including five newly detected frequencies, comparing to the literature data. We also detected the possible signs of additional frequencies around some of the G 207-9 modes, but their separations being close to the 1d$^{-1}$ value makes their detection uncertain. Multi-site or space-based observations could verify or disprove their presence. In the case of LP 133-144, we detected three new normal modes out of the six derived, and revealed that at least at three modes there are actually triplet frequencies with frequency separations of $\sim4\mu$Hz. All the pulsation modes of LP 133-144 and most of the modes of G 207-9 are found to be below 330s, with amplitudes up to $\sim10$mmag. This fits to the well-known trend observed at ZZ Ceti stars that at higher effective temperatures we see lower amplitude and shorter period light variations than closer to the red edge of the instability strip (see e.g. @2008PASP..120.1043F). We also found that on the five-month time scale of our observations there were no significant amplitude variations in either stars. This suits to their location in the instability domain again, as short time scale large amplitude variations are characteristics of ZZ Cetis with lower effective temperatures. However, in the case of G 207-9, the different frequency content of the 1975 and 2007 observations shows that amplitude variations do occur on decade-long time scale. In addition, similar pulsational feature of the two stars is that both show light variations with one dominant mode ($A=10-11$mmag) and several lower amplitude frequencies. The extended list of known modes allowed to perform new asteroseismic fits for both objects, in which we compared the observed and calculated periods both with and without any restrictions on the $l$ values of modes. The best-matching models of G 207-9 have found to be close to the spectroscopic effective temperature and stellar mass, predicting $T_{\rmn{eff}}=12\,000$ or $12\,400$K and $M_*=0.855-0.870\,M_{\sun}$. For LP 133-144, the best-fitting models prefer more than $0.1\,M_{\sun}$ larger stellar masses than the spectroscopic measurements and $T_{\rmn{eff}}=11\,800$K effective temperatures. The main sources of the differences in our model solutions and the models presented by @2009MNRAS.396.1709C, even though they also used the <span style="font-variant:small-caps;">wdec</span>, can arise from the different periods utilized for the fits, the different core composition profiles applied, and the different way they determined the best-fitting models utilizing the amplitudes of observed periods as weights to define the goodness of the fits. At last, we derived the rotational period of LP 133-144 based on the observed triplets and obtained $P_{\rmn{rot}}\simeq42$h. Note that the results of the asteroseismic fits presented in this manuscript are preliminary findings, and both objects deserve more detailed seismic investigations utilizing the extended period lists, similarly to the modelling presented for other hot DAV stars, GD 165 and Ross 548 [@2016ApJS..223...10G]. In the case of these objects, the authors could identify models reproducing the observed periods quite well while staying close to the spectroscopic stellar parameters, and also verified the credibility of the selected models in many other ways, including the investigation of rotationally split frequencies. Acknowledgements {#acknowledgements .unnumbered} ================ The authors thank the anonymous referee for the constructive comments on the manuscript. The authors thank Agnès Bischoff-Kim for providing her version of the <span style="font-variant:small-caps;">wdec</span> and the <span style="font-variant:small-caps;">fitper</span> program. The authors also thank the contribution of E. Bokor, Á. Győrffy, Gy. Kerekes, A. Már and N. Sztankó to the observations of the stars. The financial support of the Hungarian National Research, Development and Innovation Office (NKFIH) grants K-115709 and PD-116175, and the LP2014-17 Program of the Hungarian Academy of Sciences are acknowledged. P.I.P. is a Postdoctoral Fellow of the The Research Foundation – Flanders (FWO), Belgium. L.M. and Á.S. was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. {#app:g207} Normalized differential light curves of G 207-9 obtained in 2007 at Piszkéstető mountain station of Konkoly Observatory. {#app:lp133} Normalized differential light curves of LP 133-144 obtained in 2007 at Piszkéstető mountain station of Konkoly Observatory. \[lastpage\] [^1]: E-mail: [email protected] (Zs.B.) [^2]: <span style="font-variant:small-caps;">iraf</span> is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. [^3]: http://astroutils.astronomy.ohio-state.edu/time/utc2bjd.html

--- abstract: 'For second countable locally compact Hausdorff groupoids, the property of possessing a Haar system is preserved by equivalence.' address: | Department of Mathematics\ Dartmouth College\ Hanover, NH 03755\ USA author: - 'Dana P. Williams' date: 15 January 2015 title: Haar Systems on Equivalent Groupoids --- [^1] Introduction {#sec:introduction} ============ Beginning with the publication of Renault’s seminal paper [@ren:groupoid], locally compact groupoids have been an especially important way to construct operator algebras. Just as with the time honored group [$\mathcs$]{}-algebra construction, this is done by turning $C_{c}(G)$ into a convolution algebra and then completing. In the group case, there is always a (left) Haar measure on $G$ which allows us to form the convolution product. In the groupoid case, the natural convolution formula requires a family of measures $\lambda^{u}$ with support $G^{u}=\set{x\in G:r(x)=u}$ for each $u\in{G^{(0)}}$. We want the family to be left-invariant in that $x\cdot \lambda^{s(x)} = \lambda^{r(x)}$ where $x\cdot \lambda^{s(x)}(E)=\lambda^{s(x)}(x^{-1}E)$. In order that the convolution formula return a continuous function, we need the continuity condition that $$u\mapsto \int_{G}f(x)\,d\lambda^{u}(x)$$ be continuous for all $f\in C_{c}(G)$ (the necessity is the main result in [@sed:pams86]). Such a family ${\{\lambda^{u}\}}_{u\in{G^{(0)}}}$ is called a (continuous) *Haar system* for $G$. An annoying gap in the theory is that there is no theorem guaranteeing Haar systems exist. The only significant positive existence result I am aware of is that if ${G^{(0)}}$ is open in $G$ and the range map is open (and hence the source map as well), then the family consisting of counting measures is always a Haar system. Groupoids with ${G^{(0)}}$ open and for which the range map is open are called *étale*. (It is also true that Lie Groupoids necessarily have Haar systems [@pat:groupoids99]\*[Theorem 2.3.1]{}, but this result is crucially dependent on the manifold structure and hence not in the spirit of this note.) It is also well known that if $G$ is any locally compact groupoid with a Haar system, then its range and source maps must be open. (This is a consequence of Remark \[rem-haar-r-sys\] and Lemma \[lem-pi-sys-open\].) Thus if a locally compact groupoid has a range map which is not open, then it can’t possess a Haar system. Such groupoids do exist; for example, see [@sed:pams86]\*[§3]{}. However, to the best of my knowledge, there is no example of a locally compact groupoid with open range and source maps which does not possess a Haar system. I have yet to find an expert willing to conjecture (even off the record) that all such groupoids need have Haar systems, but the question remains open. The purpose of this note is to provide some additional examples where Haar systems must exist. The main result being that if $G$ and $H$ are equivalent second countable locally compact groupoids (as defined in [@mrw:jot87] for example), and if $G$ has a Haar system, then so does $H$. Since equivalence is such a powerful tool, this result gives the existence of Haar systems on a great number of interesting groupoids. For example, every transitive groupoid with open range and source maps has a Haar system (Proposition \[prop-trans\]). The proof given here depends on several significant results from the literature. The first is that if $\pi:Y\to X$ is a continuous, open surjection with $Y$ second countable, then there is a family of Radon measures ${\{\beta^{x}\}}$ on $Y$ such that $\operatorname*{supp}\beta^{x}=\pi^{-1}(x)$ and $$x\mapsto \int_{Y} f(y)\,d\beta^{x}(y)$$ is continuous for all $f\in C_{c}(Y)$. (This result is due to Blanchard who makes use of a Theorem of Michael’s [@mic:am56].) The second is the characterization in [@kmrw:ajm98]\*[Proposition 5.2]{} of when the imprimitivity groupoid of free and proper $G$-space has a Haar system. The third is the concept of a Bruhat section or cut-off function. These are used in [@bou:integrationII04]\*[Chapter 7]{} to construct invariant measures. They also appear prominently in [@ren:xx14]\*[Lemma 25]{} and [@tu:doc04]\*[§6]{}. Since Blanchard’s result requires separability, we can only consider second countable groupoids here. I would like to thank Marius Ionescu, Paul Muhly, Erik van Erp, Aidan Sims, and especially Jean Renault for helpful comments and discussions. The Theorem {#sec:main-theorem} =========== \[thm-main\] Suppose that $G$ is a second countable, locally compact Hausdorff groupoid with a Haar system ${\{\lambda^{u}\}}_{u\in{G^{(0)}}}$. If $H$ is a second countable, locally compact groupoid which is equivalent to $G$, then $H$ has a Haar system. As in [@ren:jot87]\*[p. 69]{} or [@anaren:amenable00]\*[Definition 1.1.1]{}, if $\pi:Y\to X$ is a continuous map between locally compact spaces $Y$ and $X$, then a *$\pi$-system* is a family of (positive) Radon measures $\set{\beta^{x}:x\in X}$ on $Y$ such that $\operatorname*{supp}\beta^{x}\subset \pi^{-1}(x)$ and for every $f\in C_{c}(Y)$, the function $$x\mapsto \int_{Y}f(y)\,d\beta^{x}(y)$$ is continuous. We say that $\beta$ is *full* if $\operatorname*{supp}\beta^{x}=\pi^{-1}(x)$ for all $x\in X$. If $Y$ and $X$ are both (left) $G$-spaces and $\pi$ is equivariant, then we say $\beta$ is *equivariant* if $\gamma\cdot \beta^{x} = \beta^{\gamma\cdot x}$ where $\gamma\cdot \beta^{x}(E)=\beta^{x}(\gamma^{-1}\cdot E)$ for all $(\gamma,x)\in G*X=\set{(\gamma,x):s(\gamma)=r(x)}$. Alternatively, $$\int_{Y}f(\gamma\cdot y)\,d\beta^{x}(y)=\int_{Y} f(y)\,d\beta^{\gamma\cdot x}(y)$$ for all $f\in C_{c}(Y)$ and $(\gamma,x)\in G*X$. \[rem-haar-r-sys\] It is useful to keep in mind that a Haar system on $G$ is a full, equivariant $r$-system on $G$ for the range map $r:G\to{G^{(0)}}$. In many cases, such as [@kmrw:ajm98]\*[§5]{}, $\pi$-systems are reserved for continuous *and open* maps $\pi:Y\to X$. In the case of full systems, the next lemma implies that there is no loss in generality. (This part of the result does not require second countability.) Conversely, if $Y$ is second countable $\pi$ is an open surjection, then Blanchard has proved that full $\pi$ systems must exist. Blanchard’s result will be crucial in the proof of the main result. \[lem-pi-sys-open\] Suppose that $\pi:Y\to X$ is a continuous surjection between second countable locally compact Hausdorff spaces. Then $\pi$ is open if and only if it admits a full $\pi$ system. Suppose that $\beta$ is a full $\pi$ system. To show that $\pi$ is open, we appeal to the usual lifting argument as in [@wil:crossed]\*[Proposition 1.15]{}. Thus we assume that $x_{i}\to \pi(y)$ is a convergent net. It will suffice to produce a subnet ${\{x_{j}\}}_{j\in J}$ and elements $y_{j}\in \pi^{-1}(x_{j})$ such that $y_{j}\to y$. To this end, let $$J=\set{(i,V):\text{$V$ is an open neighborhood of $y$ and $\pi^{-1}(x_{i})\cap V\not=\emptyset$}}.$$ We need to see that $J$ is directed in the expected way: $(i,V)\ge (j,U)$ if $i\ge j$ and $V\subset U$. So let $(k,V)$ and $(j, U)$ be in $J$. Let $f\in C_{c}^{+}(G)$ be such that $f(y)=1$ and $\operatorname*{supp}f\subset V\cap U$. Then $$\beta(f)(x_{i})=\int_{Y} f(y)\,d\beta^{x_{i}}(y) \to \int_{Y} f(y) \, \beta^{\pi(y)}(y)=\beta(f)(\pi(y)).$$ The latter is nonzero since $\beta^{\pi(y)}$ has full support. Hence there is a $k\ge i$ and $k\ge j$ such that $$\int_{Y}f(y)\,d\beta^{x_{i}}(y)\not=0.$$ It follows that $(i,U\cap V)\in J$. Thus if we let $x_{(i,V)}=x_{i}$, then ${\{x_{(i,V)}\}}_{(i,V)\in J}$ is a subnet. If we let $y_{(i,V)}$ be any element of $\pi^{-1}(x_{i})\cap V$, then $y_{(i,V)}\to y$. This suffices. The converse is much more subtle, and is due to Blanchard [@bla:bsmf96]\*[Proposition 3.9]{}. We also need what is sometimes called a *Bruhat section* or *cut-off* function for $\pi$. The construction is modeled after Lemma 1 in Appendix I for [@bou:integrationII04]\*[Chapter 7]{}. Recall that a subset $A\subset Y$ is called $\pi$-compact if $A\cap \pi^{-1}(K)$ is compact whenever $K$ is compact in $X$. We write $C_{c,\pi}(Y)$ for the set of continuous functions on $Y$ with $\pi$-compact support. \[lem-phi-exists\] Let $\pi:Y\to X$ be a continuous open surjection between second countable locally compact Hausdorff spaces. Then there is a $\phi\in C_{c,\pi}^{+}(Y)$ such that $\pi\bigl(\set{y\in Y:\phi(y)>0} \bigr) = X$. Let $\mathscr V=\set{V_{i}}$ be a countable, locally finite cover of $X$ by pre-compact open sets $V_{i}$. Let ${\{\alpha_{i}\}}$ be a partition of unity on $X$ subordinate to $\mathscr V$. Let $\phi_{i}\in C_{c}^{+}(Y)$ be such that $\pi\bigl(\set{y\in Y:\phi_{i}>0}\bigr) \supset V_{i}$. Then we can define $$\phi(y) =\sum_{i} \phi_{i}(y) \alpha_{i}\bigl(\pi(y)\bigr).$$ Since $\mathscr V$ is locally finite, the above sum is finite in a neighborhood of any $y\in Y$. Hence $\phi$ is well-defined and continuous. Local finiteness also implies that every compact subset of $X$ meets at most finitely many $V_{i}$. Since ${\{\alpha_{i}\}}$ is subordinate to $\mathscr V$, it follows that $\phi$ has $\pi$-compact support. If $x\in X$, then there is an $i$ such that $\alpha_{i}(x)>0$. Then there is a $y$ such that $\phi_{i}(y)>0$ and $\pi(y)=x$. Hence the result. Let $Z$ be a $(G,H)$-equivalence. Then the opposite module, ${Z^{\text{op}}}$, is a $(H,G)$ equivalence. Therefore, in view of [@kmrw:ajm98]\*[Proposition 5.2]{}, it will suffice to produce a full $G$-equivariant $s_{{Z^{\text{op}}}}$-system for the structure map $s_{{Z^{\text{op}}}}:{Z^{\text{op}}}\to{G^{(0)}}$. Equivalently, we need a full equivariant $r_{Z}$-system for the map $r_{Z}:Z\to{G^{(0)}}$.[^2] Hence the main Theorem is a consequence of Proposition \[prop-jean-proper-map\] below. The following proposition is even more that what is called for in the proof of Theorem \[thm-main\]: it shows that every proper $G$-space has a full equivariant $r$-system for the moment map whether the action is free or not. It should be noted that pairs $(X,\alpha)$ where $X$ is a proper $G$-space and $\alpha$ and equivariant $r$-system play an important role in the constructions in [@ren:jot87] and [@holren:xx14]. This makes the assertion that such $\alpha$’s always exist even more interesting. \[prop-jean-proper-map\] Let $G$ be a locally compact Hausdorff groupoid with a Haar system ${\{\lambda^{u}\}}_{u\in{G^{(0)}}}$. Suppose that $Z$ is a proper $G$-space. Then there is a full equivariant $r_{Z}$-system ${\{\nu^{u}\}}_{u\in{G^{(0)}}}$ for the moment map $r_{Z}:Z\to{G^{(0)}}$. Blanchard’s Lemma \[lem-pi-sys-open\] implies that there is a full $r_{Z}$-system $\beta={\{\beta^{u}\}}_{u\in{G^{(0)}}}$. The idea of the proof is to use the Haar system on $G$ to average this system to create and equivariant system. The technicalities are provided by the next lemma. Notice that since $G$ acts properly, the orbit map $q:Z\to G\backslash Z$ is a continuous open surjection between locally compact Hausdorff spaces. \[lem-phi-fcns\] Let $G$, $Z$, $\lambda$ and $\beta$ be as above. 1. If $F\in C_{c}(G\times Z)$, then $$\Phi(F)(g,u)=\int_{Z} F(g,z)\,d\beta^{u}(z)$$ defines an element of $C_{c}(G\times{G^{(0)}})$. 2. If $f\in C_{c}(Z)$ and $\phi\in C_{c,q}(Z)$, then $$\Psi_{\phi}(f)(g)=\int_{Z} f(g\cdot z)\phi(z) \,d\beta^{s(g)}(z)$$ defines an element of $C_{c}(G)$. \(a) This is straightforward if $F(g,z)=f(g)\phi(z)$ with $f\in C_{c}(G)$ and $\phi\in C_{c}(Z)$. But we can approximate $F$ in the inductive limit topology with sums of such functions. \(b) Let $L=\operatorname*{supp}\phi\cap q^{-1}\bigl(q(\operatorname*{supp}f)\bigr)$. By assumption on $\phi$, $L$ is compact. Since $G$ acts properly on $Z$, the set $$P(\operatorname*{supp}f,L)=\set{g\in G:g\cdot L\cap \operatorname*{supp}f\not=\emptyset}$$ is compact. It follows that $$F(g,z)=f(g\cdot z)\phi(z)$$ defines an element of $C_{c}(G\times Z)$. Then $$\Psi_{\phi}(f)(g)=\Phi(F)(g,s(g)).$$ The assertion follows. Using Lemma \[lem-phi-exists\], we fix $\phi\in C_{c,q}^{+}(Z)$ such that $q\bigl(\set{z:\phi(z)>0} \bigr)=G\backslash Z$. Then we define a Radon measure on $C_{c}(Z)$ by $$\label{eq:1} \nu^{u}(f)=\int_{G}\int_{Z} f(g\cdot z)\phi(z)\,d\beta^{s(g)}(z) \,d\lambda^{u}(g) =\int_{G}\Psi_{\phi}(f)(g) \,d\lambda^{u}(g).$$ Since $\lambda$ is a Haar system and $\Psi_{\phi}(f)\in C_{c}(G)$, we see immediately that $$u\mapsto \nu^{u}(f)$$ is continuous. Clearly, $\operatorname*{supp}\nu^{u}\subset r_{Z}^{-1}(u)$. Suppose $r_{Z}(w)=u$ and $f\in C_{c}^{+}(Z)$ is such that $f(w)>0$. Then there is a $z'\in\set{z:\phi(z)>0}$ such that $q(z')=q(w)$. Hence there is a $g\in G$ such that $g\cdot z'=w$. Note that $r_{Z}(z')=s(g)$ and $r(g)=r_{Z}(g\cdot z)=r_{Z}(w)=u$. Since $\beta^{s(g)}$ has full support and since everything in sight is continuous and non-negative, $$\Psi_{\phi}(g) =\int_{Z} f(g\cdot z)\phi(z)\,\beta^{s(g)}(z)>0.$$ Hence $\nu^{u}(f)>0$ and we have $$\operatorname*{supp}\nu^{u} = r_{Z}^{-1}(u).$$ Hence to complete the proof of the theorem, we just need to establish equivariance. But $$\begin{aligned} \int_{Z} f(g'\cdot z)\,d\nu^{s(g')}(z) &= \int_{G}\int_{Z} f(g'g\cdot z)\phi(z)\,d\beta^{s(g)}(z)\,d\lambda^{s(g')}(g) \\ &= \int_{G}\Psi_{\phi}(f)(g'g)\,d\lambda^{s(g')}(g) \\ \intertext{which, since $\lambda$ is a Haar system on $G$, is} &= \int_{G}\Psi_{\phi}(g) \,d\lambda^{r(g')}(g) \\ &=\int_{G}\int_{Z} f(g\cdot z)\phi(z)\,d\beta^{s(g)}(z) \,d\lambda^{r(g')}(g) \\ \intertext{which, since $g'\cdot s(g')=r(g')$, is} &=\int_{Z}f(z)\,d\nu^{g'\cdot s(g')}(z).\end{aligned}$$ This completes the proof. Proposition \[prop-jean-proper-map\] is interesting even for a group action. The result itself is no doubt known to experts, but is amusing none-the-less. \[cor-jean-group\] Suppose that $G$ is a locally compact group acting properly on a space $X$. Then $X$ has at least one invariant measure with full support. Examples and Comments {#sec:exampl-comm-future} ===================== As pointed out in the introduction, any étale groupoid $G$ has a Haar system. As a consequence of Theorem \[thm-main\], any second countable groupoid equivalent to $G$ has a Haar system (provided $G$ is second countable). In this section, I want to look at some additional examples. In some cases it is possible and enlightening to describe the Haar system in finer detail. Proper Principal Groupoids {#sec:prop-princ-group} -------------------------- Recall that $G$ is called principal if the natural action of $G$ on ${G^{(0)}}$ given by $x\cdot s(x)=r(x)$ is free. We call $G$ proper if this action is proper in that $(x,s(x))\mapsto (r(x),s(x))$ is proper from $G\times {G^{(0)}}\to{G^{(0)}}\times{G^{(0)}}$. If $G$ is a proper principal groupoid with open range and source maps, then the orbit space $G\backslash {G^{(0)}}$ is locally compact Hausdorff, and it is straightforward to check that ${G^{(0)}}$ implements an equivalence between $G$ and the orbit space $G\backslash {G^{(0)}}$. Since the orbit space clearly has a Haar system, the following is a simple corollary of Theorem \[thm-main\]. \[prop-proper-prin\] Every second countable proper principle groupoid with open range and source maps has a Haar system. \[rem-blanchard\] If $G$ is a second countable proper principle groupoid with open range and source maps, then the orbit map $q:{G^{(0)}}\to G\backslash {G^{(0)}}$ sending $u$ to $\dot u$ is continuous and open. Hence Blanchard’s Lemma \[lem-pi-sys-open\] implies there is a full $q$-system ${\{\beta^{\dot u}\}}_{\dot u\in G\backslash {G^{(0)}}}$. It is not hard to check that $\lambda^{u}=\delta_{u}\times \beta^{\dot u}$ is a Haar system for $$G_{q}:=\set{(u,v)\in{G^{(0)}}\times{G^{(0)}}:\dot u = \dot v}.$$ Since $x\mapsto (r(x),s(x))$ is a groupoid isomorphism of $G$ and $G_{q}$, we get an elementary description for a Haar system on $G$. While there certainly exist groupoids that fail to have open range and source maps — and hence cannot have Haar systems — most of these examples are far from proper and principal. In fact the examples I’ve seen are all group bundles which are as a far from principal as possible. This poses an interesting question. Must a second countable, locally compact, proper principal groupoid have open range and source maps? Transitive Groupoids {#sec:transitive-groupoids} -------------------- Recall that a groupoid is called *transitive* if the natural action of $G$ on ${G^{(0)}}$ given by $x\cdot s(x):=r(x)$ is transitive. If $G$ is transitive and has open range and source maps, then $G$ is equivalent to any of its stability groups $H=G_{v}^{v}=\set{x\in G:r(x)=u=s(x)}$ for $v\in{G^{(0)}}$; the equivalence is given by $G_{v}$ with the obvious left $G$-action and right $H$ action. Second countability is required to see that the restriction of the range map to $G_{v}$ onto ${G^{(0)}}$ is open.[^3] Since locally compact groups always have a Haar measure, the following is an immediate consequence of Theorem \[thm-main\]. (Similar assertions can be found in [@sed:pria76].) \[prop-trans\] If $G$ is a second countable, locally compact transitive groupoid with open range and source maps, then $G$ has a Haar system. As before, I don’t know the answer to the following. Must a second countable, locally compact, transitive groupoid have open range and source maps? Blowing Up the Unit Space {#sec:blowing-up-unit} ------------------------- While there are myriad ways groupoid equivalences arise in applications, one standard technique deserves special mention (see [@txlg:acens12] for example). Suppose that $G$ is a second countable locally compact groupoid with a Haar system (or at least open range and source maps). Let $f:Z\to {G^{(0)}}$ be a continuous and open map. Then we can form the groupoid $$\label{eq:2} G[Z]=\set{(z,g,w)\in Z\times G\times Z: \text{$f(z)=r(g)$ and $s(g)=f(w)$}} .$$ (The operations are as expected: $(z',g',z)(z,g,w)=(z',g'g,w)$ and $(z,g,w)^{-1} = (w,g^{-1},z)$.) The idea being that we use $f$ to “blow-up” the unit space of $G$ to all of $Z$. If $\phi:G[Z]\to G$ is the homormorphism $(z,g,w)\mapsto g$, then we get a $(G[Z],G)$-equivalence given by “the graph of $\phi$” (see [@kmrw:ajm98]\*[§6]{}): $$W = \set{(z,g)\in Z\times G:f(z)=r(g)}.$$ The left $G[Z]$-action is given by $(z,g,w)\cdot (w,g')=(z,gg')$ and the right $G$-action by $(w,g')\cdot g = (w,g'g)$. The openness of the range map for $G$ are required to see that the structure map $r_{W}:W\to Z$ is open, while the openness of $f$ is required to see that $s_{W}:W\to {G^{(0)}}$ is open. Assuming $G$ has a Haar system and $Z$ is second countable, Theorem \[thm-main\] implies $G[Z]$ has a Haar system. However in this case we can do a bit better and write down a tidy formula for the Haar system on the blow-up. We still require Blanchard’s Lemma \[lem-pi-sys-open\] that there is a full $f$-system for any continuous open map $f:Z\to {G^{(0)}}$ (provided that $Z$ is second countable). \[prop-blow-up-haar\] Suppose that $G$ is a locally compact Hausdorff groupoid with a Haar system ${\{\lambda^{u}\}}_{u\in{G^{(0)}}}$, and that $Z$ is second countable. Let $f:Z\to{G^{(0)}}$ be a continuous open map, and let $$G[Z]=\set{(w,g,z)\in Z\times G\times Z: \text{$f(w)=r(g)$ and $s(g)=f(z)$}}$$ be the “blow-up” of $G$ by $f$. If ${\{\beta^{u}\}}_{u\in{G^{(0)}}}$ is a full $f$-system, then we get a Haar system ${\{\kappa^{z}\}}_{z\in Z}$ on $G[Z]$ given by $$\kappa^{z}(f)=\int_{G}\int_{Z} f(z,g,w)\,d\beta^{s(g)}(w)\,d\lambda^{f(z)}(g).$$ The proof is relatively straightforward. Imprimitivity Groupoids {#sec:impr-group} ----------------------- If $X$ is a free and proper right $G$-space, then assuming $G$ has open range and source maps, we can form the imprimitivity groupoid $G^{Z}$ as in [@muhwil:plms395]\*[pp. 119+]{}. Specifically we let $G^{Z}$ be the quotient of $X*_{s}X=\set{(x,y)\in X\times X:s(x)=s(y)}$ by the diagonal right $G$-action. Then $G^{Z}$ is a groupoid with respect to the operations $[x,y][y,z]=[x,z]$ and $[x,y]^{-1}=[y,z]$. Furthermore $X$ implements an equivalence between $G^{Z}$ and $G$. Again we can apply Theorem \[thm-main\]. \[prop-imprimitivity\] Suppose that $G$ is a second countable, locally compact Hausdorff groupoid with a Haar system. Let $X$ be a free and proper right $G$-space. Then the imprimitivity groupoid $G^{Z}$ has a Haar system. \[rem-kmrw\] In [@kmrw:ajm98]\*[§§9-10]{}, we associated a group $\operatorname{Ext}(G,\T)$ to any second countable, locally compact Hausdorff groupoid $G$. In [@kmrw:ajm98]\*[Theorem 10.1]{} we showed that $\operatorname{Ext}(G,\T)$ was naturally isomorphic to the Brauer group $\operatorname{Br}(G)$. The definition of $\operatorname{Ext}(G,\T)$ required we consider the space $\mathcal{P}(G)$ of all free and proper right $G$-spaces $X$ *such that $G^{X}$ has a Haar system*. In view of Proposition \[prop-imprimitivity\], $\mathcal{P}(G)$ becomes simply the collection of all free and proper right $G$-spaces. \#1 [^1]: This work was supported in part by a grant from the Simons Foundation. [^2]: The gymnastics with the opposite space is just to accommodate a preference for left-actions. This has the advantage of making closer contact with the literature on $\pi$-systems. [^3]: Proving the openness of $r\restr{G_{v}}$ is nontrivial. It follows from [@ram:jfa90]\*[Theorem 2.1]{} or Theorems 2.2A and 2.2B in [@mrw:jot87]. The assertion and equivalence fail without the second countability assumption as observed in [@mrw:jot87]\*[Example 2.2]{}. It should be noted that in both [@ram:jfa90] and [@mrw:jot87] openness of the range and source maps on a topological groupoid is a standing assumption.

--- abstract: 'We consider a mechanism to generate controllable qudit-qudit interactions in a charge-position paradigm for a quantum computer, through the use of auxiliary states. By controlling the tunneling rates onto these auxiliaries from the qudits proper, we can controllably switch the entangling operations. We consider a practical architecture in which to realize such a computer and examine the associated Hilbert space dimension.' author: - 'S. G. Schirmer' - 'Andrew D. Greentree' - 'D. K. L. Oi' title: 'Implementation of controlled multi-qudit operations for a solid-state quantum computer based on charge qudits' --- Quantum computing [@bib:NielsenBook] has been identified as an important field recently, and significant work is being undertaken to find suitable systems in which to observe scalable, coherent interactions. Recent advances in semi-conductor technology have opened up new possibilities for silicon-based solid-state realizations of quantum computers, which are highly attractive due to their compatibility with conventional Silicon metal-oxide-semiconductor technology [@bib:ClarkReview2003]. All scalable quantum computing proposals require some particle that can be placed into a superposition of several distinguishable quantum states, and for which particle-particle entanglement can be achieved. The Hilbert space dimension of an $N$ particle quantum register obtained by entangling all the particles simultaneously has been shown to be a good measure for the power of a quantum computer [@bib:Blume-Kohout2002]. One model for a scalable quantum computer is based on localization of an electronic charge in potential wells, the so-called charge-qubit approach. This has been discussed by Ekert [@bib:Ekert1995] and appears in various flavors in the literature, see for example Refs [@bib:EkertRMP1996; @bib:BarencoPRL1995]. A generic problem with charge-based quantum computing schemes is that, although it is relatively easy to generate single particle operations, i.e., to realize the Hadamard transform gate, it is usually quite difficult to obtain *controllable* particle-particle interactions. The mechanism normally suggested is the Coulomb interaction, which has the problem of being difficult to turn off. It may be possible to perform universal operations with an “always on” interaction, and schemes for realizing global operations in this setting have been considered by Benjamin [@bib:BenjaminPRL2003] and Pachos and Vedral [@bib:PachosPreprint2003]. However, for practical purposes it seems highly desirable to have controllable multi-particle interactions. Here we suggest ways to achieve this by making use of local tunneling to auxiliary states, which has the effect of switching the Coulomb action by changing the effective distance between components of the electronic wavefunction. The basic schemes we propose are in principle not limited to a particular charge-based implementation, and may in fact be useful for a variety of charge-based proposals including, for instance, Cooper-pair box schemes [@bib:MakhlinNature1999]. However, for concreteness we consider a generalization of a specific recent proposal by Hollenberg *et al.* [@bib:Hollenberg2003] where the confining potentials are obtained from individual Phosphorus implants in a Silicon matrix. We envisage this being fabricated via a “bottom up” approach to nanofabrication such as has been recently realized [@bib:OBrienPRB2001]. Hollenberg’s charge proposal deserves particular consideration in our opinion since it takes advantage of existing fabrication technologies and medium-scale realizations of such an architecture appear to be within experimental reach in the near future. Although the original proposal involved charge qubits, we shall consider a generalization to qudits (i.e., systems whose single particle Hilbert space has dimension $D\ge 2$), and concentrate on qutrit structures ($D=3$) as they have been shown to optimize the total Hilbert space dimension of the composite system [@bib:GreentreePrePrint2003]. We propose concrete scalable architectures that permit efficient, controlled multi-particle gates in this setting, which is essential for the operation of quantum error-correction algorithms. The donor impurities in the Hollenberg *et al.* scheme can be regarded as quantum dots that create an electric potential with $D$ local minima located at the sites of the donor impurities, which confines the shared electron as shown in Fig. \[figure1\]. The $D$ ground states $\ket{d}$ for $d=1,\ldots,D$, corresponding to the various localizations of the electron at the donor impurities, form a basis for the Hilbert space $\H$ of a single charge qudit. The height of the potential barrier between a pair of adjacent quantum dots belonging to the same qudit can be manipulated by applying an external electric potential via surface electrodes located between the two sites. By adjusting the voltages applied we can manipulate the height of the potential barrier and therefore the rate of tunneling between adjacent sites. We follow the notation of Ref [@bib:Hollenberg2003] and refer to these surface electrodes as barrier or $B$-gates. Quantum tunneling through the potential barriers leads to the creation of coherent superposition states of the localized charge qudit states $\ket{d}$. Furthermore, by applying electric potentials to surface electrodes located directly above the donor impurities we can create asymmetries in the potential, and thus change the ground state energy $E_d$ of state $\ket{d}$, introducing local energy shifts. We shall call these electrodes (energy) shift gates, or $S$-gates. It can be shown that by combining $S$-gates and $B$-gates, arbitrary single qudit operations (local unitary operations) can be performed. One of the main advantages of the proposed charge qudit architecture compared to the original Kane proposal [@bib:Kane1998] using the nuclear spins of the donor atoms as qubits is the possibility of *easy readout* of quantum information via single electron transistors [@bib:GrabertNATO1992], the suitability of which has already been shown for quantum computing applications when run in RF mode [@bib:RFSET]. Moreover, charge qudit quantum logic operations can be implemented using voltage gates only, without the need for additional radio frequency control pulses. Besides reducing the operational complexity, this should largely avoid problems of non-selective excitation of multiple sites, which may complicate the implementation of selective single and multi-qubit quantum logic operations in the original Kane proposal. One of the main disadvantages of charge qudits compared to nuclear spin qubits is their much shorter coherence lifetime. However, local operations can be performed much faster for charge qudits than for nuclear spin qubits, on the order of $10^{-11}$ seconds, and estimates suggest that the decoherence lifetime of a charge qudit should be much longer than that, at least on the order of $10$ ns [@bib:Hollenberg2003; @bib:Barrett2003]. It also seems likely that advances in technology and control will lead to further reductions of the gate operation time, and that the coherence lifetime might be increased though decoherence control measures. For instance, continuous quantum error correction by means of quantum feedback control has recently been shown to be able to reduce spontaneous emission errors [@bib:Ahn2003], and similar control techniques might be able to reduce decoherence. We present our proposal for a scalable charge-qudit quantum computer in Fig. \[figure3a\]. As mentioned above, we differentiate it from previous models by the use of (a) qutrits to optimise the Hilbert-space dimensionality, and (b) auxillary dots to effectively switch the Coulomb interaction to mediate particle-particle interactions. The qudits are arranged vertically, alternatingly above and below a center row of auxiliary quantum dots which mediate interactions between multiple adjacent qudits. The alternating arrangement of the qudits above and below the row of auxiliary quantum dots reduces unwanted Coulomb interactions between quantum dots belonging to different qudits. The efficiency of this shielding effect can be further enhanced by adding “trenches” filled with a conducting material between adjacent qudits as shown in the figure. Since the donor impurities are buried, realizing such shielding trenches would require a 3D structure. One way to realize such a structure would be to metalize a sheet of delta-doped Silicon, which could be achieved with a bottom-up strategy [@bib:OBrienPRB2001]. The staggering of the qudits further reduces unwanted interactions between diagonally facing qudits. The efficiency of multi-qudit operations might be enhanced by embedding the auxiliary quantum dots into a high-permittivity material to facilitate the Coulomb interaction. This would require that the silicon substrate be replaced by another material in the region occupied by the auxiliary quantum dots. This would be difficult to realize given current technology, however it may well become feasible with further advances in fabrication technology. The high permittivity material would reduce the time needed to implement multi-qudit gates, increasing the number of such operations that can be performed within the coherence lifetime of the qudit register, potentially improving the performance of a working quantum device. To activate the Coulomb interaction between two or more adjacent qudits, we coherently transfer the populations of state $\ket{1}$ of the corresponding qudits to the auxiliary state $\ket{0}$ by *simultaneously* lowering the height of the potential barriers between the corresponding quantum dots by applying suitable gate voltages to the auxiliary tunneling gates. Then we restore the potential barriers and let the Coulomb interaction between the auxiliary quantum dots create the necessary controlled phase gate. After a certain predetermined time, the populations of the auxiliary quantum dots are coherently transferred back to the original qudit states by simultaneously lowering the corresponding auxiliary potential barriers, thus switching off the Coulomb interaction. The details of how to implement a sufficient set of elementary gates required for universal quantum computation and efficient qudit error correction will be discussed in a forthcoming, more detailed paper. Note that this arrangement of the qudits is scalable and permits not only the implementation of arbitrary two-qudit gates between any pair of adjacent qudits by combining controlled phase gates with single qudit operations, but, in principle, the implementation of arbitrary gates involving a set of $k\le N$ adjacent qudits. Furthermore, since the effects of the control fields (voltage gates) are strictly spatially confined, unlike for radio-frequency control fields for instance, it should be easily possible to implement multi-qudit interactions on disjoint subsets of qudits simultaneously. One drawback of the proposed architecture is that it requires a comparatively large number of auxiliary sites to mediate the interactions. In principle, it is possible to reduce the number of auxiliary quantum dots required by modifying the architecture as shown in Fig. \[figure3b\], for instance. In this modified arrangement the auxiliary sites are horizontally offset such that each auxiliary site is shared by four qudits. This reduces the number of auxiliary sites required by half. However, a potential drawback of this architecture is that, while it should be easy to implement four-qudit gates, it would appear to be difficult to implement controlled *two*-qudit interactions since the Coulomb interaction will always affect all four qudits surrounding an auxiliary quantum dot. Finally, we consider the effect of the auxiliary quantum dots on the Hilbert space dimension of the register. As mentioned earlier, in absence of auxiliary quantum dots, partitioning into qutrits will always optimize the Hilbert space dimension of the quantum register. Clearly, given any fixed number of quantum dots, the need for auxiliary quantum dots reduces the dimension of the Hilbert space of the quantum register. If one auxiliary site is needed for each qudit, as in the architecture shown in Fig. \[figure3a\] for instance, then the dimension of the Hilbert space of a register of $K$ quantum dots partitioned into $D$-qudits will be $D^{K/(D+1)}$ instead of $D^{K/D}$. If only a single auxiliary site for every two qudits is needed as in Fig. \[figure3b\] then the Hilbert space dimension of the register will be $D^{K/(D+0.5)}$ instead. Fig. \[figure4\] shows the Hilbert space dimension of a qudit register comprised of a fixed number of quantum dots as a function of qudit size for the three architectures discussed. The graph clearly shows that partitioning into qutrits maximizes the Hilbert space dimension of the quantum register if either no auxiliary quantum dots or only a single dot per pair of qudits is required. If one auxiliary dot for each qudit is used then partitioning into four-qudits increases the Hilbert space dimension slightly. However, this increase is rather small compared to the increase from qubits to qutrits and may be offset by other considerations such as the increased complexity of single qudit gates or the number of measurements required to extract a comparable amount of quantum information for $D>3$. In summary, we have proposed concrete scalable architectures for a charge qudit quantum computer that allow the efficient implementation of controlled multi-qudit gates by making use of auxiliary sites to mediate multi-qudit interactions. Our emphasis has been on the implementation of such schemes for a practical realization of a charge qudit quantum computer based on donor impurities embedded in a Silicon matrix, but they are not fundamentally limited to this specific case. SGS and DKLO are supported by the Cambridge-MIT Institute project on quantum information. ADG is supported by the Australian Research Council.\ [99]{} M. A. Nielsen and I. L. Chuang, [*Quantum Computation and Quantum Information*]{} (Cambridge University Press, Cambridge, England, 2000). R. G. Clark *et al.* *Progress in silicon-based quantum computing*, submitted to Phil. Trans. R. Soc. Lond. A. R. Blume-Kohout, C. M. Caves, and I. H. Deutsch, Found. Phys. [**32**]{}, 1641 (2002). A. K. Ekert, *Quantum Cryptography and Computation* in Advances in Quantum Phenomena (Plenum Press, New York, 1995), pp243-262. A. Barenco, D. Deutsch, A. Ekert and R. Jozsa, Phys. Rev. Lett. [**74**]{}, 4083 (1995) A. Ekert and R. Jozsa, Rev. Mod. Phys. [68]{}, 733 (1996). S. C. Benjamin, Phys. Rev. Lett. [**88**]{}, 017904 (2002). J. K. Pachos and V. Vedral, *Topological Quantum Gates with Quantum Dots*, eprint arXiv:quant-ph/0302077. L. C. L. Hollenberg, C. Wellard, A. R. Hamilton, D. J. Reilly, G. J. Milburn, and R. G. Clark, *Charge-based quantum computing using single donors in semiconductors*, to be submitted. Yu. Makhlin, G. Schön, A. Shnirman, Nature [**398**]{}, 305 (1999). B. E. Kane, *A Silicon-based Nuclear Spin Quantum Computer*, Nature [**393**]{}, 133 (1998). J. L. O’Brien, *et al.*, Phys. Rev. B [**64**]{}, 161401 (2001). A. D. Greentree, S. G. Schirmer, F. Green, L. C. L. Hollenberg, A. R. Hamilton, and R. G. Clark, submitted and eprint arXiv:quant-ph/0304050. H. Grabert, and M. H. Devoret, NATO Adv. Study Inst. Ser., Ser. B 294 (1992). M. H. Devoret and R. J. Schoelkopf, Nature (London) [**406**]{}, 1039 (2000); A. A. Clerk, S. M. Girvin, A. Nguyen, and A. D. Stone, Phys. Rev. Lett. [**89**]{}, 176804 (2002); T. M. Buehler *et al.*, eprint arXiv:cond-mat/0304384. S. D. Barrett and G. J. Milburn, eprint arXiv:cond-mat/0302238. C. Ahn, H. M. Wiseman and G. J. Milburn, eprint arXiv:quant-ph/0302006.

--- abstract: 'I show that a particle structure in conformal field theory is incompatible with interactions. As a substitute one has particle-like exitations whose interpolating fields have in addition to their canonical dimension an anomalous contribution. The spectra of anomalous dimension is given in terms of the Lorentz invariant quadratic invariant (compact mass operator) of a conformal generator $R_{\mu }$ with pure discrete spectrum. The perturbative reading of $R_{0\text{ }}$as a Hamiltonian in its own right i.e. associated with an action in a functional integral setting naturally leads to the AdS formulation. The formal service role of AdS in order to access CQFT by a standard perturbative formalism (without being forced to understand first massive theories and then taking their scale-invariant limit) vastly increases the realm of conventionally accessible 4-dim. CQFT beyond those for which one had to use Lagrangians with supersymmetry in order to have a vanishing Beta-function.' author: - | Bert Schroer\ presently CBPF, Rua Dr. Xavier Sigaud, 22290-180 Rio de Janeiro, Brazil\ email: [email protected]\ Prof. emeritus of the Institut für Theoretische Physik\ FU-Berlin, Arnimallee 14, 14195 Berlin, Germany date: 'May 9, 2000' title: Particle versus Field Structure in Conformal Quantum Field Theories --- A few introductory remarks ========================== Ideas about the use of conformal quantum field theory entered particle physics for the first time at the height of the Kramers-Kronig dispersion relations [@Kastrup]. They were met with reactions ranging from doubts to outright rejection and the subject lay dormant for another 10 years when it reemerged on the statistical mechanics side in connection with second order phase transitions. In the next section we will show that these early doubts of the old-time particle physicists were partially justified, because the particle structure in CQFT is indeed incompatible with interactions. However far from supplying a coffin nail for its utility in high energy physics, this no-go theorem also contains the message that one must use finer concepts in order preserve the usefulness of conformal quantum field theory as a theoretical laboratory for particle physics. There are massive particle-like objects (“infraparticles” [@Bu]) which have a continuous mass distribution with an accumulation of spectral weight at $p^{2}=m^{2}$ whose generating local fields have an anomalous non-integer (non-semi-integer in the case of Fermion fields) contribution to their long distance behavior. In a CQFT long and short distance behavior coalesce and the accumulation of spectral weight at $p^{2}=0\,$ which becomes related to the anomalous dimension of operators is the vestige of the particle interaction in the massive parent theory from which the CQFT arose by taking the scale-invariant limit. This structure is the collective effect of a total collapse of all multiparticle thresholds on top of each other. The standard LSZ large time scattering limit does not commute with this scaling limit, in fact the LSZ limit of such fields vanishes. It is believed that in order to re-extract from such a situation anything which resembles particle physics one has to apply a more general form of scattering theory [@Bu] which is based on expectation values and probabilities for inclusive cross sections (where outcoming “stuff” below a prescribed energy-momentum resolution is not registered) rather than on amplitudes. But it is presently not clear how one can achieve this. In the case of infraparticles (the electron in QED which is inexorably linked to its photon-cloud) where one also meete a situation of coalescing thresholds, this generalized scattering theory is known to be very useful [@Bu]. Recently there has been a quite different and conceptually[^1] less ambitious but formally quite attractive idea which promises to strengthen the utility of CQFT for particle physics and which is presented in the third section. It basically consists in finding a theory which radically reprocesses the spacetime interpretation and degrees of freedom of CQFT in such a way that now the “energy momentum vector” $R_{\mu }$ of the Dirac-Weyl compactified world $\bar{M}$ becomes the bona fide energy momentum instead of $P_{\mu }$ which in standard canonical or functional terminology means that $R_{\mu }$ is the one related to an action and not $P_{\mu }$. If one insists that this total reshuffling of physical interpretation should leave the basic mathematical building blocks (a certain generating set of algebras and the symmetry group structure) untouched, then there is only one answer: an associated anti De Sitter (AdS) theory [@Wit]. The nontrivial reprocessing leads to a mathematical isomorphism as described in [@Reh1] i.e. it goes far beyond that picture about the AdS-CQFT correspondence which is limited to the (infinitely remote) boundary of AdS (see in particular the remarks at the end of [@Reh2]). The AdS appearance of the AdS structure as a kind of reprocessed CQFT is less surprizing if one recalls the 6-dimensional lightcone formalism which one uses in order to obtain an efficient description of the conformal compactification $\bar{M}$ of Minkowski space $M $ and the construction of its covering $\tilde{M}$ [@Schroer]. In this way one obtains a (perturbative) new constructive non-Lagrangian access to CQFT which opens a new window into the realm of CQFT beyond those few 4-dimensional Lagrangian candidates for which one had to use a combination of gauge theory with supersymmetry. This means that one has no guaranty that the conformal side at all permits a description in terms of an action. Particle Structure and Triviality ================================= We start with recalling an old theorem which clarifies the relation between the particle-versus-field content of conformal field theories. To be more precise the following statement is a result of the adaptation of a combination of several theorems [@BF][@Pohl] The existence of one-particle states in conformally invariant theories forces the associated interpolating fields to be canonical free fields. The only particle-like structures consistent with interactions are hidden in the structure of those interpolating fields which have anomalous dimensions and whose mass spectrum is continuous with an accumulation of weight at $p^{2}=0,\,\,p_{0}>0.$ The easiest way to get a first glimpse at this situation is to look at conformal two-point functions $$\left\langle \psi (x)\psi ^{\ast }(y)\right\rangle =\left\{ \begin{array}{c} c\frac{1}{-\left( x-y\right) ^{2}},\,\,dim\psi =1 \\ c(\frac{1}{-\left( x-y\right) ^{2}})^{d_{\psi }},\,\,dim\psi =d_{\psi }>1 \end{array} \right. \label{an}$$ In the first case the application of the LSZ large time scattering limit yields $$\left\langle \psi (x)\psi ^{\ast }(y)\right\rangle =\left\langle \psi _{in}(x)\psi _{in}^{\ast }(y)\right\rangle$$ which preempts the equality $\psi =\psi ^{in}=\psi ^{out},$ whereas in the anomalous case the large distance fall-off is too strong in order to be reconcilable with the mass shell structure of a zero mass particle which means $$\psi (x)\overset{LSZ}{\rightarrow }0$$ It is worthwhile to reconsider the argument which leads to the absence of interaction in the space created by the interpolating field $\psi .$ The crucial observation is that the presence of a zero mass scalar particle state vector $\left| p\right\rangle $ with $$\left\langle p\left| \psi \right| 0\right\rangle \neq 0$$ forces $\psi $ to have a two-point function with a canonical scale dimension dim$\psi =1.$ The special feature of conformal invariance is that this implies that the two-point function is free i.e. $$\left\langle 0\left| \psi ^{\ast }(x)\psi (y)\right| \right\rangle =c\frac{1}{\left[ -(x-y-i\varepsilon )\right] ^{2}}$$ Such a conclusion relating canonical short distance dimension with absence of interactions cannot be drawn in the massive case. However the following theorem which was proven in the late 50$^{ies}$ by Jost and the present authors, and can be found in [@St-Wi], holds for both cases: The freeness of the $\psi $ two-point function implies the field $\psi $ to be a free field in Fock space. The guiding idea is to show that a localized operator or pointlike field which vanishes on the vacuum, vanishes automatically on all states i.e. is the zero operator. This is a consequence of the Reeh-Schlieder theorem [@St-Wi] which in conformal field theory is also known under the name state-field relation). It says that the operators from a region with a nontrivial causal complement (or fields smeared with test functions with support in such a region) act cyclically on the vacuum (and on any other finite energy state). If we denote by $\mathcal{A}(\mathcal{O})$ either the polynomial $^{\ast }$-algebra of unbounded smeared fields with supports of testfunctions in $\mathcal{O}$ or the affiliated bounded operator algebra, this cyclicity property reads $$\overline{\mathcal{A}(\mathcal{O})\Omega }=\mathcal{H}$$ where the bar denotes the closure and $H$ is the Hilbert space generated by all fields (bosonic and fermionic). Since (for fermionic $\psi $ there will be a change of sign) $$\psi (x)\mathcal{A}(\mathcal{O})\Omega =\mathcal{A}(\mathcal{O})\psi (x)\Omega$$ if we choose $O$ spacelike with respect to $x,$ the vanishing of the “current” $j(x)=(\partial _{\mu }\partial ^{\mu }+m^{2})\psi (x)$ on the vacuum implies the vanishing on the dense set $\mathcal{A}(\mathcal{O})\Omega $ and hence (operators in physics are closable) on all $\mathcal{H}.\,$The next step consists in proving that the commutator of two $\psi s$ on the vacuum is a c-number $$\left( \left[ \psi (x),\psi (y)\right] -i\Delta (x-y)\right) \Omega =0$$ It then follows according to the previous argument that the bracket vanishes identically. We prove this last relation by using the frequency decomposition $\psi =\psi ^{(-)}+\psi ^{(+)}$ (which follows from $j\equiv 0) $ in the commutator $$\left[ \psi (x),\psi (y)\right] \Omega =(\left[ \psi ^{(+)}(x),\psi ^{(+)}(y)\right] +\psi ^{(-)}(x),\psi ^{(+)}(y)-\psi ^{(-)}(y),\psi ^{(+)}(x))\Omega$$ where we omitted all annihilation terms. The on-shell creation with subsequent on-shell annihilation as in the last two terms and the physical spectrum condition only admits the vacuum as its energy momentum content and therefore they yield a c-number which, by a finite renormalization of $\psi $ if necessary, yields $$(\psi ^{(-)}(x),\psi ^{(+)}(y)-\psi ^{(-)}(y),\psi ^{(+)}(x))\Omega =i\Delta (x-y)\mathbf{1}\Omega$$ Since this and the full commutator is causal, the first term on the right hand side has to vanish all by itself. But on the other hand it is the separate Fouriertransform of momenta which lie on the forward mass shell and hence it is the boundary value of an analytic function in two complex 4-vectors of the form $z=\xi -i\eta ,\eta $ from the forward light cone. However an analytic function which vanish on an open set on its boundary vanished identically (generalized Schwartz reflection principle). The resulting relation on the vacuum holds according to the previous arguments for the operators and therefore we obtained the characterizing relation for a free field. The generalization to any spin including half-integer values is now a routine matter. A closer look at the zero mass situation reveals that contrary to the massive case where the difference of two on-shell vectors is either spacelike or zero, the difference of two lightlike vectors may in addition be lightlike but this only happens for parallel vectors. Since this special configurations should not matter in the sense of L$^{2}$-integrability of zero mass particle wave functions one again expects at least for $d>1+1$ the above result. However a mathematical proof of this result turned out to be quite nontrivial [@Pohl]. It is very helpful to place the above theorem into the setting of a more general theorem relating interactions and particle properties in general local quantum physics which states that operators localized in sub-wedge regions in interacting theories which possess nontrivial matrix elements between vacuum and one-particle states necessarily show the phenomenon of vacuum polarization i.e. operators which create polarization-free one-particle states exist only in interaction free field theories. Polarization-free-generators (PFG) which create pure one-particle states from the vacuum do however exist in any QFT if their localization region is a semi-infinite wedge region or larger [@Essay][@BBS]. Since in conformal theories the wedge region is conformally equivalent to a compact double cone, a conformal one-particle structure according to this more general theorem is only possible in conformal free field theories. The above argument is typical for a real-time structure which cannot be unraveled in the euclidean formulation. Trying to make the best out of it ================================= The negative result on the compatibility of zero mass particle structure with nontriviality of conformal theories should not be misread as an incompatibility with an intuitive idea about what constitutes particle-like excitations. The point here is that conformal theories in particle physics should be considered as the zero mass (scaling) limits of massive theories with mass gaps for which the LSZ scattering theory can be derived. In the scaling limit all the multiparticle thresholds in momentum space coalesce on top of each other and build up the possibly anomalous dimension. In this limit the Wigner particle theory (irreducible representation of the Poincaré group) and with it the prerequisite of the LSZ scattering theory gets lost in the presence of interactions, a fact which we have demonstrated above where it was shown that the field is either free or the LSZ limits are zero. So the right question would be: can one think of a more general scattering theory which may recuperate some of the lost structure in the aforementioned collapse of multiparticle cuts on top of each other? There is indeed another particle concept (“infraparticle”) which goes together with a generalized scattering theory build on inclusive scattering probabilities instead of amplitudes [@Bu]. This concept is expected to distinguish those anomalous dimensional fields which are of relevance in particle physics (which originate from the previous collapse in the scaling limit) from mere mathematical constructs as e.g. generalized free fields with anomalous dimensions. But we think that for the problem at hand, namely the formulation of a theory of anomalous dimension, we do not need to enter this deep and difficult issue of particle-like interpretation since here we restrict our interests in conformal theories as a simplified theoretical laboratory for field- and algebra- aspects and not for the study of particles and their scattering theory. We believe that the setting of local observable algebras which fulfill in addition to Einstein causality also Huygens principle for timelike distances [@Sch] contains all scale limits of theories which are of interest for particle physics and that interaction in this setting is characterized by the appearance of charge-carrying fields with anomalous dimensions. In view of the above No-Go theorem we will consider the noncanonical (anomalous dimension) nature of those fields as our pragmatic definition of interaction in this conformal setting. But we defer this analysis to a following longer paper which contains the relevant mathematical machinery [@Sch]. As a consequence the observable algebra of an interacting conformal field theory (conserved currents etc.) should not have the structure of composites of free fields (e.g. free currents) since otherwise the fields carrying the superselected charges may not have anomalous dimensions. Apart from normalization constants the 2- and 3-point functions of conformal observable fields (currents) are indistinguishable from those formed with free composites with the same integer dimensions. If all correlations would be indistinguishable from those of free composites (total protection) then also the charge-carrying fields associated with such observables can be shown to be free. A weak form of what in the case of conformal SYM theories has been called (partial) “protection” would be one where the relative normalization between 2-and 3-point functions is that of free composites (partial protection). Apparently perturbative supersymmetry causes partial protections [@prot]. Although such models hardly represent realistic particle physics, they are the only *Lagrangian* candidates for d=1+3 nontrivial conformal field theories and may yet turn out to be the first 4-dimensional mathematically completely controllable models. The interest and fascination in conformal field theories originates to a large part from the well-founded belief that the simplest nontrivial 4-dimensional conformal field theories which will break the age old existence deadlock[^2] for nontrivial quantum field theories in physical spacetime. For this one wants to have as much protection as possible without ending with a free conformal theory. Instead of entering an ambitious program in order to extract the particle physics “honey” from CQFT which requires a heavy conceptual investment in the area of a generalized scattering theory, there is another way which is more faithful to the formal aspects with which QFT is often identified (erroneously in my opinion, if one uses them for a definition of QFT) namely canonical formalism and/or functional integrals. It starts from the observation that in addition to the translation generator $P_{\mu }$ there is another translation-analogue described by a Lorentz-vector $R_{\mu }.$ It has a timelike purely discrete spectrum and the L-invariant “mass” $m_{c}$ with $m_{c}^{2}=R_{\mu }R^{\mu }$ plays a similar role as the rigid rotation operator $L_{0}$ in chiral theories. In fact it describes a generalized rotation around the Dirac-Weyl compactified Minkowski space $\bar{M}\simeq S^{3}\times S^{1}.$ Therefore it is not surprising that the bottom of the spectrum of $m_{c}$ is the anomalous part of the scaling dimension common to a whole equivalence class of fields which carry the same superselected charge. But despite all analogies to $P_{\mu }$ this operator is not related to an imagined functional integral action of CQFT. Nevertheless one can ask the question: is there a theory whose Lagrangian can be associated with a Hamiltonian interpretation of $R_{0}?$ In order for this new theory to be useful for particle physics it should keep the same algebraic and group-theoretical building blocks as CQFT i.e. one seeks a mathematical isomorphism which goes hand in hand with that total physical reprocessing which is necessary to accomplish such an impossible looking task. The unique answer is the AdS-CQFT correspondence [@Wit] which was proven to be a such a “radical” isomorphism [@Reh1]. Although this step does not completely answer the question posed at the beginning of how to extract and analyze the particle content of CQFT, it goes a long way to open up conformal field theory as a genuine theoretical laboratory for particle physics. And last not least it facilitates the unsolved problem number one: find a nontrivial physically relevant (i.e. one which fits at least the conceptual framework of local quantum physics, even if it falls short in describing nature) and mathematically controllable model in 4-dimensional QFT. The presented arguments suggest strongly that there exists a whole world of non-Lagrangian non-supersymmetric CQFT (in the sense that they cannot be accessed in the standard perturbative way) besides the Lagrangian SYM family. In fact the perturbative calculations in the literature already give some support in this direction. This is most visible in [@Ruehl] although these authors, evidently under the strong spell of the string-theoretic origin of the AdS-CQFT, do not interprete their calculations from this viewpoint. The possible non-Lagrangian nature of most CQFT is in a certain way explained by Rehren’s deep observation [@Reh1][@Reh2] that due to the isomorphic nature of the AdS-CQFT relation there must be degrees of freedom on the conformal side which cannot be described in terms of local fields namely those which originate from the AdS bulk (and not from the boundary) and which are necessary in order to return $CQFT\rightarrow AdS$. This leaves the interesting question of what should one make of the original observation by which the protagonists of the AdS-CQFT correspondence found this relation which is the relation between two Lagrangian field theories namely the conformal SYM model with some form of AdS supergravity [@Wit]. Since this is based on consistency checks within string theory which owes its widespread acceptance to perturbative mathematical consistency and a kind of globalized social contract but certainly not to its harmonious coexistence with the principles underlying particle physics, there is reason for some scepticism; in particular because such degrees of freedom would be easily overlooked in perturbative calculations on the CQFT side. It cannot be overstressed that this correspondence is very different and much more radical then those which arise from a different choice of  “field coordinates”. It is impossible to understand its full content in terms of pointlike physical fields. Some concluding remarks ======================= If, as argued in this letter, the AdS theories are a useful new calculational tool which open up CQFT to particle physics studies within the standard Lagrangian quantization framework, than perhaps with an additional conceptual investment one could directly understand the structure underlying the anomalous dimension spectra within CQFT i.e. without the described reprocessing on the AdS side. This turns out to be true and will be the subject of a subsequent paper [@Sch] since the necessary conceptual investment does not fit the format of a letter like this. *Acknowledgements*: I am indebted to Detlev Buchholz and Karl-Henning Rehren for a helpful exchange of emails. Furthermore I would like to thank Francesco Toppan for interesting questions which helped in shaping the presentation. [99]{} H.A. Kastrup, Ann. Physik 7, (1962) 388 J. Maldacena, Adv. Theor. Math. Phys. **2 (**1998**)**, 231 S.S. Gubser, I. R. Klebanov and A.M. Polyakov, Phys. Lett. **B448**, (1998) 253 E. Witten, Adv. Theor. Math. Phys. **2** (1998) 253 D. Buchholz, “Mathematical Physics Towards the 21st Century”, Proceedings Beer-Sheva 1993, Ben Gurion University Press 1994 K-H Rehren, “Algebraic Holography”, hep-th/9905179 K-H Rehren, “Local Quantum Observables in the Anti-deSitter - Conformal QFT Correspondence”, hepth/0003120 D. Buchholz and K. Fredenhagen, JMP **18**, Vol.5 (1977) 1107 K. Pohlmeyer, Commun. Math. Phys. **12**, (1969) 201 R.F. Streater and A.S. Wightman, *PCT, Spin and Statistics and all That*, Benjamin 1964 B. Schroer, “Facts and Fictions about Anti de Sitter Spacetimes with Local Quantum Matter”, hep-th/9911100 B. Schroer, “Particle Physics and QFT at the Turn of the Century: Old principles with new concepts, (an essay on local quantum physics)”, Invited contribution to the Issue 2000 of JMP, in print, to appear in the June issue H-J Borchers, D. Buchholz and B. Schroer, “Polarization-Free Generators and the S-Matrix”, hep-th/0003243 B. Schroer, “A Theory of Anomalous Scale-Dimensions”, hep-th/0005134 for example: J. Erdmenger, M. Perez-Victoria, “Non-renormalization of next-to-extremal correlators in N=4 SYM and the AdS/CFT correspondence” and literature quoted therein L. Hoffmann, A.C. Petkou and W. Ruehl, “Aspects of Conformal Operator Product Expansion in AdS/CFT Correspondence”, hep-th/0002154 [^1]: The attribute “conceptually” here refers to the local quantum physical aspects and not to differential-geometric ones. [^2]: In any area of Theoretical Physics there always have been plenty of nontrivial mathematically controllable illustrations which demonstrate the nontrivial physical content of the conceptual basis of those areas, not so in 4-dim. QFT. This annoying totally singular situation has been sometimes overemphasized at the cost of practical calculations, but most of the time it went totally ignored.

--- abstract: 'We study the fine scale $L^{2}$-mass distribution of toral Laplace eigenfunctions with respect to random position, in $2$ and $3$ dimensions. In $2$d, under certain flatness assumptions on the Fourier coefficients and generic restrictions on energy levels, both the asymptotic shape of the variance is determined and the limiting Gaussian law is established, in the optimal Planck-scale regime. In $3$d the asymptotic behaviour of the variance is analysed in a more restrictive scenario (“Bourgain’s eigenfunctions"). Other than the said precise results, lower and upper bounds are proved for the variance, under more general flatness assumptions on the Fourier coefficients.' address: - | IW: Department of Mathematics\ King’s College London\ Strand\ London WC2R 2LS\ England, UK - | NY: Department of Mathematics\ King’s College London\ Strand\ London WC2R 2LS\ England, UK author: - Igor Wigman - Nadav Yesha title: CLT for Planck scale mass distribution of toral Laplace eigenfunctions --- Introduction ============ Given a smooth compact $d$-manifold ${\mathcal{M}}$ we are interested in the spectral properties of the Laplace-Beltrami operator $\Delta$ on ${\mathcal{M}}$. It is well-known that the eigenvalue spectrum of $\Delta$ is purely discrete, i.e., the set of numbers $E$ admitting a solution to the Helmholtz equation $$\Delta \phi + E\phi = 0$$ is a sequence $\{E_{j}\}_{j\ge 1}$ of numbers ordered with multiplicity in a non-decreasing order such that $ E_j \to \infty $. We denote the corresponding sequence $\{\phi_{j} \}_{j\ge 1}$ of (real-valued) eigenfunctions constituting an orthonormal basis of the square-integrable functions $L^{2}({\mathcal{M}})$ on ${\mathcal{M}}$; the sequence $\{\phi_{j} \}_{j\ge 1}$ is uniquely determined up to the spectral degeneracies (i.e., up to orthogonal transformations in each eigenspace of dimension $\ge 2$). Shnirelman’s Theorem and Small-Scale Equidistribution ----------------------------------------------------- Assuming w.l.o.g. that ${\mathcal{M}}$ is unit volume ${\operatorname{Vol}}({\mathcal{M}})=1$, the celebrated Shnirelman’s Theorem  [@Sn; @Ze; @CdV] asserts that if ${\mathcal{M}}$ is chaotic (i.e., the geodesic flow on $ \mathcal{M} $ is ergodic), then “most" of the $\{\phi_{j}\}$ are $L^{2}$-equidistributed. In particular, they are equidistributed in position space, i.e., there exists a density $1$ sequence $j_{k}$ such that for all “nice" domains ${\mathcal{A}}\subseteq{\mathcal{M}}$ we have $$\label{eq:L2 mass phi->A/M} \lim\limits_{k\rightarrow\infty}\int\limits_{{\mathcal{A}}}\phi_{j_{k}}(x)^{2}dx= {\operatorname{Vol}}({\mathcal{A}}).$$ Beyond Shnirelman’s Theorem, Berry’s universality conjecture  [@Berry1; @Berry2] implies that for a [*generic*]{} chaotic manifold holds for ${\mathcal{A}}$ shrinking with $k$, slower than the Planck’s scale $E_{j_{k}}^{-1/2}$. More precisely, it states that there exists a density $1$ sequence $\{j_{k}\}_{k}$ so that if $r_{0}(E):{\mathbb{R}}_{> 0}\rightarrow{\mathbb{R}}_{> 0}$ satisfies $r_{0}(E)\cdot E^{1/2}\rightarrow\infty$ diverging arbitrarily slowly, then, for $B_{x}(r)$ the radius $r$ geodesic ball in ${\mathcal{M}}$ centred at $x$, we have $$\label{eq:|mass-exp|=o(r^d)} \left|\int\limits_{B_{x}(r)}\phi_{j_{k}}(y)^{2}dy - {\operatorname{Vol}}(B_{x}(r)) \right| = o_{k\rightarrow\infty} (r^{d})$$ uniformly for all $x\in{\mathcal{M}}$ and $r>r_{0}(E_{j_{k}})$, i.e., $$\label{eq:L2mass shrinking uniform} \sup\limits_{\substack{x\in{\mathcal{M}}\\ r>r_{0}(E_{j_{k}})}} \left|\frac{\int\limits_{B_{x}(r)}\phi_{j_{k}}(y)^{2}dy}{{\operatorname{Vol}}(B_{x}(r))} - 1\right| \rightarrow 0.$$ The following recent results are rigorous manifestations of the small-scale (“shrinking balls") statement . Luo and Sarnak  [@Luo-Sarnak Theorem 1.2] established the small-scale equidistribution for Laplace eigenfunctions on the modular surface (assuming in addition that they are Hecke eigenfunctions) where $r>E^{-\alpha}$ with a small $ \alpha>0 $, and Young [@Young], conditionally on GRH, refined this estimate for $r>E^{-1/6+o(1)}$ holding for *all* such eigenfunctions. Hezari and Rivière  [@Hezari-Riviere1], and independently Han  [@Han1] established the equidistribution for Laplace eigenfunctions on manifolds of negative curvature on logarithmic scale (i.e., $r>(\log{E})^{-\alpha}$, for some $\alpha>0$), and Han  [@Han2] considered random Laplace eigenfunctions on “symmetric" manifolds, of high spectral degeneracy; here the higher the spectral degeneracy is the smaller the allowed scale is. More recently, Han and Tacy [@HanTacy] proved small-scale equidistribution for random Gaussian combinations of eigenfunctions on compact manifolds for $ r>E^{-1/2+o(1)} $, and de Courcy-Ireland [@DeCourcyIreland] showed that, with high probability, the $L^{2}$-mass of random Gaussian spherical harmonics is, up to a small error, equidistributed, slightly above Planck scale. Toral Laplace eigenfunctions ---------------------------- For the $d$-dimensional torus ${\mathbb{T}}^{d}={\mathbb{R}}^{d}/{\mathbb{Z}}^{d}$, $d\ge 2$, there are high spectral degeneracies; in this case Lester and Rudnick  [@LeRu Theorem 1.1] proved that the small-scale equidistribution is satisfied by a generic Laplace eigenfunction (also considered by Hezari and Rivière  [@Hezari-Riviere2]). More precisely, they showed that every o.n.b. $\{\phi_{j}\}$ admits a density one subsequence $\{\phi_{j_{k}}\}$ of Laplace eigenfunctions obeying , with $r_{0}(E)=E^{-\alpha(d)}$, where $\alpha(d)$ is any number smaller than $$\label{eq:alpha(d) LeRu} \alpha(d)<\frac{1}{2(d-1)},$$ an (almost) optimal Planck-scale result for $d=2$, yet somewhat weaker than Berry’s conjecture for $d> 2$. One can express the real toral Laplace eigenfunctions explicitly as a sum of exponentials $$\label{eq:fn sum exp} f_{n}\left(x\right)=\sum_{\lambda\in\mathcal{E}_{n}}c_{\lambda}e\left(\left\langle x,\lambda\right\rangle \right), \hspace{10pt} (c_{-\lambda}=\overline{c_\lambda})$$ for $$\label{eq:S_d} n\in S_{d}:=\{n=a_{1}^{2}+\ldots +a_{d}^{2}:\: a_{1},\ldots,a_{d}\in{\mathbb{Z}}\}$$ expressible as a sum of $d$ integer squares, and the corresponding frequencies $\lambda$ are the standard lattice points $$\label{eq:E_n} {\mathcal{E}}_{n} = {\mathcal{E}}_{d;n}=\{\lambda\in{\mathbb{Z}}^{d}:\: \|\lambda\|^{2}=n\}$$ lying on the $(d-1)$-dimensional sphere (a circle for $ d=2 $) of radius-$\sqrt{n}$; in this case the energy is $E=E_{n}=4\pi^{2}n$. We will assume w.l.o.g. that $f_{n}$ is $L^{2}$-normalised, equivalent to $$\label{eq:BasicNormalization} \|f_{n}\|_{L^{2}({\mathbb{T}}^{d})}^{2} = \sum_{\lambda\in\mathcal{E}_{n}}\left|c_{\lambda}\right|^{2}=1.$$ For every $ n\in S_d $, denote $$\label{eq:N} N=N_{d;n}=\#\mathcal{E}_n.$$ When $d=2$, by Landau’s theorem, $ \left\{ n\le x:\,n\in S_{2}\right\} \sim K\frac{x}{\sqrt{\log x}} $ where $ K>0 $ is the “Landau-Ramanujan constant". On average $N=N_{2;n}$ is of order of magnitude $\sqrt{\log n}$; however, for a density one sequence in $S_{2}$ we have $ N=\left(\log n\right)^{\log2/2+o\left(1\right)}. $ In general, for $n\in S_{2}$ we have $$N=n^{o\left(1\right)}.$$ For $d=3$, Siegel’s theorem asserts that for $n\not\equiv0,4,7\,\left(8\right)$, $$N=N_{3;n}=n^{1/2+o\left(1\right)};$$ since $x\mapsto2^{a}x$ is a bijection between the solutions to $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=n$ and $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=4^{a}n$, we can always assume that $n\not\equiv0,4,7\,\left(8\right)$ with no loss of generality. Granville and Wigman  [@GranvilleWigman Theorem 1.2] refined the aforementioned estimate by Lester-Rudnick for $d=2$. They proved that in this case, is valid slightly above Planck-scale $r_{0}(E)=E^{-1/2+o(1)}$, for [*all*]{} eigenfunctions $f_{n}$ as in , corresponding to numbers $n$ so that the lattice points ${\mathcal{E}}_{n}$ are well-separated (“Bourgain-Rudnick sequences"), a condition satisfied  [@Bourgain-Rudnick Lemma 5] by “generic" integers $n\in S_{2}$ in a strong quantitative sense, subsequently refined in  [@GranvilleWigman Theorem 1.4], see section \[sec:quasi corr\]. Averaging mass w.r.t. ball centre --------------------------------- For both the $2$-dimensional and the higher-dimensional tori it is possible to construct exceptional examples of sequences of toral eigenfunctions where the equidistribution condition is not satisfied: for $d\ge 2$ thin sequences  [@LeRu Theorem $3.1$] $\{\phi_{j_{k}}\}$ of eigenfunctions violating condition at Planck-scale $r \cdot E_{j_{k}}^{1/2} \rightarrow \infty$, around the origin $x=0$, and even stronger, for $d\ge 3$  [@LeRu Theorem $4.1$ (construction by J. Bourgain)] eigenfunctions violating with $r \gg E^{-\alpha(d)}$ where $\alpha(d)> \frac{1}{2(d-1)}$, again around the origin $x=0$. In these cases, rather than keeping the ball centre $x=0$ at the origin, one may vary $x$, and study whether the “typical" discrepancy on the l.h.s. of is [*small*]{}, even if the existence of $x$ so that the l.h.s. of is [*not small*]{} is known, so that, in particular, is not satisfied. A natural way to vary $x$ is to think of $x$ as [*random*]{}, drawn uniformly in ${\mathbb{T}}^{d}$. We define the random variable $$\label{eq:X_RV} X_{f_{n},r}=X_{f_{n},r;x}:= \int\limits_{B_{x}(r)}f_{n}(y)^{2}dy,$$ and are interested in the distribution of $X_{f_{n},r}$ where $x$ is drawn randomly uniformly in ${\mathbb{T}}^{d}$. The relevant moments are: expectation $$\label{eq:Expectation} {\mathbb{E}}[X_{f_{n},r}] = \int\limits_{{\mathbb{T}}^{d}}X_{f_{n},r;x}dx,$$ higher centred moments $$\label{eq:centred moments} {\mathbb{E}}[(X_{f_{n},r}-{\mathbb{E}}[X_{f_{n},r}])^{k}] = \int\limits_{{\mathbb{T}}^{d}}\left(X_{f_{n},r;x}-{\mathbb{E}}[X_{f_{n},r}]\right)^{k}dx, \hspace{1em}k\ge2,$$ and in particular the variance $$\label{eq:Variance} {\mathcal{V}}(X_{f_{n},r}) = {\mathbb{E}}[(X_{f_{n},r}-{\mathbb{E}}[X_{f_{n},r}])^{2}].$$ This approach of averaging the $L^{2}$-mass with respect to the ball centre (and keeping $f_{n}$ fixed) was pursued by Granville-Wigman  [@GranvilleWigman] in the $2$-dimensional case, again slightly above the Planck scale $r>E^{-1/2+o(1)}$. In this regime, by proving an upper bound for ${\mathcal{V}}(X_{f_{n},r})$ beyond $({\mathbb{E}}[X_{f_{n},r}])^{2} = O(r^4)$, valid for [*all*]{} $n\in S_{2}$, under some flatness assumption on $f_{n}$ (cf. Definition \[def:ultraflat\] below), they established for [*“typical"*]{}, if [*not all*]{} $x\in{\mathbb{T}}^{2}$. It would be desirable to find a regime where it is possible to analyse the precise asymptotic behaviour of the variance ${\mathcal{V}}(X_{f_{n},r})$ of $X_{f_{n},r}$, and, if possible, determine the limit distribution law for $X_{f_{n},r}$; our principal results below achieve both of these in the $2$-dimensional case, and the former in the $3$-dimensional one (see theorems \[thm:VarMain\] and \[thm:Var3D\]). Such an approach of bounding the discrepancy variance while averaging over ball centres was recently used by Sarnak  [@Sa] for mass distribution of forms on symmetric spaces, and P. Humphries  [@Humphries] for mass distribution of automorphic forms. Statement of the main results for $d=2,3$: asymptotics for the variance, CLT ---------------------------------------------------------------------------- Our principal results below are applicable to “flat" functions for $d=2,3$, understood in suitable, more and less restrictive, senses. For example, “Bourgain’s eigenfunction" [@Bourgain] $$\label{eq:BourgainEF} f_{n}\left(x\right)=\frac{1}{\sqrt{N}}\sum_{\lambda\in\mathcal{E}_{n}}\varepsilon_{\lambda}e\left(\left\langle x,\lambda\right\rangle \right)$$ with $\varepsilon_{\lambda}=\pm1$ for every $\lambda\in\mathcal{E}_{n}$, i.e. corresponding to $\left|c_{\lambda}\right|=N^{-1/2},$ satisfies any of the flatness conditions in the most restrictive sense. Denote ${\mathcal{B}}_{n}$ to be the class of Bourgain’s eigenfunctions. Our first principal result determines the precise asymptotic behaviour of the variance ${\mathcal{V}}(X_{f_n,r})$ for the $2$-dimensional case, and moreover asserts that the moments of the standardized random $L^{2}$-mass of $f_{n}$ are asymptotically Gaussian; we subsequently deduce a Central Limit Theorem (see Corollary \[cor:CLT\_result\]). For the sake of elegance of presentation, it is formulated for Bourgain’s eigenfunctions ; below we formulate a more general result which holds for a larger class of flat eigenfunctions (see Theorem \[thm:VarMainGeneralized\] in section \[sec:statement results strong\]), and later a result where the averaging over the ball centre $x$ is itself restricted to shrinking balls (Theorem \[thm:VarMainExplRestricted\] in section \[sec:RestrictedAverages\]). \[thm:VarMain\] There exists a density one sequence $S_{2}'\subseteq S_{2}$ so that the following holds. Let $r_{0}=r_{0}\left(n\right)=n^{-1/2}T_{0}\left(n\right)$ with $T_{0}\left(n\right)\to\infty $. 1. Fix a number $\epsilon>0$, and suppose that $ T_0(n) < \left(\log n\right)^{\frac{1}{2}\log\frac{\pi}{2}-\epsilon} $. Then as $n\to\infty$ along $S_{2}'$ we have $$\label{eq:var asympt d=2 Bourgain} {\mathcal{V}}\left(X_{f_{n},r}\right)\sim\frac{16}{3 \pi}r^{4}T^{-1}$$ uniformly for all $$\label{eq:r0<r<n^-1/2*discr} r_{0} < r <n^{-1/2}\left(\log n\right)^{\frac{1}{2}\log\frac{\pi}{2}-\epsilon}$$ and $ f_n \in {\mathcal{B}}_{n} $, where $T:=n^{1/2}r.$ 2. Under the above notation let $$\label{eq:standardizedX} \hat{X}_{f_{n},r}:=\frac{X_{f_{n},r}-{\mathbb{E}}[X_{f_{n},r}]}{\sqrt{{\mathcal{V}}\left(X_{f_{n},r}\right)}}$$ be the standardized random $L^{2}$-mass of $f_{n}$, $r_{1}=r_{1}(n)=n^{-1/2}T_{1}\left(n\right)$, and suppose further that the sequence of numbers $T_{1}(n)>T_{0}(n)$ satisfies $T_{1}(n)=O\left(N^{\xi}\right)$ for every $\xi>0$. Then for all $k\ge 3$ the $k$-th the moment of $\hat{X}_{f_{n},r}$ converges, for $n\rightarrow\infty$ along $S_{2}'$, to the standard Gaussian moment $$\label{eq:moments Gaussian lim} {\mathbb{E}}[\hat{X}_{f_{n},r}^{k}] \rightarrow {\mathbb{E}}[Z^{k}],$$ uniformly for $r_{0}<r<r_{1}$ and $ f_n \in {\mathcal{B}}_{n} $, where $Z\sim N(0,1)$ is the standard Gaussian variable. The claimed uniform asymptotics of the variance means explicitly that, as $n\rightarrow\infty$ along $S_{2}'$, one has $$\label{eq:var unif asymp d=2} \sup\limits_{\substack{r_0 < r < \left(\log n\right)^{\frac{1}{2}\log\frac{\pi}{2}-\epsilon} \\ f_n \in {\mathcal{B}}_{n}}} \left|\frac{{\mathcal{V}}\left(X_{f_{n},r}\right)}{\frac{16}{3 \pi}r^{4}T^{-1}} - 1\right| \rightarrow 0$$ and the uniform convergence of the moments means that for every $k\ge 3$, $$\sup\limits_{\substack{r_0 < r < r_1 \\ f_n \in {\mathcal{B}}_{n}}} \left|{\mathbb{E}}[\hat{X}_{f_{n},r}^{k}] - {\mathbb{E}}[Z^{k}]\right| \rightarrow 0.$$ Concerning the restricted range in Theorem \[thm:VarMain\] (and ) for the possible radii, it is directly related to a well-known result on the angular distribution of lattice points in ${\mathcal{E}}_{n}$, for generic $n\in S_{2}$. Namely, it was shown  [@ErdosHall] that ${\mathcal{E}}_{n}$, projected by homothety to the unit circle, is equidistributed, and moreover, a quantitative measure for the discrepancy is asserted (see section \[sec:ang distr\] below, and, in particular, ), satisfied by [*generic*]{} $n\in S_{2}$. Bourgain  [@Bourgain] observed that $f_{n}\in {\mathcal{B}}_{n}$, when averaged over $x\in {\mathbb{T}}^{d}$, exhibits Gaussianity in the following sense. Let $T>0$ be a fixed number, and define the scaled function $\varphi_{x}:[-1,1]^{2}\rightarrow{\mathbb{R}}$ around $x$ as $$\label{eq:varphi rand x def} \varphi_{x}(y):= f_{n}\left( x+ \frac{T}{\sqrt{n}}\cdot y\right),$$ i.e. the trace of $f_{n}$ on the side-$2\frac{T}{\sqrt{n}}$ square centred at $x$. It was found  [@Bourgain], that, upon thinking of $x\in{\mathbb{T}}^{2}$ as [*random*]{}, and $\varphi_{x}(\cdot)$ as a [*random field*]{} indexed by $[-1,1]^{2}$, it converges, in a suitable sense, to a particular [*Gaussian*]{} field (“monochromatic isotropic waves") on ${\mathbb{R}}^{2}$, restricted to $[-1,1]^{2}$. This allows one to infer some results on the (deterministic) functions $f_{n}\in {\mathcal{B}}_{n}$ from the analogous results on the limit Gaussian random field. We may then reinterpret the quantitative version of the angular equidistribution of lattice points as allowing the parameter $T$ in to grow as a (positive) logarithmic power of $n$, while still retaining the said asymptotic Gaussianity, also allowing for the comparison between the mass distribution of $f_{n}$ w.r.t. the position and mass distribution of monochromatic isotropic waves. Our intuition regarding the possibility of carrying on the explained “de-randomisation" argument for establishing results of similar nature to the presented results was recently validated by Sartori  [@Sartori]. An application of the standard theory [@Feller §XVI.3 Lemma 2] allows us to infer a uniform Central Limit Theorem for the random variables $\hat{X}_{f_{n},r}$ from the convergence of their respective moments to the Gaussian ones. \[cor:CLT\_result\] In the setting of Theorem \[thm:VarMain\] part (2), the distribution of the random variables $\{\hat{X}_{f_{n},r}\}$ converges uniformly to the standard Gaussian distribution: as $n\rightarrow\infty$ along $S_{2}'$ $${\operatorname{meas}}\{ x\in \mathbb{T}^2:\: \hat{X}_{f_{n},r;x} \le t\} \rightarrow \frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^{t}e^{-z^{2}/2}dz,$$ uniformly for $t\in{\mathbb{R}}$, $r_{0}<r<r_{1}$ and $ f_n \in {\mathcal{B}}_{n} $. For the $3$-dimensional case, for Bourgain’s eigenfunctions, we only claim precise asymptotic result on $\text{\ensuremath{\mathcal{V}}}\left(X_{f_{n},r}\right)$, the good news being that the claimed results are valid for [*all*]{} energies satisfying the natural congruence assumptions. \[thm:Var3D\] There exists a number $\eta>0$ such that if $r_{0}=r_{0}(n)=n^{-1/2}T_{0}(n)$ with $T_{0}(n)\rightarrow\infty$, then for all $n\not\equiv0,4,7\,\left(8\right)$ we have $$\text{\ensuremath{\mathcal{V}}}\left(X_{f_{n},r}\right)\sim r^{6}T^{-2},$$ uniformly for $r_{0} < r < n^{-1/2+\eta}$ and $ f_n \in {\mathcal{B}}_{n} $. The meaning of the uniform statement in Theorem \[thm:Var3D\] is that $$\sup_{\begin{subarray}{c} r_{0} < r < n^{-1/2+\eta} \\ f_n \in {\mathcal{B}}_{n} \end{subarray}}\left|\frac{\text{\ensuremath{\mathcal{V}}}\left(X_{f_{n},r}\right)}{r^{6}T^{-2}}-1\right|\to0\label{eq:AympVar3D}$$ as $n\to\infty$ along $n\not\equiv0,4,7\,\left(8\right)$, cf. in the $2$-dimensional case. Statement of the main results for $d=2,3$: more general upper and lower bounds {#sec:statement results weak} ------------------------------------------------------------------------------ Let $f_{n}$ be as in , and consider the vector $$\label{eq:v_def} \underline{v}:=(|c_{\lambda}|^{2})_{\lambda \in{\mathcal{E}}_{n}} \in {\mathbb{R}}^{\mathcal{E}_n} $$ of the squared absolute values of its coefficients; we denote its normalised $\ell_{\infty}$-norm $$\label{eq:vnorm inf} [\underline{v}]_{\infty} := N \cdot \max\limits_{\lambda\in{\mathcal{E}}_{n}}|c_{\lambda}|^{2}.$$ \[def:ultraflat\] We say that an eigenfunction $f_{n}$ in is $\epsilon$-ultraflat if its coefficients satisfy $$\label{eq:ultra_flat_cond} [\underline{v}]_{\infty} \le N^{\epsilon}.$$ Denote ${\mathcal{U}}_{n;\epsilon}$ to be the class of $\epsilon$-ultraflat functions. The following couple of theorems establish more general upper and lower bounds on ${\mathcal{V}}(X_{f_n,r})$ in the $2$ and $3$-dimensional cases respectively. \[thm:UpperBound2d\] There exists a density $1$ sequence $S_{2}'\subseteq S_{2}$ and an absolute constant $C>0$ such that for every $ \epsilon>0 $, $ \eta >0 $, $r_{0}=r_{0}(n)=n^{-1/2}T_{0}(n)$ with $T_{0}(n)\rightarrow\infty$ arbitrarily slowly, and $r=n^{-1/2}T>r_{0}$, as $n\to\infty$ along $S_{2}'$ we have $$\label{eq:bounds var ultraflat d=2} T^{-1}N^{-2\epsilon}\ll \frac{{\mathcal{V}}(X_{f_n,r})}{r^{4}} \ll N^{\epsilon}\cdot \left(T^{-1}+(\log{n})^{-\frac{1}{2}\log{\frac{\pi}{2}}+\eta} \right)$$ uniformly for $r_{0} < r< Cn^{-1/2}N^{1-\epsilon}$ and $f_{n}\in {\mathcal{U}}_{n;\epsilon}$, with the constant involved in the “$\ll$"-notation in is absolute for the lower bound, and depends only on $\eta$ for the upper bound. Moreover, the upper bound is valid for the extended range $r > r_{0}$ (with no upper bound on $r$ imposed), and the lower bound is valid for every $ n\in S_2 $. \[thm:UpperBound3d\] There exists a number $\eta>0$ and a constant $C>0$ such that for every $\epsilon>0$, $r_{0}=r_{0}(n)=n^{-1/2}T_{0}(n)$ with $T_{0}(n)\rightarrow\infty$ arbitrarily slowly, $r=n^{-1/2}T>r_{0}$, and $n\not\equiv 0,4,7\,\left(8\right)$ we have $$\label{eq:bounds var ultraflat d=3} T^{-2}N^{-2\epsilon} \ll \frac{{\mathcal{V}}(X_{f_{n},r})}{r^{6}} \ll N^{\epsilon} \left( T^{-2}+n^{-\eta}\right),$$ uniformly for $r_{0} < r < Cn^{-1/2}N^{1-\epsilon}$ and $f_{n}\in {\mathcal{U}}_{n;\epsilon}$, where the constants involved in the “$\ll$"-notation are absolute. Moreover, the upper bound in is valid for the extended range $r > r_{0}$. For Bourgain’s eigenfunctions, the proofs of Theorem \[thm:UpperBound2d\] and Theorem \[thm:UpperBound3d\] yield slightly stronger bounds compared to and , namely $$T^{-1}\ll \frac{{\mathcal{V}}(X_{f_n,r})}{r^{4}} \ll T^{-1}+(\log{n})^{-\frac{1}{2}\log{\frac{\pi}{2}}+\epsilon}$$ for $ d=2 $, and $$T^{-2} \ll \frac{{\mathcal{V}}(X_{f_{n},r})}{r^{6}} \ll T^{-2}+n^{-\eta}$$ for $ d=3 $. Outline of the paper -------------------- The rest of the paper is organised as follows. In section \[sec:statement results strong\] we formulate Theorem \[thm:VarMainGeneralized\], which, on one hand generalizes Theorem \[thm:VarMain\] for a larger class of flat eigenfunctions, and on the other hand, explicates a sufficient condition on $ n\in S_2 $ for its statements to hold; a few examples of application of Theorem \[thm:VarMainGeneralized\], corresponding to different asymptotic behaviour of the variance , are also discussed. Section \[sec:Proof\_Main\_thm\_part1\] is dedicated to giving a proof of the first part of Theorem \[thm:VarMain\] (resp. $1$st part of Theorem \[thm:VarMainGeneralized\]), whereas the second part of Theorem \[thm:VarMain\] (resp. $2$nd part of Theorem \[thm:VarMainGeneralized\]) is proved in section \[sec:Proof\_Main\_thm\_part2\]. Theorem \[thm:Var3D\], claiming the precise asymptotics for the $L^{2}$-mass variance for Bourgain’s eigenfunctions in $3$d, is proved in section \[sec:Proof\_3d\_theorem\]. In section \[sec:ProofOfBoundsThm\] we prove the various upper and lower bounds asserted by theorems \[thm:UpperBound2d\] and \[thm:UpperBound3d\]. A refinement of Theorem \[thm:VarMainGeneralized\], where rather than draw $x$ w.r.t. the uniform measure on the full torus, $x$ is drawn on balls slightly above Planck scale, is presented in section \[sec:RestrictedAverages\], and the additional subtleties of its proof as compared to the proof of Theorem \[thm:VarMainGeneralized\] are highlighted. Finally, section \[sec:AuxLemmasProof\] contains the proofs of all auxiliary lemmas, postponed in course of the proofs of the various results. Acknowledgements {#acknowledgements .unnumbered} ---------------- The authors of this manuscript wish to express their gratitude to J. Benatar, A. Granville, P. Kurlberg, Z. Rudnick, P. Sarnak and M. Sodin for numerous stimulating and fruitful discussions concerning various aspects of our work, and their interest in our research. It is a pleasure to thank the anonymous referee for his comments on an earlier version of this manuscript. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013), ERC grant agreement n$^{\text{o}}$ 335141. On Theorem \[thm:VarMain\]: CLT for mass distribution, $d=2$ {#sec:statement results strong} ============================================================ In this section we focus on Theorem \[thm:VarMain\]. Our first goal is to formulate a result, that on one hand generalises the statement of Theorem \[thm:VarMain\] to a larger class of eigenfunctions, and, on the other hand, provides a more explicit control over the generic numbers $n\in S_{2}$. To this end we discuss the angular distribution of $\lambda\in{\mathcal{E}}_{n}$ (section \[sec:ang distr\]), and the spectral correlations (section \[sec:quasi corr\]), also used in the course of the proof of the $3$-dimensional Theorem \[thm:Var3D\]; we will be able to formulate Theorem \[thm:VarMainGeneralized\], as prescribed above, by appealing to these. In section \[eq:examples varying theta\] we consider a few scenarios when Theorem \[thm:VarMainGeneralized\] is applicable, prescribing different asymptotic behaviour for the variance . Angular equidistribution of lattice points {#sec:ang distr} ------------------------------------------ For every $\lambda=\left(\lambda_{1},\lambda_{2}\right)\in\mathcal{E}_{n}$, write $\lambda_{1}+i\lambda_{2}=\sqrt{n}e^{i\phi}$, and denote the various angles by $$0\le\phi_{1}<\phi_{2}<\dots<\phi_{N}<2\pi.$$ Recall that the discrepancy of the sequence $\phi_{j}$ is defined by $$\label{eq:Discrepancy_2d} \Delta\left(n\right)=\sup_{0\le a\le b\le2\pi}\left|\frac{1}{N}\cdot \#\left\{ 1\le j\le N:\,\phi_{j}\in\left[a,b\right]\,\text{mod\,}2\pi\right\} -\frac{\left(b-a\right)}{2\pi}\right|.$$ For every $ \epsilon>0 $, we say that $ n\in S_2 $ satisfies the hypothesis $ \mathcal{D}(n,\epsilon) $ if $$\label{eq:D_n_epsilon} \Delta\left(n\right)\le \left(\log n\right)^{-\frac{1}{2}\log\frac{\pi}{2}+\epsilon}.$$ By Erdős-Hall [@ErdosHall Theorem 1], there exists a density one sequence $S_2'(\epsilon)\subseteq S_2 $ such that $ \mathcal{D}(n,\epsilon) $ is satisfied for every $n\in S_2'(\epsilon) $. By a standard diagonalization argument, there exists a density one sequence $ S_2'\subseteq S_2 $ such that $\mathcal{D}(n,\epsilon) $ is satisfied for *every* $ \epsilon>0 $ and $n\in S_2' $ sufficiently large. In particular, the angles $\left\{ \phi_{j}\right\} $ are equidistributed mod $2\pi$ along this sequence, i.e., the lattice points are equidistributed on the corresponding circles. Spectral correlations in $2d$ (and $3d$) {#sec:quasi corr} ---------------------------------------- For $d=2$, while computing the moments of $X_{f_{n},r}$ (e.g. for Bourgain’s eigenfunction ), with $x$ drawn uniformly on the whole of ${\mathbb{T}}^{2}$, one exploits the orthogonality relations $$\int\limits_{{\mathbb{T}}^{2}}e(\langle \lambda , x \rangle)dx = \begin{cases} 0 &\lambda\ne 0 \\ 1 &\lambda=0 \end{cases}$$ for $\lambda\in{\mathbb{Z}}^{2}$ to naturally encounter the length-$l$ spectral correlation problem. That is, for $l\ge 2$ and $n\in S_{2} $ one is interested in the size of the length-$ l $ spectral correlation set $$\label{eq:Sc correlations def} {\mathcal{S}}_{n}(l) = \left\{ (\lambda^{1},\ldots,\lambda^{l})\in({\mathcal{E}}_{n})^{l}:\: \sum\limits_{i=1}^{l}\lambda^{i}=0 \right\},$$ which, by an elementary congruence obstruction argument modulo $ 2 $, is only non-empty for $l=2k$ even. In this case $l=2k$ we further define the [*diagonal*]{} correlations set to be all the permutations of tuples of the form $(\lambda^{1},-\lambda^{1},\ldots, \lambda^{k},-\lambda^{k})$: $$\label{eq:Dc diag def} {\mathcal{D}}_{n}(l) = \left\{ \pi(\lambda^{1},-\lambda^{1},\ldots,\lambda^{k},-\lambda^{k}): \lambda^{1},\ldots,\lambda^{k}\in ({\mathcal{E}}_{n})^{k},\,\pi\in S_{l}\right\}.$$ The set ${\mathcal{D}}_{n}$ is dominated by non-degenerate tuples (i.e. $\lambda^{i}\ne \pm\lambda^{j}$ for $ i\ne j $), hence its size is asymptotic to $$|{\mathcal{D}}_{n}(l)|= \frac{(2k)!}{2^{k}\cdot k!}N^{k}\cdot \left(1 + O_{N\rightarrow\infty}\left( \frac{1}{N} \right) \right).$$ Clearly, ${\mathcal{D}}_{n}(l)\subseteq {\mathcal{S}}_{n}(l)$ so that, in particular ${\mathcal{S}}_{n}(l) \gg N^{l/2}$. To the other end, we have ${\mathcal{S}}_{n}(2) = {\mathcal{D}}_{n}(2)$ by the definition, and both the precise statement $$\label{eq:Zygmund 4-corr} {\mathcal{S}}_{n}(4) = {\mathcal{D}}_{n}(4)$$ (used for the variance computation below) and the bound $$|{\mathcal{S}}_{n}(l)| = O_{N\rightarrow\infty}(N^{l-2})$$ follow from Zygmund’s elementary observation  [@Zygmund]. For $ l=6 $, Bourgain (published in  [@K-K-W]) improved Zygmund’s bound to $$|{\mathcal{S}}_{n}(6)| = o_{N\rightarrow\infty}(N^{4});$$ this was improved  [@BombieriBourgain] to $$|{\mathcal{S}}_{n}(6)| = O_{N\rightarrow\infty}(N^{7/2}),$$ valid for [*all*]{} $n\in S_{2}$. If one is willing to excise a thin sequence in $S_{2}$, then the more striking estimate [@BombieriBourgain] $$|{\mathcal{S}}_{n}(6)| = |{\mathcal{D}}_{n}(6)| + O(N^{3-\gamma}),$$ with some $\gamma >0$, is valid for a density $1$ sequence $S_{2}'\subseteq S_{2}$. More generally [@Bourgain], for every $l\ge 6$ even, there exists a density $1$ sequence $S_{2}'(l)\subseteq S_{2}$ and a number $\gamma_{l}>0$ such that $$\label{eq:corr diag dom l} |{\mathcal{S}}_{n}(l)| = |{\mathcal{D}}_{n}(l)| + O(N^{l/2-\gamma_l})$$ along $n\in S_{2}'(l)$. A standard diagonal argument then yields the existence of a density $1$ sequence $S_2'\subseteq S_{2}$ so that is valid for [*all*]{} even $l\ge 4$. Given an even number $l=2k\ge 2$ we say that a sequence $S_{2}'\subseteq S_{2}$ satisfies the length-$l$ [**diagonal domination**]{} assumption if there exists a number $\gamma=\gamma_l >0$ so that holds. For the $3$-dimensional case under the consideration of Theorem \[thm:Var3D\] the analogous estimates to are required to evaluate the relevant moments of $X_{f_{n},r}$. We define ${\mathcal{S}}_{3;n}$ and ${\mathcal{D}}_{3;n}$ analogously to and respectively, this time the $\lambda^{i}$ are lying on the $2$-sphere of radius $\sqrt{n}$. Unlike the lattice points lying on circles, Zygmund’s argument is not applicable for the $2$-sphere, so that an analogue of is not valid; luckily the asymptotic statement $$\label{eq:3d 4-corr} |{\mathcal{S}}_{3;n}(4)| = |{\mathcal{D}}_{3;n}(4)| + O\left( N^{7/4 +\epsilon} \right),$$ a key input to the variance computation in Theorem \[thm:Var3D\], was recently established  [@BenatarMaffucci]. It was also shown in [@BenatarMaffucci] that the asymptotic diagonal domination for the higher length correlations sets does not hold in the $ 3 $-dimensional case. A more general version of Theorem \[thm:VarMain\], with explicit control over $S_{2}'$ {#sec:VarMainExpl} -------------------------------------------------------------------------------------- We are interested in extending Theorem \[thm:VarMain\] to a larger class of eigenfunctions. To this end, we introduce the following notation: Let $f_{n}$ be an eigenfunction on the $2$-torus corresponding to coefficients $(c_{\lambda})_{\lambda\in{\mathcal{E}}_{n}}$ via , and $\underline{v}\in{\mathbb{R}}^{\mathcal{E}_n}\simeq {\mathbb{R}}^{N}$ as above. 1. Denote $$\label{eq:A4Def} A_{4} = A_{4}(\underline{v}) = N\sum\limits_{\lambda\in{\mathcal{E}}_{n}} |c_{\lambda}|^{4} = N\cdot \|\underline{v}\|^{2}.$$ 2. Given $\lambda\in{\mathcal{E}}_{n}$ let $\lambda_{+}$ be the clockwise nearest neighbour of $\lambda$ on $\sqrt{n}\mathcal{S}^{1}$, and $$\label{eq:BasicVariablesV} V\left(\underline{v} \right):= N\sum_{\lambda\in\mathcal{E}_{n}}\left|\left|c_{\lambda_{+}}\right|^{2}-\left|c_{\lambda}\right|^{2}\right|.$$ 3. Let $$\label{eq:alphaDef} \widetilde{V}(\underline{v}) = \frac{[\underline{v}]_{\infty} \cdot V(\underline{v})}{A_{4}(\underline{v})}.$$ The following Lemma, proved in section \[sec:AuxLemmasProof\], summarizes some basic properties of the quantities in (\[eq:vnorm inf\]), (\[eq:A4Def\]), (\[eq:BasicVariablesV\]) and (\[eq:alphaDef\]): \[lem:BasicVarProp\] We have 1. $1\le A_{4} \le[\underline{v}]_{\infty}.$ 2. $[\underline{v}]_{\infty} \le1+V\left(\underline{v} \right)$. 3. $V\left(\underline{v} \right) \le\widetilde{V}(\underline{v})\le V\left(\underline{v} \right)\left(1+V\left(\underline{v} \right)\right)$. By we have that $$\label{eq:A4<->theta} A_{4} = \cos(\theta)^{-2},$$ where $\theta = \theta_{f_n} = \theta(\underline{v},\underline{v_{0}})$ is the angle between $\underline{v}$ and the vector $\underline{v_{0}}= (\frac{1}{N})_{\lambda\in{\mathcal{E}}_{n}}$ corresponding to Bourgain’s eigenfunctions, hence $ \theta $ reflects the proximity of $f_{n}$ to Bourgain’s eigenfunction; by the first part of Lemma \[lem:BasicVarProp\], the angle $\theta$ is restricted to the interval $ \left[0,\arccos \left(1/\sqrt{N}\right)\right] \subseteq [0,\pi/2) $. Given a sequence $T(n)\rightarrow\infty$ and a sequence $\eta(n)>0$ we define: 1. A sequence $\{{\mathcal{F}}_{1}(n;T(n),\eta(n))\}_{n}$ of families of functions consisting for $n\in S_2$ of all functions $f_{n}$ as in satisfying $$\label{eq:F_1_def}{\mathcal{F}}_{1}(n;T(n),\eta(n)) = \left\{f_{n}:\: \widetilde{V}(\underline{v})< \eta(n) \cdot \frac{T(n)}{\log T(n)} \right\}.$$ 2. A sequence $\{{\mathcal{F}}_{2}(n;T(n),\eta(n))\}_{n}$ of families of functions consisting for $n\in S_2$ of all functions $f_{n}$ as in satisfying $$\label{eq:F_2_def}{\mathcal{F}}_{2}(n;T(n),\eta(n)) = \left\{f_{n}:\: [\underline{v}]_{\infty} < T(n)^{\eta(n)} \right\},$$ where we recall the notation for $[\underline{v}]_{\infty}$. We are now in a position to state the generalized version of Theorem \[thm:VarMain\]: \[thm:VarMainGeneralized\] Let $r_{0}=r_{0}\left(n\right)=n^{-1/2}T_{0}\left(n\right)$ with $T_{0}\left(n\right)\to\infty$, and $\eta(n)>0$ any vanishing sequence $\eta(n)\rightarrow 0$. 1. Fix a number $\epsilon>0$, and suppose that $ T_0(n) < \left(\log n\right)^{\frac{1}{2}\log\frac{\pi}{2}-\epsilon} $. Then, if $S_{2}'\subseteq S_{2}$ is a sequence satisfying $ \mathcal{D}(n,\epsilon/2)$ for all $n\in S_{2}'$, as $n\rightarrow\infty$ along $S_{2}'$, we have $$\label{eq:var asympt d=2 precise} {\mathcal{V}}\left(X_{f_{n},r}\right)\sim\frac{16}{3 \pi \cos^{2}\theta_{f_{n}}}r^{4}T^{-1}$$ with $\theta_{f_{n}}$ as in , uniformly for all $r_{0} < r <n^{-1/2}\left(\log n\right)^{\frac{1}{2}\log\frac{\pi}{2}-\epsilon}$ and $f_{n}\in{\mathcal{F}}_{1}(n;T(n),\eta(n))$, where $T:=T(n)=n^{1/2}r.$ 2. Let $k\ge 3$ be an integer, $r_{1}=r_{1}(n)=n^{-1/2}T_{1}\left(n\right)$, and suppose further that the sequence of numbers $T_{1}(n)>T_{0}(n)$ satisfies $T_{1}(n)=O\left(N^{\xi}\right)$ for every $\xi>0$. Suppose that $S_{2}'\subseteq S_{2}$ is a sequence satisfying the length-$2k$ diagonal domination assumption and the hypothesis $\mathcal{D}(n,\epsilon)$ for all $n\in S_{2}'$. Then the $k$-th the moment of $\hat{X}_{f_{n},r}$ converges, as $n\rightarrow\infty$ along $S_{2}'$, to the standard Gaussian moment $${\mathbb{E}}[\hat{X}_{f_{n},r}^{k}] \rightarrow {\mathbb{E}}[Z^{k}],$$ uniformly for $r_{0}<r<r_{1}$ and $f_{n}\in {\mathcal{F}}_{2}(n;T(n),\eta(n))$ where $Z\sim N(0,1)$ is the standard Gaussian variable. Section \[eq:examples varying theta\] exhibits a few scenarios when Theorem \[thm:VarMainGeneralized\] is applicable; as in these the true asymptotic behaviour of the variance genuinely varies together with $\theta_{f_{n}}$, this demonstrates that $\theta_{f_{n}}$ (and hence $A_{4}$) is the proper flatness measure of $f_{n}$, see also examples \[ex:Bourgain\] and \[ex:flat vs nonflat\]. In the setting of Theorem \[thm:VarMainGeneralized\] part (2), the distribution of the random variables $\{\hat{X}_{f_{n},r}\}$ converges uniformly to the standard Gaussian distribution: as $n\rightarrow\infty$ along $S_{2}'$ $${\operatorname{meas}}\{ x\in\mathbb{T}^2 :\: \hat{X}_{f_{n},r;x} \le t\} \rightarrow \frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^{t}e^{-z^{2}/2}dz,$$ uniformly for $t\in{\mathbb{R}}$, $r_{0}<r<r_{1}$, and $f_{n}\in {\mathcal{F}}_{2}(n;T(n),\eta(n))$. Some examples of application of Theorem \[thm:VarMainGeneralized\] {#eq:examples varying theta} ------------------------------------------------------------------ \[ex:Bourgain\] Let $ f_n $ be Bourgain’s eigenfunction, so that $ [\underline{v}]_{\infty} =A_4 =1 $ and $ V\left(\underline{v} \right) = \widetilde{V}(\underline{v}) =0 $. For every $ \eta(n)>0,T(n)>1 $ we have $${\mathcal{B}}_{n} \subseteq {\mathcal{F}}_{1}(n;T(n),\eta(n)) \cap {\mathcal{F}}_{2}(n;T(n),\eta(n)).$$ Hence Theorem \[thm:VarMainGeneralized\] implies Theorem \[thm:VarMain\]. The following example exhibits a scenario when an application of Theorem \[thm:VarMainGeneralized\] yields a Central Limit Theorem for $X_{f_{n},r}$, corresponding to asymptotic behaviour of the respective variance ${\mathcal{V}}(X_{f_{n},r})$ which is very different from the behaviour in Theorem \[thm:VarMain\]. \[ex:flat vs nonflat\] Let $\epsilon>0$, $r_{0}$, and $ T_0(n) $ as in Theorem \[thm:VarMainGeneralized\], and $r_{1}= r_1(n)=n^{-1/2} T_{1}(n) > r_{0}$ with $T_{1}(n) \le (\log{n})^{\frac{1}{2}\log{\frac{\pi}{2}}-\epsilon}$. There exists a density $1$ sequence $S_{2}'\subseteq S_{2}$ so that the following holds. Let $t=t(n)\in (0,1)$ be a number satisfying $t(n) \gg \frac{1}{T_{0}(n)^{\xi}}$ for every $\xi>0$, such that $N\cdot t$ is an integer. We choose an ordering $\lambda^{1},\lambda^{2},\ldots \lambda^{N}\in{\mathcal{E}}_{n}$ such that for every $1\le i \le N-1$ we have that $\lambda^{i+1}$ is the (clockwise) nearest neighbour $\lambda^{i+1}=\lambda^{i}_{+}$, and set $$\left(\left|c_{\lambda^{1}}\right|^{2},,\dots,\left|c_{\lambda^{N}}\right|^{2}\right)=(\underset{\begin{subarray}{c} Nt\end{subarray}\text{\,times}}{\underbrace{\left(Nt\right)^{-1},\dots\dots,\left(Nt\right)^{-1}}},0\dots,0).$$ Then $$\label{eq:var asympt nonflat} {\mathcal{V}}(X_{f_{n},r}) \sim \frac{16}{3 \pi}r^{4}t^{-1}T^{-1},$$ uniformly for $r_{0}<r=n^{-1/2}T<r_{1}$, and $f_{n}$ with coefficients $c_{\lambda}$ as above. If, in addition, we have $T_{1}(n)=O(N^{\xi})$ for every $\xi>0$, then the distribution of the standardised random variable $\hat{X}_{f_{n},r}$ converges to standard Gaussian uniformly. Comparing to we observe that the asymptotic behaviour of the variance for the flat and the non-flat functions respectively is genuinely different, provided that we choose $t(n)\rightarrow 0$; we infer that the proposed flatness measure is the natural choice for this problem. One can also generalise Theorem \[thm:VarMain\] as follows: \[cor:VarAsympGen\] Let $\epsilon$, $r_{0}$, $T_{0}(n)$, $r_{1}$ and $T_{1}(n)$ be as in Theorem \[thm:VarMainGeneralized\], and $g:{\mathcal{S}}^{1}\rightarrow{\mathbb{R}}$ a non-negative function of bounded variation such that $ \|g\|_{L^{1}({\mathcal{S}}^{1})}=1$. For $n\in S_{2}$ and $\lambda\in {\mathcal{E}}_{n}$ we set $|\widetilde{c_{\lambda}}|^{2} := g(\lambda/\sqrt{n})$, and normalise the vector $\widetilde{\underline{v}}:=(|\widetilde{c_{\lambda}}|^{2})_{\lambda\in{\mathcal{E}}_{n}}$ by setting $\underline{v} := \frac{\widetilde{\underline{v}}}{\|\widetilde{\underline{v}}\|_{1}}$, i.e. $$\label{eq:v BV norm} v:=(|{c_{\lambda}}|^{2})_{\lambda\in{\mathcal{E}}_{n}} = \left(\frac{|\widetilde{c_{\lambda}}|^{2}}{\sum\limits_{\mu\in{\mathcal{E}}_{n}}|\widetilde{c_{\mu}}|^{2}}\right)_{\lambda\in{\mathcal{E}}_{n}}.$$ Then along a generic sequence $S_{2}'\subseteq S_{2}$ we have $${\mathcal{V}}(X_{f_{n},r}) \sim \frac{16}{3 \pi}\|g\|_{L^{2}({\mathcal{S}}^{1})}^{2}r^{4}T^{-1},$$ uniformly for $r_{0}<r=n^{-1/2}T<r_{1}$, and $f_{n}$ with coefficients $c_{\lambda}$ as in . If, in addition, we have $T_{1}(n)=O(N^{\xi})$ for every $\xi>0$, then the distribution of the standardised random variable $\hat{X}_{f_{n},r}$ converges to standard Gaussian. By Koksma’s inequality (see e.g. [@KuipersNiederreiter]), $A_{4}\left(\underline{v} \right)\sim\left\Vert g\right\Vert _{2}^{2}$ along a density one sequence in $S_{2}$. Also note that $$V(\underline{v}) \ll V\left(g\right),$$ with the l.h.s. as in , and r.h.s. the variation of $g$ on ${\mathcal{S}}^{1}$. In light of Lemma \[lem:BasicVarProp\], both parts of Corollary \[cor:VarAsympGen\] follow from Theorem \[thm:VarMainGeneralized\]. Notation ======== For the convenience of the reader, we summarize here the notation used in our paper. $ S_d=\{n=a_{1}^{2}+\ldots +a_{d}^{2}:\: a_{1},\ldots,a_{d}\in{\mathbb{Z}}\} $: the set of integers expressible as a sum of $ d $ squares, see .\ ${\mathcal{E}}_{n} = {\mathcal{E}}_{d;n}=\{\lambda\in{\mathbb{Z}}^{d}:\: \|\lambda\|^{2}=n\}$: the standard lattice points lying on the $(d-1)$-dimensional sphere (a circle for $ d=2 $) of radius-$\sqrt{n}$, see .\ $f_{n}\left(x\right)=\sum\limits_{\lambda\in\mathcal{E}_{n}}c_{\lambda}e\left(\left\langle x,\lambda\right\rangle \right)$: the toral Laplace eigenfunctions, see .\ $ N=N_{d;n}=\#\mathcal{E}_n$: the number of lattice points on the $(d-1)$-dimensional sphere (a circle for $ d=2 $) of radius-$\sqrt{n}$, see .\ $B_{x}(r)$: the radius $r$ geodesic ball in $\mathbb{T}^d$ centred at $x$.\ $X_{f_{n},r}=X_{f_{n},r;x}= \int\limits_{B_{x}(r)}f_{n}(y)^{2}dy$: the $L^{2}$-mass of $ f_n $ restricted to $ B_{x}(r) $, where $ x $ is drawn randomly uniformly in $ \mathbb{T}^d $, see .\ ${\mathbb{E}}[X_{f_{n},r}] = \int\limits_{{\mathbb{T}}^{d}}X_{f_{n},r;x}dx$: the expected value of $X_{f_{n},r} $, see .\ ${\mathcal{V}}(X_{f_{n},r}) = {\mathbb{E}}[(X_{f_{n},r}-{\mathbb{E}}[X_{f_{n},r}])^{2}]$: the variance of $X_{f_{n},r} $, see .\ $\hat{X}_{f_{n},r}:=\frac{X_{f_{n},r}-{\mathbb{E}}[X_{f_{n},r}]}{\sqrt{{\mathcal{V}}\left(X_{f_{n},r}\right)}}$: the standardized random $L^{2}$-mass of $ f_n $, see .\ $ T = n^{1/2}r $.\ $\underline{v}=(|c_{\lambda}|^{2})_{\lambda \in{\mathcal{E}}_{n}} \in {\mathbb{R}}^{\mathcal{E}_n} $: the vector of the squared absolute values of the coefficients of $f_n $, see .\ $[\underline{v}]_{\infty} = N \cdot \max\limits_{\lambda\in{\mathcal{E}}_{n}}|c_{\lambda}|^{2}$: the normalised $\ell_{\infty}$-norm of $ \underline{v} $, see .\ $\mathcal{B}_n$: the class of Bourgain’s eigenfunctions $f_{n}\left(x\right)=\frac{1}{\sqrt{N}}\sum\limits_{\lambda\in\mathcal{E}_{n}}\varepsilon_{\lambda}e\left(\left\langle x,\lambda\right\rangle \right)$, where $\varepsilon_{\lambda}=\pm1$ for every $\lambda\in\mathcal{E}_{n}$, see .\ ${\mathcal{U}}_{n;\epsilon}$: the class of $ \epsilon $-ultraflat functions, where $ [\underline{v}]_{\infty} \le N^{\epsilon} $, see .\ $A_{4} = A_{4}(\underline{v}) = N\sum\limits_{\lambda\in{\mathcal{E}}_{n}} |c_{\lambda}|^{4} = N\cdot \|\underline{v}\|^{2}$, see .\ $\theta = \theta_{f_n} = \theta(\underline{v},\underline{v_{0}})$: the angle between $\underline{v}$ and the vector $\underline{v_{0}}= (\frac{1}{N})_{\lambda\in{\mathcal{E}}_{n}}$ corresponding to Bourgain’s eigenfunctions, see .\ $V\left(\underline{v} \right)= N\sum\limits_{\lambda\in\mathcal{E}_{n}}\left|\left|c_{\lambda_{+}}\right|^{2}-\left|c_{\lambda}\right|^{2}\right|$, where $\lambda_{+}$ is the clockwise nearest neighbour of $\lambda$ on $\sqrt{n}\mathcal{S}^{1}$, see .\ $\widetilde{V}(\underline{v}) = \frac{[\underline{v}]_{\infty} \cdot V(\underline{v})}{A_{4}(\underline{v})}$, see .\ ${\mathcal{F}}_{1}(n;T(n),\eta(n)) = \left\{f_{n}:\: \widetilde{V}(\underline{v})< \eta(n) \cdot \frac{T(n)}{\log T(n)} \right\}$, see .\ ${\mathcal{F}}_{2}(n;T(n),\eta(n)) = \left\{f_{n}:\: [\underline{v}]_{\infty} < T(n)^{\eta(n)} \right\}$, see .\ $\widehat{\lambda}=\lambda/\sqrt{n}$: the projection of $\lambda \in \mathcal{E}_n$ onto $\mathcal{S}^{d-1}.$\ $\Delta\left(n\right)=\sup\limits_{0\le a\le b\le2\pi}\left|\frac{1}{N}\cdot \#\left\{ 1\le j\le N:\,\phi_{j}\in\left[a,b\right]\,\text{mod\,}2\pi\right\} -\frac{\left(b-a\right)}{2\pi}\right|$: the discrepancy of the angles $ \phi_j $ corresponding to the lattice points $ {\mathcal{E}}_{2;n} $, see .\ Hypothesis $ \mathcal{D}(n,\epsilon) $ holds if $\Delta\left(n\right)\le \left(\log n\right)^{-\frac{1}{2}\log\frac{\pi}{2}+\epsilon}$, see .\ $\Delta_{3}\left(n\right)=\sup\limits_{\begin{subarray}{c} x\in\mathcal{S}^{2}\\ 0<r\le2 \end{subarray}}\left|\frac{1}{N} \cdot \#\left\{ \lambda\in\mathcal{E}_{3;n}:\,\left|\widehat{\lambda}-x\right|\le r\right\} -\frac{r^{2}}{4}\right|$: the spherical cap discrepancy of the points $ \mathcal{E}_{3;n} $, see .\ ${\mathcal{S}}_{n}(l) = \left\{ (\lambda^{1},\ldots,\lambda^{l})\in({\mathcal{E}}_{n})^{l}:\: \sum\limits_{i=1}^{l}\lambda^{i}=0 \right\}$: the length-$ l $ spectral correlation set, see .\ ${\mathcal{D}}_{n}(l) = \left\{ \pi(\lambda^{1},-\lambda^{1},\ldots,\lambda^{k},-\lambda^{k}): \lambda^{1},\ldots,\lambda^{k}\in ({\mathcal{E}}_{n})^{k},\,\pi\in S_{l}\right\}$: the diagonal correlations set, see .\ $\mathcal{A}_n (2k) = \left\{\left(\lambda_{1},\dots,\lambda_{2k}\right)\in{\mathcal{D}}_{n}(2k): \; \forall 1\le i\le k \; \lambda_{2i-1}\ne-\lambda_{2i} \right\}$: the set of “admissible” $ 2k $-tuples of lattice points, see .\ $S\left(\lambda_{1},\dots,\lambda_{2k}\right)$: the structure set of an admissible $ 2k $-tuple $\left(\lambda_{1},\dots,\lambda_{2k}\right)$, see .\ $J_{\alpha}\left(x\right)$: the Bessel function of the first kind of order $\alpha$.\ $g_{d}\left(x\right)=\frac{J_{d/2}\left(2 \pi x\right)}{(2 \pi x)^{d/2}}$: the Fourier transform of the characteristic function of the unit ball in $\mathbb{R}^{d}$, see .\ $ h_{2}\left(x\right)=\frac{J_{1}\left(2 \pi x\right)^{2}}{(2\pi x)^{2}}$, see .\ $h_{3}\left(x\right)=2\pi^{-1}(2 \pi x)^{-4}\left(\frac{\sin 2 \pi x}{2\pi x}-\cos 2 \pi x\right)^{2}$, see .\ $F_{\lambda_{0}}\left(s\right)=\frac{1}{N}\cdot \#\left\{ \lambda\in\mathcal{E}_{2;n}:\,\left\Vert \widehat{\lambda}-\widehat{\lambda_{0}}\right\Vert \le s\right\}$, see .\ $F\left(s\right)=F_{f_{n}}\left(s\right)=\sum\limits_{\begin{subarray}{c} \lambda,\lambda'\in\mathcal{E}_{2;n}\\ 0<\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s \end{subarray}}\left|c_{\lambda}\right|^{2}\left|c_{\lambda'}\right|^{2}$, see .\ $F_{3}\left(s\right)=\frac{1}{N^{2}} \cdot \#\left\{ \lambda\ne\lambda'\in\mathcal{E}_{3;n}:\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\right\}$, see .\ ${\mathbb{E}}_{x_{0},\rho}[X_{f_{n},r}] = \frac{1}{{\operatorname{Vol}}(B_{x_0}(\rho))}\int\limits_{B_{x_{0}}(\rho)}X_{f_{n},r;x}dx$: the “restricted” expected value of $X_{f_{n},r} $, see .\ ${\mathcal{V}}_{x_{0},\rho}(X_{f_{n},r}) = {\mathbb{E}}_{x_{0},\rho}[(X_{f_{n},r}-{\mathbb{E}}_{x_{0},\rho}[X_{f_{n},r}])^{2}]$: the restricted variance of $X_{f_{n},r} $, see .\ $\mathcal{C}_{n}(l;K) = \left\{(\lambda^{1},\ldots,\lambda^{l})\in\mathcal{E}_{n}^{l}:\: 0 < \left\| \sum\limits_{j=1}^{l}\lambda^{j} \right\| \le K \right\}$: the set of length-$l$ spectral quasi-correlations, see .\ Hypothesis $\mathcal{A}(n;l,\delta)$ holds if $\mathcal{C}_{n}(l;n^{1/2-\delta}) = \varnothing$, see . Proof of Theorem \[thm:VarMainGeneralized\], part 1: asymptotics for the variance, $d=2$. {#sec:Proof_Main_thm_part1} ========================================================================================= Expressing the variance ----------------------- We begin with some preliminary expressions for the variance. Note that if $x$ is drawn randomly, uniformly on $\mathbb{T}^{d}$, then $$\mathbb{E}\left[X_{f_{n},r}\right]=\frac{\pi^{d/2}}{\Gamma\left(d/2+1\right)}r^{d},\label{eq:ExpectationEquality}$$ and therefore in this case, we have $$\label{eq:variance_integral_form} {\mathcal{V}}(X_{f_{n},r}) = \int\limits_{{\mathbb{T}}^{d}}\left(\int_{B_{x}\left(r\right)}f_{n}\left(y\right)^{2}\,\text{d}y- \frac{\pi^{d/2}}{\Gamma\left(d/2+1\right)}r^{d} \right)^{2}dx.$$ Let $J_{\alpha}\left(x\right)$ be the Bessel function of the first kind of order $\alpha$. The following lemma, proved in section \[sec:AuxLemmasProof\], explicates the inner integral in : \[lem:InnerIntegral\]We have $$\begin{aligned} & \int_{B_{x}\left(r\right)}f_{n}\left(y\right)^{2}\,\text{d}y-\frac{\pi^{d/2}}{\Gamma\left(d/2+1\right)}r^{d} =\left(2\pi\right)^{d/2}r^{d}\sum_{\begin{subarray}{c} \lambda,\lambda'\in\mathcal{E}_{n}\\ \lambda\ne\lambda' \end{subarray}}c_{\lambda}\overline{c_{\lambda'}}e\left(\left\langle x,\lambda-\lambda'\right\rangle \right)g_{d}\left(r\left\Vert \lambda-\lambda'\right\Vert \right),\label{eq:IntegrandVar} \end{aligned}$$ where $$\label{eq:g_d} g_{d}\left(x\right):=\frac{J_{d/2}\left(2 \pi x\right)}{(2 \pi x)^{d/2}}$$ is the Fourier transform of the characteristic function of the unit ball in $\mathbb{R}^{d}$. The following formula for the variance follows from Lemma \[lem:InnerIntegral\], and : \[lem:VarExpd2\]  1. (Granville-Wigman [@GranvilleWigman Lemma 2.1]) For $d=2$ we have $$\label{eq:VarFormula2d} \text{\ensuremath{\mathcal{V}}}\left(X_{f_{n},r}\right)=8\pi^{2}r^{4}\sum_{\begin{subarray}{c} \lambda,\lambda'\in\mathcal{E}_{n}\\ \lambda\ne\lambda' \end{subarray}}\left|c_{\lambda}\right|^{2}\left|c_{\lambda'}\right|^{2}h_{2}\left(r\left\Vert \lambda-\lambda'\right\Vert \right)$$ where $$\label{eq:h_2} h_{2}\left(x\right):=\frac{J_{1}\left(2 \pi x\right)^{2}}{(2\pi x)^{2}}.$$ 2. For $d=3$ and for every $\epsilon>0$, we have $$\begin{aligned} \text{\ensuremath{\mathcal{V}}}\left(X_{f_{n},r}\right) & =16\pi^{3}r^{6}\sum_{\begin{subarray}{c} \lambda,\lambda'\in\mathcal{E}_{n}\\ \lambda\ne\lambda' \end{subarray}}\left|c_{\lambda}\right|^{2}\left|c_{\lambda'}\right|^{2}h_{3}\left(r\left\Vert \lambda-\lambda'\right\Vert \right)\label{eq:VarFormula3d}\\ & +O\left([\underline{v}]_{\infty}^2 r^{6}N^{-1/4+\epsilon}\right),\nonumber \end{aligned}$$ where $$\label{eq:h_3} h_{3}\left(x\right):=2\pi^{-1}(2 \pi x)^{-4}\left(\frac{\sin 2 \pi x}{2\pi x}-\cos 2 \pi x\right)^{2}.$$ Note that functions $g_{2}$ and $h_{2}$ satisfy the following properties: \[lem:H2Formulas\]We have 1. $\int_{0}^{\infty}h_{2}\left(s\right)\,\text{d}s=\frac{2}{3\pi^2}$. 2. $g_{2}\left(s\right)\sim\frac{1}{2}\hspace{1em}\left(s\to0\right)$. 3. $g_{2}\left(s\right)\ll s^{-3/2}\hspace{1em}\left(s\to\infty\right)$. 4. $g_{2}'\left(s\right)=-\frac{J_{2}\left(2\pi s\right)}{s}\ll\left(1+s\right)^{-3/2}.$ Proof of Theorem \[thm:VarMainGeneralized\], part 1: ---------------------------------------------------- For $\lambda\in\mathcal{E}_{n}$ let $\widehat{\lambda}=\lambda/\sqrt{n}$ be the projection of $\lambda$ onto the unit circle $\mathcal{S}^{1}.$ 1. For $ \lambda_0 \in \mathcal{E}_n $ and $0\le s\le2$, denote $$\label{eq:F_Lambda_Def} F_{\lambda_{0}}\left(s\right)=\frac{1}{N}\cdot \#\left\{ \lambda\in\mathcal{E}_{n}:\,\left\Vert \widehat{\lambda}-\widehat{\lambda_{0}}\right\Vert \le s\right\} .$$ 2. For $0\le s\le2$ denote $$F\left(s\right)=F_{f_{n}}\left(s\right)=\sum_{\begin{subarray}{c} \lambda,\lambda'\in\mathcal{E}_{n}\\ 0<\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s \end{subarray}}\left|c_{\lambda}\right|^{2}\left|c_{\lambda'}\right|^{2}.\label{eq:F_Function}$$ Recall that $\widetilde{V}(\underline{v})=\cos^{2}\theta \cdot [\underline{v}]_{\infty} V(\underline{v})$ by and . \[prop:DiscreteProp\]We have $$\begin{aligned} F\left(s\right) & =\frac{s}{\pi\cos^{2}\theta}\left(1+O\left(s^{2}+s^{-1}\Delta\left(n\right)+\widetilde{V}(\underline{v}) s+\widetilde{V}(\underline{v}) s^{-1}\Delta\left(n\right)^{2}\right)\right). \end{aligned}$$ We postpone the proof of Proposition \[prop:DiscreteProp\] until section \[sec:ProofOfLemmaDiscrete\] to present the proof of the first part of Theorem \[thm:VarMainGeneralized\] (that yields the first part of Theorem \[thm:VarMain\]): Assume that $ n\in S_2 $ satisfies the hypothesis $ \mathcal{D}(n,\epsilon/2) $. We may rewrite (\[eq:VarFormula2d\]) as $$\mathcal{V}\left(X_{f_{n},r}\right)=8\pi^{2}r^{4}\int_{0}^{2}h_{2}\left(Ts\right)\,\text{d}F\left(s\right).\label{eq:VarianceintegralForm}$$ We apply integration by parts to (\[eq:VarianceintegralForm\]) twice, in opposite directions: first, by integration by parts and Proposition \[prop:DiscreteProp\], we get $$\begin{aligned} 8\pi^{2}r^{4}\int_{0}^{2}h_{2}\left(Ts\right)\,\text{d}F\left(s\right) & =8\pi^{2}r^{4}h_{2}\left(2T\right)F\left(2\right)-8\pi^{2}r^{4}\int_{0}^{2}F\left(s\right)\,\text{d}h_{2}\left(Ts\right)\label{eq:VarEqAfterIntByParts}\\ & =8\pi^{2}r^{4}h_{2}\left(2T\right)F\left(2\right)-8\pi r^{4}\cos^{-2}\theta\int_{0}^{2}s\,\text{d}h_{2}\left(Ts\right)\nonumber \\ & +Err\left(X_{f_n,r}\right)\nonumber \end{aligned}$$ where $$\begin{aligned} & Err\left(X_{f_n,r}\right)\ll r^{4}\cos^{-2}\theta\int_{0}^{2}\left(s^{3}+\Delta\left(n\right)+\widetilde{V}(\underline{v}) s^{2}+\widetilde{V}(\underline{v})\Delta\left(n\right)^{2}\right)T\left|h_{2}'\left(Ts\right)\right|\,\text{d}s. \end{aligned}$$ Integrating by parts again, the first two terms on the r.h.s of (\[eq:VarEqAfterIntByParts\]) satisfy $$\begin{aligned} & 8\pi^{2}r^{4}h_{2}\left(2T\right)F\left(2\right)-8\pi r^{4}\cos^{-2}\theta\int_{0}^{2}s\,\text{d}h_{2}\left(Ts\right)=8\pi^{2}r^{4}h_{2}\left(2T\right)F\left(2\right)\label{eq:MainTermsVar}\\ & -16\pi r^{4}h_{2}\left(2T\right)\cos^{-2}\theta+8\pi r^{4}\cos^{-2}\theta\int_{0}^{2}h_{2}\left(Ts\right)\,\text{d}s.\nonumber \end{aligned}$$ By the first and the third parts of Lemma \[lem:H2Formulas\], $$\int_{0}^{2}h_{2}\left(Ts\right)\,\text{d}s=\frac{1}{T}\int_{0}^{2T}h_{2}\left(s\right)\,\text{d}s=\frac{2}{3\pi^2}T^{-1}+O\left(T^{-3}\right),\label{eq:h2Integral}$$ and therefore, substituting (\[eq:h2Integral\]) into (\[eq:MainTermsVar\]), we obtain $$\begin{aligned} \label{eq:VarianceMainTerms} 8\pi^{2}r^{4}h_{2}\left(2T\right)F\left(2\right)-8\pi r^{4}\cos^{-2}\theta\int_{0}^{2}s\,\text{d}h_{2}\left(Ts\right) & =\frac{16}{3\pi}\cos^{-2}\theta r^{4}T^{-1} +O\left(\cos^{-2}\theta r^{4}T^{-3}\right). \end{aligned}$$ By the fourth part of Lemma \[lem:H2Formulas\], $$\begin{aligned} \int_{0}^{2}T\left|h_{2}'\left(Ts\right)\right|\,\text{d}s & =\int_{0}^{2T}\left|h_{2}'\left(s\right)\right|\,\text{d}s\le\int_{0}^{\infty}\left|h_{2}'\left(s\right)\right|\,\text{d}s<\infty, \end{aligned}$$ $$\int_{0}^{2}s^{2}T\left|h_{2}'\left(Ts\right)\right|\,\text{d}s=T^{-2}\int_{0}^{2T}s^{2}\left|h_{2}'\left(s\right)\right|\,\text{d}s\ll T^{-2}\log T$$ and $$\int_{0}^{2}s^{3}T\left|h_{2}'\left(Ts\right)\right|\,\text{d}s=T^{-3}\int_{0}^{2T}s^{3}\left|h_{2}'\left(s\right)\right|\,\text{d}s\ll T^{-2},$$ and therefore for $ n $ satisfying $ \mathcal{D}(n,\epsilon/2), $ $$\begin{aligned} Err\left(X_{f_n,r}\right) & \ll\cos^{-2}\theta r^{4}\left(T^{-2}+\Delta\left(n\right)+\widetilde{V}(\underline{v}) T^{-2}\log T+\widetilde{V}(\underline{v})\Delta\left(n\right)^{2}\right)\nonumber \\ & \ll\cos^{-2}\theta r^{4}\left(T^{-2}+\left(\log n\right)^{-\frac{1}{2}\log\frac{\pi}{2}+\frac{\epsilon}{2}}+\widetilde{V}(\underline{v}) T^{-2}\log T+\widetilde{V}(\underline{v})\left(\log n\right)^{-\log\frac{\pi}{2}+\epsilon}\right),\label{eq:ErrorTerm} \end{aligned}$$ and follows from , and . Note that by (\[eq:ErrorTerm\]), for Bourgain’s eigenfunctions, for almost all $n\in S_{2}$ we have $$\sup_{\begin{subarray}{c} r > r_{0} \\ f_n\in {\mathcal{B}}{n} \end{subarray}}\left|\frac{\text{\ensuremath{\mathcal{V}}}\left(X_{f_{n},r}\right)}{r^{4}}-\frac{16}{3\pi}T^{-1}\right|=O\left(T_{0}^{-2}+\left(\log n\right)^{-\frac{1}{2}\log\frac{\pi}{2}+\epsilon}\right)$$ for every $\epsilon>0$, and in particular $$\label{eq:o_r4_2d} \text{\ensuremath{\mathcal{V}}}\left(X_{f_{n},r}\right)=o\left(r^{4}\right)$$ uniformly for $r > r_{0}$ for a density one sequence in $S_{2}$. Therefore, serves as a refinement of [@GranvilleWigman Corollary 1.10] for this specific case (for a density one sequence in $S_{2}$), since [@GranvilleWigman Corollary 1.10] yields $\mathcal{V}\left(X_{f_n,r}\right)=o\left(r^{4}\right)$ under the additional assumption $T_{0}\gg n^{4\epsilon}$. Proof of Proposition \[prop:DiscreteProp\] {#sec:ProofOfLemmaDiscrete} ------------------------------------------ In this section we prove Proposition \[prop:DiscreteProp\]. First, we define a binary relation on $\mathcal{E}_n$: for $\lambda\ne-\lambda'\in\mathcal{E}_{n}$, we say that $\lambda\prec\lambda'$ if the arc on the circle $\sqrt{n}\mathcal{S}^{1}$ that connects $\lambda$ to $\lambda'$ counter-clockwise to $\lambda'$ is shorter than the arc that connects them clockwise to $\lambda'$. Recall that $\lambda_{+}$ is the clockwise nearest neighbour of $\lambda$ on $\sqrt{n}\mathcal{S}^{1}$. The proof of Proposition \[prop:DiscreteProp\] employs the following auxiliary lemma to be proved at section \[sec:AuxLemmasProof\], establishing (\[eq:F\_Function\]) in the particular case $\left|c_{\lambda}\right|^{2}=1$ for every $\lambda\in\mathcal{E}_{n}$: \[lem:CosToDist\] Fix $\lambda'\in\mathcal{E}_{n}.$ For $0\le s<2$, we have $$\frac{1}{N}\cdot \#\left\{ \lambda\in\mathcal{E}_{n}:\,\lambda\succeq\lambda',\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\right\} =\frac{s}{2\pi}+O\left(s^{3}+\Delta\left(n\right)\right)\label{eq:CosToDistEq}$$ where the constant involved in the ’O’-notation in (\[eq:CosToDistEq\]) is absolute. The estimate (\[eq:CosToDistEq\]) is also valid with either ‘$\succ$’, ‘$\preceq$’ or ‘$\prec$’ in place of ‘$\succeq$’. We are now in a position to prove Proposition \[prop:DiscreteProp\]: First, we write $$\begin{aligned} F\left(s\right) & =\sum_{\lambda'\in\mathcal{E}_{n}}\left|c_{\lambda'}\right|^{2}\sum_{\begin{subarray}{c} \lambda\in\mathcal{E}_{n}\\ \left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\\ \lambda\preceq\lambda' \end{subarray}}\left|c_{\lambda}\right|^{2}+\sum_{\lambda'\in\mathcal{E}_{n}}\left|c_{\lambda'}\right|^{2}\sum_{\begin{subarray}{c} \lambda\in\mathcal{E}_{n}\\ \left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\\ \lambda\succeq\lambda' \end{subarray}}\left|c_{\lambda}\right|^{2}\label{eq:PartialSummation} +O\left(\frac{A_{4}}{N}\right). \end{aligned}$$ Using summation by parts, we get that for every $\lambda'\in\mathcal{E}_{n}$ $$\begin{aligned} \sum_{\begin{subarray}{c} \lambda\in\mathcal{E}_{n}\\ \left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\\ \lambda\preceq\lambda' \end{subarray}}\left|c_{\lambda}\right|^{2} & =\left|c_{\lambda'}\right|^{2} \cdot \#\left\{ \lambda\in\mathcal{E}_{n}:\,\lambda\preceq\lambda',\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\right\} \label{eq:PartialSumEq}\\ & -\sum_{\begin{subarray}{c} \lambda\in\mathcal{E}_{n}\\ \left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\\ \lambda\prec\lambda' \end{subarray}}\left(\left|c_{\lambda_{+}}\right|^{2}-\left|c_{\lambda}\right|^{2}\right) \cdot \#\left\{ \mu\in\mathcal{E}_{n}:\,\mu\preceq\lambda,\,\left\Vert \widehat{\mu}-\widehat{\lambda'}\right\Vert \le s\right\} .\nonumber \end{aligned}$$ By Lemma \[lem:CosToDist\], the contribution of the first term on the r.h.s of (\[eq:PartialSumEq\]) to $F\left(s\right)$ is $$\begin{aligned} \label{eq:first_term_discrete} \sum_{\lambda'\in\mathcal{E}_{n}}\left|c_{\lambda'}\right|^{4} \cdot \#\left\{ \lambda\in\mathcal{E}_{n}:\,\lambda\preceq\lambda',\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\right\} =A_{4} \cdot \left(s/2\pi+O\left(s^{3}+\Delta\left(n\right)\right)\right). \end{aligned}$$ The contribution of the sum on the r.h.s of (\[eq:PartialSumEq\]) to $F\left(s\right)$ is $$\begin{aligned} \label{eq:second_term_discrete} & \sum_{\lambda'\in\mathcal{E}_{n}}\left|c_{\lambda'}\right|^{2}\sum_{\begin{subarray}{c} \lambda\in\mathcal{E}_{n}\\ \left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\\ \lambda\prec\lambda' \end{subarray}}\left(\left|c_{\lambda_{+}}\right|^{2}-\left|c_{\lambda}\right|^{2}\right) \cdot \#\left\{ \mu\in\mathcal{E}_{n}:\,\mu\preceq\lambda,\,\left\Vert \widehat{\mu}-\widehat{\lambda'}\right\Vert \le s\right\} \\ & \ll N\left(s+\Delta\left(n\right)\right)\sum_{\begin{subarray}{c} \lambda\in\mathcal{E}_{n}\end{subarray}}\left|\left|c_{\lambda_{+}}\right|^{2}-\left|c_{\lambda}\right|^{2}\right|\sum_{\begin{subarray}{c} \lambda'\in\mathcal{E}_{n} \nonumber \\ \left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\\ \lambda\prec\lambda' \end{subarray}}\left|c_{\lambda'}\right|^{2}\ll\left(s+\Delta\left(n\right)\right)^{2}[\underline{v}]_{\infty} V(\underline{v}) . \end{aligned}$$ By and , we have $$\label{eq:first_summation_discrete} \sum_{\begin{subarray}{c} \lambda\in\mathcal{E}_{n}\\ \left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\\ \lambda\preceq\lambda' \end{subarray}}\left|c_{\lambda}\right|^{2} = A_{4} \cdot \left(s/2\pi+O\left(s^{3}+\Delta\left(n\right)\right)\right) + O\left(\left(s+\Delta\left(n\right)\right)^{2}[\underline{v}]_{\infty}V(\underline{v}) \right).$$ By symmetry, the second summation in (\[eq:PartialSummation\]) obeys with ‘$ \succ $’, ‘$ \succeq $’ and $ |c_{\lambda_-}|^2 $ in place of ‘$ \prec $’, ‘$ \preceq $’ and $ |c_{\lambda_+}|^2 $, where $ \lambda_- $ is the counter-clockwise nearest neighbour to $ \lambda $. The statement of Proposition \[prop:DiscreteProp\] follows from the analogues of the estimates , and . Proof of Theorem \[thm:VarMainGeneralized\], part 2: Gaussian moments, $d=2$. {#sec:Proof_Main_thm_part2} ============================================================================= In this section we study the higher moments of $\hat{X}_{f_{n},r}$ defined in , and prove the second part of Theorem \[thm:VarMainGeneralized\], also implying the second part of Theorem \[thm:VarMain\]. The proof of the following lower bound for $ {\mathcal{V}}\left(X_{f_{n},r}\right) $ with $f_{n}\in {\mathcal{F}}_{2}(n;T(n),\eta(n))$ goes along the same lines as the proof of the lower bound in Theorem \[thm:UpperBound2d\] below: \[lem:Lower\_Bound\_F2\] In the setting of Theorem \[thm:VarMainGeneralized\] part (2), we have $$\frac{{\mathcal{V}}(X_{f_n,r})}{r^{4}} \gg T(n)^{-1-2\eta(n)}$$ uniformly for $r_{0} < r < r_1$ and $f_{n}\in {\mathcal{F}}_{2}(n;T(n),\eta(n))$. Before proceeding to the proof of Theorem \[thm:VarMainGeneralized\], we introduce some notation: 1. Define the set of “admissible” $2k$-tuples of lattice points by $$\label{eq:admissible_tuples} \mathcal{A}_n (2k) = \left\{\left(\lambda_{1},\dots,\lambda_{2k}\right)\in{\mathcal{D}}_{n}(2k): \; \forall 1\le i\le k \; \lambda_{2i-1}\ne-\lambda_{2i} \right\}.$$ 2. Given an admissible $2k$-tuple of lattice points $\left(\lambda_{1},\dots,\lambda_{2k}\right)\in \mathcal{A}_n (2k)$, let $\sim$ be the equivalence relation on the set $\left\{ 1,\dots,2k\right\} $, generated by: 1. $2i-1\sim2i$ for every $1\le i\le k$. 2. $j\sim j'$ if $\lambda_{j}+\lambda_{j}'=0.$ Let $\left\{ \Lambda_{1},\dots,\Lambda_{m}\right\} $ be the partition of $\left\{ 1,\dots,2k\right\} $ into equivalence classes of $\sim$, and denote $l_{j}=\#\Lambda_{m}/2$ for $1\le j\le m$, so that $\sum_{j=1}^{m}l_{j}=k$; clearly, $2\le l_{j}\in\mathbb{Z}$ for every $1\le j\le m$. We call the multiset $$\label{eq:structure_set} S\left(\lambda_{1},\dots,\lambda_{2k}\right):=\left\{ l_{1},\dots,l_{m}\right\}$$ the structure set of the $2k$-tuple $\left(\lambda_{1},\dots,\lambda_{2k}\right).$ Recall that the moments of a standard Gaussian random variable $Z\sim N(0,1)$ are $${\mathbb{E}}[Z^{k}]=\begin{cases} \left(k-1\right)!! & k\,\text{even}\\ 0 & k\,\text{odd}. \end{cases}$$ We are now in a position to prove the second part of Theorem \[thm:VarMainGeneralized\]. By the length-$2k$ diagonal domination assumption, we have $$\begin{aligned} {\mathbb{E}}[\hat{X}_{f_{n},r}^{k}] & =(2\pi)^k r^{2k}{\mathcal{V}}\left(X_{f_{n},r}\right)^{-k/2}\sum_{\begin{subarray}{c} \left(\lambda_{1},\dots,\lambda_{2k}\right)\in \mathcal{A}_n (2k)\end{subarray}}\label{eq:kthMoment}\prod_{j=1}^{k}c_{\lambda_{2j-1}}c_{\lambda_{2j}}g_{2}\left(r\left\Vert \lambda_{2j-1}+\lambda_{2j}\right\Vert \right) \\ &+O\left({\mathcal{V}}\left(X_{f_{n},r}\right)^{-k/2} [\underline{v}]_{\infty}^k r^{2k}N^{-\gamma}\right)\nonumber \end{aligned}$$ for some $ \gamma>0. $ We can rearrange the summation in (\[eq:kthMoment\]), first summing over all possible structure sets $\mathcal{L}=\left\{ l_{1},\dots,l_{m}\right\} $ and then summing over the admissible $2k$-tuples $\left(\lambda_{1},\dots,\lambda_{2k}\right)\in\mathcal{E}_{n}^{2k}$ with the given structure set $S\left(\lambda_{1},\dots,\lambda_{2k}\right)=\mathcal{L} $: let $$\begin{aligned} S_{\mathcal{L}} & :=\sum_{\begin{subarray}{c} \begin{subarray}{c} \left(\lambda_{1},\dots,\lambda_{2k}\right)\in \mathcal{A}_n (2k)\\ S\left(\lambda_{1},\dots,\lambda_{2k}\right)=\mathcal{L} \end{subarray}\end{subarray}}\prod_{j=1}^{k}c_{\lambda_{2j-1}}c_{\lambda_{2j}}g_{2}\left(r\left\Vert \lambda_{2j-1}+\lambda_{2j}\right\Vert \right), \end{aligned}$$ so that we may rewrite the summation on the r.h.s. of as $$\begin{aligned} \label{eq:Inner_Sum_with_SL} & \sum_{\begin{subarray}{c} \left(\lambda_{1},\dots,\lambda_{2k}\right)\in \mathcal{A}_n (2k)\end{subarray}}\prod_{j=1}^{k}c_{\lambda_{2j-1}}c_{\lambda_{2j}}g_{2}\left(r\left\Vert \lambda_{2j-1}+\lambda_{2j}\right\Vert \right)=\sum_{\begin{subarray}{c} l_{1}+\dots+l_{m}=k\\ l_{1},\dots,l_{m}\ge2 \end{subarray}}\sum_{\begin{subarray}{c} \begin{subarray}{c} \left(\lambda_{1},\dots,\lambda_{2k}\right)\in \mathcal{A}_n (2k)\\ S\left(\lambda_{1},\dots,\lambda_{2k}\right)=\mathcal{L} \end{subarray}\end{subarray}}S_{\mathcal{L}}. \end{aligned}$$ For a fixed structure set $\mathcal{L}=\left\{ l_{1},\dots,l_{m}\right\} $, we have $$\begin{aligned} S_{\mathcal{L}} =a\left(\mathcal{L}\right)\prod_{j=1}^{m}\sum_{\lambda_{1},\dots,\lambda_{l_{j}}\in\mathcal{E}_{n}}\left|c_{\lambda_{1}}\right|^{2}g_{2}\left(r\left\Vert \lambda_{l_{j}}-\lambda_{1}\right\Vert \right)\label{eq:RearrangeInPartition}\prod_{i=1}^{l_{j}-1}\left|c_{\lambda_{i+1}}\right|^{2}g_{2}\left(r\left\Vert \lambda_{i}-\lambda_{i+1}\right\Vert \right)+O\left([\underline{v}]_{\infty}^k N^{-1}\right) \end{aligned}$$ where $a\left(\mathcal{L}\right)$ is a constant depending on $\mathcal{L}$; omitting the condition that the lattice points are distinct on the r.h.s of (\[eq:RearrangeInPartition\]) is absorbed within the error term in (\[eq:RearrangeInPartition\]). Thus, $$\begin{aligned} \label{eq:SL_upper_bound} S_{\mathcal{L}} & \ll [\underline{v}]_{\infty}^k N^{-k}\prod_{j=1}^{m}\sum_{\lambda_{1}\in\mathcal{E}_{n}}\sum_{\lambda_{2}\in\mathcal{E}_{n}}\left|g_{2}\left(r\left\Vert \lambda_{2}-\lambda_{1}\right\Vert \right)\right|\cdots\sum_{\lambda_{l_{j}}\in\mathcal{E}_{n}}\left|g_{2}\left(r\left\Vert \lambda_{l_{j}-1}-\lambda_{l_{j}}\right\Vert \right)\right|+[\underline{v}]_{\infty}^k N^{-1}. \end{aligned}$$ Recall the definition of $ F_{\lambda_0} $ in . By Lemma \[lem:CosToDist\], we have $$\label{eq:f_lambda_zero} F_{\lambda_{0}}\left(s\right)=\frac{s}{\pi}+O\left(s^{3}+\Delta\left(n\right)\right)=O\left(s+\Delta\left(n\right)\right).$$ Thus, by Lemma \[lem:H2Formulas\] and , we have that $$\begin{aligned} \label{eq:abs_g2_bound} \frac{1}{N}\sum_{\lambda\in\mathcal{E}_{n}}\left|g_{2}\left(r\left\Vert \lambda-\lambda_{0}\right\Vert \right)\right| & =\int_{0}^{2}\left|g_{2}\left(Ts\right)\right|\,\text{d}F_{\lambda_{0}}\left(s\right)\\ & =\left|g_{2}\left(2T\right)\right|-\frac{1}{2N}+O\left(\int_{0}^{2}\left(s+\Delta\left(n\right)\right)T\left|g_{2}'\left(Ts\right)\right|\,\text{d}s\right) \nonumber \\ & =O\left(T^{-3/2}+\left(\Delta\left(n\right)+T^{-1}\right)\int_{0}^{2T}\left|g_{2}'\left(s\right)\right|\,\text{d}s\right) \nonumber \\ & =O\left(T^{-1}\right) \nonumber \end{aligned}$$ for $ n $ satisfying the hypothesis $ \mathcal{D}(n,\epsilon) $. Applying to each of the $l_{j}-1$ inner summations in , we obtain $$\begin{aligned} S_{\mathcal{L}} & \ll [\underline{v}]_{\infty}^k N^{-k+m}\prod_{j=1}^{m}\left(NT^{-1}\right)^{l_{j}-1} + [\underline{v}]_{\infty}^k N^{-1} \ll [\underline{v}]_{\infty}^k T^{-k+m}. \end{aligned}$$ Let $ \mathcal{L}_0 = \left\{ 2,2,\dots2,\right\} $. Note that if $\mathcal{L}\ne \mathcal{L}_0 $ then $m\le\frac{k-1}{2}$ and therefore $$\label{eq:SL_Estimate} S_{\mathcal{L}}=O\left([\underline{v}]_{\infty}^k T^{-\frac{k+1}{2}}\right).$$ If $\mathcal{\mathcal{L}}=\mathcal{L}_0 $ (this is a viable option for $k$ even), then $$\label{eq:SL_formula} S_{\mathcal{L}_0}=2^{k/2}\left(k-1\right)!!\left[\sum_{\lambda_{1}\ne\lambda_{2}\in\mathcal{E}_{n}}\left|c_{\lambda_{1}}\right|^{2}\left|c_{\lambda_{2}}\right|^{2}h_{2}\left(r\left\Vert \lambda_{1}-\lambda_{2}\right\Vert \right)\right]^{k/2}+O\left([\underline{v}]_{\infty}^k N^{-1}\right).$$ By (\[eq:VarFormula2d\]), $$\label{eq:SL_Variance_asympt} \sum_{\lambda_{1}\ne\lambda_{2}\in\mathcal{E}_{n}}\left|c_{\lambda_{1}}\right|^{2}\left|c_{\lambda_{2}}\right|^{2}h_{2}\left(r\left\Vert \lambda_{1}-\lambda_{2}\right\Vert \right)=\frac{{\mathcal{V}}\left(X_{f_{n},r}\right)}{8\pi^2 r^4}.$$ Hence, and yield $$\label{eq:SL_final_form} S_{\mathcal{L}_0}=\left(k-1\right)!!\left(\frac{{\mathcal{V}}\left(X_{f_{n},r}\right)}{4\pi^2r^4}\right)^{k/2}+O\left([\underline{v}]_{\infty}^k N^{-1}\right).$$ Substituting and into and applying Lemma \[lem:Lower\_Bound\_F2\], we finally obtain that for $k$ even $$\begin{aligned} \left|{\mathbb{E}}[\hat{X}_{f_{n},r}^{k}]-\left(k-1\right)!!\right| & \ll T^{k\eta(n)} [\underline{v}]_{\infty}^k\left(T^{-1/2}+T^{k/2}N^{-\min\{1,\gamma \}}\right) \ll T^{-1/2+2k\eta(n)} \end{aligned}$$ and since for $k$ odd $ \mathcal{L}=\mathcal{L}_0 $ is not a viable option, we obtain $$\begin{aligned} {\mathbb{E}}[\hat{X}_{f_{n},r}^{k}] & \ll T^{k\eta(n)} [\underline{v}]_{\infty}^k\left(T^{-1/2}+T^{k/2}N^{-\min\{1,\gamma \}}\right) \ll T^{-1/2+2k\eta(n) }, \end{aligned}$$ and the second part of Theorem \[thm:VarMainGeneralized\] follows. Proof of Theorem \[thm:Var3D\]: asymptotics for the variance, $d=3$ {#sec:Proof_3d_theorem} =================================================================== Proof of Theorem \[thm:Var3D\] ------------------------------ Denote $$\label{eq:F_3} F_{3}\left(s\right)=\frac{1}{N^{2}} \cdot \#\left\{ \lambda\ne\lambda'\in\mathcal{E}_{n}:\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\right\}$$(cf. ), and recall that the spherical cap discrepancy for the points in $\mathcal{E}_{n}$ is defined by $$\label{eq:Discrepancy_3d} \Delta_{3}\left(n\right)=\sup_{\begin{subarray}{c} x\in\mathcal{S}^{2}\\ 0<r\le2 \end{subarray}}\left|\frac{1}{N} \cdot \#\left\{ \lambda\in\mathcal{E}_{n}:\,\left|\widehat{\lambda}-x\right|\le r\right\} -\frac{r^{2}}{4}\right|.$$ \[cor:DistMeasure3d\]We have $$\label{eq:3d_Discrepancy} F_{3}\left(s\right)=\frac{s^{2}}{4}+O\left(\Delta_{3}\left(n\right)\right).$$ The estimate follows immediately from the definition of spherical cap discrepancy, since $$\begin{aligned} F_{3}\left(s\right) & =\frac{1}{N}\sum_{\lambda'\in\mathcal{E}_{n}}\#\left\{ \lambda\in\mathcal{E}_{n}:\,0<\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\right\} =\frac{s^{2}}{4}+O\left(\Delta_{3}\left(n\right)\right). \end{aligned}$$ The discrepancy $\Delta_{3}\left(n\right)$ satisfies $ \Delta_3(n) \le n^{-\eta} $ for some small $\eta>0$, see [@BourgainRudnickSarnak]. We are now in a position to prove Theorem \[thm:Var3D\]: By we have $$\begin{aligned} \label{eq:var_3d_asymp_formula} \text{\ensuremath{\mathcal{V}}}\left(X_{f_n,r}\right) & =16\pi^{3}r^{6}\frac{1}{N^{2}}\sum_{\begin{subarray}{c} \lambda,\lambda'\in\mathcal{E}_{n}\\ \lambda\ne\lambda' \end{subarray}}h_{3}\left(T\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \right)+O\left(r^{6}N^{-1/4+\epsilon}\right). \end{aligned}$$ For the summation in we have, $$\frac{1}{N^{2}}\sum_{\begin{subarray}{c} \lambda,\lambda'\in\mathcal{E}_{n}\\ \lambda\ne\lambda' \end{subarray}}h_{3}\left(T\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \right)=\int_{0}^{2}h_{3}\left(Ts\right)\,\text{d}F_{3}\left(s\right).$$ Thus, integrating by parts and using Lemma \[cor:DistMeasure3d\], $$\begin{aligned} \frac{1}{N^{2}}\sum_{\begin{subarray}{c} \lambda,\lambda'\in\mathcal{E}_{n}\\ \lambda\ne\lambda' \end{subarray}}h_{3}\left(T\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \right) & =h_{3}\left(2T\right)F_{3}\left(2\right)-\int_{0}^{2}F_{3}\left(s\right)\,\text{d}h_{3}\left(Ts\right)\label{eq:VarAftIntParts3D}\\ & =h_{3}\left(2T\right)F_{3}\left(2\right)-\frac{1}{4}\int_{0}^{2}s^{2}\,\text{d}h_{3}\left(Ts\right)+Err\left(X_{f_n,r}\right)\nonumber \end{aligned}$$ where $$\label{eq:error_term_3d} Err\left(X_{f_n,r}\right)\ll\Delta_{3}\left(n\right)\int_{0}^{2}T\left|h_{3}'\left(Ts\right)\right|\,\text{d}s.$$ Note that $h_{3}\left(s\right)\ll s^{-4}$ as $s\to\infty.$ Thus, integrating by parts, the main term on the r.h.s of (\[eq:VarAftIntParts3D\]) satisfies $$\label{eq:main_terms_after_intbyparts_3d} h_{3}\left(2T\right)F_{3}\left(2\right)-\frac{1}{4}\int_{0}^{2}s^{2}\,\text{d}h_{3}\left(Ts\right)=\frac{1}{2}\int_{0}^{2}s \cdot h_{3}\left(Ts\right)\,\text{d}s+O\left(T^{-4}\right),$$ so that $$\label{eq:main_int_after_intbyparts} \int_{0}^{2}s \cdot h_{3}\left(Ts\right)\,\text{d}s=\frac{1}{T^{2}}\int_{0}^{2T}s \cdot h_{3}\left(s\right)\,\text{d}s=\frac{1}{T^{2}}\int_{0}^{\infty}s \cdot h_{3}\left(s\right)\,\text{d}s+O\left(T^{-4}\right).$$ A direct computation shows that $$\label{eq:final_calc_main_term_3d} \int_{0}^{\infty}s \cdot h_{3}\left(s\right)\,\text{d}s=\frac{1}{2\pi^3}\int_{0}^{\infty}\frac{1}{s^{3}}\left(\frac{\sin s}{s}-\cos s\right)^{2}\,\text{d}s=\left(2\pi\right)^{-3},$$ and therefore, substituting into and then into we get $$\label{eq:main_term_final_form_3d} h_{3}\left(2T\right)F_{3}\left(2\right)-\frac{1}{4}\int_{0}^{2}s^{2}\,\text{d}h_{3}\left(Ts\right)=\frac{1}{16\pi^3}T^{-2}+O\left(T^{-4}\right).$$ Note that $h_{3}'\left(s\right)\ll\left(1+s^{4}\right)^{-1}$. Thus, $$\label{eq:err_upper_bound_3d} \int_{0}^{2}T\left|h_{3}'\left(Ts\right)\right|\,\text{d}s=\int_{0}^{2T}\left|h_{3}'\left(s\right)\right|\,\text{d}s\le\int_{0}^{\infty}\left|h_{3}'\left(s\right)\right|\,\text{d}s<\infty$$ and therefore, substituting into we obtain $$Err\left(X_{f_n,r}\right)=O\left(\Delta_{3}\left(n\right)\right).\label{eq:3DErrorVar}$$ Substituting into and finally into we obtain (\[eq:AympVar3D\]). Note that by (\[eq:3DErrorVar\]), $$\sup_{\begin{subarray}{c} r > r_{0} \\ f_n\in {\mathcal{B}}{n} \end{subarray}}\left|\frac{\text{\ensuremath{\mathcal{V}}}\left(X_{f_{n},r}\right)}{r^{6}}-T^{-2}\right|=O\left(T_{0}^{-4}+n^{-\eta}\right)$$ for every $n\not\equiv0,4,7\,\left(8\right)$, and in particular $$\text{\ensuremath{\mathcal{V}}}\left(X_{f_{n},r}\right)=o\left(r^{6}\right)$$ uniformly for $r > r_{0}$ for every $n\not\equiv0,4,7\,\left(8\right)$. Proofs of Theorem \[thm:UpperBound2d\] and Theorem \[thm:UpperBound3d\] {#sec:ProofOfBoundsThm} ======================================================================= By substituting the bound $\left|c_{\lambda}\right|^{2}\le N^{-1+\epsilon}$ into (\[eq:VarFormula2d\]), we have $$\begin{aligned} \label{eq:var_ultraflat_bound} \text{\ensuremath{\mathcal{V}}}\left(X_{f_{n},r}\right) & \ll r^{4} N^{-1+\epsilon}\sum_{\lambda_0 \in \mathcal{E}_n}\left|c_{\lambda_0}\right|^{2}\sum_{\lambda \in \mathcal{E}_n} h_{2}\left(r\left\Vert \lambda-\lambda_0\right\Vert \right).\end{aligned}$$ By Lemma \[lem:H2Formulas\] and by , we have $$\begin{aligned} \label{eq:H2_sum} \frac{1}{N}\sum_{\lambda\in\mathcal{E}_{n}}h_{2}\left(r\left\Vert \lambda-\lambda_{0}\right\Vert \right) & =\int_{0}^{2}h_{2}\left(Ts\right)\,\text{d}F_{\lambda_{0}}\left(s\right)\\ & =h_{2}\left(2T\right)-\frac{1}{4N}+O\left(\int_{0}^{2}\left(s+\Delta\left(n\right)\right)T\left|h_{2}'\left(Ts\right)\right|\,\text{d}s\right) \nonumber \\ & =O\left(T^{-1}+\left(\log n\right)^{-\frac{1}{2}\log\frac{\pi}{2}+\epsilon}\right) \nonumber\end{aligned}$$ for $ n $ satisfying the hypothesis $ \mathcal{D}(n,\epsilon) $. Substituting in , we get the upper bound in Theorem \[thm:UpperBound2d\]. The upper bound in Theorem \[thm:UpperBound3d\] follows along similar lines. We now turn to proving the claimed lower bounds for the variance of $X_{f_n,r}$. First, we need the following lemma, proved at the end of section \[sec:ProofOfBoundsThm\]: \[lem:LowerBoundPairs\] 1. Let $\left\{ x_{m}\right\} _{m=1}^{M}$ be $M$ points on the unit circle $\mathcal{S}^{1}.$ For every $1<T<M/2$ we have $$\#\left\{ x_{i}\ne x_{j}:\,\left|x_{i}-x_{j}\right|\le1/T\right\} \gg M^{2}/T.$$ 2. Let $\left\{ x_{m}\right\} _{m=1}^{M}$ be $M$ points on the unit sphere $S^{2}.$ For every $1<T<\sqrt{M}/2$ we have $$\#\left\{ x_{i}\ne x_{j}:\,\left|x_{i}-x_{j}\right|\le1/T\right\} \gg M^{2}/T^{2}.$$ We are now in a position to prove the lower bounds , of Theorem \[thm:UpperBound2d\] and Theorem \[thm:UpperBound3d\]: For $d=2$, we let $$R=\#\left\{ \lambda\in\mathcal{E}_{n}:\,\left|c_{\lambda}\right|^{2}\ge \frac{1}{2N}\right\},$$ so that $$1=\sum_{\lambda\in\mathcal{E}_{n}}\left|c_{\lambda}\right|^{2}=\sum_{\lambda\in R}\left|c_{\lambda}\right|^{2}+\sum_{\lambda\notin R}\left|c_{\lambda}\right|^{2}\le N^{-1+\epsilon} \cdot \#R+1/2,$$ and hence $\#R\ge 2N^{1-\epsilon}.$ By the second part of Lemma \[lem:H2Formulas\], for $c>0$ sufficiently small we have $$\begin{aligned} {\mathcal{V}}(X_{f_n,r}) & =8\pi^{2}r^{4}\sum_{\begin{subarray}{c} \lambda,\lambda'\in\mathcal{E}_{n}\\ \lambda\ne\lambda' \end{subarray}}\left|c_{\lambda}\right|^{2}\left|c_{\lambda'}\right|^{2}h_{2}\left(T\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \right) \gg r^{4}N^{-2}\sum_{\lambda\ne\lambda'\in R}h_{2}\left(T\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \right)\\ & \gg r^{4}N^{-2}\cdot \#\left\{ \lambda\ne\lambda'\in R:\,\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le c/T\right\} . \end{aligned}$$ By the first part of Lemma \[lem:LowerBoundPairs\], $${\mathcal{V}}(X_{f_n,r})\gg r^{4}N^{-2}\left(\#R\right)^{2}T^{-1}\gg r^{4}N^{-2\epsilon}T^{-1}.$$ The lower bound of Theorem \[thm:UpperBound3d\] follows along the same lines as the above, this time using the second part of Lemma \[lem:LowerBoundPairs\] in place of the first one. Note that in the proof of of the lower bound in Theorem \[thm:UpperBound2d\] we have used the abundance of close-by pairs of lattice points with **$\left|c_{\lambda}\right|^{2}\ge \frac{1}{2N}$**; in the absence of such close-by lattice points, the bound does not hold. For example, for $d=2$, fix $\lambda_{0}\in\mathcal{E}_{n}$ and let $\left|c_{\pm\lambda_{0}}\right|^{2}=1/2$ and $c_{\lambda}=0$ for every $\lambda\ne\pm\lambda_{0}.$ Then $${\mathcal{V}}(X_{f_{n},r})=4\pi^{2}r^{4}h_{2}\left(2T\right)\ll r^{4}T^{-3}.$$ For the first part of Lemma \[lem:LowerBoundPairs\], divide $S^{1}$ into $k=O\left(T\right)$ arcs $I_{1},I_{2},\dots,I_{k}$ of length $<1/T$. For every $1\le j\le k,$ let $n_{j}=\#\left\{ m:\,x_{m}\in I_{j}\right\} ,$ so $\sum_{j=1}^{k}n_{j}=M$. By the Cauchy-Schwarz inequality, $$M^{2}=\left(\sum_{j=1}^{k}n_{j}\right)^{2}\le k\sum_{j=1}^{k}n_{j}^{2}\ll T\sum_{j=1}^{k}n_{j}^{2}.$$ Thus, $$\begin{aligned} \#\left\{ x_{i}\ne x_{j}:\,\left|x_{i}-x_{j}\right|\le1/T\right\} & =\#\left\{ x_{i},x_{j}:\,\left|x_{i}-x_{j}\right|\le1/T\right\} -M\\ & \gg\sum_{j=1}^{k}n_{j}^{2}-M\gg M^{2}/T-M\gg M^{2}/T. \end{aligned}$$ The second part of Lemma \[lem:LowerBoundPairs\] is proved similarly. Restricted averages {#sec:RestrictedAverages} =================== Restricted moments ------------------ For $ d=2 $, most of our principal results above are also valid in the more difficult scenario where $x$ is drawn in $B_{x_{0}}(\rho)$ for some $x_{0}\in{\mathbb{T}}^{2}$ and $\rho\gg n^{-1/2+o(1)}$. In this case, the restricted moments are: expectation $$\label{eq:restricted_expectation} {\mathbb{E}}_{x_{0},\rho}[X_{f_{n},r}] = \frac{1}{{\operatorname{Vol}}(B_{x_0}(\rho))}\int\limits_{B_{x_{0}}(\rho)}X_{f_{n},r;x}dx,$$ higher centred moments $$\label{eq:centred moments rest} {\mathbb{E}}_{x_{0},\rho}[(X_{f_{n},r}-{\mathbb{E}}_{x_{0},\rho}[X_{f_{n},r}])^{k}] = \frac{1}{{\operatorname{Vol}}(B_{x_0}(\rho))}\int\limits_{B_{x_{0}}(\rho)}\left(X_{f_{n},r;x}-{\mathbb{E}}_{x_{0},\rho}[X_{f_{n},r}]\right)^{k}dx, \hspace{1em}k\ge2,$$ and in particular the variance $$\label{eq:restricted_variance} {\mathcal{V}}_{x_{0},\rho}(X_{f_{n},r}) = {\mathbb{E}}_{x_{0},\rho}[(X_{f_{n},r}-{\mathbb{E}}_{x_{0},\rho}[X_{f_{n},r}])^{2}].$$ We reinterpret the statement of Granville-Wigman’s  [@GranvilleWigman Theorem 1.2] as evaluating the expected mass $${\mathbb{E}}_{x_{0},\rho}[X_{f_{n},r}] \sim \pi r^{2},$$ valid for almost all $n\in S_{2}$, uniformly for $\rho\gg n^{-1/2+o(1)}$, $x_{0}\in{\mathbb{T}}^{2}$, and $ r>0 $ (see the first part of Lemma \[lem:ExpVarShrinking\]). Quasi-correlations ------------------ For the restricted moments of $X_{f_{n},r}$ one also needs to cope with [*quasi-correlations*]{}, i.e. tuples $(\lambda^{1},\ldots,\lambda^{l}) \in {\mathcal{E}}_{n}^{l}$ with the sum $\sum\limits_{i=1}^{l}\lambda^{i}$ unexpectedly small, e.g. given a (small) fixed number $\delta >0$, $$\label{eq:sum tuple small l,delta} \left\|\sum\limits_{i=1}^{l}\lambda^{i}\right\| < n^{1/2-\delta};$$ unlike the correlations , here there are no congruence obstructions, so that makes sense with $l$ odd or even. \[def:separatedness Ac\] 1. For $n\in S_2$, $l\in\mathbb{Z}_{\ge 2}$, and $0 < K=K(n) < l\cdot n^{1/2}$ define the set of length-$l$ spectral quasi-correlations $$\label{eq:quasi_correlations} \mathcal{C}_{n}(l;K) = \left\{(\lambda^{1},\ldots,\lambda^{l})\in\mathcal{E}_{n}^{l}:\: 0 < \left\| \sum\limits_{j=1}^{l}\lambda^{j} \right\| \le K \right\}.$$ 2. Given $\delta>0$ we say that $n \in S_{2}$ satisfies the $(l,\delta)$-separateness hypothesis $\mathcal{A}(n;l,\delta)$ if $$\label{eq:sep_hypothesis} \mathcal{C}_{n}(l;n^{1/2-\delta}) = \varnothing.$$ For example, ${\mathcal{A}}(n;2,\delta)$ is equivalent to the aforementioned Bourgain-Rudnick separateness, satisfied  [@Bourgain-Rudnick Lemma 5] by a density $1$ sequence $S_{2}'\subseteq S_{2}$. More generally, it was shown in the forthcoming paper  [@BenatarBuckleyWigman], that for every $\delta>0$ and $l\ge 2$, the assumption ${\mathcal{A}}(n;l,\delta)$ is satisfied by generic $n\in S_{2}'(l,\delta)$, and hence a standard diagonal argument yields a density $1$ sequence $S_{2}'\subseteq S_{2}$ so that ${\mathcal{A}}(n;l,\delta)$ is satisfied for [*all*]{} $l\ge 2$ and $ \delta>0 $ for $ n\in S_2' $ sufficiently large. \[thm:quasi-corr small\] For every fixed $l\ge 2$ and $\delta>0$ there exist a set $S_2'(l,\delta)\subseteq S_2$ such that: 1. The set $S_2'(l,\delta)$ has density $1$ in $S_2$. 2. For every $n\in S_2'(l,\delta)$ the length-$l$ spectral quasi-correlation set $$\mathcal{C}_{n}(l;n^{1/2-\delta})=\varnothing$$ is empty, i.e., the $(l,\delta)$-separateness hypothesis $\mathcal{A}(n;l,\delta)$ is satisfied. A version of Theorem \[thm:VarMainGeneralized\] with restricted averages ------------------------------------------------------------------------ We have the following analogue of Theorem \[thm:VarMainGeneralized\]: \[thm:VarMainExplRestricted\] 1. If $S_{2}'\subseteq S_{2}$ is a sequence satisfying the hypotheses $ \mathcal{D}(n,\epsilon/2),$ $ {\mathcal{A}}(n;2,\epsilon)$, and ${\mathcal{A}}(n;4,\epsilon)$ for all $n\in S_{2}'$, then in the setting of Theorem \[thm:VarMainGeneralized\] part (1), $${\mathcal{V}}_{x_{0},\rho}\left(X_{f_{n},r}\right)\sim\frac{16}{3\pi \cos^{2}\theta_{f_{n}}}r^{4}T^{-1}$$ uniformly for all $x_{0}\in\mathbb{T}^{2}$, $n^{-1/2+\delta}\le\rho\le 1$ and $r_{0}< r< r_{1}$, and $f_{n}\in{\mathcal{F}}_{1}(n;T(n),\eta(n))$. 2. Let $k\ge 3$ be an integer. If $S_{2}'\subseteq S_{2}$ is a sequence satisfying the length-$2k$ diagonal domination assumption and the hypotheses $ \mathcal{D}(n,\epsilon),$ $ {\mathcal{A}}(n;2,\epsilon)$, $ {\mathcal{A}}(n;4,\epsilon)$, and ${\mathcal{A}}(n;2k,\epsilon)$ for all $n\in S_{2}'$, then in the setting of Theorem \[thm:VarMainGeneralized\] part (2), $$\mathbb{E}_{x_{0},\rho}\left[\hat{X}_{f_{n},r}^{k}\right] \to {\mathbb{E}}[Z^{k}]$$ uniformly for $x_{0}\in{\mathbb{T}}^{2}$, $r_{0} < r <r_{1}$, $n^{-1/2+\delta} \le \rho \le 1$, and $f_{n}\in {\mathcal{F}}_{2}(n;T(n),\eta(n))$, where $Z\sim N(0,1)$ is the standard Gaussian variable. Theorem \[thm:VarMainExplRestricted\] follows along similar lines as the proof of Theorem \[thm:VarMainGeneralized\], where we use the expressions for the restricted moments below (cf. equation , Lemma \[lem:VarExpd2\] and equation ). We remark that Theorem \[thm:UpperBound2d\] can also be extended to $ {\mathcal{V}}_{x_{0},\rho}(X_{f_{n},r}) $, however the lower bound will only hold for a generic $ n\in S_2 $. \[lem:ExpVarShrinking\] For $d=2$ let $0<\delta<1/2$, $0<\epsilon<\delta/5$, and $S_{2}'\subseteq S_{2}$. 1. If $n\in S_{2}'$ satisfy the hypothesis ${\mathcal{A}}(n;2,\epsilon) $, then $$\mathbb{E}_{x_{0},\rho}\left[X_{f_{n},r}\right]=\pi r^{2}+O\left(r^{2}n^{-\frac{3}{5}\delta+3\epsilon}\right)$$ uniformly for $x_{0}\in\mathbb{T}^{2},$ $n^{-1/2+\delta}\le\rho\le1$ and $r>0$. 2. \[lem:VarFormulaShrinking\] If $n\in S_{2}'$ satisfy the hypotheses $ {\mathcal{A}}(n;2,\epsilon) $ and ${\mathcal{A}}(n;4,\epsilon)$, then $$\begin{aligned} \text{\ensuremath{\mathcal{V}}}_{x_{0},\rho}\left(X_{f_{n},r}\right) & =8\pi^{2}r^{4}\sum_{\begin{subarray}{c} \lambda,\lambda'\in\mathcal{E}_{n}\\ \lambda\ne\lambda' \end{subarray}}\left|c_{\lambda}\right|^{2}\left|c_{\lambda'}\right|^{2}h_{2}\left(r\left\Vert \lambda-\lambda'\right\Vert \right) +O\left(r^{4} n^{-\frac{3}{5}\delta+4\epsilon}\right) \end{aligned}$$ uniformly for $x_{0}\in\mathbb{T}^{2},$ $n^{-1/2+\delta}\le\rho\le1$ and $r>0$. \[lem:KthMoment\]For $d=2$ let $ k\ge3 $, $0<\delta<1/2$, $0<\epsilon< \delta/5$, and $ S_2' \subseteq S_2 $ satisfying $ {\mathcal{A}}(2;n,\epsilon), $ $ {\mathcal{A}}(4;n,\epsilon), $ and ${\mathcal{A}}(n;2k,\epsilon) $ for every $ n\in S_2' $ . We have $$\begin{aligned} {\mathbb{E}}_{x_{0},\rho}[\hat{X}_{f_{n},r}^{k}] & =(2\pi)^k r^{2k}{\mathcal{V}}_{x_{0},\rho}\left(X_{f_{n},r}\right)^{-k/2}\sum_{\begin{subarray}{c} \forall1\le i\le k,\,\lambda_{i}\ne\lambda_{i}'\in\mathcal{E}_{n}\\ \sum_{i=1}^{k}\left(\lambda_{i}-\lambda_{i}'\right)=0 \end{subarray}}\prod_{j=1}^{k}c_{\lambda_{j}}\overline{c_{\lambda_{j}'}}g_{2}\left(r\left\Vert \lambda_{j}-\lambda_{j}'\right\Vert \right)\\ & +O\left({\mathcal{V}}_{x_{0},\rho}\left(X_{f_{n},r}\right)^{-k/2} r^{2k} n^{-\frac{3}{5}\delta+4\epsilon}\right)\nonumber \end{aligned}$$ uniformly for $x_{0}\in\mathbb{T}^{2},$ $n^{-1/2+\delta}\le\rho\le1$ and $r>0$. Proofs of Lemma \[lem:ExpVarShrinking\] and Lemma \[lem:KthMoment\] ------------------------------------------------------------------- We have $$\label{eq:exp_basic_formula} {\mathbb{E}}_{x_{0},\rho}\left[X_{f_{n},r}\right]=\frac{1}{\pi\rho^{2}}\int_{B_{x_{0}}\left(\rho\right)}\int_{B_{x}\left(r\right)}f_{n}\left(y\right)^{2}\,\text{d}y\,\text{d}x.$$ Granville-Wigman’s [@GranvilleWigman Theorem 1.2] asserts that for $ \epsilon_1 > \epsilon_2 > 0$, $ 0 < \epsilon_3 < \epsilon_1 - \epsilon_2 $ and $ n\in S_2 $ satisfying ${\mathcal{A}}(n;2,\epsilon_2) $, we have $$\label{eq:Granville_Wigman_theorem} \int_{B_{x}\left(r\right)}f_{n}\left(y\right)^{2}\,\text{d}y = \pi r^2 \left(1+ O\left(n^{-3\epsilon_3 /2}\right)\right)$$ uniformly in $ x\in \mathbb{T}^2 $ and $ r>n^{-1/2 + \epsilon_1} $. If $r>n^{-1/2+\frac{2}{5}\delta}$, then by substituting with $ \epsilon_1 = \frac{2}{5}\delta $, $ \epsilon_2 = \epsilon $ and $ \epsilon_3 = \frac{2}{5}\delta - 2\epsilon $ into , we have $${\mathbb{E}}_{x_{0},\rho}\left[X_{f_{n},r}\right]=\pi r^{2}\left(1+O\left(n^{-\frac{3}{2}\left( \frac{2}{5}\delta - 2\epsilon\right)}\right)\right)$$ for every $\rho$. Otherwise, note that $${\mathbb{E}}_{x_{0},\rho}\left[X_{f_{n},r}\right]=\frac{1}{\pi\rho^{2}}\int_{B_{x_{0}}\left(\rho+r\right)}f_{n}\left(y\right)^{2}\int_{B_{x_{0}}\left(\rho\right)\cap B_{y}\left(r\right)}\,\text{d}x\,\text{d}y,$$ so $$\label{eq:Exp_upper_lower_bnds} \frac{r^{2}}{\rho^{2}}\int_{B_{x_{0}}\left(\rho-r\right)}f_{n}\left(y\right)^{2}\,\text{d}y\le{\mathbb{E}}_{x_{0},\rho}\left[X_{f_{n},r}\right]\le\frac{r^{2}}{\rho^{2}}\int_{B_{x_{0}}\left(\rho+r\right)}f_{n}\left(y\right)^{2}\,\text{d}y.$$ Since $r/\rho \le n^{-\frac{3}{5}\delta}$, we can use with $ \epsilon_1 = \delta $, $ \epsilon_2 = \epsilon $ and $ \epsilon_3 = \delta - 2\epsilon $ to deduce that $$\label{eq:Exp_Inner_integral} \int_{B_{x_{0}}\left(\rho\pm r\right)}f_{n}\left(y\right)^{2}\,\text{d}y=\pi\rho^{2}\left(1+O\left(n^{-\frac{3}{5}\delta}\right)\right),$$ and the statement of the first part of Lemma \[lem:ExpVarShrinking\] follows upon substituting into . We have $${\mathcal{V}}_{x_{0},\rho}(X_{f_{n},r}) = \frac{1}{\pi\rho^{2}}\int_{B_{x_{0}}\left(\rho\right)}\left(\int_{B_{x}\left(r\right)}f_{n}\left(y\right)^{2}\,\text{d}y- {\mathbb{E}}_{x_{0},\rho}\left[X_{f_{n},r}\right]\right)^{2}\,\text{d}x.$$ By (\[eq:IntegrandVar\]), $$\begin{aligned} & \frac{1}{\pi\rho^{2}}\int_{B_{x_{0}}\left(\rho\right)}\left(\int_{B_{x}\left(r\right)}f_{n}\left(y\right)^{2}\,\text{d}y-\pi r^{2}\right)^{2}\,\text{d}x\\ & =4\pi^{2}r^{4}\sum_{\begin{subarray}{c} \lambda,\lambda',\lambda'',\lambda'''\in\mathcal{E}_{n}\\ \lambda\ne\lambda'\\ \lambda''\ne\lambda''' \end{subarray}}c_{\lambda}\overline{c_{\lambda'}}c_{\lambda''}\overline{c_{\lambda'''}}g_{2}\left(r\left\Vert \lambda-\lambda'\right\Vert \right)g_{2}\left(r\left\Vert \lambda''-\lambda'''\right\Vert \right)\\ & \times\frac{1}{\pi\rho^{2}}\int_{B_{x_{0}}\left(\rho\right)}e\left(\left\langle x,\lambda-\lambda'+\lambda''-\lambda'''\right\rangle \right)\,\text{d}x\\ & =8\pi^{2}r^{4}\sum_{\begin{subarray}{c} \lambda,\lambda'\in\mathcal{E}_{n}\\ \lambda\ne\lambda' \end{subarray}}\left|c_{\lambda}\right|^{2}\left|c_{\lambda'}\right|^{2}g_{2}\left(r\left\Vert \lambda-\lambda'\right\Vert \right)^{2}\\ & +8\pi^{2}r^{4}\sum_{\begin{subarray}{c} \lambda,\lambda',\lambda'',\lambda'''\in\mathcal{E}_{n}\\ \lambda\ne\lambda'\\ \lambda''\ne\lambda'''\\ \lambda-\lambda'+\lambda''-\lambda'''\ne0 \end{subarray}}c_{\lambda}\overline{c_{\lambda'}}c_{\lambda''}\overline{c_{\lambda'''}}g_{2}\left(r\left\Vert \lambda-\lambda'\right\Vert \right)g_{2}\left(r\left\Vert \lambda''-\lambda'''\right\Vert \right)\\ & \times e\left(\left\langle x_{0},\lambda-\lambda'+\lambda''-\lambda'''\right\rangle \right)g_{2}\left(\rho\left\Vert \lambda-\lambda'+\lambda''-\lambda'''\right\Vert \right). \end{aligned}$$ By the hypothesis ${\mathcal{A}}(n;4,\epsilon)$ and Lemma \[lem:H2Formulas\], we have $$\begin{aligned} & \sum_{\begin{subarray}{c} \lambda,\lambda',\lambda'',\lambda'''\in\mathcal{E}_{n}\\ \lambda\ne\lambda'\\ \lambda''\ne\lambda'''\\ \lambda-\lambda'+\lambda''-\lambda'''\ne0 \end{subarray}}c_{\lambda}\overline{c_{\lambda'}}c_{\lambda''}\overline{c_{\lambda'''}}g_{2}\left(r\left\Vert \lambda-\lambda'\right\Vert \right)g_{2}\left(r\left\Vert \lambda''-\lambda'''\right\Vert \right)\\ & \times e\left(\left\langle x_{0},\lambda-\lambda'+\lambda''-\lambda'''\right\rangle \right)g_{2}\left(\rho\left\Vert \lambda-\lambda'+\lambda''-\lambda'''\right\Vert \right)\\ & \ll\left(\sum_{\lambda\in\mathcal{E}_{n}}\left|c_{\lambda}\right|\right)^{4}\frac{1}{\left(n^{\delta-\epsilon}\right)^{3/2}}\ll N^{2}n^{-\frac{3}{2}(\delta-\epsilon)} \ll n^{-\frac{3}{2}\delta + 2\epsilon}. \end{aligned}$$ Next, note that $$\begin{aligned} \label{eq:inner_int_estimate} \int_{B_{x}\left(r\right)}f_{n}\left(y\right)^{2}\,\text{d}y-\pi r^{2} &= 2\pi r^{2}\sum_{\lambda\ne\lambda'\in\mathcal{E}_{n}} c_{\lambda}\overline{c_{\lambda'}}g_{2}\left(r\left\Vert \lambda-\lambda'\right\Vert \right) \ll r^2 \left(\sum_{\lambda\in\mathcal{E}_{n}}\left|c_{\lambda}\right|\right)^{2} \ll N r^2. \end{aligned}$$ By and the first part of Lemma \[lem:ExpVarShrinking\], $${\mathcal{V}}_{x_{0},\rho}(X_{f_{n},r}) = \frac{1}{\pi\rho^{2}}\int_{B_{x_{0}}\left(\rho\right)}\left(\int_{B_{x}\left(r\right)}f_{n}\left(y\right)^{2}\,\text{d}y-\pi r^{2}\right)^{2}\,\text{d}x + O\left(r^4 n^{-\frac{3}{5}\delta+4\epsilon}\right)$$ and the statement of Lemma \[lem:VarExpd2\] follows. We have $${\mathbb{E}}_{x_{0},\rho}[\hat{X}_{f_{n},r}^{k}] = {\mathcal{V}}_{x_{0},\rho}\left(X_{f_{n},r}\right)^{-k/2} \cdot \frac{1}{\pi\rho^{2}}\int_{B_{x_{0}}\left(\rho\right)}\left(\int_{B_{x}\left(r\right)}f_{n}\left(y\right)^{2}\,\text{d}y- {\mathbb{E}}_{x_{0},\rho}\left[X_{f_{n},r}\right]\right)^{k}\,\text{d}x.$$ By (\[eq:IntegrandVar\]), we have $$\begin{aligned} \frac{1}{\pi\rho^{2}}&\int_{B_{x_{0}}\left(\rho\right)} \left(\int_{B_{x}\left(r\right)}f_{n}\left(y\right)^{2}\,\text{d}y-\pi r^{2}\right)^{k}\,\text{d}x = \left(2\pi\right)^{k}r^{2k} \sum_{\begin{subarray}{c} \forall1\le i\le k,\,\lambda_{i}\ne\lambda_{i}'\in\mathcal{E}_{n}\\ \sum_{i=1}^{k}\left(\lambda_{i}-\lambda_{i}'\right)=0 \end{subarray}}\prod_{j=1}^{k}c_{\lambda_{j}}\overline{c_{\lambda_{j}'}}g_{2}\left(r\left\Vert \lambda_{j}-\lambda_{j}'\right\Vert \right)\\ & +\left(2\pi\right)^{k}r^{2k}\sum_{\begin{subarray}{c} \forall1\le i\le k,\,\lambda_{i}\ne\lambda_{i}'\in\mathcal{E}_{n}\\ \sum_{i=1}^{k}\left(\lambda_{i}-\lambda_{i}'\right)\ne0 \end{subarray}}\prod_{j=1}^{k}c_{\lambda_{j}}\overline{c_{\lambda_{j}'}}g_{2}\left(r\left\Vert \lambda_{j}-\lambda_{j}'\right\Vert \right)\\ & \times2e\left(\left\langle x_{0},\sum_{j=1}^{k}\left(\lambda_{j}-\lambda_{j}'\right)\right\rangle \right)g_{2}\left(\rho\left\Vert \sum_{j=1}^{k}\left(\lambda_{j}-\lambda_{j}'\right)\right\Vert \right). \end{aligned}$$ By the hypothesis ${\mathcal{A}}(n;2k,\epsilon)$, $$\begin{aligned} \sum_{\begin{subarray}{c} \forall1\le i\le k,\,\lambda_{i}\ne\lambda_{i}'\in\mathcal{E}_{n}\\ \sum_{i=1}^{k}\left(\lambda_{i}-\lambda_{i}'\right)\neq0 \end{subarray}}& \prod_{j=1}^{k}c_{\lambda_{j}}\overline{c_{\lambda_{j}'}}g_{2}\left(r\left\Vert \lambda_{j}-\lambda_{j}'\right\Vert \right) e\left(\left\langle x_{0},\sum_{j=1}^{k}\left(\lambda_{j}-\lambda_{j}'\right)\right\rangle \right)g_{2}\left(\rho\left\Vert \sum_{j=1}^{k}\left(\lambda_{j}-\lambda_{j}'\right)\right\Vert \right)\\ & \ll\left(\sum_{\lambda\in\mathcal{E}_{n}}\left|c_{\lambda}\right|\right)^{2k}\frac{1}{\left(n^{\delta - \epsilon}\right)^{3/2}}\ll N^k n^{-\frac{3}{2}(\delta-\epsilon)} \ll n^{-\frac{3}{2}+2\epsilon}. \end{aligned}$$ By and the first part of Lemma \[lem:ExpVarShrinking\], $$\begin{aligned} \mathbb{E}_{x_{0},\rho}[\hat{X}_{f_{n},r}^{k}] & = {\mathcal{V}}_{x_{0},\rho}\left(X_{f_{n},r}\right)^{-k/2} \cdot \frac{1}{\pi\rho^{2}}\int_{B_{x_{0}}\left(\rho\right)}\left(\int_{B_{x}\left(r\right)}f_{n}\left(y\right)^{2}\,\text{d}y-\pi r^2 \right)^{k}\,\text{d}x \\ &+ O\left({\mathcal{V}}_{x_{0},\rho}\left(X_{f_{n},r}\right)^{-k/2} r^{2k} n^{-\frac{3}{5}\delta+4\epsilon}\right), \end{aligned}$$ and the statement of Lemma \[lem:KthMoment\] follows. \[sec:AuxLemmasProof\]Proofs of auxiliary lemmas ================================================ In this section we provide the proofs for lemmas \[lem:BasicVarProp\], \[lem:InnerIntegral\], and \[lem:CosToDist\]:   1. The upper bound is straightforward, and the lower bound follows from (\[eq:BasicNormalization\]) by invoking the Cauchy-Schwarz inequality on . 2. By partial summation, for every $\lambda_{0}\in\mathcal{E}_{n}$ we have $$1=\sum_{\lambda\in\mathcal{E}_{n}}\left|c_{\lambda}\right|^{2}=N\left|c_{\lambda_{0}}\right|^{2}+E,$$ where $\left|E\right|\le V\left(\underline{v} \right)$. Since $\lambda_{0}$ is arbitrary, we deduce that $$[\underline{v}]_{\infty} \le1+V\left(\underline{v} \right).$$ 3. Follows directly from parts $1$ and $2$ of this lemma. We have $$\begin{aligned} \label{eq:inner_int_expansion} \int_{B_{x}\left(r\right)}f_{n}\left(y\right)^{2}\,\text{d}y&=\int_{B_{x}\left(r\right)}\sum_{\lambda,\lambda'\in\mathcal{E}_{n}}c_{\lambda}\overline{c_{\lambda'}}e\left(\left\langle y,\lambda-\lambda'\right\rangle \right)\,\text{d}y\\ & =\frac{\pi^{d/2}}{\Gamma\left(d/2+1\right)}r^{d}+\sum_{\begin{subarray}{c} \lambda,\lambda'\in\mathcal{E}_{n}\\ \lambda\ne\lambda' \end{subarray}}c_{\lambda}\overline{c_{\lambda'}}\int_{B_{x}\left(r\right)}e\left(\left\langle y,\lambda-\lambda'\right\rangle \right)\,\text{d}y.\nonumber \end{aligned}$$ Transforming the variables $y=rz+x$, we obtain $$\label{eq:var_transformation} \int_{B_{x}\left(r\right)}e\left(\left\langle y,\lambda-\lambda'\right\rangle \right)\,\text{d}y=r^{d}e\left(\left\langle x,\lambda-\lambda'\right\rangle \right)\int_{B_{0}\left(1\right)}e\left(\left\langle z,r\left(\lambda-\lambda'\right)\right\rangle \right)\,\text{d}z.$$ Note that $$\begin{aligned} \label{eq:Fourier_ball} \int_{B_{0}\left(1\right)}e\left(\left\langle z,r\left(\lambda-\lambda'\right)\right\rangle \right)\,\text{d}z & =\frac{\left(2\pi\right)^{d/2}J_{d/2}\left(2 \pi r\left\Vert \lambda-\lambda'\right\Vert \right)}{\left(2 \pi r\left\Vert \lambda-\lambda'\right\Vert \right)^{d/2}}, \end{aligned}$$ and (\[eq:IntegrandVar\]) follows upon substituting into and finally into . Let $\theta_{\lambda}$ be the angle between $\lambda$ and $\lambda'$. Then $$\begin{aligned} \frac{1}{N} \cdot \#\left\{ \lambda\in\mathcal{E}_{n}:\,\lambda\succeq\lambda',\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\right\} & =\frac{1}{N}\cdot \#\left\{ \lambda\in\mathcal{E}_{n}:\,\theta_{\lambda}\ge0,\,\sqrt{2\left(1-\cos\theta_{\lambda}\right)}\le s\right\} \\ & =\frac{1}{N}\cdot \#\left\{ \lambda\in\mathcal{E}_{n}:\,\theta_{\lambda}\in\left[0,\arccos\left(1-s^{2}/2\right)\right]\right\} \\ & =\frac{1}{2\pi}\arccos\left(1-s^{2}/2\right)+O\left(\Delta\left(n\right)\right)\\ & =\frac{s}{2\pi}+O\left(s^{3}+\Delta\left(n\right)\right) \end{aligned}$$ which is the statement (\[eq:CosToDistEq\]) of Lemma \[lem:CosToDist\]. [10]{} Benatar, J., Buckley, J., Wigman, I., *Two applications of Bourgain’s de-randomization method on toral eigenfunctions and spectral quasi-correlations*, in preparation. Benatar, J., Marinucci, D., Wigman, I. *Planck-scale distribution of nodal length of arithmetic random waves*. Benatar, J., Maffucci R. W., *Random waves on $\mathbb{T}^{3}$: nodal area variance and lattice point correlations.* Int. Math. Res. Not., to appear. Berry, Michael. [*Regular and irregular semiclassical wavefunctions.*]{} J. Phys. A 10 (1977), no. 12, 2083–2091. Berry, Michael. [*Semiclassical mechanics of regular and irregular motion.*]{} Chaotic behavior of deterministic systems (Les Houches, 1981), 171–271, North-Holland, Amsterdam, 1983. Bombieri, E., Bourgain, J., *A problem on sums of two squares*. Int. Math. Res. Not. IMRN (2015), no. 11, 33433407. Bourgain, J. *On toral eigenfunctions and the random wave model*. Israel J. Math. **201** (2014), no. 2, 611630. Bourgain J., Rudnick Z., Sarnak P., *Spatial statistics for lattice points on the sphere I: Individual results*. The Bulletin of the Iranian Mathematical Society (BIMS), special issue in honor of Freydoon Shahidi’s 70’th birthday, to appear. Bourgain, Jean; Rudnick, Zeév. [*On the geometry of the nodal lines of eigenfunctions of the two-dimensional torus.*]{} Ann. Henri Poincaré 12 (2011), no. 6, 1027–1053. Y. Colin de Verdière, *Ergodicité et fonctions propres du Laplacien*, Comm. Math. Phys. [**102**]{} (1985), 497-502. de Courcy-Ireland, Matthew. [*Small-scale equidistribution for random spherical harmonics.*]{} Preprint. available online, <https://arxiv.org/abs/1711.01317> Erdős, P., Hall, R. R., *On the angular distribution of Gaussian integers with fixed norm*. Discrete Math. **200** (1999), no. 1-3, 8794. Feller W., *An introduction to probability theory and its applications, Vol. 2.* Second edition. Wiley, New York (1971). Gradhsteyn I. S., Rizhik I.M., *Table of integrals, series and products,* 6th Edition, Academic Press (2000). Granville, A., Wigman, I., *Planck-scale mass equidistribution of toral Laplace eigenfunctions,* Communications in Mathematical Physics, 355(2), 2017, pp.767-802. Han, Xiaolong. [*Small scale quantum ergodicity in negatively curved manifolds.*]{} Nonlinearity 28 (2015), no. 9, 3263–3288. Han, Xiaolong. [*Small Scale Equidistribution of Random Eigenbases.*]{} Comm. Math. Phys., to appear, available online <https://arxiv.org/abs/1511.01195> Han, Xiaolong, Tacy, Melissa. [*Equidistribution of random waves on small balls.*]{} Preprint. available online, <https://arxiv.org/abs/1611.05983> Hezari, Hamid; Rivière, Gabriel. [*$L^{p}$ norms, nodal sets, and quantum ergodicity.*]{} Adv. Math. 290 (2016), 938–966. Hezari, Hamid; Rivière, Gabriel. [Quantitative equidistribution properties of toral eigenfunctions.]{} J. Spec. Thr., to appear, available online <https://arxiv.org/abs/1503.02794> Humphries, P., 2017. Equidistribution in Shrinking Sets and $L^4$-Norm Bounds for Automorphic Forms. arXiv preprint arXiv:1705.05488. Kuipers L., Niederreiter H., *Uniform distribution of sequences.* Wiley, New York (1974). Krishnapur, M., Kurlberg, P., Wigman, I. *Nodal length fluctuations for arithmetic random waves*, Annals of Mathematics (2) 177, no. 2, 699–737 (2013). Lester, Stephen; Rudnick, Zeév. [*Small scale equidistribution of eigenfunctions on the torus.*]{} Comm. Math. Phys. 350 (2017), no. 1, 279–-300. Luo, Wen Zhi; Sarnak, Peter. [*Quantum ergodicity of eigenfunctions on $PSL_{2}({\mathbb{Z}})\setminus \mathbb{H}^{2}$.*]{} Inst. Hautes Études Sci. Publ. Math. No. 81 (1995), 207–237. Sarnak, P. [*Variance sums on symmetric spaces.*]{} Private communication. Sartori, A. [*Mass distribution for toral eigenfunctions via Bourgain’s de-randomisation.*]{} available online <https://arxiv.org/abs/1812.00962> A. Snirel’man, *Ergodic properties of eigenfunctions*, Uspekhi Mat. Nauk [**180**]{} (1974), 181-182. Young, M. *The quantum unique ergodicity conjecture for thin sets*. Adv. Math. **286** (2016), 958-1016. S. Zelditch, *Uniform distribution of eigenfunctions on compact hyperbolic surfaces*, Duke Math. J. [**55**]{} (1987), 919-941. Zygmund, A. *On Fourier coefficients and transforms of functions of two variables*. Studia Math. 50 (1974), 189201.

--- abstract: 'We give the emission function of the axially symmetric Buda-Lund hydro model and present its simultaneous, high quality fits to identified particle spectra, two-particle Bose-Einstein or HBT correlations and charged particle pseudorapidity distributions as measured by BRAHMS and PHENIX in 0-30 % central, $\sqrt{s_{\NN}} = 200$ GeV Au+Au collisions at RHIC. The best fit is achieved when the most central region of the particle emitting volume is superheated to $T_0 = 200 \pm 9$ MeV $ \ge T_c =172 \pm 3$ MeV, a preliminary, 3 $\sigma$ effect.' author: - | M. Csanád$^1$, T. Csörgő$^2$, B. Lörstad$^3$ and A. Ster$^2$\ $^1$Dept. Atomic Phys., ELTE, H-1117 Budapest, Pázmány P. 1/a, Hungary\ $^2$MTA KFKI RMKI, H - 1525 Budapest 114, P.O.Box 49, Hungary\ $^3$Dept. Physics, University of Lund, S - 22362 Lund, Sweden title: | [**A hint at quark deconfinement\ in 200 GeV Au+Au data at RHIC**]{} --- Introduction ============ The Buda-Lund hydro model is successful in describing the identified single particle spectra and the transverse mass dependent Bose-Einstein or HBT radii as well as the pseudorapidity distribution of charged particles in Au + Au collisions at $\sqrt{s_{\NN}} = 130 $ GeV [@ster-ismd03], as measured by the BRAHMS, PHENIX, PHOBOS and STAR collaborations. The result of the simultaneous fit to all these datasets indicate the existence of a very hot region, with a temperature significantly greater than 170 MeV [@mate-ell1]. Recently, Fodor and Katz calculated the phase diagram of lattice QCD at finite net barion density [@Fodor:2001pe]. These lattice results, obtained with light quark masses four times heavier than the physical value, indicated that in the $0 \le \mu_B \le 700$ MeV region the transition from confined to deconfined matter is a cross-over, with $T_c \simeq 172 \pm 3$ MeV. This value is, within one standard deviation, independent of the bariochemical potential in the $0 \le \mu_B \le 300$ MeV region. The Buda-Lund fits, combined with these lattice results, provide an indication for quark deconfinement in Au + Au collisions with $\sqrt{s_{\NN}} = 130 $ GeV colliding energies at RHIC. This observation was confirmed [@mate-ell1] by the analysis of the transverse momentum and rapidity dependence of the elliptic flow as measured by the PHENIX and PHOBOS collaborations. Here we investigate what happens if a similar analysis is performed on the final, published Au+Au collision data at RHIC at the maximum, $\sqrt{s_{\NN}} = 200$ GeV bombarding energies. The emission function of the Buda-Lund hydro model ================================================== The Buda-Lund hydro model was introduced in refs. [@Csorgo:1995bi; @Csorgo:1995vf]. This model was defined in terms of its emission function $S(x,k)$, for axial symmetry, corresponding to central collisions of symmetric nuclei. The observables are calculated analytically, see refs. [@cs-rev; @ster-ismd03] for details and key features. Here we summarize the Buda-Lund emission function in terms of its fit parameters. The presented form is equivalent to the original shape proposed in refs. [@Csorgo:1995bi; @Csorgo:1995vf], however, it is easier to fit and interpret it. The single particle invariant momentum distribution, $N_1(k_1)$, is obtained as N\_1(k\_1) = \^4 x S(x,k\_1). For chaotic (thermalized) sources, in case of the validity of the plane-wave approximation, the two-particle invariant momentum distribution $N_2(k_1,k_2) $ is also determined by $S(x,k)$, the single particle emission function, if non-Bose-Einstein correlations play negligible role or can be corrected for, see ref. [@cs-rev] for a more detailed discussion. Then the two-particle Bose-Einstein correlation function, $ C_2 (k_1,k_2) = { N_2(k_1,k_2)}/\left[{ N_1(k_1) N_1(k_2) }\right] $ can be evaluated in a core-halo picture [@Csorgo:1994in], where the emission function is a sum of emission functions characterizing a hydrodynamically evolving core and a surrounding halo of decay products of long-lived resonances, $S(x,k) = S_c(x,k) + S_h(x,k)$. Consequently, the single particle spectra can also be given as a sum, $N_1(k) = N_{1,c}(k) + N_{1,h}(k)$. In the correlation function, an effective intercept parameter $\lambda \equiv \lambda_*(K)$ appears and its relative momentum dependence can be calculated directly from the emission function of the core, C\_2 (k\_1,k\_2) = 1+ 1+\_[\*]{}(K), where the relative and the momenta are $q = k_1-k_2$, $K = 0.5 (k_1+k_2)$, and the Fourier-transformed emission function is defined as $\tilde S(q,K) = \int \d^4 x S(x,K) \exp(i q x).$ The measured $\lambda_*$ parameter of the correlation function is utilized to correct the core spectrum for long-lived resonance decays [@Csorgo:1994in]: $ N_1(k) = N_c(k)/{\sqrt{\lambda_{*}(k)}}. $ The emission function of the core is assumed to have a hydrodynamical form, $$S_c(x,k) d^4 x = \frac{g}{(2 \pi)^3} \frac{ k^\nu d^4\Sigma_\nu(x)}{B(x,k) +s_q},$$ where $g$ is the degeneracy factor ($g = 1$ for pseudoscalar mesons, $g = 2$ for spin=1/2 barions). The particle flux over the freeze-out layers is given by a generalized Cooper–Frye factor: the freeze-out hypersurface depends parametrically on the freeze-out time $\tau$ and the probability to freeze-out at a certain value is proportional to $H(\tau)$, $ k^\nu d^4\Sigma_\nu(x) = m_t \cosh(\eta - y) H(\tau) d\tau \, \tau_0 d\eta \, dr_x \, dr_y. $ Here $\eta = 0.5 \log[(t + r_z)/(t-r_z)]$, $\tau=\sqrt{t^2 - r_z^2}$, $ y = 0.5 \log[(E + k_z)/(E-k_z)]$ and $m_t=\sqrt{E^2 - k_z^2}$. The freeze-out time distribution $H(\tau)$ is approximated by a Gaussian, $ H(\tau) = \frac{1}{(2 \pi \Delta\tau^2)^{3/2}} \exp\left[-\frac{(\tau - \tau_0)^2} {2 \Delta \tau^2} \right], $ where $\tau_0$ is the mean freeze-out time, and the $\Delta\tau$ is the duration of particle emission, satisfying $\Delta\tau \ll \tau_0$. The (inverse) Boltzmann phase-space distribution, $B(x,k)$ is given by $$B(x,k)= \exp\left( \frac{ k^\nu u_\nu(x)}{T(x)} -\frac{\mu(x)}{T(x)} \right),$$ and the term $s_q$ is $ 0$, $-1$, and $+1$ for Boltzmann, Bose-Einstein and Fermi-Dirac statistics, respectively. The flow four-velocity, $u^\nu(x)$, the chemical potential, $\mu(x)$, and the temperature, $T(x)$ distributions for axially symmetric collisions were determined from the principles of simplicity, analyticity and correspondence to hydrodynamical solutions in the limits when such solutions were known [@Csorgo:1995bi; @Csorgo:1995vf]. Recently, the Buda-Lund hydro model lead to the discovery of a number of new, exact analytic solutions of hydrodynamics, both in the relativistic [@relsol-cyl; @relsol-ell] and in the non-relativistic domain [@nr-sol; @nr-ell; @nr-inf]. The expanding matter is assumed to follow a three-dimensional, relativistic flow, characterized by transverse and longitudinal Hubble constants, u\^(x) = ( , H\_t r\_x, H\_t r\_y, H\_z r\_z ), where $\gamma$ is given by the normalization condition $u^\nu(x) u_\nu(x) = 1$. In the original form, this four-velocity distribution $u^\nu(x)$ was written as a linear transverse flow, superposed on a scaling longitudinal Bjorken flow . The strength of the transverse flow was characterized by its value $\langle u_t\rangle$ at the “geometrical" radius $R_G$, see refs. [@Csorgo:1995bi; @Chapman:1994ax; @Ster:1998hu]: u\^(x) & = & ( , , , ),\ & = & r\_t / R\_G, with $ r_t = (r_x^2 + r_y^2)^{1/2}$. Such a flow profile, with a time-dependent radius parameter $R_G$, was recently shown to be an exact solution of the equations of relativistic hydrodynamics of a perfect fluid at a vanishing speed of sound, see refs. [@Biro:1999eh; @Biro:2000nj]. The Buda-Lund hydro model characterizes the inverse temperature $1/T(x)$, and fugacity, $\exp\left[\mu(x)/T(x)\right]$ distributions of an axially symmetric, finite hydrodynamically expanding system with the mean and the variance of these distributions, in particular & = & - -[ (- y\_0)\^2 2 \^2 ]{}, \[e:mu\]\ [1 T(x)]{} & = & [1 T\_0 ]{} ( 1 + [r\_t\^2 2 R\_s\^2]{} ) ( 1 + [(- \_0)\^2 2 \_s\^2 ]{} ). Here $R_G$ and $\Delta\eta$ characterize the spatial scales of variation of the fugacity distribution, $\exp\left[\mu(x)/T(x)\right]$, that control particle densities. Hence these scales are referred to as geometrical lengths. These are distinguished from the scales on which the inverse temperature distribution changes, the temperature drops to half if $r_x = r_y = R_s$ or if $\tau = \tau_0 + \sqrt{2} \Delta\tau_s$. These parameters can be considered as second order Taylor expansion coefficients of these profile functions, restricted only by the symmetry properties of the source, and can be trivially expressed by re-scaling the earlier fit parameters. The above is the most direct form of the Buda-Lund model. However, different combinations may also be used to measure the flow, temperature and fugacity profiles [@Csorgo:1995bi; @cs-rev]: $ H_t \equiv {b}/{\tau_0} \, = \, \ave{u_t} / R_G \, = \, \ave{u_t^\prime} / R_s$ , $H_l \equiv \gamma_t /\tau_0$, where $ \gamma_t = \sqrt{ 1 + H_t^2 r_t^2}$ is evaluated at the point of maximal emittivity, and & = & = \_r = ,\ & = & = \_s = . Buda-Lund fits to Au+Au data at $\sqrt{s_{\NN}}= 200$ GeV ========================================================== In this section, we present new fit results to BRAHMS data on charged particle pseudorapidity distributions [@Bearden:2001qq], and PHENIX data on identified particle momentum distributions and Bose-Einstein (HBT) radii [@Adler:2003cb; @Adler:2004rq] in Au+Au collisions at $\sqrt{s_{\NN}}= 200$ GeV. The analysis codes and methods are identical to the ones used to fit the BRAHMS [@Bearden:2001xw], PHENIX [@Adcox:2001mf; @Adcox:2002uc], PHOBOS [@Back:2001bq], and STAR [@Adler:2001zd] data in 0- 5% most central Au+Au collisions at $\sqrt{s_{\NN}} = 130$ GeV, see ref. [@ster-ismd03]. The applied Buda-Lund 1.5 fitting package can be downloaded, together with the detailed fit results, from ref. [@Csorgo-blhome]. This calculation determines the position of the saddle point exactly in the beam direction, but in the transverse direction, the saddle point equations are solved only approximately, as summarized in ref. [@cs-rev]. The new results for $\sqrt{s_{\NN}} = 200$ GeV Au+Au collisions in the 0-30% centrality class are shown in the first column of Table 1. For comparison, we also show the results of an identical fit to $\sqrt{s_{\NN}} = 130$ GeV Au+Au collisions in the 0- 5% centrality class. ------------------------------ ------ ----------- ------- ----------- -- -- -- -- Buda-Lund v1.5 0 - 30 % 0 - 5(6) % $T_0$ \[MeV\] 200 $\pm$ 9 214 $\pm$ 7 $T_{\mbox{\rm e}}$ \[MeV\] 127 $\pm$ 13 102 $\pm$ 11 $\mu_B$ \[MeV\] 61 $\pm$ 40 77 $\pm$ 38 $R_{G}$ \[fm\] 13.2 $\pm$ 1.3 28.0 $\pm$ 5.5 $R_{s}$ \[fm\] 11.6 $\pm$ 1.0 8.6 $\pm$ 0.4 $\langle u_t^\prime \rangle$ 1.5 $\pm$ 0.1 1.0 $\pm$ 0.1 $\tau_0$ \[fm/c\] 5.7 $\pm$ 0.2 6.0 $\pm$ 0.2 $\Delta\tau$ \[fm/c\] 1.9 $\pm$ 0.5 0.3 $\pm$ 1.2 $\Delta\eta$ 3.1 $\pm$ 0.1 2.4 $\pm$ 0.1 $\chi^2/\mbox{\rm NDF}$ 132 / 208 158.2 / 180 ------------------------------ ------ ----------- ------- ----------- -- -- -- -- : The first column shows the source parameters from simultaneous fits of final BRAHMS and PHENIX data for 0 - 30 % most central $Au+Au$ collisions at $\sqrt{s_{\NN}} = 200$ GeV, as shown in Figs. 1 and 2, as obtained with the Buda-Lund hydro model, version 1.5. The errors on these parameters are still preliminary. The second column is the result of an identical analysis of BRAHMS, PHENIX, PHOBOS and STAR data for 0 - 5 % most central Au+Au collisions at $\sqrt{s_{\NN}}=130$ GeV, ref. [@ster-ismd03]. []{data-label="tab:results"} ![ \[fig:spectra\] [Solid line shows the simultaneous Buda-Lund v1.5 fit to final Au+Au data at $\sqrt{s_{\NN}} = 200$ GeV. The transverse mass distributions of identified particles are measured by PHENIX [@Adler:2003cb] the pseudorapidity distributions of charged particles are measured by BRAHMS [@Bearden:2001qq], the transverse mass dependence of the radius parameters are data of PHENIX [@Adler:2004rq]. Note that the identified particle spectra are published in more detailed centrality classes, but we recombined the 0-30% most central collisions so that the fitted spectra and radii be obtained in the same centrality class.]{} ](fig1.eps "fig:"){width="2.7in"} ![ \[fig:spectra\] [Solid line shows the simultaneous Buda-Lund v1.5 fit to final Au+Au data at $\sqrt{s_{\NN}} = 200$ GeV. The transverse mass distributions of identified particles are measured by PHENIX [@Adler:2003cb] the pseudorapidity distributions of charged particles are measured by BRAHMS [@Bearden:2001qq], the transverse mass dependence of the radius parameters are data of PHENIX [@Adler:2004rq]. Note that the identified particle spectra are published in more detailed centrality classes, but we recombined the 0-30% most central collisions so that the fitted spectra and radii be obtained in the same centrality class.]{} ](fig2.eps "fig:"){width="2.7in"} ![ \[fig:radii\] [Top row shows the transverse mass dependence of the side, out and longitudinal HBT radii, the central line shows their pairwise ratio (usually only $R_{\mbox{\rm out}}/R_{\mbox{\rm side}}$ is shown) together with the Buda-Lund fits, vers. 1.5. The bottom line shows the inverse of the squared radii. The intercept of the curves in this row is within errors zero for the two transverse components, so the fugacity is within errors independent of the transverse coordinates. However, the intercept is nonzero in the longitudinal direction, which makes the fugacity (hence particle ratios) rapidity dependent. See also ref. [@ster-ismd03] for a similar plot at $\sqrt{s_{\NN}} = 130$ GeV.]{} ](fig3.eps){width="4.3in"} Let us clarify first the meaning of the parameters shown in Table 1. The temperature at the center of the fireball at the mean freeze-out time is denoted by $T_0 \equiv T(r_x=r_y=0, \tau=\tau_0)$. The surface temperature is also a characteristic, kind of an average temperature, and its value is always $T_s \equiv T(r_x=r_y=R_s, \tau=\tau_0) = T_0/2$. In fact this relationship defines the “surface" radius $R_s$. During the particle emission, the system may cool due to evaporation and expansion, this is measured by the “post-evaporation temperature" $T_e \equiv T(r_x=r_y=0, \tau = \tau_0 + \sqrt{2} \Delta\tau)$. In the presented cases, the strength of the transverse flow is measured by $\ave{u_t^\prime}$, its value at the “surface radius" $R_s$. The “mean freeze-out time" parameter is denoted by $\tau_0$ and the “duration" of particle emission, or the width of the freeze-out time distribution is measured by $\Delta\tau$. The fugacity distribution varies on the characteristic transverse scale given by the “geometrical radius" $R_G$. Finally, the width of the space-time rapidity distribution, or the longitudinal variation scale of the fugacity distribution is measured by the parameter $\Delta\eta$. Perhaps it could be more appropriate to directly fit the transverse Hubble constant, $H_t = \ave{u_t^\prime}/R_s$ to the data, as this value is not sensitive to the length-scale chosen to evaluate the “average" transverse flow $\ave{u_t^\prime}$. In the case of parameters shown in Table 1, the density drop in the transverse direction is dominated by the cooling of the local temperature distribution in the transverse direction, and not so much by the change of the fugacity distribution. That is why we fitted here $\ave{u_t^\prime}$ at the “surface radius" $R_s$. Note also that $\tau_0$ could more properly be interpreted as the inverse of the longitudinal Hubble constant $H_l$, which is only an order of magnitude estimate of the mean freeze-out time, similarly to how the inverse of the present value of the Hubble constant in astrophysics provides only an order of magnitude estimate of the life-time of our Universe. The feasibility of directly fitting the transverse and longitudinal Hubble constants to data will be investigated in a subsequent publication. Let us also note, that we have fitted the absolute normalized spectra for identified particles, and the normalization conditions were given by central chemical potentials $\mu_0$ that were taken as free normalization parameters for each particle species. All these directly fitted parameters are made public at [@Csorgo-blhome]. From these values, we have determined the net bariochemical potential as $\mu_B = \mu_p - \mu_{\overline{p}}$. Although this parameter is not directly fitted but calculated, we have included $\mu_B$ in Table 1, so that our results could be compared with other successful models of two-particle Bose-Einstein correlations at RHIC, namely the AMPT cascade [@Lin:2002gc], Tom Humanic’s cascade [@Humanic:2002iw], the blast-wave model [@Retiere:2003qb; @Retiere:2003kf], the Hirano-Tsuda numerical hydro [@Hirano:2002hv] and the Cracow “single freeze-out thermal model"  [@Broniowski:2002wp; @Florkowski:2002wn; @Broniowski:2001we]. Now, we are ready for the discussion of the results in Table 1. In case of more central collisions at the lower RHIC energies, a well defined minimum was found, with accurate error matrix and a statistically acceptable fit quality, $\chi^2$/NDF= 158/180, that corresponds to a confidence level of 88 %. (These fit results were shown graphically on Figs. 1 and 2 of ref. [@ster-ismd03], and the parameters are summarized in the second column of Table 1.) In the case of the less central but more energetic Au+Au collisions, the obtained $\chi^2/\mbox{\rm NDF}$ fit is [*too small*]{}. Note that in these fits we added the systematic and statistical errors in quadrature, and this procedure is preliminary and has to be revisited before we can report on the final values of the fit parameters and determine their error bars. It could also be advantageous to analyze a more central data sample, or the centrality dependence of the radius parameters and the pseudorapidity distributions, or to fit additional data of STAR and PHOBOS too, so that the parameters of the Buda-Lund hydro model could be determined with smaller error bars. At present, we find that $T_0 > T_c = 172 \pm 3$ MeV [@Fodor:2001pe] by 3 $\sigma$ in case of the 0-30 % most central Au+Au data at $\sqrt{s_{\NN}} = 200$ GeV, while $T_0 > T_c$ by more than 5 $\sigma$ in case of the 0-5(6) % most central Au+Au data at $\sqrt{s_{\NN}} = 130 $ GeV. Thus this signal of a cross-over transition to quark deconfinement is not yet significant in the more energetic but less central Au+Au data sample, while it is significant at the more central, but less energetic sample. In this latter case of 130 GeV Au+Au data, $R_G$ obviously became an irrelevant parameter, with $1/R_G\approx 0.$ . This is explicitly visible in Fig. 2 of ref. [@ster-ismd03], where the last row indicates that the correlation radii are in the scaling limit and the fugacity distribution, $\exp\left[\mu(x)/T(x)\right]$ is independent of the transverse coordinates. The Buda-Lund model predicted, see eqs. (53-58) in ref. [@Csorgo:1995bi] and also eqs. (26-28) in [@nr-inf], that the linearity of the inverse radii as a function of $m_t$ can be connected to the Hubble flow and the temperature gradients. The slopes are the same for side, out and longitudinal radii if the Hubble flow (and the temperature inhomogeneities) become direction independent. The intercepts of the linearly extrapolated $m_t$ dependent inverse squared radii at $m_t=0$ determine $1/R_G^2$, or the magnitude of corrections from the finite geometrical source sizes, that stem from the $\exp[\mu(x)/T(x)]$ terms. We can see on Fig. 2, that these corrections within errors vanish also in $\sqrt{s_{\NN}} = 200$ Au+Au collisions at RHIC. This result is important, because it explains, why thermal and statistical models are successful at RHIC: if $\exp[\mu(x)/T(x)] = \exp(\mu_0/T_0)$, then this factor becomes an overall normalization factor, proportional to the particle abundances. Indeed, we found that when the finite size in the transverse direction is generated by the $T(x)$ distribution, the quality of the fit increased and we had no degenerate parameters in the fit any more. This is also the reason, why we interpret $R_s$, given by the condition that $T(r_x=r_y=R_s) = T_0/2$, as a “surface" radius: this is the scale where particle density drops. Note that we have obtained similarly good description of these data if we require that the four-velocity field is a fully developed, three-dimensional Hubble flow, with $u^\nu = x^\nu/\tau$, however, we cannot elaborate on this point here due to the space limitations [@mate-ell1]. Conclusions =========== Table 1, Figures 1 and 2 indicate that the Buda-Lund hydro model works well both at the lower and the higher RHIC energies, and gives a good quality description of the transverse mass dependence of the HBT radii. For the dynamical reason, see refs. [@nr-inf] and [@Csorgo:1995bi]. In fact, even the time evolution of the entrophy density can be solved from the fit results, $s(\tau) = s_0 (\tau_0/\tau)^3$, which is the consequence of the Hubble flow, $u^\nu = x^\nu/\tau$, a well known solution of relativistic hydrodynamics, see also ref. [@relsol-ell]. This is can be considered as the resolution of the RHIC HBT “puzzle", although a careful search of the literature indicates that this “puzzle" was only present in models that were not tuned to CERN SPS data [@Csorgo:2002ry]. We also observe that the central temperature is $T_0 = 214 \pm 7$ MeV in the most central Au+Au collisions at $\sqrt{s_{\NN}} = 130$ GeV, and we find here a net bariochemical potential of $\mu_B = 77 \pm 38$ MeV. Recent lattice QCD results indicate [@Fodor:2001pe], that the critical temperature is within errors a constant of $T_c = 172 \pm 3$ MeV in the $0 \le \mu_B \le 300$ MeV interval. Our results clearly indicate $(T,\mu_B)$ values above this critical line, which is a significant, more than 5 $\sigma$ effect. The present level of precision and the currently fitted PHENIX and BRAHMS data set does not yet allow a firm conclusion about such an effect at $\sqrt{s_{\NN}} = 200$ GeV, however, a similar behavior is seen on a 3 standard deviation level. This can be interpreted as a hint at quark deconfinement at $\sqrt{s_{\NN}} = 200$ GeV at RHIC. Finding similar parameters from the analysis of the pseudorapidity dependence of the elliptic flow, it was estimated in ref. [@mate-ell1] that 1/8th of the total volume is above the critical temperature in Au+Au collisions at $\sqrt{s_{\NN}} = 130$ GeV, at the time when pions are emitted from the source. We interpret this result as an indication for quark deconfinement and a cross-over transition in Au+Au collisions at $\sqrt{s_{\NN}} = 130 $ GeV at RHIC. This result was signaled first in ref. [@Csorgo:2002ry] in a Buda-Lund analysis of the final PHENIX and STAR data on midrapidity spectra and Bose-Einstein correlations, but only at a three standard deviation level. By including the pseudorapidity distributions of BRAHMS and PHOBOS, the $T_0 \gg T_c$ effect became significant in most central Au+Au collisions at $\sqrt{s_{\NN}} = 130 $ GeV. We are looking forward to observe, what happens with the present signal in Au+Au collisions at $\sqrt{s_{\NN}} = 200$ GeV, if we include STAR and PHOBOS data to the fitted sample. The above observation of temperatures, that are higher than the critical one, is only an indication, with other words, an indirect proof for the production of a new phase, as the critical temperature is not extracted directly from the data, but taken from recent lattice QCD calculations. More data are needed to clarify the picture of quark deconfinement at the maximal RHIC energies, for example the centrality dependence of the Bose-Einstein (HBT) radius parameters could provide very important insights. Acknowledgments {#acknowledgments .unnumbered} =============== T. Cs. and M. Cs. would like to the Organizers for their kind hospitality and for their creating an inspiring and fruitful meeting. The support of the following grants are gratefully acknowledged: OTKA T034269, T038406, the OTKA-MTA-NSF grant INT0089462, the NATO PST.CLG.980086 grant and the exchange program of the Hungarian and Polish Academy of Sciences. [99]{} M. Csanád, T. Csörgő, B. Lörstad, A. Ster, Act. Phys. Pol. B[**35**]{}, 191 (2004), nucl-th/0311102. M. Csanád, T. Csörgő and B. Lörstad, nucl-th/0310040. Z. Fodor and S. D. Katz, JHEP [**0203**]{} (2002) 014 T. Csörgő and B. Lörstad, Phys. Rev. C [**54**]{} (1996) 1390 T. Csörgő and B. Lörstad, Nucl. Phys. A [**590**]{} (1995) 465C T. Csörgő, Heavy Ion Phys.  [**15**]{} (2002) 1, hep-ph/0001233. T. Csörgő, B. Lörstad and J. Zimányi, Z. Phys. C [**71**]{} (1996) 491 T. Csörgő, F. Grassi, Y. Hama, T. Kodama, Phys. Lett. B[**565**]{}, 107 (2003) T. Csörgő, L. Csernai, Y. Hama, T. Kodama, nucl-th/0306004. T. Csörgő, nucl-th/9809011. T. Csörgő, hep-ph/0111139. T. Csörgő and J. Zimányi, Heavy Ion Phys.  [**17**]{}, 281 (2003), nucl-th/0206051. S. Chapman, P. Scotto and U. Heinz, Heavy Ion Phys.  [**1**]{} (1995) 1, hep-ph/9409349. A. Ster, T. Csörgő and J. Beier, Heavy Ion Phys.  [**10**]{} (1999) 85, hep-ph/9810341. T. S. Biró, Phys. Lett. B [**474**]{} (2000) 21 T. S. Biró, Phys. Lett. B [**487**]{} (2000) 133 S. S. Adler [*et al.*]{} \[PHENIX Collaboration\], \[arXiv:nucl-ex/0307022\]. I. G. Bearden [*et al.*]{} \[BRAHMS Collaboration\], Phys. Rev. Lett.  [**88**]{} (2002) 202301 S. S. Adler \[PHENIX Collaboration\], arXiv:nucl-ex/0401003. I. G. Bearden [*et al.*]{} \[BRAHMS Collaborations\], Phys. Lett. B [**523**]{} (2001) 227 B. B. Back [*et al.*]{} \[PHOBOS Collaboration\], Phys. Rev. Lett.  [**87**]{} (2001) 102303 K. Adcox [*et al.*]{} \[PHENIX Collaboration\], Phys. Rev. Lett.  [**88**]{} (2002) 242301 K. Adcox [*et al.*]{} \[PHENIX Collaboration\], Phys. Rev. Lett.  [**88**]{} (2002) 192302 C. Adler [*et al.*]{} \[STAR Collaboration\], Phys. Rev. Lett.  [**87**]{} (2001) 082301 Z. W. Lin, C. M. Ko and S. Pal, Phys. Rev. Lett.  [**89**]{} (2002) 152301, T. J. Humanic, Nucl. Phys. A [**715**]{} (2003) 641 F. Retiere, J. Phys. G [**30**]{} (2004) S335 F. Retiere and M. A. Lisa, arXiv:nucl-th/0312024. T. Hirano and K. Tsuda, Nucl. Phys. A [**715**]{} (2003) 821 W. Broniowski, A. Baran and W. Florkowski, AIP Conf. Proc.  [**660**]{} (2003) 185 W. Florkowski and W. Broniowski, AIP Conf. Proc.  [**660**]{} (2003) 177 W. Broniowski and W. Florkowski, Phys. Rev. Lett.  [**87**]{} (2001) 272302 T. Csörgő and A. Ster, Heavy Ion Phys.  [**17**]{} (2003) 295, nucl-th/0207016.

--- abstract: 'We investigate the problem of divergences appearing in the two-particle irreducible vertex functions of many-fermion systems with [*attractive*]{} on-site interactions. By means of dynamical mean-field theory calculations we determine the location of singularity lines in the phase diagram of the attractive Hubbard model at half-filling, where the local Bethe-Salpeter equations are non invertible. We find that divergences appear [*both*]{} in the magnetic [*and*]{} in the density scattering channels. The former affect a sector of suppressed fluctuations and are consistent with the mapping of the physical susceptibilities of the repulsive case. The appearance of singularities in the density channel demonstrate, instead, how vertex divergences can also plague the “dominant" scattering sectors associated with [*enhanced*]{} local susceptibilities, differently as observed for repulsive interactions. By introducing an insightful graphical representation of generalized susceptibilities and exploiting the underlying physical symmetries, we elucidate the relation between the two-particle vertices and the local response of the system, discussing algorithmic and physical implications of their singular behavior in the non-perturbative regime.' author: - 'D. Springer' - 'P. Chalupa' - 'S. Ciuchi' - 'G. Sangiovanni' - 'A. Toschi' bibliography: - 'VERTEX.bib' title: | Interplay between local response and vertex divergences\ in many-fermion systems with on-site attraction --- Introduction {#sec:intro} ============ The large number of degrees of freedom controlling the physics of correlated electron systems requires, in most cases, treatments based on quantum field theory (QFT), such as the Green’s function approach and the Feynman diagrammatic technique. Due to the complexity of the microscopic processes considered, the QFT formalism is mostly applied to the conceptually simplest case, i.e., at the “one-particle" level. This corresponds to the propagation of one extra electron (or hole) added to the interacting system or -more physically- to the description of (direct/inverse) photoemission experiments. The widespread application of the QFT to one-particle processes is reflected in a fully structured textbook[@Abrikosov1975; @Mahan2000] description and a clear physical interpretation of the different quantities appearing in the formalism, such as the electronic self-energy which can be accessed experimentally by ARPES experiments. However, a complete understanding of the physical response in correlated systems often[@Liu2012; @Toschi2012; @Galler2015; @Hausoel2017; @Kauch2019] requires to work at the next level of complexity, namely at the [*“two-particle level"*]{}. This represents also a fundamental prerequisite for several cutting-edge many-body schemes[@Maier2005; @Metzner2012; @Rohringer2018], which explains the increasing efforts [@Kunes2011; @Rohringer2012; @Hafermann2014; @Gunnarsson2015; @Wentzell2016; @Kaufmann2017; @Tagliavini2018; @Thunstroem2018; @Reza2019; @Krien2019SBE] for extending our knowledge in this direction. Ideally, one would like to handle the QFT description of two-particle processes at the same level of confidence we have for the one-particle ones, including a comparable understanding of their mathematical and physical properties. In this paper, we take a step in this direction by analyzing one surprising property which characterizes the two-particle analog of the self-energy, i.e. the irreducible vertex function. In particular, we refer here to the occurrence of multiple divergences displayed by this two-particle quantity, in the Matsubara frequency domain. In this respect, we recall that the self-energy expressed as function of Matsubara diverges only in the “extreme" case of a Mott-insulating phase, reflecting the complete suppression of the one-particle Green’s function. On the contrary, an ubiquitous presence of divergences in the irreducible vertex functions has been recently demonstrated in all fundamental models of many-electron physics: from the Hubbard atom[@Schaefer2016c; @Thunstroem2018] to the Falicov-Kimball model[@Janis2014; @Schaefer2016c], the Anderson Impurity model[@Chalupa2018], and the Hubbard model[@Schaefer2013; @Gunnarsson2016; @Schaefer2016c]. These divergences are a manifestation of the breakdown of self-consistent perturbation expansions in QFT and, as it was recently demonstrated, are also directly related[@Gunnarsson2017] to the intrinsic multivaluedness[@Kozik2015; @Stan2015; @Tarantino2018] of the Luttinger-Ward functional for interacting many electron systems. Mathematically, they correspond to a non-invertibility of the Bethe-Salpeter equation, through which the irreducible vertex functions are defined. The physical processes controlling these divergences are, instead, not fully clarified yet. In fact, they do [*not*]{} appear to be associated to any phase-transition in the systems considered: At low $T$, they take place well inside of the metallic, Fermi-liquid phases in the AIM[@Chalupa2018] and the dynamical mean-field theory solution of the Hubbard model[@Schaefer2013; @Schaefer2016c]. Heuristically, their occurrence has at first been related[@Schaefer2013] to the appearance of kinks in the spectral functions[@Byczuk2007] and in the specific heat[@Toschi2009; @Held2013] or to underlying non-equilibrium properties[@Eckstein2010; @Schiro2011]. A recent, more convincing interpretation[@Gunnarsson2016; @Gunnarsson2017], however, associates the vertex divergences (occurring in a given channel) to the suppression of the corresponding physical susceptibility caused by the electronic interaction. This interpretation works quite satisfactorily in all the cases studied hitherto and it can be regarded, to a good extent, as a two-particle generalization of the suppression of the Green’s function. In this paper, we study how the divergences of the irreducible vertex functions of the half-filled Hubbard model are transformed by changing the sign of the interaction from $U$ to $-U$. We will interpret our numerical calculations of two-particle susceptibilities and vertex functions, performed by means of the dynamical mean-field theory (DMFT)[@Georges1996], extending the existing mapping to treat also generalized two-particle quantities. This will allow us to relate our results to the underlying physical symmetries of the model considered, and to investigate the multifaceted aspects of “coupling” of two-particle vertex properties and their possible divergences to the behavior of the physical local response of the system. These considerations do not only improve our understanding of the physics responsible for the breakdown of the (bold) perturbation expansion[@Kozik2015], but allows us to make predictions about which kind of vertex divergences can be expected -on general grounds- in different physical situations. Beyond the conceptual progress of an improved mathematical and physical understanding of the two-particle QFT formalism, our results will be also of particular interest for the future development and applications of several cutting-edge many-electrons algorithms (e.g., as all those based on the parquet formalism[@Toschi2007; @Ayral2016], the diagrammatic Monte Carlo[@Kozik2015], the nested cluster scheme[@Vucicevic2018]) [*beyond*]{} the weak-coupling, perturbative regime. The paper is organized as follows: In Sec. II, we introduce the basic two-particle formalism needed for this study; in Sec. III, we present our numerical results for the two-particle vertex functions and their divergences in the attractive Hubbard model, as well as an interpretation of our observations, based on the mapping of the repulsive case and on the high-$T$ behavior; in Sec. IV we exploit a properly chosen graphical representation to improve the immediate physical readability of the generalized two-particle susceptibilities while in Sec. V we discuss possible physical and algorithmic implications of our findings. Our conclusions are concisely summarized in Sec. VI. Model and formalism {#sec:model} =================== In this work we will compute, by means of the dynamical mean-field-theory (DMFT)[@Georges1996], the local two-particle susceptibilities and (irreducible) vertex functions of both, the attractive and the repulsive Hubbard model, $$\begin{aligned} \label{eq:Hub} \mathcal{H} = - t \sum_{\langle i, j \rangle,\sigma} c^{\dagger}_{i,\sigma} c^{\phantom \dagger}_{j,\sigma} + U \sum_{i} n_{i,\uparrow} n_{i,\downarrow}\end{aligned}$$ where $c (c^{\dagger})$ are the annihilation (creation) fermionic operators at lattice position $i$ and spin $\sigma$, $t$ is the hopping between next-neighboring sites on a Bethe lattice (with semielliptic DOS of half-bandwidth $D= 2t =1$), and the local Hubbard interaction $U$ can take both positive (repulsive interaction) and negative (attractive interaction) values. The chemical potential is kept fixed to $\frac{U}{2}$ to preserve the particle-hole symmetry of the model. In order to extract irreducible quantities, one has to invert the Dyson equation at the one- and the Bethe-Salpeter equation (BSE) as well as the parquet equation at the two-particle level. At the one-particle level the self-energy $\Sigma(\nu)$ may be computed from the inversion $$\Sigma(\nu) = {\mathcal G}_0^{-1}(\nu) - G^{-1}(\nu), \label{eq:Dysoninv}$$ of the non-interacting Green’s function $ {\mathcal G}_0$ and the interacting impurity Green’s function $$G(\nu) = - \int_0^\beta \, d\tau \, \mbox{e}^{i\nu \tau} \, \langle T_\tau c(\tau) c^\dagger(0) \rangle$$ of the auxiliary AIM associated to the DMFT solution (here $\nu = \pi T (2n+1)$ is a fermionic Matsubara frequency). Eq.  illustrates that a divergence of $\Sigma(\nu)$ is associated to a complete suppression of $G(\nu)$, which only occurs in the Mott-insulating regime for $T , \nu \rightarrow 0$ . Though more complex, the formalism is extendable to the two-particle level[@Bickers2004; @Rohringer2012]. The analog of $\Sigma$ at the two-particle level is the irreducible vertex function $\Gamma_{r}$, given in a specific scattering channel $r$ (e.g. density, magnetic, see below). $\Gamma_r$ is obtained by inverting the corresponding BSE $$\label{eq:BSEinv} \Gamma_{r}^{\nu \nu'}(\Omega) = \beta^2 \big( [\chi_{r}^{\nu \nu'}(\Omega)]^{-1} - [ \chi_{0}^{\nu \nu'}(\Omega)]^{-1} \big) \ ,$$ where the explicit expression of the generalized susceptibility of the impurity-site reads in particle-hole notation [@Rohringer2012; @Rohringer2018] $$\begin{aligned} \label{equ:form_gen_chi} \chi_{\sigma \sigma'}^{\nu \nu'} (\Omega) &=& \int \limits_0^\beta d \tau_1 d\tau_2 d\tau_3 \, e^{-i\nu \tau_1} e^{i(\nu + \Omega)\tau_2} e^{-i(\nu' + \Omega)\tau_3} \nonumber \\ &\times & [ \langle T_{\tau} c_{\sigma}^{\dagger} (\tau_1) c_{\sigma}^{\phantom \dagger}(\tau_2) c_{\sigma'}^{\dagger}(\tau_3) c_{\sigma'}^{\phantom \dagger}(0) \rangle \\ & -& \langle T_{\tau} c_{\sigma}^{\dagger} (\tau_1) c_{\sigma}^{\phantom \dagger}(\tau_2) \rangle \langle T_{\tau} c_{\sigma'}^{\dagger}(\tau_3) c_{\sigma'}^{\phantom \dagger}(0) \rangle \nonumber] \ .\end{aligned}$$ Here, $\sigma$ and $\sigma'$ denote the spin directions of the impurity electrons, and $\nu$, $\nu'$ and $\Omega$ represent two fermionic and one bosonic Matsubara frequency, respectively. $\chi_{0}^{\nu \nu'}$ corresponds to the bare bubble given by $-\beta G(\nu) G(\nu + \Omega) \delta_{\nu \nu'}$. In the case of SU(2) symmetry, the BSE can be diagonalized in the spin sector defining the density ($r= d$) and magnetic ($r=m$) channel: \[$\chi_{d[m]}^{\nu \nu'} (\Omega) = \chi_{\uparrow \uparrow}^{\nu \nu'} (\Omega) +[-] \chi_{\uparrow \downarrow}^{\nu \nu' } (\Omega)$\]. Similar considerations apply to the particle-particle ($pp$)-sector: the expression of the generalized susceptibilities in the corresponding ($pp$) notation can be obtained[@Rohringer2012; @Rohringer2013a] via a frequency shift of the particle-hole expressions $\chi_{pp,\uparrow\downarrow}^{\nu\nu'}(\Omega) = \chi_{\uparrow\downarrow}^{\nu\nu'}(\Omega -\nu-\nu')$. [^1] At the two-particle level the inversion of $\chi_r^{\nu \nu'}$ in Eq. \[eq:BSEinv\] written in terms of its eigenvalue decomposition takes the form$$\label{eq:InvChiEVSpectrum} [\chi_{r}^{\nu \nu'}]^{-1} = \sum_\ell V^r_\ell(\nu)^* [\lambda^r_\ell]^{-1} V^r_\ell(\nu') \ .$$ Similar to the one-particle level, where $\Sigma(\nu)\!\rightarrow\!\infty$ directly corresponds to a zero of $G(\nu)$, a divergence of the “two-particle" self-energy, the irreducible vertex $\Gamma_r^{\nu \nu'}$, is related to a vanishing eigenvalue $\lambda_\ell^r$ in Eq. . Note, that this is merely an analogy, since a single vanishing eigenvalue $\lambda_\ell^r$ does not imply a vanishing of the whole $\chi_r^{\nu \nu'}$ matrix. Hence, a divergence of $\Gamma_r$ does not cause the corresponding static ($\Omega=0$) physical susceptibility $$\chi_{r}= \frac{1}{\beta^2} \sum_{\nu, \nu'} \, \chi_{r}^{\nu \nu'}(\Omega=0) \label{eq:chiphys}$$ to vanish as well. However, after crossing a divergence line the corresponding eigenvalue $\lambda_\ell^r$ becomes negative, resulting in a [*negative*]{} contribution in the eigenvalue decomposition of the physical susceptibility $$\label{eq:ChiSpectrum} \chi_{r}= \sum_\ell \lambda^r_\ell | \sum_\nu V^r_\ell(\nu) |^2 \ .$$ eventually causing a [*progressive*]{} suppression of the physical fluctuations in the respective channel. Therefore, a divergence of $\Gamma_r^{\nu \nu'}$ followed by the presence of a negative eigenvalue in $\chi_r^{\nu \nu'}$ can be interpreted as the analog of the suppression of the single-particle Green’s function by the single-particle self-energy [@Gunnarsson2016; @Gunnarsson2017]. Indeed, in all previous studies of models with [*repulsive*]{} interactions, negative eigenvalues have exclusively occurred in physical channels that are [*suppressed*]{} upon increasing the interaction strength $U$, namely in the charge and in the particle-particle sectors. According to this observation, one may expect that vertex divergences in models with [*attractive*]{} interaction will occur in the [*magnetic*]{} channel only. This would heuristically be consistent with the known “mapping" of the physical degrees of freedom (D.o.F.) of the half-filled Hubbard model. Due to the intrinsic $O(4) = SU(2) \times SU(2)$ symmetry, the partial particle-hole, or Shiba, transformation[@Shiba1972; @Micnas1990] $$c_{i \uparrow} \rightarrow c_{i \uparrow} \ \ \text{and} \ \ c_{i\downarrow} \rightarrow (-1)^{i} c_{i \downarrow}^\dagger \label{eq:shiba}$$ acts as a mapping of all physical observables between $U>0$ and $U<0$. In particular, the two SU(2) spin ($\vec{S}$) and pseudospin ($\vec{S}_p$) sectors, which are related to the respective suppressed channels on the attractive and repulsive side, are transformed into each other $$\begin{aligned} \mathcal{S}_x &=& \frac{1}{2}[c_\uparrow^\dagger c_\downarrow + c_\downarrow^\dagger c_\uparrow] \leftrightarrow -\frac{1}{2}[c_\uparrow^\dagger c_\downarrow^\dagger + c_\downarrow c_\uparrow] = \mathcal{S}_{p,x} \nonumber \\ \mathcal{S}_y &=& \frac{i}{2}[c_\uparrow^\dagger c_\downarrow - c_\downarrow^\dagger c_\uparrow] \leftrightarrow \ \ \frac{i}{2}[c_\uparrow^\dagger c_\downarrow^\dagger - c_\downarrow c_\uparrow] = \mathcal{S}_{p,y} \label{Eq:SpinMap} \\ \mathcal{S}_z &=& \frac{1}{2}[c_\uparrow^\dagger c_\uparrow - c_\downarrow^\dagger c_\downarrow] \leftrightarrow \frac{1}{2}[c_\uparrow^\dagger c_\uparrow + c_\downarrow^\dagger c_\downarrow - 1] = \mathcal{S}_{p,z} \ . \nonumber \end{aligned}$$ This mapping of physical D.o.F. suggests that a similar “transformation" may as well apply to the vertex-divergences. However, as already noted in [@Rohringer2012; @DelRe2019], the mapping of [*generalized*]{} two-particle quantities, and especially of dynamical irreducible vertices, is more complex than Eq.  would imply. We will see in the next section, how this is reflected in the appearance and the nature of the vertex divergences in attractive Hubbard model. {width="85.00000%"} Vertex divergences of the attractive Hubbard model {#sec:phasediag} ================================================== DMFT results {#subsec:DMFTResults} ------------ We start our analysis of the vertex functions and their divergences in the attractive Hubbard model by presenting our DMFT calculations at the two-particle level[^2] performed with a continuous time quantum Monte Carlo (CTQMC) impurity solver in the hybridization expansion[@Gull2011a], precisely, the *w*$2$*dynamics*-package [@w2dynamics]. The main outcome of our DMFT calculations are summarized in Fig. \[Fig:1\], where we report the location of the vertex divergences found for different values of the local attraction $U <0$ and the temperature $T$ (left side), compared against the corresponding results for the repulsive case $U >0$ (right side). In the large $|U|$ regime our numerical results are consistent with analytical calculations[@Thunstroem2018] in the atomic limit. Furthermore, in the whole repulsive sector, we also reproduce [^3] the outcome of previous DMFT studies[@Schaefer2013; @Schaefer2016c], finding multiple lines in the $U$-$T$ plane, where the irreducible vertex diverges. As already observed[@Schaefer2016c], the first divergences are located at moderate repulsion values, well before the Mott-Hubbard MIT. With increasing interaction the occurrence of divergence lines becomes more dense, and the lines occur in alternating order starting with a divergence in the density channel (red lines) followed by a simultaneous divergence in the density and $pp$ channel (orange lines). In the case of [*attractive*]{} interaction, our DMFT results show the following: We find vertex divergences in the density channel ([*red*]{} lines), which are perfectly mirrored with respect to the repulsive side. These occur in alternating order with lines of divergences in the magnetic channel ([*green*]{} lines), which mirror, instead, the [*orange*]{} divergence lines of the repulsive model. As a consequence, the overall location of the vertex divergences looks highly symmetric when comparing the repulsive and the attractive sides of the phase diagram. At first sight this symmetry may appear rather surprising, because the physical properties of a given scattering channel in the repulsive and the attractive model are very different[@Micnas1990; @Taranto2012; @Tagliavini2016], as dictated by the mapping of the physical degrees of freedom (cf. Eq.  and Fig. \[Fig:1\]). At a closer look, we can distinguish the situation of the three-times degenerate divergences found at the orange and green lines, respectively, from that of the single degenerate divergences found at the red lines, occurring in the density sector only. Specifically, the mapping of the combined divergences in the $pp$ and density sector (orange lines) into divergences of the magnetic sector (green lines) is fully matching our physical expectations that (i) divergences play a role in the suppression of a scattering channel and that (ii) they are mapped consistently with the physical D.o.F., i.e. according Eq. . At the same time, the perfect mirroring of the [*density*]{} divergence lines (red) under the $U \leftrightarrow - U$ transformation looks puzzling, because (i) for $U < 0$, these divergences affect a scattering channel associated to a physical susceptibility, which is not suppressed but [*enhanced*]{} by the attractive interaction, and (ii) the physical degrees of freedom associated to the density channel is mapped onto one of the three spin-components. A first understanding of this apparent discrepancy is provided by the analysis of the symmetry of the eigenvectors associated to a vanishing eigenvalue ($\lambda^r_{\alpha} =0 ~ \text{for} ~ \ell=\alpha$ in Eq. (\[eq:InvChiEVSpectrum\])). In Figure \[Fig:2\] we compare the shape of eigenvectors following the first and second divergence lines at different temperatures for $U\lessgtr0$. Evidently, the perfect mirroring of divergence lines is also reflected in identical shapes of the eigenvectors associated to a vanishing eigenvalues for $U\lessgtr0$. The singular eigenvectors associated to [*all*]{} divergences in the density sector only (red lines), display a completely [*antisymmetric*]{} frequency structure \[$V_\ell(-\nu) \! = \! - V_\ell(\nu)$\]. In contrast, all other divergence lines (green and orange lines) are associated to frequency [*symmetric*]{} singular eigenvectors \[$V_\ell(-\nu) \! = \! V_\ell(\nu)$\]. The symmetry of eigenvectors is clearly essential in the calculation of the physical susceptibility, as can be seen quickly in Eq.. Due to the summation over Matsubara frequencies, the actual value of $\chi_r$ is [*independent*]{} from any antisymmetric eigenvector, irrespective of whether associated to a positive or a negative eigenvalue. Hence, the appearance of negative eigenvalues in a channel is [*not*]{} necessarily associated to a suppression of the respective physical susceptibility. While in the [*repulsive*]{} model the occurrence of divergences and the suppression of the respective channel, maybe incidentally, coincide, our calculations of the [*attractive*]{} model provide a clear-cut counter-example: the crossing of several divergence lines in the density sector is accompanied by an [*enhanced*]{} susceptibility. To rationalize the results of our two-particle DMFT calculations on more general grounds, we investigate the effect of the attractive-repulsive mapping on generalized two-particle quantities and its relation the physical symmetries of the system under consideration. ![ Comparison of the singular eigenvectors $V^r_\alpha$ in the repulsive and the attractive case, plotted as a function of the Matsubara index $N=\nu \frac{\beta}{\pi}$. The upper \[lower\] panel shows perfectly identical singular antisymmetric \[symmetric\] eigenvectors located at different temperatures along the first \[second\] attractive (left) and first \[second\] repulsive (right) divergence line.[]{data-label="Fig:2"}](DMFT_F-infty_Attractive_Eigenvectors_FirstLine_Density_final "fig:"){width="23.00000%"} ![ Comparison of the singular eigenvectors $V^r_\alpha$ in the repulsive and the attractive case, plotted as a function of the Matsubara index $N=\nu \frac{\beta}{\pi}$. The upper \[lower\] panel shows perfectly identical singular antisymmetric \[symmetric\] eigenvectors located at different temperatures along the first \[second\] attractive (left) and first \[second\] repulsive (right) divergence line.[]{data-label="Fig:2"}](DMFT_F-infty_Repulsive_Eigenvectors_FirstLine_Density_final "fig:"){width="23.00000%"} ![ Comparison of the singular eigenvectors $V^r_\alpha$ in the repulsive and the attractive case, plotted as a function of the Matsubara index $N=\nu \frac{\beta}{\pi}$. The upper \[lower\] panel shows perfectly identical singular antisymmetric \[symmetric\] eigenvectors located at different temperatures along the first \[second\] attractive (left) and first \[second\] repulsive (right) divergence line.[]{data-label="Fig:2"}](DMFT_F-infty_Attractive_Eigenvectors_SecondLine_Magnetic_final "fig:"){width="23.00000%"} ![ Comparison of the singular eigenvectors $V^r_\alpha$ in the repulsive and the attractive case, plotted as a function of the Matsubara index $N=\nu \frac{\beta}{\pi}$. The upper \[lower\] panel shows perfectly identical singular antisymmetric \[symmetric\] eigenvectors located at different temperatures along the first \[second\] attractive (left) and first \[second\] repulsive (right) divergence line.[]{data-label="Fig:2"}](DMFT_F-infty_Repulsive_Eigenvectors_SecondLine_PP_final "fig:"){width="23.00000%"} The role of the underlying symmetries {#subsec:symmetries} ------------------------------------- As mentioned at the end of Sec.\[sec:model\], the mapping of the [*generalized*]{} two-particle quantities is less obvious than the mapping of the physical D.o.F.. When considering purely local quantities, the single-particle Green’s function $G(\tau_1,\tau_2)$ is identical for the repulsive ($U>0$) and attractive ($U<0$) half-filled model, while the two-particle Green’s function $G_{\uparrow \downarrow}(\tau_1,\tau_2,\tau_3,\tau_4)$, i.e. the first time-ordered product appearing on the right hand side of Eq., with anti-parallel spin orientation transforms[@Rohringer2012; @Rohringer2013a] according to $$G^{(U)}_{\uparrow \downarrow}(\tau_1,\tau_2,\tau_3,\tau_4) = -G^{(-U)}_{\uparrow \downarrow}(\tau_1,\tau_2,\tau_4,\tau_3),$$ which, after Fourier transformation of all fermionic variables, reads $$G^{(U)}_{\uparrow \downarrow}(\nu_1,\nu_2,\nu_3) = -G_{\uparrow \downarrow}^{(-U)}(\nu_1,\nu_2,-\nu_4)$$ with $\nu_4 = \nu_1 - \nu_2 + \nu_3$. After changing to the $ph$-notation, as defined in Eq. , ($\nu_1=\nu$, $\nu_2=\nu+\Omega$, $\nu_3=\nu'+\Omega$, $\nu_4=\nu'$) one can easily see how the transformation maps the generalized static ($\Omega=0$) susceptibility, $\chi^{\nu,\nu'}_{\uparrow \downarrow}= G_{\uparrow \downarrow}(\nu, \nu, \nu')$ of the $\uparrow\downarrow$ sector according to $$\label{eq:mapping_chi_U} \chi^{\nu \nu'}_{\uparrow \downarrow}\, \overset{U\leftrightarrow(-U)}{\Longleftrightarrow} \, -\chi^{\nu (-\nu')}_{\uparrow \downarrow},$$ while $\chi_{\uparrow \uparrow}$ is obviously invariant under a partial particle-hole transformation. Hence, in general, the Shiba transformation on the two-particle level will [*mix*]{} the different (particle-hole) channels of generalized susceptibilities and the associated irreducible vertices. In particular, one sees that [*only*]{} the mapping of the generalized susceptibility expressed in the $pp$ notation $$\chi^{\nu (-\nu')}_{pp,\uparrow\downarrow} - \chi_{0,pp}^{\nu \nu'} \, \overset{U\leftrightarrow(-U)}{\Longleftrightarrow} \, \chi^{\nu \nu'}_{m} \label{eq:mapping_chi_pp}$$ reflects[@Rohringer2012; @Rohringer2013a; @DelRe2019] the transformation of the physical (spin/pseudospin) degrees of freedoms, discussed in Eq., in a direct fashion. As the location of divergence lines is directly encoded in the generalized susceptibilities, it will be also subject to the mixing of channels, explaining the differences w.r.t. the mapping of the physical degrees of freedom, discussed in Sec.\[sec:model\]. To fully rationalize the results observed in Sec.\[subsec:DMFTResults\], we will focus on the symmetry properties of the generalized susceptibilities. In this respect we note, that Eq. (\[eq:mapping\_chi\_pp\]) shows already why the mirrored divergences of the particle-particle $\uparrow\downarrow$ channel for $U>0$ are observed in the magnetic channel for $U<0$. Hence, the main question concerns the behavior of the particle-hole channels. We start by considering the (spin resolved) generalized susceptibility $\chi^{\nu \nu' \Omega}_{\sigma \sigma'}$, as defined in Eq. (\[equ:form\_gen\_chi\]). Due to the particle-hole (PH) symmetry of the system considered here, $\chi^{\nu \nu' \Omega}_{\sigma \sigma'}$ has all real entries. Exploiting the time-reversal (TR)- and the SU(2)-symmetry of the problem $$\begin{aligned} \big( \chi_{\sigma\sigma'}^{\nu \nu' \Omega} \big)^* \overset{\scalebox{0.6}{PH}}{=} \chi_{\sigma\sigma'}^{\nu \nu' \Omega } \overset{\scalebox{0.6}{TR}}{=} \chi_{\sigma'\sigma}^{\nu'\nu \Omega} \overset{\scalebox{0.6}{SU(2)}}{=} \chi_{\sigma\sigma'}^{\nu' \nu \Omega} \ . \label{eq:chisym1}\end{aligned}$$ it is evident that $\chi^{\nu \nu' \Omega}_{\sigma \sigma'}$ is a symmetric matrix of $\nu$ and $\nu'$. Relation ensures that all matrix entries and all eigenvalues remain real for any $\Omega$. Another symmetry relation can be obtained by exploiting the complex conjugation (CC) of $\chi^{\nu \nu' \Omega}_{\sigma \sigma'}$. For $\Omega\!=\!0$ it can be shown that the generalized susceptibility is invariant under the rotation of the matrix along both of its cardinal axes ($\nu\!\rightarrow\!-\nu, \nu'\!\rightarrow\!-\nu'$) $$\begin{aligned} \label{eq:symm_chi} \chi_{\sigma\sigma'}^{\nu \nu'} \overset{\scalebox{0.6}{PH}}{=} \big( \chi_{\sigma\sigma'}^{\nu \nu'} \big)^* &\overset{\scalebox{0.6}{CC}}{=}& {\chi_{\sigma'\sigma}^{(-\nu') (-\nu)} } \overset{\scalebox{0.6}{TR}}{=} {\chi_{\sigma\sigma'}^{(-\nu) (-\nu')}} \ .\end{aligned}$$ A matrix obeying the conditions $$\begin{aligned} \label{eq:centrosymm_chi} \chi_{\sigma\sigma'}^{\nu \nu'} = \chi_{\sigma\sigma'}^{(-\nu) (-\nu')} \quad \text{and} \quad \chi_{\sigma\sigma'}^{\nu \nu' } = \chi_{\sigma\sigma'}^{\nu' \nu} \end{aligned}$$ is a so-called [*bisymmetric*]{} matrix, where the matrix elements are symmetric with respect to [*both*]{} the main diagonal ($\nu = \nu'$) as well as the secondary diagonal ($\nu=-\nu'$). Essentially, this particular symmetry is at the core to understand the mapping of divergence lines. A bisymmetric matrix can always be diagonalized in blocks (here associated to positive/negative Matsubara frequencies), by applying an orthogonal matrix $Q$, defined in terms of the counteridentity $(J^{\nu,\nu'}\!=\!\delta_{\nu (-\nu')}$) and identity submatrix $\mathbb{1}$ (see Appendix \[Appendix:BlockDiag\] for more details) $$Q=\frac{1}{\sqrt{2}} \begin{pmatrix} \mathbb{1} & -J \\ \mathbb{1} & J \end{pmatrix} \quad , \quad Q\chi_rQ^T = \begin{pmatrix} \text{A} & \vline & 0 \\ \hline 0 & \vline & \text{S} \\ \end{pmatrix} \ . \label{Eq:Subspaces}$$ The block-diagonalization of $\chi_r$ is associated with precise symmetry properties: the subspace denoted by A represents a submatrix with exclusively antisymmetric eigenvectors, while S is the subspace of purely symmetric eigenvectors. As a consequence, one can attribute, unambiguously, the occurrence of a red divergence line in $\chi_d$ to the purely antisymmetric subspace A, while all other divergence lines will be accounted for by the symmetric subspace S. {width="43.00000%" height="33.00000%"} {width="43.00000%" height="33.00000%"} A crucial ingredient for connecting the bisymmetry of the generalized susceptibilities to the mapping of divergence lines lies in the equivalence of the Shiba transformation for $\chi^{\nu\nu'}_{\uparrow \downarrow}$ to a matrix multiplication with the negative counteridentity matrix $(-J)$ $$\begin{aligned} \label{eq:mapping_chi_U} \chi^{\nu\nu'}_{\uparrow \downarrow,(U)}(-J) = -\chi^{\nu(-\nu')}_{\uparrow \downarrow,(U)} = \chi^{\nu\nu'}_{\uparrow \downarrow,(-U)} \ . \label{eq:JMap}\end{aligned}$$ Combining Eq. , and the fact that $J^2=\mathbb{1}$ one can prove (see Appendix \[App:equality\]) the remarkable result, that the antisymmetric sector A remains [*invariant*]{} under $U\!\leftrightarrow\!-U$ for all $\chi_r$. This explains why the red divergence lines ($\chi_d$) and their associated antisymmetric eigenvectors are perfectly mirrored on both sides of the phase-diagram in Fig.\[Fig:1\] and Fig. \[Fig:2\]. At the same time one finds that the symmetric parts (S) of $\chi_d$ and $\chi_m$ are [*mapped into one-another*]{} for $U\!\leftrightarrow\!-U$, therefore connecting the symmetric divergences, and the corresponding eigenvectors, appearing in $\chi_d^{U>0}$ (orange) and in $\chi_m^{U<0}$ (green). Let us stress at this point that the proof made in Appendix \[App:equality\] applies not only to singular eigenvalues, which are connected to divergence lines, but to [*all*]{} eigenvalues and eigenvectors of $\chi^{\nu\nu'}_{r}$. In this way, we have extended the mapping relation known for $\chi^{\nu\nu'}_{pp,\uparrow\downarrow}$ to the whole particle-hole sector, clarifying the relation with the mapping of the physical D.o.F.: the antisymmetric subspace A, not contributing to the sum for the physical susceptibility in Eq., is [*invariant*]{} under the Shiba transformation, while the symmetric subspace is found to transform in accordance with Eq.. As we have illustrated, the particle-hole symmetry plays a central role in determining the mirroring properties of the generalized susceptibilities. If one relaxes this constraint, the relations in Eq.  no longer hold in the particle-hole sector and, therefore, the bisymmetry is lost and eigenvalues are not necessarily real. This implies, in turn, that the eigenvectors of the corresponding $\chi_r$ are not necessarily symmetric or antisymmetric any longer. At the same time, it is important to stress, that even in the absence of PH-symmetry (e.g. out of half-filling) $\chi^{\nu \nu'}_{pp, \uparrow\downarrow}$ continues to fulfill[@Thunstroem2018] both relations in Eq. , ensuring the validity of all associated properties (i.e. real eigenvalues as well as bisymmetry and associated properties). High-Temperature Limit ---------------------- To exemplify the concepts discussed in the previous section, we performed DMFT calculations in the high-temperature regime ($\beta\!=\!5$), where the frequency structure of the two-particle generalized susceptibilities strongly simplifies. Due to the large stepsize on the Matsubara frequency grid, most information of the system is encoded in the central $2\times2$ matrix. The analysis of the divergences can be then restricted[@Gunnarsson2016] to the innermost $2\! \times \! 2 $ matrix defined by the smallest Matsubara frequecies ($\nu, \nu'= - \frac{\pi}{\beta}, \frac{\pi}{\beta}$). For a $2 \times 2$ case, the bisymmetry condition (see III B) poses significant constraints on the matrix elements and a singularity can be realized only in two ways: $$\begin{aligned} \chi_r^{\lambda_A=0}&=& \left(\begin{array}{cccc} \colorbox{red!15}{\color{black!100}{a}} & \colorbox{red!15}{\color{black!100}{a}} \\ \colorbox{red!15}{\color{black!100}{a}} & \colorbox{red!15}{\color{black!100}{a}} \\ \end{array}\right) \label{Eq:Centro1}\end{aligned}$$ which corresponds to the (anti-symmetric) singular eigenvector $V_A (\nu) \propto \, \delta_{\nu,\frac{\pi}{\beta}} - \delta_{\nu, -\frac{\pi}{\beta}}$, and $$\begin{aligned} \chi_r^{\lambda_S=0}&=& \left(\begin{array}{cccc} \colorbox{blue!15}{\color{black}{$\mp b$}} & \colorbox{red!15}{\color{black}{$\pm b$}} \\ \colorbox{red!15}{\color{black}{$\pm b$}} & \colorbox{blue!15}{\color{black}{$\mp b$}} \\ \end{array}\right) \scriptsize \normalsize \label{Eq:Centro2}\end{aligned}$$ with $a, b> 0$, corresponding to a (symmetric) singular eigenvector $V_S (\nu) \propto \, \delta_{\nu,\frac{\pi}{\beta}} + \delta_{\nu, -\frac{\pi}{\beta}}$. On the basis of these considerations, we analyze the $U$-dependence of the diagonal ($\chi_r^D$) and off-diagonal ($\chi_r^O$) elements of the $2\!\times\!2$ lowest frequency-submatrix of the generalized susceptibility, extending the study of Ref. \[\] to the attractive case. The corresponding data are reported in Fig. \[Fig:MatElTrendb5\] for the density (left) and the magnetic/pp sectors (right). A general trend can be readily identified: Upon increasing $|U|$ all diagonal matrix elements ($\chi_r^D$) eventually [*decrease*]{}, while the off-diagonal elements ($\chi_r^O$) mostly [*increase*]{} in absolute values, for the considered interaction regime. The decrease of $\chi_r^D$ upon increasing $|U|$ is dominated by the bubble term ($\propto - \beta \,G(\nu)G(\nu') \, \delta_{\nu \nu'}$), reflecting the suppression of the single particle Green’s function $G(\nu)$ at low-frequencies. Vertex corrections are responsible for the asymmetry of the damping effects on $\chi_r^D$ with respect to $\pm U$ as well as for differentiating its size between the different sectors. In particular, we find the following behavior for the [*diagonal*]{} entries: (i) the decrease-rate with $|U|$ of $\chi_r^D$ is stronger in those channels that correspond to a suppressed susceptibility, (ii) $\chi_d^D$ decreases faster compared to the other two channels and even turns negative for large $U>0$, where density fluctuations are suppressed. The [*off-diagonal*]{} matrix elements are obviously zero in the non-interacting case ($U=0$) and for small values of $U$ yield positive/negative corrections to the enhanced/suppressed susceptibilities. For large $U$ values this behavior is preserved in the $m$ and $pp$ channel. An exception is the suppressed density channel where $\chi_d^O$ displays a strong increase, becoming positive again. From these observations, we conclude that the suppression/enhancement of a static physical susceptibility is controlled by the interplay of suppressed diagonal entries and the enhanced magnitude of the (positive/negative) off-diagonal terms. Due to the considerably milder damping of the diagonal entries in the magnetic and the $pp$ sector, one always finds that $\chi_r^D > \chi_r^O$, with $r=m,pp$. Therefore only singularities of the second kind ($\chi_r^D = - \chi_r^O$, s. Eq. (\[Eq:Centro2\])) can occur in these channels. This implies that singularities of the second kind can occur exclusively in sectors of suppressed susceptibilities. On the contrary, the much stronger damping of $\chi_d^D$ plays a crucial role in suppressing the density fluctuations for $U>0$. For $U<0$ this decrease of $\chi_d^D$ is outperformed by an even stronger increase of $\chi_d^O$ in order to describe the corresponding enhancement of $\chi_d$. As one can easily see in Fig. \[Fig:MatElTrendb5\], these conditions allow divergences of the first kind with $\chi_d^D = \chi_d^O$ (compare Eq. ), to occur specularly on [*both*]{} sides of the phase-diagram. In fact, frequency-antisymmetric divergences are the only one to be expected in sectors of enhanced physical susceptibilities, because in this regime, [*both*]{} diagonal and off-diagonal components of $\chi_r^{\nu,\nu'}$ have the same (positive) sign. Spectral representations of physical susceptibilities {#sec:spectral} ===================================================== The relation between generalized and physical susceptibilities emerging from our numerical and analytical analysis can be illustrated in a physically more insightful way. As all eigenvalues in Eq. (\[eq:ChiSpectrum\]) are real, we introduce a [*susceptibility density*]{} ($\rho(\chi)$) defined as $$\rho_r(\chi) = \sum_\ell \left\vert \sum_{\nu} V^r_\ell(\nu) \right\vert^2 \delta(\chi - \lambda^r_\ell) \geq 0 \label{e1}$$ from which the local physical susceptibility is readily obtained as an average over $\rho_r(\chi)$: $$\langle \chi_r \rangle = \int \chi \, \, \rho_r(\chi) \, d\chi \ . \label{eq2}$$ This representation has several advantages: Equations (\[e1\]) and (\[eq2\]) enable to distinguish immediately between positive ($\lambda^\ell>0, \rho(\lambda^\ell)>0$), negative ($\lambda^\ell<0, \rho(\lambda^\ell)>0$) and vanishing ($\lambda^\ell=0$ or $\rho(\lambda^\ell)=0$) contributions to the static response $\chi_r$. Further, its graphical conciseness will allow to comprehend, at a single glance, how the mapping of the generalized susceptibilities works for the different cases, highlighting the most relevant physical implications. {width="32.00000%"} {width="32.00000%"} {width="32.00000%"} The introduced representation is applied here to analyze our susceptibility data after crossing four divergence lines at two mirrored positions in the phase diagram (light-blue stars in Fig.\[Fig:1\]). The corresponding results are shown in the three plots of Fig. \[Fig:Densities\], representing the three scattering channels. Here, the positions of all eigenvalues $\lambda_\ell^r$ are shown as bars in the light-blue shaded innermost panels of the three plots: [*Gray bars*]{} indicate eigenvalues associated to antisymmetric eigenvectors and thus to a vanishing $\rho_r$ which does [*not*]{} contribute to $\chi_r$, [*Colored bars*]{} account for eigenvalues associated to finite $\rho_r(\lambda_\ell^r)$ values, corresponding to symmetric eigenvectors whose weighted sum builds up the full $\chi_r$. The actual value of the susceptibility-density $\rho_r$ for a given eigenvalue is indicated by the circle-symbols in the outermost panels of the plots in Fig. \[Fig:Densities\]. The color-shaded regions slightly above $\chi\!\sim \! 0$ represent an increasingly denser distribution of small positive eigenvalues, arising from the high-frequency behavior of $\chi_r^{\nu \nu} \propto \, \frac{1}{\nu^2} \, \delta^{\nu\nu'}$. It can be shown that this (essentially non-interacting) large-$\nu$ feature induces a van Hove singularity in the $T \rightarrow 0$ behavior of $\rho_r(\chi) \simeq 1/\chi^{-3/2}$ for $\chi \rightarrow 0$ (See Appendix \[App:BM\]). The three plots of Fig. \[Fig:Densities\] graphically combine [*all*]{} aspects of the attractive-repulsive mapping of the generalized susceptibilities and allow a comprehensive understanding at a single glance. The location of the colored bars together with the corresponding values of $\rho_r(\chi)$ are transformed fully consistently with the mapping of the physical D.o.F.. In accordance with our results in Sec. IIIB, not only the physical susceptibility, but the entire distribution $\rho_r(\chi)$ of the identical density and $pp$ (pseudospin) sectors are mapped onto the magnetic (spin) sector and vice versa. On the contrary, the positions of the gray bars of each channel are [*unchanged*]{} in the $+U$ and $-U$ cases, reflecting the perfect [*invariance*]{} of the antisymmetric subspaces of all generalized $\chi_r$ under the mapping. We note that the identical location of the gray bars in the magnetic and the $pp$ channel reflects the fact that the [*entire*]{} generalized susceptibility sectors are transformed exactly as the physical degrees of freedom (compare Eq. ). On the other hand, the different locations of gray bars in the density sector compared to the other channels explain the non-trivial mapping properties of $\chi_{d}^{\nu\nu'}$ and of the corresponding irreducible vertices. These general observations allow for a remarkable rationalization of the problem. Any suppressed local physical susceptibility can be associated to a [*unique*]{} susceptibility-density $$\rho_{sup} (U)\!=\!\rho_{m}^{U<0}\!=\!\rho_{d}^{U>0}\!=\!\rho_{pp}^{U>0} \ . \label{eq:rhosup}$$ Obviously, by replacing $U$ with $-U$ in Eq. , a similar property holds for all enhanced susceptibility densities $$\rho_{enh}\!(U)= \rho_{sup}(-U) =\!\rho_{m}^{U>0}\!=\!\rho_{d}^{U<0}\!=\!\rho_{pp}^{U<0} \ . \label{eq:rhoenh}$$ The comparison of the attractive and repulsive panels of each channel in Fig. \[Fig:Densities\] indicates as an overall trend, that the suppression of a susceptibility is associated to a systematic shift of the [*colored*]{} bars towards smaller values, as well as with a change of the weight distribution, where the highest values of $\rho_{sup}$ are associated with the lowest eigenvalues. This supports the physical picture that an interaction-driven suppression of a static local susceptibility is connected to an increasing number of negative eigenvalues and therefore with the crossing of multiple vertex divergences. This corresponds to a loose generalization of the self-energy behavior at the 2P level, as discussed in Sec. \[sec:model\]. At the same time, this demonstrates why the “reverse" implication of the above physical picture is not correct. The perfect [*invariance*]{} of the gray bars under the mapping, whose physical content is totally decoupled from the static susceptibility, implies the perfect mirroring of all red-lines where only the density channel is singular (Fig. \[Fig:1\]). Hence, the occurrence of red divergence lines is [*independent*]{} of the behavior of the corresponding susceptibility as well as of the SU(2)$\times$SU(2) symmetry properties of the model considered. Finally, important quantitative information can be also gained from Fig. \[Fig:Densities\]. By analyzing the behavior of the enhanced susceptibilities, it is evident that $\rho_{enh}$ is dominated by the contribution of a [*single*]{} term: the one associated to [*largest*]{} eigenvalue $\lambda^{max}$. This property is illustrated in Fig. \[Fig:ChargeTrend\], where we compare the actual values of $\chi_d $ and $\chi_m$ obtained from Eq.  with the case where the summation in Eq.  is reduced to the largest eigenvalue only. The contribution from the largest eigenvalue $\lambda_{max}$ very well reproduces the trend across the entire repulsive and attractive regime and even well approximates the actual value of the static susceptibilities $ \chi_d$ and $\chi_m$ in their respective enhanced regions. Since the relation $V_{enh}^{max}=V_m^{max}=V_{d}^{max}=V_{pp}^{max}$ follows from the proof in Appendix \[App:equality\] and Eq. \[eq:mapping\_chi\_pp\], the value of [*all*]{} physical susceptibilities in their respective enhanced regions can be well approximated by $$\langle \chi_{r} \rangle \sim \lambda^{max} \left\vert \sum_{\nu} V^{max}_{enh}(\nu) \right\vert^2\ .$$ According to this relation, the Curie-Weiss behavior of any static local susceptibility in the strong-coupling regime can be ascribed to the evolution of the corresponding $\lambda^{max}$ and the associated eigenvector. ![Comparison of the static density $\chi_d(\Omega=0)$ (red) and magnetic $\chi_m(\Omega=0)$ (green) susceptibility with the contribution of the largest eigenvalue only, as a function of the attractive/repulsive Hubbard interaction $U$ at $T=0.2$. In the bottom of the plot, the lowest eigenvalue of $\chi_d^{\nu \nu'}$ is shown. The evolution of the lowest eigenvalues ($\lambda^{min}_d$, in dark gray) is completely decoupled from the behaviour of the static susceptibility.[]{data-label="Fig:ChargeTrend"}](DMFT_F-infty_RepAtt_Trends_Eigenvalue_Contributions_Charge){width="45.00000%"} Physical and algorithmic consequences {#sec:consequences} ===================================== The numerical and analytical results of the previous sections allow us to draw some relevant conclusions on algorithmic and physics implications of vertex divergences. As we have seen, [*only*]{} divergences associated to symmetric singular eigenvectors reflect an interaction-driven suppression of the corresponding static susceptibility. Our DMFT study of the attractive Hubbard model provides, indeed, a clear example where vertex divergences associated with antisymmetric singular eigenvectors [*do* ]{} affect also the [*dominant*]{} scattering channel. This observation has direct implications for the usage of parquet-based schemes in the non perturbative regime, such as, e.g., D$\Gamma$A[@Toschi2007] and QUADRILEX[@Ayral2016]. In fact, if the occurrence of vertex divergences could be completely confined to the secondary scattering channels, with suppressed scattering and fluctuations, their appearance could be exploited as an useful “indicator" that this channel can be safely neglected. This would considerably simplify the parquet treatment of the problem under investigation (e.g. reducing the parquet treatment to some effective BSE-based algorithm). Evidently, the occurrence of divergences in the dominant channels prevents a straightforward implementation of this idea. Hence, other ways to address this problem should be followed, such as the combination of fRG and DMFT, ($\text{DMF}^2\text{RG}$) [@Taranto2014] or the single-boson exchange (SBE) approach[@Krien2019SBE]. At the same time, the antisymmetric nature of the divergences occurring in the dominant channels will not hinder the applicability of post-processing schemes of non-perturbative results based on the parquet equations (e.g. the parquet-decomposition of the self-energy[@Gunnarsson2016]), since the potentially dangerous effects of such divergences will be cancelled out by the internal summation over fermionic variables. This might suggest alternative strategies to circumvent the divergences occurring in the major channels, even at the level of parquet solvers[@Tam2013; @Valli2015; @Wentzell2016; @Li2016; @Kauch2019], by exploiting the odd symmetry properties of their frequencies (and/or momentum[@Gunnarsson2016]) structures. We should also note that the divergences associated to antisymmetric eigenvectors are the [*first*]{} to be encountered upon increasing the interaction, independent of the interaction sign. As they affect the density channel, diagrammatic Monte Carlo algorithms based on [*bold resummations*]{} are likely going to encounter difficulties of formal convergence towards unphysical solutions for repulsive[@Kozik2015] as well as attractive interactions. It is important to stress that our results (and in particular that of Sec. III B and Sec. IV) do not only apply to the singular eigenvalues and eigenvectors. Instead, they fully define the effects of the Shiba mapping on all generalized two-particle quantities: The symmetric subspaces of $\chi_r$ are transformed exactly [*in the same way*]{} as the physical D.o.F., while the antisymmetric subspaces remain [*invariant*]{}. On the basis of these considerations, and consistently with the total decoupling of the antisymmetric eigenvectors from the static susceptibilities (see Secs. IIIB and IV), one would be tempted to associate the whole physically relevant information with the symmetric subspace of the generalized susceptibilities. However, this is not true in general. In fact, while the antisymmetric subspace of $\chi_{r}$ does not contribute at all to the corresponding static susceptibility, it can affect the behavior of other physical quantities. A pertinent example are energy-energy correlation-functions, i.e. response functions which explicitly contain first-order time-derivatives (i.e., $i\hbar \frac{d}{d t} = - \hbar \frac{d}{d \tau} = \hat{H}$) such as the thermal conductivity[^4]. By Fourier transforming the (imaginary) time-derivative, one gets an additional linear dependence of the generalized response on the two fermionic Matsubara frequencies $\nu$, $\nu'$. This additional frequency dependence essentially inverts the symmetry effects in the final fermionic frequency summations, hence allowing for contributions arising from the antisymmetric subspace. Finally, we note that if symmetries of the problem are lifted (e.g. by doping the system, considering further hopping terms or applying a magnetic field, etc.), correspondent changes must be expected. The high-symmetry case we considered in this work will represent then a good “compass" to interpret the observed deviation. For instance, doping the model with electron/holes will break the SU(2) symmetry of the pseudospin D.o.F., and one will observe a corresponding splitting of the degeneracy of the “orange" (pseudospin) divergences lines, with a different location of the singularities in the density and in the $pp$ channel. For the $pp$ channel however, the internal symmetry subdivision into fully symmetric and anti-symmetric subspaces will continue to hold. Similarly, the three-fold degenerate divergences in the magnetic channel will be split, if the SU(2) symmetry is lifted by applying a magnetic field. Conclusion ========== In the present work we conducted a comparative DMFT analysis to understand the location and physical role of vertex divergences occurring in the two-particle vertex correlation functions of the repulsive and attractive Hubbard model. Our calculations show that the location of divergences of two-particle irreducible vertices is perfectly symmetric in the attractive and repulsive Hubbard model. This result partly contradicts the expectation from the one-particle picture, where a divergence of the self-energy is accompanied by the suppression of the one-particle Green’s function. In particular the symmetric occurrence of singular eigenvalues in $\chi^{\nu \nu'}_d$ for $U\!\lessgtr \! 0$ shows, that divergences of the two-particle self-energy $\Gamma$ do not necessarily occur in physically suppressed channels. By considering the specific symmetries that apply in the presently considered system, we show that the antisymmetric and symmetric subspaces behave differently under the $U\!\leftrightarrow\!-U$ transformation. The antisymmetric part of the generalized susceptibilities is [*invariant*]{} under the Shiba transformation, hence explaining the perfectly mirrored divergence (“red") lines in the density sector, while for the symmetric subspace on the other hand, the density, particle-particle and magnetic channels are mapped into each other for $U\leftrightarrow-U$. Therefore, we confirm that the interaction-driven suppression of a static local susceptibility is generally accompanied by an increasing number of negative eigenvalues, [*if*]{} they are associated to symmetric eigenvectors, which actively contribute to the suppression of the channel. However, the reversed implication, that the occurrence of negative eigenvalues is in general indicative of the suppression of a channel, is [*not*]{} valid, because of the antisymmetric divergence lines being invariant under $U\leftrightarrow -U$. This suggests to represent the physically relevant information in terms of a susceptibility density distribution which naturally distinguish the symmetric from the vanishing antisymmetric eigenvector subspace. This representation allows to summarize the $U\!\leftrightarrow\!-U$ mapping behavior of the generalized susceptibilities and its relation to the mapping of the physical (spin and pseudospin) degrees of freedom at a single glance. Moreover, since the associated spectral distribution is identical for all suppressed as well as all enhanced channels, the introduced representation provides a universal description of [*all*]{} physical susceptibilities relevant for this problem. Further studies are required to clarify the role of the [*antisymmetric*]{} subspace for other physical quantities such as the thermal conductivity, the effect of a progressive reduction of the symmetry conditions and, on a broader perspective, the relation with the non-equilibrium properties of the system under investigation. [*Acknowledgments:*]{} We are indebted for insightful discussions with Sabine Andergassen, Massimo Capone, Lorenzo Del Re, James Freericks, Anna Kauch, Olle Gunnarsson, Andreas Hausoel, Cornelia Hille, Friedrich Krien, Erik van Loon, Matthias Reitner, Georg Rohringer, Thomas Schäfer, Agnese Tagliavini, Patrik Thunström, and Angelo Valli. We acknowledge financial support from the Austrian Science Fund (FWF) through the projects: SFB ViCoM F41 (DS) and I 2794-N35 (PC, AT). Calculations have been performed on Vienna Scientific Cluster (VSC). Bisymmetric Matrices {#App:CentrosymmetricMatrices} ==================== The following part is a short summary of mathematical literature on bisymmetric and centrosymmetric matrices. It is reported here to present the reader the possibility to follow more easily the proof in part \[App:equality\]. Note that at this point we focus on the matrix properties related to Eq. , without taking into account that the matrix is also symmetric. In this case one speaks of [*centrosymmetric*]{} matrices. In the following we consider a centrosymmetric matrix $H$, a $2n\times 2n$ matrix, where $n$ is the number of positive/negative fermionic Matsubara frequencies. As $H$ is a centrosymmetric matrix it fulfills the following condition: $$\label{equ:def} JHJ=H$$ where $J$ is the counteridentity matrix ($J^2=\mathbb{1}$), given in Eq. (\[equ:appix\_centrosymm\_J\]). $$\label{equ:appix_centrosymm_J} J= \begin{pmatrix} 0 & \dots & 0 & 1 \\ \vdots &\reflectbox{$\ddots$} & \reflectbox{$\ddots$} & 0 \\ 0 & 1 & \reflectbox{$\ddots$} & \vdots \\ 1 & 0 & \dots & 0 \end{pmatrix} = \begin{pmatrix} \mathbb{0} & J \\ J & \mathbb{0} \end{pmatrix}$$ If $J$ is multiplied from the right it inverts the columns of a matrix, if it is multiplied from left the rows are inverted. As one can easily see, for $\chi_{\sigma\sigma'}^{\nu \nu'}$ this means $$J\chi_{\sigma\sigma'}^{\nu \nu'}J=J\chi_{\sigma\sigma'}^{\nu (-\nu')} = \chi_{\sigma\sigma'}^{(-\nu) (-\nu')} = \chi_{\sigma\sigma'}^{\nu \nu'} \, ,$$ which is true for our case, see Eq. (\[eq:centrosymm\_chi\]) in the main text. If H is a centrosymmetric matrix, the following condition holds, where the submatrices $A,B,C,D$ are $n\times n$ matrices. $$\begin{aligned} H= \begin{pmatrix} A & \vline & B \\ \hline C & \vline & D \end{pmatrix} &\stackrel{(\ref{equ:def})}{=}&JHJ \\ \begin{pmatrix} A & \vline & B \\ \hline C & \vline & D \end{pmatrix} &=& \begin{pmatrix} \mathbb{0} & J \\ J & \mathbb{0} \end{pmatrix} \begin{pmatrix} A & \vline & B \\ \hline C & \vline & D \end{pmatrix} \begin{pmatrix} \mathbb{0} & J \\ J & \mathbb{0} \end{pmatrix} \\ &=& \begin{pmatrix} \mathbb{0} & J \\ J & \mathbb{0} \end{pmatrix} \begin{pmatrix} BJ & \vline & AJ \\ \hline DJ & \vline & CJ \end{pmatrix} \\ &=& \begin{pmatrix} JDJ & \vline & JCJ \\ \hline JBJ & \vline & JAJ \end{pmatrix}\end{aligned}$$ $$\Rightarrow D=JAJ \quad\&\quad B=JCJ$$ This means that the centrosymmetric matrix H can be written in the following form: $$H= \begin{pmatrix} A & \vline & JCJ \\ \hline C & \vline & JAJ \end{pmatrix}$$ Eigenvalues and Eigenvectors {#eigenvalues-and-eigenvectors .unnumbered} ---------------------------- Centrosymmetric matrices have a very useful property. Their eigenvalues can be obtained from the diagonalization of specific combinations of the submatrices $A$ and $C$, corresponding to either symmetric or antisymmetric eigenvectors. This can be seen as follows: Consider $\textbf{v}$, an eigenvector of $H$ $$\begin{aligned} H\textbf{v} &=& \lambda\textbf{v} \quad \quad | \cdot J \rightarrow \\ JH\textbf{v} &=& \lambda J\textbf{v} \\ HJ\textbf{v} &=& \lambda J\textbf{v} \quad ,\end{aligned}$$ where we used Eq.  and $J^2=\mathbb{1}$. From this it follows that $J\textbf{v}$ is also an eigenvector of H corresponding to the eigenvalue $\lambda$, i.e. $$\begin{aligned} J\textbf{v} = a\textbf{v} \quad ,\end{aligned}$$ with $a \neq 0$, being the eigenvalue of J and since J is an orthogonal matrix, $a=\pm 1$. This leads to antisymmetric or symmetric eigenvectors $\mathbf{v}$. In our terms this means that: $$\textbf{v} = \begin{pmatrix} v \\ \hline Jv \end{pmatrix} \quad or \quad \begin{pmatrix} v \\ \hline -Jv \end{pmatrix} \quad with \quad \begin{pmatrix} \text{\small{neg. Matsubara}} \\ \text{\small{frequencies}} \\ \hline \text{\small{pos. Matsubara}} \\ \text{\small{frequencies}} \end{pmatrix} \quad ,$$ where $\textbf{v}$ is a $2n\times 1$ vector and $v$ is a $n\times 1$ subpart of it. Next, we consider $\lambda_S$, an eigenvalue corresponding to a symmetric eigenvector $H \textbf{v}_S = \lambda_S \textbf{v}_S$: $$\begin{aligned} \begin{pmatrix} A & JCJ \\ C & JAJ \end{pmatrix} \begin{pmatrix} v \\ Jv \end{pmatrix} &=& \lambda_S \begin{pmatrix} v \\ Jv \end{pmatrix} \\ &\Downarrow& \nonumber \\ (A+JC)v &=& \lambda_S v\end{aligned}$$ In a similar fashion for $\lambda_A$, corresponding to an antisymmetric eigenvector: $$\begin{aligned} \begin{pmatrix} A & JCJ \\ C & JAJ \end{pmatrix} \begin{pmatrix} v \\ -Jv \end{pmatrix} &=& \lambda_A \begin{pmatrix} v \\ -Jv \end{pmatrix} \\ &\Downarrow& \nonumber \\ (A-JC)v &=& \lambda_A v\end{aligned}$$ This shows that the centrosymmetric matrix $H$ has eigenvalues $\lambda_S$ obtained from diagonalizing $A+JC$, which also gives the non-trivial parts $v$ of the symmetric eigenvectors $\textbf{v}_S$. In our case they correspond to the orange and green divergence lines, for $\lambda_S=0$. On the other hand we observe that $\lambda_A$ corresponds to antisymmetric eigenvectors obtained from the diagonalization of the submatrices $A-JC$ - the red divergence lines. In the following a very elegant way to see this block structure of $H$ is presented, which will be used later in the proof. Block-diagonalization {#Appendix:BlockDiag .unnumbered} --------------------- Using the following orthogonal matrix $Q$ ($QQ^T=\mathbb{1}$) one can block-diagonalize a centrosymmetric matrix $H$: $$Q=\frac{1}{\sqrt{2}} \begin{pmatrix} \mathbb{1} & -J \\ \mathbb{1} & J \end{pmatrix}$$ $$\begin{aligned} QHQ^{T}&=& \frac{1}{2} \begin{pmatrix} \mathbb{1} & -J \\ \mathbb{1} & J \end{pmatrix} \begin{pmatrix} A & JCJ\\ C & JAJ \end{pmatrix} \begin{pmatrix} \mathbb{1} & \mathbb{1}\\ -J & J \end{pmatrix} \\ &=& \frac{1}{2} \begin{pmatrix} \mathbb{1} & -J \\ \mathbb{1} & J \end{pmatrix} \begin{pmatrix} A -JC & A + JC\\ C-JA & C+JA \end{pmatrix} \\ &=& \frac{1}{2} \begin{pmatrix} 2(A-JC) & \mathbb{0} \\ \mathbb{0} & 2(A +JC) \end{pmatrix} \\ &=& \begin{pmatrix} A -JC & \mathbb{0} \\ \mathbb{0} & A +JC \end{pmatrix} \label{equ:block}\end{aligned}$$ Where immediately the block-structure described before is found. Bisymmetric Matrices {#bisymmetric-matrices .unnumbered} -------------------- As stated in the main text, due to the SU(2)- and the time-reversal-symmetry the centrosymmetric matrix $H$ considered is in fact bisymmetric. This has important consequences for the submatrices $A$ and $C$ introduced earlier: $$\begin{aligned} H&=&H^T \\ \begin{pmatrix} A & JCJ\\ C & JAJ \end{pmatrix} &=& \begin{pmatrix} A^T & C^T\\ (JCJ)^T & (JAJ)^T \end{pmatrix} \quad ,\end{aligned}$$ as $J=J^T$ one finds $A=A^T$ immediately. For $C$ the following equation holds: $$\label{eq:skewC} C^T = JCJ \rightarrow C^TJ^T = JC \rightarrow (JC)^T = JC$$ This means that the combination of submatrices yielding the eigenvalues and the corresponding symmetric or antisymmetric eigenvectors is symmetric, ensuring together with the particle-hole symmetry that the obtained eigenvalues are real. $$(A\pm JC)^T = A^T \pm (JC)^T \overset{\ref{eq:skewC}}{=} A \pm JC \quad ,$$ The mapping of divergence lines {#App:equality} =============================== Because of the specific mapping from $U>0$ to $U<0$ of $\chi_{\uparrow \uparrow}$ and $\chi_{\uparrow \downarrow}$, it is possible to show, that the red divergence lines for $U<0$ [*have to be*]{} the mirrored ones of $U>0$. As it turns out it also follows that the symmetric density divergences $(U>0)$ are mapped to symmetric divergences in the magnetic channel for $U<0$. The starting point is to consider the bisymmetric $\chi_{\uparrow \uparrow}$ and $\chi_{\uparrow \downarrow}$ matrices, where the fermionic Matsubara frequency indices will be omitted in the following. $\chi_{\uparrow \uparrow}$ and $\chi_{\uparrow \downarrow}$ fulfill the following relations, discussed in the main text in Sec. \[subsec:symmetries\], when mapped from positive to negative $U$. $$\begin{aligned} \chi^{U>0}_{\uparrow \uparrow} &=& \chi^{U<0}_{\uparrow \uparrow} = \chi_{\uparrow \uparrow} = \begin{pmatrix} A & JBJ \\ B & JAJ \end{pmatrix} \\ \chi^{U>0}_{\uparrow \downarrow} &=& \begin{pmatrix} C & JDJ \\ D & JCJ \end{pmatrix} \quad \\ \chi^{U<0}_{\uparrow \downarrow} &=& \chi^{U>0}_{\uparrow \downarrow} (-J) = \begin{pmatrix} C & JDJ \\ D & JCJ \end{pmatrix} \begin{pmatrix} \mathbb{0} & -J \\ -J & \mathbb{0} \end{pmatrix} \nonumber \\ &=& \begin{pmatrix} -JD & -CJ \\ -JC & -DJ \end{pmatrix}\end{aligned}$$ Now, block-diagonalization of $\chi_{\uparrow \uparrow}$ and $\chi_{\uparrow \downarrow}$ for both cases leads to: $$\begin{aligned} Q\chi_{\uparrow \uparrow}Q^T&=& \begin{pmatrix} A-JB & \mathbb{0} \\ \mathbb{0} & A+JB \end{pmatrix}\\ Q\chi^{U>0}_{\uparrow \downarrow}Q^T&=& \begin{pmatrix} C-JD & \mathbb{0} \\ \mathbb{0} & C+JD \end{pmatrix}\\ Q\chi^{U<0}_{\uparrow \downarrow}Q^T&=& \begin{pmatrix} -JD-J(-JC) & \mathbb{0} \\ \mathbb{0} & -JD+J(-JC) \end{pmatrix}\nonumber \\ &=& \begin{pmatrix} C-JD & \mathbb{0} \\ \mathbb{0} & -[C+JD] \end{pmatrix} \end{aligned}$$ This shows immediately that the antisymmetric block of $\chi_{\uparrow\downarrow}$, ($C-JD$), is unchanged, whereas the symmetric one changes sign for $U>0 \leftrightarrow U<0$. Considering $\chi_d$ and $\chi_m$ for $U<0$ and $U>0$ the following conclusions can be drawn, where we use the trivial relation: $$Q\chi_{d^+,m^-}Q^T = Q(\chi_{\uparrow\uparrow} \pm \chi_{\uparrow\downarrow})Q^T = Q\chi_{\uparrow\uparrow}Q^T \pm Q\chi_{\uparrow\downarrow}Q^T$$ $$\begin{aligned} \label{eq:mapping_dens} Q\chi_{d}^{U\gtrless 0}Q^{T}\!&=&\! \begin{pmatrix} [A\!-\!JB]\!+\![C\!-\!JD] & \mathbb{0} \\ \mathbb{0} & [A\!+\!JB]\!\pm\![C\!+\!JD] \end{pmatrix} \nonumber \\\end{aligned}$$ $$\begin{aligned} \label{eq:mapping_mag} Q\chi_{m}^{U\gtrless 0}Q^{T} &=& \begin{pmatrix} [A\!-\!JB]\!-\![C\!-\!JD] & \mathbb{0} \\ \mathbb{0} & [A\!+\!JB]\!\mp\![C\!+\!JD] \end{pmatrix} \ , \nonumber \\ \label{equ:neg_U}\end{aligned}$$ where in the density case the $+$ sign corresponds to $U>0$ and the $-$ to $U<0$, for the magnetic case it is the other way around. From Eqs.(\[eq:mapping\_dens\],\[eq:mapping\_mag\]) three things can be learned: $(i)$: The antisymmetric block of $Q\chi_{d}^{U\gtrless 0}Q^{T}$ is independent of the sign of $U$. The diagonalization of $[A-JB] + [C-JD]$ will yield the eigenvalues and the corresponding antisymmetric eigenvectors of $\chi_d$. Their singularity corresponds to a red divergence line - independent of the sign of $U$. This is the mathematical reason for the perfect mapping of the red divergence lines reported in Fig. \[Fig:1\] and the equality of the singular eigenvectors seen in Fig. \[Fig:2\] of the main text. Note that this statement is crucially dependent on the perfect particle-hole symmetry of the problem analyzed - otherwise the bisymmetry property is lost. $(ii)$: The antisymmetric block of $Q\chi_{m}^{U\gtrless 0}Q^{T}$ is also independent of the sign of $U$. This means that, irrespective of the sign of $U$, the eigenvalues corresponding to antisymmetric eigenvectors of $\chi_m$ can be calculated by diagonalizing $[A-JB] - [C-JD]$. However, so far none of these eigenvalues were found to be singular. $(iii)$: The symmetric parts of $\chi_d$ and $\chi_m$ are mapped in the following way: $[A+JB]+[C+JD]$ is the symmetric blockmatrix of $\chi^{U>0}_d$ [*and*]{} $\chi^{U<0}_m$. This explains why the symmetric density channel divergences for $U>0$ are mapped to divergences with symmetric eigenvectors in the magnetic channel for $U<0$. Analogously, $[A+JB]-[C+JD]$ is the symmetric blockmatrix of $\chi^{U<0}_d$ [*and*]{} $\chi^{U>0}_m$ , exactly the parameter regime where these channels exhibit the dominant, non suppressed, physics. Here the bisymmetry explains the mapping of the eigenvalues, as discussed in Sec. \[sec:spectral\]. Finally we note that also the matrix causing the divergences in the particle-particle up-down channel $\chi^{\nu (-\nu')}_{pp,\uparrow\downarrow} - \chi_{0,pp}^{\nu \nu'} $, is bisymmetric, having hence the same properties as mentioned above. Combining this insight with Eq. \[eq:mapping\_chi\_pp\] we have now fully clarified how the mapping of the generalized susceptibilities works and its exact relation with the physical degrees of freedom. The antisymmetric sectors are not mapped along the lines of Eq. \[Eq:SpinMap\], but they cancel in the sum in Eq. \[eq:ChiSpectrum\]. The symmetric subparts on the other hand follow the mapping of the physical D.o.F.. Susceptibility density in the binary mixture disordered model {#App:BM} ============================================================= We calculate analytically the $\rho_d(\chi)$ in the Binary Mixture (BM) disordered case defined by the Hamiltonian $$H=-t\sum\limits_{<ij>}{c^{\dagger}_{i}c_{j}} + \sum_{i} \epsilon_i {c^{\dagger}_{i}c_{i}} \ . \label{eqn:BM}$$ Here spin indices can be omitted and we can safely consider spinless electrons moving in a random background with equal probability for $\epsilon_i=\pm W/2$. ![Eigenvalues from Eq. (\[eq:chi\]) evaluated a $T=0.0$.[]{data-label="fig:chivsmatsu"}](Chi_vs_Matsu.pdf){width="47.00000%"} The Green’s function at half-filling can be easily calculated within DMFT $$G(\nu)= \frac{1}{2} \left( \frac{1}{G^{-1}_0(\nu)-\frac{W}{2}}+\frac{1}{G^{-1}_0(\nu)+\frac{W}{2}}\right) \ , \label{eq:BM}$$ where $G^{-1}_0(\nu)=\nu-D^2 G(\nu) / 4$ in the Bethe lattice case. This result is in perfect analogy with the Hubbard III (CPA) approximation for the Hubbard model, where $W$ must be understood as $U$. The BM shows divergences in the irreducible vertex function as well as negative eigenvalues in the generalized susceptibility for the density channel. They appear at sufficiently large $W$ beyond the Mott-like transition in which the DOS at vanishes at the Fermi level[@Schaefer2016c]. The susceptibility $\chi^{\nu,\nu^\prime}_d$ can be easily calculated as $$\chi_{d}^{\nu \nu'}=\frac{2}{W^2}\sqrt{1+W^2G^2(\nu)}\left [ \sqrt{1+W^2G^2(\nu^\prime)}\mp 1\right] \delta_{\nu \nu'} \label{eq:chi}$$ where the $\pm$ sign is a consequence of the multivaluedness of the electronic self-energy and must hence be taken into account properly, in order to access the physical solution[@Schaefer2016c]. Eq. (\[eq:chi\]) states that $\chi^{\nu,\nu^\prime}_d$ is diagonal in Matsubara frequency space. This is a consequence of the locality of the functional relation which relates the self-energy and the local single-particle propagator $\Sigma[G]$[@Schaefer2016c]. The phase-diagram of the BM model shows an accumulation point of vertex divergences at $T=0$ located at $W_c/D=1/\sqrt{2}$, before the Mott-like transition. As we shall see below, this implies a continuous eigenvalue distribution that exhibits a tail towards negative values above $W_c$. Eigenvalues of $\chi^{\nu,\nu^\prime}_d$ can be directly obtained from Eq.  once the self-consistency condition has been enforced. A singular eigenvalue occurs when $$1+W^2G^2=0 \label{eq:FKdivergence}$$ at zero frequency. The schematic behavior of eigenvalues as a function of Matsubara frequencies is shown in Fig. \[fig:chivsmatsu\]. The behavior of the distribution of eigenvalues which turns out to be continuous in the $T=0$ limit is shown in Fig. \[fig:MapChiDistrib\]. Notice the logarithmic scale in Fig. \[fig:MapChiDistrib\], which means that the weight associated to negative eigenvalues is very small. However, the zero-crossing of a small amount of eigenvalues marks the onset of the strong disorder limit at $W_c$ before the Mott-like transition where the local $\chi$ vanishes. ![Map of the eigenvalue distribution at $T=0$. The green curve is the local charge susceptibility.[]{data-label="fig:MapChiDistrib"}](MapChiDistrib.png){width="50.00000%"} [^1]: With this definitions, a BSE, formally similar to Eq.(\[eq:BSEinv\]), can be written for the quantity $\tilde{\chi}^{\nu \nu'}(\Omega) = \chi_{pp}^{\nu \nu'}(\Omega) - \chi_{0,pp}^{\nu \nu'}(\Omega)$ with $ \chi_{0,pp}^{\nu \nu'}= -\beta G(\nu) G(\Omega- \nu') \delta_{\nu \nu'}$. [^2]: For previous DMFT studies of the [*attractive*]{} Hubbard model at the [*one*]{}-particle level see, e.g., \[\]. [^3]: The small differences arise from the different lattice (here: Bethe lattice) used in the DMFT calculations [^4]: Note that the thermal conductivity expression would also contains the spatial derivatives of the heat-current. This would preventing any kind of vertex-corrections in the single-orbital model at the DMFT level[@Georges1996]. The latter are possible, however, at the level of DCA[@Maier2005], where vertex-divergences with antisymmetric structure have been also reported[@Gunnarsson2016]. Furthermore, the correlation functions containing time-derivatives, where the effects of the antisymmetric part of the vertex functions are important, are the most suited to establish a connection with non-equilibrium phenomena.

--- abstract: 'By relating the number of images of a function with finite domain to a certain parameter, we obtain both an upper and lower bound for the image set. Even though the arguments are elementary, the bounds are, in some sense, best possible. These bounds are then applied in several contexts. In particular, we obtain the first non-trivial upper bound for the image set of a planar function over a finite field.' address: - | Ewing Hall\ Department of Mathematical Sciences\ University of Delaware\ Newark, DE 19716, USA - | Ewing Hall\ Department of Mathematical Sciences\ University of Delaware\ Newark, DE 19716, USA author: - 'Robert S. Coulter' - Steven Senger title: On the number of distinct values of a class of functions with finite domain --- Introduction ============ Let $A$ and $B$ be sets, with $A$ finite of order $n$, and let $f:A\rightarrow B$. We define the following notation, which will be used throughout this article. - The number of distinct images of $f$ is denoted by $V(f)$. That is, $V(f)=|f(A)|$. - For $r\in{\mathbb N}$, $M_r(f)$ is the number of $y\in B$ for which $f(x)=y$ has $r$ solutions. - Since $A$ is finite, clearly $M_r(f)=0$ for all sufficiently large $r$. We therefore define $m$ to be the largest integer for which $M_m>0$. - For each integer $r\ge 2$, $N_r(f)$ is the number of $r$-tuples $(x_1,\ldots,x_r)$ with $x_i=x_j$ if and only if $i=j$ which satisfy $f(x_1)=f(x_2)=\cdots=f(x_r)$. Several identities follow immediately from these definitions. 1. $V(f) = \sum_{r=1}^m M_r(f)$. 2. $n = \sum_{r=1}^m r M_r(f)$. 3. $N_s(f) = \sum_{r=s}^m P(r,s) M_r(f)$. (Here $P(r,s)$ denotes the number of $s$-permutations from $r$ distinct objects. Recall $P(r,s)=0$ when $r<s$.) In this paper we are interested in the relationship between $V(f)$ and $N_s(f)$ for a fixed $s$. Intuitively, knowledge of $N_s(f)$ should imply some knowledge on $V(f)$, and knowledge of $N_s(f)$ should yield more knowledge concerning $V(f)$ than $N_{s'}(f)$ would for $s'>s$. Our main result is to obtain bounds for $V(f)$ in terms of $N_s(f)$ which confirm this intuition. Moreover, when $s=2$, our lower bound is tight for any value of $N_2(f)$, while our upper bound is tight in infinitely many cases. Our main theorem can be given in the following form. \[mainthm\] Let $f:A\rightarrow B$ with $|A|=n$. Then $$\frac{1}{s-1}\left(n - \frac{N_s(f)}{s!}\right) \le V(f) \le n - N_s(f)^{1/s} + O(N_s(f)^{1/(s+1)}).$$ We pay particular attention to the case $s=2$ because it is more likely that one has information on pairs of elements with the same image than, say, $3$-tuples or $4$-tuples. In addition, the upper bound can be made explicit in this case. \[seq2thm\] Let $f:A\rightarrow B$ with $|A|=n$ and set $N_2(f)=t$. Then $M_1(f)\ge {{\rm Max}}(0,n-t)$ and $$n - \frac{t}{2} \le M_1(f)+M_2(f)\le V(f) \le n - \frac{2t}{1+\sqrt{4t+1}}$$ Interestingly, the upper bound in Theorem \[seq2thm\] is related to triangular numbers, and a slight improvement of this bound, in some cases, could be obtained by resolving a problem on them. Theorems \[mainthm\] and \[seq2thm\] can be applied in a variety of settings. We choose to limit ourselves to just one main application – to polynomials over finite fields. Let $q$ be a positive power of some prime $p$. We use the standard notation of $\ff{q}$ for the finite field of $q$ elements, $\ffs{q}$ for the non-zero elements of $\ff{q}$, and $\ffx{q}$ for the ring of polynomials over $\ff{q}$ in $X$. We prove that for a polynomial $f\in\ffx{q}$, the expected value of $N_2(f)$ is $q-1$. Consequently, we obtain the following corollary to Theorem \[seq2thm\]. \[polyversion\] Suppose $f\in\ffx{q}$ is a polynomial for which $N_2(f)=q-1$, the expected value. Then $$\frac{q+1}{2} \le V(f) \le q - \frac{2(q-1)}{1+\sqrt{4q-3}}.$$ Several classes of polynomials which obtain the expected value for $N_2(f)$ are then described; these include the class of planar polynomials (for further definitions, see Section \[polysection\]). Planar polynomials are closely related to affine planes [@coulter97a; @dembowski68], semifields [@coulter08], and difference sets [@ding06; @qiu07]. Consequently, they have received a significant amount of attention. However, the bound given by Theorem \[polyversion\] constitutes the first non-trivial upper bound obtained on the size of the image set of a planar function. We suspect that, for planar functions, our upper bound can still be improved as we do not utilise the full set of restrictions implied by the planar property. The lower bound is, for planar functions, tight, and has been derived previously by several authors, see [@coulter11; @kyureghyan08; @qiu07]. Our result, in this sense, constitutes a generalisation of the respective results given in each of those three papers. The paper is set out as follows. In the next section we prove Theorems \[mainthm\] and \[seq2thm\]. We also discuss briefly the connection between Theorem \[seq2thm\] and triangular numbers. In Section 3 we apply our results to polynomials over finite fields. The paper ends with some observations in arithmetic combinatorics and coding theory. Bounding $V(f)$ when $N_s(f)$ is known ====================================== For convenience, we set $N_s(f) = t$. By the definitions above, $$\sum_{r=1}^{s-1} r M_r = n - t + \sum_{r=s}^m \left(P(r,s)-r\right) M_r. \label{aneq}$$ (We note that, since the sum on the right is at least $m(m-2)$, we must have $\sum_{r=1}^{s-1} r M_r \ge {{\rm Max}}(0, n - t + m(m-2))$.) We may manipulate (\[aneq\]) as follows: $$\begin{aligned} \sum_{r=1}^{s-1} r M_r &= n - t + \sum_{r=s}^m \left(P(r,s)-r\right) M_r\\ &= n - t + (s! -s) M_s + \sum_{r=s+1}^m \left(P(r,s)-r\right) M_r\\ &\ge n - t + (s!-s) M_s + (s!-1) \sum_{r=s+1}^m r M_r\\ &= n - t + (s!-s) M_s\\ &\quad+ (s!-1) \sum_{r=1}^m r M_r - (s!-1)\sum_{r=1}^{s-1} rM_r - (s!-1) sM_s\\ &= s!\, n - t + s!\, (1-s) M_s - (s! -1) \sum_{r=1}^{s-1} r M_r.\end{aligned}$$ Rearranging, we find $$\begin{aligned} s! \, n - t &\le s! \sum_{r=1}^{s-1} rM_r + s!\,(s-1) M_s\\ &\le s!\, (s-1) \sum_{r=1}^{s-1} M_r + s!\,(s-1)M_s\\ &= s!\, (s-1) \sum_{r=1}^s M_r\\ &\le s!\, (s-1) \, V(f),\end{aligned}$$ which establishes the lower bound in Theorem \[mainthm\]. (We mention, in passing, that this proof is a generalisation of the lower bound obtained by Matthews and the first author [@coulter11]; it was that note that formed the motivation for this article.) We now move to determine the upper bound. First, we note that $M_m>0$, and so $P(m,s)\leq t$, which yields $$\label{tUpper} m\leq t^\frac{1}{s}+O(t^\frac{1}{s+1}).$$ Now, we apply the definitions above to obtain $$\begin{aligned} t &= N_s(f)\\ &= \sum_{r=s}^m P(r,s)M_r = \sum_{r=1}^m P(r,s) M_r\\ &\leq m\sum_{r=1}^m P(r-1,s-1) M_r\\ &\leq m\cdot P(m-2,s-2) \sum_{r=1}^m (r-1)M_r,\\\end{aligned}$$ from which we deduce $$\label{mUpper} \sum_{r=1}^m (r-1) M_r \geq \frac{t}{m\cdot P(m-2,s-2)}.$$ Combining and , we get $$\label{sumBound} \sum_{r=1}^m (r-1) M_r \geq t^\frac{1}{s} - O(t^\frac{1}{s+1}).$$ We can now estimate $V(f)$ using this sum: $$\begin{aligned} V(f) &= n - n + V(f)\\ &= n - \sum_{r=1}^m r M_r - \sum_{r=1}^m M_r\\ &= n - \sum_{r=1}^m (r-1) M_r\\\end{aligned}$$ Applying yields $$\label{mainUpper} V(f) \leq n - t^\frac{1}{s} + O(t^\frac{1}{s+1}),$$ as claimed. The proof of Theorem \[seq2thm\] is no more difficult; in fact, the lower bound is precisely that from before, while the upper bound follows from a careful re-working of the proof of the upper bound. We omit the details. It is easy to see that, provided $N_2(f) < 2n$, this lower bound is tight, as one can easily construct functions that meet this bound. Set $N_2(f)=t$. Randomly choose $t$ distinct elements $x_1,x_2,\ldots,x_t\in A$ and $t/2$ distinct elements $y_1,y_2,\ldots,y_{t/2}\in B$. For $1\le i\le t/2$, assign $f(x_{2i-1})=f(x_{2i})=y_i$. At this point, $N_2(f)=t$, so that $f$ must be 1-1 on $A\setminus\{x_1,\ldots,x_t\}$. It follows that $V(f)=\frac{t}{2} + n-t = n - \frac{t}{2}$, which is the lower bound. It is clear from symmetry that $N_2(f)=t$ is necessarily even. Set $t=2k$. Then the bounds read $$n-k\le V(f) \le n - \frac{4k}{1+\sqrt{8k+1}}.$$ It is natural to ask when is $\sqrt{8k+1}\in{\mathbb Z}$? Interestingly, $8k+1$ is a square precisely when $k$ is a triangular number. In such cases, we have $k=u(u-1)/2$ for some integer $u$, $8k+1=\delta^2$ where $\delta=2u-1$, and the upper bound simplifies neatly to $$V(f) \le n - \frac{\delta-1}{2} = n + 1 - u.$$ In all cases where $k$ is a triangular number, there exist functions which attain this bound. To construct such a function, choose $u$ elements $x_1,x_2,\ldots,x_u\in A$ and set $f(x_1)=f(x_2)=\cdots=f(x_u)$. Now set $f$ to behave 1-1 on the remaining elements of $A$. It can be seen that $N_2(f)=2k$ and that the upper bound is attained. In all cases where $k$ is not a triangular number, our upper bound is not exact. To make our upper bound tight, one needs to solve the following problem: > Let $T_r=\binom{r}{2}$ for any $r\in{\mathbb N}$, and fix $k\in{\mathbb N}$. By a [*triangular sum of length $l$ for $k$*]{} we mean any instance of the equation $$k = \sum_{i=1}^l T_{r_i},$$ where $r_1\ge r_2\ge\cdots\ge r_l$. The [*weight*]{} of a given triangular sum is given by $-l+(\sum_{i=1}^l r_i)$. Given $k$, we define $B_k$ to be the smallest weight among all triangular sums for $k$. Find a formula for $B_k$. Clearly, when $k=T_u$, $B_k=u-1$, but we do not know of a general formula for $B_k$. While Gauss famously proved that there exists a triangular sum for any $k$ with length at most 3, it may not necessarily be the case that one such instance will provide the value for $B_k$. The connection to our bound should be clear: If $N_2(f)=2k$, then $V(f)\le n - B_k$, with equality always possible. Polynomials over finite fields and $N_2(f)$ {#polysection} =========================================== We now look to apply these bounds on $V(f)$ to polynomials over finite fields. It is, of course, well known that every function over $\ff{q}$ can be represented uniquely, via Lagrange interpolation, by a polynomial of degree less than $q$. By the [*reduced form*]{} of a polynomial $f\in\ffx{q}$ we shall mean the polynomial $g(X)$ given by $g(X) = f(X) \bmod (X^q-X)$. A polynomial $f\in\ffx{q}$ is a [*permutation polynomial*]{} over $\ff{q}$ if $V(f)=q$. Research concerning the value of $V(f)$ for polynomials over finite fields is extensive; we restrict ourselves to discussing a few outstanding general results. It is clear that, for lower bounds, there are obvious limits to the results you can expect to obtain – obviously $V(f)\ge 1$ with equality possible, while for polynomials of given degree $d$, $V(f)\ge 1+\frac{q-1}{d}$ is clear. That said, we have the following deep result by Cohen [@cohen73] concerning the average lower bound of $V(f)$. \[cohensthm\] Let $f\in\ffx{q}$ be of the form $$f(X) = X^d + \sum_{i=1}^{d-1} a_i X^i.$$ Let $t$ be any integer such that $0\le t\le d-2$ and let $a_{d-1},\ldots,a_{d-t}$ be fixed. Define $v(d,t)=\sum V(f)/q^{d-t-1}$, where the sum is over all $a_1,\ldots,a_{d-t-1}$. Set $m=\lfloor (d-t)/2\rfloor$. Then $v(d,t) > c(q,m) q$, where $$c(q,m) = 1 - \left(\sum_{r=0}^m \binom{q}{r} (q-1)^{-r}\right)^{-1}.$$ Setting $t=d-2$ in Cohen’s result, we find that, in particular, on average, $V(f)>\frac{q^2}{2q-1}>\frac{q}{2}$. A specific lower bound was obtained by Wan, Shiue, and Chen [@wan93b] under an additional condition on the polynomial. For $f\in\ffx{q}$, define $u_p(f)$ to be the smallest positive integer $k$ such that $\sum_{x\in\ff{q}} f(x)^k\ne 0$. If no such $k$ exists, define $u_p(f)=\infty$. If $u_p(f)<\infty$, then $V(f)\ge u_p(f) + 1$. The authors note that $u_p(f)\ge \lfloor\frac{q-1}{{{\rm Degree}}(f)}\rfloor$, so that under the conditions, their bound is at least as good as the obvious bound noted above. In terms of an upper bound, there is the following general bound by Wan [@wan93a], given in terms of the degree of the polynomial. \[wansthm\] Let $f\in\ffx{q}$. If $f$ is not a permutation polynomial over $\ff{q}$, then $$V(f)\le q - \left\lfloor\frac{q-1}{{{\rm Degree}}(f)}\right\rfloor.$$ A better bound was obtained in [@wan93b] using $p$-adic techniques. To avoid unnecessary technical details, we simply refer the interested reader to [@wan93b], Theorem 3.1. Integral to applying our bounds is having knowledge of $N_s(f)$ for some $s$. For simplicity, we only discuss the case $s=2$ here. We do not feel this is particularly limiting as, of the values of $N_s(f)$, knowledge of $N_2(f)$ seems most likely. We approach this issue by first establishing the expected value of $N_2(f)$ for any polynomial $f\in\ffx{q}$ and applying our bounds to polynomials with this expected value. We then consider classes of polynomials which meet this expected value. Denote the standard trace mapping from $\ff{q}{}$ to $\ff{p}{}$ by ${{\text {Tr}}}$. Let $\omega$ be a primitive $p$th root of unity. Recall that the canonical additive character, $\chi_1$, of $\ff{q}{}$ is defined by $\chi_1(x)=\omega^{{{\text {Tr}}}(x)}$ for any $x\in\ff{q}{}$, and that all additive characters of $\ff{q}{}$ are given by $\chi_h(x)=\chi_1(hx)$ for any $h\in\ff{q}{}$. The following result is a straight generalisation of a result of Carlitz [@carlitz55]. \[averagelemma\] Given a random polynomial $f\in \ffx{q}$, the expected value of $N_2(f)$ is $q-1$. Equivalently, for any $f\in\ffx{q}$, $$\label{Navg} \sum_{a\in \ff{q}} N_2(f(X)+aX) = q(q-1).$$ Fix a polynomial $f\in \ffx{q}$. By the definitions above, $$\begin{aligned} q(N_2(f)+ q) &= q (|\lbrace (x,y): f(x)=f(y), x,y\in \ff{q}, x\neq y \rbrace| \\ &\quad + |\lbrace (x: f(x)=f(x), x\in \ff{q} \rbrace|)\\ &=\sum_{h\in\ff{q}} \sum_{x,y\in \ff{q}}\chi_h(f(x)-f(y)).\end{aligned}$$ To generate our average value for $N_2(f)$, we consider the average over the set $\{ f(X)+aX \,:\, a\in\ff{q}\}$. We have $$\begin{aligned} \sum_{a\in\ff{q}} q&(N_2(f(X)+aX)+q)\\ &= \sum_{a\in\ff{q}}\sum_{h\in\ff{q}} \sum_{x,y\in \ff{q}}\chi_h(f(x)-f(y)+a(x-y))\\ &=q^3+ \sum_{h\in\ffs{q}} \sum_{x,y\in \ff{q}} \chi_h(f(x)-f(y)) \sum_{a\in\ff{q}} \chi_h(a(x-y))\\ &=q^3+ \sum_{h\in\ffs{q}} \sum_{x\in\ff{q}} q\\ &= q^3+q^2(q-1),\end{aligned}$$ where, in the second to last line, we have exploited the fact $\sum_{a\in\ff{q}} \chi(a(x-y))=0$ unless $x=y$. Comparing the left and right hand sides yields $$\label{a_1Avg} \sum_{a\in\ff{q}} N_2(f(X)+aX) = q(q-1).$$ The claimed expected value of $N_2(f)$ now follows at once, for we can, of course, partition the set of polynomials into equivalence classes, with two polynomials being equivalent if they differ only by a linear term $aX$: the average value of $N_2(f)$ for the polynomials in any equivalence class is $q-1$ by (\[a\_1Avg\]). Theorem \[polyversion\] now follows at once from Theorem \[seq2thm\] and Lemma \[averagelemma\]. Now suppose $f\in\ffx{q}$ is a polynomial for which $N_2(f)=q-1$, the expected value. For our lower bound, we find $V(f)\ge \frac{q+1}{2}$, which is more or less the same as that obtained by Cohen’s result. In the other direction, applying our upper bound to $f$, we find $$V(f) \le q - \frac{2(q-1)}{1+\sqrt{4q-3}}.$$ However, this cannot be compared directly to the result of Wan, for we do not know if $N_2(f)=q-1$ has any direct implication on ${{\rm Degree}}(f)$. Given Lemma \[averagelemma\], one obvious question arises: Is it possible to describe classes of polynomials for which the expected value for $N_2(f)$ is obtained? Are there natural conditions on $f$ which force $N_2(f)=q-1$? We now discuss, for $q$ odd, several such conditions (the case $q$ even is clearly impossible for $N_2(f)$ is necessarily even). For any $a\in\ffs{q}$, we define the [*difference polynomial*]{}, $\Delta_{f,a}(X)=\Delta_a(X)$, to be the polynomial given by $\Delta_a(X)=f(X+a)-f(X)$. A polynomial $f\in\ffx{q}$ is [*planar*]{} over $\ff{q}$ if, for every $a\in\ffs{q}$, the polynomial $\Delta_a(X)$ is a permutation polynomial over $\ff{q}$. An equivalent definition for planarity is that $|S_h(f(X)+aX)|=|\sum_{x\in\ff{q}} \chi_h(f(x)+ax)|=\sqrt{q}$ for all $a,h\in\ff{q}$, $h\ne 0$. Consider the following conditions on a polynomial $f\in\ffx{q}$: 1. $f$ is planar over $\ff{q}$. 2. For $h\in\ffs{q}$, $|S_h(f)|=|\sum_{x\in\ff{q}} \chi_h(f(x))|=\sqrt{q}$. 3. For all $a\in\ffs{q}$, the polynomial $\Delta_{f,a}(X)$ has a unique root. 4. $N_2(f)=q-1$. Clearly, $C_1 \rightarrow C_2$ and $C_1 \rightarrow C_3\rightarrow C_4$. It is shown in the proof of [@coulter11], Theorem 1, that $C_2\rightarrow C_4$, while a counting argument, also given in [@coulter11], shows $C_1 \not\equiv C_2$. The relationship between $C_2$ and $C_3$ is less clear. Computations show that they are almost certainly inequivalent for sufficiently large $q$. Over $\ff{3}$, they are equivalent; over $\ff{5}$, they are not, though $(C_2\land C_3)\rightarrow C_1$. For $q\in\{7,9\}$, they are inequivalent, and - there exist polynomials which satisfy both $C_2$ and $C_3$ but not $C_1$; for example, $f(X)=X^4+2X^2\in\ffx{7}$; and - there exist polynomials which satisfy one or other but not both conditions; for example, with $g$ a primitive element of $\ff{9}$, $X^7+gX^2$ satisfies $C_2$ but not $C_3$, while $X^8+gX^2$ satisfies $C_3$ but not $C_2$. This also shows $C_2\not\equiv C_4$ and $C_3\not\equiv C_4$. We suspect that the following statement is true, though we have no direct idea of how to establish it. For any finite field of any characteristic, the number of polynomials satisfying $C_3$ is greater than or equal to the number of polynomials satisfying $C_2$. Two further settings where the bounds apply =========================================== We end by describing two settings where our results can be applied, and where we suspect some refinements of our methods might lead to stronger results than those we give here. Arithmetic combinatorics ------------------------ Here, we present a setting where $N_2$ arises rather naturally. Let $G$ be a (not necessarily abelian) group. For subsets $A,B \subset G$, define the product set of $A$ and $B$ to be $$A\cdot B = \lbrace ab : a\in A, b \in B \rbrace.$$ Much interest revolves around the relative sizes of $A, B$, and $A\cdot B$. Some examples are the Cauchy-Davenport Theorem, the Plünnecke-Rusza inequalities, and Freiman’s Theorem; see the books by Nathanson [@bnathanson96ii] or Tao and Vu [@btaovu]. One useful tool for these questions is the concept of energy. Various types of energy bounds have been the key ingredient in many recent results, such as the current best known sums and products bound due to Solymosi [@solymosi08]. Given $G, A, $ and $B$ as above, we define the multiplicative energy, $E(A,B)$, to be $$E(A,B)= |\lbrace(a,a',b,b')\in A\times A\times B\times B:ab=a'b' \rbrace|.$$ If we consider $f:A\times B \rightarrow G, f:(a,b) \mapsto ab,$ we get a very close relationship between $N_2(f)$ and $E(A,B)$, namely $$N_2(f) = E(A,B) - |A|\cdot |B|,$$ which we obtain by removing the “diagonal" elements of the form $(a,a,b,b)$ from the energy count. With this in mind, the following is a direct application of Theorem \[seq2thm\]. Let $G$ be a group, $A,B \subset G$ and set $n = |A| \cdot |B|$. Then we have $$\label{energyBounds} \frac{3n-E(A,B)}{2} \leq |A \cdot B|\leq n - \frac{2(E(A,B)-n)}{1+\sqrt{4(E(A,B)-n)+1}}$$ Notice that these bounds are most effective when energy is small. Coding theory ------------- Our second setting is in coding theory. Much is known about the interplay between the redundancy of a given code and the amount of information that can be communicated per unit time; see Hall’s notes on coding [@HallCoding], for a good introduction. Here, we investigate messages transmitted through a noisy medium. Consider a function $f:\mathcal{C}\rightarrow \mathcal{M}$, where $\mathcal{C}$ is the codespace and $\mathcal{M}$ is the message space. In order to increase the likelihood that a message is decoded properly, even with errors in transmission, we will often give a single message word more than one code word. That is, it will often be the case that $f(c) = f(c')$ for distinct $c,c'\in\mathcal{C}$. By definition, $V(f)$ will be precisely the number of distinct words in $\mathcal{M}$, and $N_2(f)$ will be the number of times that two code words represent the same message. There are situations in which one has a particularly uneven message space, where a small number of messages have high priority, and need the best chances of being decoded correctly, while all remaining messages are less important, and their incorrect decodings would have very little consequence. For example, a message space between fire towers in a forest could have a small number of special words about the existence or severity of a fire, and the other words could describe other, less important details, like the weather, in the case that there is no fire. Similar applications exist in a variety of different contexts such as operations in hostile environments. In such situations, an application of Theorem \[seq2thm\] yields the following. In a code with a codespace $\mathcal{C},$ a message space $\mathcal{M}$, an assignment function $f:\mathcal{C} \rightarrow \mathcal{M}$, and $t=|\lbrace f(c)=f(c'): c,c'\in \mathcal{C}, c\neq c' \rbrace|$, we have $$n - \frac{t}{2} \le |\mathcal{C}| \le n - \frac{2t}{1+\sqrt{4t+1}}$$ In this setting, our bounds can be viewed as providing a guide for balancing between levels of redundancy and flexibility within the code. [10]{} L. Carlitz, *[On the number of distinct values of a polynomial with coefficients in a finite field]{}*, Proc. Japan Acad. **31** (1955), 119–120. S.D. Cohen, *The values of a polynomial over a finite field*, Glasgow Math. J. **14** (1973), 205–208. R.S. Coulter and M. Henderson, *Commutative presemifields and semifields*, Adv. Math. **217** (2008), 282–304. R.S. Coulter and R.W. Matthews, *Bent polynomials over finite fields*, Bull. Austral. Math. Soc. **56** (1997), 429–437. [to3em]{}, *[On the number of distinct values of a class of functions over a finite field]{}*, Finite Fields Appl. **17** (2011), 220–224. P. Dembowski and T.G. Ostrom, *Planes of order $n$ with collineation groups of order $n^2$*, Math. Z. **103** (1968), 239–258. C. Ding and J. Yuan, *[A family of skew Hadamard difference sets]{}*, J. Combin. Theory Ser. A **113** (2006), 1526–1535. J. Hall, *[Notes on Coding Theory]{}*,\ http://www.mth.msu.edu/$\sim$hall/classes/codenotes/coding-notes.html, 2010. G.M. Kyureghyan and A. Pott, *Some theorems on planar mappings*, Arithmetic of Finite Fields: Proceedings of the 2nd International Workshop, WAIFI 2008 (J. von zur Gathen, J.L. Iman[ã]{}, and C.K. Koç, eds.), Lecture Notes in Computer Science, vol. 5130, 2008, pp. 117–122. M. Nathanson, *[Additive Number Theory: Inverse Problems and the Geometry of Sumsets]{}*, Graduate Texts in Mathematics, vol. 165, Springer-Verlag, 1996. W. Qiu, Z. Wang, G. Weng, and Q. Xiang, *[Pseudo-Paley graphs and skew Hadamard difference sets from presemifields]{}*, Des. Codes Cryptogr. **44** (2007), 49–62. J. Solymosi, *[Bounding multiplicative energy by the sumset]{}*, Adv. Math. **222** (2009), 402–408. T. Tao and V. Vu, *Additive combinatorics*, Cambridge University Press, Cambridge, 2006. D. Wan, *[A $p$-adic lifting lemma and its applications to permutation polynomials]{}*, Finite Fields, Coding Theory, and Advances in Communications and Computing (New York), Lecture Notes in Pure and Applied Mathematics, vol. 141, Marcel Dekker, 1993, pp. 209–216. D. Wan, P.J-S. Shiue, and C-S. Chen, *Value sets of polynomials over finite fields*, Proc. Amer. Math. Soc. **119** (1993), 711–717.

--- abstract: 'The dynamical stability of nonstationary states of homogeneous spin-2 rubidium Bose-Einstein condensates is studied. The states considered are such that the spin vector remains parallel to the magnetic field throughout the time evolution, making it possible to study the stability analytically. These states are shown to be stable in the absence of an external magnetic field, but they become unstable when a finite magnetic field is introduced. It is found that the growth rate and wavelength of the instabilities can be controlled by tuning the strength of the magnetic field and the size of the condensate.' author: - 'H. Mäkelä and E. Lundh' title: 'Stability of nonstationary states of spin-$2$ Bose-Einstein condensates' --- Introduction ============ The physics of $F=2$ spinor Bose-Einstein condensates (BECs) started to gain the attention of both theorists and experimentalists during the last decade. The interest was motivated by the structure of $F=2$ condensates: being more complex than that of $F=1$ condensates, it made possible properties and phenomena which are not present in an $F=1$ system. One example of this can been seen in the structure of the ground states. The energy functional of an $F=2$ condensate is characterized by one additional degree of freedom compared to the $F=1$ case. This leads to a rich ground state manifold as now there are two free parameters parametrizing the ground states [@Ciobanu00; @Zheng10]. This should be contrasted with an $F=1$ condensate, where the ground state is determined by the sign of the spin-dependent interaction term [@Ho98; @Ohmi98]. Another difference can be seen in the structure of topological defects. It has been shown that non-commuting vortices can exist in an $F=2$ condensate [@Makela03], while these are not possible in an $F=1$ BEC [@Ho98; @Makela03]. The topological defects of $F=2$ condensates have been studied further by the authors of Refs. [@Makela06; @Huhtamaki09; @Kobayashi09]. Experimental studies of $F=2$ BECs have been advancing in the past ten years. Experiments on $F=2$ ${}^{87}$Rb atoms cover topics such as spin dynamics [@Schmaljohann04; @Chang04; @Kuwamoto04; @Kronjager06; @Kronjager10], creation of skyrmions [@Leslie09a], spin-dependent inelastic collisions [@Tojo09], amplification of fluctuations [@Klempt09; @Klempt10], spontaneous breaking of spatial and spin symmetry [@Scherer10], and atomic homodyne detection [@Gross11]. An $F=2$ spinor condensate of ${}^{23}$Na atoms has been obtained experimentally [@Gorlitz03], but it has a much shorter lifetime than $F=2$ rubidium condensates. In this work, we study the dynamical stability of nonstationary states of homogeneous $F=2$ spinor condensates. The stability of stationary states has been examined both experimentally [@Klempt09; @Klempt10; @Scherer10] and theoretically [@Martikainen01; @Ueda02]. Interestingly, the experimental studies show that the observed instability of the $|m_F=0\rangle$ state can be used to amplify vacuum fluctuations [@Klempt10] and to analyze symmetry breaking [@Scherer10] (see Refs. [@Lamacraft07; @Leslie09b] for related studies in an $F=1$ system). The stability of nonstationary states of spinor condensates, on the other hand, has received only little attention. Previous studies on the topic concentrate on $F=1$ condensates [@Matuszewski08; @Matuszewski09; @Matuszewski10; @Zhang05; @Makela11]. Here we extend the analysis of the authors of Ref. [@Makela11] to an $F=2$ rubidium condensate and present results concerning the magnetic field dependence of the excitation spectrum and stability. Although we concentrate on the stability of ${}^{87}$Rb condensates, many of the excitation spectra and stability conditions given in this article are not specific to rubidium condensates but have a wider applicability. We show that, in comparison with an $F=1$ system, the stability analysis of an $F=2$ condensate is considerably more complicated. This is partly due to the presence of a spin-singlet term in the energy functional of the latter system, but the main reason for the increased complexity is seen to be the much larger number of states available in an $F=2$ condensate. This article is organized as follows. Section \[sec:overview\] introduces the system and presents the Hamiltonian and equations of motion. In Sec. \[sec:stability\] the Bogoliubov analysis of nonstationary states is introduced. This method is applied to study the stability both in the presence and absence of a magnetic field. In this section it is also described how Floquet theory can be used in the stability analysis. In Sec. \[sec:g2not0\] the stability is studied under the (physically motivated) assumption that one of the interaction coefficients vanishes. Finally, Sec. \[sec:conclusions\] contains the concluding remarks. Theory of a spin-2 condensate {#sec:overview} ============================= The order parameter of a spin-$2$ Bose-Einstein condensate can be written as $\psi=(\psi_2,\psi_{1},\psi_{0},\psi_{-1},\psi_{-2})^T$, where $T$ denotes the transpose. The normalization is $\sum_{m=-2}^2|\psi_{m}|^2=n$, where $n$ is the total particle density. We assume that the trap confining the condensate is such that all the components of the hyperfine spin can be trapped simultaneously and are degenerate in the absence of magnetic field. This can be readily achieved in experiments [@Stamper-Kurn98]. If the system is exposed to an external magnetic field which is parallel to the $z$ axis, the energy functional reads $$\begin{aligned} \label{energy} &E[\psi] =\!\! \int d{\bm r} \left[ \langle \hat{h}\rangle +\frac{1}{2}\left(g_0 n^2 + g_1 \langle{\hat{\mathbf{F}}}\rangle^2 + g_2 |\Theta|^2\right)\right],\end{aligned}$$ where ${\hat{\mathbf{F}}}=({\hat{F}_x},{\hat{F}_y},{\hat{F}_z})$ is the (dimensionless) spin operator of a spin-2 particle. $\Theta$ describes singlet pairs and is given by $\Theta=2\psi_2\psi_{-2}-2\psi_1\psi_{-1}+\psi_0^2$. It can also be written as $\Theta=\psi^T e^{-i\pi \hat{F}_y}\psi$. The single-particle Hamiltonian $\hat{h}$ reads $$\begin{aligned} \label{h} \hat{h}= -\frac{\hbar^2 \nabla^2}{2m} + U(\mathbf{r}) -\mu-p{\hat{F}_z}+q{\hat{F}_z}^2. \end{aligned}$$ Here $U$ is the external trapping potential, $\mu$ is the chemical potential, and $p=-g\mu_{\rm B}B$ is the linear Zeeman term. In the last of these $g$ is the Landé hyperfine $g$-factor, $\mu_{\rm B}$ is the Bohr magneton, and $B$ is the external magnetic field. The last term in Eq. (\[h\]) is the quadratic Zeeman term, $q=-(g\mu_{\rm B}B)^2/E_{\rm hf}$, where $E_{\rm hf}$ is the hyperfine splitting. The sign of $q$ can be controlled experimentally by using a linearly polarized microwave field [@Gerbier06]. In this article we consider both positive and negative values of $q$. The strength of the spin-independent interaction is characterized by $g_0=4\pi \hbar^2(4a_2+3a_4)/7m$, whereas $g_1=4\pi \hbar^2(a_4-a_2)/7m$ and $g_2=4\pi\hbar^2[(a_0-a_4)/5-2(a_2-a_4)/7]$ describe spin-dependent scattering. Here $a_F$ is the $s$-wave scattering length for two atoms colliding with total angular momentum $F$. In the case of $^{87}$Rb, we calculate $g_0$ using the scattering lengths given in Ref. [@Ciobanu00], and $g_2$ and $g_4$ are calculated using the experimentally measured scattering length differences from Ref. [@Widera06]. Two important quantities characterizing the state $\psi$ are the spin vector $$\begin{aligned} {\mathbf{f}}(\mathbf{r})= \frac{\psi^\dag(\mathbf{r}) {\hat{\mathbf{F}}}\psi(\mathbf{r})}{n(\mathbf{r})},\end{aligned}$$ and the magnetization in the direction of the magnetic field $$\begin{aligned} \label{Mz} M_z= \frac{\int d\mathbf{r}\,n(\mathbf{r}) f_z (\mathbf{r})}{\int d\mathbf{r}\,n(\mathbf{r})}.\end{aligned}$$ The length of ${\mathbf{f}}$ is denoted by $f$. For rubidium the magnetic dipole-dipole interaction is weak and consequently the magnetization is a conserved quantity. The Lagrange multiplier related to the conservation of magnetization can be included into $p$. The time evolution equation obtained from Eq. (\[energy\]) is $$\begin{aligned} i\hbar \frac{\partial }{\partial t}\psi =\hat{H}[\psi] \psi,\end{aligned}$$ where $$\begin{aligned} \label{H} \hat{H}[\psi]= \hat{h}+ g_0 \psi^\dag\psi +g_1 \langle{\hat{\mathbf{F}}}\rangle\cdot{\hat{\mathbf{F}}}+ g_2 \Theta \hat{{\mathcal{T}}}. \end{aligned}$$ Here $\hat{{\mathcal{T}}}=e^{-i\pi \hat{F}_y}\hat{C}$ is the time-reversal operator, where $\hat{C}$ is the complex conjugation operator. Stability of nonstationary states when $g_2\not=0$ {#sec:stability} ================================================== The stability analysis is performed in a basis where the state in question is time independent. This requires that the time evolution operator of the state is known. As we are interested in analytical calculations, an analytical expression for this operator has to be known. To calculate the time evolution operator analytically, the Hamiltonian has to be time independent. In particular, the singlet term $\Theta$ should not depend on time. This is clearly the case if the time evolution of the state is such that $\Theta$ vanishes at all times, and we now study this case. We define a state $$\begin{aligned} \label{eq:psiparallel} \psi_{2;-1}= \sqrt{\frac{n}{3}} \begin{pmatrix} \sqrt{1+f_z}\\ 0\\ 0\\ \sqrt{2- f_z}\\ 0 \end{pmatrix},\quad -1\leq f_z\leq 2.\end{aligned}$$ For this state $\Theta=0$, $\langle {\hat{F}_x}\rangle=\langle {\hat{F}_y}\rangle=0$, and $\langle{\hat{F}_z}\rangle =f_z$. Furthermore, the populations of the state $\psi_{2;-1}$ remain unchanged during the time evolution determined by the Hamiltonian (\[H\]). Consequently, $\Theta=0$ throughout the time evolution. The state $\psi_{2;-1}$ with $f_z=0$, called the cyclic state, is a ground state at zero magnetic field [@Ciobanu00]. The creation of vortices with fractional winding number in states of the form $\psi_{2;-1}$ has been discussed by the authors of Ref. [@Huhtamaki09]. The stability properties of the state $\psi_{1;-2}=\sqrt{n}(0,\sqrt{2+f_z},0,0,\sqrt{1-f_z})^T/\sqrt{3}$ are similar to those of $\psi_{2;-1}$ and will therefore not be studied separately. The Hamiltonian giving the time evolution of $\psi_{2;-1}$ is $$\begin{aligned} \label{eq:Hparallel} \hat{H}[\psi_{2;-1}]=g_0 n-\mu +(g_1 n f_z - p){\hat{F}_z}+q{\hat{F}_z}^2,\end{aligned}$$ where we have set $U=0$ as the system is assumed to be homogeneous. This is of the same form as the Hamiltonian of an $F=1$ system discussed by the authors of Ref. [@Makela11]. The time evolution operator of $\psi_{2;-1}$ is given by $$\begin{aligned} \label{eq:Uparallel} \hat{U}_{2;-1}(t)= e^{-i t \hat{H}[\psi_{2;-1}]/\hbar}.\end{aligned}$$ We analyze the stability in a basis where the state $\psi_{2;-1}$ is time independent. In the new basis, the energy of an arbitrary state $\phi$ is given by [@Makela11] $$\begin{aligned} \label{Erot} E^{\textrm{new}}[\phi]&=E[{\hat{U}}_{2;-1}\phi]+i\hbar\langle\phi|\left(\frac{\partial}{\partial t}{\hat{U}}^{-1}_{2;-1}\right){\hat{U}}_{2;-1}\phi\rangle, \end{aligned}$$ and the time evolution of the components of $\phi$ can be obtained from the equation $$\begin{aligned} \label{variE} i\hbar\frac{\partial\phi_\nu}{\partial t}=\frac{\delta E^{\textrm{new}}[\phi]}{\delta\phi_\nu^*},\quad \nu =-2,-1,0,1,2. \end{aligned}$$ We replace $\phi\rightarrow \psi_{2;-1} +\delta\psi$ in the time evolution equation (\[variE\]) and expand the resulting equations to first order in $\delta\psi$. The perturbation $\delta\psi=(\delta\psi_2,\delta\psi_1,\delta\psi_0,\delta\psi_{-1},\delta\psi_{-2})^T$ is written as $$\begin{aligned} \delta\psi_j=\sum_{\mathbf{k}} \left[ u_{j;\mathbf{k}}(t)\,e^{i\mathbf{k}\cdot\mathbf{r}} -v_{j;\mathbf{k}}^{*}(t)\, e^{-i\mathbf{k}\cdot\mathbf{r}}\right],\end{aligned}$$ where $j=-2,-1,0,1,2$. Straightforward calculation gives the differential equation for the time evolution of the perturbations as $$\begin{aligned} i\hbar\frac{\partial}{\partial t} \begin{pmatrix} \mathbf{u}_\mathbf{k}\\ \mathbf{v}_\mathbf{k} \end{pmatrix} &=\hat{B}_{2;-1} \begin{pmatrix} \mathbf{u}_\mathbf{k}\\ \mathbf{v}_\mathbf{k} \end{pmatrix},\\ \label{HBG} \hat{B}_{2;-1} &=\begin{pmatrix} {\hat{X}}&- {\hat{Y}}\\ {\hat{Y}}^* & -{\hat{X}}^* \end{pmatrix},\end{aligned}$$ where $\mathbf{u}_\mathbf{k}=(u_{2;\mathbf{k}},u_{1;\mathbf{k}},u_{0;\mathbf{k}},u_{-1;\mathbf{k}}, u_{-2;\mathbf{k}})^T$, $\mathbf{v}_\mathbf{k}$ is defined similarly, and the $5\times 5$ matrices ${\hat{X}}$ and ${\hat{Y}}$ are $$\begin{aligned} \nonumber {\hat{X}}=&\,\, \epsilon_k + g_0 |\psi_{2;-1}\rangle\langle\psi_{2;-1}| +g_1 \!\!\!\!\sum_{j=x,y,z} |\psi_{2;-1}^j (t)\rangle\langle\psi_{2;-1}^j (t)| \\ \label{X} &+2g_2 |\psi_{2;-1}^\textrm{s} (t)\rangle\langle\psi_{2;-1}^\textrm{s} (t)|, \\ \label{Y} {\hat{Y}}=&\,\, g_0 |\psi_{2;-1}\rangle\langle\psi^*_{2;-1}| +g_1 \!\!\!\!\sum_{j=x,y,z} |\psi_{2;-1}^j (t)\rangle\langle (\psi_{2;-1}^{j})^*(t)|.\end{aligned}$$ Here we have defined $$\begin{aligned} \label{epsilonk} \epsilon_k &= \, \frac{\hbar^2 k^2}{2m},\\ \psi_{2;-1}^j (t) &= \, {\hat{U}}_{2;-1}^\dag(t)\hat{F}_j U_{2;-1} (t)\psi_{2;-1},\quad j=x,y,z,\\ \psi_{2;-1}^\textrm{s} (t) &= \, {\hat{U}}_{2;-1}^T(t)e^{-i\pi \hat{F}_y} U_{2;-1} (t)\psi_{2;-1}. \end{aligned}$$ In the rest of the article we call the operator determining the time evolution of the perturbations the Bogoliubov matrix. In the present case, $\hat{B}_{2;-1}$ is the Bogolibov matrix of $\psi_{2;-1}$. It is possible to write $\hat{B}_{2;-1}$ as a direct sum of three operators $$\begin{aligned} \hat{B}_{2;-1} (t) &={\hat{B}_{2;-1}^{4}}\oplus {\hat{B}_{2;-1}^3}(t) \oplus {\hat{B}_{2;-1}^{3'}}(t),\\ {\hat{B}_{2;-1}^{3'}}&=-({\hat{B}_{2;-1}^3})^*. \end{aligned}$$ Here ${\hat{B}_{2;-1}^{4}}$ is a time independent $4\times 4$ matrix and ${\hat{B}_{2;-1}^3}$ is a time-dependent $3\times 3$ matrix. The bases in which these operators are defined are given in Appendix \[sec:appendixa\]. The time-dependent terms of ${\hat{B}_{2;-1}^3}$ are proportional to $e^{\pm i k q t/\hbar}$, where $k=2,4$, or $6$, and consequently the system is periodic with minimum period $T=\pi\hbar/q$. Hence it is possible to use Floquet theory to analyze the stability of the system [@Makela11]. In the following we first calculate the eigenvalues of ${\hat{B}_{2;-1}^{4}}$, then those of ${\hat{B}_{2;-1}^3}$ and $ {\hat{B}_{2;-1}^{3'}}$ in the case $q=0$, and finally we discuss the general case $q\not =0$ using Floquet theory. Eigenvalues of ${\hat{B}_{2;-1}^{4}}$ ------------------------------------- First we calculate the eigenvalues and eigenvectors of ${\hat{B}_{2;-1}^{4}}$. This operator is independent of $q$. The eigenvalues are $$\begin{aligned} \nonumber \label{psi2m1omega1234} \hbar\omega_{1,2,3,4} &=\pm\Big[\epsilon_k\Big(\epsilon_k +g_0 n+g_1 n(2+f_z)\\ &\pm n\sqrt{[g_0-g_1 (2+f_z)]^2+4 g_0 g_1 f_z^2}\Big)\Big]^{1/2}.\end{aligned}$$ Here we use a labeling such that $++,-+,+-$, and $--$ correspond to $\omega_1,\omega_2,\omega_3$, and $\omega_4$, respectively. Now $\omega_{1,2}$ have a non-vanishing imaginary part only if $g_0$ and $g_1$ are both negative, while $\omega_{3,4}$ have an imaginary component if $g_0$ and $g_1$ are not both positive. Consequently, these modes are stable for rubidium for which $g_0,g_1>0$. The eigenvectors can be calculated straightforwardly, see Appendix \[sec:appendixa\]. The eigenvectors, like the eigenvalues, are independent of $g_2$. The perturbations corresponding to the eigenvectors of ${\hat{B}_{2;-1}^{4}}$ can be written as $$\begin{aligned} \delta\psi^{1,2,3,4}(\mathbf{r},\mathbf{k};t) = C_{1,2,3,4}(\mathbf{r},\mathbf{k};t)\,\psi_{2;-1}, \end{aligned}$$ where the $C_j$’s include all position, momentum, and time dependence. These change the total density of the condensate and are therefore called density modes. Eigenvalues of ${\hat{B}_{2;-1}^3}$ and ${\hat{B}_{2;-1}^{3'}}$ at $q=0$ ------------------------------------------------------------------------ In the absence of an external magnetic field ${\hat{B}_{2;-1}^3}$ is time independent. The eigenvalues of ${\hat{B}_{2;-1}^{3'}}$ can be obtained from those of ${\hat{B}_{2;-1}^3}$ by complex conjugating and changing the sign. For this reason we give only the eigenvalues of ${\hat{B}_{2;-1}^3}$: $$\begin{aligned} \label{psi2m1omega5} \hbar\omega_{5} &= \epsilon_k +2 g_2 n,\\ \label{psi2m1omega67} \hbar\omega_{6,7} &=\frac{1}{2}\Big[g_1 n f_z \pm \sqrt{(2\epsilon_k-g_1 n f_z)^2+ 16 g_1 n\epsilon_k}\Big].\end{aligned}$$ The eigenvalues $\hbar\omega_6$ and $\hbar\omega_7$ have a non-vanishing complex part if $g_1<0$. For rubidium all eigenvalues are real. There are two gapped excitations: at $\epsilon_k=0$ we get $\hbar\omega_5=2g_2 n$ and $\hbar\omega_{6} \,(\hbar\omega_{7})=g_1 nf_z$ if $g_1 f_z>0$ $(g_1 f_z<0)$. The eigenvectors are given in Appendix A. The corresponding perturbations become $$\begin{aligned} \label{omega3} \delta\psi^{5}(\mathbf{r},\mathbf{k};t) &= C_5\, \begin{pmatrix} 0\\ \sqrt{2-f_z}\\ 0\\ 0\\ -\sqrt{1+f_z} \end{pmatrix},\\ \delta\psi^{6,7}(\mathbf{r},\mathbf{k};t) &= \sum_\mathbf{k} C_{6,7} \, \begin{pmatrix} 0\\ e^{i \mathbf{k}\cdot \mathbf{r}} g_1 n\sqrt{(2-f_z)(1+f_z)}\\ e^{-i \mathbf{k}\cdot \mathbf{r}}\sqrt{\frac{3}{2}}(\epsilon_k+2 g_1 n-\hbar\omega_{6,7})\\ 0\\ e^{i \mathbf{k}\cdot \mathbf{r}} g_1 n (2-f_z), \end{pmatrix}\end{aligned}$$ where $C_{5,6,7}$ are functions of $\mathbf{r},\mathbf{k}$, and $t$. These modes change both the direction of the spin and magnetization and are therefore called spin-magnetization modes. Non-vanishing magnetic field ---------------------------- If $q\not =0$, the stability can be analyzed using Floquet theory due to the periodicity of ${\hat{B}_{2;-1}^3}$ [@Makela11]. We denote the time evolution operator determined by ${\hat{B}_{2;-1}^3}$ by $\hat{U}_{2;-1}^3$. According to the Floquet theorem (see, e.g., Ref. [@Chicone]), $\hat{U}_{2;-1}^3$ can be written as $$\begin{aligned} {\hat{U}}_{2;-1}^3(t)=\hat{M}(t) e^{-i t \hat{K}},\end{aligned}$$ where $\hat{M}$ is a periodic matrix with minimum period $T$ and $\hat{M}(0)=\hat{\textrm{I}}$, and $\hat{K}$ is some time-independent matrix. At times $t=nT$, where $n$ is an integer, we get ${\hat{U}}_{2;-1}^3(nT)=e^{-i n T \hat{K}}$. The eigenvalues of $\hat{K}$ determine the stability of the system. We say that the system is unstable if at least one of the eigenvalues of $\hat{K}$ has a positive imaginary part. We calculate the eigenvalues $\{\hbar\omega\}$ of $\hat{K}$ from the equation $$\begin{aligned} \hbar\omega=\hbar\omega^\textrm{r}+i \hbar\omega^\textrm{i}=i\frac{\ln\lambda}{T}, \end{aligned}$$ where $\{\lambda\}$ are the eigenvalues of ${\hat{U}}_{2;-1}^3(T)$. ![(Color online) The positive imaginary part $\omega^\textrm{i}$ related to $\hat{U}_{2;-1}(T)$ for different values of the magnetic field parameter $q$. The unit of $\omega^\textrm{i}$ is $|g_1|n/\hbar$. Note that the scales of $\epsilon_k$ and $\omega^\textrm{i}$ are not equal in the top and bottom rows. Note also that the scale of the $q=5|g_1|n$ figure is shifted with respect to the scale of the $q=|g_1|n$ case. The solid white line gives the approximate location of the fastest-growing instability, and the dashed white line corresponds to the largest possible size of a stable condensate, see Eq. (\[lambda2m1\]) and Table \[table\]. \[fig:psi2m1\]](psi2m1fig_rev.eps) We plot $\omega^\textrm{i}$ for several values of the magnetic field in Fig. \[fig:psi2m1\]. By comparing this to the case of a rubidium condensate with $g_2=0$, we found that the instabilities are essentially determined by $g_1$, the effect of $g_2$ is negligible. The eigenvectors of $\hat{U}_{2;-1}^3(T)$ correspond to perturbations which affect both spin direction and magnetization. With the help of numerical results we find that a good fitting formula is given by $$\begin{aligned} \hbar\omega^\textrm{i} \approx &\textrm{Im}\Big\{ \sqrt{(\epsilon_k+q)[\epsilon_k+q+\frac{5}{3}(2-f_z) g_1 n]}\Big\}, q<0,\\ \nonumber \approx &\textrm{Im}\Big\{\sqrt{(\epsilon_k-2q+ g_1 n)^2-\frac{4}{9}|(f_z-2)(f_z+1)g_1n|}\Big\},\\ & q>0.\end{aligned}$$ We see that for $q>0$ the fastest-growing instability is located approximately at $\epsilon_k=\max\{0,2q-g_1 n\}$ regardless of the value of $f_z$. For $q<0$ the location of this instability becomes magnetization dependent and is approximately given by $\epsilon_k=\max \{0, |q|-5(2-f_z)|g_1|n/6\}$. The values of $\epsilon_k$ corresponding to unstable wavelengths are bounded above approximately by the inequality $\epsilon_k\leq (3|q|+q)/2$. Therefore, the state $\psi_{2;-1}$ is stable if the condensate is smaller than the shortest unstable wavelength $$\begin{aligned} \lambda_{2;-1} =\frac{2\pi\hbar}{\sqrt{m(3|q|+q)}}. \label{lambda2m1}\end{aligned}$$ At $q=0$ the system is stable regardless of its size. Figure \[fig:psi2m1\] shows that the shape of the unstable region depends strongly on the sign of $q$. This can be understood qualitatively with the help of the energy functional of Eq. (\[energy\]). We choose $\psi_{\textrm{ini}}=\sqrt{n}|m_F=-1\rangle$ to be the initial state of the system and assume that the final state is of the form $$\begin{aligned} \psi_{\textrm{fin}}(x,y,z)= \begin{cases} \sqrt{n}|0\rangle, & x\, \textrm{mod}\, 2L \in [0,L),\\ \sqrt{n}|-2\rangle, & x\, \textrm{mod}\, 2L \in [L,2L). \end{cases}\end{aligned}$$ Then the energy of the initial state is $E_{\textrm{ini}}=g_1 n/2+q$ (dropping constant terms), while the energy of the final configuration is $E_{\textrm{fin}}=g_1 n+ 2q$. If $g_1,q>0$, $E_{\textrm{ini}}<E_{\textrm{fin}}$ and domain formation is suppressed for energetic reasons. If, on the other hand, $g_1>0$ and $q<0$, the energy of the final state is smaller than the energy of the initial state if $q<-g_1 n/2$ and domain formation is possible. Stability of nonstationary states when $g_2=0$ {#sec:g2not0} ============================================== For rubidium the value of $g_2$ is small in comparison with $g_0$ and $g_1$. Consequently, it can be assumed that this term has only a minor effect on the stability of the system. This assumption is supported by the results of the previous section. In the following we will therefore study the stability in the limit $g_2=0$. This makes it possible to obtain an analytical expression for the time evolution operator also for states other than $\psi_{2;-1}$. First we discuss a state that has three nonzero components, and then two states that have two nonzero components. Nonzero $\psi_2$, $\psi_0$, and $\psi_{-2}$ ------------------------------------------- We consider a state of the form $$\begin{aligned} \psi_{2;0;-2}= \frac{\sqrt{n}}{2}\begin{pmatrix} \sqrt{2-2 \rho_0+f_z}\\ 0\\ 2e^{i\theta} \sqrt{\rho_0}\\ 0\\ \sqrt{2-2 \rho_0-f_z} \end{pmatrix},\quad |f_z|\leq 2-2\rho_0\end{aligned}$$ For this $\langle\hat{F}_x\rangle=\langle\hat{F}_y\rangle=0$ and $\langle\hat{F}_z\rangle= nf_z$. The Hamiltonian and time evolution operator of this state are given by Eqs. (\[eq:Hparallel\]) and (\[eq:Uparallel\]), respectively. The equations determining the time evolution of the perturbations can be obtained from Eqs. (\[X\]) and (\[Y\]) by replacing $\psi_{2;-1}$ with $\psi_{2;0;-2}$ and setting $g_2=0$. In this way, we obtain a time dependent Bogoliubov matrix $\hat{B}_{2;0;-2}$, which is a function of the population of the zero component $\rho_0$. The Bogoliubov matrix can now be written as $$\begin{aligned} \label{eq:HB20m2} \hat{B}_{2;0;-2} (t) =\hat{B}_{2;0;-2}^6\oplus \hat{B}_{2;0;-2}^4(t), \end{aligned}$$ where $\hat{B}_{2;0;-2}^6$ is time independent and $\hat{B}_{2;0;-2}^4$ is periodic in time with period $T=\pi/q$. The bases in which these operators are defined are given in Appendix B. The eigenvalues of $ \hat{B}_{2;0;-2}^6$ are $$\begin{aligned} \hbar\omega_{1,2} =& \pm\epsilon_k,\\ \nonumber \hbar\omega_{3,4,5,6} =& \pm\Big[\epsilon_k^2 +\epsilon_k [g_0+4g_1(1-\rho_0)]n \\ &\pm \epsilon_k n\sqrt{[g_0-4g_1(1-\rho_0)]^2+4 g_0 g_1 f_z^2}\Big]^{1/2}.\end{aligned}$$ Here $++,-+,+-$, and $--$ correspond to $\omega_3,\omega_4,\omega_5$, and $\omega_6$, respectively. These eigenvalues are always real if $g_0$ and $g_1$ are positive. From the eigenvectors given in Appendix \[sec:appendixb\] we see that $\omega_{3,4}$ are density modes and $\omega_{1,2}$ and $\omega_{5,6}$ are magnetization modes. All these are gapless excitations. Note that the eigenvalues are independent of $\theta$. We discuss next the stability properties determined by $\hat{B}_{2;0;-2}^4$. We consider first the special case $\rho_0=0$ and proceed then to the case $\rho_0>0$. ### Stability at $\rho_0=0$ In the case $\rho_0=0$ a complete analytical solution of the excitation spectrum can be obtained. In Appendix \[sec:appendixb\] we show that by a suitable choice of basis the time dependence of the Bogoliubov matrix can be eliminated. The eigenvalues are $$\begin{aligned} \nonumber \label{psi20m2omega78910} &\hbar\omega_{7,8,9,10} = \frac{1}{2}\Big[\pm g_1 n f_z + 6q \\ &\pm\sqrt{4(\epsilon_k+g_1 n-3 q)^2 -(4-f_z^2)(g_1 n)^2} \Big]. \end{aligned}$$ These are gapped excitations and correspond to spin-magnetization modes (see Appendix \[sec:appendixb\]). If $g_1>0$, these eigenvalues have a non-vanishing complex part when $3q-2g_1 n\leq \epsilon_k \leq 3q$. This is possible only if $q$ is positive. The location of the fastest-growing unstable mode, determined by $\epsilon_k=\max\{0,3q - g_1 n\}$, is independent of $f_z$. The maximal width of the unstable region in the $\epsilon_k$ direction, obtained at $f_z=0$, is $2|g_1|n$. The state is stable if the system is smaller than the size given by $$\begin{aligned} \label{lambda20m2} \lambda_{2;0;-2} =\frac{2\pi\hbar}{\sqrt{6mq}},\quad q>0. \end{aligned}$$ If $q<0$, the state is stable regardless of the size of the condensate. In Fig. \[fig:psi20m2\] we plot the positive imaginary part of the eigenvalues (\[psi20m2omega78910\]) for various values of $q$. ![The positive imaginary part $\omega^\textrm{i}$ related to the eigenvalues (\[psi20m2omega78910\]) and to $\hat{U}_{2;0;-2}(T)$ for various values of the quadratic Zeeman term $q$ and population $\rho_0$. The unit of $\omega^\textrm{i}$ is $|g_1|n/\hbar$. We have chosen $f_z=0$ as this choice gives the fastest-growing instabilities and the smallest size of a stable condensate. In the top row the dashed, dotted, and solid lines correspond to $q=1,2,3$, respectively, while in the bottom row they correspond to $q=-1,-2,-3$, respectively. We have set $\theta=0$ in $\psi_{2;0;-2}$ as the stability was found to be independent of $\theta$. \[fig:psi20m2\]](psi20m2fig_rev.eps) ### Stability when $\rho_0>0$ In the case $\rho_0>0$ the stability can be studied using Floquet theory. The stability properties can be shown to be independent of the sign of $f_z$. At $q=0$ the operator $\hat{B}_{2;0;-2}^4$ is time independent. The eigenvalues can be obtained analytically but are not given here. The eigenvalues show that in the absence of magnetic field the state is stable in a rubidium condensate regardless of the value of $\rho_0$. Figure \[fig:psi20m2\] illustrates how the stability depends on the value of $q$ and the population $\rho_0$. We plot only the case $f_z=0$ as it gives the fastest-growing instabilities and the smallest size of a stable condensate. We found numerically that the stability properties are independent of the value of $\theta$. We have set $\theta=0$ in the calculations described here. If $q>0$, the amplitude $\omega^\textrm{i}$ of the short-wavelength instabilities is suppressed as $\rho_0$ increases. This can be understood with the help of the energy functional $$\begin{aligned} E_{2;0;-2} = \frac{1}{2} g_1 n f_z^2 + 8q (1-\rho_0).\end{aligned}$$ If $q>0$, the energy decreases as $\rho_0$ increases. Therefore there is less energy available to be converted into the kinetic energy of the domain structure. From the top row of Fig. \[fig:psi20m2\] it can be seen that Eq. (\[lambda20m2\]) gives an upper bound for the size of a stable condensate also when the value of $\rho_0$ is larger than zero. If $q<0$, the state is stable at $\rho_0=0$. The bottom row of Fig. \[fig:psi20m2\] shows that now $\psi_{2;0;-2}$ becomes more unstable as $\rho_0$ grows. This is natural because the energy $E_{2;0;-2}$ grows as $\rho_0$ increases, the energy surplus can be converted into kinetic energy of the domains. Figure \[fig:psi20m2\] shows that Eq. (\[lambda20m2\]) gives an upper bound for the size of a stable condensate also in the case $q<0$. Nonzero $\psi_1$ and $\psi_{-1}$ -------------------------------- As the next example we consider a state of the form $$\begin{aligned} \psi_{1;-1}= \sqrt{\frac{n}{2}}\begin{pmatrix} 0\\ \sqrt{1+f_z}\\ 0\\ \sqrt{1-f_z}\\ 0 \end{pmatrix},\quad |f_z|\leq 1.\end{aligned}$$ Also for this state $\langle\hat{F}_x\rangle=\langle\hat{F}_y\rangle=0$ and $\langle\hat{F}_z\rangle= nf_z$ and therefore the Hamiltonian and time evolution operator are given by Eqs. (\[eq:Hparallel\]) and (\[eq:Uparallel\]), respectively. The Bogoliubov matrix reads $$\begin{aligned} \label{eq:HB1m1} \hat{B}_{1;-1} (t) =\hat{B}_{1;-1}^6(t)\oplus \hat{B}_{1;-1}^4. \end{aligned}$$ Here $\hat{B}_{1;-1}^6(t)$ is time dependent with period $T=\pi/q$ and $\hat{B}_{1;-1}^4$ is time independent. The eigenvalues of $\hat{B}_{1;-1}^4$ are $$\begin{aligned} \nonumber \hbar\omega_{1,2,3,4} =&\pm\Big[\epsilon_k\Big(\epsilon_k+ (g_0+g_1)n\\ & \pm n\sqrt{(g_0-g_1)^2+4 g_0 g_1 f_z^2}\Big)\Big]^{1/2}.\end{aligned}$$ Now $++,-+,+-$, and $--$ correspond to $\omega_1,\omega_2,\omega_3$, and $\omega_4$, respectively. These are all gapless modes. For rubidium the eigenvalues are real. In Appendix C we show that $\omega_{1,2}$ are density modes and $\omega_{3,4}$ are magnetization modes. We now turn to the eigenvalues of $\hat{B}_{1;-1}^6$. At $q=0$ $\hat{B}_{1;-1}^6$ becomes time independent and the eigenvalues are $$\begin{aligned} \hbar\omega_{5,6} &= \pm\epsilon_k,\\ \nonumber \hbar\omega_{7,8,9,10} &= \pm\frac{1}{\sqrt{2}} \Big[2\epsilon_k^2+10\epsilon_k g_1 n +(g_1 n f_z)^2 \\ &\pm g_1 n \sqrt{(6\epsilon_k+g_1 n f_z^2)^2-8\epsilon_k f_z^2(4\epsilon_k-g_1 n)}\Big]^{1/2}. \end{aligned}$$ For rubidium these are all real. One of the eigenvalues $\hbar\omega_{4,5}$ has an energy gap $|g_1 n f_z|$. These eigenvalues describe spin-magnetization modes. For non-zero $q$ the stability can be analyzed using Floquet theory. As in the previous section, the fastest growing instabilities are obtained at $f_z=0$. This case can be studied analytically by changing basis as described in Appendix \[sec:appendixc\]. The eigenvalues for the case $f_z=0$ are $$\begin{aligned} \label{omega1m1a} &\hbar\omega_{5,6} = -3 q\pm \sqrt{(\epsilon_k+3 q)(\epsilon_k+ 2 g_1 n+3 q)},\\ \nonumber \label{omega1m1b} &\hbar\omega_{7,8,9,10} = 3q \pm \Big[(\epsilon_k+q)^2+4(\epsilon_k g_1 n+q^2)\\ &\pm 4\sqrt{[q^2+\epsilon_k (g_1 n+q)]^2-3 g_1 n q (\epsilon_k^2-q^2)}\Big]^{1/2}.\end{aligned}$$ These are gapped excitations with a magnetic-field-dependent gap. In more detail, at $\epsilon_k=0$ we get $\hbar\omega_{5,6}=-3q\pm\sqrt{3q(2 g_1 n+3q)}$ and $\hbar\omega_{7,8,9,10}=3q\pm\sqrt{5q^2\pm 4q\sqrt{q(3 g_1 n+q)}}$. For positive $q$, the fastest-growing instability is determined by $\omega_8^{\textrm{i}}$ and is located approximately at $\epsilon_k =\textrm{max}\{0,-3+q\}.$ For negative $q$ there are three local maxima for $\omega^{\textrm{i}}$. The one with the largest amplitude is given by $\omega^{\textrm{i}}_{7}$ and $\omega^{\textrm{i}}_{10}$ and is located at $\epsilon_k\approx \textrm{max}\{0,q^2(|q|-1)/(q^2+|q|+1)\}$. The second largest is given by $\omega^{\textrm{i}}_5$ and is at $\epsilon_k\approx \textrm{max}\{0,3|q|+1\}$. Finally, the instability with the smallest amplitude is related to $\omega^{\textrm{i}}_8$ and is at $\epsilon_k\approx \textrm{max}\{0,29 (10 |q|-1)/100\}$. In Fig. \[fig:psi1m1fig\] we plot the behavior of $\omega^\textrm{i}$ for $q=6$ and $q=-3$. From Eqs. (\[omega1m1a\]) and (\[omega1m1b\]) it can be seen (see also Fig. \[fig:psi1m1fig\]) that the state is stable if the size of the condensate is smaller than $$\begin{aligned} \label{lambda1m1} \lambda_{1;-1} =\frac{2\pi\hbar}{\sqrt{2m(2|q|-q)}}.\end{aligned}$$ ![The positive imaginary part $\omega^\textrm{i}$ of the eigenvalues (\[omega1m1a\]) and (\[omega1m1b\]) related to $\psi_{1;-1}$ for $q=3$ and $q=-6$. The unit of $\omega^\textrm{i}$ is $|g_1|n/\hbar$. We have chosen $f_z=0$ as it gives the fastest-growing instabilities and the smallest size of a stable system. The solid and dashed lines correspond to $\omega_8^\textrm{i}$ and $\omega^{\textrm{i}}_5$, respectively, while the dotted line gives $\omega^{\textrm{i}}_7$ and $\omega^{\textrm{i}}_{10}$ \[see Eqs. (\[omega1m1a\]) and (\[omega1m1b\])\]. \[fig:psi1m1fig\]](psi1m1fig_rev.eps) Nonzero $\psi_2$ and $\psi_{0}$ ------------------------------- As the final example we consider a state $$\begin{aligned} \psi_{2;0}= \sqrt{\frac{n}{2}}\begin{pmatrix} \sqrt{f_z}\\ 0\\ \sqrt{2-f_z}\\ 0\\ 0 \end{pmatrix},\quad 0\leq f_z\leq 2.\end{aligned}$$ As for other states considered in this article, now $\langle\hat{F}_x\rangle=\langle\hat{F}_y\rangle=0$ and $\langle\hat{F}_z\rangle= nf_z$ and the Hamiltonian and time evolution operator are given by Eqs. (\[eq:Hparallel\]) and (\[eq:Uparallel\]), respectively. We note that the stability properties of the states $\psi_{2;0}$ and $\psi_{0;-2}=\sqrt{n}(0,0,\sqrt{2-f_z},0,\sqrt{f_z})/\sqrt{2}$ are similar. Therefore the latter state will not be discussed in more detail. The Bogoliubov matrix of $\psi_{2;0}$ reads $$\begin{aligned} \label{eq:HB20} \hat{B}_{2;0} (t) =\hat{B}_{2;0}^2\oplus\hat{B}_{2;0}^4\oplus \hat{B}_{2;0}^{4'}(t),\end{aligned}$$ where only $\hat{B}_{2;0}^{4'}$ is time dependent (with period $T=\pi/q$). The eigenvalues of $\hat{B}_{0;2}^2$ and $\hat{B}_{0;2}^4$ are $$\begin{aligned} \hbar\omega_{1,2} = &\pm \epsilon_k, \\ \nonumber \hbar\omega_{3,4,5,6} =& \pm\Big[\epsilon_k^2+ \epsilon_k( g_0 n+2 g_1 n f_z)\\ &\pm \epsilon_k n\sqrt{(g_0-2 g_1 f_z)^2+4 g_0 g_1 f_z^2} \Big]^{1/2}.\end{aligned}$$ In the lower equation, $++, -+, +-$, and $--$ correspond to $\omega_3,\omega_4,\omega_5$, and $\omega_6$, respectively. These are gapless excitations. In Appendix \[sec:appendixd\] we show that $\omega_{3,4}$ correspond to density modes, while $\omega_{1,2,5,6}$ are magnetization modes. For rubidium, these are all stable modes. After a suitable change of basis the Bogoliubov matrix $\hat{B}_{2;0}^{4'}$ becomes time independent, see Appendix \[sec:appendixd\]. The eigenvalues of the new matrix are found to be $$\begin{aligned} \label{omega20} &\hbar\omega_{7,8,9,10} = \pm\frac{1}{\sqrt{2}}\sqrt{s_1 \pm \sqrt{(2\epsilon_k+g_1 n f_z +2 q)s_2}},\end{aligned}$$ where $$\begin{aligned} \nonumber &s_1 = 2 \epsilon_k^2+(g_1 n f_z)^2 +4 \epsilon_k [(3-f_z)g_1 n +q]\\ \nonumber &\hspace{0.7cm} - 8 f_z g_1 n q +2 q (6 g_1 n+5q),\\ \nonumber & s_2 = f_z (g_1 n)^2[24(\epsilon_k+q)-10 \epsilon_k f_z -18 q f_z+ g_1 n f_z^2]\\ \nonumber &\hspace{0.7cm} +32 q^2 [q+\epsilon_k +3 g_1 n (2- f_z)] - 16 g_1 n q \epsilon_k f_z.\end{aligned}$$ Now $++,-+,+-$, and $--$ are related to $\omega_7,\omega_8,\omega_9$, and $\omega_{10}$, respectively. These are gapped excitations and correspond to spin-magnetization modes. These modes can be unstable for rubidium; an example of the behavior of the positive imaginary component of $\omega_{7,8,9,10}$ is shown in Fig. \[fig:psi20fig\]. An upper bound for the size of a stable condensate is the same as in the case of $\psi_{1;-1}$, see Eq. (\[lambda1m1\]). With the help of Eq. (\[omega20\]) it can be seen that the fastest-growing instability is approximately at $\epsilon_k=\textrm{max}\{0,-2+0.9 q+0.04 f_z (1+q)\}$ when $q>0$ and at $\epsilon_k=\textrm{max}\{0,|q|-3+1.3 f_z-0.16 f_z^2\}$ when $q<0$. ![(Color online) The positive imaginary part $\omega^\textrm{i}$ of the eigenvalues (\[omega20\]) related to $\psi_{2;0}$ for $q=5$ and $q=-2$. The unit of $\omega^\textrm{i}$ is $|g_1|n/\hbar$. The solid white line gives the approximate location of the fastest-growing instability, and the dashed white line corresponds to the largest possible size of a stable condensate, see Table (\[table\]). \[fig:psi20fig\]](psi20fig_rev.eps) State $q$ Stable size Fastest-growing instability ($\epsilon_k$) ----------------- ------ ---------------------------------- -------------------------------------------------- $\psi_{2;-1},$ $>0$ $\frac{2\pi\hbar}{\sqrt{4mq}}$ $2q-g_1 n$ $\psi_{1;-2}$ $<0$ $\frac{2\pi\hbar}{\sqrt{2m|q|}}$ $|q|-\frac{5}{6}(2\mp f_z)|g_1|n$ $\psi_{2;0;-2}$ $>0$ $\frac{2\pi\hbar}{\sqrt{6mq}}$ $3q-g_1 n$ $<0$ $\infty$ - $\psi_{1;-1}$ $>0$ $\frac{2\pi\hbar}{\sqrt{2mq}}$ $q-3g_1 n$ $<0$ $\frac{2\pi\hbar}{\sqrt{6m|q|}}$ $\frac{q^2(|q|-g_1 n)}{q^2+|q| g_1 n+(g_1 n)^2}$ $\psi_{2;0},$ $>0$ $\frac{2\pi\hbar}{\sqrt{2mq}}$ $-2+0.9 q+0.04 |f_z| (1+q)$ $\psi_{0;-2}$ $<0$ $\frac{2\pi\hbar}{\sqrt{6m|q|}}$ $|q|-3+1.3 |f_z| -0.16 f_z^2$ : Summary of the results. Stable size gives the largest possible size of a stable homogeneous condensate and the fastest-growing instability indicates the approximate value of $\epsilon_k$ corresponding to the fastest growing instability. If $q$ is such that the $\epsilon_k$ given in the table is negative, the fastest-growing instability is at $\epsilon_k=0$. On the second line of the table, the $-$ sign holds for $\psi_{2;-1}$ and the $+$ sign for $\psi_{1;-2}$. \[table\] Conclusions {#sec:conclusions} =========== In this article, we have studied the dynamical stability of some nonstationary states of homogeneous $F=2$ rubidium BECs. The states were chosen to be such that the spin vector remains parallel to the magnetic field throughout the time evolution, making it possible to study the stability analytically. The stability analysis was done using the Bogoliubov approach in a frame of reference where the states were stationary. The states considered had two or three spin components populated simultaneously. These types of states were found to be stable in a rubidium condensate in the absence of a magnetic field, but a finite magnetic field makes them unstable. The wavelength and the growth rate of the instabilities depends on the strength of the magnetic field. The locations of the fastest-growing instabilities and the upper bounds for the sizes of stable condensates are given in Table \[table\]. For positive $q$, the most unstable state, in the sense that its upper bound for the size of a stable condensate is the smallest, is $\psi_{2;0;-2}$. However, this is the only state that is stable when $q$ is negative. For $q<0$, the states giving the smallest size of a stable condensate are $\psi_{1;-1}$ and $\psi_{2;0}$. In comparison with $F=1$ condensates, the structure of the instabilities is much richer in an $F=2$ condensate. In an $F=1$ system, there is only one type of a state whose spin is parallel to the magnetic field. The excitations related to this state can be classified into spin and magnetization excitations [@Makela11]. In the present system, there are many types of states which are parallel to the magnetic field; we have discussed six of these. In addition to the spin and magnetization excitations, there exist also modes which change spin and magnetization simultaneously. The increase in the complexity can be attributed to the number of components of the spin vector. The stability properties of the states discussed in this article can be studied experimentally straightforwardly. These states had two or three non-zero components, a situation which can be readily achieved by current experimental means [@Ramanathan11]. Furthermore, the stability of these states does not depend on the relative phases of the populated components, making it unnecessary to prepare states with specific relative phases. Finally, we note that the lifetime of an $F=2$ rubidium condensate is limited by hyperfine changing collisions [@Schmaljohann04]. Consequently, the instabilities are visible only if the their growth rate is large enough compared to the lifetime of the condensate. We also remark that the stability analysis was performed for a homogeneous condensate, whereas in experiments an inhomogeneous trapping potential is used. The stability properties can be sensitive to the shape of this potential [@Klempt09]. Eigenvectors of $\hat{B}_{2;-1}$ {#sec:appendixa} ================================ Here we give the (unnormalized) eigenvectors of ${\hat{B}_{2;-1}^{4}}$, ${\hat{B}_{2;-1}^3}$, and ${\hat{B}_{2;-1}^{3'}}$. Unlike ${\hat{B}_{2;-1}^{4}}$, the operators ${\hat{B}_{2;-1}^3}$ and ${\hat{B}_{2;-1}^{3'}}$ depend on the magnetic field. The eigenvectors of the latter two are given at $q=0$. The operators ${\hat{B}_{2;-1}^{4}}$, ${\hat{B}_{2;-1}^3}$, and ${\hat{B}_{2;-1}^{3'}}$ will not be given here explicitly as they can obtained straightforwardly from Eqs. (\[X\]) and (\[Y\]). However, we give the bases with respect to which these operators and their eigenvectors are defined. The matrix ${\hat{B}_{2;-1}^{4}}$ is given in the basis $\{\mathbf{b}_1,\mathbf{b}_4,\mathbf{b}_6,\mathbf{b}_9\}$, where $\mathbf{b}_j$ is a ten-component vector with the $j$:th element equal to one and all other elements equal to zero. The eigenvectors of ${\hat{B}_{2;-1}^{4}}$ are $$\begin{aligned} \mathbf{x}_j &=((\epsilon_k +\hbar\omega_j)\alpha_j, (\epsilon_k +\hbar\omega_j),(\epsilon_k -\hbar\omega_j)\alpha_j,\epsilon_k -\hbar\omega_j),\end{aligned}$$ where $j=1,2,3,4$ and $$\begin{aligned} \alpha_j &\equiv \sqrt{\frac{1+f_z}{2-f_z}}\frac{g_0+4 g_1}{g_0-2 g_1} \left(1+\frac{6 \epsilon_k g_0 g_1 n(2-f_z)}{(g_0+4 g_1)[\epsilon_k^2-(\hbar\omega_j)^2]}\right). \end{aligned}$$ The matrix ${\hat{B}_{2;-1}^3}$ is defined in the basis $\{\mathbf{b}_2,\mathbf{b}_5,\mathbf{b}_8\}$. At $q=0$ the eigenvectors are $$\begin{aligned} \mathbf{x}_5 =&(\sqrt{2-f_z},-\sqrt{1+f_z},0),\\ \nonumber \mathbf{x}_j =&(g_1 n\sqrt{(2-f_z)(1+f_z)},g_1 n (2-f_z),\\ &-\sqrt{\frac{3}{2}}(\epsilon_k+2 g_1 n-\hbar\omega_j)),\quad j=6,7.\end{aligned}$$ By defining ${\hat{B}_{2;-1}^{3'}}$ with respect to the basis $\{\mathbf{b}_7,\mathbf{b}_{10},\mathbf{b}_3\}$ we get ${\hat{B}_{2;-1}^{3'}}=-({\hat{B}_{2;-1}^3})^*$. Therefore, the eigenvectors of ${\hat{B}_{2;-1}^{3'}}$ can be obtained from those of ${\hat{B}_{2;-1}^3}$ by complex conjugating. Eigenvectors of $\hat{B}_{2;0;-2}$ {#sec:appendixb} ================================== The operator $\hat{B}_{2;0;-2}^6$ appearing in Eq. (\[eq:HB20m2\]) is given in the basis $\{\mathbf{b}_1,\mathbf{b}_3,\mathbf{b}_5,\mathbf{b}_{6},\mathbf{b}_{8},\mathbf{b}_{10}\}$. The eigenvectors of $\hat{B}_{2;0;-2}^6$ corresponding to $\omega_{1,2}$ are $$\begin{aligned} \nonumber \mathbf{x}_1 =& (\sqrt{\rho_0 \rho_{-2}},-e^{i \theta}\sqrt{\rho_2\rho_{-2}},\sqrt{\rho_0\rho_2},0,0,0),\\ \mathbf{x}_2 =& (0,0,0,\sqrt{\rho_0 \rho_{-2}},-e^{i \theta}\sqrt{\rho_2\rho_{-2}},\sqrt{\rho_0\rho_2}). \end{aligned}$$ These are magnetization modes as they change the magnetization but not the spin direction. The exact eigenvectors corresponding to $\omega_{3,4,5,6}$ are too complicated to be given here. Therefore we approximate $g_1\approx 0$ (for rubidium $g_1/g_0\approx 0.01$) and obtain $$\begin{aligned} \nonumber \mathbf{x}_{3,4} \approx & (\sqrt{\rho_2},e^{i \theta}\sqrt{\rho_0},\sqrt{\rho_{-2}},\\ &-\gamma_\pm \sqrt{\rho_{2}},-\gamma_\pm \, e^{-i \theta} \sqrt{\rho_0}, -\gamma_\pm \sqrt{\rho_{-2}}),\\ \mathbf{x}_5 \approx & [(2-f_z)\sqrt{\rho_2},-e^{i\theta} f_z\sqrt{\rho_{0}}, -(2+f_z)\sqrt{\rho_{-2}},0,0,0],\\ \mathbf{x}_6 \approx & [0,0,0,(2-f_z)\sqrt{\rho_2},-e^{i\theta} f_z\sqrt{\rho_{0}}, -(2+f_z)\sqrt{\rho_{-2}}],\end{aligned}$$ where $$\begin{aligned} \rho_{\pm 2} = & \frac{1}{4}(2-2 \rho_0 \pm f_z),\\ \label{gammapm} \gamma_\pm = & \frac{1}{g_0 n} \left[\epsilon_k+g_0 n\pm \sqrt{\epsilon_k (\epsilon_k+2 g_0 n)}\right]. \end{aligned}$$ Of these $\mathbf{x}_{3,4}$ are density modes and $\mathbf{x}_{5,6}$ are magnetization modes. The operator $\hat{B}_{2;0;-2}^4$ is given in the basis $\{\mathbf{b}_2,\mathbf{b}_4,\mathbf{b}_7,\mathbf{b}_9\}$. $\hat{B}_{2;0;-2}^4$ is time dependent, but at $\rho_0=0$ the time evolution determined by $\hat{B}_{2;0;-2}^4$ can be solved analytically. With the help of the unitary basis transformation $$\begin{aligned} V = \frac{1}{\sqrt{2}} \begin{pmatrix} e^{-3 i t q} & e^{-3 i t q} & 0 & 0 \\ 0 & 0 & e^{-3 i t q} & e^{-3 i t q} \\ 0 & 0 & e^{3 i t q} & -e^{3 i t q}\\ e^{3 i t q} & -e^{3 i t q} & 0 & 0 \end{pmatrix},\end{aligned}$$ we obtain a new Bogoliubov operator $$\begin{aligned} \hat{\bar{B}}_{2;0;-2}^4\Big|_{\rho_0=0} \equiv V^\dag \hat{B}_{2,0;-2}^4\Big|_{\rho_0=0} V +i\hbar\left(\frac{\partial}{\partial t} V^\dag\right)V, \end{aligned}$$ which is time independent. The eigenvectors of this operator are $$\begin{aligned} \nonumber \mathbf{x}_{7,8} = &(2\hbar\omega_{4,5} -g_1 n f_z-6 q,\\ & 2\epsilon_k+[2+\sqrt{4-f_z^2}] g_1 n-6 q,0,0),\\ \nonumber \mathbf{x}_{9,10} = &(0,0,2\hbar\omega_{6,7} + g_1 n f_z -6 q,\\ & 2\epsilon_k+[2+\sqrt{4-f_z^2}] g_1 n-6 q). \end{aligned}$$ These modes change both magnetization and spin direction. Eigenvectors of $\hat{B}_{1;-1}$ {#sec:appendixc} ================================ The operator $\hat{B}_{1;-1}^4$ is defined in the basis $\{\mathbf{b}_2,\mathbf{b}_4,\mathbf{b}_7,\mathbf{b}_9\}$. The eigenvectors are (in the limit $g_1=0$) $$\begin{aligned} \mathbf{x}_1 &=(\sqrt{1+f_z},\sqrt{1-f_z}, -\gamma_-\sqrt{1+f_z},-\gamma_-\sqrt{1-f_z}),\\ \mathbf{x}_2 &=(-\gamma_- \sqrt{1+f_z},-\gamma_- \sqrt{1-f_z}, \sqrt{1+f_z},\sqrt{1-f_z}),\\ \mathbf{x}_3 &=(\sqrt{1-f_z},-\sqrt{1+f_z},0,0),\\ \mathbf{x}_4 &=(0,0,\sqrt{1-f_z},-\sqrt{1+f_z}).\end{aligned}$$ Here $\gamma_\pm$ is defined as in Eq. (\[gammapm\]). Clearly $\mathbf{x}_{1,2}$ are density modes and $\mathbf{x}_{3,4}$ are magnetization modes. The operator $\hat{B}_{2;0;-2}^6$ is defined in the basis $\{\mathbf{b}_1,\mathbf{b}_3,\mathbf{b}_5,\mathbf{b}_6,\mathbf{b}_8,\mathbf{b}_{10}\}$. Here we give the eigenvectors at $f_z=0$. $$\begin{aligned} \mathbf{x}_{5,6} &= (\epsilon_k+ g_1 n-\hbar\omega_{5,6}, g_1 n,0,0,0,0),\\ \mathbf{x}_{j} &= (0,0,\alpha_j,\beta_j,\gamma_j,\delta_j),\quad j=7,8,9,10,\end{aligned}$$ where $\alpha_j,\beta_j,\gamma_j,\delta_j$ are too complex to be given here. These modes change both spin direction and magnetization. Eigenvectors of $\hat{B}_{2;0}$ {#sec:appendixd} =============================== The operators $\hat{B}_{2;0}^2,\hat{B}_{2;0}^4$, and $\hat{B}_{2;0}^{4'}$ are defined in the bases $\{\mathbf{b}_5,\mathbf{b}_{10}\}$,$\{\mathbf{b}_1,\mathbf{b}_3,\mathbf{b}_6,\mathbf{b}_8\}$, and $\{\mathbf{b}_2,\mathbf{b}_4,\mathbf{b}_7,\mathbf{b}_9\}$, respectively. The eigenvectors of $\hat{B}_{2;0}^2$ and $\hat{B}_{2;0}^4$ read $$\begin{aligned} \mathbf{x}_{1,2} =& \mathbf{b}_{5,10},\\ \mathbf{x}_{3,4} = &(-\gamma_\pm\sqrt{f_z},-\gamma_\pm\sqrt{2-f_z},\sqrt{f_z},\sqrt{2-f_z}), \\ \mathbf{x}_{5} = & (\sqrt{2-f_z},\sqrt{f_z},0,0),\\ \mathbf{x}_{6} = & (0,0,\sqrt{2-f_z},\sqrt{f_z}).\end{aligned}$$ Here $\gamma_\pm$ is defined as in Eq. (\[gammapm\]) and the index $3$ corresponds to $\gamma_+$ and the index $4$ to $\gamma_-$. Of these $\mathbf{x}_{3,4}$ are density modes and $\mathbf{x}_{1,2,5,6}$ are magnetization modes. When calculating $\mathbf{x}_{3,4,5,6}$ we have set $g_1=0$. The time dependence of the operator $\hat{B}_{2;0}^{4'}$ can be eliminated with the help of the basis transformation $$\begin{aligned} V = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & 0 & e^{-i t q} & 0 \\ 0 & 0 & 0 & e^{- i t q} \\ e^{i t q} & 0 & 0 & 0\\ 0 & e^{-3 i t q} & 0 & 0 \end{pmatrix}\end{aligned}$$ The eigenvectors $\mathbf{x}_{7,8,9,10}$ of the resulting time-independent operator describe spin-density modes and are too complicated to be given here. [1]{} C. V. Ciobanu, S.-K. Yip, and T.-L. Ho, Phys. Rev. A [**61**]{}, 033607 (2000). G.-P. Zheng, Y.-G. Tong, and F.-L. Wang, Phys. Rev. A [**81**]{}, 063633 (2010). T.-L. Ho, Phys. Rev. Lett. [**81**]{}, 742 (1998). T. Ohmi and K. Machida , J. Phys. Soc. Jap. [**67**]{}, 1822 (1998). H. Mäkelä, Y. Zhang, and K.-A. Suominen, J. Phys. A: Math. Gen. [**36**]{}, 8555 (2003). H. Mäkelä, J. Phys. A: Math. Gen. [**39**]{}, 7423 (2006). J. A. M. Huhtamäki, T. P. Simula, M. Kobayashi, and K. Machida, Phys. Rev. A [**80**]{}, 051601(R) (2009). M. Kobayashi, Y. Kawaguchi, M. Nitta, and M. Ueda, Phys. Rev. Lett. [**103**]{}, 115301 (2009). H. Schmaljohann, M. Erhard, J. Kronjäger, M. Kottke, S. van Staa, L. Cacciapuoti, J.J. Arlt, K. Bongs, and K. Sengstock, Phys. Rev. Lett. [**92**]{}, 040402 (2004). M.-S. Chang, C.D. Hamley, M.D. Barrett, J.A. Sauer, K.M. Fortier, W. Zhang, L. You, and M.S. Chapman, Phys. Rev. Lett. [**92**]{}, 140403 (2004). T. Kuwamoto, K. Araki, T. Eno, and T. Hirano, Phys. Rev. A [**69**]{}, 063604 (2004). J. Kronjäger, C. Becker, P. Navez, K. Bongs, and K. Sengstock, Phys. Rev. Lett. [**97**]{}, 110404 (2006). J. Kronjäger, C. Becker, P. Soltan-Panahi, K. Bongs, and K. Sengstock, Phys. Rev. Lett. [**105**]{}, 090402 (2010). L.S. Leslie, A. Hansen, K.C. Wright, B.M. Deutsch, and N.P. Bigelow, Phys. Rev. Lett. [**103**]{}, 250401 (2009). S. Tojo, T. Hayashi, T. Tanabe, T. Hirano, Y. Kawaguchi, H. Saito, and M. Ueda, Phys. Rev. A [**80**]{}, 042704 (2009). C. Klempt, O. Topic, G. Gebreyesus, M. Scherer, T. Henninger, P. Hyllus, W. Ertmer, L. Santos, and J. J. Arlt, Phys. Rev. Lett. [**103**]{}, 195302 (2009). C. Klempt, O. Topic, G. Gebreyesus, M. Scherer, T. Henninger, P. Hyllus, W. Ertmer, L. Santos, and J. J. Arlt, Phys. Rev. Lett. [**104**]{}, 195303 (2010). M. Scherer, B. Lücke, G. Gebreyesus, O. Topic, F. Deuretzbacher, W. Ertmer, L. Santos, J. J. Arlt, and C. Klempt, Phys. Rev. Lett. [**105**]{}, 135302 (2010). C. Gross, H. Strobel, E. Nicklas, T. Zibold, N. Bar-Gill, G. Kurizki, and M. K. Oberthaler, Nature (London) [**480**]{}, 219 (2011). A. Görlitz, T.L. Gustavson, A.E. Leanhardt, R. Löw, A.P. Chikkatur, S. Gupta, S. Inouye, D.E. Pritchard, and W. Ketterle, Phys. Rev. Lett. [**90**]{}, 090401 (2003). J.-P. Martikainen and K.-A. Suominen, J. Phys.B [**34**]{}, 4091 (2001). M. Ueda and M. Koashi, Phys. Rev. A [**65**]{}, 063602 (2002). A. Lamacraft, Phys. Rev. Lett. [**98**]{}, 160404 (2007). S. R. Leslie, J. Guzman, M. Vengalattore, J. D. Sau, M. L. Cohen, and D. M. Stamper-Kurn, Phys. Rev. A [**79**]{}, 043631 (2009). M. Matuszewski, T. J. Alexander, and Y. S. Kivshar, Phys. Rev. A [**78**]{}, 023632 (2008). M. Matuszewski, T. J. Alexander, and Y. S. Kivshar, Phys. Rev. A [**80**]{}, 023602 (2009). M. Matuszewski, Phys. Rev. Lett. [**105**]{}, 020405 (2010). W. Zhang, D. L. Zhou, M.-S. Chang, M. S. Chapman and L. You, Phys. Rev. Lett. [**95**]{}, 180403 (2005). H. Mäkelä, M. Johansson, M. Zelan, and E. Lundh, Phys. Rev. A [**84**]{}, 043646 (2011). D. M. Stamper-Kurn, M. R. Andrews, A. P. Chikkatur, S. Inouye, H.-J. Miesner, J. Stenger, and W. Ketterle, Phys. Rev. Lett. [**80**]{}, 2027 (1998). F. Gerbier, A. Widera, S. Fölling, O. Mandel, and I. Bloch, Phys. Rev. A [**73**]{}, 041602(R) (2006). A. Widera, F. Gerbier, S. Fölling, T. Gericke, O. Mandel, and I. Bloch, New. J. Phys. [**8**]{}, 152 (2006). C. Chicone, [*Ordinary Differential Equations with Applications*]{} (Springer, New York, 1999). A. Ramanathan, K. C. Wright, S. R. Muniz, M. Zelan, W. T. Hill, III, C. J. Lobb, K. Helmerson, W. D. Phillips, and G. K. Campbell, Phys. Rev. Lett. [**106**]{}, 130401 (2011).

--- abstract: 'The cardiovascular system is composed of the heart, blood and blood vessels. Regarding the heart, cardiac conditions are determined by the electrocardiogram, that is a noninvasive medical procedure. In this work, we propose autoregressive process in a mathematical model based on coupled differential equations in order to obtain the tachograms and the electrocardiogram signals of young adults with normal heartbeats. Our results are compared with experimental tachogram by means of Poincaré plot and dentrended fluctuation analysis. We verify that the results from the model with autoregressive process show good agreement with experimental measures from tachogram generated by electrical activity of the heartbeat. With the tachogram we build the electrocardiogram by means of coupled differential equations.' address: - '$^1$Instituto Federal de Educação, Ciência e Tecnologia do Paraná, Telêmaco Borba, PR, Brazil.' - '$^2$Pós-Graduação em Ciências, Universidade Estadual de Ponta Grossa, Ponta Grossa, PR, Brazil.' - '$^3$Departamento de Matemática e Estatística, Universidade Estadual de Ponta Grossa, Ponta Grossa, PR, Brazil.' - '$^4$Instituto de Física, Universidade de São Paulo, São Paulo, SP, Brazil.' - '$^5$Institute for Complex Systems and Mathematical Biology, Aberdeen, Scotland, UK.' - '$^6$Departamento de Física, Universidade Federal do Paraná , Curitiba, PR, Brazil.' - '$^7$Faculdade de Medicina de São José do Rio Preto, São José do Rio Preto, SP, Brazil.' author: - 'Ronaldo M Evaristo$^{1,2}$, Antonio M Batista$^{2,3,4,5}$, Ricardo L Viana$^6$, Kelly C Iarosz$^{4,5}$, José D Szezech Jr$^{2,3}$, Moacir F de Godoy$^7$' title: Mathematical model with autoregressive process for electrocardiogram signals --- heartbeat ,autoregressive model ,electrocardiogram Introduction ============ The cardiovascular system (CVS) is responsible for supplying the human organs with blood. It is composed by the heart, the arteries, and the veins. The heart has as function to pump blood throughout the body, that is realised by means of contractions [@mohrman14]. The human heart beats an average $72$ beats per minute and pumps 0.07 liters of blood per beat [@curtis89; @hall11]. The contraction and relaxation of the heart is obtained by a single cycle of the electrocardiogram signal (ECG), namely the ECG records of the electrical activity of the heart. Waller in 1887 [@sykes87] measured for the first time the electrical activity from the heart, and the first practical electrocardiograph was invented by Einthoven in 1901 [@ruiz08] that it was used as a tool for the diagnosis of cardiac abnormalities. In the recent past, several theoretical investigations pertaining to CVS have been carried out to analyse electrocardiogram signal [@gaetano09; @kudinov15; @schenone16]. Mathematical models have been developed to understand physiological function and disfunction in CVS. A mathematical model which have been used to generate ECG signals is the Van der Pol oscillator [@pol26]. Gois and Savi considered three modified Van der Pol oscillators connected by time delay coupling to describe heart rhythm behaviour [@gois09]. The coupled Van der Pol oscillators was also used in studies about the control of irregular behaviour in pathological heart rhythms [@ferreira14]. McSharry and collaborators [@mcsharry03] introduced a dynamical model to describe generating synthetic electrocardiogram signals. This model is based on a set of three ordinary differential equations in that it is incorporated the respiratory sinus arrhythmia (RSA) by means of a bimodal power spectrum consisting of the sum of two Gaussian distributions. In this work, instead two Gaussian distributions we propose an autoregressive (AR) process for the RSA to obtain the tachograms and, consequently, the ECGs of young adults with normal heartbeats. AR model can be used to quantify gains and delays by which cardiac interval, lung volume, blood pressure and sympathetic activity affect each other [@cohen02]. Cardiologists utilise AR model when they are interested in fit tachograms through mathematical regressions softwares. The tachogram is a signal that allows the study of heart rate variability (HRV) [@yasuma04; @acharya06]. Boardman and collaborators [@boardman02] study autoregressive model for the HRV. They found the optimum order of autoregressive model which can be used for spectral analysis of short segments of tachograms. We compare the results obtained from two Gaussian distributions and AR process with experimental tachograms of healthy young adults. To do that, we use as diagnostic tools the Poincaré plot [@guzik07] and detrended fluctuation analysis (DFA) [@peng94]. We have verified that the result with AR process agrees with the experimental tachogram more closely than the result with two Gaussian distributions. This way, we generate the ECG by means of coupled differential equations considering the AR process to obtain the tachogram and the frequency of the ECG. The frequency is an important parameter in the mathematical model for the ECG signal, and variation in the frequency produces variation in the times elapsed between successive heartbeats. This article is organised as follows: in Section 2 we introduce the ECG model with autoregressive process for tachograms, in Section 3 we compare our results with experimental data, and in the last Section we draw our conclusions. The mathematical model ====================== ![(Colour online) ECG following a normal hearbeat.[]{data-label="fig1"}](fig1.eps){height="5.5cm" width="11cm"} The ECG is a noninvasive method used to measure electrical activity of the heart through electrodes placed on the surface of the skin. Figure \[fig1\] shows the relationship between the cardiac conduction and the ECG. In the sinoatrial node (SA), known as natural pacemaker, the heartbeat starts. The atrioventricular node (AV) is responsible for the passage of electrical signals from the atria to the ventricles. At last, the signal arrives at the Purkinje fibers and makes the heart contract to pump blood, where the R-peak occurs. The time between successive R-peaks is the RR-interval and the series of RR-intervals is known as RR tachogram. McSharry and collaborators [@mcsharry03] argued that the heartbeat can be described by means of three coupled ordinary differential equations with the inclusion of RSA at the high frequencies ($f_{\rm RSA}$) and Mayer waves (MW) at the low frequencies ($f_{\rm MW}$). The equations are given by $$\begin{aligned} \label{eqgaus} {\dot x} & = & \alpha x-\omega y,\nonumber \\ {\dot y} & = & \alpha y+\omega x, \\ {\dot z} & = & z_0-z-\sum_i a_i\Delta\theta_i {\rm e}^{-\frac{\Delta\theta_i^2}{2b_i^2}}, \nonumber\end{aligned}$$ where $i\in\{P,Q,R,S,T\}$, $\alpha=1-\sqrt{x^2+y^2}$, $\Delta\theta_i=\theta-\theta_i$ (mod $2\pi$), $\theta={\rm atan}2(y,x)$ ($-\pi\leq {\rm atan}2(y,x)\leq \pi$), $z_0(t)=A\sin(2\pi f_{\rm RSA}t)$, and $A=0.15$mV. Visual analysis of a ECG from a normal individual is used to obtain the times, as well as the angles $\theta_i$, the $a_i$ and $b_i$ values for the PQRST points. The parameters values are given in Table \[pareq\] according to Reference [@mcsharry03]. index (i) time (s) $\theta_i$ (rad) $a_i$ $b_i$ ----------- ---------- ------------------ ------- ------- P -0.2 $-\pi/3$ 1.2 0.25 Q -0.05 $-\pi/12$ -5.0 0.1 R 0 0 30.0 0.1 S 0.05 $\pi/12$ -7.5 0.1 T 0.3 $\pi/2$ 0.75 0.4 : Parameters for Equation (\[eqgaus\]).[]{data-label="pareq"} The frequency $\omega(t)$ controls the variations in the RR-intervals, and it is given by $$\label{omega} \omega(t)=\frac{2\pi}{r(t)},$$ where $r(t)$ is the continuous version of the $r(n)$ time series which is obtained from the inverse discrete-time Fourier transform (DTFT) [@oppenheim10; @proakis07] of the power spectrum $$\label{eqbimodal} |H_G(f)|^2=\frac{\sigma^2_{\rm MW}}{\sqrt{2\pi c_{\rm MW}^2}} \exp \frac{(f-f_{\rm MW})^2}{2c^2_{\rm MW}}+ \frac{\sigma^2_{\rm RSA}}{\sqrt{2\pi c_{\rm RSA}^2}} \exp \frac{(f-f_{\rm RSA})^2}{2c^2_{\rm RSA}},$$ with phase randomly distributed from $0$ to $2\pi$ [@mcsharry03]. The tachogram exhibits similarity with a real one when the phase is randomly distributed. Figure \[fig2\](a) exhibits the power spectrum $|H_G(f)|$ for $f_{\rm MW}=0.1$Hz, $f_{\rm RSA}=0.25$Hz, $c_{\rm MW}=0.01$, $c_{\rm RSA}=0.01$, and $\sigma^2_{\rm MW}/\sigma^2_{\rm RSA}=0.5$. The values of the frequencies $f_{MW}$ and $f_{RSA}$ are motivated by the power spectrum of a real RR tachogram. The $f_{\rm MW}$ value is approximately equal to $0.1$ in humans, and it is related to arterial pressure occurring spontaneously in conscious subjects. RSA is characterised by the presence of oscillations in the tachogram considering the parasympathetic activity. It is synchronous with the breathing rate which for normal subjects is equal to $15$ breaths per minute, i.e., $f_{\rm RSA}=0.25$Hz. The spectrum has a bimodal form, where one peak is located in the low frequency range $0.04\leq f< 0.15$Hz and the other is located in the high frequency range $0.15\leq f\leq 0.4$Hz. These two bands appear due to the effects of both Mayer waves and RSA. The tachogram $r(n)$ is generated by the inverse DTFT from the power spectrum $|H_G(f)|$ with random phase. To obtain the continuous signal $r(t)$, first we increase the sample rate of $r(n)$ to the same sample rate of the desired ECG by interpolation [@mcsharry03; @oppenheim10; @proakis07]. Then, a R-peak detection algorithm [@mcsharry03] is applied to the interpolated signal to determine $r(t)$, as a result $\omega(t)$ (Eq. \[omega\]) is calculated and the ECG is built by means of Equation (\[eqgaus\]). In this work, in order to model electrocardiogram signals we propose an autoregressive (AR) stochastic process [@boardman02] to determine $\omega(t)$. The AR process of order $p$ is defined as $$R(n)=\sum_{l=1}^p d_lR(n-l)+\epsilon(n),$$ where $\epsilon(n)$ is a white noise with zero mean and unit variance. Boardman and collaborators [@boardman02] found that $p=16$ is an optimum order of autoregressive model for heart rate variability. The AR power spectrum density is $$\label{HAR} |H_{AR}(f)|=\frac{1}{\left|1-\sum_{l=1}^p d_l{\rm e}^{-{\rm i}2\pi fl}\right|},$$ with the coefficients values given in Table \[tabd\]. The coefficients values for the AR power spectrum density have been adjusted to be used for the set of data that we obtained from healthy individuals. The presence of arrhythmia can influence the coefficients values, and as a consequence it would be necessary to calculate the new coefficients values. ----------------- ------------------ ------------------ ----------------- $d_{1}=-0.9099$ $d_{2}=0.5188$ $d_{3}=-0.2840$ $d_{4}=-0.2063$ $d_{5}=0.0382$ $d_{6}=0.0709$ $d_{7}=0.0305$ $d_{8}=-0.1533$ $d_{9}=0.0009$ $d_{10}=-0.0070$ $d_{11}=-0.0218$ $d_{12}=0.0043$ $d_{13}=0.0316$ $d_{14}=0.0155$ $d_{15}=-0.0591$ $d_{16}=0.0252$ ----------------- ------------------ ------------------ ----------------- : Coefficients values for the AR power spectrum density (Eq. \[HAR\]).[]{data-label="tabd"} Figure \[fig2\](b) shows the power spectrum calculated from the tachogram generated by means of the AR process. In placed of the two separated Gaussian distributions, the AR process produces a damped in the power spectrum, and as consequence the separation between the frequency components cannot be exactly identified, as in real situation. This way, with the tachogram $r(n)$ we find $r(t)$ (interpolation followed by R-peak detection) and $\omega(t)$ to build the ECG using Equation (\[eqgaus\]). ![Power spectrum from (a) Equation (\[eqbimodal\]) and (b) Equation (\[HAR\]).[]{data-label="fig2"}](fig2.eps){height="7cm" width="8cm"} Results and discussions ======================= In this work, the power spectrum is considered to obtain the theoretical tachogram and consequently the frequency (Eq. \[omega\]) that is used in Equation (\[eqgaus\]) to build the ECG. We generate $124$ theoretical ECG signals, being $62$ from the Gaussian spectrum and $62$ from the AR spectrum. Then, we obtain their respective tachograms and compare them with $62$ experimental tachograms collected from healthy adults. The experimental protocol consisted of $20$min of monitoring of the heartbeat frequency in resting from patients without sound and visual stimulations in a supine rest position. The heartbeats were recording with a sample rate of $1000$Hz, and $1000$ RR intervals were analysed. The experimental tachograms were filtered to remove ectopic beats and noise effects. We calculate the power spectrum from Gaussian distributions (blue), AR process (red), and experimental data (green), as shown in Figure \[fig3\](a). The green dashed lines exhibit the confidence interval of the experimental power spectrum that are calculated from $62$ experimental tachograms. ![(Colour online) (a) Power spectrum and (b) dentrended fluctuation analysis (DFA) for Gaussian distributions (blue), AR process (red), and experimental data (green). The green dashed lines exhibit the confidence interval of the experimental power spectrum that are calculated from $62$ experimental tachograms.[]{data-label="fig3"}](fig3.eps){height="6.5cm" width="8cm"} With regard to Figure \[fig3\](a), we see that the power spectrum from AR process has a shape closer the experimental result than the power spectrum from the Gaussian distributions. This way, in order to verify the agreement among the experimental tachogram and the tachograms obtained from theoretical power spectra we have utilised the dentrended fluctuation analysis (DFA) [@hu01; @chen02]. DFA yields a fluctuation function $F(k)$ as a function of $k$, given by [@peng94] $$F(k)=\sqrt{\frac{1}{N}\sum_{n=0}^{N-1}[r_I(n)-r_k(n)]^2},$$ where $N=1000$, $r_I(n)=\sum_{l=0}^{n}r(l)$ is the cumulative sum of the $r(n)$, $k$ is the box size that partitions the time interval of the tachogram, and $r_k(n)$ is the local trend in each box. DFA is a nonlinear dynamical analysis that have been used for the understanding of biological systems [@peng94]. Moreover, the DFA allows the detection of long-range correlations embedded in a patchy landscape. Figure \[fig3\](b) shows the mean DFAs for $4<k<100$, where linear regressions present slopes equal to $0.1955\pm 0.0150$, $0.7034\pm 0.0686$, and $0.6817\pm 0.2448$ for Gaussian distributions, AR process and experimental data, respectively. The experimental data are collected from $62$ healthy young adults. Each one with length equal to $1000$ heartbeats without trend removal. In general, ectopic beats or noise effects are excluded of time series through filters [@santos13]. As a result, the DFA for the tachogram generated by the experimental data and AR process are in close agreement with each other. Whereas DFA for the Gaussian distributions exhibits a good agreement only for $k<20$. ![(Colour online) Tachograms generated by (a) the sum of two Gaussian distributions, (b) the AR process, (c) experimental data, and respective Poincaré plots in (d), (e), and (f).[]{data-label="fig4"}](fig4.eps){height="7.5cm" width="12cm"} Through the Gaussian power spectrum with phases randomly distributed between $0$ and $2\pi$ we build a ECG and extract the tachogram $r_G(n)$ shown in Figure \[fig4\](a). A tachogram extracted of ECG generated by the AR spectrum $r_{AR}(n)$ is illustrated in Figure \[fig4\](b) and an experimental tachogram $r_{EXP}(n)$ of the healthy adult is in Figure \[fig4\](c). In Figures \[fig4\](d), \[fig4\](e), and \[fig4\](f) we calculate the respective Poincaré plots. The Poincaré plot is a visualising technique to analyse RR intervals, where it is computed the standard deviation of points perpendicular to the axis (SD1) and points along (SD2) the axis of line-of-identity. Table \[tabSD\] exhibits the mean SD1 and SD2 values of the tachograms shown in Figure \[fig4\]. Comparing the Poincaré plots we see that both SD1 and SD2 for the AR process agree with the experimental results better than the method based on the Gaussian distribution. In addition, we calculate the $p$-values according to the two-sided Wilcoxon rank sum test [@wilcoxon45] of the SD1 and SD2 time series from the simulated and experimental data. This statistical test verifies if two paired time series have the same distribution when the data cannot be assumed to be normally distributed. We find that the $p$-values of SD1 and SD2 are greater than $0.05$ for the AR process, consequently the time series of SD1 and SD2 in the AR process and the experimental data have the same distributions. However, the same does not happen for the Gaussian distributions, where the $p$-values are less than $0.05$. RR-intervals $r_G(62)$ \[A\] $r_{AR}(62)$ \[B\] $r_{EXP}(62)$ \[C\] $p$-value -------------- ----------------- -------------------- --------------------- ---------------------- SD1 \[ms\] $63.9\pm 1.9$ $32.8\pm 10.5$ $38.8\pm 19.6$ $0.0$ \[A$\times$C\] $0.2$ \[B$\times$C\] SD2 \[ms\] $103.4\pm 2.6$ $82.0\pm 26.2$ $79.0\pm 28.5$ $0.0$ \[A$\times$C\] $0.6$ \[B$\times$C\] : Mean SD1 and SD2 values for the theoretical and experimental tachograms with $p$-values of the Wilcoxon rank sum test.[]{data-label="tabSD"} ![ECG signal generated by means of Equation (\[eqgaus\]) considering AR process.[]{data-label="fig5"}](fig5.eps){height="5cm" width="8cm"} All in all, we obtain the tachogram $r(n)$ (discrete-time) by means of the AR process. Then, an interpolation followed by a R-peak detection allows us to determine the signal $r(t)$ (continuous-time). As a result, we calculate $\omega(t)$ and solve the ordinary differential equations to build the ECG signal. We use the fourth order Runge-Kutta method to solve the ordinary differential equations, where we consider a fixed time step $\Delta t=1/f_s$ and sampling frequency $f_s=256$Hz. Figure \[fig5\] shows the ECG signal in the time interval $0\leq t\leq 20$s, where $z(t)$ yields a synthetic ECG with a realistic PQRST morphology according to Figure \[fig1\]. Conclusions =========== In conclusion, we have studied a mathematical model given by coupled differential equations that describes electrocardiogram signals and it was proposed by McSharry and collaborators [@mcsharry03]. In the original model, the frequency is calculated from a power spectrum with two Gaussian distributions that incorporates both respiratory sinus arrhythmia and Mayer waves. The use of two Gaussian distributions is in disagreement with our experimental data obtained from healthy adults, due to the fact that the two distributions are not well separated in the power spectrum. This way, we propose to calculate the frequency by means of the AR process. We believe that our model also allows the study of respiratory sinus arrhythmia. We verify that the power spectrum from AR process has a good agreement with the power spectrum from experimental data. We have also compared the tachograms generated from Gaussian distributions and AR process with the experimental tachogram using DFA and Poincaré plot. As a result, in both DFA and Poincaré plot, the tachogram generated considering the AR process is in closer agreement with experimental data than the two Gaussian. As a consequence, with the tachogram, the frequency is calculated and the ECG signal can be built utilising coupled differential equations with AR process. We believe that the mathematical model with autoregressive process constitutes an important step toward developing strategies to simulate electrocardiogram signals. We have considered many experimental tachograms from healthy adults to obtain the parameters, this way our simulations allow to obtain and also to do forecast of electrocardiogram signals of individuals in the same situation analysed in this work. We have tested our results using experimental tachograms obtained from $62$ healthy patients without sound and visual stimulations in a supine rest position. In future works, we plan to do the same analyse considering patients with different clinical conditions. Acknowledgements {#acknowledgements .unnumbered} ================ This study was possible by partial financial support from the following agencies: Fundação Araucária, Brazilian National Council for Scientific and Technological Development (CNPq), Coordination for the Improvement of Higher Education Personnel (CAPES), and São Paulo Research Foundation (FAPESP) process numbers 2011/19296-1, 2015/07311-7, and 2016/16148-5. [00]{} Mohrman DM, Heller LJ. Cardiovascular physiology. New York: McGraw-Hill Education; 2014. Curtis H. Biology. New York: Worth; 1989. Hall J. Guyton and hall textbook of medical physiology. Philadelphia: Elsevier; 2011. Sykes AH. A D Waller and the electrocardiogram. Br Med J 1987; 294:1396-1398. Rivera-Ruiz M, Cajavilca C, Varon J. Einthoven’s string galvanometer. Tex Heart Inst J 2008;35:174-178. de Gaetano A, Panunzi S, Rinaldi F, Risi A, Sciandrone M. A patient adaptable ECG beat classifier based on neural networks. Appl Math Comput 2009;213:243-249. Kudinov AN, Lebedev DY, Tsvetkov VP, Tsvetkov IV, Mathematical model of the multifractal dynamics and analysis of heart rates. Math Models Comput Simul 2015;7:214-221. Schenone E, Colin A, Gerbeau J-F. Numerical simulation of electrocardiograms for full cardiac cycles in healthy and pathological conditions. Int J Numer Meth Biomed Engng 2016;32:e02744. Van der Pol B. On relaxation oscillations. Philos Mag 1926;7:978-992. Gois SRFSM, Savi MA. An analysis of hearth rhythm dynamics using a three-coupled oscillator model. Chaos Solitons Fractals 2009;41:2553-2565. Ferreira BB, Savi MA, de Paula AS. Chaos control applied to cardiac rhythms represented by ECG signals. Phys Scripta 2014;89:105203. McSharry PE, Clifford GD, Tarassenko L, Smith LA. A dynamical model for generating synthetic electrocardiogram signals. IEEE Trans Biomed Eng 2003;50:289-294. Cohen MA, Taylor JA. Short-term cardiovascular oscillations in man: measuring and modelling the physiologies. J Physiol 2002;542:669-683. Yasuma F, Hayano J. Why does the heartbeat synchronize with respiratory rhythm? Chest 2004;125:683-690. Acharya UR, Joseph KP, Kannathal N, Lim CM, Suri JS. Heart rate variability: a review. Med Bio Eng Comput 2006;44:1031-1051. Boardman A, Schlindwein FS, Rocha AP, Leite A. A study on the optimum order of autoregressive models for heart rate variability. Physiol Meas 2002;23:325-336. Guzik P, Piskorski J, Krauze T, Schneider R, Wesseling KH, Wykretowicz A, Wysocki H. Correlations between the Poincaré plot and conventional heart rate variability parameters assessed during paced breathing. J Physiol Sci 2007;57:63-71. Peng C-K, Buldyrev SV, Havlin S, Simons M, Stanley HE, Goldberger AL. Mosaic Organization of DNA Nucleotides. Phys Rev E 1994;49:1685-1689. Oppenheim AV, Schafer RW. Discrete-time signal processing. New Jersey: Pearson Prentice Hall; 2010. Proakis JG, Manolakis DG. Digital signal processing. New Jersey: Pearson Prentice Hall; 2007. Hu K, Ivanov PCh, Chen Z, Carpena P, Stanley HE. Effect of trends on detrended fluctuation analysis. Phys Rev E 2001;64:011114. Chen Z, Ivanov PCh, Hu K, Stanley HE. Effect of nonstationarities on detrended fluctuation analysis. Phys Rev E 2002;65:041107. Santos L, Barroso JJ, Macau EEN, Godoy MF. Application of an automatic adaptive filter for Heart Rate Variability analysis. Med Eng Phys 2013;35:1778-1785. Wilcoxon F. Individual comparisons by ranking methods. Biometrics Bulletin 1945;1:80-83.

--- abstract: 'Collimated supersonic flows in laboratory experiments behave in a similar manner to astrophysical jets provided that radiation, viscosity, and thermal conductivity are unimportant in the laboratory jets, and that the experimental and astrophysical jets share similar dimensionless parameters such as the Mach number and the ratio of the density between the jet and the ambient medium. When these conditions apply, laboratory jets provide a means to study their astrophysical counterparts for a variety of initial conditions, arbitrary viewing angles, and different times, attributes especially helpful for interpreting astronomical images where the viewing angle and initial conditions are fixed and the time domain is limited. Experiments are also a powerful way to test numerical fluid codes in a parameter range where the codes must perform well. In this paper we combine images from a series of laboratory experiments of deflected supersonic jets with numerical simulations and new spectral observations of an astrophysical example, the young stellar jet HH 110. The experiments provide key insights into how deflected jets evolve in 3-D, particularly within working surfaces where multiple subsonic shells and filaments form, and along the interface where shocked jet material penetrates into and destroys the obstacle along its path. The experiments also underscore the importance of the viewing angle in determining what an observer will see. The simulations match the experiments so well that we can use the simulated velocity maps to compare the dynamics in the experiment with those implied by the astronomical spectra. The experiments support a model where the observed shock structures in HH 110 form as a result of a pulsed driving source rather than from weak shocks that may arise in the supersonic shear layer between the Mach disk and bow shock of the jet’s working surface.' author: - 'P. Hartigan , J. M. Foster B. H. Wilde R. F. Coker P. A. Rosen J. F. Hansen B. E. Blue R. J. R. Williams R. Carver A. Frank' title: 'Laboratory Experiments, Numerical Simulations, and Astronomical Observations of Deflected Supersonic Jets: Application to HH 110' --- Introduction ============ Collimated supersonic jets originate from a variety of astronomical sources, including active galactic nuclei [@agnref], several kinds of interacting binaries [@binaryjetref], young stars [@ysojetref], and even planetary nebulae [@pnjetref]. Most current jet research focuses on how accretion disks accelerate and collimate jets, or on understanding the dynamics of the jet as it generates shocks along its beam and in the surrounding medium. Both areas of research have broad implications for astrophysics. Models of accretion disks typically employ magnetized jets to remove the angular momentum of the accreting material. The distribution and transport of the angular momentum in an accretion disk affects its mass accretion rate, mixing, temperature profile, and density structure, and in the case of young stars, also helps to define the characteristics of the protoplanetary disk that remains after accretion ceases. At larger distances from the source, shock waves in jets clear material from the surrounding medium, provide insights into the nature of density and velocity perturbations in the flow, and enable dynamical studies of mixing, turbulence and shear. Jets from young stars are particularly good testbeds for investigating all aspects of the physics within collimated supersonic flows [see @ray07 for a review]. Shock velocities within stellar jets are low enough that the gas cools by radiating emission lines rather than by expanding. Relative fluxes of the emission lines determine the density, temperature, and ionization of the postshock gas, while the observed Doppler shifts and emission line profiles define the radial velocities and nonthermal motions within the jet [see @hartigan08 for a review]. Moreover, many stellar jets are located relatively close to the Earth, so that one can observe proper motions in the plane of the sky from observations separated by several years [@heathcote92]. Combining this information with radial velocity measurements gives the orientation of the flow to the line of sight. Using the Hubble Space Telescope, one can observe morphological changes of knots within jets, and follow how these changes evolve in real time [@hartigan01]. Results from these studies show that internal shock waves, driven by velocity variations in the flow, sweep material in jets into a series of dense knots. Typical internal shock velocities are $\sim$ 40 km$\,$s$^{-1}$, or $\sim$ 20% of the flow speed. In several cases it is easy to identify both the bow shock and the Mach disk from emission line images. Because stellar jets are mostly neutral, strong H$\alpha$ emission occurs at the shock front where neutral H is collisionally excited [@heathcote96]. Forbidden line radiation occurs in a spatially-extended cooling zone in the postshock material. Temperatures immediately behind the internal shocks can exceed $10^5$K, but the gas in the forbidden-line-emitting cooling zone is typically 8000 K. Young stars show a strong correlation between accretion and outflow, leading to the idea that accretion disks power the outflows [@heg95; @cabrit07]. Most current models use magnetic fields in the disk to launch a fraction of the accreting material from the disk into a collimated magnetized jet [@ferreira06]. Stellar jets often precess, and there is some evidence that they rotate [@ray07], although rotation signatures are difficult to measure because the rotational velocities are typically only a few percent of the flow speeds and precession can mimic rotational signatures [@cerquieriaref]. In all cases proper motion measurements show that jets move radially away from the source. Jets show no dynamical evidence for kink instabilities, and in fact while magnetic fields may dominate in the acceleration regions of jets, they appear to play a minor role in the dynamics at the distances where most jet knots are observed [@hartigan07]. In the cooling zones behind the shock waves the plasma $\beta$ can drop below unity, so that magnetic fields dominate thermal pressure in those areas. However, the magnetic pressure is small compared with the ram pressure of the jet. While more unusual than internal working surfaces produced by velocity perturbations, shocks also occur when jets collide with dense obstacles such as a molecular cloud. When the obstacle is smaller than the jet radius, it becomes entrained by the jet, and a reverse bow shock or ‘cloudlet shock’ forms around the obstacle [@schwartz78; @l1551]. Alternatively, when a large obstacle like a molecular cloud deflects the jet, a quasi-stationary deflection shock at the impact point forms, followed by a spray of shocked jet material downstream. The classic example of such a jet is HH 110 [@riera03; @lopez05]. Though the observations summarized above provide a great deal of information about stellar jets, several important questions remain unanswered. The basic mechanism by which disks load material onto field lines (assuming the MHD disk scenario is correct), and the overall geometry of this wind is unknown, and the roles of reconnection and ambipolar diffusion in heating the jet close to the star are unclear. At larger distances, the magnetic geometry and its importance in shaping the internal working surfaces is poorly-constrained, as are the time scales and spatial scales associated with mixing in supersonic shear layers and working surfaces. The inherently clumpy nature of jets also affects the flow dynamics and observed properties of jets in uncertain ways, and the degree to which fragmentation and turbulence influence the morphologies of jets is unknown. Developing laboratory analogs of stellar jets could help significantly in addressing the questions above. Observations of a specific astronomical jet are restricted to a small range of times and to a particular observing angle, while laboratory experiments have no such restrictions. In principle, one could explore a wide range of initial collimations, velocity and density structures within the jet, as well as densities, geometries, and magnetic field configurations with laboratory experiments. The experiments also provide a powerful and flexible way to test 3-D numerical fluid codes, and to investigate how real flows develop complex morphologies in 3-D. The challenge is to design an experiment that is relevant to the astrophysical case of interest. Laboratory experiments differ by 15 $-$ 20 orders of magnitude in size, density and timescale from stellar jets, but because the Euler equations that govern fluid dynamics involve only three variables, time, density, and velocity, the behavior of the fluid is determined primarily by dimensionless numbers such as the Mach number (supersonic or subsonic), Reynolds number (viscous or inertial), and Peclet number (importance of thermal conduction). If the experiments behave as a fluid and have similar dimensionless fluid numbers as those of stellar jets, then the experiments should scale well to the astrophysical case [@ryutov99]. Other parameters, such as magnetic fields and radiational cooling are more difficult to match, and it is impossible to study the non-LTE excitation physics in the lab because the critical density for collisional deexcitation is not scalable. In any case, the materials are markedly different between the experiments and stellar jets, so it is not possible to study emission line ratios in any meaningful way with current laboratory capabilities. Hence, at present the main utility of laboratory experiments of jets is to clarify how complex supersonic flows evolve with time. Laboratory work relevant to stellar jets is an emerging area of research, and several papers have appeared recently which address various aspects of supersonic flows in the appropriate regime [see @remington06 for a review]. @hansen07 observed how a strong planar shock wave disrupts a spherical obstacle and tested numerical models of the process, and @loupias generated a laboratory jet with a Mach number similar to that of a stellar jet. In a different approach, @lebedev04 and @ampleford07 used a conical array of wires at the Magpie facility to drive a magnetized jet, and explored the geometry of the deflection shock formed as a jet impacts a crosswind. Laboratory experiments have also recently studied the physics associated with instabilities along supernova blast waves [@snrref; @drake09], and the dynamics within supernovae explosions [@snrcoreref]. In this paper we present the results of a suite of experiments which deflect a supersonic jet from a spherical obstacle, where the dimensionless fluid parameters are similar to those present in stellar jets. In section 2, we describe our experimental design, consider how the experiments scale to astrophysical jets, demonstrate that the experiments are reproducible, and report how the observed flows change as one varies several parameters, including the distance between the axis of the jet and the obstacle, the time delay, the density probed with different backlighters, and the viewing angle. The numerical work is summarized in section 3. Detailed calculations with the 3-D RAGE code reproduce all of the major observed morphologies well. In section 4, we present new high-resolution optical spectra of the shocked wake of the HH 110 protostellar jet, and a new wide-field H$_2$ image of the region. These observations quantify how the internal dynamics of the gas behave as material flows away from the deflection shock and show how the jet entrains material from the molecular cloud core. Finally, in section 5 we consider how the experiments and the simulations from RAGE provide new insights into the internal dynamics of deflected supersonic jets, especially in the regions of the working surface and at the interface where jet material entrains and accelerates the obstacle. Laser Experiments of Deflected Supersonic Jets ============================================== Experimental Design ------------------- Figure 1 shows the experimental assembly we used to collide a supersonic jet into a spherical obstacle. The design consists of a 125 $\mu$m thickness titanium disk in direct contact with a 700 $\mu$m thickness titanium washer with a central, 300 $\mu$m diameter hole. The surface of the disk is heated by the thermal (soft x-ray) radiation from a hohlraum laser target, which itself is heated by 12 beams of the Omega laser at the University of Rochester [@soures96]. X-ray driven ablation of the surface of the disk creates a near-planar shock within the disk and washer assembly, and the subsequent breakout of this shock from the inner surface of the disk results in the directed outflow of a plug of dense, shock-heated titanium plasma through the cylindrical hole in the washer. This outflow is further collimated by the hole in the washer, and directed into an adjacent block of low-density (0.1 g$\,$cm$^{-3}$) polymer foam within which it propagates to a distance of $\sim$ 2 mm in $\sim$ 200 ns. After the primary jet forms, the ablation-driven shock continues to progress through the titanium target assembly and along the sides of the hole in the washer. As the hole collapses inward a secondary jet of material forms by a process analogous to that occurring in a shaped-charge explosive. The secondary jet forms after the primary jet, and propagates into the high-pressure cocoon of material already within the polymer foam. At later times, the shock propagates from the surrounding titanium washer into the foam, and causes the interface between the titanium washer and the hydrocarbon foam to move. A bow shock runs ahead of the jets into the foam. Both jets form as a result of hydrodynamic phenomena alone; there is no significant magnetic field and associated magneto-hydrodynamics, and the temperature of the jet is sufficiently low for thermal-conduction and radiative energy losses to be insignificant. Hence, the hydrodynamic phenomena which determine how the jet forms and evolves are scalable (see Section 2.2) to jets of very different dimensions that evolve over very different timescales. The experimental assembly described above is identical in many respects to that used in experiments we have reported previously [@foster05], but with the following two significant differences [see @rosen06; @coker07 for further details]: (1) in the present case we use indirect (hohlraum) drive, instead of the direct laser drive used in our earlier work; and (2) the foam medium through which the jet propagates contains an obstacle (a ball of CH). The hohlraum drive enables us to obtain greater spatial uniformity of the ablation of the titanium surface, and thereby generates a jet of improved cylindrical symmetry. The (optional) addition of an obstacle in the foam along the path of the jet medium makes it possible to set up, and in a controlled manner test, how the flow behaves when the two-dimensional cylindrical symmetry is broken. A pinhole-apertured, laser-produced-plasma, x-ray backlighting source projects an image of the jet onto radiographic film for later, detailed analysis. Details of the experimental setup are as follows. A 1600 $\mu$m diameter, 1200 $\mu$m length (internal dimensions) cylindrical gold hohlraum target with a single 1200 $\mu$m diameter laser-entry hole [see also @foster02] generates the radiation drive. The experimental package mounts over an 800 $\mu$m diameter hole in the end wall of the hohlraum, immediately opposite the laser entry hole. This axisymmetric configuration enables us to model the assembly at high resolution using two-dimensional radiation hydrocodes during the stages of formation and early-time evolution of the jet; the later stages of three-dimensional hydrodynamics are modeled by linking to a three-dimensional hydrocode, albeit at lower spatial resolution. The hohlraum is heated by 12 beams of the Omega laser with a total energy of 6 kJ in a 1 ns duration, constant power laser pulse of 0.35 $\mu$m wavelength. The resulting peak radiation temperature in the hohlraum measured with a filtered x-ray diode diagnostic (Dante) is 190 $-$ 200 eV [@foster02]. The titanium experimental assembly is made of an alloy with 90% titanium, 6% aluminium, and 4% vanadium. The diamond-polished surfaces of the hole and the planar surfaces of the components each have a 0.05 $-$ 0.3 $\mu$m peak-valley, and 0.01 $-$ 0.03 $\mu$m RMS, surface finish. We place a 100 $\mu$m thickness, 500 $\mu$m diameter gold ‘cookie-cutter’ disk between the hohlraum and the titanium disk to control the area of the titanium disk illuminated by x-rays, and to control the time it takes shocks to propagate into the titanium washer. By this means we are able to adjust, to some extent, the relative importance to the overall hydrodynamics of the primary and secondary jets, and the late-time motion of the titanium/foam interface that surround both jets. The medium through which the jets propagate is a 4 mm diameter, 6 mm length cylinder of resorcinol-formaldehyde (C$_{15}$H$_{12}$O$_4$; hereafter RF) foam, of 0.1 g$\,$cm$^{-3}$ density. We used RF foam because it has a very small ($<$ 1 $\mu$m) pore size. The obstacle in the foam is a 1 mm diameter, solid polystyrene (1.03 g$\,$cm$^{-3}$ density) sphere, supported by a small-diameter, silicon-carbide-coated tungsten stalk. The axial position and radial offset (impact parameter) of the sphere within the monomer material is set (approximately) before polymerisation of the foam, but is determined accurately after polymerisation by inspecting each experimental assembly radiographically. Typically, the axial position of the center of the ball, relative to the titanium-to-foam interface is 800 $-$ 1000 $\mu$m, and the impact parameter (perpendicular distance from the axis of the jet to the centre of the ball) is in the range 300 $-$ 500 $\mu$m. We could determine these quantities with an accuracy of approximately $\pm$ 10 $\mu$m, by analyzing pre-shot radiographs of the experimental assembly. The process of target fabrication is to some extent non-repeatable, and necessitates a separate hydrocode simulation of each experimental shot once the target dimensions are known. A sequence of experimental shots, diagnosed at different times, thus measures the hydrodynamic behavior of several very similar (but not strictly identical) target assemblies. A thin-foil, transition-metal, laser target illuminated with 2 $-$ 5 beams of the Omega laser creates the x-ray point backlighting source we use to diagnose the hydrodynamics. Each beam provides 400 J of energy in a 1 ns duration laser pulse, focused into a 600 $\mu$m diameter spot (spot size determined by use of a random-phase plate). X-ray emission from this laser-produced plasma passes through a laser-machined pinhole of typically 10 $-$ 20 $\mu$m diameter in 50 $\mu$m thickness tantalum foil, and generates a backlighting source of size comparable to the pinhole aperture. He-like resonance-line radiation of the backlighter target material dominates the spectrum of this x-ray backlighting source, and the choice of backlighter targets of typically titanium, iron or zinc results in He-like resonance line radiation of, respectively, 4.75, 6.7 and 9.0 keV. Radiation from the point x-ray backlighting source creates a point-projection (shadow) image of the experimental assembly on Kodak DEF x-ray film, with approximately 10-times magnification. Temporal resolution is determined by the duration of the x-ray backlighting source (very nearly equal to the laser pulse length). Motion blurring is insignificant for the 1 ns duration backlighting source. The time delay between the laser beams heating the hohlraum target, and the laser beams incident on the x-ray backlighting target is varied from shot to shot to build up a sequence of x-ray images of the jet hydrodynamics. Scaling the Experiments to Stellar Jets --------------------------------------- The size scales, times, densities and pressures within the Omega experiments differ markedly from those present in stellar jets. For the laboratory results to be meaningful the fluid dynamical variables must scale well, and the experiment should resemble the overall density and velocity structures present in astrophysical jets. We can connect the fluid dynamics of the astrophysical and experimental cases through the Euler equations for a polytropic gas [e.g. @ll87] $$\label{eq:mass} {{\partial \rho}\over{\partial t}} + \nabla \cdot \left(\rho\bf{v}\right) = 0$$ $$\label{eq:momentum} \rho\left({{\partial\bf{v}}\over{\partial t}} + \bf{v}\cdot\bf{\nabla v} \right) + \nabla {\rm P} = 0$$ $$\label{eq:energy} {{\partial {\rm P}}\over{\partial t}} + \gamma {\rm P} \nabla \cdot \bf{v} + \bf{v} \cdot \nabla {\rm P} = 0,$$ where $\rho$ is the density, $\bf{v}$ is the velocity, and P is the gas pressure. @ryutov99 showed that these equations are invariant to the rescaling $$r^\prime = ar ;\ \ \ \rho^\prime = b\rho ;\ \ \ P^\prime = cP$$ where a, b, and c are constants, provided one also rescales the time as $$t^\prime = a\sqrt{{b\over c}}t.$$ With this scaling, the velocity transforms as $$V^\prime = \sqrt{{c\over b}}V.$$ Solutions to the Euler equations will be identical in a dimensionless sense provided equation 6 holds. Hence, to verify that the experiment scales to the astrophysical case we need to estimate the pressures, temperatures, and velocities in both the experiment and in the HH 110 system. The parameters in the jet differ from those in the working surface where the jet deflects from the obstacle, and from those in the working surface where the bow shock impacts the ambient medium, so we must consider these regions separately. The density and temperature vary throughout stellar jets as material encounters weak shocks, heats, and cools, so we are interested in order of magnitude estimates of these quantites for the scaling estimates. As the discussion below will show, the two working surfaces scale well in our experiment but the jet scales less well. Consider the parameters in the jet first. Unlike most other jets from young stars, the HH 270 jet that collides with the molecular cloud to produce HH 110 is rather ill-defined, and consists of several faint wisps that resemble weak bow shocks [@choi06]. The electron density and ionization fraction in this part of the flow is poorly-constrained by observations, but an electron density of $10^3$ cm$^{-3}$ is typical for faint shocked structures of this kind. Taking a typical ionization fraction of 10% we obtain a total density of $10^4$ cm$^{-3}$, or about $2\times 10^{-20}$ g$\,$cm$^{-3}$. These values refer to the material in the radiating bow shocks; densities (and temperatures) between the bow shocks are likely to be lower. The temperature where \[S II\] radiates in the HH 270 bow shocks will be $\sim$ 7000 K. Material in the jet is mostly H, so using the ideal gas law we obtain 1$\times 10^{-8}$ dyne$\,$cm$^{-2}$ for the pressure. As shown in Table 1, a $\sim$ $5\times 10^{16}$, b $\sim$ $2\times 10^{-20}$, and c $\sim$ $3\times 10^{-19}$. With this scaling, 200 ns in the laboratory experiments corresponds to $\sim$ 80 years for an HH flow. Bow shocks in HH objects typically move their own diameter in $\sim$ 20 years, so the agreement with the observed timescales of HH objects and the experiment is ideal. However, the jet velocity transforms less well, with 10 km$\,$s$^{-1}$ in the experiment scaling to $\sim$ 40 km$\,$s$^{-1}$ in the stellar jet, where the actual velocity is a factor of four higher. This difference arises in part because the stellar jet cools radiatively, lowering the pressure and therefore the value of c. Another way to look at the velocity scaling is to consider the Euler number V/V$_E$, where V$_E$ = (P/$\rho$)$^{0.5}$ is the sound speed for $\gamma$=1. The Euler number in the HH 270 jet is about 20, and for other stellar jets may range up to 40. In the experiment this number is only $\sim$ 6. The main effect of the difference in Euler numbers is that the experimental jet has a wider opening angle than the stellar jet does. However, both numbers are significantly larger than unity, so both jets are highly supersonic. As we discuss in section 2.3, collapse and subsequent rebound of the washer along the axis of the flow determines to a large extent how the experimental jet is shaped, and has no obvious astrophysical analog. For this reason we will focus our analysis primarily on the leading bow shock and on entrainment of material in the obstacle rather than on the collimation of the jet. The working surfaces are generally modeled well by the experiment. In these regions the critical parameters are the timescales, which match very well, the Mach numbers in the shocks, and the density contrasts between the jet and the material ahead of the working surface. In stellar jets, the velocity of the bow shock into the preshock medium (equivalently, the velocity of the preshock medium into the bow shock) is similar to that of the jet, $\sim$ 200 km$\,$s$^{-1}$, because the jet is much denser than the ambient medium. The sound speed in the ambient medium can range from 1 $-$ 10 km$\,$s$^{-1}$, depending on how much ambient ultraviolet light heats the preshock gas. So the Mach number of the leading bow shock can range from about 20 $-$ 200. In our experiments, the sound speed ahead of the bow shock is low ($\sim$ 0.03 km$\,$s$^{-1}$), so the Mach number of the leading bow shock in the experiments is also very large, $\sim$ 200. When a stellar jet encounters a stationary obstacle like a molecular cloud, the preshock sound speed is that of the molecular cloud, $\lesssim$ 1 km$\,$s$^{-1}$, so the Mach number of material entering the shock into the molecular cloud is $\gtrsim$ 200. The Mach number of this shock in the experiments depends on the impact parameter, but is typically $\gtrsim$ 100. The ratio $\eta$ of the density in the jet to that in the ambient medium to a large extent determines the morphology of the working surface that accelerates the ambient medium. Overdense jets with $\eta$ $>$ 1 act like bullets, and have strong bow shocks and weak Mach disks, while the opposite case of $\eta$ $<$ 1 produces jets that ‘splatter’ from strong Mach disks (sometimes called hot spots) and create large backflowing cocoons [@krause]. Stellar jets are overdense, with $\eta$ $\sim$ 10 while extragalactic jets are underdense with $\eta$ $\lesssim$ $10^{-3}$ [@krause]. In our experiments, $\eta$ ranges from 8 $-$ 1 between 50 ns and 200 ns, respectively, in good agreement with the overdense stellar jet case. For the obstacle, the molecular cloud density increases from at least an order of magnitude less than that of the jet in the periphery of the cloud (essentially the ambient medium density), to $\gtrsim$ 2 orders of magnitude larger than the jet at the center of the core. So the equal densities of the jet and obstacle in the experiment cover a relevant astrophysical regime. The main difference between the experimental and astrophysical cases is the uniform density of the experimental obstacle as compared with the $\sim$ r$^{-2}$ density falloff in an isothermal molecular cloud core. Internal shock waves are by far the most common kind of shock in a stellar jet. Here, velocity perturbations of 30 $-$ 60 km$\,$s$^{-1}$ produce forward and reverse shock waves within a jet as it flows outward from the source at 150 $-$ 300 km$\,$s$^{-1}$ [e.g. @hartigan01]. Gas behind these shock waves cools rapidly to $\sim$ 4000 K by emitting permitted and forbidden line radiation [e.g. @hrh87], and then more slowly thereafter. Typically the temperature falls to $\sim$ 2000 K before the gas encounters another shock front, so the sound speed of the preshock gas is $\sim$ 5 km$\,$s$^{-1}$. Hence, the internal Mach number is $\sim$ 10 for these shock waves. Unlike the astrophysical case, our experiments do not produce multiple velocity pulses, and so are less relevant to studying the internal shocks within the beams of stellar jets. However, when jets are deflected obliquely from an obstacle the velocities in the deflected flow can remain supersonic, and may form weaker shocks in the complex working surface area that lies between the leading bow shock and the deflection shock (Mach disk analog) within the jet. Our experiments are very useful for studying the dynamics of this region, and for investigating how a jet penetrates and accelerates an obstacle along its path. For scaling to hold, several additional conditions must apply. First, dissipation mechanisms such as viscosity and heat conduction must be negligible. Hence, the Reynolds number $Re \sim VL/(\nu_{mat}+\nu_{rad})$ and Peclet number $Pe \sim VL/\chi$, need to be much larger than unity, where $\nu_{mat}$ and $\nu_{rad}$ are kinematic viscosities for matter and radiation, respectively, and $\chi$ is the thermal diffusivity. Using expressions for $\nu_{mat}, \nu_{rad}$, and $\chi$ from @ryutov99 and @drake06, we find thermal conductivity and viscosity are unimportant for both the experimental and astrophysical cases (Table 1). The Euler equations above implicitly assume that the gas behaves as a polytrope. In reality, stellar jets cool by radiating emission lines from shock-heated regions, and so are neither adiabatic ($\gamma$ = 5/3) nor isothermal ($\gamma$ = 1). Existing numerical simulations provide some insight into how jet morphologies change between the limiting cases of adiabatic and isothermal. In the adiabatic case, material heated by strong shocks cools by expanding, so working surfaces of jets tend to have rounder and more extended bow shocks, while working surfaces collapse into a dense plug for strongly-cooling jets [e.g. @blondin90]. The effect is less pronounced for weaker oblique shocks like those we are studying here in the wakes of the deflected flows. However, cooling should affect the morphologies of the stronger shocks as described above. In order to use the Euler equations, the laboratory jet must act as a fluid. Hence, the material mean free path ($\lambda_{mat} \sim v_{th}/\nu_c$ where $v_{th} \sim \sqrt(k_BT/m)$ is the particle thermal velocity and $\nu_c$ is the sum of ion and electron collisional frequencies) of the electrons and ions must be short compared with the size of the system ($\tau_{mat} \sim L/\lambda_{mat} \gg 1$). Table 1 shows that this condition is easily satisfied in both stellar jets and the laboratory experiments. The Euler equations do not include the radiative energy flux, and so we must verify that this number is small compared with the hydrodynamical energy flux. When, as in our experiments, the minimum radiation mean free path $\lambda_{rad}$ from Thomson scattering or thermal bremsstrahlung is short compared to $L$ (i.e., $\tau_{rad} \sim L/\lambda_{rad} \gg 1$), the appropriate dimensionless parameter for determining the importance of radiation is the Boltzmann number Bo\# $\sim \rho U^3 / f_{rad}$, where $U$ is a velocity scale and $f_{rad}$ is the radiative flux. In the Omega experiments, Bo\# $>>$ 1, implying that the radiative energy flux is unimportant in the flow dynamics. Radiative fluxes are also negligible in optically thin astrophysical shocks like those in stellar jets. Even in the brightest jets the observed radiative luminosity is $\lesssim$ $10^{-3}$ of the energy of the bulk flow [@bbm81], so the observations and experiments are consistent with regard to radiative energy flux. Finally, stellar jets have magnetic fields while our experiments do not. Field strengths within stellar jets are difficult to measure, but could dominate the flow dynamics close to the acceleration region [i.e., V$_{Alfven}$ $<$ V$_{flow}$ @hartigan07]. At the larger distances of interest to these experiments, fields are weaker and act mainly to reduce compression in the postshock regions of radiatively-cooling flows [@morse92]. In summary, the experiments are good analogs of jets from young stars. The Mach numbers of the jet relative to the ambient medium and to the obstacle are high in both cases, the density contrasts between the jet and the obstacle are similar, and the scaled times match almost exactly. Both systems behave like fluids, and viscosity, thermal conduction, and radiative fluxes are unimportant to the dynamics in both cases. The main differences are that stellar jets cool radiatively and the experimental jets do not, stellar jets may have magnetic fields that are not present in the experimental jets, and the density profile within the experimental jet is unlikely to have an astrophysical analog. Results ------- The complete suite of our experimental radiographs are available on the web (http:$//$ sparky.rice.edu$/\sim$hartigan$/$LLE\_shots.html). Because our experiment generates only a single image for each shot, if we wish to investigate how the flow changes with time or compare experiments that have different offsets of the ball from the axis of the jet (we refer to this distance as the ‘impact parameter’ in what follows), we must first quantify the degree to which target fabrication affects the radiographs of the flow. To this end, we obtained several groups of laser shots that had identical backlighters and delay times. One such set appears in Fig. 2, where the overall morphology of the flow and position of the bow shock is reproduced well between the shots, but irregularities appear in the bow shock shape that are specific to each target. In the case of the Ti backlighter image the bow shock resolves into a collection of nearly cospatial shells. The level of difference between the two images at the left of Fig. 2 indicates a typical variation caused by target fabrication and alignment. We varied the backlighter type, exploring V, Fe, Zn, and Ni. Each backlighter has a different opacity through the material, making it possible to probe the different depths within the structure of the flow at any given time. For example, the Ti backlighter image in Fig. 2 has significantly less opacity than that of the V images. We also tested four different types of foam, normal RF, large-pore RF, TPX (poly-4-methyl-1-pentene), and DVB (divinylbenzene), and found that the neither the foam pore size nor the foam type had any effect on the results. Fig. 3 summarizes the results from our experiments. The time sequence at the top shows how the jet evolves in a uniform medium without an obstacle. As described in section 2.1, the experiment first accelerates a plug of Ti into the foam, followed by a secondary jet of Ti that originates primarily from the collapse of the washer. At 50 ns, the radiograph shows a flat-topped profile that defines the shape of the plug, and as the plug proceeds into the foam the leading shock becomes bow-shaped. Irregularities in the shape of the bow shock are similar in size and shape to those caused by variations in the manufacture of the target (Fig. 2). By 100 ns the secondary jet has also formed, but it does not become well-collimated until about 200 ns. At 150 ns the end of the jet has a flute-shape, with a less-dense interior (see also Fig. 5, below). This shape is not a particularly good analog for a stellar jet, so we place less emphasis on this area in our analysis. The second row in Fig. 3 illustrates the principal shock structures that form when the jet encounters a sperical obstacle (labeled ‘Ball’ in the Figure) along its path. In general we expect two shocks to form when a continuous supersonic jet impacts ambient material $-$ a forward bow shock that accelerates the ambient medium, and a reverse shock, sometimes called a Mach disk, that decelerates the jet. The area between the forward and reverse shocks is known as the working surface, and within the working surface there is a boundary known as the contact discontinuity which separates shocked jet material from shocked ambient (or shocked ball) material. The two forward bow shocks, one into the ball and the other from the deflected jet into the foam, are clearly visible in the radiographs. However, the Mach disks for these bow shocks are more difficult to see in Fig. 3. At 150 ns the Ti plug is the primary driver of the deflected bow shock, and the RAGE simulations discussed below show that the shocked plug is located near the head of the bow shock at this time (Fig. 4). Fig. 5 shows that a disk-shaped area of high temperature material exists in the shocked plug and at the end of the flute, and it is tempting to associate these areas of hot gas with the Mach disk. The situation is complicated by the fact that the plug jet is impulsive, so one would expect the Mach disk to disappear once material in the plug has passed through it, though the temperature will remain elevated in this area. The last two rows of images in Fig. 3 depict two sequences of radiographs taken at common times after the deposition of the laser pulse, but with impact parameters that increase from left to right. As expected, the jet burrows into the ball more when the impact parameter is smaller, and is deflected more when the impact parameter is larger. In all cases the forward bow shock into the ball is quite smooth, with no evidence for any fluid dynamical instabilities. In contrast, the contact discontinuity between the shocked jet and shocked ball material is highly structured. As we discuss further in section 5, the irregularity of the contact discontinuity plays a major role in breaking up the ball. Images obtained with the Zn backlighter are best for revealing the morphology of the secondary jet, which at late times appears to fragment into a complex filamentary structure in the region of the flute. The ability to observe a flow from an arbitrary viewing angle is an attractive feature of the laboratory experiments that is not available to astrophysicists who study stellar jets. Fig. 6 shows two pairs of identical laser shots, one with a 150 ns time delay the other with a 200 ns delay, where the viewing angle changed by 90 degrees within each pair. The outline of the ball is clearly visible through the deflected jet in the symmetrical view ($\theta$=0) at 150 ns. Discerning the true nature of the deflected flow is much more difficult in the symmetrical view, where the radiograph resembles a single, less-collimated bow shock. Numerical Simulations of the Laboratory Experiments =================================================== The design of these experiments and their post-shot analysis was done with the RAGE (Radiation Adaptive Grid Eulerian) simulation code. We have also adapted the astrophysical MHD code AstroBEAR to model laser experiments by including a laser drive and real equation of state for different materials, but we summarize the results of this work elsewhere [@carver09]. RAGE Simulations ---------------- RAGE is a multi-dimensional, multi-material Eulerian radiation-hydrodynamics code developed by Los Alamos National Laboratory and Science Applications International (SAIC) [@gittings08]. RAGE uses a continuous (in space and time) adaptive-mesh-refinement (CAMR) algorithm to follow interfaces and shocks, and gradients of physical quantities such as material densities and temperatures. At each cycle, the code automatically determines whether to subdivide or recombine Eulerian cells. The user also has the option to de-zone (that is, reduce the resolution of the mesh) as a function of time, space, and material. Adjacent square cells may differ by only one level of resolution, that is, by a factor of 2 in cell size. The code has several interface-steepening options and easily follows contact discontinuities with fine zoning at the material interfaces. This CAMR method speeds calculations by as much as two orders of magnitude over straight Eulerian methods. RAGE uses a second-order-accurate Godunov hydrodynamics scheme similar to the Eulerian scheme of @colella85. Mixed cells are assumed to be in pressure and temperature equilibrium, with separate material and radiation temperatures. The radiation-transfer equation is solved in the grey, flux-limited-diffusion approximation. Given the placement of the ball with respect to the symmetry axis of the undeflected jet, these experiments are inherently three-dimensional. However, before the jet impacts the ball, the hohlraum and the jet are two-dimensional. This allows us to perform highly resolved, two-dimensional simulations in cylindrically-symmetric geometry to best capture the ablation of the titanium, the acceleration of the titanium plug through the vacuum free-run region in the washer, the collapse of the titanium hole onto the symmetry axis, and the subsequent jet formation. The two-dimensional simulations are initialized by imposing the measured radiation-drive temperature in a region that is inside the hohlraum. To save computer time, we have determined that it is sufficient to eliminate the hohlraum and just use the measured temperature profile as the source of energy that creates the ablation pressure to drive the jet. Neglecting asymmetries that arise from initial perturbations and unintentional misalignment of the hohlraum with the vacuum free-run region (the hole in the washer), the early-time experimental data show that the jet remains cylindrically symmetric until at least 50 ns. At approximately this time we link the two-dimensional, cylindrically-symmetric simulation into three dimensions and add the 1 mm diameter ball at the different impact parameters. While the two-dimensional simulations are run with a resolution of 1.5 $\mu$m to capture the radiation ablation of the titanium correctly, the three-dimensional simulations are typically run at lower resolution, especially during the design of the experiments. Fig. 4 shows the model density, composition, and expected radiograph for a deflected jet at 200 ns. Material labeled as blue (Ti plug) comes from the Ti disk, while orange (Ti sleeve) originates in the Ti washer and constitutes most of the secondary jet. The flute-shape of the secondary jet is clear in this figure. The areas of greatest interest are the region where the jet is creating a cavity in the ball, because this shows how jets destroy obstacles and entrain material, and the region of filaments in the working surface of the leading bow shock, because these filaments could in principle propagate downstream as weak shocks like those seen in HH 110. The processes of jet formation and propagation are illustrated in Fig. 5, which show ‘snapshots’ of the temperature and density distribution within the jet taken from a RAGE hydrocode simulation (single choice of axial position and impact parameter for the polystyrene sphere) at different times. The primary (plug) and secondary (‘shaped-charge’ hole collapse) jets are identified in Fig. 5, as well as the late-time motion at the titanium-to-foam interface that results in the formation of an opaque pedestal-like feature at the base of the jet in the synthetic radiographs. All these features are clearly discernible in the experimental data. Fig. 7 shows post-processed, zinc-backlit (9 keV x-ray backlighter energy) radiographic images from two perpendicular views that represent the time evolution of the jet, and how it subsequently deflects from the ball, in the RAGE simulation. Data Analysis and Comparisons with RAGE Simulations --------------------------------------------------- The experimental data recording the jet hydrodynamics are in the form of radiographic images recorded on x-ray-sensitive film. Fig. 3 shows a composite of experimental radiographs from several laser shots, to illustrate the formation and deflection of the jets, in a series of shots where the position and impact parameter of the polystyrene sphere inevitably vary somewhat from shot to shot. We digitized the film data with a Perkin-Elmer PDS scanning microdensitometer and converted the film density to exposure using calibration data for the Kodak DEF film that recorded the images [@henke86]. Ideally, the pinhole-apertured x-ray backlighting source would provide spatially uniform illumination of the experimental assembly, although in practice this is not the case because of laser intensity variations at the backlighting target and vignetting resulting from the specific size and shape of the pinhole aperture. Regions of the image resulting from x-ray transmission through the undisturbed foam (that is, outside the jet-driven bow shock) provide a means to determine the uniformity of backlighting intensity, after we allow for the known x-ray attenuation resulting from the undisturbed foam. Starting from this measured intensity distribution we use a polynomial fitting procedure to infer the unattenuated backlighter intensity that underlies the image of the jet and bow shock. We divide the backlit image data by the inferred (unattenuated) backlighter intensity to obtain a map of x-ray transmission through the experiment. We compare this x-ray transmission data with post-processed hydrocode calculations that simulate the experimental radiographs. Because the absolute variation of backlighter intensity across the image is small (typically, 10 $-$ 20% across the entire image), the polynomial fitting procedure enables the absolute x-ray transmission to be inferred with second-order accuracy. We use three principal metrics for comparing the experimental radiographs with synthetic x-ray images obtained by post-processing of our hydrocode simulations. These are: (1) the large-scale hydrodynamic motion (determined by comparison of the positions of the bow shock in experiment and simulation); (2) the spatial distribution of mass of the titanium jet material (obtained from the spatial integral of optical depth throughout the image, or from sub-regions of the image); and (3) the small-scale structure in the deflected jet (quantified by the two-dimensional discrete Fourier transform of sub-regions of the image, and its corresponding power spectral density function of spatial frequency). In making comparison of the experimental with hydrocode simulation, two specific points of detail require attention: (i) each laser-target assembly has its own specific location of the polystyrene sphere within the polymer foam cylinder (because of target-to-target variations arising in fabrication), and (ii) the angular orientation of the polymer-foam cylinder attached to the laser-heated hohlraum target determines the x-ray backlighting line of sight, relative to the plane in which the polystyrene sphere is displaced from the axis of the jet. Ideally all laser targets would be identical, and all backlit images would be recorded orthogonal to the axis of the target, and either in the plane of radial displacement of the polystyrene sphere or orthogonal to that plane. The geometry of the target chamber of the Omega laser facility determines the possibilities for backlighting orthogonal to the axis of the experimental assembly, and is well-characterized. However, to model a specific experiment fully, we must run three-dimensional hydrocode simulations specific to that experiment which include target-to-target differences of the fabrication assembly, followed by post-processing specific to the backlighter line of sight used in each experiment (which may differ by up to 10 degrees from the two preferred directions dictated by the symmetry of the experiment). The three-dimensional hydrocode simulations are expensive in their use of computing time and resources, and we therefore make detailed comparison of the experimental data with simulation for only a small number of representative cases. ### Large-Scale Hydrodynamics and Bow Shock Position Figs. 8 and 9 compare images of the deflected jet at 200 ns after the onset of radiation drive to the experimental assembly, and for backlighting lines of sight orthogonal to the plane of jet deflection (Fig. 8) and within the plane of jet deflection (Fig. 9). In each case, the data are compared with post-processed simulations from the RAGE hydrocode. The x-ray backlighting source was the 9.0 keV resonance line of He-like zinc (Fig. 8) or the 6.7 keV line radiation of He-like iron (Fig. 9). In each case, we corrected for spatial variations of incident backlighter intensity (as described above) and the images are therefore maps of absolute backlighter transmission through the experiment. The spatial resolution of both experiments was 15 $\mu$m. The bow shock in the hydrocarbon foam ahead of the jet is clearly visible, as is the late-time hydrodynamic behavior of the primary (outflow) jet (the mushroom-like feature lying off axis, resulting from deflection of the jet) and the secondary (hole-collapse) jet (the dense, near-axis stem apparently penetrating the initial position of the polystyrene sphere). Also evident in the radiographs is the mound-shaped pedestal that arises from motion of the titanium-to-foam interface following shock transit across this interface at late time. In the case of each experiment (the two images were obtained from different experimental shots) the radial offset of the center of the sphere from the axis (impact parameter) was close to 350 $\mu$m , and the axial position of the center of the sphere was close to 920 $\mu$m . A single RAGE simulation is shown for purposes of comparison, in which the impact parameter was 350 $\mu$m, and the axial position for the sphere 915 $\mu$m. The spatial resolution of this simulation was 3.1 $\mu$m. We make quantitative comparisons of experiment and simulation by comparison of lineouts of x-ray transmission in Fig. 10 and Fig. 11. The simulation reproduces the large-scale hydrodynamics of the experiment (bow-shock position, formation of the deflected jet, motion of the pedestal and the creation of a Mach-stem-like feature where it meets the bow shock). However, the simulation does less well with other features of the hydrodynamics, including the “clumpiness” of the deflected jet material and its proximity to the bow shock running ahead of the deflected jet, and the small-scale structure at the interface of the titanium washer and foam (apparently at the surface of the pedestal feature). The finely-resolved simulations capture more fine-scale structure $-$ the jet does indeed break up into structure similar to that seen in the experiment with simulations at 1 $\mu$m spatial resolution, although at a somewhat later time than observed experimentally. These small-scale structures tend to have larger local Mach numbers and are therefore closer to the bow shock. The filigree structure at the pedestal probably arises from Richtmyer-Meshkov growth of small-scale machining or polishing marks on the surface of the titanium washer, following shock transit across this interface. There are also multiple shock interactions that form Mach stems which are not captured by these simulations owing to the reduced computational resolution in the pedestal area. ### Spatial Distribution of Mass To proceed further with quantitative comparison with simulation, we consider the spatial distribution of mass of the materials present in the experiment. At each point in the experimental and synthetic images, the optical depth at the photon energy of the backlighter radiation is given by $$\tau = \kappa_1\sigma_1 + \kappa_2\sigma_2 + \kappa_3\sigma_3$$ where $\tau$ and $\sigma$ are the opacity and areal density (integral of density along the line of sight) and subscripts distinguish the titanium jet (1), the hydrocarbon foam (2) and the polystyrene sphere (3) materials, respectively. The opacity of the titanium jet is significantly greater than that of either the RF foam or the polystyrene sphere, and their temperatures are sufficiently low for the opacity of these materials to be essentially constant throughout the volume of the experiment. For example, at 9 keV (the photon energy of the zinc backlighting source) the opacities of titanium, RF foam and polystyrene are, respectively, 150, 4.12 and 2.83 cm$^2$g$^{-1}$. We divide the experimentally measured and simulated images, arbitrarily, into 500 $\mu$m square regions, and for each region we calculate the mean optical depth $\bar\tau$ using $\tau$ = $-$ln(I/I$_\circ$), and $$\bar\tau = {{\int\tau dA}\over{\int dA}}$$ where I/I$_\circ$ is the measured (or simulated) x-ray transmission, dA is the pixel area, and the integration extends over the area of each specific region of the image. A comparison of mean optical depth defined in this way, for both experiment and simulation, appears in Fig. 12. The grouping together of adjacent, square regions of the image enables the mean to be calculated for rather larger areas encompassing all of, or the majority of, the mass of the deflected jet. In particular, we consider two larger areas of the images shown in Fig. 12: the rectangular region composed of six adjacent 500 $\mu$m squares and labeled A, and the L-shaped region composed of three adjacent 500 $\mu$m squares and labeled B. These encompass essentially all of the mass of the jet that has interacted with the polystyrene sphere (region A), and all except the mass of the primary jet deflected by the sphere (in the case of the L-shaped region B). In each case, the simulation reproduces the experimentally measured optical depth to $\sim$ 10%: for the rectangular region A, mean optical depths are 0.94 for the experiment and 0.86 for the simulation; for the L-shaped region B, mean optical depths are 1.15 for the experiment and 1.10 for the simulation. The greatest difference between experiment and the simulation arises in the magnitude and distribution of mass of the primary (outflow) jet deflected by the obstacle. Material from this primary jet resides mainly in a further L-shaped region, labeled C in Fig. 12. Although the mean optical depths for region C differ by only 15% (0.72 in the case of experiment, 0.62 in the case of simulation), the distribution of mass shows significant variation (mean optical depths of 1.14 and 0.65 in the case of the 500 $\mu$m square region where the difference is most evident). Fig. 12 shows that the RF foam makes only a relatively small contribution to the measured optical depth, but we may assess the magnitude of this contribution simply by setting the opacity of the other components of the experiment (titanium jet and polystyrene sphere) to zero in the post-processing of the hydrocode simulation. We conclude that for the various 500 $\mu$m square regions of the images shown in Fig. 12, the optical depth of the foam lies in the range 0.10 $-$ 0.17 (this variation arises because of chord-length effects, and because density variations of the foam at the bow shock and within the cocoon). Hence, RF foam contributes a negligible optical depth in these experiments. ### Discrete Fourier Transform and Power Spectral Density To compare the experimental data with the small-scale structure of the jet in the simulations, we use the two-dimensional discrete Fourier transform (DFT) of optical depth. Starting from the experimental (or simulated) images of the experiment, we obtain maps of optical depth ($\tau$ = -ln(I/I$_\circ$)) and use the fast Fourier transform algorithm to obtain the DFT of selected regions of the experimental (or simulated) data and then proceed to calculate the power spectral density (PSD). Fig. 13 shows an image of the deflected jet within which we have identified two separate regions (green and dark red boxes), as well as an area of undisturbed hydrocarbon foam (bright red box), and part of the fiducial grid attached to the surface of the foam (blue box). For each region, we show the PSD of optical depth. We define the PSD as the sum of amplitude-squared of all Fourier components whose spatial frequency lies in the range k $-$ k + dk, where k = (k$_x^2$ + k$_y^2$)$^{0.5}$ and k$_x$ and k$_y$ are orthogonal spatial frequency components of the two-dimensional DFT. Fig. 13 clearly shows the fundamental spatial frequency of the grid and its harmonics, the flat spectrum of white noise of the backlighter transmission through the undisturbed foam, and a spectrum arising from the clumpy, perhaps near turbulent, structure arising within the deflected titanium jet. Fig. 14 shows an analogous PSD analysis, for three different RAGE simulations of the experiment. Our approach is the same for the simulation as it is for the experimental data: we obtain a map of optical depth from the simulated radiograph and then the PSD of regions whose size and position is identical to those chosen in analysis of the experimental data. In the case of Fig. 14, we show the result of RAGE simulations for three different levels of AMR calculational resolution: 12.5, 6.25, and 3.125 $\mu$m. Although the calculational resolution (dimension of the smallest Eulerian cell) differs in these three cases, we first obtain (by interpolating the post-processed data) a simulated radiograph with the same spatial resolution as the experimental data before proceeding to calculate the PSD. This procedure enables us to avoid any potential uncertainty of the scaling of PSD amplitude and frequency spacing when comparing with the experimental data. Astronomical Observations and Numerical Models of the Internal Dynamics within the Deflected Jet HH 110 ======================================================================================================= Spectral Maps of HH 110 ----------------------- We observed HH 110 on 9 Jan 2008 UT with the echelle spectrograph on the 4-m Mayall telescope at Kitt-Peak National Observatory in order to quantify the dynamics present in a shocked, deflected astrophysical jet. The 79-63 grating and 226-1 cross disperser combined with the T2KB CCD and 1.5 arcsecond slit gave a spectral resolution of 3.0 pixels, or 11.1 km$\,$s$^{-1}$ as measured from the FWHM of night sky emission lines. The CCD, binned by two along the spatial direction, produced a plate scale of 0.52 arcseconds per pixel. The position angle of the slit was 13 degrees, and the length of the slit was limited by vignetting to be about 140 arcseconds. Seeing was 1.3 arcseconds, and skies were stable with light cirrus. A plot of the slit position superposed on an archival HST image of HH 110 appears in Fig. 15. We employed an unusual mode of observation, long slit but with a wide order separating filter (GG 435). This setup causes orders to overlap at different spatial positions along the slit, but in the case of HH 110 there is no ambiguity because no continuum sources are present. The advantage of this setup is that one can obtain position-velocity diagrams simultaneously for all the bright emission lines, including \[S II\] 6716, \[S II\] 6731, \[N II\] 6583, \[N II\] 6548, and H$\alpha$. The \[O I\] 6300 and 6363 lines were also present, but the signal-to-noise of these faint lines was too low to warrant any profile analysis. Distortion in the spectrograph optics causes each emission line to be imaged in a curved arc whose shape varies between orders. Fortunately, there is strong background line emission in each of the emission lines from the HH 110 molecular cloud, and we used this emission to correct for distortion and to define zero radial velocity for the object. A third degree polynomial matched the distortion of the night sky emission lines within an rms of about 0.2 pixels (0.74 km$\,$s$^{-1}$). It is important to remove night sky emission lines as well as the background emission lines from the molecular cloud, as these lines contaminate multiple orders in our instrumental setup. To this end, we imaged blank sky near HH 110 frequently and subtracted this component from the object spectra after aligning the night sky emission between the object and sky frames to account for flexure in the spectrograph. Cosmic rays and hot pixels were removed using a routine described by @hartigan04, and flatfielding, bias correction, and trimming were accomplished using standard IRAF tasks[^1]. Alignment of individual exposures was accomplished in two ways. First, we compared the position of a star relative to the slit that was captured from the guide camera, whose plate scale of 0.16 arcseconds per pixel we determined by measuring the length of the slit image when a decker of known size truncated the slit. More precise positioning in the direction along the slit is possible by extracting the spatial H$\alpha$ trace and performing a spatial cross correlation between frames. The uncertainty in the positional measurements from the guide camera image inferred from the secondary corrections required by the cross-correlations is about 1.0 arcseconds. To align and compare different emission lines we must also register the position-velocity diagrams to account for the spatial positions of the orders and for the tilt of the spectrum across the CCD within each order, but this is easy to do with a continuum lamp exposure through a small decker to define the spectral trace. The guide camera images indicate, and the spectra confirm, that the last three object exposures drifted by 1.5 arcseconds relative to the position shown in Fig. 15, so these were not used in the final analysis. In all, the total exposure time for the spectra is 140 minutes. There were no differences between the \[S II\] 6716 and \[S II\] 6731 position-velocity diagrams, so we combined these to produce a single \[S II\] spectrum. The \[N II\] 6548 line is fainter by a factor of three than the companion line \[N II\] 6583, and the fainter line is also contaminated by faint residuals from a bright night sky emission line from an adjacent order, so we simply use the 6583 line for the \[N II\] line profiles. Internal Dynamics of HH 110 --------------------------- The image in Fig. 15 divides the position-velocity diagram into ten distinct emitting regions along the slit. Spectra for each of these positions appear in Fig. 17 for H$\alpha$ and for the sum of \[N II\] + \[S II\]. All regions have well-resolved emission line profiles in H$\alpha$, \[N II\], and in \[S II\]. Only object 9 showed any difference between the \[S II\] and \[N II\] profiles, with \[N II\] blueshifted by 12 km$\,$s$^{-1}$ relative to \[S II\]. In low-excitation shocks like HH 110, the forbidden line emission peaks when the gas temperature is $\sim$ 8000 K [@hartigan95], which corresponds to a spread in radial velocity owing to thermal motions of HWHM = ((2ln2)kT/m)$^{0.5}$, or 2.6 km$\,$s$^{-1}$ for N, and 1.7 km$\,$s$^{-1}$ for S. These thermal line widths will be unresolved with the Kitt-Peak observations, which have a spectral resolution of 11 km$\,$s$^{-1}$. Therefore, the observed widths of the forbidden lines measure nonthermal line broadening in the jet. This line broadening most likely arises from nonplanar shock geometry or clumpy morphologies on small scales, as the HST images show structure down to at least a tenth of an arcsecond. However, other forms of nonthermal line broadening, such as magnetic waves or turbulence, may also contribute to the line widths. Figs. 16 and 17 show that the H$\alpha$ emission line widths are larger than those of the forbidden lines. This behavior is expected because a component of H$\alpha$ occurs from collisional excitation immediately behind the shock front where the temperature is highest. The thermal FWHM of H$\alpha$ is given by $$V_{th} = \left(V_{OBS}^2 - V_{forb}^2\right)^{0.5}$$ where V$_{forb}$ = (V$_{NT}^2$ + V$_{INST}^2$)$^{0.5}$ is the observed FWHM of the forbidden lines, V$_{NT}$ the nonthermal FWHM and V$_{INST}$ the instrumental FWHM. We can use the thermal line width observation to measure the shock velocity in the gas. The temperature immediately behind a strong shock is given by $$T = {3\over 16}{\mu m_H V_S^2\over k}$$ where $\mu$ is the mean molecular weight of the gas and V$_S$ is the shock velocity. The mean molecular weight depends on the preshock ionization fraction of the gas, and the postshock gas will have different ion and electron temperatures immediately behind the shock until the two fluids equilibrate [@mckee74], but the equilibration distance should be unresolved for HH 110. For simplicity we take the preshock gas to be mostly neutral, so $\mu$ $\sim$ 1. For thermal motion along the line of sight, the FWHM of a hydrogen emission line profile is then $$V_{th} = 2.354\left({k T\over {m_H}}\right)^{0.5}$$ Combining these two equations we obtain $$V_S = 0.98 V_{th}$$ so the shock velocity is closely approximated by the observed thermal FWHM. Fig. 18 summarizes the kinematics and dynamics within HH 110. The radial velocity gradually becomes more negative at distances greater than about $2\times 10^{17}$ cm. The radial velocities in the different emission lines track one another well. However, the same is not true for the line widths: both Fig. 17 and Fig. 18 show clearly that H$\alpha$ is broader than the forbidden line profiles. This behavior is expected because when the preshock gas is neutral, much of the H$\alpha$ comes from collisional excitation immediately behind the shock front where the temperature is high. The low atomic mass of H also increases its line width relative to those of N and S. The graph shows that the nonthermal component of the line profiles stays approximately constant at $\sim$ 40 km$\,$s$^{-1}$, and the thermal component of H$\alpha$ and the shock velocity are $\sim$ 50 km$\,$s$^{-1}$. There are two epochs of HST images available (Reipurth PI), separated by about 22 months, and these data show intriguing and complex proper motions. Regions 1 through 4 have multiple shock fronts some of which move in the direction of the jet while others move along the deflected flow. Fig. 18 shows that this impact zone has higher nonthermal line widths than present in the flow downstream, consistent with the HST data. Throughout this rather broad region, denoted as such in Fig. 15, jet material impacts the molecular cloud. Hints of this behavior are evident in the ground based proper motion data of @lopez05, which show proper motion vectors directed midway between the direction of the jet and that of the deflected flow. Downstream from region 4, the material all moves in the direction of the deflected flow and gradually expands in size. The electron density of the shocked gas declines as the flow expands [@ro91]. The slow increase in the blueshifted radial velocity could arise if the observer sees a concave cavity that gradually redirects the deflected flow towards our line of sight. One gets the impression from the HST images of a series of weak bubbles that emerges from the impact zone. Published proper motion measurements [@lopez05] suggest tangential velocities of $\sim$ 150 km$\,$s$^{-1}$ along the deflected flow. Hence, the nonthermal broadening is $\sim$ 25% of the bulk flow speed in the deflected jet, while the internal shock speeds are typically 30% of the flow speed. The temperature immediately behind a 50 km$\,$s$^{-1}$ shock is $\sim$ $5.7\times 10^4$ K, and will drop to $\lesssim$ 5000 K in areas where forbidden lines have cooled. The corresponding sound speeds are $\sim$ 20 km$\,$s$^{-1}$ and 8 km$\,$s$^{-1}$, respectively, so the bulk flow speed is $\sim$ Mach 10 in the deflected flow, while the internal shocks there are $\sim$ Mach 3 $-$ 6. The magnetosonic Mach numbers will be lower, depending on the field strength. Wide-Field H$_2$ Images of the HH 110 Region -------------------------------------------- In order to better define how the HH 110 jet entrains material from the molecular cloud, and to verify that the deflected jet model is appropriate for this object, we obtained new wide-field near-infrared images of the region. In Figs. 19 and 20 we present a portion of an H$_2$ image taken 21 Sept., 2008 with the NEWFIRM infrared camera attached to the 4-m telescope at Kitt Peak National Observatory. The image was constructed from 20 individual dithers of 2 minutes apiece for a total exposure time of 40 minutes. The wide-field image in Fig. 19 shows VLA 1, the driving source of the HH 110 flow [@choi06]. The jet from this source, known as HH 270, does not radiate in H$_2$ until it strikes the molecular cloud, although the jet is visible in deep \[S II\] images [@choi06]. Our H$_2$ image clearly shows two other jets that emanate from sources embedded within the molecular cloud core. IRS 1 (aka IRAS 05487+0255) is a bright near-IR source, while IRS 2 appears to be obscured by a flared disk seen nearly edge-on (as in HH 30, @burrows96). These sources drive a molecular outflow along the direction of the jets we see in the H$_2$ image [@ro91]. Archival Spitzer images of the region at mid-infrared (3.6 $\mu$m $-$ 8 $\mu$m), and far-infrared (24 $\mu$m, 70 $\mu$m and 160 $\mu$m) wavelengths reveal three very bright sources that persist in all bands in the region, VLA 1, IRS 1, and IRS 2. The spectral energy distribution of IRS 1 is still rising at 100 $\mu$m, indicative of a heavily embedded source. The morphology of the HH 30 clone IRS 2 splits into two pieces at shorter wavelengths in the Spitzer images, consistent with an obscuring disk seen edge-on. Epoch 2000 coordinates for the midpoint of the HH 30-like disk in IRS 2 are 5:51:22.70 +2:56:05, and for IRS 1 are 5:51:22.60 +2:55:43. The ubiquity of jets in this region is a common occurrence, as most star-forming regions have multiple sources that drive jets. However, it does potentially bring into question whether or not what we and others [e.g. @rrh96] interpret as a deflected jet for HH 110 may simply be a distinct flow generated by some other embedded source to the northeast of the emission. The counter to this argument is that Fig. 19 does not show a source near the apex of HH 110, and the Spitzer images of the region also show nothing there at mid-infrared wavelengths. If the hot H$_2$ is dragged out from the molecular cloud core by the impact of the jet, there should be a spatial offset between the H$_2$ from the core and the H$\alpha$ in the jet, with the H$_2$ located on the side of the spray closest to the core. The color composite in Fig. 20 demonstrates this effect very well, as noted previously for a small portion of the jet [@nc96]. Connecting the Laboratory Experiments With Astrophysical Jets ============================================================= Our laboratory results highlight two aspects of the fluid dynamics that are particularly useful for interpreting astronomical images and spectra $-$ entrainment of ambient material and the dynamics within contact discontinuities. The experiments also serve as a reminder that viewing angle affects how a bow shock appears in an image. We discuss each of these ideas below. Interpretation of Astronomical Images ------------------------------------- Keeping in mind that magnetic fields may play some role in the dynamics, we can look to the experiments as a guide to how material from an obstacle like a molecular cloud core becomes entrained by a jet in a glancing collision. As noted above, in the astronomical images the H$_2$ emission in Fig. 20 must arise from the cloud core to be consistent with the observed spatial offset of the H$\alpha$ and H$_2$, and the lack of H$_2$ in the jet before it strikes the cloud. In the experiments, the jet entrains material in the ball in part because the flute penetrates into the ball and ‘scoops up’ whatever material falls within the flute (Fig. 4). This type of entrainment is probably of little interest astrophysically because its origin is unique to the relatively hollow density structure within the experimental jet. However, material from the ball is also lifted into the flow because the jet penetrates into the ball along the contact discontinuity. Both the radiographs and the RAGE simulations show this region to be highly structured (Figs. 3-5). The experiments indicate that once a part of the jet becomes deflected into the ball along the contact discontinuity it creates a small cavity where the jet material lifts small fragments of the ball into the flow. We see the same morphology in the astrophysical case. This type of entrainment is one that occurs as a natural outgrowth of the complex 3-D structure along a contact discontinuity. As Fig. 6 shows, the deflected bow shock appears much wider when the bow shock has a significant component along the line of sight. While this result is rather elementary, Fig. 6 provides a graphic example that is important to keep in mind when interpreting astronomical images of bow shocks when the shocked gas has a large redshift or blueshift. In such cases a flow may appear to be much less collimated than if one were to observe the bow shock perpendicular to its direction of motion. An example of a wide bow shock oriented at a large angle from the plane of the sky is HH 32A, which has been thoroughly studied at optical wavelengths [@beck04]. Filamentary Structures in the Experiment and the Astronomical Observations -------------------------------------------------------------------------- The working surface of the deflected bow shock in the experiment exhibits an intriguing filamentary structure in the observed radiographs and in the RAGE simulations (Figs. 3 and 21) that resembles the filamentary shock waves seen in HH 110 (Fig. 15). If differential motions between adjacent filaments in the working surface are supersonic, then weak shocks like those seen in HH 110 could form as the filaments interact at later times. To test this idea, we generated synthetic images of the Mach number and the velocity along a plane that contains the center of the ball in Fig. 21. Together, these two images indicate whether shock waves are likely to form within the working surface at later times. Neither image alone provides this information: a constant velocity flow with two adjecent regions of different temperature will show markedly different Mach numbers in close proximity but will not create a shock; similarly, a shock will only form between adjacent fluid elements with differing velocities and the same temperature if the difference between the Mach numbers exceeds unity. Velocity differences between the filaments in the working surface in Fig. 21 are typically 1 $-$ 2 km/s, or only about 10% of the initial jet velocity. In contrast, the shock waves in HH 110 have higher velocities, about 30% of the jet speed (section 4.2; Fig. 18). Moreover, Mach numbers in the filaments range from $\sim$ 1 $-$ 2, so differences between the filaments are $\lesssim$ 1, indicating that the relative motions between the filaments are subsonic. We conclude that the velocity differences between the filaments in the working surface are too low to form shock waves unless the filaments cool. Even if that were to occur, the shock velocities will be about a factor of three smaller relative to the jet speed than those seen in HH 110. Hence, the best explanation for the shocks observed in HH 110 is that the source is impulsive, where each pulse impacts the molecular cloud in a similar manner to our experimental setup. It is instructive to consider what one might observe in a position-velocity diagram in the experiment, were it possible to align a slit down the axis of the deflected flow and obtain a velocity-resolved spectrum, as is possible with astronomical observations. We show this exercise in Fig. 22, where we have simply assumed the emissivity of the gas to be proportional to its density. For a real emission line the emissivity depends on the temperature, density, and ionization state of the gas in a complex manner determined by the atomic physics of the line. However, it is still instructive to see what sort of morphologies appear in a p-v diagram of this sort. The synthetic p-v diagram in Fig. 22 contains a series of arcs that resemble those present in astronomical slit spectra of spatially-resolved bow shocks [@raga86; @hrm90]. These arcs occur when the slit crosses the jet or an interface such as the cavity evacuated by the jet. In the case of spatially resolved bow shocks, arcs in the p-v diagram result from the motion of a curved shell of material, where the orientation of the velocity vector relative to the observer changes along the slit. The lesson from the experiment seems to be that in highly structured flows like our deflected jet, curved cavities and filaments naturally produce p-v diagrams that contain multiple arcuate features. As in the case of resolved bow shocks, the velocity amplitude of the arcs in the p-v diagram is on the order of the shock velocity responsible for the arc. In an optically thin astrophysical nebula one can measure five out of the six phase space dimensions: x and y position on the sky from images, proper motion velocities V$_x$ and V$_y$ from images taken at two times, and the radial velocity V$_z$ along the line of sight from spectroscopy. As we have shown above, V$_z$ is an extremely useful diagnostic of the dynamics of a flow, but one that is currently not possible to measure in laboratory experiments. If such a diagnostic instrument could be developed it would open up a wide range of possibilities for new studies of supersonic flows. Summary and Future Work ======================= The combination of experimental, numerical and astronomical observational data from this study demonstrates the potential of the emerging field of laboratory astrophysics. In this paper we have studied how a supersonic jet behaves with time as it deflects from an obstacle situated at various distances from the axis of the jet. The laboratory analog of this phenomenon scales very well to the astrophysical case of a stellar jet which deflects from a molecular cloud core. An important component to our study was to expand the observational database of best astrophysical example, HH 110, by obtaining new spatially-resolved high-spectral resolution observations capable of distinguishing thermal motions from turbulent motions, and by acquiring a new deep infrared H$_2$ image that can be compared with existing optical emission line images from the Hubble Space Telescope. The laboratory experiments span a range of times, spatial offsets between the axis of the jet and the center of the ball (impact parameters), viewing angles, opacities (backlighters), and materials. The experiments are reproducible and do not depend on composition or structure of foam, or on the pinhole diameter of the backlighter (spatial resolution of the experiment). Synthetic radiographs of the experiment from RAGE match the experimental data extremely well, both qualitatively as images and quantiatively with Fourier analysis. In fact, the agreement is so good that we have used the synthetic velocity maps from RAGE to compare the internal dynamics of the experiment with those that we measure from the new spectral maps of HH 110. A new wide-field H$_2$ image supports a scenario where HH 110 represents the shocked ‘spray’ that results from a glancing collision of the HH 270 jet with a molecular cloud core. The H$_2$ in HH 110 is offset from the H$\alpha$ toward the side closest to the molecular core, consistent with the deflected jet model. The H$_2$ images also uncovered two sources within the core that drive collimated jets, one a bright near-infrared source, and the other a highly-obscured source that appears to be a dense protostellar disk observed nearly edge-on, as in HH 30. The experiments provide several important insights into how deflected supersonic jets like HH 110 behave. In the experiment, entrainment of material in the obstacle occurs in part because the morphology of the contact discontinuity between the shocked jet and shocked obstacle easily develops a complex 3-D structure of cavities that enables the jet to isolate clumps of obstacle material and entrain them into the flow. A similar process likely operates in HH 110. The experiments also reveal filamentary structure in the working surface area of the deflected bow shock, but the relative motion between these filaments is subsonic. Hence, while this dynamical process will generate density fluctuations in the outflowing gas, it cannot produce the filamentary structure and $\sim$ Mach 5 shocks shown by the new velocity maps of HH 110. For this reason the best model for HH 110 remains that of a pulsed jet which interacts with a molecular cloud core. Synthetic position-velocity maps along the deflected jet from the RAGE simulations of the experiments appear as a series of arcs, similar to those observed in astronomical observations of resolved bow shocks. A close examination of the experimental data shows that these arcs correspond to regions where the slit crosses different regions of the flow, such as cavities evacuated by the jet, the jet itself, or entrained material from the ball. This correspondance between the appearance of a p-v diagram and the actual morphology of a complex flow is an intuitive, although perhaps unexpected result of studying the dynamics within the experimental flow. Finally, observations of the deflected bow shock from different viewing angles emphasize that the observed morphology and collimation properties of bow shocks depend strongly upon the orientation of the flow with respect to the observer. As one would expect, a bow shock deflected toward the observer appears less collimated than one that is redirected into the plane of the sky. The impact parameter of the jet and obstacle determines how much the jet deflects from the obstacle and how rapidly the obstacle becomes disrupted by the jet. While experimental analogs of astrophysical jets are highly unlikely to ever reproduce accurate emission line maps, laser experiments can provide valuable insights into how the dynamics of complex flows behave. Our study of a deflected supersonic jet is only one example of how the fields of astrophysics, numerical computation, and laboratory laser experiments can compliment one another. We are currently embarking on a similar program to study the dynamics within supersonic flows that are highly clumpy, and other investigations are underway related to the launching and collimation of jets [@bellan09]. We are grateful to Dean Jorgensen, Optimation Inc., Burr-Free Micro Hole Division, 6803 South 400 West, Midvale, Utah 84047, USA, for supplying the precision-machined titanium-alloy components for these experiments. We thank K. Dannenberg for her assistance in manufacturing the initial targets for these experiments, the staff of General Atomics for their dedication in developing new targets and delivering them on time, the staff at Omega for their efficient operation of the laser facility, and an anonymous referee for useful comments regarding scaling. This research was made possible by a DOE grant from NNSA as part of the NLUF programs DE-PS52-08NA28649 and DE-FG52-07NA28056. Ampleford, D., et al. 2007, Ap&SS 307, 29 Balick, B., & Frank, A. 2002, ARA&A 40, 439 Beck, T., Riera, A., Raga, A., & Aspin, C. 2004, AJ 127, 408 Bellan, P. et al. 2009, Phys. Plasmas 16, 41005 Blondin, J., Fryxell, B., & Königl, A. 1990, ApJ 360, 370 Brugel, E., Böhm, K.-H., & Mannery, E. 1981, ApJS 47, 117 Burrows, C., et al. 1996, ApJ 473, 437 Cabrit, S. 2007, Proceedings of the IAU Symposium 243, J. Bouvier & I. Appenzeller eds., (Cambridge:Cambridge University Press), 203 Carver, R., et al. 2009, in preparation. Cerqueira, A., Velazquez, P., Raga, A., Vasconcelos, M. & de Colle, F. 2006, A&A 448, 231 Choi, M., & Tang, Y.-W. 2006, ApJ 648, 504 Coker, R. F. et al. 2007, Ap&SS 307, 57 Colella, P. 1985, SIAM J. Comput. 6, 104 Cunningham, A.J., Frank, A., & Blackman, E.G. 2006, ApJ, 646, 1059 Drake, R.P., Davison, L., & Horie, Y. 2006, “High-Energy-Density Physics: Fundamentals, Inertial Fusion, and Experimental Astrophysics” (Berlin:Springer). Drake, P., Kuranz, C., Miles, A., Muthsam, H., & Plewa, T. 2009, Phys. Plasmas 16, 41004 Cunningham, A. J., Frank, A., Varniere, P., Mitran, S., & Jones, T. W. 2008, ArXiv e-prints, 710, arXiv:0710.0424 Ferreira, J., Dougados, C., & Cabrit, S. 2006, A&A 453, 785 Foster, J. M. et al. 2002, Phys. Plasmas 9, 2251 Foster, J. M. et al. 2005, ApJ 634, L77 Gittings, M., et al. 2008, Comput. Sci. Disc. 1, 15005 Hansen, J. F., Robey, H. F., Klein, R. I., & Miles, A. R. 2007, ApJ 662, 379 Harris, D., & Krawzynski, H. 2006, ARA&A 44, 463 Hartigan, P. 2008, Lecture Notes in Physics 742, 15 Hartigan, P., Edwards, S., & Ghandour, L. 1995, ApJ 452, 736 Hartigan, P., Edwards, S., & Pierson, R. 2004, ApJ 609, 261 Hartigan, P., Frank, A., Varniere, P., & Blackman, E. 2007, ApJ 661, 910 Hartigan, P., Morse, J., & Raymond, J. 1995, ApJ 444, 943 Hartigan, P., Morse, J., Palunas, P., Bally, J., & Devine, D. 2000, AJ 119, 1872 Hartigan, P., Morse, J., Reipurth, B., Heathcote, S. & Bally, J. 2001, ApJ 559, L157 Hartigan, P., Raymond, J., & Hartmann, L. 1987, ApJ 316, 323 Hartigan, P., Raymond, J., & Meaburn, J. 1990, ApJ 362, 624 Heathcote, S., Morse, J., Hartigan, P., Reipurth, B., Schwartz, R., Bally, J., & Stone, J. 1996, AJ 112, 1141 Heathcote, S., & Reipurth, B. 1992, AJ 104, 2193 Henke, B. L., Uejio, J., Y., Stine, G., F., Dittmore, C., H., & Fujiwara, F. G. 1986, J. Opt. Soc. Am. B 3, 1540 Krause, M. 2003, A&A 398, 113 Kuranz, C. et al. 2007, Ap&SS 307, 115 Landau, L., & Lifshitz, E. 1987, Fluid Mechanics, (Oxford:Pergamon) Lebedev, S., et al. 2004, ApJ 616, 988 Loupias, B. et al. 2007, PRL 99, 265001 Lopez, R., Estalella, R., Raga, A., Reira, A., Reipurth, B., & Heathcote, S. 2005, A&A 432, 567 McKee, C. 1974, ApJ 188, 335 Mirabel, I., & Rodriguez, L. 1999, ARA&A 37, 409 Morse, J., Hartigan, P., Cecil, G., Raymond, J., & Heathcote, S. 1992, ApJ 399, 231 Mottura, L. 1997, J. Comp. Phys. 138, 354 Noriega-Crespo, A., Garnavich, P., Raga, A., Canto, J., & Böhm, K.-H. 1996, ApJ 462, 804 Raga, A., & Böhm, K.-H. 1986, ApJ 308, 829 Ray, T., Dougados, C., Bacciotti, F., Eislöffel, J., & Chrysostomou, A. 2007, in “Protostars & Planets V”, B. Reipurth, D. Jewitt, & K. Keil eds., (Tucson:Univ. of Arizona Press) Reighard, A. & Drake, R. 2007, Ap&SS 307, 121 Remington, B., Drake, P., & Ryutov, D. 2006, Rev. Mod. Phys. 78, 755 Riera, A., Raga, A., Reipurth, B., Amram, P., Boulesteix, J., Canto, J., & Toledano, O. 2003, AJ 126, 327 Reipurth, B., & Olberg, M. 1991, A&A 246, 535 Reipurth, B., & Bally, J. 2001, ARA&A 39, 403 Reipurth, B., Raga, A., & Heathcote, S. 1996, A&A 311, 989 Ryutov, D., Drake, R., Kane, J., Liang, E., Remington, B. & Wood-Vasey, W. 1999, ApJ 518, 821 Rosen, P. A. et al. 2006, J. Phys. IV France 133, 1019 Soures, J., et al. 1996, Phys. Plasmas 3, 218 Schwartz, R. 1978, ApJ 223, 884 [lcc]{} = 0.08in =0em L (cm) \[jet size\] & 1e15 & 0.02\ V (km$\,$s$^{-1}$) \[jet velocity\] & 150 & 10\ $\rho$ (g$\,$cm$^{-3}$) \[jet density\] & 2e-20 & 1\ P (dyne cm$^{-2}$) \[jet pressure\] & 1e-8 & 3e10\ t$_{flow}$ (sec) \[flow timescale\] & 3e9 & 1e-7\ Composition & H & Ti\ T (eV) \[temperature of wake\] & 0.6 & 1\ Euler number in jet &22 & 6\ $\nu_{mat}$ (cm$^2$s$^{-1}$) \[viscosity\] & 9e13 & 2e-5\ $\nu_{rad}$ (cm$^2$s$^{-1}$) \[viscosity\] & 2e21 & 1e-12\ $\nu_c$ (s$^{-1}$) \[collision freq\] & 0.002 & 2e13\ $\chi$ (cm$^2$s$^{-1}$) \[diffusivity\] & 5e24 & 5e-18\ Re$_{mat}$ \[Reynolds number\] & 7e8 & 1e9\ Pe$_{mat}$ \[Peclet number\] & 1e7 & 3e4\ $\lambda_{mat}$ (cm) (mean-free path) & 3e8 & 8e-9\ $\lambda_{rad}$ (cm) (mean-free path) & $>>$L$^a$ & 3e-5\ $\tau_{mat}$ \[optical depth\] & 9e6 & 2e6\ $\tau_{rad}$ \[optical depth\] & $<<$1$^a$ & 7e2\ Bo\# & $>$1e3 & 2e4\ 6.9in 0.0in 0.0in 0.0in 0.0in [^1]: IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy Inc., under cooperative agreement with the National Science Foundation.

--- abstract: 'We study electrostatic interaction between molecules of the DNA in which a number of phosphate groups of the sugar-phosphate backbone are exchanged for the pyrophosphate ones. We employ a model in which the DNA is considered as a one-dimensional lattice of dipoles and charges corresponding to base pairs and (pyro)phosphate groups, respectively. The interaction between molecules of the DNA is described by a pair potential $U$ of electrostatic forces between the two sets of dipoles and charges belonging to respective lattices describing the molecules. Minima of potential $ U$ indicate orientational ordering of the molecules and thus liquid crystalline phases of the DNA. We use numerical methods for finding the set of minima in conjunction with symmetries verified by potential $U$ . The symmetries form a noncommutative group of 8-th order, ${\cal S}$. Using the group ${\cal S}$ we suggest a classification of liquid crystalline phases of the DNA, which allows of several cholesteric phases, that is polymorphism. Pyrophosphate forms of the DNA could clarify the part played by charges in its liquid crystalline phases, and make for experimental research, important for nano-technological and bio-medical applications.' author: - 'V.L. Golo$^1$' - 'E.I.Kats$^2$' - 'S.A. Kuznetsova$^3$' - 'Yu.S. Volkov$^1$' date: 'August 3, 2008' title: Pyrophosphate Groups in Liquid Crystalline Phases of the DNA --- Introduction ============ According to the familiar legend the discovery of the cholesteric phase of the DNA was due to a happy chance that occurred to C.Robinson, who was sharp enough to see it, [@robinson]. The story resembles that of Fleming’s discovering the penicilin. Since then cholesteric phases of the DNA have been the subject of numerous experimental and theoretical investigations owing to their variety and regularity, [@livolant1]. It has been established that the formation of the phases depends on chemical properties of an ambient solution and ions, the ingenious experimental technique has been worked out, [@livolant1], [@livolant2], and liquid crystalline phases of the DNA have become instrumental in studying the DNA itself. Another important development began surfacing in chemical physics of nucleic acids early in the 90-th. The group led by Z.A.Shabarova at the Lomonosov Moscow University, [@s1] — [@s5] succeeded in synthesizing the DNA in which some inter-nucleotide phosphate groups are exchanged for the pyro-phosphate ones, and thus considerably extended the field of research, providing new insights into the chemistry of nucleic acids, as well as new possible bio-medical applications. In this paper we aim at making it clear that pyrophosphate forms of the DNA could be helpful in studying liquid crystalline phases of the DNA. Theoretical work on the physics of liquid crystalline phases dates from the seminal paper by Onsager, [@onsager], which relies on the picture of hard rods representing molecules in solvent. Applications of the model require the use of phenomenological constants and theoretical assumptions, difficult to verify in specific situations. Cholesteric phases need even more careful investigating owing to their chirality. In a series of papers Kornyshev, Leikin, and their collaborators[@lk_model], [@lk1] - [@lk3], [@kim], [@osipov], put forward the theory of cholesteric liquid crystalline phases of the DNA that relies on the helical distribution of charges of the DNA. Within the framework of this theory one can employ various approaches and approximations and investigate specific conformations. Generally, a molecule of the DNA is considered as a charged rod or cylinder, the charge being distributed continuously over the surface of the rod, complying with the helical symmetry, theoretical tools employed being of analytical character. In the present paper we use a discrete approximation for the charge’s distribution and rely a computer simulation for finding molecular conformations. It should be noted that the distribution of charge in the DNA molecule is essentially discrete being caused by (1) the dipole moments of the base pairs, (2) the charges of the phosphate groups, (3) counterions which are not uniformly distributed round the DNA molecule. The electrostatic interaction between two DNA molecules is due to this essentially non-uniform distribution of charges. Our approach, still remaining within the framework of papers [@lk_model], [@lk1], provides new details of the phenomenon. It is worth noting that we aim only at a qualitative description, which could be useful for explaining experimental data. Since the current theory considers electrostatic interaction as a cause for the formation of liquid crystalline phases the DNA , it is interesting to investigate opportunities that can be provided by the use of the DNA containing a number of pyrophosphate groups, PP-forms, instead of the usual phosphate ones, P-forms, see Figure 1, so as to have a means of changing the charge conformation of the molecule. It is important that synthetic forms of the DNA can contain PP-groups in the duplex of the DNA instead of the usual phosphate ones in such a way that the structures of the pyro-modified and usual phosphate molecules, remain rather close,[@s5], the inter-nucleotide distance, the stacking, and the Watson-Crick interaction suffering no change. Synthetic forms of the DNA are instrumental in the study of fundamental problems in molecular biology, biochemistry, medicine, ferments’ activity in nucleotide exchange, protein-nucleic acids complexes, structural functioning of biopolymers, and regulation of the genetic expression. The modification of inter-nucleotide groups is of particular importance owing to its preserving the ability of molecules of the DNA to penetrate cell membranes and regulate gene expression, while retaining the basic function of the DNA, that is to interact with the complimentary sequences of nucleotides. It is possible to synthesize the DNA so as to have the exchanged pyrophosphate groups located at prescribed sites of the sugar-phosphate spine, each pyrophosphate group bringing forth an additional negative charge. Minima of the potential [U]{} for pair-interaction of molecules of the DNA should correspond to orientational ordering of the molecules in solvent and therefore liquid, or solid, crystalline phases. Special means are required to find the minima of $ U$. At this point the symmetry of [U]{} provides valuable information. As was found in the previous paper [@gkv], $ U$ is invariant under the action of a group of discrete transformations, and therefore its minima form a set having the same symmetry. The circumstance reduces the amount of numerical work, which is quite large. But we feel that the symmetry of the pair-interaction $ U$ is by far of more general importance for understanding the physics of liquid crystalline phases of the DNA than one can infer from its numerical applications. Preliminaries. =============== We shall recall certain basic facts of the DNA. A molecule of the DNA can attain several hundred $\mu m$ in length. If we neglect details that have a size of one thousand Å, or more, we can visualize it as a soft shapeless line and conclude that on this scale it behaves like an ordinary polymer. In contrast, looking at its smaller segments, of one hundred Åor less, we observe that it tends to be straight. Thus, borrowing a comparison from everyday life, we may say that a molecule of the DNA looks like a piece of steel wire whose long segments are flexible and the short ones are stiff. The elastic properties of the DNA are intimately related to its being a double helix. The latter imposes severe constraints on deformations which can be effected without destroying the molecule and to a large extent determines its mechanical properties. In fact, the two strands comprising the molecule of DNA have just small bending rigidities, just as usual polymers. But the formation of the two-stranded structure drastically changes the DNA by making it both stiff and capable of forming sophisticated spatial shapes. As was mentioned above, the double helix of DNA consists of long chains, or strands, which have the backbones composed of sugar and phosphate residues, and special chemicals, bases, keeping the two strands together (the structure is illustrated in Figure 2). The fundamental building blocks of the strands are nucleotides, joined to each other in polynucleotide chains. The nucleotide consists of a phosphate joined to a sugar (2’-deoxyribose), to which a base is attached. The sugar and base alone are called a nucleoside. The chains, or strands, of the DNA wind round each other in a spiral forming a double helix, the bases being arranged in pairs: adenine - thymine (AT), guanine - cytosine (GC), so that the sequence of bases in one strand determines the complimentary sequence of bases in the other and constitutes the genetic code stored by the molecule of DNA. There are several forms of the DNA, denoted by A,B, and Z. The most common one in nature, is the so-called B-form. One turn of the helix of the B-form, corresponds approximately to $10.5$ base-pairs, and the distance between adjacent pairs of bases is approximately $3.4$ Å. In real life there are considerable deviations from the canonical B-form of the DNA. Therefore, there is a need for a special nomenclature for describing its conformations (see [@calladine] for the details),and generally a considerable set of parameters is required. It is worth noting that the deviations from the canonical form are by no means small, and may have a size of tens of degrees. Synthetic analogues of nucleic acids (NA) containing modified internucleotide groups are useful for solving various problems of molecular biology, biotechnology, and medicine. Shabarova et al, [@s1], [@s2] ,developed a novel type of modified DNA duplexes containing pyrophosphate ( PP) and substituted pyrophosphate ( SPP) internucleotide groups at the definite position of the sugar-phosphate backbone [@s1] - [@s5] (see Figure 1). The PP-group bears one additional negative charge in comparison with a natural internucleotide group; SPP group contains no additional charges. The introduction of PP-groups into DNA leads to an increase of total negative charge of a molecule of the DNA. The study of oligonucleotide duplex containing a PP and SPP groups has revealed that stacking and Watson-Crick interactions are not significantly affected. By flipping out of the disubstituted phosphate these groups fit into the helix structure without elongation of the internucleotide distance. The analysis of helical parameters of base-pairs , internucleotide distances, and overall global structure, reveals a close similarity of the initial and modified duplexes. The location of PP-groups of the DNA are to verify certain conditions: 1. their total number does not exceed 10 % of the total number of phosphates;\ 2. they are not allowed to be located opposite each other;\ 3. they are not allowed to occupy the ends of a molecule;\ 4. two adjacent pyrophosphate groups are to be separated by at least 10 phosphate ones. ![Phosphate, pyrophosphate and substituted pyrophosphate group.[]{data-label="fig:p-groups"}](fig1.eps){width="50.00000%"} ![DNA with a pyrophosphate group.[]{data-label="fig:p-dna"}](fig2.eps){width="50.00000%"} \[sec: Main\] Model. ==================== Theoretical study of liquid crystalline phases of the DNA generally uses models that are necessarily based on very crude simplifications. The first point at issue is the right choice of a potential of interaction. In this paper we model the molecular of the DNA on a 1-dimensional lattice of charges and dipoles with an elementary cell of size $3.4$ Å. It mimics the spatial conformation of charges of phosphate groups and dipoles of base-pairs. We consider short segments of the DNA, approximately $500$ Å, that is of the size of persistence length, so that to a good approximation they are segments of straight lines, and assume that both molecules have the same number, 151, of base-pairs that can be visualized as points on a straight line parallel to the axis of the molecule, one base-pair being located at the center of a corresponding molecule (see Figure 3). The centers of the straight lines belong to a straight line perpendicular to plane x-y which is parallel to either of them. We shall denote by $\xi$ the angle between the straight lines describing the molecules. Both molecules are of the same helicity, which is determined by the rotation of the frame of the dipole moments. Thus, we model a molecule of the DNA on a one-dimensional lattice having at its sites either vectors of dipoles of the base pairs or scalars of the phosphate charges. It is important that the values of the dipoles and charges are renormalized owing to screening effects caused by counterions and ions adsorbed at the molecule. Therefore, we consider effective phosphate charges and dipoles of base pairs. The case of total neutralization of phosphate charges was considered in paper [@kik]. The dipoles are suggested to have the helix symmetry with $\pi / 5$ rotation / bp, corresponding to the structure of the ideal double helix of the DNA. Of course, it is necessary to take into account the structure of DNA being not uniform and the relative positions of the base pairs varying slightly from base pair to base pair. Hence, the dipole moments of the base pairs do not form a precise lattice structure. Even more so they should depend on the local DNA sequence. Therefore, our assumption of the regular dipole positions is a crude [*approximation*]{}. The distance, $\kappa$, between the centers of the lattices, which is fixed, is an important parameter of the model. In what follows we use the distance between adjacent base-pairs, that is $3.4$ Å, as a unit of length, take a unit of charge for which the dipole moment of $1 \; Debye$ equals $1$, and perform calculations in the dimensionless units generated by these quantities. The energy of electrostatic interaction of two molecules can be cast in the sum $$\label{U} \epsilon \, U = U_0 + u_{dd} + u_{dc} + u_{cd} + u_{cc}$$ in which $\epsilon$ is the dielectric permeability of solvent and $U_0$ is the self energy of the pair, which does not influence its conformation, the first term describes the interaction between dipoles of the first molecule and those of the second; the second term - dipoles of the first and phosphate charges of the second; the third - charges of the first and dipoles of the second; the fourth - charges of the first and the second. The interactions are given by the equations $$\begin{aligned} u_{dd}(\rho) &=& e^{-\nu \,\rho} \, \left[ g(\rho) \frac{1}{\rho^3} (\vec p \cdot \vec p^{\; \prime}) - 3 h(\rho) \frac{ [\vec p \cdot (\vec r - \vec r^{\; \prime})] [\vec p^{\; \prime} \cdot (\vec r - \vec r^{\; \prime})] } {\rho^5} \right] \label{Udd} \\ u_{dc}(\vec r, \vec r^{\; \prime}) &=& e^{-\nu \,\rho} k(\rho) Q^{\prime} \frac{\vec p \cdot \vec r^{\; \prime}}{\rho} \label{Ucd}\\ u_{cd} (\vec r^{\; \prime}, \vec r) &=& e^{-\nu \,\rho} k(\rho) Q \frac{\vec p^{\; \prime} \cdot \vec r}{\rho} \label{Udc}\\ u_{cc}(\vec r, \vec r^{\; \prime}) &=& e^{-\nu \,\rho} \frac{ Q Q^{\prime}}{\rho} \label{Ucc}\end{aligned}$$ in which $\nu$ is the inverse Debye length $\nu = \lambda^{-1}$, and $$\rho = |\vec r - \vec r^{\; \prime}|$$ We shall take the screening functions $k(\rho), g(\rho), h(\rho)$ in Schwinger’s form $$k = g = 1 + \nu \, \rho, \quad h = 1 + \nu \, \rho + \frac{1}{3} \, \nu^2 \rho^2$$ The important point about the electro-statical interaction between molecules of the DNA is a wise choice of the screening factor. The common practice is to employ the Debye-Hückel theory, or its modifications that might accommodate the dipole charges, the so-called Schwinger screening, [@podg]. The full treatment of this problem requires a separate investigation. In this paper we confine ourselves to the Debye-Hückel and the Schwinger theories, [@podg]. It is worth noting that the pair potential $U$ is invariant: if we change the sign of angle $\xi$ between the axes of the two molecules, at the same time as the sign of helicity, the potential $U$ remains the same. There are symmetry rules for the helixes of the same kind. One may convince oneself that the following transformations $$\begin{aligned} \label{t1} t_1 : \; (\phi_1, \; \phi_2, \; \xi) & \rightarrow & (\phi_1, \; \pi - \phi_2, \; \xi + \pi) \\ \label{t2} t_2 : \; (\phi_1, \; \phi_2, \; \xi) & \rightarrow & ( \pi - \phi_1, \; \phi_2, \; \xi + \pi) \\ \label{t3} t_3 : \; (\phi_1, \; \phi_2, \; \xi) & \rightarrow & (\phi_2 + \pi, \; \phi_1 + \pi, \; \xi)\end{aligned}$$ leave the potential $U$ invariant. The angles are defined within limits $$- \pi \le \phi_1 \le \pi, \quad - \pi \le \phi_2 \le \pi, \quad - \pi \le \xi \le \pi$$ values $\pm \pi$ corresponding to the same configurations of the molecules. The transformations given by equations (\[t1\]-\[t3\]) verify the equations $$t^2_1 = t^2_2 = t^2_3 = id, \quad t_2 \, t_3 = t_3 \, t_1, \quad t_1 \, t_2 = t_2 \, t_1,$$ where $id$ is a transformation that leaves all $\phi_1, \phi_2, \xi$ invariant. Using the above equations one can easily convince oneself that $t_1, t_2, t_3$ generate a [*non-commutative group*]{} of 8-th order, ${\cal S}$. Its maximal subgroup ${\cal H}$ is a normal subgroup of 4-th order, commutative, and generated by the transformations $$\label{f12} f_1 = t_3, \quad f_2 = t_1 \, t_2 \, t_3$$ Elements $f_1, \; f_2$ in its turn generate subgroups ${\cal H}_1$ and ${\cal H}_2$ of ${\cal H}$, respectively. It is worth noting that ${\cal H}_1, \;{\cal H}_2$ are of second order, both. They are conjugate subgroups of [S]{}, that is for an element $g$ of ${\cal S}$ we have $f_1 = g^{-1} \, f_2 \, g$, or we may state ${\cal H}_1 = g^{-1} \, {\cal H}_2 \, g$, in the notations of group theory, which can be cast in the form of the diagram $$\label{conjug} {\cal H}_1 \longleftrightarrow {\cal H}_2$$ The element $$\label{f3} f_3 = t_1 \, t_2$$ generates subgroup ${\cal H}_3$ of ${\cal H}$. It is important that ${\cal H}_3$ is a normal subgroup of ${\cal S}$, that is $g^{-1} \, {\cal H}_3 \, g = {\cal H}_3 $ for any element $g$ of ${\cal S}$. Thus, we have the diagram of subgroups inside the symmetry group ${\cal S}$ $$\label{groupdiagram} \begin{array}{lllll} {\cal H}_1 & & & \\ & \; \searrow & & & \\ {\cal H}_3 & \longrightarrow & {\cal H} & \longrightarrow & {\cal S} \\ & \; \nearrow & & & \\ {\cal H}_2 & & & \end{array}$$ in which the arrows signify the embedding of subgroups. The group of symmetries, ${\cal S}$, plays the key role in finding minima of the potential $U$. The following general arguments, based on the theory of groups, are quite useful in this respect. Consider a point $\mu$ of space ${\cal X}$ of the angles $\phi_1, \, \phi_2, \, \xi$. Suppose that $\mu$ is a minimum of $U$. Then points $$\mu^{\prime} = g \cdot \mu,$$ called the orbit of the point $\mu$ under the action of the group ${\cal S}$, are also minima of $U$. The number of points $\mu^{\prime}$ of the orbit can vary. In fact, let us consider all transformations $g$ of ${\cal S}$ that leave $\mu$ invariant, that is $\mu^{\prime} = g \cdot \mu = \mu$. It is alleged to be known that the transformations form a subgroup of ${\cal S}$, called stationary subgroup ${\cal H}_{\mu}$. The stationary subgroups, ${\cal H}_{\mu}$ and ${\cal H}_{\nu}$ , for points $\mu$ and $\nu$ of an orbit, are conjugate, that is $ {\cal H}_{\mu} = g^{-1}\, {\cal H}_{\nu} \, g$ for an element $g$ of ${\cal S}$. The number of different points $\mu^{\prime}$ equals to the ratio of the orders of ${\cal S}$ and ${\cal H}_{\mu}$, that is to 2 or 4, depending on the choice of point $\mu$. To be specific, consider a point $\mu$ having a stationary subgroup ${\cal H}_{\mu}$ that coincides with the subgroup ${\cal H}$. The latter is a normal subgroup of ${\cal S}$ of index 2, that is the factor set ${\cal S} /{\cal H}$ consists of two elements. Thus, the orbit of $\mu$ under the action of ${\cal S}$ consists of only two points that correspond to the same value of $U$ and have the same stationary subgroup ${\cal H}$, because the latter is a normal subgroup of ${\cal S}$. The situation is quite different if we take a point $\nu$ having stationary subgroup ${\cal H}_1$, which is different from ${\cal H}_2.$. The subgroups do not coincide in ${\cal S}$, even though they are conjugate. The orbit of ${\cal \mu}$ under the action of ${\cal S}$ indicated above consists of four points that we may sort out as follows: two points having the stationary subgroup ${\cal H}_1$ and two points having ${\cal H}_2$. This is due to the fact that for one thing the subgroup ${\cal H}$ is commutative and therefore its elements generate points of the orbit but with the same stationary subgroup, that is ${\cal H}_1$, and for another there is an element $g$ that gives points of the orbit having the stationary subgroup ${\cal H}_2$. In contrast, a point $\mu$ having the stationary subgroup ${\cal H}_3$ has the orbit consisting of four points which have the same stationary subgroup ${\cal H}_3$, because the latter is a normal subgroup of ${\cal S}$.\[statsubgr\] Numerical simulation ==================== ![ Scheme for the locations of dipoles and charges of base pairs and P-groups, respectfully, in a molecule of the DNA. The conformation of a pair of molecules is determined by angle $\xi$ between the axes of the molecules, and angles $\phi_1, \; \phi_2$ of rotations of the molecules about their axes. []{data-label="fig:model"}](fig3.eps){width="40.00000%"} It is to be noted that numerical evaluation of the minima runs across a poor convergence of standard algorithms for minimization, owing to flat surfaces of constant value for the function of three variables, $U(\phi_1, \; \phi_2, \; \xi)$. To some extent, one may get round the difficulty by observing that for points remaining fixed with respect to a subgroup ${\cal G}$ of ${\cal S}$, the minimization problem is reduced to that for a smaller number of variables. This is due to the necessary conditions for extremum being verified automatically for degrees of freedom perpendicular to the set of invariant points, so that one needs only to study the conditions for longitudinal variables, that is to solve a smaller system of equations. To see the point let us consider a function $f(x,y,z)$ of variables $x, y, z$ even in $x$, so that $f(x,y,z) = f(- \, x, y,z)$. The set of invariant points is $y-z$ plane, and we may look for minima of the function $f(x=0,y,z)$, thus we need to solve only two equations $$\frac{\partial}{\partial y}f(x=0, y,z) = 0, \quad \frac{\partial }{\partial z}f(x=0, y,z) = 0$$ The number of variables necessary for calculations can be reduced even further in case of larger groups of symmetries. It is easy to convince oneself that the sets of fixed points $(\phi_1, \phi_2, \xi)$, that is invariant under the action of a subgroup of ${\cal S}$, read as follows $$\begin{aligned} {\cal F}_1 &:& (\phi_1 = \phi, \; \phi_2 = \phi + \pi, \; \xi) \label{F1} \\ {\cal F}_2 &:& (\phi_1 = \phi, \; \phi_2 = - \, \phi , \; \xi) \label{F2} \\ {\cal F}_3 &:& (\phi_1 = \pm \, \frac{\pi}{2}, \; \phi_2 = \pm \,\frac{\pi}{2} , \; \xi) \label{F3}\end{aligned}$$ in which the ${\cal F}_i$ are invariant under the action of subgroups ${\cal H}_1, \; {\cal H}_2, \; {\cal H}_3$, respectively. The above symmetries are illustrated in Figure 4. ![ \[symmetrygraph\] (a) Cube of the symmetries indicating sets in space $\phi_1, \phi_2, \xi$ invariant with respect to subgroups of ${\cal S}$. Main diagonal plane, $B$, corresponding to subgroup ${\cal H}_2$; two rectangles perpendicular $A$ to ${\cal H}_1$; solid lines $\gamma_1, \; \gamma_4$, and $\gamma_2, \; \gamma_3$ corresponding, to ${\cal H}_3$ and ${\cal H}$, respectfully. (b) Cube of the symmetries view from the top. Dotted line describes invariant points ${\cal F}_2$, corresponding to subgroup ${\cal H}_2$; dashed line points ${\cal F}_1$ and subgroup ${\cal H}_1$; solid circles $\gamma_1$ and $\gamma_4$ to subgroup ${\cal H}_3$; $\gamma_2$ and $\gamma_3$ to subgroup ${\cal H}$. ](fig4.eps){width="1.\textwidth"} The analysis of symmetries of $U$ given above, cf. p. , enables us to sort out the minima according to the effective value of the phosphate charge $Q$. The dependence of the values of minima on the effective charge is illustrated in Figure 5. ![ Minima of $U$ in units of $k_B T, \; T = 300 K$: (a) cholesteric angle $\xi > 0$; (b) $\xi < 0$; against effective charge $q$ in units of the electron charge. Values of $U(\xi, \phi_1, \phi_2)$ and $U(- \; \xi, \phi_1, \phi_2)$ are equal to within $0.01 \; k_B T$. Minima that could correspond to cholesteric phase with $\xi > 0$ vanish at $q = 0.013$ []{data-label="fig:chargeagainstenergy-separated"}](fig5.eps){width="60.00000%"} ![\[q\_E\] Cholesteric angle $\xi$ against effective charge $q$ in units of the electron charge. PP-groups are located symmetrically: (A) no PP-groups; (B) one PP-group at either end of a molecule; (C) two PP-groups at either end of a molecule; (D) three PP-groups; (E) every tenth P-group is exchanged for the PP-one. ](fig6.eps){width="60.00000%"} It should be noted that minima of the pair-interaction $U$ depend on the distance between molecules $\kappa$, and the effective phosphate charge $Q$. The latter is the control parameter we employ in numerical simulation. It is also useful for the description of possible experimental results. In this paper we are considering $\kappa$ to within $10.2 \pm 34$ Å. Effective charge $Q$ of phosphate groups determines the neutralization; it varies to within $0 \pm .6$, in dimensionless units, $Q=0$ corresponding to the total neutralization. Charges $Q$ that correspond to the charge inversion, have not been considered. The Debye length, $\lambda$, has been varied to within $7 \pm 35$ Å, depending on the ion strength of solution. The value of effective charge $Q$ is determined by its electrostatic surrounding. It depends not only on ion charges in solvent, but also on those adsorbed by a molecule of the DNA.It seems that the conventional Debye — Hückel theory does not work in this situation, [@podg]. At any rate, it does not accommodate the adsorbed charges.According to [@livolant2] the effective charge is small. The numerical data and the symmetry analysis given above suggest that there should be the following three types of the minima. 1. Type I characterized by molecules having a cross-like conformation, “ snowflakes”, that is $\xi$ being close to $\pi /2$. It exists for $Q$ large enough. Its symmetry subgroup depends on $Q$ and may take values ${\cal H}_1, \; {\cal H}_2, \; {\cal H}_3, \;{\cal H}$. Therefore, we may claim that there exist four sub-types of minima I: $\rm{I}_{{\cal H}_1}, \; I_{{\cal H}_2}, \;I_{{\cal H}_3}, \;I_{{\cal H}}$, each of them consisting of two subtypes which are given by specific conformations of the angle variables.\ 2. Type II for which $\xi$ takes values to within $0.1^{o}$; the symmetry subgroups are ${\cal H}_1$ and ${\cal H}_2$, either type consists of two sub-types. 3. Type III for which $\xi$ is to within $1^{o}$, that is larger than for II. The symmetry subgroup is ${\cal H}_3$, and there are four constituent types of the same symmetry. As can be inferred from the considerations given above, the study of the pair-interactions between molecules of the DNA requires a means to vary the charges of a molecule and their positions in it. By now the only method available to that end is to vary the ion composition of solvent, the molecules itself being intact. The use of the DNA containing PP-groups, should provide new opportunities for the research, for it could change in a prescribed way the conformation of charges of the molecule. Thus one may compare the formation of liquid crystalline phases for the same solutions but for a different charge conformation of the DNA. Our numerical simulation suggests that the effect could be tangible enough to be looked in experiment. The dependence of the minima on effective charge taking into account PP-groups is indicated in Figure 6. The behavior of $U$ is illustrated in Figure 7 by means of iso-energy surfaces. Another point in favor of working with the pyro-phosphate forms of the DNA is that one can vary the effective charges of a molecule. In the case of the usual phosphate DNA the phosphate charges are all equal, and therefore one may suggest that the effective charges, which enter in our simulation, are also equal. Using the pyro-phosphate forms we may expect to achieve even the regime for which all the phosphate charges are neutralized whereas the pyro-phosphate ones still remain, even though being small. Such an experiment would be helpful in determining the nature of intra-molecular pair interaction that leads, according to the accepted physical picture, [@lk_model], to the formation of cholesteric phases of the DNA. The main point about our numerical simulation is the choice of values for effective charges and dipoles. In case there are PP-groups we may consider the effective charges in a way described above. The situation is more subtle as far as dipoles are concerned. We assume their numerical values being of the first order in the units indicated above. This is as much as to say that the charges of base pairs that constitute dipoles, are screened much less than the phosphate ones. If we proceed otherwise and take small values of the dipoles, there are no minima small but nonzero value of $\xi$, that is no cholesteric phases. In fact, they are, [@livolant1]. Thus, it is reasonable to assume that the charges of base pairs are screened in a manner different from that for the phosphate ones. One may suggest to the effect that the renormalization of charges is due mainly to adsorption of ions from the solvent, and not to the screening clouds of ions in the solvent. If so, it is likely that the charges of P-groups, due to ions of $O^{-}$, are in a different positions than those of the base pairs, and the renormalization of charges of P-groups and base pairs follow different paths. For this argument we are indebted to Yu.M. Yevdokimov. The important point is that PP-groups make for the formation of cholesteric angles different from those of P-groups, and thus provide a means of the identification of new phases. Summing up: - the phase of “snowflakes” sustains the presence of PP-groups; - the PP-groups may result in splitting energy levels of minima, so that minima corresponding to the same values of $U$ become different, when PP-groups are present; - minima of $U$ are separated by low potential barriers; iso-energy surfaces of constant values of $U$, being like galleries between halls that illustrate the minima; - minima corresponding to cholesteric phases are very narrow, whereas those of snowflakes are broad and sloping; energy barriers separating minima corresponding to snowflakes and cholesteric phases, respectively, are of the order $k_{B}T$; - values of angles $\phi_1, \; \phi_2$ are subject to constraints inside galleries joining two minima, ![ Surfaces of constant value of $U$ near cholesteric minima. (a) Screened charge $q = 0.0086$ $q_e$, energy level is $0.05 \; kT$. (b) Screened charge $q = 0.0086$ $q_e$, energy level is $0.2 \;kT$. \[fig:cube1\] ](fig7.eps){width="1.\textwidth"} Conclusions: New opportunities for studying the liquid crystalline phases of the DNA. ====================================================================================== The pyro-phosphate DNA may serve a valuable probe into the physics of cholesteric phases of the DNA. providing a unique opportunity for changing the effective charge of a molecules of the DNA. The use of PP-forms may result in appreciable experimental effects, which in its turn could throw light on the nature of intramolecular interaction in solution of the DNA. The charge screening still poses a number of questions. The usual Debye - Hückel theory does not seem to be an adequate solution, [@podg], especially as the screening of electrical dipole moments is concerned. A theory that could give a reasonable agreement with experiment, should be that of finite number of particles, whereas the Debye - Hückel theory relies on the use of macroscopical considerations. The study of the cholesteric phases of the DNA with pyrophosphate inter-nucleotide insertions could provide a means to find characteristics that indicate a way to understanding the phenomenon. An important issue is the different screening of the phosphate charges and the dipoles of the base pairs. It is should be noted that the screening is caused by a non-uniform adsorption of counterions at a molecule of the DNA, so that the phosphate charges and the base pair dipoles are not screened in the same manner. Our calculations rely on a model that is based on general and qualitative assumptions of the helical charge distribution of the DNA. We feel that it accommodates a picture of the DNA, without going into finer details, and agrees with the approach of paper [@lk_model] in which the continuous approximation plays the directive part. The choice of the pair-potential for the intramolecular interaction is important. The shape of the pair potential chosen in this paper enabled us to accommodate the two different sets of charges — the Coulomb and the dipole ones, and also take into account a finer detail of the pyrophosphate charges, which could turn out to be a valuable instrument for further investigating the cholesteric phases. The symmetry constraints have played an important part in finding the minima of the pair-potential. It seems that their meaning could be greater than a simple arithmetic device for simplifying calculations, and could indicate certain symmetry law peculiar to the cholesteric phases of the DNA. It is reasonable to expect the polymorphism liquid crystalline phases of the DNA. New artificially synthesized phases of the DNA could be a fruitful instrument to that end. We are thankful to F.Livolant for the useful correspondence. This work was done within the framework of the Interdisciplinary Programme of the Lomonosov Moscow State University, and partially supported by RFBR Grant \# NS—660.2008.1. [99]{} C.Robinson, Tetrahedron, [**13**]{}, 219 (1961). F.Livolant, J.de Phys. (Paris), [**47**]{}, 1605 (1986). F.Livolant, Biophys. J., [**65**]{}, 56 (1993). Onsager L., Ann.Acad.Sci. NY, [bf 51]{}, 621 (1999). A.A. Purmal, V.L. Druca, Z.A. Shabanova, Bio-Org.Chem. [**10**]{}, 394 (1984) (in Russian). S.A. Kuznetsova, C. Clusel, E. Ugarte, I. Elias, M. Vasseur, M. Blumenfeld, and Z.A. Shabanova, Nucl.Acids Res. [**24**]{}, 4783 (1996). G.Ya. Sheflyan, E.A. Kubareva, S.A. Kuznetsova, A.S. Karyagina, I.I. Nikol’skaya, E.S. Gromova, and Z.A. Shabanova, FEBS Lett. [**390**]{}, 307 (1996). A.A. Purmal, Z.A. Shabanova, R.I. Gumport, Nucl.Acids Res. [**20**]{}, 3713 (1992). M.V. Rogacheva, A.V. Bochenkova, S.A. Kuznetsova, M.K. Saparbaev, and A.V. Nemukhin, J.Phys.Chem. [**111**]{}, 432 (2007). A.A.Kornyshev and S.Leikin, J.Chem.Phys. [**107**]{}, 3656 (1997). A.A.Kornyshev and S.Leikin, Phys.Rev.Lett., [**84**]{}, 2537 (2000). A.A.Kornyshev, D.J.Lee, S.Leikin, A.Wynveen, and S.B.Zimmerman, Phys.Rev.Lett., [**95**]{}, 148102-1, (2005). A.A.Kornyshev and S.Leikin, Proc.Natl.Acad.Sci.U.S.A. [**95**]{}, 13579 (1998). Y.H.Kim, J. de Phys.(Paris) [**43**]{}, 559 (1982). B.Samorĭ, M.Osipov, I.Domini, and A.Bartolini, Int.J.Macromol., [**15**]{}, 353 (1993). V.L.Golo, E.I.Kats, and I.P.Kikotx, Pisma ZhETF [**84**]{}, 334 (2006). V.L. Golo, E.I.Kats and, Yu.S.Volkov, Pis’ma ZhETF [**86**]{}, 311 (2007), M.A.El Hassan and C.R.Calladine 1995 J.Mol.Biol. [**251**]{} 648 Punkkinen1 O., Naji A. , Podgornik R., . Vattulainen1 I., and Hansen  P.-L., EPL, [**82**]{}, 48001, (2008).

\ <span style="font-variant:small-caps;">[W[ł]{}adys[ł]{}aw A. Majewski]{}\ Institute of Theoretical Physics and Astrophysics\ Gda[ń]{}sk University\ Wita Stwosza 57\ 80-952 Gda[ń]{}sk, Poland</span>\ *E-mail address:* `[email protected]`\ <span style="font-variant:small-caps;">Abstract.</span> We outline the scheme for quantization of classical Banach space results associated with some prototypes of dynamical maps and describe the quantization of correlations as well. A relation between these two areas is discussed. : Primary: 46L53, 46L60: Secondary: 46L45, 46L30 *Key words and phrases:* $C^*$-algebras, positive maps, separable states, entanglement, quantum stochastic dynamics, quantum correlations. INTRODUCTION ============ The aim of this paper is to bring together two areas, theory of positive maps on [[[$\hbox{\bf C}^*$]{}]{}]{}-algebras and theory of entanglement considered as a peculiar feature of non-commutative Radon measures. Both topics are at the heart of quantum theory, thus in particular, in the foundations of quantum information theory. It will be shown that such pure quantum features as: peculiar behaviour of positive maps, quantum correlations, entanglement, and quantum stochastic dynamics, can be easily obtained within the framework of the [[[$\hbox{\bf C}^*$]{}]{}]{}-algebraic approach to Quantum Mechanics. This approach sheds new light on entanglement and quantum features of correlations of non-commutative systems. In particular, some emphasis will be put on evolution of entanglement. The paper is organized as follows. Section 2 provides sufficient preparation for the concept of “quantization” of classical results related to prototypes of dynamical maps. Section 3 is concerned with entanglement and the coefficient of quantum correlations. The latter is again an example of “quantization” of a classical concept. The last section contains a brief discussion of applications of the presented results to the description of quantum dynamical systems. We will discuss the evolution of entanglement for some selected models as well as relations between classification of positive maps and measures of entanglement. POSITIVE MAPS ============= In this section we compile some basic facts on the theory of positive maps on [[[$\hbox{\bf C}^*$]{}]{}]{}-algebras. To begin with, let ${{\mathcal A}}$ and ${{\mathcal B}}$ be [[[$\hbox{\bf C}^*$]{}]{}]{}-algebras (with unit), ${{\mathcal A}}_h = \{ a \in {{\mathcal A}}; a = a^* \}$, ${{\mathcal A}}^+ = \{ a \in {{\mathcal A}}_h; a \ge 0 \}$ - the set of all positive elements in ${{\mathcal A}}$, and ${{\mathcal S}}({{\mathcal A}})$ the set of all states on ${{\mathcal A}}$. In particular $$({{\mathcal A}}_h, {{\mathcal A}}^+)\quad is \quad an \quad ordered \quad Banach \quad space.$$ We say that a linear map $\alpha : {{\mathcal A}}\to {{\mathcal B}}$ is positive if $\alpha({{\mathcal A}}^+) \subset {{\mathcal B}}^+$. The theory of positive maps on non-commutative algebras can be viewed as a jig-saw-puzzle with pieces whose exact form is not well known. Therefore, as we address this paper to a readership interested in quantum mechanics and quantum information theory, we will focus our attention on “quantization” procedure of some classical (Banach space) results in order to facilitate access to some main problems of that theory. We begin with the classical Banach-Stone result ([@B], [@S]) [*if a unital linear map $T: C(X) \to C(Y)$, where $X,Y$ are compact Hausdorff spaces, is either isometric or an order-isomorphism then it is also an algebraic isomorphism.*]{} Thus, even in the Banach space setting, the order and algebraic structures are strongly related. The Banach-Stone theorem has the following non-commutative generalization (Kadison, [@K]): [*a unital isometric or order isomorphic linear map $\alpha : {{\mathcal A}}_h \to {{\mathcal B}}_h$ must preserve the Jordan product ( $(a,b) \mapsto {1/2}(ab + ba)$)*]{}. In other words, this result indicates the role of a specific algebraic structure - the Jordan structure - in operator algebras and that remark will be frequently used throughout the paper. Moreover, such a result makes it legitimate to study and to classify [[[$\hbox{\bf C}^*$]{}]{}]{}-algebras ${{\mathcal A}}$ by a detailed analysis of ordered Banach spaces ${{\mathcal A}}_h$. However, this is [*a very difficult task*]{}. In particular, it was soon realized that one of the basic problems is the answer to the following question: which compact convex sets can arise as the state spaces of unital $C^*$-algebras (again a very difficult task!). To describe the next result we need some preliminaries. Let $(\Omega, \mu)$ be a measure space. Here and subsequently, $\mu$ stands for a probability measure. The triple (semigroup $\{S_t\}$, $\Omega$, $\mu$) will denote the classical dynamical system where $ S_t : \Omega \to \Omega$ is a one parameter family of measure preserving maps. The phase functions $f : \Omega \to {{\mathbb{C}}}$ evolve according to the Koopman operators $$V_tf(\omega)=f(S_t\omega) \ , \ \ \omega\in\Omega \,.$$ It is known that the Koopman operators $V_t$ are isometries on the Banach space $L^p=L^p(\Omega, \mu)$, $p\ge 1$ of $p$-integrable functions and unitary operators when restricted to the Hilbert space $L^2$ and the transformations $S_t$ are automorphisms. The relation of the point dynamics with the Koopman operators is clarified by asking the question: [*what types of isometries on $L^p$ spaces are implementable by point transformations?*]{} For $L^p$ spaces $p\not=2$, all isometries induce underlying point transformations, i.e. if $||Vf|| = ||f||$ for all $f \in L^p$, then $V$ is given by an underlying measurable point transformation $S$ and a certain function $h$ according to $(Vf)(x) = h(x)f(Sx)$. Such theorems on the implementability of isometries on $L^p$ spaces, $p\neq 2$, are known as [*Banach-Lamperti theorems* ]{} [@B], [@La]. They are of great importance for the Misra-Prigogine-Courbage theory [@MPC] which is trying [*to reconcile irreversible phenomena with the basic dynamical laws*]{}. Again, one may “quantize” Banach-Lamperti theorems [@Ye], see also [@AMS]. To this end one should use the so called non-commutative (quantum) $L_p$-spaces. Namely, using the “quantized” measure theory, let $\{ {{\mathcal A}}, \varphi \}$ be a von Neumann algebra with faithful normal trace and let $L_p({{\mathcal A}}, \varphi)$, $p \ge 1$, be the corresponding quantum $L_p$-space, i.e. a Banach space of operators which is closed under an appropriate norm. Assume that $T: L_p({{\mathcal A}}, \varphi) \to L_p({{\mathcal A}}, \varphi)$ is a linear map. [*Then $T$ is an $L_p$-isometry if and only if* ]{} $$T(x) = W B J(x), \quad x \in L_p({{\mathcal A}}, \varphi) \cap {{\mathcal A}}$$ where $W \in {{\mathcal A}}$ is a partial isometry, $B$ a selfadjoint operator affiliated with ${{\mathcal A}}$, $J$ a normal Jordan isomorphism mapping ${{\mathcal A}}$ into a weakly closed $^*$-subalgebra of ${{\mathcal A}}$ such that $ W^*W = J({{\mathbb{I}}}) = supp(B)$ and $B$ commutes strongly with $ J({{\mathcal A}})$. Again, we can see the importance of the Jordan structure. The third example we wish to recall is associated with a very strong notion of positivity: the so called complete positivity (CP). Namely, a linear map $\tau : {{\mathcal A}}\to {{\mathcal B}}$ is CP iff $$\tau_n : M_n({{\mathcal A}}) \to M_n({{\mathcal B}}); [a_{ij} \mapsto [\tau(a_{ij})]$$ is positive for all n. To explain the basic motivation for that concept we need the following notion: [*an operator state of ${{\mathcal A}}$ on a Hilbert space ${{\mathcal K}}$ is a CP map $\tau : {{\mathcal A}}\to {{\mathcal B}}({{\mathcal K}})$*]{}. Having that concept we can recall the Stinespring result, [@Sti], which is the generalization of GNS construction and which was the starting point for a general interest in the concept of complete positivity. [*For operator state $\tau$ there is a Hilbert space ${{\mathcal H}}$, a $^*$-representation $\pi : {{\mathcal A}}\to {{\mathcal B}}({{\mathcal H}})$ and a partial isometry $V : {{\mathcal K}}\to {{\mathcal H}}$ for which*]{} $$\tau(a) = V^* \pi(a) V.$$ Following the quantization “route”, it was shown - (Choi, [@Ch]) if $\tau : {{\mathcal A}}\to {{\mathcal B}}$ is a CP order isomorphism then it is a $^*$-isomorphism. This can be considered as a final “quantization” of the Banach-Stone theorem. - (Arverson, [@Ar]) the Hahn-Banach theorem and its order-theoretical version (due to Krein) has a nice generalization for non-commutative structures in terms of CP maps: [*Let ${\mathcal N}$ be a closed self-adjoint subspace of [[[$\hbox{\bf C}^*$]{}]{}]{}algebra ${{\mathcal A}}$ containing the identity and let $\tau: {\mathcal N} \to {{\mathcal B}}({{\mathcal H}})$ be a CP map. Then $\tau$ possesses an extension to a CP map $\tilde{\tau} : {{\mathcal A}}\to {{\mathcal B}}({{\mathcal H}})$.*]{} It is worth pointing out that plain positivity is not enough for these generalizations. Moreover, Arverson’s extension theorem is the basis of the CP ideology in open system theory. Up to now we considered linear positive maps on an algebra without entering into the (possible) complexity of the underlying algebra. The situation changes when one is dealing with composed systems (for example in the framework of open system theory). Namely, there is a need to use the tensor product structure. In particular, again, we wish to consider positive maps but now defined on the tensor product of two [[[$\hbox{\bf C}^*$]{}]{}]{}-algebras, $\tau : {{\mathcal A}}\otimes {{\mathcal B}}\to {{\mathcal A}}\otimes {{\mathcal B}}$. But now the question of order is much more complicated. Namely, there are various cones determining the order structure in the tensor product of algebras (cf. [@W]) $${{\mathcal C}}_{inj} \equiv ({{\mathcal A}}\otimes {{\mathcal B}})^+ \supseteq, ..., \supseteq {{\mathcal C}}_{\beta} \supseteq,..., \supseteq {{\mathcal C}}_{pro}\equiv conv({{\mathcal A}}^+\otimes{{\mathcal B}}^+)$$ and correspondingly in terms of states (cf [@MM]) $${{\mathcal S}}({{\mathcal A}}\otimes {{\mathcal B}}) \supseteq,..., \supseteq {{\mathcal S}}_{\beta} \supseteq, ..., \supseteq conv({{\mathcal S}}({{\mathcal A}})\otimes{{\mathcal S}}({{\mathcal B}})).$$ Here, ${{\mathcal C}}_{inj}$ stands for the injective cone, ${{\mathcal C}}_{\beta}$ for a tensor cone, while ${{\mathcal C}}_{pro}$ for the projective cone. The tensor cone ${{\mathcal C}}_{\beta}$ is defined by the property: the canonical bilinear mappings $\omega :{{\mathcal A}}_h \times {{\mathcal B}}_h \to ({{\mathcal A}}_h \otimes {{\mathcal B}}_h, {{\mathcal C}}_{\beta})$ and $\omega^* : {{\mathcal A}}^*_h \times {{\mathcal B}}^*_h \to ({{\mathcal A}}^*_h \otimes {{\mathcal B}}^*_h, {{\mathcal C}}_{\beta}^*)$ are positive. The connes ${{\mathcal C}}_{inj}, C_{\beta}, C_{pro}$ are different unless either ${{\mathcal A}}$, or ${{\mathcal B}}$, or both ${{\mathcal A}}$ and ${{\mathcal B}}$ are abelian. This feature is the origin of various positivity concepts for non-commutative composed systems and it was Stinespring who used the partial transposition (transposition tensored with identity map) for showing the difference among $C_{\beta}$ and ${{\mathcal C}}_{inj}$ and ${{\mathcal C}}_{pro}$. Clearly, in dual terms, the mentioned property corresponds to the fact that separable states are different from the set of all states and that there are various special subsets of states if both subsystems are truly quantum. To summarize one can say that contrary to the plain positivity, CP property plays a dominant role in the programme of quantization of classical results for composed systems. However, as it will be discussed in the final section, other types of positivity are helpful for better understanding the relations between various subsets of states, the algebraic and the order structure. QUANTUM CORRELATIONS ==================== Now we wish to discuss problems associated with the partial order structure of tensor product of [[[$\hbox{\bf C}^*$]{}]{}]{}-algebras which are related to quantum information theory. Our first remark is the observation that quantum information theory relies on the fact that the restriction of a pure state of a composed system to a subsystem is, in general, not pure. Moreover, if the restriction of a pure state is pure then the state of the composed system is of the product form. This leads to the observation that the coupling of observables of a composed system by an entangled state offers additional possibilities for information exchange as well as a chance to reproduce states. All that follows from the fact that entangled states exhibit non-classical correlations. To be more precise, let ${{\mathcal A}}_1 \subseteq {{\mathcal B}}({{\mathcal H}}_1)$ and ${{\mathcal A}}_2 \subseteq {{\mathcal B}}({{\mathcal H}}_2)$ be two concrete [[[$\hbox{\bf C}^*$]{}]{}]{}-algebras and define, for a state $\omega$ on ${{\mathcal A}}_1 \otimes {{\mathcal A}}_2$, the following map: $$(r_1 \omega)(A) \equiv \omega(A \otimes {\bf 1}) \quad ( (r_2\omega(B) = \omega( {\bf 1}\otimes B) )$$ where $A \in {{\mathcal A}}_1$ ($B \in {{\mathcal A}}_2$). $r_{1(2)} \omega$ is a state on ${{\mathcal A}}_{1(2)}$ . Moreover: [*Let $(r_1 \omega)$ be a pure state on ${{\mathcal A}}_1$. Then $\omega$ can be written as a product state on ${{\mathcal A}}_1 \otimes {{\mathcal A}}_2$.*]{} Let $\omega$ be a state on ${\mathcal A}_1 \otimes {{\mathcal A}}_2$. [*The entanglement of formation*]{}, EoF, of $\omega$ can be defined as ( [@Mjp], see also [@Ben]) $${E}(\omega) = \inf_{\mu \in M_{\omega}({{\mathcal S}})} \int_{{{\mathcal S}}} d\mu(\varphi) S(r\varphi)$$ where $S(\cdot)$ stands for the von Neumann entropy, i.e. $S(\varphi) = - Tr \varrho_{\varphi} \log \varrho_{\varphi}$ where $\varrho_{\varphi}$ is the density matrix determining the state $\varphi$. We want to stress that other entropy-functions can be used! The given definition of EoF is based on the decomposition theory and in particular $M_{\omega}({{\mathcal S}}) \equiv \{ \mu: \omega = \int_{{{\mathcal S}}}\nu d\mu(\nu)\}$. We recall that the separable states are those which are in the closure of the convex hull of simple tensors (so tensor products of subsystem states) while an entangled state stands for a non-separable one. One can prove [@Mjp] A state $\omega \in {{\mathcal S}}$ is separable if and only if $E(\omega)$ is equal to 0. Let us denote the set of all states on ${{\mathcal A}}\equiv {{\mathcal A}}_1 \otimes {{\mathcal A}}_2$ (${{\mathcal A}}_1$, ${{\mathcal A}}_2$) by ${{\mathcal S}}({{\mathcal A}})$ (${{\mathcal S}}({{\mathcal A}}_1)$, ${{\mathcal S}}({{\mathcal A}}_2)$ respectively). Obviously, $r_i \omega$ is in ${{\mathcal S}}({{\mathcal A}}_i)$, $i=1,2$. Next, take a measure $\mu$ on ${{\mathcal S}}({{\mathcal A}})$. Then, using the restriction maps $r_i$ one can define measures $\mu_i$ on ${{\mathcal S}}({{\mathcal A}}_i)$ in the following way: for a Borel subset $F_i \subset {{\mathcal S}}({{\mathcal A}}_i)$ we put $$\mu_i(F_i) = \mu(r_i^{-1}(F_i)), \quad i =1,2.$$ Having measures $\mu_1$ and $\mu_2$, both coming from the given measure $\mu$ on ${{\mathcal S}}({{\mathcal A}})$, one can define new measure $\boxtimes \mu$ on ${{\mathcal S}}({{\mathcal A}}_1) \times {{\mathcal S}}({{\mathcal A}}_2)$ which encodes classical correlations between the two subsystems described by ${{\mathcal A}}_1$ and ${{\mathcal A}}_2$ respectively (see [@Ma2] for details). The measure $\boxtimes \mu$ leads to the concept of coefficient of local (quantum) correlations for $\phi \in {{\mathcal S}}({{\mathcal A}}), a_1 \in {{\mathcal A}}_1, a_2 \in {{\mathcal A}}_2$, which is defined as $$\begin{aligned} d(\phi, a_1, a_2)& = & \inf_{\mu \in M_{\phi}({{\mathcal S}}({{\mathcal A}}))} |\phi(a_1 \otimes a_2) \nonumber \\ && - (\int \xi d(\boxtimes \mu)(\xi))(a_1 \otimes a_2)| \nonumber\end{aligned}$$ The crucial property of the coefficient of quantum correlations is that $d(\phi, \cdot \cdot)$ is equal to $0$ if and only if the state $\phi$ is separable ([@Ma2], [@Ma3]). The advantage of using $d(\cdot)$ lies in the fact that that concept looks more operational and that it does not use an entropy function. Moreover, $d(\cdot)$ is nothing else but the “quantization” of the classical concept of coefficient of independence. Hence, we got a strong indication that entangled states contain new type of correlations which are called quantum. SOME APPLICATIONS ================= QUANTUM STOCHASTIC DYNAMICS ([@MZ1]-[@MZ5], [@MOZ]) --------------------------------------------------- It is well known that in the theory of classical particle systems one of the basic objectives is to produce, describe and analyze dynamical systems with an evolution originated from stochastic processes in such a way that their equilibrium states are given Gibbs states (cf. [@Ligget]). A well known illustration of such an approach are systems with the so called Glauber dynamics [@Rx]. To carry out the analysis of dynamical systems with evolution originated from stochastic processes, it is convenient to use the theory of Markov processes in the framework of $L_p$-spaces. In particular, for the Markov-Feller processes, using the unique correspondence between the process and the corresponding dynamical semigroup, one can give a recipe for the construction of Markov generators (see [@Ligget]). The correspondence uses the concept of conditional expectation which can be nicely characterized within the (classical) $L_p$-space framework (cf. [@Moya]). Furthermore, (classical) $L_p$ spaces are extremely useful in a detailed analysis of the ergodic properties of the evolution. However, as contemporary science is based on [*quantum mechanics*]{}, it is again legitimate to look for a quantization of the above approach. That task was carried out in the setting of quantum mechanics and the main ingredient of the quantization was the concept of generalized conditional expectation and Dirichlet forms defined in terms of non-commutative (quantum) $L_p$-spaces. We already met these spaces in the description of quantized Banach-Lamperti theorems. The advantage of using quantum $L_p$-spaces for the quantization of stochastic dynamics lies in the fact that we can follow the traditional “route” of analysis of dynamical systems and that it is possible to have a single scheme for the quantum counterparts of stochastic dynamics of both jump and diffusive type. Turning to concrete dynamical systems, for example to jump type evolutions, we recall that one of the essential ingredients of the $L_p$-space approach to the analysis of such evolutions, is the usage of local knowledge. To illustrate that idea let us consider a region $\Lambda_I$ (usually finite) and its environment $\Lambda_{II}$. Then, performing an operation over $\Lambda_I$ (e.g. a block-spin flip or a symmetry transformation) one is changing locally the reference state. Such a change can be expressed in terms of generalized conditional expectations. Guided by the classical theory, one can define, now in terms of generalized conditional expectations, the infinitesimal generator of quantum dynamics. It is important to note that such a dynamics is the result of local operations (associated with the mentioned local knowledge about the system).\ Then, having defined the dynamics, we should pose the natural question of its nontriviality. By this we understand, first of all, that the infinitesimal generator of the dynamics is [*not*]{} a function of the hamiltonian defining the reference Gibbs state. This requirement arises in a natural way from the methodology of constructing stochastic dynamics as sketched in the preceding paragraph. In fact, it has been shown [@MOZ] that generators defined within the $L_p$-space setting satisfy the above requirement. On the other hand, [*in order to confirm that the constructed dynamics are interesting, and the genuine quantum counterparts of classical dynamical maps it is necessary to study the evolution of entanglement and correlations as measures of coupling between two subsystems*]{} caused by local (e.g. block-spin flip) operations. Going in that direction, an analysis of stochastic quantum models based on reference systems determined by Ising type and XXZ hamiltonians ([@KM2], see also [@KM1]) was done. It has shown the tendency of enhancement of quantum correlations. In the first example, based on one dimensional Ising model with nearest neighbor interactions, the lack of production of quantum correlations was shown. This is to be expected because the Ising model illustrates a behaviour typical of classical interactions (cf [@BR]). The second example, based on the quantum XXZ model with more interesting and complicated features of propagation, provides [*clear signatures of production of quantum correlations.*]{} POSITIVE MAPS VERSUS ENTANGLEMENT --------------------------------- The analysis of evolution of entanglement which was described in the previous subsection indicates that there is a need for an operational measure of entanglement. This demand is strenghtened by the observation that the amount of states that can be used for quantum information is measured by the entanglement. On the other hand, the programme of classification of entanglement seems to be a very difficult task. In particular, it was realized that the first step must presumably take the full classification of all positive maps. To see this let us take a positive map $\alpha_{1,t}: {{\mathcal A}}_1 \to {{\mathcal A}}_1$, $t$ being the time, and consider the evolution of a density matrix $\varrho$ ($\varrho$ determines the state $\phi \in {{\mathcal S}}({{\mathcal A}}\otimes {{\mathcal B}})$), i.e. we wish to study $(\alpha_{1,t} \otimes id_2)^d\varrho$. Here $(\alpha_{1,t} \otimes id_2)^d$ stands for the dual map, i.e. for the dynamical map in the Schrödinger picture. Then, if $\varrho$ is an entangled state, $(\alpha_{1,t} \otimes id_2)^d\varrho$ may develop negative eigenvalues and thus lose consistency as a physical state. That observation was the origin of rediscovery, now in the physical context, of Stinespring’s argument saying that the tensor product of transposition with the identity map can distinguish various cones in the tensor product structure (see Section 2). This led to the criterion of separability ([@P], [@H]) saying that only separable states are globally invariant with respect to the familly of all positive maps. However, criterions of that type are not operational. Even worse, they are strongly related to a classification of positive maps. In particular, the old open problem concerning the description of non-decomposable maps was revived. To describe that problem we need (cf [@St]): Let $\tau : {{\mathcal A}}\to {{\mathcal B}}({{\mathcal H}})$ be a linear, positive map. $\tau$ is called decomposable if there exists a Hilbert space ${{\mathcal K}}$, a bounded linear map $V: {{\mathcal H}}\to {{\mathcal K}}$ and a [[[$\hbox{\bf C}^*$]{}]{}]{}-homomorphism $\pi : {{\mathcal A}}\to {{\mathcal B}}({{\mathcal K}})$ such that $\tau = V^* \pi V$. [[[$\hbox{\bf C}^*$]{}]{}]{}-homomorphism means that $\pi (\{a,b\}) = \{\pi(a), \pi(b)\}$ where $\{ \cdot, \cdot \}$ stands for anticommutator, i.e. $\pi$ preserves the Jordan structure! A more subtle notion is the following: $\tau$ is locally decomposable if for $0 \ne x \in {{\mathcal H}}$, there exists a Hilbert space ${{\mathcal K}}_x$, $V_x:{{\mathcal K}}_x \to {{\mathcal H}}$ and a [[[$\hbox{\bf C}^*$]{}]{}]{}-hommomorphism $\pi_x$ of ${{\mathcal A}}$ to ${{\mathcal B}}({{\mathcal K}}_x)$ such that $$V_x \pi_x(a) V^*_x x = \tau (a)x$$ for all $a \in {{\mathcal A}}$. It is known ([@Wor], [@Choi]) that for the case $M_k({{\mathbb{C}}}) \otimes M_l({{\mathbb{C}}})$ with $k=2=l$ and $k=2$, $l=3$ all positive maps are decomposable. Then, the criterion for separability simplifies significantly. Namely, to verify separability it is enough to analyse $(\tau \otimes id)^d$, with $\tau$ being the transposition, as other positive maps are just convex combinations of CP maps (they always map states into states) and the composition of CP map with $\tau \otimes id$. The situation changes dramatically when both $k$ and $l$ are larger than 2. In that case there are plenty of non-decomposable maps (see [@Kos] and the references given there) and to analyse entanglement one cannot restrict oneself to study $\tau \otimes id$. Thus, a full description of positive maps is needed. Furthermore, one can constuct examples of entangled states using concrete non-decomposable maps (see [@HKP]). However, the classification of non-decomposable maps is a difficult task which is still not completed ([@St3], [@LMM]). We want to close the section with an important remark. Namely, if $d(\phi, A)= 0$ for any $A \in {{\mathcal A}}_1 \otimes {{\mathcal A}}_2$ then, using the description of locally decomposable maps, one can show that the state $\phi$ is separable [@Ma3]. This result shows how strong the interplay between separability and certain subtle features of positive maps is. However, this is not unexpected as the full correspondence between Schrödinger and Heisenberg picture relies on the underlying algebraic structure and geometry of the state space, see [@A], [@C], and [@E]. Acknowledgements ================ The author would like to thank the organisers of the XXXV Symposium on Mathematical Physics, Torun, Poland for a very nice and interesting conference where the main topics of this paper were presented. Thanks also to Mark Fannes for careful reading the manuscript. The support of BW grant 5400-5-0255-3is gratefully acknowledged. [99]{} E. M. Alfsen and F. W. Shultz, [*State spaces of operator algebras*]{}, Birkhauser, Boston, 2001 I. Antoniou, W. A. Majewski and Z. Suchanecki, Implementability of Liouville evolution, Koopman and Banach-Lamperti theorems in classical and quantum dynamics, [*Open Sys $\&$ Information Dyn.*]{} [**9**]{}, 1-12 (2002) W. B. Arverson, Subalgebras of $C^*$-algebras, [*Acta Math.*]{} [**123**]{} (1969), 141-224 S. Banach, [*Théorie des Opérations Linéares*]{}, Warszawa, 1932 Ch. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wotters, Phys. Rev. A [**54**]{} 3824 (1996) O. Bratteli and D. W. Robinson [*Operator Algebras and Quantum Statistical Mechanics*]{}, Springer Verlag, New York-Heidelberg-Berlin, vol. I (1979), vol. II (1980) M.-D., Choi, [Completely Positive Maps on Complex Matrices]{}, *Lin. Alg. Appl.* **10** (1975), 285–290. M.-D. Choi, A Schwarz inequality for positive linear maps on $C^*$-algebras, *Illinois J. Math.* **18**(4) (1974), 565–574. A. Connes, Charactérisation des espaces vectoriels ordonnés sous-jacents aux algèbres de von Neumann, [*Ann. Inst. Fourier, Grenoble*]{} [**24**]{} (1974), 121-155 E. G. Effros, Aspects of non-commutative order, [*The Second U.S. Japan Seminar on $C^*$-algebras and Applications to Physics*]{}, 1977 R.J. Glauber, Time dependent statistics of the Ising model, [*J. Mat. Phys.*]{}, [**4**]{}, 294, (1963) K-Ch. Ha, S-H. Kye, Y. S. Park, Entangled states with positive partial transposes arising from indecomposable positive linear maps, Phys. Lett. A, [**313**]{}, 163-174, (2003) M. Horodecki, P. Horodecki, R. Horodecki, Separability of mixed states: necessary and sufficient conditions, [*Phys. Lett A*]{} [**223**]{}, 1-8 (1996) R. V. Kadison, Isometries of operator algebras, [*Annals of Math*]{}, [**54**]{} (1951) 325-338 A. Kossakowski, A class of linear positive maps in matrix algebras, [*Open Sys $\&$ Information Dyn.*]{}, to appear S. Kozie[ł]{}, A.W. Majewski, On quantum correlations for stochastic dynamics of XXZ type,[*Acta Phys. Polonica B*]{}, [**34**]{}, 3731 (2003) S. Kozie[ł]{}, A.W. Majewski, Evolution of quantum correlations for jump-type quantum stochastic dynamics, [*Acta Phys. Polonica B*]{}, [**33**]{}, 1103 (2002) L. E. Labuschagne, W. A. Majewski and M. Marciniak, [*On k-decomposability of positive maps*]{}, LANL preprint, math-phys/0306017 J. Lamperti, On the isometries of certain function spaces, [*Pacific J. Math.*]{} [**8**]{} 459-466 (1958). T.M. Ligget, [*Interacting Particle Systems*]{}, Springer Verlag, (1985) A. W. Majewski, On entanglement of states and quantum correlations, in [*Operator algebras and Mathematical Physics*]{}, Eds. J.M. Combes, J. Cuntz, G.A. Elliott, G. Nenciu, H. Siedentop, S. Stratila; [*Theta*]{}, Bucharest, 2003 LANL preprint, math-phys/0202030 A. W. Majewski, On entanglement of states and quantum correlations, to appear A., W., Majewski, On entanglement of formation, [*J. Phys. A; Math. Gen.,*]{} [**35**]{}, 123-134 (2002) A. W., Majewski, M. Marciniak, On a characterization of positive maps, [*J. Phys. A; Math. Gen.,*]{} [**34**]{}, 5863-5874 (2001) A.W. Majewski, B. Zegarlinski, Quantum Stochastic Dynamics I: Spin Systems on a Lattice, [*Math. Phys. Electronic J.*]{} [**1**]{}, Paper 2 (1995) A.W. Majewski, B. Zegarlinski, On Quantum Stochastic Dynamics and Noncommutative [$L_p$]{} Spaces, [*Lett. Math. Phys.*]{} [**36**]{}, 337 (1996) A.W. Majewski, B. Zegarlinski, Quantum Stochastic Dynamics II, [*Rev. Math. Phys.*]{} [**8**]{}, 689 (1996) A.W. Majewski, B. Zegarlinski, On quantum stochastic dynamics, [*Markov Proc. and Rel. Fields*]{} [**2**]{}, 87 (1996) A.W. Majewski, B. Zegarlinski, “On quantum stochastic dynamics for quantum spin systems on a lattice; Low temperature problem” in [*Mathematical Results in Statistical Mechanics*]{}, ed. S. Miracle-Solé, J. Ruiz, V. Zagrebnov, World Scientific, 1999 A.W. Majewski, R. Olkiewicz, B. Zegarlinski, Dissipative dynamics for quantum spin systems on a lattice, [*J. Phys. A: Math. Gen.*]{} [**31**]{}, 2045 - 2056, 1998. B. Misra, I. Prigogine and M. Courbage, From deterministic dynamics to probabilistic descriptions, [*Physica*]{} [**98 A**]{} 1-26 (1979). S-T. C. Moy, [ Characterization of conditional expectation as a transformation on function spaces]{}, [*Pacific J. Math.*]{} [**4**]{}, 47-64 (1954) A. Peres, Separability criterion for density matrices, [*Phys. Rev. Lett*]{} [**77**]{}, 1413 (1996) M. Stone, Applications of the theory of Boolean rings to general topology, [*Trans. Amer. Math. Soc.*]{} [**41**]{} (1937), 375-481 E. St[ø]{}rmer, Positive linear maps of operator algebras, *Acta Math.* **110** (1963), 233–278. E. St[ø]{}rmer, [Decomposable positive maps on $C^*$-algebras]{}, *Proc. Amer. Math. Soc.* **86** (1980), 402–404. W. Stinespring, Positive functions on $C^*$-algebras, [*Proc. Amer. Math.Soc.* ]{} [**6**]{} (1955), 211-216 G. Wittstock, [*Ordered Normed Tensor Products*]{} in “Foundations of Quantum Mechanics and Ordered Linear Spaces” (Advanced Study Institute held in Marburg) A. Hartkämper and H. Neumann eds. [*Lecture Notes in Physics*]{} vol. [**29**]{}, Springer Verlag 1974. S. L. Woronowicz, [Positive maps of low dimensional matrix algebras]{}, *Rep. Math. Phys.* **10** (1976), 165–183. F.J. Yeadon, Isometries of non-commutative $L^p$-spaces, [*Math. Proc. Camb. Phil. Soc.*]{} [**90**]{} 41-50 (1981).

--- abstract: 'Recent papers that have studied variants of the Peyrard-Bishop model for DNA, have taken into account the long range interaction due to the dipole moments of the hydrogen bonds between base pairs. In these models the helicity of the double strand is not considered. In this particular paper we have performed an analysis of the influence of the helicity on the properties of static and moving breathers in a Klein–Gordon chain with dipole-dipole interaction. It has been found that the helicity enlarges the range of existence and stability of static breathers, although this effect is small for a typical helical structure of DNA. However the effect of the orientation of the dipole moments is considerably higher with transcendental consequences for the existence of mobile breathers.' address: - 'Departamento de Física Aplicada I, Universidad de Sevilla. Av. Reina Mercedes s/n, 41012-Sevilla, Spain' - 'Facultad de Física, Universidad de Sevilla. Avda. Reina Mercedes s/n, 41012-Sevilla, Spain' author: - 'B. Sánchez-Rey, JFR Archilla, F Palmero' - FR Romero date: Mars 2002 title: 'Breathers in a system with helicity and dipole interaction.' --- \#1\#2 [2]{} Introduction ============ A great deal of attention has been paid to the interplay between geometry and nonlinearity in locating problems in recent years. The relationship between geometry and nonlinearity has an important role in the functions of some biomolecules, such as DNA, where the localization of energy has been put forward as a precursory mechanism of the transcription bubble [@Peyrard], and moving localized excitations as a method of transporting information along the double strand [@Salerno]. The fact that hydrogen bonds that link each pair of bases in DNA have a finite dipole moment, has brought about the introduction of models [@Christiansen; @Mingaleev; @Archilla; @Cuevas] with long range dipole–dipole interaction. Apart from its theoretical interest, this interaction becomes relevant when the secondary structure of DNA is considered. The shape of the molecule can influence the localization and transport properties of energy, which is thought to play a biological function [@DNA]. Some of these models [@Christiansen] study the effects of the curvature in a chain of nonlinear oscillators using the discrete nonlinear Schrödinger equation. Other models consider Klein-Gordon systems to study kinks [@Mingaleev], breathers in curved chains [@Archilla] or breathers with two competing interactions [@Cuevas]. However, all these models with long range interaction fail to take into account the peculiar helicoidal structure of the DNA chain, although this has been considered in some models [@Barbi] without the dipole interaction. In this paper, we study the effect of helicity on the properties of breathers in a Klein–Gordon model with dipole–dipole interaction. These periodic nonlinear localized oscillations in discrete systems are very localized excitations that appear as a consequence of the nonlinearity and discreteness of the system [@Mackay]. They are specially suitable for biomolecules when considering excitations that involve a few units, that is, far from the continuous limit. They can be static but, under certain conditions, also move and transport energy along the system [@Mobile]. We have found that the introduction of helicity enhances the stability of static breathers, although this effect is relatively small for the typical helicoidal structure of the DNA. On the other hand, the profile of the static breathers and the properties of moving ones are strongly dependent on the relative orientation between the dipole moments. The model {#sec:mod} ========= The model is inspired by the primary structure of DNA, with dipole moments perpendicular to the helix axis, and where the stretching of the hydrogen bonds within base pairs is described as a variation of the dipole moments. More detailed justification of the model can be found in [@Cuevas]. ![Sketch of the model at equilibrium. The arrows represent the dipoles moments, perpendicular to the helix axis.[]{data-label="fig1"}](figure1.eps){width="6cm"} We denote $\phi_n$ the angle of the n-dipole with respect to a reference axis perpendicular to the helix axis. Then, the angle between the nearest neighbouring dipoles is $\theta_{tw}=\phi_n-\phi_{n-1}$. We have considered this neighbouring angle constant along the chain, and it will be called the twisting angle. Thus, $\phi_{n+m}-\phi_n=m \theta_{tw}$, and, therefore, $2 \pi/ \theta_{tw}$ dipoles are needed to complete a turn of screw. In DNA, for example, the twisting angle is $36^{\mathrm{o}}$ and a turn of screw requires ten base pairs. Figure \[fig1\] shows a sketch of the model, where it can be appreciated that the system of dipoles have an helicoidal structure. In the appropriate dimensionless variables, the Hamiltonian of our system becomes $$\begin{aligned} H & =& \sum_{n=1}^{N} \bigg( \frac{1}{2} \dot{u}_n^2+ V(u_n) \nonumber\\ & & +\frac{1}{2} J \sum_{m=n-N/2}^{n+N/2}\frac{u_n u_m}{|n-m|^3}\cos [\theta_{tw} (n-m)] \bigg)\;\; ,\end{aligned}$$ where $N$ is the number of variables. The variables $\{u_n\}_{n=1}^N$, where $u_{n\pm N}=u_n$, represent, in the context of the Peyrard-Bishop model for DNA [@Peyrard], the transversal displacements of the two complementary nucleotides in the n-th pair with respect to the molecular axis. In our model, they describes the stretching of the dipoles with respect to their equilibrium length. $V(u_n)$ is the on site potential, which, in DNA models, describes the hydrogen bonds linking the two bases, and the parameter $J$ measures the strength of the long range dipole-dipole interaction. We have chosen the on–site potential as the Morse potential, given by $$V(u_n)=\frac{1}{2}(e^{-u_n}-1)^2$$ The reason for this, is that it is a suitable potential for representing chemical bonds, being asymmetric, with a hard part, modeling the repulsion between atoms or molecules, and a soft part that becomes flat, modeling the breakage of the bond. The dynamical equations become $$\label{eq:dyn} \ddot{u}_n+V'(u_n)+ J \sum_{m=n-N/2}^{n+N/2} \frac{\cos[\theta_{tw} (n-m)]} {|n-m|^3}u_m=0,$$ where n=1…N. To study the linear modes of the system we replace $V'(u_n)$ in equations \[eq:dyn\] with the linear term $u_n$, which implies that the time has been scaled so that the linear frequency $\omega_0=1$. Considering solutions of the form $u_n= e^{iqn-iwt}$ the following dispersion relation is obtained: $$\label{eq:fonon} w_k=\sqrt{1+2J \sum_{m=1}^{N/2} \frac{\cos(m\,\theta_{tw})}{m^3} \cos(m\,q_k )}$$ where $q_k=\frac{2 \pi k}{N}$, with $k=1\dots N$ due to the periodic boundary conditions. The variation of the phonon band with the helicity is shown in Figure \[fig2\], where the frequencies of the linear modes are represented as a function of the twisting angle, $\theta_{tw}$, for a fixed value of the coupling parameter $J=0.1$. The effect of the twisting is a narrowing of the phonon band, which will enhance the range of existence and stability of the breathers. This has been confirmed numerically. Breather existence and stability {#sec:breathers} ================================ We have studied the existence and stability of breathers in this model using the standard numerical methods described in Ref. [@Methods]. The Morse potential is a soft potential with the consequence that the frequency of a breather has to be lower than the linear frequency $\omega_0=1$. Thus, we have chosen, ${\omega_{\mathrm{b}}}=0.8$ so that the nonlinear effect will be significant but on the other hand not overtly strong, as the nonlinearities in DNA are thought to be weak. First of all, the helicity influence the breather profile. As is shown in figure \[fig3\], for a fixed value of the coupling parameter $J$, the increase of the twisting angle produces a transition from a zigzag profile (the nearest neighbour oscillating in antiphase) to a bell profile (all dipoles oscillating in phase). This effect follows from the spatial profile of the phonon state with the lowest frequency since the breather frequency is below the phonon band, and all the higher harmonics are way too high to be relevant. For $\theta_{tw}< \pi/2$ the interaction is effectively “antiferromagnetic" which leads to staggered phonons at the lower band edges. In the same way for $\theta_{tw}> \pi/2$ a “ferromagnetic" interaction is present which leads to a nonstaggered phonon at the lower band edge. The breather bifurcates from the lower band edge phonons and thus retains the property of the phonon structure. For $\theta_{tw}= \pi/2$ the system separates into two noninteracting sublattices: even and odd sites. As a result in this case, the nearest neighbors are at rest and the odd site sublattice remains unexcited. One–site breathers are stable at low coupling as was proved by Aubry [@Aubry]. For any value of the twisting angle $\theta_{tw}<90^o$ they can be continued from the anticontinuous limit till ${\omega_{\mathrm{b}}}$ enters the phonon band. Just before the breather disappears, it becomes unstable due to the occurrence of a harmonic bifurcation in the evolution of the Floquet eigenvalues. The increase of the twisting enhances the stability as is shown in Fig. \[fig4\] (circles). This can be understood if we consider only the nearest neighbor interaction (NNI). Then the influence of helicity on the stability of the breathers could be described by an effective coupling $J_{eff}=J \cos \theta_{tw}$. The one–site breather without twisting lose its stability for a coupling value of $J_c^0$. With twisting and only NNI this would occur for $J_c=J_c^0 / \cos \theta_{tw}$ (dash lines in Fig. \[fig4\]), which concurs with the numerical results. The two–site breather, which consist of two neighboring oscillators excited in phase, is also stable at low coupling. This can be understood in terms of Aubry’s band theory [@Aubry]. When coupling is increased a bubble of instability appears due to Krein crunches between the phonon band eigenvalues and a localized eigenvalue of the Floquet operator. If we continue increasing the coupling the double breather definitely becomes unstable due to the occurrence of a subharmonic bifurcation. Again, the effect of the twisting is to enlarge the range of stability toward higher values of the coupling parameter (full circles in Fig. \[fig4\]). This suggests that twisting might be a way to control the stability of the breathers in real systems. We have not considered the two–site breather in antiphase because it coincides with the one–site breather with zigzag profile, i.e., the Newton method converges to the same solution if we start at the anticontinuous limit with one non linear oscillator or with two nearest neighbor oscillators in antiphase. A rather different situation is the one with $\theta_{tw}>90^o$. First, the one–site breather is always stable until it disappears. Second, the two–site breather is unstable at low coupling but becomes stable just before its extinction. This behavior has important consequences for the mobility of these breathers as shown in the next section. For the sake of thoroughness we have also studied the effect of twisting with a hard $\phi^4$ potential $V(u_n)=u_n^2+1/4 u_n^4$, and a breather frequency ${\omega_{\mathrm{b}}}=1.2$. Qualitatively the results are similar except for the fact that breathers with $\theta_{tw}<90^o$ and breathers with $\theta_{tw}>90^o$ exchange their properties. Mobile breathers ================ Static breathers under certain conditions can be moved. The standard method to move a breather consists in perturbing its velocities with an spatially antisymmetric vector, called the marginal mode [@Mobile]. Typically, this method works within a certain range of parameters near an exchange stability bifurcation. This occurs when a one–site breather becomes unstable and a two–site breather does the opposite at a nearby point. We have looked for mobile breathers in our system both with a hard $\phi^4$ potential and with a Morse potential, but we have only had success with Morse potential and “ferromagnetic" interaction, i.e., $\theta_{tw}>90^o$. In this particular case, we found a similar situation to a stability exchange and we were able to move the breather perturbing it with the unstable localized mode of the two–site breather. This is an interesting result because this configuration is equivalent to a chain of antiparallel dipoles twisted $\pi-\theta_{tw}<\pi/2$. In fact, we can only expect parallel dipoles in synthetic DNA. A useful concept for describing the breather movement is its effective mass. If the norm of the perturbation velocity is $\lambda$, the kinetic energy added to the breather by the perturbation is $E=\lambda^2/2$. The resulting translational velocity of the breather, $v$, is found to be proportional to $\lambda$ [@Mobile]. Thus, moving breathers can be considered as a quasi-particle with a mass of $m^*$, which can be defined through the relation $m^*v^2/2=\lambda^2/2$. We have studied the dependence of the effective mass, $m^*$, with the coupling $J$. Figure \[fig5\] shows the result for antiparallel dipoles ($\theta_{tw}=180^o$). Two different behaviors were obtained depending on the initial conditions. If we perturb the two–site breather, we observe that its effective mass increases monotonically with the coupling (full squares in Figure \[fig5\]). This reflects the fact that the two–site breather becomes stable with increasing coupling. But if a static one–site breather is chosen as the initial configuration, a minimum of $m^*$ appears showing the existence of an optimal value of the coupling to move this breather (see blank squares in Figure \[fig5\]). We think that this minimum expresses a balance between the two opposite effects produced by an increase of the coupling on the stability of the one–site and two–site breathers. Similar results are obtained for other values $\theta_{tw}>\pi/2$. Conclusions {#sec:conclusion} =========== We have considered a system of oscillating dipoles with helicoidal structure in order to study the effect of helicity on the existence and properties of breathers. This study is motivated by the helicoidal structure of DNA, and the fact that it can be described by a reduced dynamic where the only degrees of freedom are the stretchings of the hydrogen bonds between base pairs, which have a finite dipole moment. In our model, the helicity produces a narrowing of the phonon band, and an enlargement of the range of existence and stability of the breathers, although this effect is small for a typical helicoidal structure of DNA. The effect of the orientation of the dipole moments, i.e., if the twisting angle is greater or not than 90 degrees, is however considerably higher. In particular, we have only found mobile breathers with a Morse potential and $\theta_{tw}>\pi/2$. Understanding the necessary conditions to move a breather is still an opened question today. Acknowledgments {#acknowledgments .unnumbered} =============== This work has been supported by the European Union under the RTN project, LOCNET, HPRN–CT–1999–00163. We acknowledge Jesús Cuevas for his useful comments. [99]{} M. Peyrard and A.R. Bishop, [*Phys. Rev. Lett.*]{}, 62:2755, 1989. M. Salerno and Yu. Kivshar, [*Phys Lett A*]{}, 193:263, 1994. P.L. Christiansen, YB. Gaididei, and S.F. Mingaleev, [*J. Phys. Condens. Matter*]{}, 13(6), 1181 (2001); YuB. Gaididei, S.F. Mingaleev, and P.L. Christiansen, [*Phys Rev E*]{}, 62:R53, (2000); S. F. Mingaleev, P. L. Christiansen, Yu. B. Gaididei, M. Johansson, and K. [Ø]{}. Rasmussen, [**]{}, 25:41–63 (1999). S. F. Mingaleev, YB. Gaididei, E. Majernikova and S. Shpyrko, [*Phys Rev E*]{}, 61(4), 4454 (2000). J.F.R. Archilla, P.L. Christiansen, S.F. Mingaleev, and YuB. Gaididei, [*J Phys A: Math. Gen.*]{}, 34:6363, 2001; J.F.R. Archilla, P.L. Christiansen, and YuB. Gaididei. [*Phys Rev E*]{}, 65(1):16609, 2002. J. Cuevas, J.F.R. Archilla, YuB. Gaididei, and F.R. Romero, [*Physica D*]{}, 2002, in press. C. Calladine and H. Drew. . Academic Press, London, 1992. M. Barbi, S. Cocco, and M. Peyrard, [*Phys Lett A*]{}, 253:358, 1999. M. Barbi, S. Cocco, M. Peyrard, and S. Ruffo, [*J. Mol. Biol.*]{}, 24:97, 1999; G. Gaeta, C. Reisss, M. Peyrard, and T. Dauxois, [*Rev. Nuovo Cimento*]{}, 17:1, 1994. R.S. MacKay and S. Aubry, [**]{}, 7:1623, 1994; S. Flach and C.R. Willis, [*Physics Reports*]{}, 295:181, 1998. Ding Chen, S. Aubry, and G.P. Tsironis, [*Phys. Rev. Lett.*]{}, 77:4776, 1996; S Aubry and T Cretegny, [*Physica D*]{}, 119:34, 1998. J.L. Marin and S. Aubry, [**]{}, 9:1501, 1996. S. Aubry, [**]{}, 103:201, 1997.

--- abstract: 'We take charged anisotropic fluid cylinder when there is no external pressure acting on the fluid. This is a cylindrical version of the Krori and Barua’s method to explore the field equations with anisotropic fluid. We discuss models with positive matter density and pressure that satisfy all the energy and stability conditions. It is found that charge does not vanish at the center of the cylinder. The equilibrium condition as well as physical conditions are discussed. Further, we highlight the connection between our solutions and the charged strange quark stars as well as with dark matter including charged massive particles. The graphical analysis of the matter variables versus charge is given which indicates a physically reasonable matter distribution.' author: - | M. Sharif[^1] and H. Ismat Fatima[^2]\ Department of Mathematics, University of the Punjab,\ Quaid-e-Azam Campus, Lahore-54590, Pakistan. title: '**Cylinder with Charged Anisotropic Source**' --- [**Keywords:**]{} Field equations; Equation of state; Charged anisotropic source.\ [**PACS:**]{} 04.40.Nr; 04.40.Dg; 04.20.Jb Introduction ============ The study of general relativistic charged compact objects is of fundamental importance in astrophysics. Strong magnetic fields, different kinds of phase transitions and solid stellar core cause anisotropy in the fluids. However, charged fluids with anisotropy complicates the solution of the field equations. Equations of state (EoS) has important consequences in such situations. Many exact solutions have been obtained [@9] by using a simple form of the energy-momentum tensor and assuming some symmetries. There have been pervious discussions of a similar nature by Evan [@9a], Bronnikov [@9b], Latelier and Tobensky [@9c], and Kramer [@9d]. They used various EoS that could be written in the form $\rho=\gamma p$ for specific positive values of $\gamma$, as well as energy conservation. Some work has been done on charged anisotropic static matter by using spherically symmetric stars with the linear, nonlinear and Chaplygin gas EoS. Ivanov [@1] showed that the field equations can be simplified by using linear EoS for a charged perfect fluid but with non-integrable equations. Sharma and Maharaj [@2] explored the field equations for static spherically symmetric uncharged anisotropic fluid with combined linear EoS and a particular mass function. Charged anisotropic fluids have been discussed in General Relativity since the pioneering work of Bonner [@3]. Ray et al. [@4] investigated charged anisotropic spheres with Chaplygin gas EoS. Thirukkanesh and Maharaj [@5] generated models for charged anisotropic spherically symmetric stars by using linear EoS as well as choosing one of the metric functions and electric field intensity. Horvat et al. [@6] studied gravastars for charged anisotropic fluid. Recently, Victor et al. [@7] explored solutions for the charged anisotropic spheres with linear or nonlinear EoS. Over the years, many authors have proposed various formulations to solve the field equations for cylindrically symmetric spacetime. Nilsson et al. [@8] investigated cylindrically symmetric perfect fluid models. One of the authors (MS) [@10] explored perfect fluid, static cylindrically symmetric solutions of the field equations by using different EoS. Sharif and Fatima [@11] worked for the charged anisotropic cylinder but they discussed gravitational collapse. Som [@12] explored the charged dust cylinder. However, there has been a little progress towards investigating charged anisotropic static cylindrically symmetric solutions with or without using an EoS. In a recent paper [@12a], we have explored exact solutions of the field equations for the charged anisotropic static cylindrically symmetric spacetime using Thirukkanesh and Maharaj [@5] approach. Here we extend this study for the charged anisotropic static cylindrically symmetric spacetime by using Victor et al. [@7] procedure. A system of differential equations for matter as well as electric field intensity and anisotropic pressures are solved on the basis of linear and nonlinear EoS. Numerical factors depending on matching conditions are used with each EoS which provide relationship among charge distribution, pressure anisotropy and EoS. The outline of the paper is as follows: In the next section, we write down the Einstein-Maxwell field equations for the static cylindrically symmetric spacetime and also express this system of equations with original Krori and Barua’s [@13] assumptions. We apply the central and boundary conditions on electric field intensity and radial pressure respectively to analyze these field equations at center and on the boundary of the cylinder. Section **3** investigates models for linear, nonlinear and Chaplygin gas EoS by taking positive matter densities and pressures corresponding to the relevant EoS. In section **4**, we match smoothly the interior and exterior metrics and bring adimesionality in the three models. Section **5** provides some physical features of these models. In particular, we discuss the stability conditions, energy conditions, the ven der Waals EoS [@14] and the equilibrium conditions for our models. The last section **6** contains concluding remarks about the results. The Field Equations =================== We take the static cylindrically symmetric spacetime given by [@15] $$\label{1} ds^{2}=e^{2\nu}dt^{2}-e^{2\mu-2\nu}dr^{2}-r^{2}e^{-2\nu}d\phi^{2}-e^{2\mu-2\nu}dz^{2},$$ where $\nu$ and $\mu$ are functions of $r$. The transformation $d\phi=e^{\nu}d\theta$ leads the above equation to the following form [@8; @11] $$\label{2} ds^{2}=e^{2\nu}dt^{2}-e^{2\mu-2\nu}dr^{2}-r^{2}d\theta^{2}-e^{2\mu-2\nu}dz^{2}.$$ The field equations for the charged anisotropic source are $$\label{3} R_{ab}-\frac{1}{2}g_{ab}R=\kappa(T^{(m)}_{ab}+T^{(em)}_{ab}),$$ where $T^{(m)}_{ab}$ and $T^{(em)}_{ab}$ are the energy-momentum tensors for anisotropic matter and electromagnetic field respectively. The energy-momentum tensor for anisotropic fluid is $$\label{4} T^{(m)}_{ab}=(\rho+p_{r})u_{a}u_{b}-p_{t}g_{ab}+(p_{t}-p_{r})\eta_{a} \eta_{b}$$ satisfying $u^{a}u_{a}=-\eta^{a} \eta_{a}=1$, where $\rho$ is the charge density, $p_{r}$ is the radial pressure, $p_{t}$ is the tangential pressure, $u_{a}=e^{\nu}\delta^{0}_{a}$ is the 4-velocity and $\eta _{a}=-e^{\mu-\nu}\delta_{a}^{1}$ is the 4-unit vector. The energy-momentum tensor for the electromagnetic field is $$\label{5} T^{(em)}_{ab}=\frac{1}{4\pi}(-g^{cd}F_{ac}F_{bd}+\frac{1}{4}g_{ab}F_{cd}F^{cd}),$$ where $F_{ab}=A_{b,a}-A_{a,b}$ is the Maxwell field tensor and $A_a$ is the 4-potential. The Maxwell field equations are given by $$\label{6} [\sqrt{-g}F^{ab}]_{,b}=4\pi J^{a}\sqrt{-g},\quad F_{[ab,c]}=0,$$ where $J^{a}=\sigma u^{a}$ is the 4-current of the fluid element and $\sigma$ is the proper charge density. The field equations for the line element (\[2\]) become $$\begin{aligned} \label{7} e^{2\nu-2\mu}(\nu''-\mu'')=8\pi\rho+E^{2},\\\label{8} \frac{e^{2\nu-2\mu}}{r}(-r\nu'^{2}+r\nu'\mu'+\mu')=8\pi p_{r}-E^{2},\\\label{9} e^{2\nu-2\mu}(\nu'~^{2}+\mu'') =8\pi p_{t}+E^{2},\\\label{10} \sigma=\frac{e^{2\nu-2\mu}}{4\pi r}(re^{\mu-\nu}E)',\end{aligned}$$ where prime denotes differentiation with respect to $r$ and $E=2\sqrt{\pi}e^{-\mu}\frac{\partial A}{\partial r}$ stands for em part. In the system of equations (\[7\])-(\[10\]), there are seven unknowns, so we make physically reasonable choices for any two of the unknowns. We take the gravitational potential $e^{2\mu-2\nu}$ and the electric field intensity $E$ as [@6] $$\begin{aligned} \label{11} e^{2\mu-2\nu}&=&\frac{1+(c_{1}-c_{2})r}{1+c_{1}r}, \\\label{12} E^{2}&=&\frac{k(3+c_{1}r)}{(1+c_{1}r)^{2}},\end{aligned}$$ where $c_1,~c_2$ and $k$ are constants. Substituting Eqs.(\[11\]) and (\[12\]) in Eq.(\[10\]), we obtain $$\label{13} \sigma\approx\frac{\sqrt{3k}}{2\pi r}.$$ This shows that there is a singularity in the charge distribution at $r=0$. However, this choice keeps the charge distribution regular at the centre of the cylinder as $E(r)$ remains finite there (\[12\]). The singularity free models for charged anisotropic static cylinder are constructed by taking [@13] $$\begin{aligned} \label{14} \mu=Ar^{2},\quad \nu=Br^{2}+C,\end{aligned}$$ where $A,~B$ and $C$ are constants. Using these values in Eqs.(\[7\])-(\[10\]), it follows that $$\begin{aligned} \label{15} e^{2r^{2}(B-A)+2C}(2B-2A)=8\pi \rho +E^{2},\\\label{16} e^{2r^{2}(B-A)+2C}(-4B^{2}r^{2}+4ABr^{2}+2A)=8\pi p_{r}-E^{2},\\\label{17} e^{2r^{2}(B-A)+2C}(4B^{2}r^{2}+2A)=8\pi p_{t}+E^{2},\\\label{18} \sigma=\frac{e^{2r^{2}(B-A)+2C}}{4\pi r}(re^{r^{2}(A-B)-C}E)'.\end{aligned}$$ Now we impose the central and boundary conditions on $E(r)$ and $p_{r}(r)$ respectively as follows: $$\begin{aligned} \label{19} E(0)=0, \quad p_{r}(a)=0,\end{aligned}$$ where $a$ is a positive constant and $r=a$ is the interface of the charged fluid and vacuum (i.e., boundary of the cylinder). We apply central conditions to the system of Eqs.(\[15\])-(\[17\]), it follows that $$\begin{aligned} \label{20} \rho (0)=\frac{(B-A)e^{2C}}{4\pi},\quad p_{r}(0)=\frac{Ae^{2C}}{4\pi},\quad p_{t}(0)=\frac{Ae^{2C}}{4\pi}.\end{aligned}$$ This shows that $p_{r}(r)=p_{t}(r)$ at $r=0$, hence the anisotropy of the cylinder vanishes at the center. Applying the boundary conditions to Eqs.(\[15\])-(\[17\]), we get $$\begin{aligned} \label{23} \rho (a)&=&\frac{e^{2a^{2}(B-A)+2C}(-2B^{2}a^{2}+B+2ABa^{2})}{4\pi},\\\label{24} p_{t}(a)&=&\frac{e^{2a^{2}(B-A)+2C}(A+ABa^{2})}{2\pi},\\\label{25} E^{2}(a)&=&e^{2a^{2}(B-A)+2C}(4ABa^{2}+2A-4B^{2}a^{2}).\end{aligned}$$ The general expressions for $p_{t}$ and $E^{2}$ from Eqs.(\[15\])-(\[17\]) are $$\begin{aligned} \label{26} p_{t}(r)&=&\frac{e^{2r^{2}(B-A)+2C}(-2B+2B^{2}r^{2}+2A)}{4\pi}+\rho ,\\\label{27} E^{2}(r)&=&e^{2r^{2}(B-A)+2C}(2B-2A)-8\pi \rho.\end{aligned}$$ Models for Equations of State ============================= The general form of EoS is $$\label{28} p_{r}=p_{r}(\rho ,a_{1},a_{2}),$$ where $a_{1}$ and $a_{2}$ are parameters constrained by $$\begin{aligned} \label{29} p_{r}(0)=p_{r}[\rho(0),a_{1},a_{2}],\quad 0=p_{r}[\rho(a),a_{1},a_{2}].\end{aligned}$$ Adding Eqs.(\[15\]) and (\[16\]), we obtain $$\label{30} \rho+p_{r}=\frac{e^{2r^{2}(B-A)+2C}}{8\pi}(-4B^{2}r^{2}+2B+4ABr^{2})\equiv l(r).$$ This equation may be used with the assumed EoS to find $\rho$ and $p_{r}$. The corresponding value of $\rho$ will be used in Eqs.(\[26\]) and (\[27\]) to evaluate $p_{t}$ and $E^{2}$ respectively. In the following we discuss three types of EoS: The Linear EoS -------------- This is given by $$\label{31} p_{r}=\alpha_{1}+\alpha_{2}\rho,$$ where $\alpha_1,~\alpha_2$ are constants. Using this EoS, we obtain expressions for $\rho,~p_{r},~p_{t}$ and $E^{2}$ as follows: $$\begin{aligned} \label{32} \rho&=&\frac{e^{2r^{2}(B-A)+2C}(-4B^{2}r^{2}+2B+4ABr^{2})-8\pi \alpha_{1} }{8\pi(1+\alpha_{2})},\\\label{33} p_{r}&=&\frac{\alpha_{2}e^{2r^{2}(B-A)+2C}(-4B^{2}r^{2}+2B+4ABr^{2})+8\pi \alpha_{1}}{1+\alpha_{2}}.\end{aligned}$$ Using Eq.(\[32\]) in Eqs.(\[26\]) and (\[27\]) successively, we get $$\begin{aligned} \label{34} p_{t}&=&\frac{e^{2r^{2}(B-A)+2C}((2B-4A-4B^{2}r^{2})(1+\alpha_{2}) +4B^{2}r^{2}-4ABr^{2}-2B)}{8\pi (1+\alpha_{2})}\nonumber\\&-&\frac{\alpha_{1}}{1+\alpha_{2}}, \\\label{35} E^{2}&=&\frac{e^{2r^{2}(B-A)+2C}((2B-2A)(1+\alpha_{2})+4B^{2}r^{2}-4ABr^{2}-2B)}{ 1+\alpha_{2}}-\frac{8\pi \alpha_{1}}{1+\alpha_{2}}.\nonumber\\\end{aligned}$$ The values of constants $\alpha_{1}$ and $\alpha_{2}$ are found by solving Eqs.(\[29\]) and (\[31\]) as $$\label{36} \alpha_{1}=-\frac{\rho(a)p_{r}(0)}{\rho(0)-\rho(a)},\quad \alpha_{2}=\frac{p_{r}(0)}{\rho(0)-\rho(a)}.$$ The Nonlinear EoS ----------------- The nonlinear EoS is given by $$\begin{aligned} \label{37} p_{r}=\beta_{1}+\frac{\beta_{2}}{\rho^{n}},\end{aligned}$$ where $n\neq-1$ and $\beta_1,~\beta_2$ are constants. It is a modification of the Chaplygin gas EoS used by Bertolami and Paramos [@16] to describe neutral dark stars. For $n=1$, Eqs.(\[30\]) and (\[37\]) lead to $$\label{38} \rho=\frac{l(r)-\beta_{1}\pm \sqrt{(l(r)-\beta_{1})^{2}-4\beta_{2}}}{2}.$$ Substituting Eq.(\[38\]) in Eqs.(\[37\]), (\[26\]) and (\[27\]) respectively, it follows that $$\begin{aligned} \label{39} p_{r}&=&\beta_{1}+\frac{2\beta_{2}}{l(r)-\beta_{1}\pm \sqrt{(l(r)-\beta_{1})^{2}-4\beta_{2}}}, \\\label{40} p_{t}&=&\frac{e^{2r^{2}(B-A)+2C}(-B+2B^{2}r^{2}+2A)}{4\pi} \nonumber\\&+&\frac{l(r)-\beta_{1}\pm \sqrt{(l(r)-\beta_{1})^{2}-4\beta_{2}}}{2} ,\\\label{41} E^{2}(r)&=&e^{2r^{2}(B-A)+2C}(2B-2A)-4\pi \left(l(r)-\beta_{1}\right.\nonumber\\ &\pm&\left.\sqrt{(l(r)-\beta_{1})^{2}-4\beta_{2}}\right),\end{aligned}$$ where $l(r)$ is given by Eq.(\[30\]). The constants $\beta_{1}$ and $\beta_{2}$ are found from Eqs.(\[37\]) and (\[29\]) as $$\label{42} \beta_{1}=\frac{\rho(0)p_{r}(0)}{\rho(0)-\rho(a)},\quad \beta_{2}=-\frac{\rho(0)\rho(a)p_{r}(0)}{\rho(0)-\rho(a)}.$$ Equation (\[38\]) implies that $\beta_{2}$ must be negative so that each root in this equation has definite sign which will correspond to a positive definite matter density. The Modified Chayplygin Gas EoS ------------------------------- This EOS has the following form $$\label{43} p_{r}=\gamma_{1}\rho+\frac{\gamma_{2}}{\rho},$$ where $\gamma_1,~\gamma_2$ are constants. This is used to describe static, neutral, phantom-like sources [@17]. Using this EoS with Eq.(\[30\]), we get $$\label{44} \rho=\frac{l(r)\pm \sqrt{l(r)^{2}-4(1+\gamma_{1})\gamma_{2}}}{2(1+\gamma_{1})}.$$ Substituting this value of $\rho$ in Eq.(\[43\]) as well as in Eqs.(\[26\]) and (\[27\]) successively, it follows that $$\begin{aligned} \label{45} p_{r}&= &\gamma_{1}\left(\frac{l(r)\pm \sqrt{l(r)^{2}-4(1+\gamma_{1})\gamma_{2}}}{2(1+\gamma_{1})}\right) \nonumber\\&+&\frac{2\gamma_{2}(1+\gamma_{1})} {l(r)\pm \sqrt{l(r)^{2}-4(1+\gamma_{1})\gamma_{2}}},\\\label{46} p_{t}&=&\frac{e^{2r^{2}(B-A)+2C}(-B+2B^{2}r^{2}+2A)}{4\pi}\nonumber\\&+&\frac{l(r)\pm \sqrt{l(r)^{2}-4(1+\gamma_{1})\gamma_{2}}}{2(1+\gamma_{1})} ,\\\label{47} E^{2}&=&e^{2r^{2}(B-A)+2C}(2B-2A)\nonumber\\&-&4\pi \left(\frac{l(r)\pm \sqrt{l(r)^{2}-4(1+\gamma_{1})\gamma_{2}}}{1+\gamma_{1}}\right),\end{aligned}$$ where $\gamma_{1}$ and $\gamma_{2}$ are $$\label{48} \gamma_{1}=\frac{\rho(0)p_{r}(0)}{\rho(0)^{2}-\rho(a)^{2}}, \quad \gamma_{2}=-\frac{\rho(0)p_{r}(0)\rho(a)^{2}}{\rho(0)^{2}-\rho(a)^{2}}.$$ For $(1+\gamma_{1})<0$ and $\gamma_{2}>0$ or $(1+\gamma_{1})>0$ and $\gamma_{2}<0$, Eq.(\[44\]) yields roots of definite sign which will again correspond to the positive matter densities. It is clear that the positive or negative matter densities depend upon the positivity or negativity of the constants $\beta_{1},~\beta_{2},~\gamma_{1}$ and $\gamma_{2}$, i.e., the roots of Eqs.(\[38\]) and (\[44\]) respectively. Matching Conditions and Adimensional Matter Sources =================================================== Here we take the charged static cylindrically symmetric spacetime as an exterior region given by [@18] $$\label{49} ds^{2}=N(r)dt^{2}-\frac{1}{N(r)}dr^{2}-r^{2}d\theta^{2}-r^{2}d\psi^{2},\quad N(r)=\frac{q^{2}}{r^{2}}-\frac{2m}{r},$$ where $q$ and $m$ are charge and mass respectively. Using the transformation, $d\psi=\frac{1}{\sqrt{N(r)}}d\phi$, this takes the form $$\label{50} ds^{2}=N(r)dt^{2}-\frac{1}{N(r)}dr^{2}-r^{2}d\theta^{2}-\frac{r^{2}}{N(r)}d\phi^{2}.$$ With the radial transformation $r=\frac{m^{2}-q^{2}}{r'}$, it becomes $$\label{51} ds^{2}=\frac{\frac{q^{2}}{r'^{2}}-\frac{2m^{3}}{r'^{3}}+\frac{2mq^{2}}{r'^{3}}} {(\frac{m^{2}-q^{2}}{r'^{2}})^{2}}dt^{2}-\frac{(\frac{m^{2}-q^{2}}{r'^{2}})^{4}} {\frac{q^{2}}{r'^{2}} -\frac{2m^{3}}{r'^{3}}+\frac{2mq^{2}}{r'^{3}}}(dr'^{2}+d\phi^{2})-r'^{2}d\theta^{2}.$$ To match the interior metric (\[2\]) with the exterior (\[51\]), we impose the continuity of $g_{00},~g_{11}$ and $\frac{\partial g_{00}}{\partial r}$ across a surface at $r'=a$ by using the procedure [@19]. In our case, this yields the following expressions for $A,~B$ and $C$ in terms of adimensional parameters $\eta=\frac{m}{a}$ and $\chi=\frac{|q|}{a}$ as $$\begin{aligned} \label{52} A&=&\frac{\ln(\eta^{2}-\chi^{2})}{a^{2}(\eta^{2}-\chi^{2})^{2}},\\\label{53} B&=&-\frac{1}{a^{2}(\eta^{2}-\chi^{2})}(\frac{1}{2} +\frac{\chi^{2}}{4(\chi^{2}-2\eta^{3}+\eta\chi^{2})}),\\\label{54} C&=&\frac{1}{2} +\frac{\chi^{2}}{4(\chi^{2}-2\eta^{3}+\eta\chi^{2})} +\ln\frac{\eta^{2}-\chi^{2}}{\sqrt{\chi^{2}-2\eta^{3}+\eta\chi^{2}}}.\end{aligned}$$ We can see from Eq.(\[14\]) that $A$ and $B$ have dimension of $length^{-2}$ and $C$ is dimensionless. It is very important that the field equations can eventually be expressed in terms of these adimensional constants and the dimensionless radial coordinate $x=\frac{r}{a}$. Here adimesionality is denoted by $hats$. We assume that the interior of the fluid cylinder is described by $x\in [0,1)$. We reformulate all models as adimensional models where we are denoting $\hat{A}=a^{2}A,~\hat{B}=a^{2}B,~\hat{\rho}=a^{2}\rho,~\hat{p}_{r}=a^{2}p_{r},~ \hat{p}_{t}=a^{2}p_{t},~\hat{E}^{2}=a^{2}E^{2}$ and $\hat{\sigma}=a^{2}\sigma$. The quantities which are originally dimensionless are denoted by the actual symbol. The central and the boundary conditions at $x\in [0,1)$ become $$\begin{aligned} \label{55} \hat{\rho} (0)&=&\frac{(\hat{B}-\hat{A})e^{2C}}{4\pi},\\\label{56} \hat{p_{r}}(0)&=&\frac{\hat{A}e^{2C}}{4\pi},\\\label{57} \hat{p_{t}}(0)&=&\frac{\hat{A}e^{2C}}{4\pi},\\\label{58} \hat{\rho} (1)&=&\frac{e^{2(\hat{B}-\hat{A})+2C}(2\hat{B}^{2} -\hat{B}-2\hat{A}\hat{B})}{4\pi},\\\label{59} \hat{p}_{t}(1)&=&\frac{e^{2(\hat{B}-\hat{A})+2C}(\hat{A}+\hat{A}\hat{B})}{2\pi},\\\label{60} \hat{E}^{2}(1)&=&e^{2(\hat{B}-\hat{A})+2C}(4\hat{A}\hat{B}+2\hat{A}-4\hat{B}^{2}).\end{aligned}$$ ### The Adimensional linear EoS model {#the-adimensional-linear-eos-model .unnumbered} The adimensional linear EoS is given by $$\begin{aligned} \label{61} \hat{p}_{r}&=&\hat{\alpha}_{1}+\alpha_{2}\hat{\rho}.\end{aligned}$$ The corresponding quantities will become $$\begin{aligned} \label{62} \hat{\rho}&=&\frac{e^{2x^{2}(\hat{B}-\hat{A})+2C}(-4\hat{B}^{2}x^{2}+2\hat{B} +4\hat{A}\hat{B}x^{2})-8\pi\hat{\alpha}_{1}}{8\pi (1+\alpha_{2})},\\\label{63} \hat{p}_{r}&=&\frac{8\pi\hat{\alpha}_{1} +\alpha_{2}e^{2x^{2}(\hat{B}-\hat{A})+2C}(-4\hat{B}^{2}x^{2}+2\hat{B} +4\hat{A}\hat{B}x^{2})}{1+\alpha_{2}},\\\label{64} \hat{p}_{t}&=&\frac{e^{2x^{2}(\hat{B}-\hat{A})+2C}}{8\pi} (-2\hat{B}+4\hat{B}^{2}x^{2}+4\hat{A})+\hat{\rho},\\\label{65} \hat{E}^{2}&=&e^{2x^{2}(\hat{B}-\hat{A})+2C}(2\hat{B}-2\hat{A})-8\pi \hat{\rho},\end{aligned}$$ where $$\begin{aligned} \label{66} \hat{\alpha}_{1}=-\frac{\hat{\rho}(1)\hat{p}(0)}{\hat{\rho}(0) -\hat{\rho}(1)},\quad \alpha_{2}=\frac{\hat{p}_{r}(0)}{\hat{\rho}(0) -\hat{\rho}(1)}.\end{aligned}$$ ### The Adimensional Nonlinear EoS Model {#the-adimensional-nonlinear-eos-model .unnumbered} Here we have $$\begin{aligned} \label{67} \hat{p}_{r}&=&\hat{\beta}_{1}+\frac{\hat{\beta}_{2}}{\hat{\rho}}.\end{aligned}$$ For this model, the corresponding quantities take the form $$\begin{aligned} \label{68} \hat{\rho}&=&\frac{\hat{l}(x)-\hat{\beta}_{1}\pm \sqrt{(\hat{l}(x)-\hat{\beta}_{1})^{2}-4\hat{\beta}_{2}}}{2},\\\label{69} \hat{p}_{t}&=&\frac{e^{2x^{2}(\hat{B}-\hat{A})+2C}(-\hat{B} +2\hat{B}^{2}x^{2}+2\hat{A})}{4\pi}+ \hat{\rho} ,\\\label{70} \hat{E}^{2}&=&e^{2x^{2}(\hat{B}-\hat{A})+2C}(2\hat{B}-2\hat{A})-8\pi \hat{\rho},\end{aligned}$$ where $\hat{\beta}_{1}$ and $\hat{\beta}_{2}$ are $$\begin{aligned} \label{71} \hat{\beta}_{1}=\frac{\hat{\rho}(0)\hat{p}_{r}(0)}{\hat{\rho}(0)-\hat{\rho}(1)}, \quad \hat{\beta}_{2}=-\frac{\hat{\rho}(0)\hat{\rho}(1)\hat{p}_{r}(0)}{\hat{\rho}(0)-\hat{\rho}(1)}.\end{aligned}$$ ### The Adimensional Modified Chaplygin EoS Model {#the-adimensional-modified-chaplygin-eos-model .unnumbered} The adimensional modified Chaplygin EoS model $$\begin{aligned} \label{72} \hat{p}_{r}&=&\gamma_{1}\hat{\rho}+\frac{\hat{\gamma}_{2}}{\hat{\rho}}\end{aligned}$$ yield the following quantities $$\begin{aligned} \label{73} \hat{\rho}&=&\frac{\hat{l}(x)\pm \sqrt{\hat{l}(x)^{2}-4(1+\gamma_{1})\hat{\gamma}_{2}}}{2(1+\gamma_{1})},\\\label{74} \hat{p}_{t}&=&\frac{e^{2x^{2}(\hat{B}-\hat{A})+2C}(-\hat{B}+2\hat{B}^{2}x^{2} +2\hat{A})}{4\pi}+\hat{ \rho} ,\\\label{75} \hat{E}^{2}&=&e^{2x^{2}(\hat{B}-\hat{A})+2C}(2\hat{B}-2\hat{A})-8\pi \hat{\rho},\end{aligned}$$ where $$\label{76} \gamma_{1}=\frac{\hat{\rho}(0)\hat{p}_{r}(0)}{\hat{\rho}^{2}(0)-\hat{\rho}^{2}(1)},\quad \hat{\gamma}_{2}=-\frac{\hat{\rho}(0)\hat{p}_{r}(0)\hat{\rho}^{2}(1)} {\hat{\rho}^{2}(0)-\hat{\rho}^{2}(1)}.$$ The adimensional proper charge density for all the above three models will become $$\label{77} \hat{\sigma}=\frac{e^{2x^{2}(\hat{B}-\hat{A})+2C}}{2\pi x}\frac{d}{dx}(xe^{x^{2}(\hat{B}-\hat{A})+2C}\hat{E}).$$ Some Features of the Models =========================== In this section, we discuss some insights of the three models. The exterior metric (\[49\]) implies that singularity occurs at $r=0,~\frac{q^2}{2m}$. It is regular everywhere except at $r=0$ [@18]. The surface with $r=\frac{q^{2}}{2m}$ describes a right singular circular cylinder. Using Eqs.(\[52\])-(\[54\]) in $\rho(0)$ of Eq.(\[20\]), it turns out to be positive. Following [@7], we have $\eta=\frac{GM}{c^{2}a}\approx 1.147$ for $M$ to be the solar mass and $a=1.48km$. For this value of $\eta$, $\hat{\rho}(0),~\hat{\rho}(1),~\hat{p}_{t}(0),~\hat{p}_{t}(1),~\hat{E}(1)$ and $\hat{p}_{r}(0)$ turn out to be functions of $\chi$ only. Similarly, the expressions for $\hat{\alpha}_{1},~\alpha_{2},~\hat{\beta}_{1},~\hat{\beta}_{2},~\gamma_{1}$ and $\hat{\gamma}_{2}$ (EoS parameters in adimensional version) also depend only on $\chi$. The analysis of $A,~B$ and $C$ implies that the values of $\chi$ are restricted by the values of $\eta$ such that $\chi<\eta$ for $\eta>0$. Figures **1** and **2** indicate that the central density $\hat{\rho}(0)$ and central pressure $\hat{p}_{r}(0)$ are monotonically decreasing with increasing values of $\chi$ for $\eta=1.147$ and $\chi\in(0,0.7]$. We see that $\hat{p}_{r}(0)>0$ only when $\chi\in[0,0.57)$, hence we take the maximum charge $\chi=0.56$ to discuss our models. Figures **3-5** show that $\hat{\beta}_{2}<0,~ 1+\hat{\gamma}_{1}>0$ and $\hat{\gamma}_{2}<0$ for the same value of $\eta$. We can have positive definite roots from Eqs.(\[68\]) and (\[73\]), hence we can analyse the models with positive matter density $\hat{\rho}(x)$. It is obvious that $\chi=0$ implies no charge. If we increase $\chi$, it increases repulsive electrostatic forces and consequently pressure and density change. Now we explore the behavior of matter sources for our models at different charges. Firstly, we investigate the sources with linear EoS. Figures **6-9** show the corresponding matter density, radial pressure, tangential pressure and electric field intensity. The graphs of density and pressure show the increasing behavior while the electric field intensity is decreasing with the increasing values of $\chi$. Further, $\hat{E}(x)$ decreases at every point in the interval $x\in(0,0.1]$ with increasing $\chi$. Figure **10** shows the behavior of charge density $\hat{\sigma}$ which is unbounded for each value of $\chi$. The analysis for models with nonlinear and Chaplygin gas EoS indicates that these models are similar to the model satisfying linear EoS. These models correspond to the decreasing matter densities and pressures. Each EoS affects the dependance of the measure of anisotropy $\hat{\delta}$ on $x$. The only difference arising from the three models is about the measure of anisotropy $\hat{\delta}=\hat{p}_{t}-\hat{p}_{r}$. Figure **11** displays the anisotropic parameter for the model corresponding to the linear EoS which is increasing with the increasing $\chi$. Figures **12** and **13** show the anisotropic parameters corresponding to the nonlinear and Chaplygin gas EoS respectively. From figures, we see that the anisotropic parameter for nonlinear EoS is increasing while for the Chaplygin gas EoS, it is decreasing with increasing $\chi$. In addition, Eq.(\[18\]) contains proper charge density $\sigma$ which implies that $$\begin{aligned} \label{79} E(r)=\frac{4\pi re^{\nu-\mu}}{r^{2}}\int_{0}^{r}r\sigma e^{2\mu-2\nu}dr=\frac{q(r)}{r^{2}},\end{aligned}$$ where $$\label{80} q(r)=4\pi r e^{\nu-\mu}\int_{0}^{r}r\sigma e^{2\mu-2\nu}dr$$ which is the net charge inside the cylinder of radius $r$. For our cylindrically symmetric fluid models, Eqs.(\[7\])-(\[10\]) represent the basic source parameters. We formulate table **I** by computing adimensional values for these basic sources. Approximate numerical values for central density $\hat{\rho}(0)$, central pressure $\hat{p}(0)$, tangential pressure $\hat{p}_{t}(1)$ and electric field intensity $\hat{E}^{2}(1)$, are shown in table **I**. We note that the maximum value of $\hat{\rho}(0)$ and the value of $\hat{p}_{r}(0)$ are smaller than maximum of $\hat{p}_{t}(1)$. These maxima correspond to zero net charge sources. The source variables $\hat{\rho}(0),~\hat{p}_{r}(0)$ with maximum charge ($\chi=0.56$) have been changed while $\hat{p}_{t}(0)$ changes its sign at the maximum charge. To compensate the stronger electric repulsion, this sign inversion of tangential pressure is necessary. This is summarized in the following table:\ \ **Table I.** Approximate numerical values of some quantities. **$\chi$** **$\hat{\rho}(0)$** **$\hat{p}(0)$** **$\hat{p}_{t}(1)$** **$\hat{E}^{2}(1)$** ------------ --------------------- ------------------ ---------------------- ---------------------- -- -- -- -- -- -- 0 0.0221364 0.00651431 0.00830257 0.266397 0.56 0.0172785 0.0000738598 -0.0000726 0.370451 Now we discuss some consequences of our results. Stability --------- Bertolami and Paramos [@16] argued that if the generalized Chaplygin gas tends to a smooth distribution over space then most density perturbations tend to be flattened within a time scale related to their initial size and the characteristic speed of sound. One of the important “physical acceptability conditions” for anisotropic matter is that the squares of radial and tangential sound speeds ($u_{r}^{2}=\frac{dp_{r}}{d\rho}$ and $u_{t}^{2}=\frac{dp_{t}}{d\rho}$) should be less than the speed of light [@20]. We explore it for the linear EoS model. The graphs of the squares of radial and tangential sound velocities are shown in Figures **14** and **15** respectively. These indicate that $u_{r}^{2}$ is independent of $x$ and decreases with increasing $\chi$ while $u_{t}^{2}$ monotonically increases with increasing $x$ and also increases with increasing $\chi$ for fixed $x$. For three particular values considered here, these parameters satisfy $0<u_{r}^{2}<1$ and $0<u_{t}^{2}<1$ everywhere within the charged fluid. Now we use Herrera [@20] and Andreasson’s [@21] approach to identify potentially unstable or stable anisotropic matter configuration. According to their approach, $|u_{t}^{2}-u_{r}^{2}|\leq1$ as shown in Figure **16**. This implies that $$(i)\quad-1\leq u_{t}^{2}-u_{r}^{2}\leq0,\quad (ii)\quad 0<u_{t}^{2}-u_{r}^{2}\leq 1.$$ The first expression corresponds to the potentially stable model which is obvious from Figure **17**, while the second expression corresponds to the potentially unstable model. We note that our model satisfy the potentially stable condition. If the graph of $u_{t}^{2}-u_{r}^{2}$ keeps the same sign everywhere within a matter distribution, there will be no cracking and the system is stable. If there is a change of sign then it is alternating potentially unstable to stable region within the matter distribution and vice versa. Energy Conditions ----------------- The energy conditions of the charged anisotropic fluid, the weak energy condition, the strong energy condition and the dominant energy condition are satisfied if and only if the following inequalities hold: $$\begin{aligned} \label{81} \hat{\rho}+\hat{p}_{r}\geq 0,\\\label{82} \hat{\rho}+\frac{\hat{E}^{2}}{8\pi}\geq 0,\\\label{83} \hat{\rho}+\hat{p}_{t}+\frac{\hat{E}^{2}}{4\pi}\geq 0,\\\label{84} \hat{\rho}+\hat{p}_{r}+2\hat{p}_{t}+\frac{\hat{E}^{2}}{4\pi}\geq 0,\\\label{85} \hat{\rho}+\frac{\hat{E}^{2}}{8\pi}-|\hat{p}_{r}-\frac{\hat{E}^{2}}{8\pi}|\geq 0,\\\label{86} \hat{\rho}+\frac{\hat{E}^{2}}{8\pi}-|\hat{p}_{t}+\frac{\hat{E}^{2}}{8\pi}|\geq 0.\end{aligned}$$ Figures **18-23** indicate that these inequalities hold for each $x\in[0,1]$. The van der Waals (VDW) EoS --------------------------- Lobo [@14] introduced this type of bounded source for the sake of cosmology. Here we express this approach with VDW EoS given by $$\label{87} p_{r}=\frac{\omega_{1}\rho}{1-\omega_{3}\rho}-\omega_{2}\rho^{2}.$$ This is used to describe dark matter and dark energy as a single fluid. It is assumed that the interior and exterior metrics are joined and $E(0)=0=p_r(a)$. Further, $p_r$ is found at the center and boundary of the charged cylinder. Using Eqs.(\[87\]) and (\[32\]), it follows that $$\label{88} \omega_{2}\omega_{3}\rho^{3}-(\omega_{2}+\omega_{3})\rho^{2} +[1+\omega_{1}+\omega_{3}l(r)]\rho-l(r)=0,$$ where $\omega_{1}$ and $\omega_{2}$ are functions of $\rho(0),~\rho(a),~p_{r}(0)$ and $\omega_{3}$. We do not expect any interesting consequences from this equation, hence we leave it here. Equilibrium Condition --------------------- Now we discuss the variation of the net charge corresponding to different forces compatible with equilibrium configuration for our models. In particular, when pressure gradients tend to zero and the charged fluid is more diluted, then what is the behavior of gravitational and other forces. Using Tolman-Oppenheimer-Volkov equation, we obtain $$\label{89} (\rho+p_{r})(e^{2\nu}(p_{r}-E^{2})-e^{4\nu-2\mu}\frac{\mu'}{r}) -\frac{dp_{r}}{dr}+\frac{\sigma qe^{\mu-\nu}}{r^{2}}+r(e^{2\mu+2\nu})(p_{t}-p_{r})=0.$$ This provides the equilibrium condition for the charged fluid elements subject to different forces. Here $q=q(r)$ as given in Eq.(\[80\]). In adimensional version, we can write $$\label{90} \hat{F}_{1}+\hat{F}_{2}+\hat{F}_{3}+\hat{F}_{4}=0,$$ where $$\begin{aligned} \label{91} \hat{F}_{1}&=&(\hat{\rho}+\hat{p}_{r})(e^{\hat{2B}x^{2}+2C} (\hat{p}_{r}-\hat{E}^{2})-2\hat{A}xe^{4\hat{B}x^{2}-2\hat{A}x^{2}+4C}),\\\label{92} \hat{F}_{2}&=&-\frac{d\hat{p}_{r}}{dx},\\\label{93} \hat{F}_{3}&=&\hat{\sigma}\hat{E}e^{x^{2}(\hat{B}-\hat{A}+C)},\\\label{94} \hat{F}_{4}&=&xe^{2x^{2}(\hat{B}-\hat{A}+2C)}(\hat{p}_{t}-\hat{p}_{r}).\end{aligned}$$ The graphs of these forces with linear EoS at $\chi=0$ and $\chi=0.56$ are shown in Figures **24** and **25** respectively which indicate that the charge has a negligible effect on these forces, hence we obtain a static equilibrium. For $\chi=0,~\hat{F_{4}}$ point outwards at every $x\in(0,1]$ and $\hat{F_{2}}$ is along the $x$-axis. The electric force $\hat{F_{3}}$ acting on the fluid elements with unbounded $\hat{\sigma}$ located at $x=1$ is infinite. This is the weakest force because it changes sign at $x\approx 0.38$ and also the force $\hat{F}_{1}$ changes sign at $x \approx 0.9$ which is due to gravity. When $\chi=0.56$, the electric force is still unbounded and infinite. This unboundedness of the force $\hat{F}_{3}$ and sign inversion of the force $\hat{F}_{1}$ is essential for the configuration of our static, charged anisotropic model with linear EoS. Outlook ======= The main purpose of this paper is to investigate the solutions of the coupled Einstein-Maxwell field equations for the static cylindrically symmetric spacetime. For this purpose, we have used charged anisotropic fluid with EoS in the light of Victor et al. [@7] procedure developed for static spherically symmetric spacetime. In particular, the linear, nonlinear and Chaplygin EoS have been used. It is mentioned here that the linear EoS corresponds to the electrically charged isotropic strange quark stars on the basis of MIT bag model [@22]. Our model with linear EoS also corresponds to the model by Victor et al. [@7] for the charged anisotropic spherically symmetric fluid with the same EoS. We know that the nonlinear and Chaplygin EoS are used to describe non-static neutral gravitational isotropic fluids. Here we are taking static charged anisotropic fluid as we would like to explore the interior regions and the fluid vacuum interfaces of the charged anisotropic cylindrically symmetric stars with these EoS. We have used the assumptions of Karori and Barua to explore the charged anisotropic static cylinder. Our models with nonlinear and Chaplygin EoS correspond to the dark matter and dark energy with constant matter densities and pressures [@23]. Delgaty and Lake [@26] proposed some physical conditions acceptable for perfect fluids, i.e., regularity of the charge at the origin, positive matter density and pressure, decreasing matter density and pressure with increasing $r$, causal sound propagation and smooth matching of internal and external metrics at boundary of the source. Burke and Hobill [@27] added one more condition that sound velocity must be monotonically decreasing with increasing $r$ which was imposed on spherical perfect fluid [@28; @29]. We have found that our cylindrical models satisfy most of the above physical conditions. In our case, conflict may arise for increasing tangential sound velocity, constant radial velocity and negative tangential pressure for higher values of $\chi$. Finally, we would like point out that we have also explained charged anisotropic static cylindrically symmetric models in our recent paper [@12a] by using Thirukkanesh and Maharaj approach. In that paper, we have found that the charge distribution as well as $E$ become singular at $r=0$. However, here we have found non-singular behavior of these quantities. [40]{} Stephani, H., Kramer, D., MacCallum, M. Hoenselaers, C. and Herlt, E.: *Exact Solutions of Einstein’s Field Equations* (Cambridge University Press, 2003). Evan, A.B.: J. Phys. **A10**(1977)1303. Bronnikov, K.A.: J. Phys. **A12**(1979)201. Laterlier, P.S. and Tabensky, R.R.: Nuovo Cimento **B28**(1975)407. Kramer, D.: Class. Quantum Grav. **5**(1988)393. Ivanov, B.V.: Phys. Rev. **D65**(2002)104001. Sharma, R. and Maharaj, S.D.: Mon. Not. R. Astron. Soc. **375**(2007)1265. Bonner, B.W.: Z. Phys. **160**(1960)59. Ray et al.: Phys. Rev. **D82**(2010)104055. Thirukkanesh, S. and Maharaj, S.D.: Class. Quantum Grav. **25**(2008)253001. Horvat, D., Ilijic, S. and Marunovic, A.: Class. Quantum Grav. **26**(2009)025003. Victor et al.: Phys. Rev. **D82**(2010)044052. Nilsson, U., Uggla, C., and Marklund, M.: J. Math. Phys. **39**(1998)3336. Sharif, M.: J. Korean Phys. Soc. **37**(2000)624. Sharif, M. and Fatima, S. Gen. Relativ. Gravit. **43**(2011)127. Som, M.M.: Foc. Phys. Soc. **90**(1967)1149. Sharif, M. and Fatima, H.I.: *Charged Anisotropic Static Cylindrically Symmetric Models*, submitted for publication. Krori, K.D. and Barua, J.: J. Phys. A Math. Gen. **8**(1975)508. Lobo, F.S.N.: Phys. Rev. **D75**(2006)024023. Prasana, A.R.: Phys. Rev. **D11**(1975)2083. Bertolami, O. and Paramos, J.: Phys. Rev. **D72**(2005)123512. Jamil, M., Farooq, U. and Rashid, M.A.: Phys. J. **C59**(2009)907. Chao-Guang, H.: Acta Physica Sinica **4**(1995)617. Junevicus, G.J.G.: J. Phys. A Math. Gen. **9**(1976)2069. Herrera, L.: Phys. Lett. **A165**(1992)206. Andr´easson, H.: Commun. Math. Phys. **288**(2009)715. Negreiros et al.: Phys. Rev. **D80**(2009)083006. Tsujikawa, S.: Phys. Rev. **D76**(2007)023514. Delgaty, M.S.R. and Lake, K.: Comput. Phys. Commun. **115**(1998)395. Burke, J. and Hobill, D.: arXiv:0910.3230. Harko, T. and Mak, M.K.: Annalen Phys. **11**(2002)3. Abreu, H. Hernandez, H. and Nunez, L.A.: Class. Quantum Grav. **24**(2007)4631. [^1]: [email protected] [^2]: [email protected]

--- abstract: | In this paper, we design and analyze a Hybrid High-Order discretization method for the steady motion of non-Newtonian, incompressible fluids in the Stokes approximation of small velocities. The proposed method has several appealing features including the support of general meshes and high-order, unconditional inf-sup stability, and orders of convergence that match those obtained for Leray–Lions scalar problems. A complete well-posedness and convergence analysis of the method is carried out under new, general assumptions on the strain rate-shear stress law, which encompass several common examples such as the power-law and Carreau–Yasuda models. Numerical examples complete the exposition.\ **Keywords:** Hybrid High-Order methods, non-Newtonian fluids, power-law, Carreau–Yasuda law, discrete Korn inequality author: - 'Michele Botti[^1]' - 'Daniel Castanon Quiroz [^2]' - 'Daniele A. Di Pietro [^3]' - 'André Harnist [^4]' bibliography: - 'pstokes.bib' title: 'A Hybrid High-Order method for creeping flows of non-Newtonian fluids' --- Introduction {#sec:introduction} ============ In this paper, we design and analyze a Hybrid High-Order (HHO) discretization method for the steady motion of a non-Newtonian, incompressible fluid in the Stokes approximation of small velocities. Notable applications include ice sheet dynamics [@Isaac.Stadler.ea:15], mantle convection [@Schubert.Turcotte.ea:01], chemical engineering [@Ko.Pustejovska.ea:18], and biological fluids rheology [@Lai.Kuei.ea:78; @Galdi.Rannacher.ea:18]. We focus on fluids with shear-rate-dependent viscosity, whose behavior is characterized by a nonlinear strain rate-shear stress function. Physical interpretations and discussions of non-Newtonian fluid models can be found, e.g., in [@Bird.Armstrong.ea:87; @Malek.Rajagopal.ea:95]. Typical examples that are frequently used in the applications include the power-law and Carreau–Yasuda model. The earliest investigations of fluids with shear-dependent viscosities date back to the pioneering work of Ladyzhenskaya [@Ladyzhenskaya:69]. For a detailed mathematical study of the well-posedness and regularity of the continuous problem, see also [@Malek.Rajagopal:05; @Ruzicka.Diening:07; @Diening.Ettwein:08; @Beirao-da-Veiga:09; @Berselli.Ruzicka:20] and references therein. Early results on the numerical analysis of non-Newtonian fluid flow problems were given in [@Sandri:93; @Barrett.Liu:94; @Glowinski.Rappaz:03]. Later, these results were improved in [@Belenki.Berselli.ea:12] and [@Hirn:13] by proving error estimates that are optimal for fluids with shear thinning behavior (described by a power law exponent ${r}\le 2$). In [@Belenki.Berselli.ea:12], the authors considered a conforming inf-sup stable finite element discretization, while in [@Hirn:13] a low-order scheme with local projection stabilization was proposed. In both works, the use of Orlicz functions is instrumental to unify the treatment of the shear thinning and shear thickening cases (also called pseudoplastic and dilatant, respectively; cf. Example \[ex:Carreau–Yasuda\]). More recently, a finite element method based on a four-field formulation of the nonlinear Stokes equations has been analyzed in [@Sandri:14]. Other notable contributions on the numerical approximation of generalized Stokes problems include [@Diening.Kreuzer.ea:13; @Isaac.Stadler.ea:15; @Kreuzer.Suli:16; @Ko.Suli:18]. The main issues to be accounted for in the numerical solution of non-Newtonian fluid flow problems are the presence of local features emerging from the nonlinear strain rate-shear stress relation, the incompressibility condition leading to indefinite systems, the roughly varying model coefficients, and, possibly, complex geometries requiring unstructured and highly-adapted meshes. The HHO method provides several advantages to deal with the complex nature of the problem, such as the support of general polygonal or polyhedral meshes, the possibility to select the approximation order, and unconditional inf-sup stability. Moreover, HHO schemes can be efficiently implemented thanks to the possibility of statically condensing a large subset of the unknowns for linearized versions of the problem encountered, e.g., when solving the nonlinear system by the Newton method. Hybrid High-Order methods have been successfully applied to the simulation of incompressible flows of Newtonian fluids governed by the Stokes [@Aghili.Boyaval.ea:15] and Navier–Stokes equations [@Di-Pietro.Krell:18; @Botti.Di-Pietro.ea:19*1], possibly driven by large irrotational volumetric forces [@Di-Pietro.Ern.ea:16; @Castanon-Quiroz.Di-Pietro:20]. Works related to the problem of creeping flows of non-Newtonian fluids are [@Botti.Di-Pietro.ea:17] and [@Di-Pietro.Droniou:17; @Di-Pietro.Droniou:17*1], respectively dealing with nonlinear elasticity and Leray–Lions problems. Going from nonlinear coercive elliptic equations to the nonlinear Stokes system involves additional difficulties arising from the pressure and the divergence constraint. Finally, we mention, in passing, that HHO methods are members of a wider family of polytopal methods that also includes, e.g., Virtual Element methods; cf., e.g., [@Beirao-da-Veiga.Lovadina.ea:17; @Beirao-da-Veiga.Lovadina.ea:18] for their application to Newtonian incompressible flows. The HHO discretization presented in this paper is inspired by the previously mentioned works. It hinges on discontinuous polynomial unknowns on the mesh and on its skeleton, from which discrete differential operators are reconstructed. The reconstruction operators are then used to define a consistency term inspired by the weak formulation of the creeping flow problem and a cleverly designed stabilization term penalizing boundary residuals. We carry out a complete analysis of the proposed method. In particular, under general assumptions on the strain rate-shear stress function, we derive error estimates for the velocity and pressure approximations. The energy-norm error estimate for the velocity given in Theorem \[thm:error.estimate\] is optimal in the sense that it yields the same convergence orders established in [@Di-Pietro.Droniou:17*1 Theorem 7] for the scalar Leray–Lions elliptic problem. A key tool in our analysis is provided by Lemma \[lem:discrete.korn.inequality\], in which we prove a generalization of the discrete Korn inequality of [@Botti.Di-Pietro.ea:19 Lemma 1] to the non-Hilbertian case. The other main contributions are a novel formulation of the requirements on the strain rate-shear stress function allowing a unified treatment of pseudoplastic and dilatant fluids and the identification of a set of general assumptions on the nonlinear stabilization function ensuring the desired consistency properties along with the well-posedness of the discrete problem. The rest of the paper is organized as follows. In Section \[sec:continuous.setting\] we introduce the strong and weak formulations of the nonlinear Stokes problem and present the assumptions on the strain rate-shear stress function. The construction of the HHO discretization is given in Section \[sec:discrete.setting\] by defining the discrete counterparts of the viscous and coupling terms. Section \[sec:discrete.setting\] also contains the proof of the discrete Korn inequality and the discussion on the nonlinear stabilization function. Section \[sec:well-posedness\] establishes the well-posedness of the discrete problem by proving the Hölder continuity and the strong-monotonicity of the viscous term, as well as the inf-sup stability of the pressure-velocity coupling. In Section \[sec:error.estimate\], we show the consistency of the discrete viscous function and coupling bilinear form. These results are then used to prove the error estimate. In Section \[sec:num.res\], we investigate the performance of the method by performing a convergence test with analytical solution on various families of refined meshes. Finally, in Appendix \[sec:properties.stress\] we provide a sufficient condition for the strain rate-shear stress law to fulfil the assumptions presented in Section \[sec:continuous.setting\]. Continuous setting {#sec:continuous.setting} ================== Let $\Omega \subset {\mathbb{R}}^d$, $d\in\{2,3\}$, denote a bounded, connected, polyhedral open set with Lipschitz boundary $\partial\Omega$. We consider a possibly non-Newtonian fluid occupying $\Omega$ and subjected to a volumetric force field $\b f : \Omega \to {\mathbb{R}}^d$. Its flow is governed by the generalized Stokes problem, which consists in finding the velocity field $\b u : \Omega \to {\mathbb{R}}^d$ and the pressure field $p : \Omega \to {\mathbb{R}}$ such that \[eq:stokes.continuous\] $$\begin{aligned} {2} -{\b\nabla{\cdot}}{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b u) + {\b\nabla}p &= \b f &\qquad& \mbox{ in } \Omega, \label{eq:stokes.continuous:momentum} \\ {\b\nabla{\cdot}}\b u &= 0 &\qquad& \mbox{ in } \Omega, \label{eq:stokes.continuous:mass} \\ \b u &= \b 0 &\qquad& \mbox{ on } \partial \Omega, \label{eq:stokes.continuous:bc} \\ \int_\Omega p(\b x){\,\mathrm{d}}\b x &= 0, \label{eq:stokes.continuous:closure} \end{aligned}$$ where ${\b\nabla{\cdot}}$ denotes the divergence operator applied to vector fields, ${\b{\nabla}_{\mathrm{s}}}$ is the symmetric part of the gradient operator ${\b\nabla}$ applied to vector fields, and, denoting by ${{\mathbb{R}}^{d \times d}_{\mathrm{s}}}$ the set of square, symmetric, real-valued $d\times d$ matrices, ${\b\sigma}: \Omega \times {{\mathbb{R}}^{d \times d}_{\mathrm{s}}} \to {{\mathbb{R}}^{d \times d}_{\mathrm{s}}}$ is the strain rate-shear stress law. In what follows, we formulate assumptions on ${\b\sigma}$ that encompass common models for non-Newtonian fluids and state a weak formulation for problem that will be used as a starting point for its discretization. Strain rate-shear stress law {#sec:strain.rate.shear.stress.law} ---------------------------- We define the Frobenius inner product such that, for all $\b\tau= (\tau_{ij})_{1 \le i,j \le d}$ and $\b\eta= (\eta_{ij})_{1 \le i,j \le d}$ in ${{\mathbb{R}}^{d \times d}}$, $\b\tau : \b\eta \coloneqq \sum_{i,j=1}^d \tau_{ij}\eta_{ij}$, and we denote by $|\b\tau|_{d \times d}\coloneqq \sqrt{\b\tau : \b\tau}$ the corresponding norm. \[ass:stress\] Let a real number ${r}\in (1,+\infty)$ be fixed, denote by ${r}' \coloneqq \frac{{r}}{{r}-1} \in (1,+\infty)$ the conjugate exponent of ${r}$, and define the singular exponent of ${r}$ by $$\label{eq:sing} {r^{\circ}}\coloneq \min({r},2) \in (1,2].$$ The strain rate-shear stress law satisfies \[eq:ass:sigma\] $$\begin{gathered} {\b\sigma}(\b x,\b 0) = \b 0 \text{ for almost every } \b x \in \Omega,\label{eq:ass-stress:0} \\ {\b\sigma}: \Omega \times {{\mathbb{R}}^{d \times d}_{\mathrm{s}}} \to {{\mathbb{R}}^{d \times d}_{\mathrm{s}}} \text{ is measurable}.\label{eq:ass-stress:power-framed} \end{gathered}$$ Moreover, there exist real numbers $\sigma_{\mathrm{de}} \in [0,+\infty)$ and $\sigma_{\mathrm{hc}},\sigma_{\mathrm{sm}} \in (0,+\infty)$ such that, for all $\b\tau,\b\eta \in {{\mathbb{R}}^{d \times d}_{\mathrm{s}}}$ and almost every $\b x \in \Omega$, we have the Hölder continuity property \[eq:power-framed:s.holder.continuity.strong.monotonicity\] $$\begin{aligned} \left| {\b\sigma}(\b x,\b\tau)-{\b\sigma}(\b x,\b\eta) \right|_{d\times d} &\le \sigma_{\mathrm{hc}} \left(\sigma_{\mathrm{de}}^{r}+|\b\tau|_{d\times d}^{r}+|\b\eta|_{d\times d}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}| \b\tau-\b\eta |_{d\times d}^{{r^{\circ}}-1},\label{eq:power-framed:s.holder.continuity} \end{aligned}$$ and the strong monotonicity property $$\begin{aligned} \left({\b\sigma}(\b x,\b\tau)-{\b\sigma}(\b x,\b\eta)\right):(\b\tau-\b\eta) \left(\sigma_{\mathrm{de}}^{r}+|\b\tau|_{d\times d}^{r}+|\b\eta|_{d\times d}^{r}\right)^\frac{2-{r^{\circ}}}{{r}} \ge \sigma_{\mathrm{sm}}|\b\tau-\b\eta|_{d\times d}^{{r}+2-{r^{\circ}}}.\label{eq:power-framed:s.strong.monotonicity} \end{aligned}$$ Some remarks are in order. Assumption can be relaxed by taking ${\b\sigma}(\cdot,\b 0) \in L^{{r}'}(\Omega,{{\mathbb{R}}^{d \times d}_{\mathrm{s}}})$. This modification requires only minor changes in the analysis, not detailed for the sake of conciseness. Inequalities – can be proved starting from the following assumptions, which correspond to the conditions characterizing an ${r}$-power-framed function: For all $\b\tau,\b\eta \in {{\mathbb{R}}^{d \times d}_{\mathrm{s}}}$ with $\b\tau \neq \b\eta$ and almost every $\b x \in \Omega$, $$\begin{aligned} |{\b\sigma}(\b x,\b\tau)-{\b\sigma}(\b x,\b\eta)|_{d \times d} &\le \sigma_{\mathrm{hc}} \left(\sigma_{\mathrm{de}}^{r}+|\b\tau|_{d \times d}^{r}+|\b\eta|_{d \times d}^{r}\right)^\frac{{r}-2}{{r}}| \b\tau-\b\eta |_{d \times d}, \\ \left({\b\sigma}(\b x,\b\tau)-{\b\sigma}(\b x,\b\eta)\right):\left(\b\tau-\b\eta\right) &\ge \sigma_{\mathrm{sm}}\left(\sigma_{\mathrm{de}}^{r}+|\b\tau|_{d \times d}^{r}+|\b\eta|_{d \times d}^{r}\right)^\frac{{r}-2}{{r}}|\b\tau-\b\eta|_{d \times d}^{2}. \end{aligned}$$ These relations are reminiscent of the ones used in [@Di-Pietro.Droniou:17*1] in the context of scalar Leray–Lions problems. The advantage of assumptions -, expressed in terms of the singular index ${r^{\circ}}$, is that they enable a unified treatment of the cases ${r}< 2$ and ${r}\ge 2$ in the proofs of Lemma \[lem:ah:holder.continuity.strong.monotonicity\], Theorem \[thm:well-posedness\], Lemma \[lem:consistency:ah\], and Theorem \[thm:error.estimate\] below. Inequalities and give $$\label{eq:power-framed:constants.bound} \sigma_{\mathrm{sm}} \leq \sigma_{\mathrm{hc}}.$$ Indeed, let $\b\tau \in {{\mathbb{R}}^{d \times d}_{\mathrm{s}}}$ be such that $|\b\tau|_{d\times d} > 0$. Using the strong monotonicity (with $\b \eta = \b 0$), the Cauchy–Schwarz inequality, and the Hölder continuity (again with $\b \eta = \b 0$), we infer that $$\begin{aligned} \sigma_{\mathrm{sm}}\left(\sigma_{\mathrm{de}}^{r}+|\b\tau|_{d\times d}^{r}\right)^\frac{{r^{\circ}}-2}{{r}}|\b\tau|_{d\times d}^{{r}+2-{r^{\circ}}} &\leq {\b\sigma}(\cdot,\b\tau):\b\tau\leq |{\b\sigma}(\cdot,\b\tau)|_{d\times d}|\b\tau|_{d\times d}\leq \sigma_{\mathrm{hc}}\left(\sigma_{\mathrm{de}}^{r}+|\b\tau|_{d\times d}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}| \b\tau|_{d\times d}^{{r^{\circ}}} \end{aligned}$$ almost everywhere in $\Omega$. Hence, $\frac{\sigma_{\mathrm{sm}}}{\sigma_{\mathrm{hc}}} \le \left(\frac{\sigma_{\mathrm{de}}^{r}+|\b\tau|_{d\times d}^{r}}{| \b\tau|_{d\times d}^{r}}\right)^\frac{|{r}-2|}{{r}}$. Letting $|\b\tau|_{d\times d} \to +\infty$ gives . \[ex:Carreau–Yasuda\] $(\mu,\delta,a,{r})$-Carreau–Yasuda fluids, introduced in [@Yasuda.Armstrong.Cohen:81] and later generalized in [@Hirn:13 Eq. (1.2)], are fluids for which it holds, for almost every $\b x\in\Omega$ and all $\b\tau \in {{\mathbb{R}}^{d \times d}_{\mathrm{s}}}$, $$\label{eq:Carreau--Yasuda} {\b\sigma}(\b x,\b\tau) = \mu(\b x)\left(\delta^{a(\b x)}+|\b\tau|_{d \times d}^{a(\b x)}\right)^\frac{{r}-2}{a(\b x)}\b\tau,$$ where $\mu : \Omega \to [\mu_-,\mu_+]$ is a measurable function with $\mu_-,\mu_+ \in (0,+\infty)$ corresponding to the local flow consistency index, $\delta \in [0,+\infty)$ is the degeneracy parameter, $a : \Omega \to [a_-,a_+]$ is a measurable function with $a_-,a_+ \in (0,+\infty)$ expressing the local transition flow behavior index, and ${r}\in (1,+\infty)$ is the flow behavior index. The Carreau–Yasuda law is a generalization of the Carreau law (corresponding to $a_- = a_+ = 2$) that takes into account the different local levels of flow behavior in the fluid. The degenerate case $\delta=0$ corresponds to the power-law model. Non-Newtonian fluids described by constitutive laws with a $(\mu,\delta,a,{r})$-structure exhibit a different behavior according to the value of ${r}$. If ${r}> 2$, then the fluid shows shear thickening behavior and is called *dilatant*. Examples of dilatant fluids are wet sand and oobleck. The case ${r}< 2$, on the other hand, corresponds to *pseudoplastic* fluids having shear thinning behavior, such as blood. Finally, if ${r}= 2$, then the fluid is Newtonian and becomes the classical (linear) Stokes problem. We show in Appendix \[sec:properties.stress\] that the strain rate-shear stress law is an ${r}$-power-framed function with $\sigma_{\mathrm{de}} = \delta$, $$\sigma_{\mathrm{hc}} = \begin{cases} \frac{\mu_+}{{r}-1}2^{\left[-\left(\frac{1}{a_+}-\frac{1}{{r}}\right)^\ominus-1\right]({r}-2)+\frac{1}{r}} & \text{if } {r}< 2, \\ \mu_+({r}-1)2^{\left(\frac{1}{a_-}-\frac{1}{{r}}\right)^\oplus({r}-2)} & \text{if } {r}\ge 2, \end{cases} \quad\text{and}\quad \sigma_{\mathrm{sm}} = \begin{cases} \mu_-({r}-1)2^{\left(\frac{1}{a_-}-\frac{1}{{r}}\right)^\oplus({r}-2)} & \text{if } {r}\le 2, \\ \frac{\mu_-}{{r}-1}2^{\left[-\left(\frac{1}{a_+}-\frac{1}{{r}}\right)^\ominus-1\right]({r}-2)-1} & \text{if } {r}> 2, \end{cases}$$ where $\xi^\oplus\coloneq\max(0,\xi)$ and $\xi^\ominus\coloneq-\min(0,\xi)$ denote, respectively, the positive and negative parts of a real number $\xi$. As a consequence, it matches Assumption \[ass:stress\]. Weak formulation {#sec:weak.formulation} ---------------- From this point on, we omit both the integration variable and the measure from integrals, as they can be in all cases inferred from the context. We define the following velocity and pressure spaces embedding, respectively, the homogeneous boundary condition for the velocity and the zero-average constraint for the pressure: $$\b U \coloneqq \left\{\b v \in W^{1,{r}}(\Omega,{\mathbb{R}}^d)\ : \ \b v{\ \!\!_{|_{\partial\Omega}}} = \b 0 \right\}, \qquad P \coloneqq L^{{r}'}_0(\Omega,{\mathbb{R}}) \coloneqq \left\{q \in L^{{r}'}(\Omega,{\mathbb{R}})\ : \ \textstyle\int_\Omega q = 0 \right\}.$$ Assuming $\b f \in L^{{r}'}(\Omega,{\mathbb{R}}^d)$, the weak formulation of problem reads: Find $(\b u,p) \in \b U \times P$ such that \[eq:stokes.weak\] $$\begin{aligned} {2} a(\b u,\b v)+b(\b v,p) &= \displaystyle\int_\Omega \b f \cdot \b v &\qquad \forall \b v \in \b U,\label{eq:stokes.weak:momentum} \\ -b(\b u,q) &= 0 &\qquad \forall q \in P, \label{eq:stokes.weak:mass} \end{aligned}$$ where the function $a : \b U \times \b U \to {\mathbb{R}}$ and the bilinear form $b : \b U \times L^{{r}'}(\Omega,{\mathbb{R}}) \to {\mathbb{R}}$ are defined such that, for all $\b v,\b w \in \b U$ and all $q \in L^{{r}'}(\Omega,{\mathbb{R}})$, $$\label{eq:a.b} a(\b w,\b v) \coloneqq \displaystyle\int_\Omega {\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w) : {\b{\nabla}_{\mathrm{s}}}\b v,\qquad b(\b v,q) \coloneqq -\displaystyle\int_\Omega ({\b\nabla{\cdot}}\b v) q.$$ The test space in can be extended to $L^{{r}'}(\Omega,{\mathbb{R}})$ since, for all $\b v \in \b U$, the divergence theorem and the fact that $\b v{\ \!\!_{|_{\partial\Omega}}} = \b 0$ yield $b(\b v,1) = -\int_\Omega {\b\nabla{\cdot}}\b v = - \int_{\partial\Omega} \b v \cdot \b n_{\partial \Omega} = 0$, with $\b n_{\partial \Omega}$ denoting the unit vector normal to $\partial\Omega$ and pointing out of $\Omega$. \[rem:a-priori\] It can be checked that, under Assumption \[ass:stress\], the continuous problem admits a unique solution $(\b u,p) \in \b U \times P$; see, e.g., [@Hirn:13 Section 2.4], where slightly stronger assumptions are considered. For future use, we also note the following a priori bound on the velocity: $$\label{eq:continuous.solution:bounds:uh} |\b u|_{W^{1,{r}}(\Omega,{\mathbb{R}}^d)} \lesssim \left(\sigma_{\mathrm{sm}}^{-1}\| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)}\right)^\frac{1}{{r}-1}+\left(\sigma_{\mathrm{de}}^{2-{r^{\circ}}}\sigma_{\mathrm{sm}}^{-1}\| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)}\right)^\frac{1}{{r}+1-{r^{\circ}}}.$$ To prove , use the strong-monotonicity of ${\b\sigma}$, sum written for $\b v=\b u$ to written for $q=p$, and use the Hölder and Korn inequalities to write $$\begin{aligned} \sigma_{\mathrm{sm}}\left( \sigma_{\mathrm{de}}^{r}+ \|{\b{\nabla}_{\mathrm{s}}}\b u\|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}\right)^\frac{{r^{\circ}}-2}{{r}} \|{\b{\nabla}_{\mathrm{s}}}\b u\|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{{r}+2-{r^{\circ}}} &\lesssim a(\b u,\b u) = \displaystyle\int_\Omega \b f \cdot \b u \lesssim \| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)}\|{\b{\nabla}_{\mathrm{s}}}\b u\|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}, \end{aligned}$$ that is, $$\label{eq:continuous.solution:bounds:uh:1} \mathcal{N}\coloneqq \left( \sigma_{\mathrm{de}}^{r}+ \|{\b{\nabla}_{\mathrm{s}}}\b u\|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}\right)^\frac{{r^{\circ}}-2}{{r}} \|{\b{\nabla}_{\mathrm{s}}}\b u\|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{{r}+1-{r^{\circ}}} \lesssim \sigma_{\mathrm{sm}}^{-1}\| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)}.$$ Observing that $\|{\b{\nabla}_{\mathrm{s}}}\b u\|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{{r}+1-{r^{\circ}}} \lesssim \max\left(\|{\b{\nabla}_{\mathrm{s}}}\b u\|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})},\sigma_{\mathrm{de}}\right)^{2-{r^{\circ}}}\mathcal{N} $, we obtain, enumerating the cases for the maximum and summing the corresponding bounds, $\|{\b{\nabla}_{\mathrm{s}}}\b u\|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})} \lesssim \mathcal{N}^\frac{1}{{r}-1}+(\sigma_{\mathrm{de}}^{2-{r^{\circ}}}\mathcal{N})^\frac{1}{{r}+1-{r^{\circ}}} $. Combining this inequality with gives . Discrete setting {#sec:discrete.setting} ================ In this section, we recall the notion of polyhedral mesh along with the definitions and properties of $L^2$-orthogonal projectors on local and broken polynomial spaces. Then, after introducing the spaces of discrete unknowns for the velocity and the pressure, we prove a discrete Korn inequality, define the discrete counterparts of the function $a$ and of the bilinear form $b$, and formulate the HHO scheme. Mesh and notation for inequalities up to a multiplicative constant ------------------------------------------------------------------ We define a mesh as a couple $\mathcal M_h\coloneq({\mathcal{T}}_h,{\mathcal{F}}_h)$, where ${\mathcal{T}}_h$ is a finite collection of polyhedral elements $T$ such that $h=\max_{T\in{\mathcal{T}}_h}h_T$ with $h_T$ denoting the diameter of $T$, while ${\mathcal{F}}_h$ is a finite collection of planar faces $F$ with diameter $h_F$. Notice that, here and in what follows, we use the three-dimensional nomenclature also when $d=2$, i.e., we speak of polyhedra and faces rather than polygons and edges. It is assumed henceforth that the mesh $\mathcal M_h$ matches the geometrical requirements detailed in [@Di-Pietro.Droniou:20 Definition 1.7]. In order to have the boundedness property for the interpolator, we additionally assume that the mesh elements are star-shaped with respect to every point of a ball of radius uniformly comparable to the element diameter; see [@Di-Pietro.Droniou:20 Lemma 7.12] for the Hilbertian case. Boundary faces lying on $\partial\Omega$ and internal faces contained in $\Omega$ are collected in the sets ${\mathcal{F}}_h^{\rm b}$ and ${\mathcal{F}}_h^{\rm i}$, respectively. For every mesh element $T\in{\mathcal{T}}_h$, we denote by ${\mathcal{F}}_T$ the subset of ${\mathcal{F}}_h$ containing the faces that lie on the boundary $\partial T$ of $T$. For every face $F \in {\mathcal{F}}_h$, we denote by ${\mathcal{T}}_F$ the subset of ${\mathcal{T}}_h$ containing the one (if $F\in{\mathcal{F}}_h^{\rm b}$) or two (if $F\in{\mathcal{F}}_h^{\rm i}$) elements on whose boundary $F$ lies. For each mesh element $T\in{\mathcal{T}}_h$ and face $F\in{\mathcal{F}}_T$, $\b n_{TF}$ denotes the (constant) unit vector normal to $F$ pointing out of $T$. Our focus is on the $h$-convergence analysis, so we consider a sequence of refined meshes that is regular in the sense of [@Di-Pietro.Droniou:20 Definition 1.9] with regularity parameter uniformly bounded away from zero. The mesh regularity assumption implies, in particular, that the diameter of a mesh element and those of its faces are comparable uniformly in $h$ and that the number of faces of one element is bounded above by an integer independent of $h$. To avoid the proliferation of generic constants, we write henceforth $a\lesssim b$ (resp., $a\gtrsim b$) for the inequality $a\le Cb$ (resp., $a\ge Cb$) with real number $C>0$ independent of $h$, of the constants $\sigma_{\mathrm{de}},\sigma_{\mathrm{hc}},\sigma_{\mathrm{sm}}$ in Assumption \[ass:stress\], and, for local inequalities, of the mesh element or face on which the inequality holds. We also write $a\simeq b$ to mean $a\lesssim b$ and $b\lesssim a$. The dependencies of the hidden constants are further specified when needed. Projectors and broken spaces ---------------------------- Given $X \in {\mathcal{T}}_h \cup {\mathcal{F}}_h$ and $l \in {\mathbb{N}}$, we denote by ${\mathbb{P}}^l(X,{\mathbb{R}})$ the space spanned by the restriction to $X$ of scalar-valued, $d$-variate polynomials of total degree $\le l$. The local $L^2$-orthogonal projector ${\pi_{X}^{l}} : L^{1}(X,{\mathbb{R}}) \to {\mathbb{P}}^l(X,{\mathbb{R}})$ is defined such that, for all $v \in L^{1}(X,{\mathbb{R}})$, $$\label{eq:proj} \displaystyle\int_X ({\pi_{X}^{l}} v-v) w = 0 \qquad \forall w \in {\mathbb{P}}^{l}(X,{\mathbb{R}}).$$ When applied to vector-valued fields in $L^1(X,{\mathbb{R}}^d)$ (resp., tensor-valued fields in $L^1(X,{{\mathbb{R}}^{d \times d}})$), the $L^2$-orthogonal projector mapping on ${\mathbb{P}}^l(X,{\mathbb{R}}^d)$ (resp., ${\mathbb{P}}^l(X,{{\mathbb{R}}^{d \times d}})$) acts component-wise and is denoted in boldface font. Let $T\in{\mathcal{T}}_h$, $n\in[0,l+1]$ and $m\in[0,n]$. The following $(n,{r},m)$-approximation properties of ${\pi_{T}^{l}}$ hold: For any $v\in W^{n,{r}}(T,{\mathbb{R}})$, \[eq:proj:app\] $$\label{eq:proj:app:T} |v-{\pi_{T}^{l}}v|_{W^{m,{r}}(T,{\mathbb{R}})} \lesssim h_T^{n-m}|v|_{W^{n,{r}}(T,{\mathbb{R}})}.$$ The above property will also be used in what follows with ${r}$ replaced by its conjugate exponent ${r}'$. If, additionally, $n\ge 1$, we have the following $(n,{r}')$-trace approximation property: $$\label{eq:proj:app:F} \|v-{\pi_{T}^{l}}v\|_{L^{{r}'}(\partial T,{\mathbb{R}})}\lesssim h_T^{n-\frac{1}{{r}'}}|v|_{W^{n,{r}'}(T,{\mathbb{R}})}.$$ The hidden constants in are independent of $h$ and $T$, but possibly depend on $d$, the mesh regularity parameter, $l$, $n$, and ${r}$. The approximation properties are proved for integer $n$ and $m$ in [@Di-Pietro.Droniou:17 Appendix A.2] (see also [@Di-Pietro.Droniou:20 Theorem 1.45]), and can be extended to non-integer vales using standard interpolation techniques (see, e.g., [@Lions.Magenes:72 Theorem 5.1]). At the global level, for a given integer $l\ge 0$, we define the broken polynomial space ${\mathbb{P}}^l({\mathcal{T}}_h,{\mathbb{R}})$ spanned by functions in $L^1(\Omega,{\mathbb{R}})$ whose restriction to each mesh element $T\in{\mathcal{T}}_h$ lies in ${\mathbb{P}}^l(T,{\mathbb{R}})$, and we define the global $L^2$-orthogonal projector ${\pi_{h}^{l}} : L^{1}(\Omega,{\mathbb{R}}) \to {\mathbb{P}}^l({\mathcal{T}}_h,{\mathbb{R}})$ such that, for all $v \in L^{1}(\Omega,{\mathbb{R}})$ and all $T \in {\mathcal{T}}_h$, $$({\pi_{h}^{l}} v){\ \!\!_{|_{T}}} \coloneq {\pi_{T}^{l}} v{\ \!\!_{|_{T}}}.$$ Broken polynomial spaces are subspaces of the broken Sobolev spaces $$W^{n,{r}}({\mathcal{T}}_h,{\mathbb{R}})\coloneq\left\{ v\in L^{r}(\Omega,{\mathbb{R}})\ : \ v{\ \!\!_{|_{T}}}\in W^{n,{r}}(T,{\mathbb{R}})\quad\forall T\in{\mathcal{T}}_h\right\}.$$ We define the broken gradient operator ${\b\nabla_h}: W^{1,1}({\mathcal{T}}_h,{\mathbb{R}}) \rightarrow L^1(\Omega,{\mathbb{R}}^d)$ such that, for all $v \in W^{1,1}({\mathcal{T}}_h,{\mathbb{R}})$ and all $T \in {\mathcal{T}}_h$, $({\b\nabla_h}v){\ \!\!_{|_{T}}} \coloneq {\b\nabla}v{\ \!\!_{|_{T}}}$. We define similarly the broken gradient acting on vector fields along with its symmetric part ${\b{\nabla}_{{\mathrm{s}},h}}$, as well as the broken divergence operator ${\b\nabla_h \cdot}$ acting on tensor fields. The global $L^2$-orthogonal projector ${\b{\pi}_{h}^{l}}$ mapping vector-valued fields in $L^1(\Omega,{\mathbb{R}}^d)$ (resp., tensor-valued fields in $L^1(\Omega,{{\mathbb{R}}^{d \times d}})$) on ${\mathbb{P}}^l({\mathcal{T}}_h,{\mathbb{R}}^d)$ (resp., ${\mathbb{P}}^l({\mathcal{T}}_h,{{\mathbb{R}}^{d \times d}})$) is obtained applying ${\pi_{h}^{l}}$ component-wise. Discrete spaces and norms ------------------------- Let an integer $k\ge 1$ be fixed. The HHO space of discrete velocity unknowns is $${{{\und{\b{U}}}}_{h}^{k}} \coloneqq \left\{ {\und{\b{v}}}_h \coloneq ((\b v_T)_{T \in {\mathcal{T}}_h},(\b v_F)_{F\in {\mathcal{F}}_h}) \ : \ \b v_T \in {\mathbb{P}}^k(T,{\mathbb{R}}^d)\ \ \forall T \in {\mathcal{T}}_h\ \mbox{ and }\ \b v_F \in {\mathbb{P}}^k(F,{\mathbb{R}}^d)\ \ \forall F \in {\mathcal{F}}_h \right\}.$$ The interpolation operator ${{\und{\b{I}}}_{h}^{k}} : W^{1,1}(\Omega,{\mathbb{R}}^d) \to {{{\und{\b{U}}}}_{h}^{k}} $ maps a function $\b v \in W^{1,1}(\Omega,{\mathbb{R}}^d)$ on the vector of discrete unknowns ${{\und{\b{I}}}_{h}^{k}}\b v$ defined as follows: $${{\und{\b{I}}}_{h}^{k}} \b v \coloneqq (({\b{\pi}_{T}^{k}} \b v{\ \!\!_{|_{T}}})_{T \in {\mathcal{T}}_h},({\b{\pi}_{F}^{k}} \b v{\ \!\!_{|_{F}}})_{F \in {\mathcal{F}}_h}).$$ For all $T \in {\mathcal{T}}_h$, we denote by ${{{\und{\b{U}}}}_{T}^{k}}$ and ${{\und{\b{I}}}_{T}^{k}}$ the restrictions of ${{\und{\b{I}}}_{h}^{k}}$ and ${{{\und{\b{U}}}}_{h}^{k}}$ to $T$, respectively and, for all ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h}^{k}}$, we let ${\und{\b{v}}}_T \coloneqq (\b v_T,(\b v_F)_{F\in {\mathcal{F}}_T}) \in {{{\und{\b{U}}}}_{T}^{k}}$ denote the vector collecting the discrete unknowns attached to $T$ and its faces. Furthermore, for all ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h}^{k}}$, we define the broken polynomial field $\b v_h\in{\mathbb{P}}^k({\mathcal{T}}_h,{\mathbb{R}}^d)$ obtained patching element unknowns, that is, $$(\b v_h){\ \!\!_{|_{T}}} \coloneqq \b v_T\qquad\forall T \in {\mathcal{T}}_h.$$ We define on ${{{\und{\b{U}}}}_{h}^{k}}$ the $W^{1,{r}}(\Omega,{\mathbb{R}}^d)$-like seminorm $\| {\cdot} \|_{{\boldsymbol{\varepsilon}},{r},h}$ such that, for all ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h}^{k}}$, \[eq:norm.epsilon.r\] $$\begin{gathered} \label{eq:norm.epsilon.r.h} \| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} \coloneqq \left(\displaystyle\sum_{T \in {\mathcal{T}}_h}\| {\und{\b{v}}}_T \|_{{\boldsymbol{\varepsilon}},{r},T}^{r}\right)^\frac{1}{{r}} \\\label{eq:norm.epsilon.r.T} \text{with $\| {\und{\b{v}}}_T \|_{{\boldsymbol{\varepsilon}},{r},T} \coloneqq \left(\| {\b{\nabla}_{\mathrm{s}}}\b v_T \|^{r}_{L^{r}(T,{{\mathbb{R}}^{d \times d}})} + \displaystyle\sum_{F \in {\mathcal{F}}_T} h_F^{1-{r}} \| \b v_F - \b v_T\|^{r}_{L^{r}(F,{\mathbb{R}}^d)}\right)^\frac{1}{{r}}$ for all $T \in {\mathcal{T}}_h$. } \end{gathered}$$ The following boundedness property for ${{\und{\b{I}}}_{T}^{k}}$ can be proved adapting the arguments of [@Di-Pietro.Droniou:20 Proposition 6.24] and requires the star-shaped assumption on the mesh elements: For all $T \in {\mathcal{T}}_h$ and all $\b v \in W^{1,{r}}(T,{\mathbb{R}}^d)$, $$\label{eq:I:boundedness} \|{{\und{\b{I}}}_{T}^{k}} \b v\|_{{\boldsymbol{\varepsilon}},{r},T} \lesssim | \b v |_{W^{1,{r}}(T,{\mathbb{R}}^d)},$$ where the hidden constant depends only on $d$, the mesh regularity parameter, ${r}$, and $k$. The discrete velocity and pressure are sought in the following spaces, which embed, respectively, the homogeneous boundary condition for the velocity and the zero-average constraint for the pressure: $${{{\und{\b{U}}}}_{h,0}^{k}} \coloneqq \left\{ {\und{\b{v}}}_h = ((\b v_T)_{T \in {\mathcal{T}}_h},(\b v_F)_{F\in {\mathcal{F}}_h}) \in {{{\und{\b{U}}}}_{h}^{k}} \ : \ \b v_F = \b 0 \quad \forall F \in {{\mathcal{F}}_h^{\mathrm{b}}}\right\},\quad {P_{h}^{k}} \coloneqq {\mathbb{P}}^k({\mathcal{T}}_h,{\mathbb{R}})\cap P.$$ By the discrete Korn inequality proved in Lemma \[lem:discrete.korn.inequality\] below, $\| {\cdot} \|_{{\boldsymbol{\varepsilon}},{r},h}$ is a norm on ${{{\und{\b{U}}}}_{h,0}^{k}}$ (the proof is obtained reasoning as in [@Di-Pietro.Droniou:20 Corollary 2.16]). Discrete Korn inequality {#sec:discrete.korn.inequality} ------------------------ We prove in this section a discrete counterpart of the following Korn inequality (see [@Geymonat.Suquet:86 Theorem 1]): For all $\b v \in\b U$. $$\label{eq:Korn} \|\b v\|_{W^{1,{r}}(\Omega,{\mathbb{R}}^d)} \lesssim \|{\b{\nabla}_{\mathrm{s}}}\b v\|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}.$$ We start by recalling a few preliminary results. The first concerns inequalities between sums of powers, and will be often used in what follows without necessarily recalling this fact explicitly each time. Let an integer $n\ge 1$ and a real number $m \in (0,+\infty)$ be given. Then, for all $a_1,\ldots,a_n \in (0,+\infty) $, we have $$\label{eq:sum-power} n^{-(m-1)^\ominus}\sum_{i=1}^n a_i^m \le \left(\sum_{i=1}^n a_i\right)^m \le n^{(m-1)^\oplus}\sum_{i=1}^n a_i^m.$$ If $m=1$, then holds with the equal sign. If $m < 1$, [@Ursell:59 Eqs. (5) and (3)] with $\alpha = 1$ and $\beta = m$ give $ n^{m-1}\sum_{i=1}^n a_i^m \le \left(\sum_{i=1}^n a_i\right)^m \le \sum_{i=1}^n a_i^m. $ If, on the other hand, $m > 1$, [@Ursell:59 Eqs. (3) and (5)] with $\alpha = m$ and $\beta = 1$ give $ \sum_{i=1}^n a_i^m \le \left(\sum_{i=1}^n a_i\right)^m \le n^{m-1}\sum_{i=1}^n a_i^m. $ Gathering the above cases yields . The second preliminary result concerns the node-averaging interpolator. Let $\mathfrak T_h$ be a matching simplicial submesh of $\mathcal M_h$ in the sense of [@Di-Pietro.Droniou:20 Definition 1.8]. The node-averaging operator ${\b{I}_{{\mathrm{av}},h}^{k}} : {\mathbb{P}}^k({\mathcal{T}}_h,{\mathbb{R}}^d) \to {\mathbb{P}}^k(\mathfrak T_h,{\mathbb{R}}^d) \cap W^{1,{r}}(\Omega,{\mathbb{R}}^d)$ is such that, for all $\b v_h \in {\mathbb{P}}^k({\mathcal{T}}_h,{\mathbb{R}}^d)$ and all Lagrange node $V$ of $\mathfrak T_h$, denoting by $\mathfrak T_V$ the set of simplices sharing $V$, $$({\b{I}_{{\mathrm{av}},h}^{k}}\b v_h)(V) \coloneq \begin{cases} \frac{1}{\rm{card}(\mathfrak T_V)}\sum_{\b\tau\in\mathfrak T_V} \b v_h{\ \!\!_{|_{\b\tau}}}(V) & \text{if }\ V \in \Omega, \\ \b 0 & \text{if }\ V \in \partial\Omega. \end{cases}$$ For all $F \in {{\mathcal{F}}_h^{\mathrm{i}}}$, denote by $T_1,T_2\in{\mathcal{T}}_h$ the elements sharing $F$, taken in an arbitrary but fixed order. We define the jump operator such that, for any function $\b v\in W^{1,1}({\mathcal{T}}_h,{\mathbb{R}}^d)$, $[\b v]_F \coloneq (\b v{\ \!\!_{|_{T_1}}}){\ \!\!_{|_{F}}}-(\b v{\ \!\!_{|_{T_2}}}){\ \!\!_{|_{F}}}$. This definition is extended to boundary faces $F\in{{\mathcal{F}}_h^{\mathrm{b}}}$ by setting $[\b v]_F \coloneq \b v{\ \!\!_{|_{F}}}$. For all $\b v_h \in {\mathbb{P}}^k({\mathcal{T}}_h,{\mathbb{R}}^d)$, it holds $$\label{eq:Iav:bound} |\b v_h-{\b{I}_{{\mathrm{av}},h}^{k}}\b v_h|_{W^{1,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{r}\lesssim \sum_{F\in{\mathcal{F}}_h} h_F^{1-{r}} \|[\b v_h]_F\|^{r}_{L^{r}(F,{\mathbb{R}}^d)}.$$ Combining [@Di-Pietro.Droniou:20 Eq. (4.13)] (which corresponds to for ${r}=2$) with the local Lebesgue embeddings of [@Di-Pietro.Droniou:20 Lemma 1.25] (see also [@Di-Pietro.Droniou:17 Lemma 5.1]) gives, for any $T\in{\mathcal{T}}_h$, $$\label{eq:Iav:bound:0} \|\b v_h-{\b{I}_{{\mathrm{av}},h}^{k}}\b v_h\|_{L^{r}(T,{\mathbb{R}}^d)}^{r}\lesssim \sum_{F\in{\mathcal{F}}_{\mathcal V,T}}h_F \|[\b v_h]_F\|^{r}_{L^{r}(F,{\mathbb{R}}^d)},$$ where ${\mathcal{F}}_{\mathcal V,T}$ collects the faces whose closure has non-empty intersection with $\overline{T}$. Using the local inverse inequality of [@Di-Pietro.Droniou:20 Lemma 1.28] (see also [@Di-Pietro.Droniou:17 Eq. (A.1)]) we can write $$\begin{aligned} |\b v_h-{\b{I}_{{\mathrm{av}},h}^{k}}\b v_h|_{W^{1,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{r}&\lesssim \sum_{T\in{\mathcal{T}}_h} h_T^{-{r}}\|\b v_h-{\b{I}_{{\mathrm{av}},h}^{k}}\b v_h\|_{L^{r}(T,{\mathbb{R}}^d)}^{r}\\ &\lesssim \sum_{T\in{\mathcal{T}}_h}\sum_{F\in{\mathcal{F}}_{\mathcal V,T}} h_F^{1-{r}} \|[\b v_h]_F\|^{r}_{L^{r}(F,{\mathbb{R}}^d)} \\ &\lesssim \sum_{F\in{\mathcal{F}}_h} \sum_{T\in{\mathcal{T}}_{\mathcal V,F}} h_F^{1-{r}} \|[\b v_h]_F\|^{r}_{L^{r}(F,{\mathbb{R}}^d)} \\ &\le\max_{F\in{\mathcal{F}}_h}\operatorname{card}({\mathcal{T}}_{\mathcal V,F}) \sum_{F\in{\mathcal{F}}_h} h_F^{1-{r}} \|[\b v_h]_F\|^{r}_{L^{r}(F,{\mathbb{R}}^d)}, \end{aligned}$$ where we have used the fact that $h_T^{-{r}} \le h_F^{-{r}}$ along with inequality to pass to the second line, and we have exchanged the sums after setting ${\mathcal{T}}_{\mathcal V,F} \coloneqq \big\{T \in {\mathcal{T}}_h : \overline F \cap \overline T \neq \emptyset\big\}$ for all $F \in {\mathcal{F}}_h$ to pass to the third line. Observing that $\max_{F\in{\mathcal{F}}_h}\operatorname{card}({\mathcal{T}}_{\mathcal V,F}) \lesssim 1$ (since, for any $F\in{\mathcal{F}}_h$, $\operatorname{card}({\mathcal{T}}_{\mathcal V,F})$ is bounded by the left-hand side of [@Di-Pietro.Droniou:20 Eq. (4.23)] written for any $T\in{\mathcal{T}}_h$ to which $F$ belongs), follows. (Discrete Korn inequality)\[lem:discrete.korn.inequality\] We have, for all ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}}$, $$\label{eq:discrete.Korn} \| \b v_h \|_{L^{r}(\Omega,{\mathbb{R}}^d)}^{r}+|\b v_h|_{W^{1,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{r}\lesssim \| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}.$$ Let ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}}$. Using a triangle inequality followed by , we can write $$\begin{aligned} |\b v_h|_{W^{1,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{r}&\lesssim |{\b{I}_{{\mathrm{av}},h}^{k}} \b v_h|_{W^{1,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{r}+|\b v_h-{\b{I}_{{\mathrm{av}},h}^{k}}\b v_h|_{W^{1,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{r}\\ &\lesssim \|{\b{\nabla}_{\mathrm{s}}}({\b{I}_{{\mathrm{av}},h}^{k}} \b v_h) \|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}+|\b v_h-{\b{I}_{{\mathrm{av}},h}^{k}}\b v_h|_{W^{1,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{r}\\ &\lesssim \|{\b{\nabla}_{{\mathrm{s}},h}}\b v_h \|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}+|\b v_h-{\b{I}_{{\mathrm{av}},h}^{k}}\b v_h|_{W^{1,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{r}\\ &\lesssim \|{\b{\nabla}_{{\mathrm{s}},h}}\b v_h \|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}+\sum_{F\in{\mathcal{F}}_h} h_F^{1-{r}} \|[\b v_h]_F\|^{r}_{L^{r}(F,{\mathbb{R}}^d)}, \end{aligned}$$ where we have used the continuous Korn inequality to pass to the second line, we have inserted $\pm{\b{\nabla}_{{\mathrm{s}},h}}\b v_h$ into the first norm and used a triangle inequality followed by to pass to the third line, and we have invoked the bound to conclude. Observing that, for any $F\in{\mathcal{F}}_h$, $|[\b v_h]_F| \leq \sum_{T\in{\mathcal{T}}_F}|\b v_F-\b v_T|$ by a triangle inequality, and using , we can continue writing $$|\b v_h|_{W^{1,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{r}\lesssim \|{\b{\nabla}_{{\mathrm{s}},h}}\b v_h \|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}+\sum_{F\in{\mathcal{F}}_h}\sum_{T\in{\mathcal{T}}_F}h_F^{1-{r}} \| \b v_F - \b v_T\|^{r}_{L^{r}(F,{\mathbb{R}}^d)} = \|{\und{\b{v}}}_h\|_{{\boldsymbol{\varepsilon}},{r},h}^{r},$$ where we have exchanged the sums over faces and elements and recalled definition to conclude. This proves the bound for the second term in the left-hand side of . Combining this result with the global discrete Sobolev embeddings of [@Di-Pietro.Droniou:17 Proposition 5.4] yields the bound for the first term in . Viscous term ------------ ### Local symmetric gradient reconstruction For all $T \in {\mathcal{T}}_h$, we define the local symmetric gradient reconstruction ${\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},T}} : {{{\und{\b{U}}}}_{T}^{k}} \to {\mathbb{P}}^{k}(T,{{\mathbb{R}}^{d \times d}_{\mathrm{s}}})$ such that, for all ${\und{\b{v}}}_T \in {{{\und{\b{U}}}}_{T}^{k}}$, $$\label{eq:G} \displaystyle\int_T {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},T}} {\und{\b{v}}}_T : \b\tau = \int_T {\b{\nabla}_{\mathrm{s}}}\b v_T : \b\tau + \sum_{F \in {\mathcal{F}}_T} \int_F (\b v_F-\b v_T)\cdot (\b\tau \b n_{TF})\qquad \forall \b\tau \in {\mathbb{P}}^{k}(T,{{\mathbb{R}}^{d \times d}_{\mathrm{s}}}).$$ This symmetric gradient reconstruction, originally introduced in [@Botti.Di-Pietro.ea:17 Section 4.2], is designed so that the following relation holds (see, e.g., [@Botti.Di-Pietro.ea:17*1 Proposition 5] or [@Di-Pietro.Droniou:20 Section 7.2.5]): For all $\b v\in W^{1,1}(T,{\mathbb{R}}^d)$, $$\label{eq:G:proj} {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},T}} ({{\und{\b{I}}}_{T}^{k}} \b v) = {\b{\pi}_{T}^{k}}({\b{\nabla}_{\mathrm{s}}}\b v).$$ The global symmetric gradient reconstruction ${\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} : {{{\und{\b{U}}}}_{h}^{k}} \to {\mathbb{P}}^{k}({\mathcal{T}}_h,{{\mathbb{R}}^{d \times d}_{\mathrm{s}}})$ is obtained patching the local contributions, that is, for all ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h}^{k}}$, $$\label{eq:Gh} ({\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{v}}}_h){\ \!\!_{|_{T}}} \coloneq {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},T}} {\und{\b{v}}}_T \qquad \forall T\in{\mathcal{T}}_h.$$ ### Discrete viscous function The discrete counterpart of the function $a$ defined by is the function ${\mathrm{a}}_h : {{{\und{\b{U}}}}_{h}^{k}} \times {{{\und{\b{U}}}}_{h}^{k}} \to {\mathbb{R}}$ such that, for all ${\und{\b{v}}}_h,{\und{\b{w}}}_h \in {{{\und{\b{U}}}}_{h}^{k}}$, $$\label{eq:ah} {\mathrm{a}}_h({\und{\b{w}}}_h, {\und{\b{v}}}_h) \coloneqq \displaystyle\int_\Omega {\b\sigma}(\cdot,{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{w}}}_h): {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{v}}}_h +\gamma {\mathrm{s}}_h({\und{\b{w}}}_h,{\und{\b{v}}}_h).$$ In the above definition, recalling , $\gamma$ is a stabilization parameter such that $$\label{eq:gamma} \gamma \in [\sigma_{\mathrm{sm}},\sigma_{\mathrm{hc}}],$$ while the stabilization function ${\mathrm{s}}_h : {{{\und{\b{U}}}}_{h}^{k}} \times {{{\und{\b{U}}}}_{h}^{k}} \to {\mathbb{R}}$ is such that, for all ${\und{\b{v}}}_h,{\und{\b{w}}}_h \in {{{\und{\b{U}}}}_{h}^{k}}$, $$\label{eq:sh} {\mathrm{s}}_h({\und{\b{w}}}_h,{\und{\b{v}}}_h) \coloneqq \displaystyle\sum_{T \in {\mathcal{T}}_h}{\mathrm{s}}_T({\und{\b{w}}}_T,{\und{\b{v}}}_T),$$ where the local contributions are assumed to satisfy the following assumption. \[ass:sT\] For all $T \in {\mathcal{T}}_h$, the local stabilization function ${\mathrm{s}}_T:{{{\und{\b{U}}}}_{T}^{k}}\times{{{\und{\b{U}}}}_{T}^{k}}\to{\mathbb{R}}$ is linear in its second argument and satisfies the following properties, with hidden constants independent of both $h$ and $T$: \[eq:ass:sT\] 1. \[ass:sT:stability.boundedness\] *Stability and boundedness.* Recalling the definition of the local $\|{\cdot}\|_{{\boldsymbol{\varepsilon}},{r},T}$-seminorm, for all ${\und{\b{v}}}_T\in{{{\und{\b{U}}}}_{T}^{k}}$ it holds: $$\label{eq:sT:stability.boundedness} \| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},T}}{\und{\b{v}}}_T \|_{L^{r}(T,{{\mathbb{R}}^{d \times d}})}^{r}+ {\mathrm{s}}_T({\und{\b{v}}}_T,{\und{\b{v}}}_T) \simeq \|{\und{\b{v}}}_T\|_{{\boldsymbol{\varepsilon}},{r},T}^{r}.$$ 2. *Polynomial consistency.*\[ass:sT:polynomial.consistency\] For all $\b w\in{\mathbb{P}}^{k+1}(T,{\mathbb{R}}^d)$ and all ${\und{\b{v}}}_T\in{{{\und{\b{U}}}}_{T}^{k}}$, $$\label{eq:sT:polynomial.consistency} {\mathrm{s}}_T({{\und{\b{I}}}_{T}^{k}} \b w,{\und{\b{v}}}_T) = 0.$$ 3. *Hölder continuity.*\[ass:sT:holder-continuity\] For all ${\und{\b{u}}}_T, {\und{\b{v}}}_T, {\und{\b{w}}}_T \in {{{\und{\b{U}}}}_{T}^{k}}$, it holds, setting ${\und{\b{e}}}_T\coloneq{\und{\b{u}}}_T - {\und{\b{w}}}_T$, $$\label{eq:sT:holder-continuity} \hspace{-0.5cm} \left|{\mathrm{s}}_T({\und{\b{u}}}_T,{\und{\b{v}}}_T)-{\mathrm{s}}_T({\und{\b{w}}}_T,{\und{\b{v}}}_T)\right| \lesssim \left({\mathrm{s}}_T({\und{\b{u}}}_T,{\und{\b{u}}}_T)+{\mathrm{s}}_T({\und{\b{w}}}_T,{\und{\b{w}}}_T)\right)^\frac{{r}-{r^{\circ}}}{{r}}{\mathrm{s}}_T({\und{\b{e}}}_T,{\und{\b{e}}}_T)^\frac{{r^{\circ}}-1}{{r}}{\mathrm{s}}_T({\und{\b{v}}}_T,{\und{\b{v}}}_T)^\frac{1}{{r}}.$$ 4. *Strong monotonicity.*\[ass:sT:strong-monotonicity\] For all ${\und{\b{u}}}_T, {\und{\b{w}}}_T \in {{{\und{\b{U}}}}_{T}^{k}}$ , it holds, setting again ${\und{\b{e}}}_T\coloneq{\und{\b{u}}}_T - {\und{\b{w}}}_T$, $$\label{eq:sT:strong-monotonicity} \left({\mathrm{s}}_T({\und{\b{u}}}_T,{\und{\b{e}}}_T) - {\mathrm{s}}_T({\und{\b{w}}}_T,{\und{\b{e}}}_T)\right)\left( {\mathrm{s}}_T({\und{\b{u}}}_T,{\und{\b{u}}}_T)+{\mathrm{s}}_T({\und{\b{w}}}_T,{\und{\b{w}}}_T)\right)^\frac{2-{r^{\circ}}}{{r}} \gtrsim {\mathrm{s}}_T({\und{\b{e}}}_T,{\und{\b{e}}}_T)^\frac{{r}+2-{r^{\circ}}}{{r}}.$$ If ${r}=2$, ${\mathrm{s}}_T$ can be any symmetric bilinear form satisfying [[(S\[ass:sT:stability.boundedness\])]{}]{}–[[(S\[ass:sT:polynomial.consistency\])]{}]{}. Indeed, property [[(S\[ass:sT:holder-continuity\])]{}]{} coincides in this case with the Cauchy–Schwarz inequality, while, by linearity of ${\mathrm{s}}_T$, property [[(S\[ass:sT:strong-monotonicity\])]{}]{} holds with the equal sign. \[lem:sT:consist\] For any $T\in{\mathcal{T}}_h$ and any ${\mathrm{s}}_T$ satisfying Assumption \[ass:sT\], it holds, for all $\b w \in W^{k+2,{r}}(T,{\mathbb{R}}^d)$ and all ${\und{\b{v}}}_T \in {{{\und{\b{U}}}}_{T}^{k}}$, $$\label{eq:sT:consist} |{\mathrm{s}}_T({{\und{\b{I}}}_{T}^{k}} \b w,{\und{\b{v}}}_T)| \lesssim h_T^{(k+1)({r^{\circ}}-1)}| \b w |_{W^{1,{r}}(T,{\mathbb{R}}^d)}^{{r}-{r^{\circ}}}|\b w|_{W^{k+2,{r}}(T,{\mathbb{R}}^d)}^{{r^{\circ}}-1}\| {\und{\b{v}}}_T \|_{{\boldsymbol{\varepsilon}},{r},T},$$ where the hidden constant is independent of $h$, $T$, and $\b w$. The proof adapts the arguments of [@Di-Pietro.Droniou:20 Propositon 2.14]. Using the polynomial consistency property [[(S\[ass:sT:polynomial.consistency\])]{}]{}, we can write $$\begin{aligned} |{\mathrm{s}}_T({{\und{\b{I}}}_{T}^{k}} \b w,{\und{\b{v}}}_T)| &= |{\mathrm{s}}_T({{\und{\b{I}}}_{T}^{k}} \b w,{\und{\b{v}}}_T)-{\mathrm{s}}_T({{\und{\b{I}}}_{T}^{k}}({\b{\pi}_{T}^{k+1}} \b w),{\und{\b{v}}}_T)| \\ &\lesssim {\mathrm{s}}_T({{\und{\b{I}}}_{T}^{k}} \b w,{{\und{\b{I}}}_{T}^{k}} \b w)^\frac{{r}-{r^{\circ}}}{{r}}{\mathrm{s}}_T({{\und{\b{I}}}_{T}^{k}}(\b w-{\b{\pi}_{T}^{k+1}} \b w),{{\und{\b{I}}}_{T}^{k}}(\b w-{\b{\pi}_{T}^{k+1}} \b w))^\frac{{r^{\circ}}-1}{{r}}{\mathrm{s}}_T({\und{\b{v}}}_T,{\und{\b{v}}}_T)^\frac{1}{{r}} \\ &\lesssim \|{{\und{\b{I}}}_{T}^{k}} \b w\|_{{\boldsymbol{\varepsilon}},{r},T}^{{r}-{r^{\circ}}}\|{{\und{\b{I}}}_{T}^{k}}(\b w-{\b{\pi}_{T}^{k+1}} \b w)\|_{{\boldsymbol{\varepsilon}},{r},T}^{{r^{\circ}}-1}\|{\und{\b{v}}}_T\|_{{\boldsymbol{\varepsilon}},{r},T}\\ &\lesssim | \b w |_{W^{1,{r}}(T,{\mathbb{R}}^d)}^{{r}-{r^{\circ}}}|\b w-{\b{\pi}_{T}^{k+1}} \b w|_{W^{1,{r}}(T,{\mathbb{R}}^d)}^{{r^{\circ}}-1}\|{\und{\b{v}}}_T\|_{{\boldsymbol{\varepsilon}},{r},T}\\ &\lesssim h_T^{(k+1)({r^{\circ}}-1)}| \b w |_{W^{1,{r}}(T,{\mathbb{R}}^d)}^{{r}-{r^{\circ}}}|\b w|_{W^{k+2,{r}}(T,{\mathbb{R}}^d)}^{{r^{\circ}}-1}\|{\und{\b{v}}}_T\|_{{\boldsymbol{\varepsilon}},{r},T}, \end{aligned}$$ where we have used the Hölder continuity [[(S\[ass:sT:holder-continuity\])]{}]{} and observed that, by the consistency property [[(S\[ass:sT:polynomial.consistency\])]{}]{}, ${\mathrm{s}}_T({{\und{\b{I}}}_{T}^{k}}({\b{\pi}_{T}^{k+1}} \b w),{{\und{\b{I}}}_{T}^{k}}({\b{\pi}_{T}^{k+1}} \b w))=0$ to pass to the second line, we have used the boundedness property [[(S\[ass:sT:stability.boundedness\])]{}]{} to pass to the third line, the boundedness of ${{\und{\b{I}}}_{T}^{k}}$ to pass to the fourth line, and the $(k+2,{r},1)$-approximation property of ${\b{\pi}_{T}^{k+1}}$ to conclude. In what follows, we will need generalized versions of the continuous and discrete Hölder inequalities, recalled hereafter for the sake of convenience. Let $X \subset {\mathbb{R}}^d$ be measurable, $n \in {\mathbb{N}}^*$, and let $t,p_1,\ldots,p_n \in (0,+\infty\rbrack$ be such that $\sum_{i=1}^n\frac{1}{p_i} = \frac{1}{t}$. The continuous $(t;p_1,\ldots,p_n)$-Hölder inequality reads: For any $(f_1,\ldots,f_n) \in \bigtimes_{i=1}^n L^{p_i}(X,{\mathbb{R}})$, $$\label{eq:holder} \left\| \prod_{i=1}^n f_i \right\|_{L^t(X,{\mathbb{R}})} \le\ \prod_{i=1}^n\| f_i \|_{L^{p_i}(X,{\mathbb{R}})}.$$ Let $m \in {\mathbb{N}}^*$. For all $f : \{1,\ldots,m\} \to {\mathbb{R}}$ and all $q \in [1,+\infty)$, setting $\|f\|_q \coloneqq \left(\sum_{i=1}^m |f(i)|^q\right)^\frac{1}{q}$, and $\|f\|_\infty \coloneqq \max_{1 \le i \le m}|f(i)|$, the discrete $(t;p_1,\ldots,p_n)$-Hölder inequality reads: For any $f_1,\ldots,f_n : \{1,\ldots,m\} \to {\mathbb{R}}$, $$\label{eq:discrete.holder} \left\| \prod_{i=1}^n f_i \right\|_{t} \le\ \prod_{i=1}^n\| f_i \|_{p_i}.$$ Let ${\mathrm{s}}_h$ be given by with, for all $T\in{\mathcal{T}}_h$, ${\mathrm{s}}_T$ satisfying Assumption \[ass:sT\]. Then it holds, for all ${\und{\b{v}}}_h \in{{{\und{\b{U}}}}_{h}^{k}}$, \[eq:sh:properties\] $$\label{eq:sh:stability.boundedness} \| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{v}}}_h \|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}+ {\mathrm{s}}_h({\und{\b{v}}}_h,{\und{\b{v}}}_h) \simeq \|{\und{\b{v}}}_h\|_{{\boldsymbol{\varepsilon}},{r},h}^{r}.$$ Furthermore, for all ${\und{\b{u}}}_h, {\und{\b{v}}}_h,{\und{\b{w}}}_h\in{{{\und{\b{U}}}}_{h}^{k}}$ it holds, setting ${\und{\b{e}}}_h\coloneq{\und{\b{u}}}_h - {\und{\b{w}}}_h$, $$\begin{gathered} \label{eq:sh:holder-continuity} \left|{\mathrm{s}}_h({\und{\b{u}}}_h,{\und{\b{v}}}_h)-{\mathrm{s}}_h({\und{\b{w}}}_h,{\und{\b{v}}}_h)\right| \lesssim \left({\mathrm{s}}_h({\und{\b{u}}}_h,{\und{\b{u}}}_h)+{\mathrm{s}}_h({\und{\b{w}}}_h,{\und{\b{w}}}_h)\right)^\frac{{r}-{r^{\circ}}}{{r}}{\mathrm{s}}_h({\und{\b{e}}}_h,{\und{\b{e}}}_h)^\frac{{r^{\circ}}-1}{{r}}{\mathrm{s}}_h({\und{\b{v}}}_h,{\und{\b{v}}}_h)^\frac{1}{{r}}, \\\label{eq:sh:strong-monotonicity} \left( {\mathrm{s}}_h({\und{\b{u}}}_h,{\und{\b{e}}}_h) - {\mathrm{s}}_h({\und{\b{w}}}_h,{\und{\b{e}}}_h) \right)\left( {\mathrm{s}}_h({\und{\b{u}}}_h,{\und{\b{u}}}_h)+{\mathrm{s}}_h({\und{\b{w}}}_h,{\und{\b{w}}}_h) \right)^\frac{2-{r^{\circ}}}{{r}} \gtrsim {\mathrm{s}}_h({\und{\b{e}}}_h,{\und{\b{e}}}_h)^\frac{{r}+2-{r^{\circ}}}{{r}} . \end{gathered}$$ Finally, for any $\b w \in \b U\cap W^{k+2,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)$, it holds $$\label{eq:sh:consist} \sup\limits_{{\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}},\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} = 1}{\mathrm{s}}_h({{\und{\b{I}}}_{h}^{k}} \b w,{\und{\b{v}}}_h) \lesssim h^{(k+1)({r^{\circ}}-1)}| \b w |_{W^{1,{r}}(\Omega,{\mathbb{R}}^d)}^{{r}-{r^{\circ}}}|\b w|_{W^{k+2,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{{r^{\circ}}-1}.$$ Above, the hidden constants are independent of $h$ and of the arguments of ${\mathrm{s}}_h$. For the sake of conciseness, we only sketch the proof and leave the details to the reader. Summing over $T \in {\mathcal{T}}_h$ immediately yields . The Hölder continuity property follows applying to the quantity in the left-hand side triangle inequalities, using , and concluding with a discrete $(1;\frac{{r}}{{r}-{r^{\circ}}},\frac{{r}}{{r^{\circ}}-1},{r})$-Hölder inequality. Moving to , starting from $|{\mathrm{s}}_h({\und{\b{e}}}_h,{\und{\b{e}}}_h)|$, we use and apply a discrete $(1;\frac{{r}+2-{r^{\circ}}}{2-{r^{\circ}}},\frac{{r}+2-{r^{\circ}}}{{r}})$-Hölder inequality to conclude. Finally, to prove we start from ${\mathrm{s}}_h({\und{\b{I}}}_h^k\b w,{\und{\b{v}}}_h)$, expand this quantity according to , use, for all $T \in {\mathcal{T}}_h$, the local consistency property together with $h_T \le h$, invoke the discrete $(1;\frac{{r}}{{r}-{r^{\circ}}},\frac{{r}}{{r^{\circ}}-1},{r})$-Hölder inequality, and pass to the supremum to conclude. ### An example of viscous stabilization function Taking inspiration from the scalar case (cf., e.g., [@Di-Pietro.Droniou:17 Eq. (4.11c)]), a local stabilization function that matches Assumption \[ass:sT\] can be obtained setting, for all ${\und{\b{v}}}_T,{\und{\b{w}}}_T \in {{{\und{\b{U}}}}_{T}^{k}}$, $$\label{eq:sT} {\mathrm{s}}_T({\und{\b{w}}}_T,{\und{\b{v}}}_T) \coloneqq \int_{\partial T} |{\b{\Delta}^{k}_{\partial T}} {\und{\b{w}}}_T|^{{r}-2}{\b{\Delta}^{k}_{\partial T}} {\und{\b{w}}}_T \cdot {\b{\Delta}^{k}_{\partial T}} {\und{\b{v}}}_T,$$ where, denoting by ${\mathbb{P}}^k({\mathcal{F}}_T,{\mathbb{R}}^d)$ the space of vector-valued broken polynomials of total degree $\le k$ on ${\mathcal{F}}_T$, the boundary residual operator ${\b{\Delta}^{k}_{\partial T}} : {{{\und{\b{U}}}}_{T}^{k}} \to {\mathbb{P}}^k({\mathcal{F}}_T,{\mathbb{R}}^d)$ is such that, for all ${\und{\b{v}}}_T \in {{{\und{\b{U}}}}_{T}^{k}}$, $$({\b{\Delta}^{k}_{\partial T}} {\und{\b{v}}}_T){\ \!\!_{|_{F}}} \coloneq h_F^{-\frac1{{r}'}}\left( {\b{\pi}_{F}^{k}}({\b{{\mathrm{r}}}^{k+1}_{T}} {\und{\b{v}}}_T-\b v_F)-{\b{\pi}_{T}^{k}}({\b{{\mathrm{r}}}^{k+1}_{T}} {\und{\b{v}}}_T-\b v_T) \right) \qquad\forall F\in{\mathcal{F}}_T,$$ with velocity reconstruction ${\b{{\mathrm{r}}}^{k+1}_{T}} : {{{\und{\b{U}}}}_{T}^{k}} \to {\mathbb{P}}^{k+1}(T,{\mathbb{R}}^d)$ such that $$\begin{gathered} \displaystyle\int_T ({\b{\nabla}_{\mathrm{s}}}{\b{{\mathrm{r}}}^{k+1}_{T}} {\und{\b{v}}}_T - {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},T}} {\und{\b{v}}}_T) : {\b{\nabla}_{\mathrm{s}}}\b w = 0\qquad\forall\b w \in {\mathbb{P}}^{k+1}(T,{\mathbb{R}}^d), \\ \text{ $\displaystyle\int_T {\b{{\mathrm{r}}}^{k+1}_{T}} {\und{\b{v}}}_T = \int_T \b v_T$, and $\int_T {\b{\nabla}_{\mathrm{ss}}}{\b{{\mathrm{r}}}^{k+1}_{T}} {\und{\b{v}}}_T = \frac{1}{2}\sum_{F \in {\mathcal{F}}_T} \int_F ({\b v_F \otimes \b n_{TF}} - {\b n_{TF} \otimes \b v_F})$. }\end{gathered}$$ Above, ${\b{\nabla}_{\mathrm{ss}}}$ denotes the skew-symmetric part of the gradient operator ${\b\nabla}$ applied to vector fields and ${ \otimes }$ is the tensor product such that, for all $\b x = (x_i)_{1 \le i \le d}$ and $\b y = (y_i)_{1 \le i \le d}$ in ${\mathbb{R}}^d$, ${\b x \otimes \b y} \coloneq (x_i y_j)_{1 \le i,j \le d} \in {{\mathbb{R}}^{d \times d}}$. \[lem:sT\] The local stabilization function defined by satisfies Assumption \[ass:sT\]. The proof of [[(S\[ass:sT:stability.boundedness\])]{}]{} for ${r}=2$ is given in [@Botti.Di-Pietro.ea:17 Eq. (25)]. The result can be generalized to ${r}\neq 2$ using the same arguments of [@Di-Pietro.Droniou:17 Lemma 5.2]. Property [[(S\[ass:sT:polynomial.consistency\])]{}]{} is an immediate consequence of the fact that ${\b{\Delta}^{k}_{\partial T}}({{\und{\b{I}}}_{T}^{k}}\b w) = \b 0$ for any $\b w\in{\mathbb{P}}^{k+1}(T,{\mathbb{R}}^d)$, which can be proved reasoning as in [@Di-Pietro.Droniou:20 Proposition 2.6]. Let us prove [[(S\[ass:sT:holder-continuity\])]{}]{}. First, we remark that, since the function $\alpha \mapsto \alpha^{{r}-2}$ verifies the conditions in , we can apply Theorem \[thm:1d.power-framed\] to infer that the function ${\mathbb{R}}^d\ni\b x \mapsto |\b x|^{{r}-2}\b x$ satisfies for all $\b x,\b y \in {\mathbb{R}}^d$, \[eq:rho.holder.strong\] $$\begin{gathered} \big| |\b x|^{{r}-2}\b x-|\b y|^{{r}-2}\b y \big| \lesssim \big( |\b x|^{r}+|\b y|^{r}\big)^\frac{{r}-{r^{\circ}}}{{r}}|\b x-\b y|^{{r^{\circ}}-1},\label{eq:rho:s.holder.continuity} \\ \big( |\b x|^{{r}-2}\b x-|\b y|^{{r}-2}\b y \big) \cdot (\b x-\b y) \big( |\b x|^{r}+|\b y|^{r}\big)^\frac{2-{r^{\circ}}}{{r}} \gtrsim |\b x-\b y|^{{r}+2-{r^{\circ}}}.\label{eq:rho:s.strong.monotonicity} \end{gathered}$$ Recalling , we can write $$\begin{aligned} \left|{\mathrm{s}}_T({\und{\b{u}}}_T,{\und{\b{v}}}_T)-{\mathrm{s}}_T({\und{\b{w}}}_T,{\und{\b{v}}}_T)\right| &\leq \int_{\partial T} \left||{\b{\Delta}^{k}_{\partial T}} {\und{\b{u}}}_T|^{{r}-2}{\b{\Delta}^{k}_{\partial T}} {\und{\b{u}}}_T-|{\b{\Delta}^{k}_{\partial T}} {\und{\b{w}}}_T|^{{r}-2}{\b{\Delta}^{k}_{\partial T}} {\und{\b{w}}}_T\right|| {\b{\Delta}^{k}_{\partial T}} {\und{\b{v}}}_T|\\ &\lesssim \int_{\partial T} \left(|{\b{\Delta}^{k}_{\partial T}} {\und{\b{u}}}_T|^{r}+|{\b{\Delta}^{k}_{\partial T}} {\und{\b{w}}}_T|^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}| {\b{\Delta}^{k}_{\partial T}} {\und{\b{e}}}_T|^{{r^{\circ}}-1}| {\b{\Delta}^{k}_{\partial T}} {\und{\b{v}}}_T|\\ &\le \left({\mathrm{s}}_T({\und{\b{u}}}_T,{\und{\b{u}}}_T)+{\mathrm{s}}_T({\und{\b{w}}}_T,{\und{\b{w}}}_T)\right)^\frac{{r}-{r^{\circ}}}{{r}}{\mathrm{s}}_T({\und{\b{e}}}_T,{\und{\b{e}}}_T)^\frac{{r^{\circ}}-1}{{r}}{\mathrm{s}}_T({\und{\b{v}}}_T,{\und{\b{v}}}_T)^\frac{1}{{r}}, \end{aligned}$$ where we have used to pass to the second line and the $(1;\frac{{r}}{{r}-{r^{\circ}}},\frac{{r}}{{r^{\circ}}-1},{r})$-Hölder inequality to conclude. Moving to [[(S\[ass:sT:strong-monotonicity\])]{}]{}, and the $(1;\frac{{r}+2-{r^{\circ}}}{2-{r^{\circ}}},\frac{{r}+2-{r^{\circ}}}{{r}})$-Hölder inequality yield $$\begin{aligned} &{\mathrm{s}}_T({\und{\b{e}}}_T,{\und{\b{e}}}_T) \\ &\quad = \int_{\partial T} |{\b{\Delta}^{k}_{\partial T}}{\und{\b{u}}}_T-{\b{\Delta}^{k}_{\partial T}}{\und{\b{w}}}_T|^{r}\\ &\quad \lesssim \int_{\partial T} \left(|{\b{\Delta}^{k}_{\partial T}} {\und{\b{u}}}_T|^{r}+|{\b{\Delta}^{k}_{\partial T}} {\und{\b{w}}}_T|^{r}\right)^\frac{2-{r^{\circ}}}{{r}+2-{r^{\circ}}}\left[ \left(|{\b{\Delta}^{k}_{\partial T}}{\und{\b{u}}}_T|^{{r}-2}{\b{\Delta}^{k}_{\partial T}}{\und{\b{u}}}_T-|{\b{\Delta}^{k}_{\partial T}}{\und{\b{w}}}_T|^{{r}-2}{\b{\Delta}^{k}_{\partial T}}{\und{\b{w}}}_T\right)\cdot {\b{\Delta}^{k}_{\partial T}}{\und{\b{e}}}_T \right]^\frac{{r}}{{r}+2-{r^{\circ}}}\\ &\quad \le \left( {\mathrm{s}}_T({\und{\b{u}}}_T,{\und{\b{u}}}_T)+{\mathrm{s}}_T({\und{\b{w}}}_T,{\und{\b{w}}}_T)\right)^\frac{2-{r^{\circ}}}{{r}+2-{r^{\circ}}}\left({\mathrm{s}}_T({\und{\b{u}}}_T,{\und{\b{e}}}_T)-{\mathrm{s}}_T({\und{\b{w}}}_T,{\und{\b{e}}}_T)\right)^\frac{{r}}{{r}+2-{r^{\circ}}}.\qedhere \end{aligned}$$ Pressure-velocity coupling -------------------------- For all $T \in {\mathcal{T}}_h$, we define the local divergence reconstruction ${{\mathrm{D}}^{k}_{T}} : {{{\und{\b{U}}}}_{T}^{k}} \to {\mathbb{P}}^{k}(T,{\mathbb{R}})$ by setting, for all ${\und{\b{v}}}_T \in {{{\und{\b{U}}}}_{T}^{k}}$, ${{\mathrm{D}}^{k}_{T}}{\und{\b{v}}}_T \coloneq {\mathrm{tr}}({\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},T}}{\und{\b{v}}}_T)$. We have the following characterization of ${{\mathrm{D}}^{k}_{T}}$: For all ${\und{\b{v}}}_T \in {{{\und{\b{U}}}}_{T}^{k}}$, $$\label{eq:D} \int_T {{\mathrm{D}}^{k}_{T}} {\und{\b{v}}}_T~ q = \int_T ({\b\nabla{\cdot}}\b v_T)~q + \sum_{F \in {\mathcal{F}}_T} \int_F (\b v_F-\b v_T)\cdot \b n_{TF}~q \qquad \forall q \in {\mathbb{P}}^{k}(T,{\mathbb{R}}),$$ as can be checked writing for $\b\tau = q\mathrm{I}_d$. Taking the trace of , it is inferred that, for all $T\in{\mathcal{T}}_h$ and all $\b v\in W^{1,1}(T,{\mathbb{R}}^d)$, ${{\mathrm{D}}^{k}_{T}} ({{\und{\b{I}}}_{T}^{k}} \b v) = {\pi_{T}^{k}}({\b\nabla{\cdot}}\b v)$. The pressure-velocity coupling is realized by the bilinear form ${\mathrm{b}}_h : {{{\und{\b{U}}}}_{h}^{k}} \times {\mathbb{P}}^k({\mathcal{T}}_h,{\mathbb{R}}) \to {\mathbb{R}}$ such that, for all $({\und{\b{v}}}_h,q_h) \in {{{\und{\b{U}}}}_{h}^{k}} \times {\mathbb{P}}^k({\mathcal{T}}_h,{\mathbb{R}})$, setting $q_T\coloneq (q_h){\ \!\!_{|_{T}}}$ for all $T\in{\mathcal{T}}_h$, $$\label{eq:bh} {\mathrm{b}}_h({\und{\b{v}}}_h,q_h) \coloneqq -\sum_{T \in {\mathcal{T}}_h}\int_T {{\mathrm{D}}^{k}_{T}} {\und{\b{v}}}_T~ q_T.$$ Discrete problem ---------------- The discrete problem reads: Find $({\und{\b{u}}}_h,p_h) \in {{{\und{\b{U}}}}_{h,0}^{k}} \times {P_{h}^{k}}$ such that \[eq:stokes.discrete\] $$\begin{aligned} {2} \label{eq:stokes.discrete:momentum} {\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{v}}}_h) + {\mathrm{b}}_h({\und{\b{v}}}_h,p_h) &= \displaystyle\int_\Omega \b f \cdot \b v_h &\qquad& \forall {\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}}, \\ \label{eq:stokes.discrete:mass} -{\mathrm{b}}_h({\und{\b{u}}}_h,q_h) &= 0 &\qquad& \forall q_h \in {P_{h}^{k}}. \end{aligned}$$ Before proceding, some remarks are in order. The space of test functions in can be extended to ${\mathbb{P}}^k({\mathcal{T}}_h,{\mathbb{R}})$ since, for all ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}}$, the divergence theorem together with the fact that $\b v_F = \b 0$ for all $F \in {{\mathcal{F}}_h^{\mathrm{b}}}$ and $\sum_{T\in {\mathcal{T}}_F} \int_F \b v_F \cdot \b n_{TF} = 0$ for all $F \in {{\mathcal{F}}_h^{\mathrm{i}}}$, yield $${\mathrm{b}}_h({\und{\b{v}}}_h,1) = -\sum_{T \in {\mathcal{T}}_h} \sum_{F \in {\mathcal{F}}_T} \int_F \b v_F \cdot \b n_{TF} = -\sum_{F \in {{\mathcal{F}}_h^{\mathrm{i}}}}\sum_{T \in {\mathcal{T}}_F} \int_F \b v_F \cdot \b n_{TF} = 0.$$ When solving the system of nonlinear algebraic equations corresponding to by a first-order (e.g., Newton) algorithm, all element-based velocity unknowns and all but one pressure unknown per element can be locally eliminated at each iteration by computing the corresponding Schur complement element-wise. As all the computations are local, this procedure is an embarrassingly parallel task which can fully benefit from multi-thread and multi-processor architectures. This implementation strategy has been described for the linear Stokes problem in [@Di-Pietro.Ern.ea:16 Section 6.2]. After further eliminating the boundary unknowns by strongly enforcing the boundary condition , we end up solving, at each iteration of the nonlinear solver, a linear system of size $d{\mathrm{card}}({{\mathcal{F}}_h^{\mathrm{i}}}){k+d-1\choose d-1} + {\mathrm{card}}({\mathcal{T}}_h)$. Well-posedness {#sec:well-posedness} ============== In this section, after studying the stability properties of the viscous function ${\mathrm{a}}_h$ and of the velocity-pressure coupling bilinear form ${\mathrm{b}}_h$, we prove the well-posedness of problem . Hölder continuity and strong monotonicity of the viscous function ----------------------------------------------------------------- \[lem:ah:holder.continuity.strong.monotonicity\] For all ${\und{\b{u}}}_h, {\und{\b{v}}}_h, {\und{\b{w}}}_h \in {{{\und{\b{U}}}}_{h}^{k}}$, setting ${\und{\b{e}}}_h\coloneq{\und{\b{u}}}_h - {\und{\b{w}}}_h$, it holds \[eq:ah:holder.continuity.strong.monotonicity\] $$\begin{gathered} \label{eq:ah:holder.continuity} \left| {\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{v}}}_h)-{\mathrm{a}}_h({\und{\b{w}}}_h,{\und{\b{v}}}_h) \right| \lesssim \sigma_{\mathrm{hc}}\left( \sigma_{\mathrm{de}}^{r}+ \| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}+\| {\und{\b{w}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}\| {\und{\b{e}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{{r^{\circ}}-1}\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}, \\\label{eq:ah:strong.monotonicity} \left({\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{e}}}_h)-{\mathrm{a}}_h({\und{\b{w}}}_h,{\und{\b{e}}}_h)\right)\left( \sigma_{\mathrm{de}}^{r}+\| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}+\| {\und{\b{w}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}\right)^\frac{2-{r^{\circ}}}{{r}} \gtrsim \sigma_{\mathrm{sm}}\| {\und{\b{e}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{{r}+2-{r^{\circ}}}. \end{gathered}$$ \(i) *Hölder continuity.* Denote by $|\Omega|_d$ the measure of $\Omega$. Using a Cauchy–Schwarz inequality followed by the Hölder continuity of ${\b\sigma}$, we can write $$\label{C2:rge2:1} \begin{aligned} &\left|\int_\Omega \big( {\b\sigma}(\cdot,{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{u}}}_h)-{\b\sigma}(\cdot,{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{w}}}_h) \big):{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{v}}}_h\right| \\ &\quad \le \sigma_{\mathrm{hc}} \int_\Omega \left(\sigma_{\mathrm{de}}^{r}+|{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{u}}}_h|_{d\times d}^{r}+|{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{w}}}_h|_{d\times d}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{e}}}_h|_{d\times d}^{{r^{\circ}}-1} |{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{v}}}_h|_{d \times d} \\ &\quad \lesssim \sigma_{\mathrm{hc}}\left(|\Omega|_d\sigma_{\mathrm{de}}^{r}+\| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{u}}}_h \|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}+\| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{w}}}_h \|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}} \\ & \qquad \times \| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{e}}}_h \|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{{r^{\circ}}-1}\| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{v}}}_h \|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}\\ &\quad \lesssim \sigma_{\mathrm{hc}}\left(\sigma_{\mathrm{de}}^{r}+ \| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}+\| {\und{\b{w}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}\| {\und{\b{e}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{{r^{\circ}}-1}\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}, \end{aligned}$$ where we have used the $(1;\frac{{r}}{{r}-{r^{\circ}}},\frac{{r}}{{r^{\circ}}-1},{r})$-Hölder inequality in the second bound and the global seminorm equivalence together with the fact that $|\Omega|_d\lesssim 1$ (since $\Omega$ is bounded) to conclude. For the stabilization term, combining the Hölder continuity of ${\mathrm{s}}_h$ and the seminorm equivalence readily gives $$\label{C2:rge2:2} \left| {\mathrm{s}}_h({\und{\b{u}}}_h,{\und{\b{v}}}_h)-{\mathrm{s}}_h({\und{\b{w}}}_h,{\und{\b{v}}}_h) \right| \lesssim \left(\sigma_{\mathrm{de}}^{r}+ \| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}+\| {\und{\b{w}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}\| {\und{\b{e}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{{r^{\circ}}-1}\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h},$$ where we have additionally noticed that $\sigma_{\mathrm{de}}^{r}\ge 0$ to add this term to the quantity inside parentheses. Using the definition of ${\mathrm{a}}_h$, a triangle inequality followed by and , and recalling that $\gamma \le \sigma_{\mathrm{hc}}$ (cf. ), follows.\ (ii) *Strong monotonicity.* Using the strong monotonicity of ${\b\sigma}$ and the $(1;\frac{{r}+2-{r^{\circ}}}{2-{r^{\circ}}},\frac{{r}+2-{r^{\circ}}}{{r}})$-Hölder inequality , we get $$\label{eq:ah:sm:1} \hspace{-0.2cm}\begin{aligned} &\sigma_{\mathrm{sm}}^\frac{{r}}{{r}+2-{r^{\circ}}}\| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{e}}}_h\|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}\\ &\leq \int_\Omega \left(\sigma_{\mathrm{de}}^{r}+|{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{u}}}_h|_{d\times d}^{r}+|{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{w}}}_h|_{d\times d}^{r}\right)^{\frac{2-{r^{\circ}}}{{r}+2-{r^{\circ}}}}\left) \left({\b\sigma}(\cdot,{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{u}}}_h)-{\b\sigma}(\cdot,{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{w}}}_h)\right):{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{e}}}_h \right)^\frac{{r}}{{r}+2-{r^{\circ}}}\\ &\lesssim \left(\sigma_{\mathrm{de}}^{r}+\| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{u}}}_h\|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}+\| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{w}}}_h\|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}\right)^\frac{2-{r^{\circ}}}{{r}+2-{r^{\circ}}} \\ &\qquad \times \left( \int_\Omega \left( {\b\sigma}(\cdot,{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{u}}}_h)-{\b\sigma}(\cdot,{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{w}}}_h) \right):{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{e}}}_h \right)^\frac{{r}}{{r}+2-{r^{\circ}}}\\ &\lesssim \left(\sigma_{\mathrm{de}}^{r}+\| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}+\| {\und{\b{w}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}\right)^\frac{2-{r^{\circ}}}{{r}+2-{r^{\circ}}} \left( \int_\Omega \left( {\b\sigma}(\cdot,{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{u}}}_h)-{\b\sigma}(\cdot,{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{w}}}_h) \right):{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{e}}}_h \right)^\frac{{r}}{{r}+2-{r^{\circ}}}, \end{aligned}$$ where the conclusion follows from the global seminorm equivalence . Additionally, using the strong monotonicity of ${\mathrm{s}}_h$ together with the fact that $\sigma_{\mathrm{sm}} \le \gamma$ (cf. ) and invoking again the seminorm equivalence , we readily obtain $$\label{eq:ah:sm:2} \sigma_{\mathrm{sm}}^\frac{{r}}{{r}+2-{r^{\circ}}}{\mathrm{s}}_h({\und{\b{e}}}_h,{\und{\b{e}}}_h) \lesssim \left(\sigma_{\mathrm{de}}^{r}+\| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}+\| {\und{\b{w}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}\right)^\frac{2-{r^{\circ}}}{{r}+2-{r^{\circ}}}\left( \gamma{\mathrm{s}}_h({\und{\b{u}}}_h,{\und{\b{e}}}_h)-\gamma{\mathrm{s}}_h({\und{\b{w}}}_h,{\und{\b{e}}}_h) \right)^\frac{{r}}{{r}+2-{r^{\circ}}}.$$ Finally, combining again the norm equivalence with and , and using yields $$\begin{aligned} \sigma_{\mathrm{sm}}^\frac{{r}}{{r}+2-{r^{\circ}}}\| {\und{\b{e}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}\lesssim &\left( \sigma_{\mathrm{de}}^{r}+\| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}+\| {\und{\b{w}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}\right)^\frac{2-{r^{\circ}}}{{r}+2-{r^{\circ}}}\left( {\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{e}}}_h)-{\mathrm{a}}_h({\und{\b{w}}}_h,{\und{\b{e}}}_h) \right)^\frac{{r}}{{r}+2-{r^{\circ}}}. \end{aligned}$$ Raising this inequality to the power $\frac{{r}-2-{r^{\circ}}}{{r}}$ yields . Stability of the pressure-velocity coupling ------------------------------------------- \[lem:bh:inf-sup\] It holds, for all $q_h \in {P_{h}^{k}}$, $$\label{eq:bh:inf-sup} \| q_h \|_{L^{{r}'}(\Omega,{\mathbb{R}})} \lesssim \sup\limits_{{\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}}, \| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} = 1} {\mathrm{b}}_h({\und{\b{v}}}_h,q_h),$$ with hidden constant depending only on $d$, $k$, ${r}$, $\Omega$, and the mesh regularity parameter. The proof follows the classical Fortin argument (cf., e.g., [@Boffi.Brezzi.ea:13 Section 8.4]), adapted here to the non-Hilbertian setting: we first prove that ${{\und{\b{I}}}_{h}^{k}}$ is a Fortin operator, then combine this fact with the continuous inf-sup condition.\ (i) *Fortin operator.* We need to prove that the following properties hold for any $\b v\in W^{1,{r}}(\Omega,{\mathbb{R}}^d)$: \[eq:fortin\] $$\begin{gathered} \| {{\und{\b{I}}}_{h}^{k}} \b v\|_{{\boldsymbol{\varepsilon}},{r},h} \lesssim | \b v |_{W^{1,{r}}(\Omega,{\mathbb{R}}^d),}\label{eq:fortin:boundedness} \\ {\mathrm{b}}_h({{\und{\b{I}}}_{h}^{k}} \b v,q_h) = b(\b v,q_h)\qquad\forall q_h\in{\mathbb{P}}^k({\mathcal{T}}_h,{\mathbb{R}}).\label{eq:fortin:consistency} \end{gathered}$$ Property is obtained by raising both sides of to the power ${r}$, summing over $T \in {\mathcal{T}}_h$, then taking the $r$th root of the resulting inequality. The proof of is given, e.g., in [@Di-Pietro.Droniou:20 Lemma 8.12].\ (ii) *Inf-sup condition on ${\mathrm{b}}_h$.* Let $q_h \in {P_{h}^{k}}$ and set $c_h \coloneqq \int_\Omega |q_h|^{{r}'-2}q_h$. Using a triangle inequality, the Hölder inequality, and the fact that $|\Omega|_d\lesssim 1$, we get $$\label{eq:inf-sup:qT} \| |q_h|^{{r}'-2}q_h-c_h \|_{L^{{r}}(\Omega,{\mathbb{R}})} \le \| q_h\|_{L^{{r}'}(\Omega,{\mathbb{R}})}^{{r}'-1}+|c_h||\Omega|_d^\frac{1}{{r}} \le \left(1+|\Omega|_d\right)\| q_h\|_{L^{{r}'}(\Omega,{\mathbb{R}})}^{{r}'-1} \lesssim \| q_h\|_{L^{{r}'}(\Omega,{\mathbb{R}})}^{{r}'-1},$$ where we have used the fact that $|c_h|\le\| q_h\|_{L^{{r}'}(\Omega,{\mathbb{R}})}^{{r}'-1} |\Omega|_d^{\frac1{{r}'}}$ along with $\frac1{r}+\frac1{{r}'}=1$ in the second bound and the fact that $|\Omega|_d\lesssim 1$ to conclude. Since $q_h \in L^{{r}'}(\Omega,{\mathbb{R}})$, bound implies that $|q_h|^{{r}'-2}q_h-c_h \in L^{{r}}_0(\Omega,{\mathbb{R}}) \coloneq \left\{ q \in L^{r}(\Omega,{\mathbb{R}}) : \int_\Omega q = 0 \right\}$ by construction. Thus, using the surjectivity of the continuous divergence operator ${\b\nabla{\cdot}}: \b U \to L^{{r}}_0(\Omega,{\mathbb{R}})$, (c.f. [@Duran.Muschietti.ea:10] and also [@Bogovski:79 Theorem 1]), we infer that there exists $\b v_{q_h} \in \b U$ such that $$\label{eq:velocity.lifting} -{\b\nabla{\cdot}}\b v_{q_h} = |q_h|^{{r}'-2}q_h-c_h \quad {\mathrm{and}} \quad | \b v_{q_h}|_{W^{1,{r}}(\Omega,{\mathbb{R}}^d)} \lesssim \| |q_h|^{{r}'-2}q_h-c_h \|_{L^{r}(\Omega,{\mathbb{R}})}.$$ Denote by $\$$ the supremum in . Using the fact that $q_h$ has zero mean value over $\Omega$, the equality in together with the definition of $b$, and the second Fortin property , we have $$\|q_h \|_{L^{{r}'}(\Omega,{\mathbb{R}})}^{{r}'} {=} \int_\Omega \big(|q_h|^{{r}'-2}q_h-c_h\big) q_h = b(\b v_{q_h},q_h) = {\mathrm{b}}_h({{\und{\b{I}}}_{h}^{k}}\b v_{q_h},q_h) \le \$ \| {{\und{\b{I}}}_{h}^{k}} \b v_{q_h} \|_{{\boldsymbol{\varepsilon}},{r},h} \lesssim \$ \| q_h \|_{L^{{r}'}(\Omega,{\mathbb{R}})}^{{r}'-1},$$ where, to conclude, we have used followed by and . Simplifying yields . Well-posedness {#well-posedness} -------------- We are now ready to prove the main result of this section. \[thm:well-posedness\] There exists a unique solution $({\und{\b{u}}}_h,p_h) \in {{{\und{\b{U}}}}_{h,0}^{k}} \times {P_{h}^{k}}$ to the discrete problem . Additionally, the following a priori bounds hold: \[eq:discrete.solution:bounds\] $$\begin{aligned} \| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} &\lesssim \left(\sigma_{\mathrm{sm}}^{-1}\| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)}\right)^\frac{1}{{r}-1}+\left(\sigma_{\mathrm{de}}^{2-{r^{\circ}}}\sigma_{\mathrm{sm}}^{-1}\| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)}\right)^\frac{1}{{r}+1-{r^{\circ}}}, \label{eq:discrete.solution:bounds:uh}\\ \| p_h \|_{L^{{r}'}(\Omega,{\mathbb{R}})} &\lesssim \sigma_{\mathrm{hc}}\left(\sigma_{\mathrm{sm}}^{-1}\| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)}+\sigma_{\mathrm{de}}^{|{r}-2|({r^{\circ}}-1)}\left(\sigma_{\mathrm{sm}}^{-1}\| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)}\right)^\frac{{r^{\circ}}-1}{{r}+1-{r^{\circ}}}\right). \label{eq:discrete.solution:bounds:ph} \end{aligned}$$ \(i) *Existence.* Denote by ${P_{h}^{k,*}}$ the dual space of ${P_{h}^{k}}$ and let $B_h : {{{\und{\b{U}}}}_{h,0}^{k}} \to {P_{h}^{k,*}}$ be such that, for all ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}}$, $$\langle B_h{\und{\b{v}}}_h, q_h \rangle \coloneqq -{\mathrm{b}}_h({\und{\b{v}}}_h,q_h) \qquad \forall q_h \in {P_{h}^{k}}.$$ Here and in what follows, $\langle{\cdot},{\cdot}\rangle$ denotes the appropriate duality pairing as inferred from its arguments. Define the following subspace of ${{{\und{\b{U}}}}_{h,0}^{k}}$ spanned by vectors of discrete unknowns with zero discrete divergence: $$\label{eq:Whk} {\und{\b{W}}}_h^k \coloneq \operatorname{Ker}(B_h) = \left\{ {\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}} : {\mathrm{b}}_h({\und{\b{v}}}_h,q_h) = 0 \quad \forall q_h \in {P_{h}^{k}} \right\},$$ and consider the following problem: Find ${\und{\b{u}}}_h\in{\und{\b{W}}}_h^k$ such that $$\label{eq:existence:auxiliary} {\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{v}}}_h) = \int_\Omega \b f\cdot \b v_h\qquad\forall {\und{\b{v}}}_h\in{\und{\b{W}}}_h^k.$$ Existence of a solution to this problem for a fixed $h$ can be proved adapting the arguments of [@Di-Pietro.Droniou:17 Theorem 4.5]. Specifically, equip ${\und{\b{W}}}_h^k$ with an inner product $(\cdot,\cdot)_{\b W,h}$ (which need not be further specified), denote by $\|{\cdot}\|_{\b W,h}$ the induced norm, and let $\b\Phi_h:{\und{\b{W}}}_h^k\to {\und{\b{W}}}_h^k$ be such that, for all ${\und{\b{w}}}_h\in{\und{\b{W}}}_h^k$, $(\b\Phi_h({\und{\b{w}}}_h),{\und{\b{v}}}_h)_{\b W,h} = {\mathrm{a}}_h({\und{\b{w}}}_h,{\und{\b{v}}}_h)$ for all ${\und{\b{v}}}_h\in{\und{\b{W}}}_h^k$. The strong monotonicity of ${\mathrm{a}}_h$ yields, for any ${\und{\b{v}}}_h\in{\und{\b{W}}}_h^k$ such that $\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} \ge \sigma_{\mathrm{de}}$, $$(\b\Phi_h({\und{\b{v}}}_h),{\und{\b{v}}}_h)_{\b W,h}\ge \sigma_{\mathrm{sm}} (\sigma_{\mathrm{de}}^{r}+ \| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r})^\frac{{r^{\circ}}-2}{{r}} \| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{{r}+2-{r^{\circ}}} \gtrsim \sigma_{\mathrm{sm}}\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}\ge C^{r}\sigma_{\mathrm{sm}}\| {\und{\b{v}}}_h\|_{\b W,h}^{r},$$ where $C$ denotes the constant (possibly depending on $h$) in the equivalence of the norms $\|{\cdot}\|_{{\boldsymbol{\varepsilon}},{r},h}$ and $\|{\cdot}\|_{\b W,h}$ (which holds since ${\und{\b{W}}}_h^k$ is finite-dimensional). This shows that $\b\Phi_h$ is coercive hence, by [@Deimling:85 Theorem 3.3], surjective. Let now ${\und{\b{w}}}_h\in{\und{\b{W}}}_h^k$ be such that $({\und{\b{w}}}_h,{\und{\b{v}}}_h)_{\b W,h}=\int_\Omega \b f\cdot \b v_h$ for all ${\und{\b{v}}}_h\in{\und{\b{W}}}_h^k$. By the surjectivity of $\b\Phi_h$, there exists ${\und{\b{u}}}_h\in {\und{\b{W}}}_h^k$ such that $\b\Phi_h({\und{\b{u}}}_h)={\und{\b{w}}}_h$ which, by definition of ${\und{\b{w}}}_h$ and $\b\Phi_h$, is a solution to the discrete problem . The proof of existence now continues as in the linear case; see, e.g., [@Boffi.Brezzi.ea:13 Theorem 4.2.1]. Denote by ${{{\und{\b{U}}}}_{h,0}^{k,*}}$ the dual space of ${{{\und{\b{U}}}}_{h,0}^{k}}$ and consider the linear mapping $\ell_h\in{{{\und{\b{U}}}}_{h,0}^{k,*}}$ such that, for all ${\und{\b{v}}}_h\in{{{\und{\b{U}}}}_{h,0}^{k}}$, $$\langle\ell_h,{\und{\b{v}}}_h\rangle\coloneq \int_\Omega \b f\cdot \b v_h - {\mathrm{a}}_h({\und{\b{u}}}_h, {\und{\b{v}}}_h).$$ Thanks to , $\ell_h$ vanishes identically for every ${\und{\b{v}}}_h\in{\und{\b{W}}}_h^k$, that is to say, $\ell_h$ lies in the polar space of ${\und{\b{W}}}_h^k$ which, denoting by $B_h^*:{P_{h}^{k}}\to{{{\und{\b{U}}}}_{h,0}^{k,*}}$ the adjoint operator of $B_h$, coincides in our case with $\operatorname{Im}(B_h^*)$ (see, e.g., [@Boffi.Brezzi.ea:13 Theorem 4.14]). Hence, $\ell_h\in\operatorname{Im}(B_h^*)$, and there exists therefore a $p_h\in {P_{h}^{k}}$ such that $B_h^* p_h = \ell_h$. This means that, for all ${\und{\b{v}}}_h\in{{{\und{\b{U}}}}_{h,0}^{k}}$, $${\mathrm{b}}_h({\und{\b{v}}}_h,p_h) = \langle B_h^* p_h,{\und{\b{v}}}_h\rangle = \langle\ell_h,{\und{\b{v}}}_h\rangle = \int_\Omega \b f\cdot \b v_h - {\mathrm{a}}_h({\und{\b{u}}}_h, {\und{\b{v}}}_h),$$ i.e., the $({\und{\b{u}}}_h,p_h)$ satisfies the discrete momentum equation . On the other hand, since ${\und{\b{u}}}_h\in{\und{\b{W}}}_h^k$, we also have, by the definition of ${\und{\b{W}}}_h^k$, $ {\mathrm{b}}_h({\und{\b{u}}}_h,q_h) = 0$ for all $q_h\in {P_{h}^{k}}$, which shows that the discrete mass equation is also verified. In conclusion, $({\und{\b{u}}}_h,p_h)\in{{{\und{\b{U}}}}_{h,0}^{k}}\times {P_{h}^{k}}$ solves .\ (ii) *Uniqueness.* We start by proving uniqueness for the velocity. Let $({\und{\b{u}}}_h,p_h),({\und{\b{u}}}'_h,p'_h) \in {{{\und{\b{U}}}}_{h,0}^{k}} \times {P_{h}^{k}}$ be two solutions of . Making ${\und{\b{v}}}_h = {\und{\b{u}}}_h - {\und{\b{u}}}'_h$ in written first for $({\und{\b{u}}}_h, p_h)$ then for $({\und{\b{u}}}'_h, p'_h)$, then taking the difference and observing that ${\mathrm{b}}_h({\und{\b{u}}}_h-{\und{\b{u}}}'_h,p_h)={\mathrm{b}}_h({\und{\b{u}}}_h - {\und{\b{u}}}'_h,p'_h)=0$ by , we infer that $${\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{u}}}_h - {\und{\b{u}}}'_h)-{\mathrm{a}}_h({\und{\b{u}}}'_h,{\und{\b{u}}}_h - {\und{\b{u}}}'_h) = 0.$$ Thus, the strong monotonicity of ${\mathrm{a}}_h$ yields $\| {\und{\b{u}}}_h - {\und{\b{u}}}'_h \|_{{\boldsymbol{\varepsilon}},{r},h} = 0$, which implies ${\und{\b{u}}}_h = {\und{\b{u}}}'_h$ since $\|{\cdot}\|_{{\boldsymbol{\varepsilon}},{r},h}$ is a norm on ${{{\und{\b{U}}}}_{h,0}^{k}}$. Moreover, using the inf-sup stability of ${\mathrm{b}}_h$ and written first for ${\und{\b{u}}}_h$ then for ${\und{\b{u}}}_h'$, we get $$\begin{aligned} \| p_h-p'_h \|_{L^{{r}'}(\Omega,{\mathbb{R}})} &\lesssim \sup\limits_{{\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}},\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} = 1} {\mathrm{b}}_h({\und{\b{v}}}_h,p_h-p'_h) \\ &= \sup\limits_{{\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}},\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} = 1} \left({\mathrm{a}}_h({\und{\b{u}}}'_h,{\und{\b{v}}}_h)-{\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{v}}}_h)\right) = 0, \end{aligned}$$ hence $p_h=p'_h$.\ (iii) *A priori estimates.* Using the strong monotonicity of ${\mathrm{a}}_h$ (with ${\und{\b{w}}}_h = {\und{\b{0}}}$), equation together with , and the Hölder inequality together with the discrete Korn inequality , we obtain $$\label{eq:well-posedness:0} \begin{aligned} \sigma_{\mathrm{sm}}\big( \sigma_{\mathrm{de}}^{r}+ \| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}\big)^\frac{{r^{\circ}}-2}{{r}} \| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{{r}+2-{r^{\circ}}} &\lesssim {\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{u}}}_h) = \displaystyle\int_\Omega \b f \cdot \b u_h \lesssim \| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)}\| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}. \end{aligned}$$ We then conclude as in the continuous case to infer (see Remark \[rem:a-priori\]). To prove the bound on the pressure, we use the inf-sup stability of ${\mathrm{b}}_h$ to write $$\begin{aligned} \| p_h \|_{L^{{r}'}(\Omega,{\mathbb{R}})} &\lesssim \sup\limits_{{\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}}, \| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} = 1} {\mathrm{b}}_h({\und{\b{v}}}_h,p_h) \\ &= \sup\limits_{{\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}}, \| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} = 1} \left(\displaystyle\int_\Omega \b f \cdot \b v_h - {\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{v}}}_h) \right) \\ &\lesssim \| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)} +\sigma_{\mathrm{hc}}(\sigma_{\mathrm{de}}^{r}+\| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r})^\frac{{r}-{r^{\circ}}}{{r}}\| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{{r^{\circ}}-1} \\ &\lesssim \sigma_{\mathrm{hc}}\left(\sigma_{\mathrm{sm}}^{-1}\| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)}+\sigma_{\mathrm{de}}^{|{r}-2|({r^{\circ}}-1)}\left(\sigma_{\mathrm{sm}}^{-1}\| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)}\right)^\frac{{r^{\circ}}-1}{{r}+1-{r^{\circ}}}\right), \end{aligned}$$ where we have used the discrete momentum equation to pass to the second line, the Hölder and discrete Korn inequalities together with the Hölder continuity of ${\mathrm{a}}_h$ to pass to the third line, and the a priori bound on the velocity together with $\frac{\sigma_{\mathrm{hc}}}{\sigma_{\mathrm{sm}}}\ge 1$ (see ) to conclude. Error estimate {#sec:error.estimate} ============== In this section, after studying the consistency of the viscous and pressure-velocity coupling terms, we prove an energy error estimate. Consistency of the viscous function {#sec:consistency:viscous.function} ----------------------------------- \[lem:consistency:ah\] Let $\b w \in \b U \cap W^{k+2,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)$ be such that ${\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w) \in W^{1,{r}'}(\Omega,{{\mathbb{R}}^{d \times d}_{\mathrm{s}}}) \cap W^{(k+1)({r^{\circ}}-1),{r}'}({\mathcal{T}}_h,{{\mathbb{R}}^{d \times d}_{\mathrm{s}}})$. Define the viscous consistency error linear form $\mathcal E_{{\mathrm{a}},h}(\b w;\cdot) : {{{\und{\b{U}}}}_{h}^{k}} \to {\mathbb{R}}$ such that, for all ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h}^{k}}$, $$\label{eq:Eah} \mathcal E_{{\mathrm{a}},h}(\b w;{\und{\b{v}}}_h) \coloneqq \int_\Omega ({\b\nabla{\cdot}}{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)) \cdot \b v_h + {\mathrm{a}}_h({{\und{\b{I}}}_{h}^{k}} \b w,{\und{\b{v}}}_h).$$ Then, under Assumptions \[ass:stress\] and \[ass:sT\], we have $$\begin{gathered} \label{eq:consistency:ah} \sup_{{\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}},\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} = 1} \mathcal E_{{\mathrm{a}},h}(\b w;{\und{\b{v}}}_h) \lesssim h^{(k+1)({r^{\circ}}-1)}\bigg[ \sigma_{\mathrm{hc}}\left(\sigma_{\mathrm{de}}^{r}+ |\b w|_{W^{1,{r}}(\Omega,{\mathbb{R}}^d)}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}|\b w|_{W^{k+2,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{{r^{\circ}}-1} \\ +|{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)|_{W^{(k+1)({r^{\circ}}-1),{r}'}({\mathcal{T}}_h,{{\mathbb{R}}^{d \times d}})} \bigg]. \end{gathered}$$ Let ${\und{\b{\hat w}}}_h \coloneqq {{\und{\b{I}}}_{h}^{k}} \b w$ and ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}}$. Expanding ${\mathrm{a}}_h$ according to its definition in the expression of $\mathcal E_{{\mathrm{a}},h}$, inserting $\pm\left( \int_\Omega{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w): {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{v}}}_h + \int_\Omega{\b{\pi}_{h}^{k}}{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w) : {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{v}}}_h \right)$, and rearranging, we obtain $$\begin{gathered} \label{eq:consistency:ah:EJ} \mathcal E_{{\mathrm{a}},h}(\b w;{\und{\b{v}}}_h) =\!\! \underbrace{ \int_\Omega ({\b\nabla{\cdot}}{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)) \cdot \b v_h {+} \int_\Omega {\b{\pi}_{h}^{k}}{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)\! :\! {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{v}}}_h }_{\mathcal T_1} +\! \underbrace{ \int_\Omega\! \left( {\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w) - {\b{\pi}_{h}^{k}}{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)\right)\! :\! {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{v}}}_h }_{\mathcal T_2} \\ {+} \underbrace{ \int_\Omega\!\left( {\b\sigma}(\cdot,{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{\hat w}}}_h) - {\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)\right)\!:\! {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{v}}}_h }_{\mathcal T_3} + \underbrace{\vphantom{\int_\Omega}\gamma {\mathrm{s}}_h({\und{\b{\hat w}}}_h,{\und{\b{v}}}_h)}_{\mathcal T_4}. \end{gathered}$$ We proceed to estimate the terms in the right-hand side. For the first term, we start by noticing that $$\label{eq:consistency:ah:null} \sum_{T \in {\mathcal{T}}_h}\sum_{F \in {\mathcal{F}}_T} \int_F \b v_F \cdot\left({\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)\b n_{TF}\right) = 0$$ as a consequence of the continuity of the normal trace of ${\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)$ together with the single-valuedness of $\b v_F$ across each interface $F\in{{\mathcal{F}}_h^{\mathrm{i}}}$ and of the fact that $\b v_F=\b 0$ for every boundary face $F\in{{\mathcal{F}}_h^{\mathrm{b}}}$. Using an element by element integration by parts on the first term of $\mathcal T_1$ along with the definitions of ${\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}$ and of ${\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},T}}$, we can write $$\begin{aligned} \mathcal T_1 &= \cancel{\int_\Omega \left({\b{\pi}_{h}^{k}}{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)- {\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)\right) : {\b{\nabla}_{{\mathrm{s}},h}}\b v_h} \\ &\qquad + \sum_{T \in {\mathcal{T}}_h}\sum_{F \in {\mathcal{F}}_T} \left(\int_F (\b v_F-\b v_T)\cdot({\b{\pi}_{T}^{k}}{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w))\b n_{TF}+\int_F\b v_T\cdot\left({\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)\b n_{TF}\right) \right) \\ &= \sum_{T \in {\mathcal{T}}_h}\sum_{F \in {\mathcal{F}}_T} \int_F (\b v_F-\b v_T)\cdot\left({\b{\pi}_{T}^{k}}{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)-{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)\right)\b n_{TF}, \end{aligned}$$ where we have used the definition of ${\b{\pi}_{h}^{k}}$ together with the fact that ${\b{\nabla}_{{\mathrm{s}},h}}\b v_h \in {\mathbb{P}}^{k-1}({\mathcal{T}}_h,{{\mathbb{R}}^{d \times d}_{\mathrm{s}}}) \subset {\mathbb{P}}^{k}({\mathcal{T}}_h,{{\mathbb{R}}^{d \times d}_{\mathrm{s}}})$ to cancel the term in the first line, and we have inserted and rearranged to conclude. Therefore, applying the Hölder inequality together with the bound $h_F \le h_T$, we infer $$\label{eq:consistency:ah:T1} \begin{aligned} \left|\mathcal T_1\right| &\le \left(\displaystyle\sum_{T \in {\mathcal{T}}_h}h_T \|{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)- {\b{\pi}_{T}^{k}}{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w) \|_{L^{{r}'}(\partial T,{{\mathbb{R}}^{d \times d}})}^{{r}'} \right)^\frac{1}{{r}'}\left(\sum_{T \in {\mathcal{T}}_h}\sum_{F \in {\mathcal{F}}_T}h_F^{1-{r}}\| \b v_F-\b v_T\|_{L^{r}(F,{\mathbb{R}}^d)}^{r}\right)^\frac{1}{{r}} \\ &\lesssim h^{(k+1)({r^{\circ}}-1)} |{\b\sigma}(\cdot, {\b{\nabla}_{\mathrm{s}}}\b w)|_{W^{(k+1)({r^{\circ}}-1),{r}'}({\mathcal{T}}_h,{{\mathbb{R}}^{d \times d}})}\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}, \end{aligned}$$ where the conclusion follows using the $((k+1)({r^{\circ}}-1),{r}')$-trace approximation properties of ${\b{\pi}_{T}^{k}}$ along with $h_T \le h$ for the first factor and the definition of the $\|{\cdot}\|_{{\boldsymbol{\varepsilon}},{r},h}$-norm for the second. For the second term, we use the Hölder inequality and the seminorm equivalence to write $$\label{eq:consistency:ah:T2} \begin{aligned} \left|\mathcal T_2\right| &= \| {\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)-{\b{\pi}_{h}^{k}}{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w) \|_{L^{{r}'}(\Omega,{{\mathbb{R}}^{d \times d}})} \| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{v}}}_h\|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})} \\ &\lesssim h^{(k+1)({r^{\circ}}-1)}|{\b\sigma}(\cdot, {\b{\nabla}_{\mathrm{s}}}\b w)|_{W^{(k+1)({r^{\circ}}-1),{r}'}({\mathcal{T}}_h,{{\mathbb{R}}^{d \times d}})} \| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}, \end{aligned}$$ where the conclusion follows from the $((k+1)({r^{\circ}}-1),{r}',0)$-approximation properties of ${\b{\pi}_{T}^{k}}$ along with $h_T\le h$ for the first factor and the global norm equivalence for the second. For the third term, using the Hölder inequality and again , we get $$\label{eq:consistency:ah:T3:0} \left|\mathcal T_3\right| \le \|{\b\sigma}(\cdot,{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{\hat w}}}_h)- {\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w) \|_{L^{{r}'}(\Omega,{{\mathbb{R}}^{d \times d}})}\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}.$$ We estimate the first factor as follows: $$\begin{aligned} &\|{\b\sigma}(\cdot,{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{\hat w}}}_h)- {\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w) \|_{L^{{r}'}(\Omega,{{\mathbb{R}}^{d \times d}})} \\ &\quad \le \sigma_{\mathrm{hc}} \left\| \left(\sigma_{\mathrm{de}}^{r}+| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{\hat w}}}_h |_{d \times d}^{r}+ | {\b{\nabla}_{\mathrm{s}}}\b w |_{d \times d}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{\hat w}}}_h - {\b{\nabla}_{\mathrm{s}}}\b w |_{d \times d}^{{r^{\circ}}-1}\right\|_{L^{{r}'}(\Omega,{\mathbb{R}})} \\ &\quad \lesssim \sigma_{\mathrm{hc}} \left(\sigma_{\mathrm{de}}^{r}+\| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{\hat w}}}_h \|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}+\| {\b{\nabla}_{\mathrm{s}}}\b w \|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}\| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{\hat w}}}_h - {\b{\nabla}_{\mathrm{s}}}\b w \|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{{r^{\circ}}-1} \\ &\quad \lesssim \sigma_{\mathrm{hc}}\left(\sigma_{\mathrm{de}}^{r}+\|{\und{\b{\hat w}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}+ |\b w|_{W^{1,{r}}(\Omega,{\mathbb{R}}^d)}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}\| {\b{\pi}_{h}^{k}}({\b{\nabla}_{\mathrm{s}}}\b w) - {\b{\nabla}_{\mathrm{s}}}\b w \|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{{r^{\circ}}-1} \\ &\quad \lesssim h^{(k+1)({r^{\circ}}-1)}\sigma_{\mathrm{hc}}\left(\sigma_{\mathrm{de}}^{r}+|\b w|_{W^{1,{r}}(\Omega,{\mathbb{R}}^d)}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}|\b w|_{W^{k+2,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{{r^{\circ}}-1}, \end{aligned}$$ where we have used the Hölder continuity of ${\b\sigma}$ in the first bound, the $({r}';\frac{{r}}{{r}-{r^{\circ}}},\frac{{r}}{{r^{\circ}}-1})$-Hölder inequality in the second, the boundedness of $\Omega$ along with and the commutation property of ${\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}$ in the third, and we have concluded invoking the $(k+1,{r},0)$-approximation property of ${\b{\pi}_{T}^{k}}$. Plugging this estimate into , we get $$\label{eq:consistency:ah:T3} \left|\mathcal T_3\right| \lesssim h^{(k+1)({r^{\circ}}-1)}\sigma_{\mathrm{hc}}\left(\sigma_{\mathrm{de}}^{r}+|\b w|_{W^{1,{r}}(\Omega,{\mathbb{R}}^d)}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}|\b w|_{W^{k+2,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{{r^{\circ}}-1} \| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}.$$ Finally, using the fact that $\gamma \le \sigma_{\mathrm{hc}}$ together with the consistency of ${\mathrm{s}}_h$ and the norm equivalence , we obtain for the fourth term $$\label{eq:consistency:ah:T4} \left|\mathcal T_4\right| \lesssim h^{(k+1)({r^{\circ}}-1)}\sigma_{\mathrm{hc}}|\b w|_{W^{1,{r}}(\Omega,{\mathbb{R}}^d)}^{{r}-{r^{\circ}}}|\b w|_{W^{k+2,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{{r^{\circ}}-1} \| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}.$$ Plug the bounds , , , and into and pass to the supremum to conclude. Consistency of the pressure-velocity coupling bilinear form ----------------------------------------------------------- Let $q \in W^{1,{r}'}(\Omega,{\mathbb{R}}) \cap W^{(k+1)({r^{\circ}}-1),{r}'}({\mathcal{T}}_h,{\mathbb{R}})$. Let $\mathcal E_{{\mathrm{b}},h}(q;\cdot) : {{{\und{\b{U}}}}_{h}^{k}} \to {\mathbb{R}}$ be the pressure consistency error linear form such that, for all ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h}^{k}}$, $$\label{eq:Ebh} \mathcal E_{{\mathrm{b}},h}(q;{\und{\b{v}}}_h) \coloneqq \int_\Omega {\b\nabla}q \cdot \b v_h - {\mathrm{b}}_h({\und{\b{v}}}_h,{\pi_{h}^{k}} q).$$ Then, we have that $$\label{eq:consistency:bh} \sup\limits_{{\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}},\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} = 1} \mathcal E_{{\mathrm{b}},h}(q;{\und{\b{v}}}_h) \lesssim h^{(k+1)({r^{\circ}}-1)} |q|_{W^{(k+1)({r^{\circ}}-1),{r}'}({\mathcal{T}}_h,{\mathbb{R}})}.$$ Let ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}}$. Integrating by parts element by element, we can reformulate the first term in the right-hand side of as follows: $$\label{eq:consistency:bh:1} \displaystyle\int_\Omega {\b\nabla}q\cdot \b v_h = - \sum_{T \in {\mathcal{T}}_h} \left( \int_T q({\b\nabla{\cdot}}\b v_T) +\sum_{F \in {\mathcal{F}}_T} \int_F q(\b v_F - \b v_T)\cdot \b n_{TF} \right),$$ where the introduction of $\b v_F$ in the boundary term is justified by the fact that the jumps of $q$ vanish across interfaces by the assumed regularity and that $\b v_F=\b 0$ on every boundary face $F \in {{\mathcal{F}}_h^{\mathrm{b}}}$. On the other hand, expanding, for each $T \in {\mathcal{T}}_h$, ${{\mathrm{D}}^{k}_{T}}$ according to its definition , we get $$\label{eq:consistency:bh:2} -{\mathrm{b}}_h({\und{\b{v}}}_h, {\pi_{h}^{k}} q) = \sum_{T \in {\mathcal{T}}_h} \left( \int_T {\pi_{T}^{k}} q~({\b\nabla{\cdot}}\b v_T) +\sum_{F \in {\mathcal{F}}_T} \int_F {\pi_{T}^{k}} q~(\b v_F - \b v_T)\cdot \b n_{TF} \right).$$ Summing and and observing that the first terms in parentheses cancel out by the definition of ${\pi_{T}^{k}}$ since ${\b\nabla{\cdot}}\b v_T \in {\mathbb{P}}^{k-1}(T,{\mathbb{R}}) \subset {\mathbb{P}}^k(T,{\mathbb{R}})$ for all $T \in{\mathcal{T}}_h$, we can write $$\begin{aligned} \mathcal E_{{\mathrm{b}},h}(q;{\und{\b{v}}}_h) &= \sum_{T \in {\mathcal{T}}_h} \left( \cancel{\int_T ({\pi_{T}^{k}} q-q) ({\b\nabla{\cdot}}\b v_T)} +\sum_{F \in {\mathcal{F}}_T} \int_F ({\pi_{T}^{k}} q-q) (\b v_F - \b v_T)\cdot \b n_{TF} \right) \\ &\le \left(\sum_{T \in {\mathcal{T}}_h} h_T\| {\pi_{T}^{k}} q-q \|_{L^{{r}'}(\partial T,{\mathbb{R}})}^{{r}'} \right)^\frac{1}{{r}'} \left(\sum_{T \in {\mathcal{T}}_h}\sum_{F \in {\mathcal{F}}_T} h_F^{1-{r}}\| \b v_F-\b v_T \|_{L^{{r}}(F,{\mathbb{R}}^d)}^{{r}} \right)^\frac{1}{{r}} \\ &\lesssim h^{(k+1)({r^{\circ}}-1)} |q|_{W^{(k+1)({r^{\circ}}-1),{r}'}({\mathcal{T}}_h,{\mathbb{R}})} \| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}, \end{aligned}$$ where we have used the Hölder inequality along with $h_F\ge h_T$ whenever $F \in {\mathcal{F}}_T$ in the second line and the $((k+1)({r^{\circ}}-1),{r}')$-trace approximation property of ${\pi_{T}^{k}}$ together with the bound $h_F \le h$ and the definition of the $\|{\cdot}\|_{{\boldsymbol{\varepsilon}},{r},h}$-norm to conclude. Passing to the supremum yields . Error estimate {#error-estimate} -------------- \[thm:error.estimate\] Let $(\b u,p) \in \b U \times P$ and $({\und{\b{u}}}_h,p_h) \in {{{\und{\b{U}}}}_{h,0}^{k}} \times {P_{h}^{k}}$ solve and , respectively. Assume the additional regularity $\b u \in W^{k+2,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)$, ${\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b u) \in W^{1,{r}'}(\Omega,{{\mathbb{R}}^{d \times d}_{\mathrm{s}}}) \cap W^{(k+1)({r^{\circ}}-1),{r}'}({\mathcal{T}}_h,{{\mathbb{R}}^{d \times d}_{\mathrm{s}}})$, and $p \in W^{1,{r}'}(\Omega,{\mathbb{R}}) \cap W^{(k+1)({r^{\circ}}-1),{r}'}({\mathcal{T}}_h,{\mathbb{R}})$. Then, under Assumptions \[ass:stress\] and \[ass:sT\], \[eq:error.estimate\] $$\begin{aligned} \label{eq:error.estimate:velocity} \| {\und{\b{u}}}_h - {{\und{\b{I}}}_{h}^{k}} \b u \|_{{\boldsymbol{\varepsilon}},{r},h} &\lesssim h^\frac{(k+1)({r^{\circ}}-1)}{{r}+1-{r^{\circ}}} \left(\sigma_{\mathrm{sm}}^{-1}\mathcal N_{\b f}^{2-{r^{\circ}}}\mathcal N_{{\b\sigma},\b u,p}\right)^\frac{1}{{r}+1-{r^{\circ}}}, \\ \label{eq:error.estimate:pressure} \| p_h - {\pi_{h}^{k}} p \|_{L^{{r}'}(\Omega,{\mathbb{R}})} &\lesssim h^{(k+1)({r^{\circ}}-1)}\mathcal N_{{\b\sigma},\b u,p} + h^{\frac{(k+1)({r^{\circ}}-1)^2}{{r}+1-{r^{\circ}}}} \sigma_{\mathrm{hc}}\mathcal N_{\b f}^{|{r}-2|({r^{\circ}}-1)} \left(\sigma_{\mathrm{sm}}^{-1}\mathcal N_{{\b\sigma},\b u,p}\right)^\frac{{r^{\circ}}-1}{{r}+1-{r^{\circ}}}, \end{aligned}$$ where we have set, for the sake of brevity, $$\begin{aligned} \mathcal N_{{\b\sigma},\b u,p} &\coloneq \sigma_{\mathrm{hc}}\left(\sigma_{\mathrm{de}}^{r}+|\b u|_{W^{1,{r}}(\Omega,{\mathbb{R}}^d)}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}|\b u|_{W^{k+2,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{{r^{\circ}}-1} \\ &\quad + |{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b u)|_{W^{(k+1)({r^{\circ}}-1),{r}'}({\mathcal{T}}_h,{{\mathbb{R}}^{d \times d}})} + |p|_{W^{(k+1)({r^{\circ}}-1),{r}'}({\mathcal{T}}_h,{\mathbb{R}})}, \\ \mathcal N_{\b f} &\coloneqq \sigma_{\mathrm{de}}+\left(\sigma_{\mathrm{sm}}^{-1}\| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)}\right)^\frac{1}{{r}-1}+\left(\sigma_{\mathrm{de}}^{2-{r^{\circ}}}\sigma_{\mathrm{sm}}^{-1}\| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)}\right)^\frac{1}{{r}+1-{r^{\circ}}}. \end{aligned}$$ \[rem:ocv\] From , neglecting higher-order terms, we infer asymptotic convergence rates of $\mathcal O_{\mathrm{vel}}^k \coloneqq \frac{(k+1)({r^{\circ}}-1)}{{r}+1-{r^{\circ}}}$ for the velocity and $\mathcal O_{\mathrm{pre}}^k \coloneqq \frac{(k+1)({r^{\circ}}-1)^2}{{r}+1-{r^{\circ}}}$ for the pressure, that is, $$\label{eq:asymptotic.order} \mathcal O_{\mathrm{vel}}^k = \begin{cases} (k+1)(r-1) & \text{if $r<2$}, \\ \tfrac{k+1}{r-1} & \text{if $r\ge 2$,} \end{cases}\quad \text{and} \quad \mathcal O_{\mathrm{pre}}^k = \begin{cases} (k+1)(r-1)^2 & \text{if $r<2$}, \\ \tfrac{k+1}{r-1} & \text{if $r\ge 2$.} \end{cases}$$ Notice that, owing to the presence of higher-order terms in the right-hand sides of , higher convergence rates may be observed before attaining the asymptotic ones; see Section \[sec:num.res\]. Let $({\und{\b{e}}}_h, \epsilon_h) \coloneqq ({\und{\b{u}}}_h - {\und{\b{\hat u}}}_h,p_h - \hat p_h) \in {{{\und{\b{U}}}}_{h,0}^{k}} \times {P_{h}^{k}}$ where ${\und{\b{\hat u}}}_h \coloneqq{{\und{\b{I}}}_{h}^{k}} \b u \in {{{\und{\b{U}}}}_{h,0}^{k}}$ and $\hat p_h \coloneqq {\pi_{h}^{k}} p \in {P_{h}^{k}}$.\ **Step 1.** *Consistency error.* Let $\mathcal E_h : {{{\und{\b{U}}}}_{h,0}^{k}} \to {\mathbb{R}}$ be the consistency error linear form such that, for all ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}}$, $$\label{eq:Eh} \mathcal E_h({\und{\b{v}}}_h) \coloneqq \int_\Omega \b f \cdot \b v_h - {\mathrm{a}}_h({\und{\b{\hat u}}}_h,{\und{\b{v}}}_h)-{\mathrm{b}}_h({\und{\b{v}}}_h,{\hat p}_h).$$ Using in the above expression the fact that $\b f = -{\b\nabla{\cdot}}{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b u)+{\b\nabla}p$ almost everywhere in $\Omega$ to write $\mathcal E_h({\und{\b{v}}}_h)=\mathcal E_{{\mathrm{a}},h}(\b u;{\und{\b{v}}}_h) + \mathcal E_{{\mathrm{b}},h}(p;{\und{\b{v}}}_h)$, and invoking the consistency properties of ${\mathrm{a}}_h$ and of ${\mathrm{b}}_h$, we obtain $$\label{eq:error.estimate:step1:eh0} \$\coloneq \sup\limits_{{\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}},\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} = 1}\mathcal E_h({\und{\b{v}}}_h) \lesssim h^{(k+1)({r^{\circ}}-1)} \mathcal N_{{\b\sigma},\b u,p}.$$\ **Step 2.** *Error estimate for the velocity.* Using the strong monotonicity of ${\mathrm{a}}_h$, we get $$\label{eq:error.estimate:step2:eh0} \begin{aligned} \| {\und{\b{e}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{{r}+2-{r^{\circ}}} &\lesssim \sigma_{\mathrm{sm}}^{-1}\left( \sigma_{\mathrm{de}}^{r}+\| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}+\|{\und{\b{\hat u}}}_h\|_{{\boldsymbol{\varepsilon}},{r},h}^{r}\right)^\frac{2-{r^{\circ}}}{{r}}\left( {\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{e}}}_h)-{\mathrm{a}}_h({\und{\b{\hat u}}}_h,{\und{\b{e}}}_h) \right) \\ &\lesssim \sigma_{\mathrm{sm}}^{-1}\mathcal N_{\b f}^{2-{r^{\circ}}}\left( {\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{e}}}_h)-{\mathrm{a}}_h({\und{\b{\hat u}}}_h,{\und{\b{e}}}_h) \right), \end{aligned}$$ where we have used the a priori bound on the discrete solution along with the boundedness of the global interpolator and the a priori bound on the continuous solution to conclude. Using then the discrete mass equation along with (written for $\b v=\b u$) and the continuous mass equation to write ${\mathrm{b}}_h({{\und{\b{I}}}_{h}^{k}} \b u,q_h) = b(\b u,q_h) = 0$, we get ${\mathrm{b}}_h({\und{\b{e}}}_h,q_h) = 0$ for all $q_h \in P^k_h$. Hence, combining this result with and the discrete momentum equation (with ${\und{\b{v}}}_h = {\und{\b{e}}}_h$), we obtain $$\label{eq:error.estimate:step1:eh1} {\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{e}}}_h)-{\mathrm{a}}_h({\und{\b{\hat u}}}_h,{\und{\b{e}}}_h) = \int_\Omega \b f \cdot \b e_h - {\mathrm{a}}_h({\und{\b{\hat u}}}_h,{\und{\b{e}}}_h)-\cancel{{\mathrm{b}}_h({\und{\b{e}}}_h,p_h)} = \mathcal E_h({\und{\b{e}}}_h).$$ Plugging into , we get $$\| {\und{\b{e}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{{r}+2-{r^{\circ}}} \le \sigma_{\mathrm{sm}}^{-1}\mathcal N_{\b f}^{2-{r^{\circ}}}\$\| {\und{\b{e}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}.$$ Simplifying, using , and taking the $({r}+1-{r^{\circ}})$th root of the resulting inequality yields .\ **Step 3.** *Error estimate for the pressure.* Using the Hölder continuity of ${\mathrm{a}}_h$, we have, for all ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}}$, $$\label{eq:error.estimate:step3:ah} \begin{aligned} \left|{\mathrm{a}}_h({\und{\b{\hat u}}}_h,{\und{\b{v}}}_h)-{\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{v}}}_h)\right| &\lesssim \sigma_{\mathrm{hc}}\left( \sigma_{\mathrm{de}}^{r}+ \| {\und{\b{\hat u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}+\| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}\| {\und{\b{e}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{{r^{\circ}}-1}\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} \\ &\lesssim \sigma_{\mathrm{hc}}\mathcal N_{\b f}^{{r}-{r^{\circ}}}\| {\und{\b{e}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{{r^{\circ}}-1}\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}, \end{aligned}$$ where the first factor is estimated as in . Thus, using the inf-sup condition , we can write $$\label{eq:error.estimate:step3:epsilonh} \begin{aligned} \| \epsilon_h \|_{L^{{r}'}(\Omega,{\mathbb{R}})} &\lesssim \sup\limits_{{\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}},\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} = 1} {\mathrm{b}}_h({\und{\b{v}}}_h,\epsilon_h) \\ &= \sup\limits_{{\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}},\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} = 1} \left(\mathcal E_h({\und{\b{v}}}_h)+{\mathrm{a}}_h({\und{\b{\hat u}}}_h,{\und{\b{v}}}_h)-{\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{v}}}_h)\right)\\ &\lesssim \$+\sigma_{\mathrm{hc}}\mathcal N_{\b f}^{{r}-{r^{\circ}}}\| {\und{\b{e}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{{r^{\circ}}-1} \\ &\lesssim h^{(k+1)({r^{\circ}}-1)}\mathcal N_{{\b\sigma},\b u,p} +h^{(k+1)({r^{\circ}}-1)^2}\sigma_{\mathrm{hc}}\mathcal N_{\b f}^{|{r}-2|({r^{\circ}}-1)} \left(\sigma_{\mathrm{sm}}^{-1}\mathcal N_{{\b\sigma},\b u,p}\right)^\frac{{r^{\circ}}-1}{{r}+1-{r^{\circ}}}, \end{aligned}$$ where we have used the definition of the consistency error together with equation to pass to the second line, to pass to the third line (recall that $\$$ denotes here the supremum in the left-hand side of ), and the bounds and (proved in Step 2) to conclude. Numerical examples {#sec:num.res} ================== We consider a manufactured solution to problem in order to assess the convergence of the method, which was implemented within the SpaFEDTe library (cf. <https://spafedte.github.io>). Specifically, we take $\Omega=(0,1)^{2}$ and consider the $(1,0,1,{r})$-Carreau–Yasuda law (corresponding to the power-law model) with Sobolev exponent ${r}\in\{1.5, 1.75, 2, 2.25, 2.5, 2.75\}$. The exact velocity $\b u$ and pressure $p$ are given by, respectively, $$\b u(x,y) = \left(\sin\left(\tfrac{\pi}{2}x\right)\cos\left(\tfrac{\pi}{2}y\right),-\cos\left(\tfrac{\pi}{2}x\right)\sin\left(\tfrac{\pi}{2}y\right)\right),\quad p(x,y) = \sin\left(\tfrac{\pi}{2} x\right)\sin\left(\tfrac{\pi}{2} y\right)-\tfrac{4}{\pi^2}.$$ The volumetric load $\b f$ and the Dirichlet boundary conditions are inferred from the exact solution. This solution matches the assumptions required in Theorem \[thm:error.estimate\] for $k = 1$, except the case $r = 1.5$ for which ${\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b u) \notin W^{1,{r}'}(\Omega,{{\mathbb{R}}^{d \times d}_{\mathrm{s}}})$. We consider the HHO scheme for $k=1$ on three mesh families, namely Cartesian orthogonal, distorted triangular, and distorted Cartesian; see Figure \[fig:meshes\]. Overall, the results are in agreement with the theoretical predictions, and in some cases the expected asymptotic orders of convergence are exceeded. Specifically, for ${r}\neq 2$, the convergence rates computed on the last refinement surpass in some cases the theoretical ones. As noticed in Remark \[rem:ocv\], this suggests that the asymptotic order is still not attained. A similar phenomenon has been observed on certain meshes for the $p$-Laplace problem; see [@Di-Pietro.Droniou:17*1 Section 3.5.2] and [@Di-Pietro.Droniou.ea:18 Section 3.7]. ![Coarsest Cartesian, distorted triangular, and distorted Cartesian mesh used in the numerical tests of Section \[sec:num.res\].\[fig:meshes\]](square.pdf "fig:") ![Coarsest Cartesian, distorted triangular, and distorted Cartesian mesh used in the numerical tests of Section \[sec:num.res\].\[fig:meshes\]](disttri.pdf "fig:") ![Coarsest Cartesian, distorted triangular, and distorted Cartesian mesh used in the numerical tests of Section \[sec:num.res\].\[fig:meshes\]](distsquare.pdf "fig:") ![Numerical results for the test case of Section \[sec:num.res\]. The slopes indicate the expected order of convergence expected from Theorem \[thm:error.estimate\], i.e. $\mathcal O_{\mathrm{vel}}^1 = 2(r-1)$ and $\mathcal O_{\mathrm{pre}}^1 = 2(r-1)^2$ for ${r}\in \{1.5,1.75,2\}$. \[tab:num.res.1\]](conv1.pdf) ![Numerical results for the test case of Section \[sec:num.res\]. The slopes indicate the expected order of convergence expected from Theorem \[thm:error.estimate\], i.e. $\mathcal O_{\mathrm{vel}}^1 = \mathcal O_{\mathrm{pre}}^1 = \frac{2}{r-1}$ for ${r}\in \{2.25,2.5,2.75\}$. \[tab:num.res.2\]](conv2.pdf) Power-framed functions {#sec:properties.stress} ====================== In the following theorem, we introduce the notion of power-framed function and discuss sufficient conditions for this property to hold. \[thm:1d.power-framed\] Let $U$ be a measurable subset of ${\mathbb{R}}^n$ with $n\ge1$, $(W,(\cdot,\cdot)_W)$ an inner product space, and $\b \sigma : U \times W \to W$. Assume that there exists a Carathéodory function $\varsigma : U \times \lbrack0,+\infty) \to {\mathbb{R}}$ such that, for all $\b\tau \in W$ and almost every $\b x \in U$, \[eq:1d.power-framed:stress\] $${\b\sigma}(\b x,\b\tau) = \varsigma(\b x,\|\b\tau\|_W)\b\tau,$$ where $\|{\cdot}\|_W$ is the norm induced by $(\cdot,\cdot)_W$. Additionally assume that, for almost every $\b x \in U$, $\varsigma(\b x,\cdot)$ is differentiable on $(0,+\infty)$ and there exist $\varsigma_{\mathrm{de}} \in \lbrack0,+\infty)$ and $\varsigma_{\mathrm{sm}},\varsigma_{\mathrm{hc}} \in (0,+\infty)$ independent of $\b x$ such that, for all $\alpha \in (0,+\infty)$, $$\begin{aligned} \varsigma_{\mathrm{sm}} (\varsigma_{\mathrm{de}}^{r}+\alpha^{r})^\frac{{r}-2}{{r}} \leq \frac{\partial(\alpha\varsigma(\b x,\alpha))}{\partial \alpha} \leq \varsigma_{\mathrm{hc}}(\varsigma_{\mathrm{de}}^{r}+\alpha^{r})^\frac{{r}-2}{{r}}. \label{eq:1d.power-framed:eta} \end{aligned}$$ Then, ${\b\sigma}$ is an *${r}$-power-framed function*, i.e., for all $(\b\tau,\b\eta) \in W^2$ with $\b\tau \neq \b\eta$ and almost every $\b x \in U$, the function ${\b\sigma}$ verifies the Hölder continuity property \[eq:od:power-framed:holder.continuity.strong.monotonicity\] $$\|{\b\sigma}(\b x,\b\tau)-{\b\sigma}(\b x,\b\eta)\|_W \le \sigma_{\mathrm{hc}} \left(\sigma_{\mathrm{de}}^{r}+\|\b\tau\|_W^{r}+\|\b\eta\|_W^{r}\right)^\frac{{r}-2}{{r}}\| \b\tau-\b\eta \|_W,\label{eq:od:power-framed:holder.continuity}$$ and the strong monotonicity property $$\left({\b\sigma}(\b x,\b\tau)-{\b\sigma}(\b x,\b\eta),\b\tau-\b\eta\right)_W \ge \sigma_{\mathrm{sm}}\left(\sigma_{\mathrm{de}}^{r}+\|\b\tau\|_W^{r}+\|\b\eta\|_W^{r}\right)^\frac{{r}-2}{{r}}\|\b\tau-\b\eta\|_W^{2},\label{eq:od:power-framed:strong.monotonicity}$$ with $\sigma_{\mathrm{de}} \coloneqq \varsigma_{\mathrm{de}}$, $\sigma_{\mathrm{hc}} \coloneqq 2^{2-{r^{\circ}}+{r}^{-1}\left\lceil\hspace{0.02cm}2-{r^{\circ}}\right\rceil}({r^{\circ}}-1)^{-1} \varsigma_{\mathrm{hc}}$, and $\sigma_{\mathrm{sm}} \coloneqq 2^{{r^{\circ}}-{r}-\left\lceil{r}^{-1}({r}-{r^{\circ}})\right\rceil}({r}+1-{r^{\circ}})^{-1} \varsigma_{\mathrm{sm}}$, where ${r^{\circ}}$ is given by and $\lceil{\cdot}\rceil$ is the ceiling function. The boldface notation for the elements of $W$ is reminescent of the fact that Theorem \[thm:1d.power-framed\] is used with $W = {{\mathbb{R}}^{d \times d}_{\mathrm{s}}}$ in Corollary \[cor:Carreau–Yasuda\] to characterize the Carreau-Yasuda law as an ${r}$-power-framed function and in Lemma \[lem:sT\] with $W = {\mathbb{R}}^d$ to study the local stabilization function ${\mathrm{s}}_T$. Let $\b x \in U$ be such that holds, and $\b\tau,\b\eta \in W$. By symmetry of inequalities and the fact that ${\b\sigma}$ is continuous, we can assume, without loss of generality, that $\|\b\tau\|_W > \|\b\eta\|_W > 0$.\ (i) *Strong monotonicity.* Let $\beta \in (0,+\infty)$ and let $g : \lbrack\beta,+\infty) \to {\mathbb{R}}$ be such that, for all $\alpha \in \lbrack\beta,+\infty)$, $$g(\alpha) \coloneqq \alpha\varsigma(\b x,\alpha)-\beta\varsigma(\b x,\beta)-C_{\mathrm{sm}}(\varsigma_{\mathrm{de}}^{r}+\alpha^{r}+\beta^{r})^\frac{{r}-2}{{r}}(\alpha-\beta), \;\text{ with }\ C_{\mathrm{sm}} \coloneqq \tfrac{2^{{r^{\circ}}-{r}}}{{r}+1-{r^{\circ}}}\varsigma_{\mathrm{sm}}.$$ Differentiating $g$ and using the first inequality in , we obtain, for all $\alpha \in \lbrack\beta,+\infty)$, $$\begin{aligned} \frac{\partial}{\partial \alpha}g(\alpha) &\geq \varsigma_{\mathrm{sm}}(\varsigma_{\mathrm{de}}^{r}+\alpha^{r})^\frac{{r}-2}{{r}}-C_{\mathrm{sm}}\left(({r}-2)(\varsigma_{\mathrm{de}}^{r}+\alpha^{r}+\beta^{r})^{-\frac{2}{{r}}}(\alpha-\beta)\alpha^{{r}-1}+(\varsigma_{\mathrm{de}}^{r}+\alpha^{r}+\beta^{r})^\frac{{r}-2}{{r}}\right) \\ &\geq \varsigma_{\mathrm{sm}}(\varsigma_{\mathrm{de}}^{r}+\alpha^{r})^\frac{{r}-2}{{r}}-({r}+1-{r^{\circ}})C_{\mathrm{sm}} (\varsigma_{\mathrm{de}}^{r}+\alpha^{r}+\beta^{r})^\frac{{r}-2}{{r}} \\ &\geq \varsigma_{\mathrm{sm}}2^{{r^{\circ}}-{r}}(\varsigma_{\mathrm{de}}^{r}+\alpha^{r}+\beta^{r})^\frac{{r}-2}{{r}}-({r}+1-{r^{\circ}})C_{\mathrm{sm}} (\varsigma_{\mathrm{de}}^{r}+\alpha^{r}+\beta^{r})^\frac{{r}-2}{{r}} = 0, \end{aligned}$$ where, to pass to the second line, we have removed negative contributions if ${r}< 2$ and used the fact that $(\alpha-\beta)\alpha^{{r}-1} \le \varsigma_{\mathrm{de}}^{r}+\alpha^{r}+\beta^{r}$ if ${r}\ge 2$, to pass to the third line we have used the fact that $t \mapsto t^{{r}-2}$ is non-increasing if ${r}< 2$, and the fact that $\beta \le \alpha$ otherwise, while the conclusion follows from the definition of $C_{\mathrm{sm}}$. This shows that $g$ is non-decreasing. Hence, for all $\alpha\in\lbrack\beta,+\infty)$, $g(\alpha)\geq g(\beta)=0$, i.e. $$\label{eq:1d.power-framed:bound:1} \alpha\varsigma(\b x,\alpha)-\beta\varsigma(\b x,\beta) \geq C_{\mathrm{sm}}(\varsigma_{\mathrm{de}}^{r}+\alpha^{r}+\beta^{r})^\frac{{r}-2}{{r}}(\alpha-\beta).$$ Moreover, for all $\alpha,\beta \in (0,+\infty)$, using (with $\beta = 0$) along with the fact that $t \mapsto t^{{r}-2}$ is decreasing if ${r}< 2$ and inequality if ${r}\ge 2$, we infer that $$\label{eq:1d.power-framed:bound:2} \begin{aligned} \varsigma(\b x,\alpha)+\varsigma(\b x,\beta) &\ge C_{\mathrm{sm}}\left((\varsigma_{\mathrm{de}}^{r}+\alpha^{r})^\frac{{r}-2}{{r}}+(\varsigma_{\mathrm{de}}^{r}+\beta^{r})^\frac{{r}-2}{{r}}\right) \ge C_{\mathrm{sm}}2^{1-\left\lceil\frac{{r}-{r^{\circ}}}{{r}}\right\rceil}(\varsigma_{\mathrm{de}}^{r}+\alpha^{r}+\beta^{r})^\frac{{r}-2}{{r}}. \end{aligned}$$ We conclude that ${\b\sigma}$ verifies by using and with $\alpha = \|\b\tau\|_W$ and $\beta = \|\b\eta\|_W$ as follows: $$\begin{aligned} &({\b\sigma}(\b x,\b\tau)-{\b\sigma}(\b x,\b\eta),\b\tau-\b\eta)_W \\ &\quad= (\b\tau\varsigma(\b x,\|\b\tau\|_W)-\b\eta\varsigma(\b x,\|\b\eta\|_W),\b\tau-\b\eta)_W \\ &\quad = \|\b\tau\|_W^2\varsigma(\b x,\|\b\tau\|_W)+\|\b\eta\|_W^2\varsigma(\b x,\|\b\eta\|_W) -(\b\tau,\b\eta)_W \left[\varsigma(\b x,\|\b\tau\|_W) + \varsigma(\b x,\|\b\eta\|_W)\right] \\ &\quad = \left[ \|\b\tau\|_W\varsigma(\b x,\|\b\tau\|_W)-\|\b\eta\|_W\varsigma(\b x,\|\b\eta\|_W) \right](\|\b\tau\|_W-\|\b\eta\|_W) \\ &\qquad + \left[ \varsigma(\b x,\|\b\tau\|_W)+\varsigma(\b x,\|\b\eta\|_W) \right](\|\b\tau\|_W\|\b\eta\|_W-(\b\tau,\b\eta)_W) \\ &\quad\geq C_{\mathrm{sm}}2^{-\left\lceil\frac{{r}-{r^{\circ}}}{{r}}\right\rceil}\left( \varsigma_{\mathrm{de}}^{r}+\|\b\tau\|_W^{r}+\|\b\eta\|_W^{r}\right)^\frac{{r}-2}{{r}}\left[ (\|\b\tau\|_W-\|\b\eta\|_W)^2+2(\|\b\tau\|_W\|\b\eta\|_W-(\b\tau,\b\eta)_W) \right] \\ &\quad= C_{\mathrm{sm}}2^{-\left\lceil\frac{{r}-{r^{\circ}}}{{r}}\right\rceil}\left( \varsigma_{\mathrm{de}}^{r}+\|\b\tau\|_W^{r}+\|\b\eta\|_W^{r}\right)^\frac{{r}-2}{{r}}\|\b\tau-\b\eta\|_W^2. \end{aligned}$$\ (ii) *Hölder continuity.* Now, setting $C_{\mathrm{hc}} \coloneqq \frac{\varsigma_{\mathrm{hc}}}{{r^{\circ}}-1}$ and reasoning in a similar way as for the proof of to leverage the second inequality in , we have, for all $\alpha \in \lbrack\beta,+\infty)$, $$\label{eq:1d.power-framed:bound:3} \alpha\varsigma(\b x,\alpha)-\beta\varsigma(\b x,\beta) \le C_{\mathrm{hc}}\left( \varsigma_{\mathrm{de}}^{r}+\alpha^{r}+\beta^{r}\right)^\frac{{r}-2}{{r}}(\alpha-\beta).$$ First, let ${r}\ge 2$. Using (with $\beta=0$) and the fact that $t \mapsto t^{{r}-2}$ is non-decreasing, we have, for all $\alpha,\beta \in (0,+\infty)$, $$\label{eq:1d.power-framed:bound:4} \varsigma(\b x,\alpha)\varsigma(\b x,\beta) \le C_{\mathrm{hc}}^2\left( \varsigma_{\mathrm{de}}^{r}+\alpha^{r}\right)^\frac{{r}-2}{{r}}\left( \varsigma_{\mathrm{de}}^{r}+\beta^{r}\right)^\frac{{r}-2}{{r}} \le \left[ C_{\mathrm{hc}}\left(\varsigma_{\mathrm{de}}^{r}+\alpha^{r}+\beta^{r}\right)^\frac{{r}-2}{{r}} \right]^2.$$ Thus, using inequalities and with $\alpha = \|\b\tau\|_W$ and $\beta = \|\b\eta\|_W$, we infer $$\label{eq:1d.power-framed:bound:5} \begin{aligned} &\|{\b\sigma}(\b x,\b\tau)-{\b\sigma}(\b x,\b\eta)\|_W^2 \\ &\quad = \left( \b\tau\varsigma(\b x,\|\b\tau\|_W)-\b\eta\varsigma(\b x,\|\b\eta\|_W), \b\tau\varsigma(\b x,\|\b\tau\|_W)-\b\eta\varsigma(\b x,\|\b\eta\|_W) \right)_W \\ &\quad = \left[\|\b\tau\|_W\varsigma(\b x,\|\b\tau\|_W)-\|\b\eta\|_W\varsigma(\b x,\|\b\eta\|_W)\right]^2 \\ &\quad \qquad + 2\varsigma(\b x,\|\b\tau\|_W)\varsigma(\b x,\|\b\eta\|_W)\left[ \|\b\tau\|_W\|\b\eta\|_W-(\b\tau,\b\eta)_W \right] \\ &\quad \le \left[ C_{\mathrm{hc}}\left(\varsigma_{\mathrm{de}}^{r}+\|\b\tau\|_W^{r}+\|\b\eta\|_W^{r}\right)^\frac{{r}-2}{{r}} \right]^2\left[ (\|\b\tau\|_W-\|\b\eta\|_W)^2+2(\|\b\tau\|_W\|\b\eta\|_W-(\b\tau,\b\eta)_W) \right] \\ &\quad = \left[ C_{\mathrm{hc}}\left(\varsigma_{\mathrm{de}}^{r}+\|\b\tau\|_W^{r}+\|\b\eta\|_W^{r}\right)^\frac{{r}-2}{{r}}\|\b\tau-\b\eta\|_W \right]^2, \end{aligned}$$ hence ${\b\sigma}$ verifies for ${r}\ge 2$. Assume now ${r}< 2$. Using a triangle inequality followed by and the left inequality in , it is inferred that $$\begin{aligned} \|{\b\sigma}(\b x,\b\tau)-{\b\sigma}(\b x,\b\eta)\|_W &\leq \varsigma(\b x,\|\b\tau\|_W)\|\b\tau\|_W+\varsigma(\b x,\|\b\eta\|_W)\|\b\eta\|_W \\ & \leq C_{\mathrm{hc}}\left((\varsigma_{\mathrm{de}}^{r}+\|\b\tau\|_W^{r})^\frac{{r}-1}{{r}}+(\varsigma_{\mathrm{de}}^{r}+\|\b\eta\|_W^{r})^\frac{{r}-1}{{r}}\right) \\ &\leq 2^\frac{1}{{r}}C_{\mathrm{hc}}(2\varsigma_{\mathrm{de}}^{r}+\|\b\tau\|_W^{r}+\|\b\eta\|_W^{r})^\frac{{r}-1}{{r}} \\ & = 2^\frac{1}{{r}} C_{\mathrm{hc}}(2\varsigma_{\mathrm{de}}^{r}+\|\b\tau\|_W^{r}+\|\b\eta\|_W^{r})^\frac{{r}-2}{{r}} (2\varsigma_{\mathrm{de}}^{r}+\|\b\tau\|_W^{r}+\|\b\eta\|_W^{r})^\frac1{r}, \\ & \leq 2^\frac{1}{{r}} C_{\mathrm{hc}}(\varsigma_{\mathrm{de}}^{r}+\|\b\tau\|_W^{r}+\|\b\eta\|_W^{r})^\frac{{r}-2}{{r}} (2\varsigma_{\mathrm{de}}+\|\b\tau\|_W+\|\b\eta\|_W), \end{aligned}$$ where the last line follows from the fact that $t \mapsto t^{{r}-2}$ is decreasing and again . If $2\varsigma_{\mathrm{de}}+\|\b\tau\|_W+\|\b\eta\|_W\le 2^{2-r}\|\b\tau-\b\eta\|_W$, from the previous bound we directly get the conclusion, i.e. with $\sigma_{\mathrm{hc}}=2^{2-r+\frac{1}{r}}C_{\mathrm{hc}}$. Otherwise, using and a triangle inequality yields $$\label{eq:est_else} \begin{aligned} (\varsigma_{\mathrm{de}}^{r}+\|\b\tau\|_W^{r})^\frac{1}{{r}}(\varsigma_{\mathrm{de}}^{r}+\|\b\eta\|_W^{r})^\frac{1}{{r}} &\ge 2^{-\frac{2}{r'}}(\varsigma_{\mathrm{de}}+\|\b\tau\|_W)(\varsigma_{\mathrm{de}}+\|\b\eta\|_W) \\ &= 2^{-2(\frac{1}{r'}+1)}\left[ \left(2\varsigma_{\mathrm{de}}+\|\b\tau\|_W+\|\b\eta\|_W\right)^2-\left(\|\b\tau\|_W-\|\b\eta\|_W\right)^2 \right] \\ &\ge 2^{-2(\frac{1}{r'}+1)}\left[ \left(2\varsigma_{\mathrm{de}}+\|\b\tau\|_W+\|\b\eta\|_W\right)^2-\|\b\tau-\b\eta\|_W^2 \right] \\ &\ge 2^{-2(\frac{1}{r'}+1)}(1-4^{r-2})\left(2\varsigma_{\mathrm{de}}+\|\b\tau\|_W+\|\b\eta\|_W\right)^2\\ &\ge 2^{\frac{2}{(r-2)r}-2}\left(\varsigma_{\mathrm{de}}^{r}+\|\b\tau\|_W^{r}+\|\b\eta\|_W^{r}\right)^\frac{2}{{r}}, \end{aligned}$$ where we concluded with together with the fact that $2^{-2(\frac{1}{r'}+1)}\left(1-4^{r-2}\right) \ge 2^{\frac{2}{(r-2)r}-2}$. Finally, raising both sides of to the power ${r}-2$, we get a relation analogous to . Hence, proceeding as in , we infer . \[cor:Carreau–Yasuda\] The strain rate-shear stress law of the $(\mu,\delta,a,{r})$-Carreau–Yasuda fluid defined in Example \[ex:Carreau–Yasuda\] is an ${r}$-power-framed function. Let $\b x \in \Omega$ and $g : (0,+\infty) \to {\mathbb{R}}$ be such that, for all $\alpha\in(0,+\infty)$, $$g(\alpha) \coloneqq \frac{\partial}{\partial \alpha}\left[ \alpha\mu(\b x)\left(\delta^{a(\b x)}+\alpha^{a(\b x)} \right)^\frac{{r}-2}{a(\b x)} \right] = \mu(\b x)\left( \delta^{a(\b x)}+\alpha^{a(\b x)} \right)^{\frac{{r}-2}{a(\b x)}-1}\left( \delta^{a(\b x)}+({r}-1)\alpha^{a(\b x)} \right).$$ We have for all $\alpha \in (0,+\infty)$, $$\mu_- ({r^{\circ}}-1) \left( \delta^{a(\b x)}+\alpha^{a(\b x)} \right)^{\frac{{r}-2}{a(\b x)}} \le g(\alpha) \leq \mu_+ ({r}+1-{r^{\circ}})\left( \delta^{a(\b x)}+\alpha^{a(\b x)} \right)^{\frac{{r}-2}{a(\b x)}},$$ and we conclude using together with Theorem \[thm:1d.power-framed\]. [^1]: [<[email protected]>]{} [^2]: [<[email protected]>]{} [^3]: [<[email protected]>]{} [^4]: [<[email protected]>]{}, corresponding author

`Cavendish-HEP-2000/11`\ `CERN-TH/2000-354`\ `MC-TH-00/11`\ `hep-ph/0011047`\ \ [J.R. Forshaw$^{1}$, D.A. Ross${^2}$ and A. Sabio Vera${^3}$]{}\ \ [*[${^1}$Department of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, U.K.\ ${^2}$ Division Théorique, CERN, CH1211-Geneva 23, Switzerland [^1]\ $^{3}$ Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge, CB3 0HE, U.K. ]{}*]{}\ [We describe a formalism for solving the BFKL equation with a coupling that runs for momenta above a certain infrared cutoff. By suitably choosing matching conditions proper account is taken of the fact that the BFKL diffusion implies that the solution in the infrared (fixed coupling) regime depends upon the solution in the ultraviolet (running coupling) regime and vice versa. Expanding the BFKL kernel to a given order in the ratio of the transverse momenta allows arbitrary accuracy to be achieved.]{} Diffractive processes are described, at least within the framework of perturbative QCD, by the BFKL equation [@bfkl]. This equation sums the leading and next-to-leading $\ln s$ [@BFKL2] terms for the amplitude involving the exchange in the $t-$channel with the quantum numbers of the vacuum. In the case of fixed coupling, the energy dependence of these diffractive processes is determined by the emergence of a branch cut in the Mellin transform plane ($\omega-$plane), with a branch point at $\omega > 0$. Lipatov [@lipatov86] has demonstrated that accounting for the running of the strong coupling together with some (non-perturbative) information about the infrared behaviour of QCD leads to a separation of the $\omega-$plane singularity structure into a series of isolated poles. More recently, Thorne [@thorne] has shown that if one uses the solution to the BFKL equation with running coupling, the factor in the amplitude which is totally calculable in perturbative QCD has an essential singularity at $\omega=0$, such that there is no power-like energy dependence. This is in accord with the recent work of Ciafaloni, Colferai and Salam, who have shown that the leading $\omega$-plane singularity is of non-perturbative origin [@ccs]. The BFKL equation is a diffusion equation and so there will always be some contribution from the infrared region. One way of treating this region is to assume that the strong coupling ceases to run below a certain scale. In this case, we expect that for sufficiently low values of the transverse momenta of the exchanged gluons, the $s-$behaviour will be dominated by the fixed coupling solution, whereas for large enough transverse momenta the solution of [@thorne] will dominate. We should like to understand the smooth extrapolation between these two extremes. In this letter we describe a formalism for solving the BFKL equation with strong coupling running above some infrared scale, in which we can see how the transition from the fixed coupling to the running coupling solution arises. [**Formalism for Solving the BFKL Equation with Running Coupling**]{} To begin with we consider only the leading order BFKL kernel and describe later how the formalism may be extended in order to include the substantial part of the higher order corrections. Since the BFKL equation is a diffusion equation there is always diffusion into the infrared regime, where renormalization group improved perturbation theory breaks down and one needs supplementary information about the behaviour of QCD beyond perturbation theory. For our purposes we simulate this infrared behaviour by assuming that the coupling freezes below a critical value of $t \equiv \ln(k^2/\Lambda_{QCD}^2), \ t=t_0$. We also make the assumption that $\alpha_s(t_0)$ is sufficiently small that perturbative results in the infrared regime maintain some level of credibility. The full BFKL equation is solved in terms of a complete set of eigenfunctions $f_\omega(t)$ of the kernel with eigenvalue $\omega$. For any $\omega$ there will be some diffusion into the infrared regime. We therefore write $f_\omega(t)$ as f\_(t)  =  f\_\^&lt;(t) (t\_0-t) + f\_\^&gt;(t) (t-t\_0) and the kernel as $$\begin{aligned} K(t,t^\prime) &=& \left[ \frac{1}{b \, t_0} {\cal K}^<(t,t^\prime) \, \theta(t_0-t') + \frac{1}{b \, t }{\cal K}^<(t,t^\prime) \, \theta(t'-t_0) \right] \, \theta(t-t^\prime) \ + \nonumber \\ & & \ \left[ \frac{1}{b \, t_0} {\cal K}^>(t,t^\prime) \, \theta(t_0-t') + \frac{1}{b \, t^\prime }{\cal K}^>(t,t^\prime) \, \theta(t'-t_0) \right] \, \theta(t^\prime-t) \end{aligned}$$ where $b=\beta_0 /12$, $\beta_0=11-2n_f/3$, being the first coefficient of the $\beta-$function for the running of the QCD coupling. We define $ {\cal K}^<(t,t^\prime)$ and $ {\cal K}^>(t,t^\prime)$ as the limits of sums, i.e. $${\cal K}^<(t,t^\prime) \ = \ \lim_{N \to \infty} {\cal K}^<_N(t,t^\prime)$$ $${\cal K}^>(t,t^\prime) \ = \ \lim_{N \to \infty} {\cal K}^>_N(t,t^\prime)$$ where \^&lt;\_N(t,t\^)  =  \_[r=0]{}\^N { e\^[(r+1/2)(t\^-t)]{}- (t-t\^) } \^&gt;\_N(t,t\^)  =  \_[r=0]{}\^N { e\^[(r+1/2)(t-t\^)]{}- (t-t\^) }. The functions $e^{\gamma \, t}$ are eigenfunctions of these kernels with eigenvalues $\chi_N^<(\gamma)$ and $\chi_N^>(\gamma)$ respectively, i.e. $$\begin{aligned} \int_{-\infty}^{t} {\cal K}^<_N(t,t^\prime) e^{\gamma t^\prime} dt^\prime &=& \chi_N^<(\gamma) e^{\gamma t} \nonumber \\ \int_{t}^{\infty} {\cal K}^>_N(t,t^\prime) e^{\gamma t^\prime} dt^\prime &=& \chi_N^>(\gamma) e^{\gamma t}\end{aligned}$$ where \_N\^&lt;()  =  \_[r=0]{}\^N { - } \_N\^&gt;()  =  \_[r=0]{}\^N { - }. Note that $\chi_{\infty}^<(\gamma) = \gamma_E - \psi(1/2+\gamma)$ and $\chi_{\infty}^>(\gamma) = \gamma_E - \psi(1/2-\gamma)$, and we have the leading order BFKL kernel. In this paper we truncate the kernel after $N$ terms. Note that the case $N=0$ reduces to the collinear model of [@ccs]. For this truncated kernel the spectrum of eigenvalues is -\_[r=0]{}\^N  b t\_0  \_[r=0]{}\^N . \[range\] The infrared part of the eigenfunction $ f_\omega^<(t)$ obeys the integral equation $$\begin{aligned} b \, t_0 \, \omega \, f_\omega^<(t) & = & \int_{-\infty}^t {\cal K}^<_N(t,t^\prime) f_\omega^<(t^\prime) \, dt^\prime \ + \ \int_t^{t_0} {\cal K}^>_N(t,t^\prime) f_\omega^<(t^\prime) \, dt^\prime \nonumber \\ & & + \ \int_{t_0}^\infty {\cal K}^>_N(t,t^\prime) \frac{t_0}{t^\prime} f_\omega^>(t^\prime)dt^\prime \label{bfkl<} \end{aligned}$$ whereas the ultraviolet part of the solution $f_\omega^>(t)$, obeys the integral equation $$\begin{aligned} b \, t \omega \, f_\omega^>(t) & = & \int_{t_0}^t {\cal K}^<_N(t,t^\prime) f_\omega^>(t^\prime) \, dt^\prime \ + \ \int_t^{\infty} {\cal K}^>_N(t,t^\prime) \frac{t}{t^\prime} f_\omega^>(t^\prime) \, dt^\prime \nonumber \\ & & + \ \int^{t_0}_{-\infty} {\cal K}^<_N(t,t^\prime) f_\omega^<(t^\prime) \, dt^\prime. \label{bfkl>} \end{aligned}$$ We note that these equations are both inhomogeneous, reflecting the fact that even for $t < t_0$, where the coupling is fixed, the BFKL equation involves diffusion into the running coupling regime and vice versa. We can convert these two equations into homogeneous integro-differential equations by operating on both sides of eq.(\[bfkl&lt;\]) with the operator $${\cal O}^N_<(t) \ = \ \frac{d}{dt} \left( \frac{d}{dt}-1 \right) \cdots \left(\frac{d}{dt}-N \right) e^{-t/2}$$ and operating on both sides of eq.(\[bfkl&gt;\]) with the operator $${\cal O}^N_>(t) \ = \ \frac{d}{dt} \left( \frac{d}{dt}+1 \right) \cdots \left(\frac{d}{dt}+N \right) e^{t/2}.$$ These equations are $(N+1)^{th}$ order in derivatives and so each one of these has solutions which contain $N+1$ arbitrary constants of integration. However, these constants are fixed by the requirement that the original integral equations (\[bfkl&lt;\],\[bfkl&gt;\]) must be obeyed, together with the requirements the eigenfunctions should be normalizable, which means that $f_\omega^<(t)$ must be square integrable as $t \to -\infty$ and $f_\omega^>(t)$ must be square integrable as $t \to \infty$. The general solution to the integro-differential equation for $f_\omega^<(t)$ is f\_\^&lt;(t)  =   \_[j=1]{}\^[N+2]{} a\_j e\^[\^0\_j t]{}, \[mistress\] where $\gamma^0_j$ are the solutions to b t\_0  =  \_N\^&lt;() + \_N\^&gt;()    \_N() \[fixed\] with $\Re e (\gamma) \geq 0$ to ensure square integrability as $t \to -\infty$. In the allowed range of eigenvalues, eq.(\[range\]), there will be $N+2$ such solutions. For example, in the case $N=0$ the solutions are \^0\_j  =   i \[n0\] where $$\lambda_0= \frac{1}{b \, \omega \, t_0 + 2}.$$ For the allowed range, $\lambda_0 \, > \, 1/4$, and so these values are purely imaginary. In the case $N=1$ we have \^0\_j  =   \[n1\] where $$\lambda_1= \frac{1}{b \, \omega \, t_0 + 3}.$$ For the allowed range, $\lambda_1 \, > \, 3/16$ and we see that two of the solutions are purely imaginary, $\pm i \nu_1$, and two purely real $\pm \beta_1$. Thus the general solution is f\_\^&lt;(t)  =  a\_1 e\^[i\_1 t]{} + a\_2 e\^[-i\_1 t]{} + a\_3 e\^[\_1t]{}. \[soln&lt;1\] For the integro-differential equation for $f_\omega^>(t)$, it is convenient to work in terms of the Laplace transform $\tilde{f}_\omega(\gamma)$ defined by  =  d \_() e\^[t]{}, \[laplace\] in which the integral over $\gamma$ is performed over some suitable contour, described below. The function $\tilde{f}_\omega(\gamma)$ obeys the differential equation b \^\_() + \_N() \^\_()+ \_N\^[&gt; ]{} \_()  =  0, where the prime indicates differentiation with respect to $\gamma$. This equation can be solved in WKB approximation yielding the result \_()  =   { - \^( \_N(\^) +) d\^}, \[soln&gt;\] where $$V(\gamma,\omega) \ = \ \chi_N^2(\gamma)-4 \, \omega \, b \, \chi_N^{> \, \prime}(\gamma).$$ Note that if we set the term $\chi_N^{> \, \prime}(\gamma)$ to zero we recover the expression obtained in [@thorne] for the case of deep-inelastic scattering in which the transverse momenta are ordered and the coupling is chosen to run always with $t$. The solution (\[soln&gt;\]) is a valid approximation except near the points where $V(\gamma)$ vanishes. We could improve the solution by expanding $V(\gamma)$ to linear order around the zeroes and matching the solutions either side to an Airy function. However, the singularities in $\tilde{f}_\omega$ at these zeroes are integrable so that such an improvement would have negligible effect on the inverse Laplace transform. For $t<t_c$ the solution is oscillatory. The value of this critical value $t_c$ can be estimated by approximating the integral of eq.(\[laplace\]) by the saddle point at $$2 \, \omega \, b \, t \ = \ \chi_N(\gamma)+\sqrt{V(\gamma,\omega)}.$$ $t_c$ is the minimum value of $t$ for which this has a solution for purely real $\gamma$. Again, if we were to neglect $\chi_N^{> \, \prime}(\gamma)$, this would occur at $\gamma=0$ and we have approximately $$t_c \ = \ \frac{1}{b \,\omega} \chi_N(0).$$ This oscillatory part is “matched” to the fixed-coupling solution by requiring consistency with the original integral equations (\[bfkl&lt;\], \[bfkl&gt;\]). To see how this is possible, we need to examine in more detail the possible contours of integration for eq.(\[laplace\]). [**The $\gamma$-contour**]{} From the positions of the simple poles in $\chi_N^<(\gamma)$ and $\chi_N^>(\gamma)$, we note that $\tilde{f}_\omega(\gamma)$ has branch points on the real axis at $\gamma=r+1/2$ and $\gamma=-1/2-r$ for $r=0, \cdots N$. The branch cuts can be “combed” in the direction of the positive or negative real axis in such a way as to leave a portion of the real axis between $n-1/2$ and $n+1/2$ for which $\tilde{f}_\omega(\gamma)$ is analytic. $n$ runs from $-N$ to $N$. For each such “combing” a contour, ${\cal C}_n$ can be chosen that crosses the real axis between $n-1/2$ and $n+1/2$. Since $t>0$ we require that the ends of the contour turn over so that at the ends of the contour $\Re e \, \gamma \to -\infty$, thus ensuring that the inverse Laplace transform integral exists. The integral over each such contour is a valid solution to the integro-differential equation for $f^>_\omega(t)$ and since this equation is linear, the most general solution which is square integrable as $t \to \infty$ is the sum of the integrals over these contours with $-N \, \le \, n \, \le 0$, with arbitrary coefficients $b_n$, i.e. \_[n=-N]{}\^0 b\_n \_[[C]{}\_n]{} d \_() e\^[t]{}, \[master\] with $\tilde{f}_\omega(\gamma)$ given by eq.(\[soln&gt;\]).[^2] As an example we consider the $N=1$ case. The four “combings” are shown in Fig. \[fig1\] together with the four contours, ${\cal C}_n$. (420,450) (170,420)[(20,20)[$\gamma$]{}]{} (0,450)[(1,0)[200]{}]{} (20,450)(20,0)[9]{}[(0,-1)[5]{}]{} (0,250)[(1,0)[200]{}]{} (20,250)(20,0)[9]{}[(0,1)[5]{}]{} (0,350)[(1,0)[200]{}]{} (0,450)[(0,-1)[200]{}]{} (0,440)(0,-20)[10]{}[(1,0)[5]{}]{} (100,450)[(0,-1)[200]{}]{} (200,440)(0,-20)[10]{}[(-1,0)[5]{}]{} (200,450)[(0,-1)[200]{}]{} (80,253)(0,10)[20]{}[(0,1)[5]{}]{} (60,270)[${\cal C}_{-1}$]{} (0,348) ------------------------------------------------------------------------ (90,348) ------------------------------------------------------------------------ (110,347) ------------------------------------------------------------------------ (130,346) ------------------------------------------------------------------------ (15,240)[-4]{}(35,240)[-3]{}(55,240)[-2]{}(75,240)[-1]{} (97,240)[0]{} (117,240)[1]{}(137,240)[2]{}(157,240)[3]{}(177,240)[4]{} (390,420)[(20,20)[$\gamma$]{}]{} (220,450)[(1,0)[200]{}]{} (240,450)(20,0)[9]{}[(0,-1)[5]{}]{} (220,250)[(1,0)[200]{}]{} (240,250)(20,0)[9]{}[(0,1)[5]{}]{} (220,350)[(1,0)[200]{}]{} (220,450)[(0,-1)[200]{}]{} (220,440)(0,-20)[10]{}[(1,0)[5]{}]{} (320,450)[(0,-1)[200]{}]{} (420,440)(0,-20)[10]{}[(-1,0)[5]{}]{} (420,450)[(0,-1)[200]{}]{} (317,253)(0,10)[20]{}[(0,1)[5]{}]{} (323,270)[${\cal C}_{0}$]{} (220,347) ------------------------------------------------------------------------ (290,348) ------------------------------------------------------------------------ (330,348) ------------------------------------------------------------------------ (350,347) ------------------------------------------------------------------------ (235,240)[-4]{}(255,240)[-3]{}(275,240)[-2]{}(295,240)[-1]{} (337,240)[1]{}(357,240)[2]{}(377,240)[3]{}(397,240)[4]{} (317,240)[0]{} (170,190)[(20,20)[$\gamma$]{}]{} (0,220)[(1,0)[200]{}]{} (20,220)(20,0)[9]{}[(0,-1)[5]{}]{} (0,20)[(1,0)[200]{}]{} (20,20)(20,0)[9]{}[(0,1)[5]{}]{} (0,120)[(1,0)[200]{}]{} (0,220)[(0,-1)[200]{}]{} (0,440)(0,-20)[10]{}[(1,0)[5]{}]{} (100,220)[(0,-1)[200]{}]{} (200,440)(0,-20)[10]{}[(-1,0)[5]{}]{} (200,220)[(0,-1)[200]{}]{} (120,23)(0,10)[20]{}[(0,1)[5]{}]{} (107,40)[${\cal C}_{1}$]{} (70,117) ------------------------------------------------------------------------ (90,118) ------------------------------------------------------------------------ (0,116) ------------------------------------------------------------------------ (130,116) ------------------------------------------------------------------------ (15,10)[-4]{}(35,10)[-3]{}(55,10)[-2]{}(75,10)[-1]{} (97,10)[0]{} (117,10)[1]{}(137,10)[2]{}(157,10)[3]{}(177,10)[4]{} (390,190)[(20,20)[$\gamma$]{}]{} (220,220)[(1,0)[200]{}]{} (240,220)(20,0)[9]{}[(0,-1)[5]{}]{} (220,20)[(1,0)[200]{}]{} (240,20)(20,0)[9]{}[(0,1)[5]{}]{} (220,120)[(1,0)[200]{}]{} (220,220)[(0,-1)[200]{}]{} (220,440)(0,-20)[10]{}[(1,0)[5]{}]{} (320,220)[(0,-1)[200]{}]{} (420,440)(0,-20)[10]{}[(-1,0)[5]{}]{} (420,220)[(0,-1)[200]{}]{} (360,23)(0,10)[20]{}[(0,1)[5]{}]{} (344,40)[${\cal C}_{2}$]{} (220,116) ------------------------------------------------------------------------ (220,115) ------------------------------------------------------------------------ (330,118) ------------------------------------------------------------------------ (310,117) ------------------------------------------------------------------------ (235,10)[-4]{}(255,10)[-3]{}(275,10)[-2]{}(295,10)[-1]{} (337,10)[1]{}(357,10)[2]{}(377,10)[3]{}(397,10)[4]{} (317,10)[0]{} Only ${\cal C}_{-1}$ and ${\cal C}_0$ give a square integrable solution, so the general solution is f\^&gt;\_(t)  = t {  b\_[-1]{} \_[-]{}\^d e\^[-t]{} \_(i-1) e\^[it]{}  +   b\_0 \_[-]{}\^d \_(i) e\^[it]{} }\[soln&gt;1\]. Returning now to the original equations (\[bfkl&lt;\]) and (\[bfkl&gt;\]), we see that these are obeyed by the solutions (\[master\]) and (\[mistress\]) provided \_[j=1]{}\^[N+2]{} a\_j -t\_0 \_[n=-N]{}\^0 b\_n \_[[C]{}\_n]{} d  =  0    (r=0 N), \[match1\] \_[j=1]{}\^[N+2]{} a\_j + \_[n=-N]{}\^0 b\_n \_[[C]{}\_n]{} d  =  0    (r=0 N). \[match2\] Thus we obtain $2N+2$ relations between the $N+2$ coefficients $a_j$ and the $N+1$ coefficients $b_k$ (the overall normalization of these coefficients is determined by the normalization condition of the eigenfunctions). This then determines uniquely the required eigenfunctions. Equations (\[bfkl&lt;\], \[bfkl&gt;\]) guarantee the continuity of the eigenfunctions at $t=t_0$, i.e. $f^<_\omega(t_0)=f^>_\omega(t_0)$, but the derivative is not continuous. This results from the fact that we have taken an expression for the coupling as a function of $t$, whose derivative is not continuous at $t=t_0$. We can now see how the inverse Mellin transform, $f(s,t)$ of the solution $f_\omega(t)$ extrapolates smoothly between the power-like behaviour for $t \sim t_0$ and the softer behaviour (with at most logarithmic dependence on $s$) for $t \gg t_0$. Examining the $\gamma-$ integral for one particular contour, ${\cal C}_n$, and for $t > t_0$, we see that this will be dominated by a saddle point $\gamma_s(\omega,t)$, which is the solution to 2 b t  =  \_N()+. \[saddle\] This is multi-valued and the dominant saddle point will be the one closest to the integration contour. We call this saddle point $\gamma^n_s(\omega,t)$. In other words, $f^>_\omega(t)$ of eq.(\[laplace\]) from the contour ${\cal C}_n$ will contain an $\omega-$dependent prefactor, which may be written (after a suitable integration by parts) $$\exp\left\{ \int^t \gamma^n_s(\omega,t^\prime) dt^\prime \right\}.$$ The essential singularity at $\omega=0$ is now encoded in $\gamma^n_s(\omega,t)$ which acquires a singularity as $\omega \to 0$, i.e. the R.H.S. of (\[saddle\]) can only tend to zero as $|\gamma| \to \infty$. Now examining eqs.(\[match1\], \[match2\]), we see that the contour integrals multiplying the coefficients $b_n$ are also dominated by a similar saddle point but with $t$ set to $t_0$. Thus these coefficients appear in eqs.(\[match1\], \[match2\]) with a prefactor $$\exp\left\{ \int^{t_0} \gamma^n_s(\omega,t^\prime) dt^\prime \right\}.$$ Therefore these coefficients also possess an essential singularity at $\omega=0$. Furthermore $b_n$ will contain factors of the form $\exp(\gamma^j_0(\omega) t_0)$ from the L.H.S. of (\[match1\], \[match2\]). For $t$ close to $t_0$ each term in the sum of eq.(\[master\]), i.e. the product of $b_n$ and the contribution to $f^>_\omega(t)$ from the integral over the contour ${\cal C}_n$ will have a dominant $\omega-$dependence of the form { \^n\_s(,t\_0)(t-t\_0)+\_0\^j()t\_0 }. \[neweq\] Note that due to the partial cancellation, the coefficient of the term that gives rise to the essential singularity vanishes as $t \to t_0$. For sufficiently large $s$, the inverse Mellin transform probes small $\omega$, where the term in the exponent proportional to $\gamma^n_s(\omega,t_0)$ will dominate even if $(t-t_0)$ is small and the $s$-behaviour is dominated by the essential singularity in the Laplace transform, $\tilde{f}_\omega(\gamma)$. However, for more moderate values of $s$ we can neglect this term when $t$ is sufficiently close to $t_0$ and the $s-$dependence will be dominated by the fixed coupling power-like behaviour. As $t$ is increased away from $t_0$, the boundary in $s$ where the soft behaviour takes over from the power-like behaviour decreases and we see that as $t \to \infty$ we recover the solution of [@thorne]. This formalism is considerably simplified if we assume that $t_0$ is sufficiently large that $\exp(-t_0) \ll 1$. In this case we only need consider the right-most contour consistent with square-integrability as $t \to \infty$. This is the contour ${\cal C}_0$. In this case, within the approximation of performing the $\gamma-$contour integral and the inverse Mellin transform by the saddle-point method, the $s-$dependence may be written as $s^{\tilde{\omega}(s,t)},$ where $\tilde{\omega}(s,t)$ is the solution to (s)  =   - { (t-t\_0) \_[t\_0]{}\^t \^0\_s(,t\^) dt\^ + \_0\^m() [Min]{}(t,t\_0) } \[plot\], and $\gamma_0^m(\omega)$ is the solution to eq.(\[fixed\]) with the largest real part. In Fig. \[fig2\] we plot $\tilde{\omega}(s,t)$ against $t$ for a large value of $s \ (\ln s =23)$. For simplicity we have taken the $N=0$ case, although we expect the result to be qualitatively the same for higher values of $N$, and we take $t_0=4$. We see that it is almost constant for $t \, < \, t_0$ and diminishes as one increases $t$ from $t_0$, indicating the smooth transition from hard to soft $s-$dependence as $t$ is increased. The discontinuity of the slope at $t=t_0$ is a reflection of the fact that we have taken a sharp discontinuity in the slope of the running coupling at $t_0$ and would be smoothed out if a smoother transition were taken. The asymptotic behaviour is only obtained for much larger values of $t$ and in Fig. \[fig3\] we show the $s$ dependence of $\tilde{\omega}(s,t)$ for $t=40$ and notice that it is much closer to zero than the case where $t \sim t_0$ and decreases as $\ln s$ increases, resulting in an $s-$dependence for the amplitude that has no fixed power of $s$. [**Accounting for Higher Order Corrections**]{} The higher order BFKL kernel [@BFKL2] does not lend itself readily to an expansion in powers of $e^{t-t^\prime}$. Furthermore, it is known that these corrections are large. However, it has been pointed out [@salam] that a substantial part of the higher order corrections consist of “collinear corrections” which are required to ensure correct behaviour when the formalism is applied to deep inelastic scattering. It was shown in [@salam] and confirmed in [@us] that once these corrections are accounted for, the remaining higher order corrections are modest. The simplest way of encoding these collinear corrections and summing them is to replace $\gamma$ by $(\gamma-\omega/2)$ in $\chi_N^<(\gamma)$ and by $(\gamma+\omega/2)$ in $\chi_N^>(\gamma)$ [@ags; @salam]. This formalism is readily adapted to the procedure described here. In the expression (\[mistress\]) for $f_\omega^<(t)$ the quantities $\gamma_j^0(\omega)$ are taken to be the solutions to the implicit equation \_N\^&lt;(-/2)+\_N\^&gt;(+/2)  =  b t\_0, \[gamm1\] and the expression for the Laplace transform of $f_\omega^>(t)$ becomes \_()  =   {- \^( \_N\^&lt;(\^-/2)+\_N\^&gt;(\^+/2) + ) d\^ }, \[soln2&gt;\]where V(,)  =  (\_N\^&lt;(-/2) +\_N\^&gt;(+/2))\^2 -4 b \_N\^[&gt; ]{} (+/2). Provided that we restrict our analysis to the case $|\omega|<1$, the singularity structure and consequently the analysis of the contours of integration remains unchanged from the leading order case, although the exact positions of the branch points will have moved. [**Acknowledgement**]{} ASV wishes to thank the University of Manchester and the CERN Theory Division where this work started and acknowledges the support of PPARC (Posdoctoral Fellowship: PPA/P/S/1999/00446). DAR wishes to thank the CERN Theory Division for its hospitality. [99]{} V.S. Fadin, E.A. Kuraev and L.N. Lipatov, [*Sov. Phys. JETP*]{} [**44**]{} 443 (1978)\ Y.Y. Balitsky and L.N. Lipatov, [*Sov. J. Nucl. Phys.*]{} [**28**]{} 822 (1978) V.S. Fadin and L.N. Lipatov, [*Phys. Lett.*]{} [**B429**]{} 127 (1998)\ G.Camici and M. Ciafaloni, [*Phys. Lett.*]{} [**B430**]{} 349 (1998) L.N. Lipatov [*Sov. Phys. JETP*]{} [**63**]{} 904 (1986) R.S. Thorne, [*Phys. Rev.*]{} [**D60**]{} 054031 (1999); [*Phys. Lett.*]{} [**B474**]{} 372 (2000) M. Ciafaloni, D. Colferai and G.P. Salam, [*JHEP*]{} [**9910:017**]{} (1999); [*JHEP*]{} [**0007:054**]{} (2000) C.P. Salam [*JHEP*]{} [**9807:019**]{} (1998) J.R. Forshaw, D.A. Ross and A. Sabio Vera, [*Phys. Lett*]{} [**B455**]{} 273 (1999) B. Andersson, G. Gustafson and J. Samuelsson, [*Nucl. Phys.*]{} [**B467**]{} 443 (1996) [^1]: On Leave of absence from:\ Department of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, U.K. [^2]: We have absorbed the factor of $2\pi i$ in the denominator in eq.(\[laplace\]) into the definition of the coefficients $b_n$.

--- title: Jet Physics with Identified Particles at RHIC and LHC --- Introduction ============ The measurement of particle identified high momentum spectra and two-particle correlations can be considered the next step in the characterization of the dense partonic phase created in relativistic heavy ion collisions at RHIC. Much emphasis is presently given to the similarities in the light and heavy quark behavior in the plasma, but these measurements are detailed in other contributions to this workshop [@dasilva; @harris]. Here I will mostly focus on the flavor dependence in light hadron production, i.e. hadrons consisting of u,d,s quarks. I will address in particular the question of baryon versus meson production through fragmentation and recombination as competing mechanisms. In order to determine the baryon production mechanism in AA collisions we need to understand the production in pp first. The main question is whether there is a difference between the light baryon production in the medium and in the vacuum. In the following section I will briefly review the latest particle identified pp results from STAR and then move on to results from AuAu collisions. The ambiguity in the production mechanism as deduced from single particle properties such as the suppression at high pt and elliptic flow v2 can be partially resolved by analyzing two particle correlations. I will show that early results of intermediate p$_{T}$ associated particle production are surprisingly featureless regarding flavor dependencies that were expected from perturbative QCD calculations in vacuum. In contrast, the single particle spectra suppression at high p$_{T}$ in AA seems very much affected by flavor suppression in the proton-proton system, in particular for strange particles. I will try to argue that this effect should be unique to the strange quark, only if the strange quark generation is gluon dominated and the heavy quark production is quark dominated. Many of the studies at RHIC lack the statistics to map out the transition from medium effects to pure vacuum fragmentation. I will show that these measurements are readily accessible at the LHC, in particular with the ALICE detector. Why is pp so important ? ======================== The latest strange and non-strange hadron production results obtained in proton-proton collisions by STAR, were presented at this workshop [@heinz; @ruan]. They show that simple leading order fragmentation codes are not sufficient to describe the baryon production at RHIC energies. Next to leading order calculations do well as soon as quark separated fragmentation functions are used [@akk], in other words there are contributions in the baryon spectra from non-valence quark fragmentation. In a leading order calculation this fact can apparently be approximated by increasing the K-factor and the multiple parton scattering contribution to the underlying event [@bellwied1]. These effects are expected to be significantly less pronounced at higher incident energies [@eskola]. In other words, next to leading order contributions are more important at RHIC energies than at LHC energies. One important new pp result is the break-down of the so called m$_{T}$-scaling [@gatoff; @schaffner] at sufficiently high transverse mass at RHIC. Fig.1a shows the measured spectra of all particle identified species obtained by STAR. After appropriate scaling of each spectrum the m$_{T}$-scaling plot (Fig.1b) can be devised. Clearly the m$_{T}$-scaling at lower m$_{T}$, which had previously been established through measurements at lower energies [@isr1; @isr2], can be confirmed. The noticeable deviation from simple scaling at sufficiently high m$_{T}$ (m$_{T}$ $>$ 3 GeV/c) is a new feature, though. The spectra seem to group into common baryon and meson curves at these momenta. This is a surprising effect in elementary pp collisions which is nevertheless described quantitatively in PYTHIA, as shown in Fig.2. {width="2.in"} \[fig:1a\] {width="2.in"} \[fig:1b\] Figs.2a and 2b break down the effect, as modeled in PYTHIA, for gluon and quark jets. The combined spectrum (Fig.2c) is dominated by the gluon jets. According to PYTHIA the ratio of gluon to quark jets is about 2:1 at RHIC energies. {width="1.5in"} \[fig:2a\] {width="1.5in"} \[fig:2b\] {width="1.5in"} \[fig:2c\] One very interesting feature in the parton separated spectra (Figs.2a and 2b) is that apparently the gluon jet fragmentation leads to a baryon/meson difference at high m$_{T}$, whereas the quark jets show a mass ordering of the high m$_{T}$ spectra. Gluon fragmentation dominates at RHIC (and even more at LHC) energies, and based on PYTHIA the baryon/meson splitting effect in the kinematic spectra is due to the di-quark formation process which is a pre-requisite for baryon formation in LUND type string fragmentation [@lund-baryon]. The initial di-quark formation leads to a lowering of the $<$p$_{T}$$>$ for the baryons (the so called di-quark suppression factor), whereas for mesons the simple quark-antiquark hadronization does not affect the $<$pT$>$. This feature in the fragmentation of elementary gluon jets could be considered the seed of the baryon meson differences in AA collisions. The effect by itself is not big enough, though, to quantitatively describe the strong peak in the baryon/meson ratio at intermediate pt in AA, but the overall feature (a bump at intermediate p$_{T}$) can already be seen in pp collisions and might be simply enhanced by additional effects in AA. In AA one needs the combined effects of radial flow and quenching to ’pile-up’ the baryons over mesons in the intermediate p$_{T}$ range. Still it is important to note that the baryon/meson differences already have their origin in the basic fragmentation process in pp collisions. This measurement also hints at the validity of the fragmentation process as modeled by PYTHIA. It is my opinion that in its nature the diquark-quark formation process is similar to a three quark coalescence process and thus exhibits equivalent features. Another conclusion from the pp modeling is that, although early pQCD calculations predicted a stronger drop in the anti-baryon over baryon ratio, our results are in accordance with PYTHIA calculations if one takes into account the strong gluon dominance at RHIC energies as shown in Fig.3. ![Measured Anti-$\Lambda$ over $\Lambda$ ratio compared to PYTHIA ratio from gluon and quark jets.[]{data-label="fig:3"}](fig3-lambdaratio.eps){width="2.5in"} In order to unambiguously determine the predicted drop, which is due to the difference between quark and gluon distribution functions and the relative contribution of these partons to the baryon and anti-baryon production, one needs measurements at higher p$_{T}$. The momentum range of the spectra will be greatly enhanced at the LHC. Fig.4 shows a projection of p$_{T}$-ranges for identified spectra based on a scaled LHC pQCD calculation. Generally the light quark spectra will reach out to at least 20 GeV/c in the first year of running, which allows for a more unambiguous study of the flavor effects at high p$_{T}$. ![Annual hard process yields expected in the ALICE Calorimeter acceptance for minimum bias Pb-Pb collisions at 5.5 TeV.[]{data-label="fig:6"}](fig4-leading.ps){width="2.5in"} ![Annual hard process yields expected in the ALICE Calorimeter acceptance for minimum bias Pb-Pb collisions at 5.5 TeV.[]{data-label="fig:6"}](fig5.eps){width="2.5in"} One interesting feature in the gluon dominated regime at RHIC and the LHC is that the $\gamma$-jet process is dominated by Compton scattering, which leads to a $\gamma$-quark-jet combination in the outgoing channel. This means not only is the tagged $\gamma$ the ’standard candle’ for the jet energy measurements, but it is also a good trigger for quark-jet events in the gluon dominated regime. The rate, based on early simulations using the ALICE calorimeter, is significantly limited compared to di-jet events, but one can still expect on the order 10,000 events with a $\gamma$ energy above 50 GeV. A similar quark-jet selection can be achieved by triggering on heavy mesons as leading particles. Projected rates for all jet measurements with the proposed ALICE EMCal are shown in Fig.5. Flavor dependencies in identified high pt AA spectra ==================================================== One of the most surprising results of the past year was the apparent difference between R$_{AA}$ and R$_{CP}$ measurements, in particular for strange baryons. Figs.6 and 7 show a direct comparison of strange and non-strange baryons and mesons. ![Nuclear suppression factor (R$_{CP}$) as measured by STAR.[]{data-label="fig:6"}](fig6-raa.eps){width="2.in"} ![Nuclear suppression factor (R$_{CP}$) as measured by STAR.[]{data-label="fig:6"}](fig7-rcp.eps){width="2.in"} The strong jet quenching effect at high p$_{T}$ and the slight baryon/meson splitting due to recombination at moderate p$_{T}$ as shown on the R$_{CP}$ plot is not visible in the R$_{AA}$ plot. The R$_{CP}$ suppression has been established for all particle species from pions to charmed mesons, and besides the already mentioned baryon/meson difference at intermediate p$_{T}$ the suppression is surprisingly flavor independent. The R$_{AA}$ pattern though exhibits a very strong flavor dependence, in particular for baryons. An enhancement of the high p$_{T}$ yield, rather than a suppression compared to the pp spectrum, actually increases as a function of the strangeness content in baryons. A similar enhancement pattern was previously measured for integrated particle yields, and is generally attributed to canonical suppression of strange quarks in small systems [@tounsi], but it is unexpected that apparently this effect of a small correlation volume in an equilibrated system should also affect the high p$_{T}$ particle production. Not only does it lead to an enhancement of the intermediate p$_{T}$ yield from pp to AA but there is also no evidence of quenching in the strange baryon R$_{AA}$ plot at high p$_{T}$. This effect is actually not seen in preliminary results of charmed meson suppression, so it could indeed be unique to the strange quarks. A measurement of a charmed baryon ($\Lambda_c$) is needed to unambiguously determine the difference between the quark flavors. Because of the difference between R$_{CP}$ and R$_{AA}$ the strange quark flavor effects need to have their origin in the particle production in the pp system. This is a good indicator that even intermediate p$_{T}$ strange baryons are predominantly produced through coalescence from a thermalized partonic system. In other words the initial gluon dominated scattering processes leads to thermalized strange quarks which coalesce into strange baryons. The effect of the correlation volume during hadronization is still dominant even at rather large (up to 3 GeV/c) transverse momentum. Beyond the intermediate p$_{T}$ range the spectrum gets quenched, but it is still enhanced in AA collisions compared to the pp system. The question is whether this strange particle production mechanism drives the pp to AA comparison even in the pure fragmentation regime above 7 GeV/c. The RHIC experiments do not have a big enough reach to establish an answer. Fig.4 shows that these measurements can be achieved at the LHC, though. We know from e$^{+}$e$^{-}$ experiments that the strangeness suppression factor in the quark condensate is about 0.4 [@e+e-], but this is just the relative quark production probability in the hadronization sea, which should also exist in the medium. In addition, the strangeness saturation factor increases from pp to AA by a factor two at RHIC energies based on measurements of integrated strange particle yields [@cleymans]. The factor in AA can be described quantitatively in lattic QCD [@gavai]. The difference between baryons and mesons in R$_{CP}$ (Fig.7), which is generally attributed to either recombination or an interplay between radial flow and jet quenching [@lamont], can also be described by the so-called Corona effect, which was shown elsewhere at this conference [@werner; @pantuev]. The principle here is that the formed medium consists of a dense core, which follows hydrodynamics and a corona of pp interactions dominated by multiple scattering. The main reason for such a distinction and the strong contribution from the corona is the relative diffuseness of the nuclear surface, which is not well described by hard spheres. The pp interactions can be modeled by codes such as EPOS which take into account the increased parton cascade activity in the low momentum sector. In these models effects such as baryon/meson splitting in v$_{2}$ occur because the corona, which carries very little v$_{2}$, has a much stronger contribution to the light particle spectrum than the heavy particle spectrum, so it pulls down the hydro v$_{2}$ for mesons to lower values than for baryons. Finally one can measure identified two particle correlations at intermediate p$_{T}$ in order to detect flavor dependencies that are expected from simple fragmentation arguments. The correlations shown in Fig.8 show surprisingly little trigger particle flavor dependence. Again it seems that non-fragmentation processes, such as recombination, dominate in this p$_{T}$ range. There is evidence for long-range correlations in $\Delta\eta$ which could be due production mechanisms that do not exhibit the flavor dependencies of simple fragmentation. These correlations lead to a significant enhancement of the associated yield for any trigger species over the associated yields measured in pp, which are in agreement with the PYTHIA simulations shown in Fig.8b. ![High p$_T$ two particle correlations using identified trigger particles and charged hadron as associated particles. a.) measurements in 0-5% centrality Au-Au collisions in STAR, b.) PYTHIA simulations of the same correlations in pp collisions.[]{data-label="fig:7"}](fig8-complete.eps){width="3.in"} Summary ======= The interpretation of our pp collision results reveal that the baryon production yields require either multiple scattering through a soft particle production model such as EPOS or NLO corrections in pQCD models such as PYTHIA. It is interesting to note that the basic string fragmentation differences in baryon and meson production lead to a breakdown of the universal m$_{T}$-scaling of identified particle spectra. Apparently this breakdown is driven by the baryon production mechanism in gluon jets and manifests itself as a slope difference at high m$_{T}$ when comparing baryon to meson spectra. This basic effect in pp is not sufficient though to describe the large baryon over meson yield enhancement at intermediate p$_{T}$ in AA collisions. Besides the baryon/meson difference there is a surprising absence of strong flavor effects in the particle to anti-particle ratios in AA, the identified two particle correlations in AA, and even the jet quenching in the medium. Thus, it is still an open question whether the partonic energy loss in AA shows the expected Casimir factor when comparing hadrons from a fragmenting gluon jet to a quark jet, i.e. is the energy loss really non-Abelian ? The only strong flavor effect is in the strangeness sector. High p$_{T}$ strange baryon production in AA is enhanced instead of suppressed compared to pp . This could be due to simple canonical suppression in pp. This thermodynamic effect, which is due to a limited strangeness phase space occupancy, has been measured for the first time as a function of transverse momentum, and it is obvious that the effect is not limited to low momentum or simply bulk properties. Surprisingly the intermediate p$_{T}$ part scales well with the canonical suppression factors, which indicates that the hadronization mechanism of strange baryons, even at higher p$_{T}$, is driven by a correlation volume, which is distinctly different from charmed meson production. The D-meson yield and high p$_{T}$ suppression factors in AA are consistent with scaled hard scattering cross section, i.e. production from string fragmentation. In identified two particle correlations in AA collisions we see a strongly enhanced associated particle yield compared to pp, independent of the trigger particle species. Long range $\Delta\eta$ correlations might account for that and they might be due to recombination [@hwa1]. A small baryon/meson trend can be found in those correlations but the effect is not very significant. Larger predicted effects for $\phi$ and $\Omega$ triggered correlations [@hwa2] are under investigation. In summary, I believe that studies of the hadronization mechanism at RHIC and LHC energies in vacuum and in medium hold the key to the puzzle of baryonic matter formation in the universe. We need to first understand the basic baryon production mechanism(s) in pp (string fragmentation vs. recombination, di-quark formation ?). Then we need to determine whether the baryon production mechanism in AA collisions is modified from the vacuum production. For a more detailed study the high pt reach and the particle identification properties of the LHC detectors are crucial. Acknowledgements {#acknowledgements .unnumbered} ================ I thank Helen Caines, Mark Heinz, and Klaus Werner for useful discussions. [99]{} C.L. DaSilva, contribution to these proceedings J. Harris, contribution to these proceedings M. Heinz, contribution to these proceedings L. Ruan, contribution to these proceedings S. Albino et al., hep-ph/0502188 R. Bellwied for the STAR collaboration, QM05 proceedings, nucl-ex/0511006 K. Eskola et al., Nucl. Phys. A713 (2003) G. Gatoff and C. Y. Wong, Phys. Rev. D 46, 997 (1992) J. Schaffner-Bielich et al., arXiv:nucl-th/0202054 P.V. Chliapnikov and V.A. Uvarov, Phys.Lett. B345, 313, 1995 M. Szczekowski, Phys.Lett. B359, 387, 1995 P. Eden, G. Gustafson, Z. Phys. C75, 41 (1997) and hep-ph/9606454 K. Redlich and A. Tounsi, Eur. Phys. J. [**C24**]{} (2002) 589 G. Abbiendi et al. (OPAL), Eur. Phys. J. C16, 407 (2000) and hep-ex/0001054 J. Cleymans, J. Phys. G28, 1575 (2002) R.V. Gavai and S. Gupta, Phys. REv. D73, 014004 (2006) J. Adams et al. (STAR), nucl-ex/0601042 K. Werner, contribution to these proceedings V. Pantuev, contribution to these proceedings C.B. Chiu, R. Hwa, Phys. Rev. C72, 034903 (2005) R. Hwa, C.B. Yang, nucl-th/0602024

--- abstract: 'The NRAO VLBA was used at 1.4 GHz to image the inner 550 mas (23 lt-years) of the nearest known Seyfert 1 nucleus, in the Sm galaxy NGC4395. One continuum source was detected, with flux density $530\pm130~\mu$Jy, diameter $d < 11$ mas (0.46 lt-year), and brightness temperature $T_{\rm b} > 2.0\times10^6$ K. The spectral power $P\/$ of the VLBA source is intermediate between those of SagittariusA and SagittariusA$^*$ in the Galactic Center. For the VLBA source in NGC4395, the constraints on $T_b$, $d\/$, and $P\/$ are consistent with an origin from a black hole but exclude an origin from a compact starburst or a supernova remnant like CassiopeiaA. Moreover, the spectral powers of NGC4395 at 1.4 and 4.9 GHz appear to be too low and too constant to allow analogy with SN1988Z, a suggested prototype for models of compact supernova remnants. The variable and warm X-ray absorber in NGC4395 has a free-free optical depth much larger than unity at 1.4 GHz and, therefore, cannot fully cover the VLBA source.' author: - 'J. M. Wrobel and C. D. Fassnacht' - 'L. C. Ho' title: The Inner Light Year of the Nearest Seyfert 1 Nucleus in NGC4395 --- MOTIVATION ========== The Seyfert 1 nucleus of NGC4395 holds three unique distinctions. First, in the optical regime it is the least luminous Seyfert nucleus known, with an absolute blue magnitude of only $M_B=-9.8$ mag. On energetics grounds it is therefore essential to explore both black hole and stellar origins for the Seyfert activity [@fil89; @fil93; @lir99]. Second, it is the only Seyfert nucleus known to be hosted by an Sm galaxy. Proving that the Seyfert traits of NGC4395 are inconsistent with a stellar origin, but consistent with a black hole origin, would thus bolster claims of black hole ubiquity in the local universe [@mag98]. Finally, NGC4395 lies at a distance $D=2.6$ Mpc [@row85], making it the nearest known Seyfert 1 nucleus. The corresponding scale is 10 mas $=$ 0.42 lt-year, so the inner light year can, in theory, be probed directly with the NRAO Very Long Baseline Array (VLBA). Direct imaging of radio emission on light-year scales can be a powerful discriminant between black hole and stellar origins: a black hole could launch jets [@fal00], while stellar processes could result in a compact starburst [@con91], a supernova remnant resembling Galactic ones [@gre84], or a compact supernova remnant [@ter95]. Recent observations of NGC4395 with the NRAO Very Large Array (VLA) showed an unresolved source at a frequency $\nu=1.4$ GHz, with a flux density $S=1680\pm94~\mu$Jy; a diameter $d < 550$ mas (23 lt-years); and a spectral index $\alpha = -0.60\pm0.08$ ($S \propto \nu^{\alpha}$) between 1.4 and 4.9 GHz, indicating synchrotron emission from a nonthermal plasma [@ho01a]. A VLA source of this strength is too weak to be imaged with the VLBA if just traditional self-calibration techniques are applied. However, the prospects for successfully imaging NGC4395 with the VLBA are good if phase-referencing techniques are employed [@wro00b]. Section 2 of this Letter reports the detection of one VLBA source at 1.4 GHz, using phase-referenced observations of a 550-mas region in NGC4395. Section 3 examines the implications of this VLBA detection, plus published radio photometry, for black hole and stellar models for the origin of the radio emission. These data are shown to be consistent with an origin from a black hole but inconsistent with an origin from a compact starburst, a supernova remnant like CassiopeiaA, or a compact supernova remnant resembling SN1988Z. OBSERVATIONS, CALIBRATION, AND IMAGING ====================================== The VLBA [@nap94] was used to observe NGC4395 and calibrators on 2000 April 11 UT. Data were acquired in dual circular polarizations with 4-level sampling and at a center frequency 1.43840 GHz with bandwidth 32 MHz. Phase-referenced observations were made in the nodding style. A 3-minute observation of NGC4395 was preceded and followed by a 2-minute observation of the phase, rate, and delay calibrator J1220+3431 [@wil98] about $1.5\arcdeg$ from NGC4395. Sources J1215+3448 and J1310+3220 were also observed, respectively, to check the astrometric accuracy and to align the phases of the independent baseband channels [@ma98]. Observation and correlation assumed a coordinate equinox of 2000. The [*a priori*]{} position adopted for NGC4395 was very close to that in @ho01a. Data editing and calibration were done using the 1999 December 31 release of the NRAO AIPS software and following the strategies outlined by @ulv00. After data deletion based on [*a priori*]{} flags, data were deleted on all baselines involving any antenna observing below an elevation of $20\arcdeg$. Such elevation-based editing minimized differential ionospheric conditions between the lines of sight to J1220+3431 and NGC4395. Despite this step, it was impossible to calibrate the phases on baselines to the VLBA antenna on Manua Kea in Hawaii, and those baselines were also deleted. This resulted in an observed baseline range of 240–5800 km and, for NGC4395, a total of 129 baseline-hours of integration. VLBA system temperatures and gains were used to set the amplitude scale to an accuracy of about 5%, after first correcting for sampler errors. No self-calibrations were performed on NGC4395. The AIPS task IMAGR was used to form and deconvolve an image of the Stokes $I\/$ emission from NGC4395, with the visibility data being naturally weighted to optimize image sensitivity. This image, given in Figure 1, was restored with an elliptical-Gaussian beam with FWHM dimensions of 13.9 mas (0.59 lt-year) by 10.7 mas (0.45 lt-year) and elongation orientation of $-20.3\arcdeg$, and has an rms noise value of 51 microjanskys ($\mu$Jy) per beam area. The left panel of Figure 1 shows that only one VLBA source was detected above 4.5 times the rms noise, within a region centered on the unresolved VLA detection and spanning 550 mas (23 lt-years), which matches the upper limit to the diameter of the VLA detection at 1.4 GHz [@ho01a]. A quadratic fit to the peak of the VLBA detection yielded a position of $\alpha(J2000) = 12^{h} 25^{m} 48^{s}.874$ and $\delta(J2000) = 33^{\circ} 32' 48''.69$. This position carries an absolute 2-dimensional error of 55 mas, set by the position error of J1220+3431 and verified with a phase-referenced image of J1215+3448. Given the signal-to-noise ratio of the VLBA detection of NGC4395, its position can, in principal, be determined with an error of 1-2 mas [@bal75] and efforts are underway to reach that limiting accuracy. The right panel of Figure 1 displays the inner 76 mas (3.2 lt-years) centered on the position of the VLBA detection. The VLBA source seems to be slightly resolved but this apparent resolution is likely to be artifical, due to the weakness of the source and/or residual errors in the phase calibration. As a gauge of the importance of the latter effect, before self-calibration an image of the strong source J1215+3448 had a peak intensity only 88% of the peak after phase self-calibration. For the VLBA detection of NGC4395, (1) image integration over $N=5.8$ beam areas gives a total flux density of $530\pm130~\mu$Jy, where the error is the quadratic sum of a 5% scale error and $\sqrt{N}$ times the rms noise; and (2) a conservative estimate to the diameter is $d < 11$ mas (0.46 lt-year). Traits (1) and (2) imply that the VLBA detection has a brightness temperature $T_{\rm b} > 2.0\times10^6$ K. IMPLICATIONS ============ Figure 2 shows that the VLA detections of NGC4395 [@ho01a] match the spectral powers and spectral index of SagittariusA in the Galactic Center [@ped89]. The VLA size limit for NGC4395, conveyed by the dimension of the left panel of Figure 1, is also comparable to the diameter of 26 lt-years for SagittariusA [@ped89]. But the VLBA detection of NGC4395 at 1.4 GHz is less powerful (Figure 2) and more compact (Figure 1, left panel) than SagittariusA. This hints that the VLBA source in NGC4395 could be an extragalactic analog to SagittariusA$^*$ [@fal98], thought to mark the massive black hole at the dynamical center of the Galaxy [@rei99]. An accurate position is not available for the dynamical center of NGC4395 [@swa99]. The position of the VLBA detection does agree with positions for the optical continuum [@cot99], X-ray continuum [@ho01b], and nuclear H$\alpha$ emission [@van98], but these agreements carry combined errors of order 1000 mas (42 lt-years). However, a stringent mass limit is available for a putative black hole in NGC4395: @fil01 report detection of the infrared triplet lines in absorption from echelle spectra taken with the Keck I telescope, yielding estimates for the strength of the stellar contribution to the nuclear light ($M_B=-7.3$ mag) and the central LOS velocity dispersion ($\sigma \sim 30$ km s$^{-1}$), which, in combination with the star cluster size from [*HST*]{} images, limit any black hole mass to $M_{\rm BH} \lesssim 80,000~M_{\sun}$. An accreting black hole could launch jets. While @fal00 use a spectral analysis to build a case for a jet origin for SagittariusA$^*$, morphological evidence for an outflow would be far more compelling. In the case of NGC4395, the VLBA source is unresolved at a linear resolution of 0.46 lt-year or 170 lt-days, and only 0.32$\pm$0.26 times as strong as the VLA source spanning 23 lt-years or less. A few nearby galaxies have had their radio nuclei probed on similar scales, or even finer scales down to a resolution of 10 lt-days (eg, NGC3031, Bietenholz, Bartel, & Rupen 2000). While jets or jet-like structures are invariably observed, those galaxies exhibit spectral powers at 1.5 GHz that are an order of magnitude, or more, above the power of the source in NGC4395 at comparable VLA resolutions [@wro01]. Still, the VLA and VLBA sources in NGC4258 at 1.5 GHz [@cec00] are only about a factor of ten more powerful than their counterparts in NGC4395. At a linear resolution of 0.94 lt-year, the VLBA source in NGC4258 is resolved into two jets elongated over 5.9 lt-years. Similar jets, if present in NGC4395, could account for some of the VLA flux density missing from Figure 1: a deeper VLBA image of NGC4395 is required to search for such jets. The present VLBA detection could then correspond to the brightest region in the jets, an hypothesis that could be tested with VLBA imaging at higher linear resolution. On energetics grounds it is reasonable also to explore a stellar origin for the Seyfert activity in NGC4395 [@fil93; @lir99]. In the radio regime, stellar processes could result in a compact starburst, a supernova remnant resembling Galactic ones, or a compact supernova remnant. For a compact starburst, @con91 use an empirical scaling relation between thermal and nonthermal radio continuum, based on more extended star-forming galaxies with thermal electron temperature $T_{\rm e} = 10^4$ K, to derive a limiting brightness temperature of $T_{\rm b} \lesssim 10^5$ K at 1.4 GHz. The VLBA source in NGC4395, with brightness temperature $T_{\rm b} > 2.0\times10^6$ K, clearly exceeds this limit, excluding a compact starburst origin. The Galactic supernova remnant CassiopeiaA, of age $t \lesssim 400$ years, has spectral powers and a spectral index [@baa77; @gre84; @fil93] very similar to those derived from the VLA detections of NGC4395 [@ho01a]. These photometric similarities are displayed in Figure 2. The upper limit $d = 23$ lt-years to the diameter of the VLA source in NGC4395 is also consistent with the diameter $d = 13$ lt-years for CassiopeiaA [@gre84]. But the VLBA source in NGC4395 is about a third as powerful (Figure 2) and considerably more compact ($d < 0.46$ lt-year) than CassiopeiaA. This VLBA size constraint strongly excludes further analogy with CassiopeiaA. Could the VLBA source in NGC4395 be a compact supernova remnant (cSNR) whose evolution is governed by a supernova expanding into dense circumstellar material [@ter95]? Such an interpretation encounters some successes and some difficulties [@fil93; @lir99]. In the optical regime, published images of NGC4395 plus the POSS revealed the presence of a starlike nucleus, visible at similar brightness levels since 1956 May 8 UT and implying an age $t \gtrsim 36$ years in 1992. @lir99 find that, while this slow blue photometric evolution could just be a signpost of a cSNR of age $t \sim 300$ years, the observed line properties roughly match those expected for a cSNR of age $t \sim 34$ years. The large-amplitude variability observed on time scales of days to months, in both the optical and X-ray regimes, remains a problem for the cSNR model. In the radio regime, the record of photometric evolution of the VLA source, while sparse, supports approximate constancy over 1–2 decades at 1.4 and 4.9 GHz, since 1982 [@mor99; @ho01a]. Also, @hee64 used the NRAO 91-m telescope in 1963 to set an upper limit of 0.2 Jy for the peak flux density of NGC4395 at 1.4 GHz. Is this record consistent with a cSNR origin? Models for cSNRs do not, as yet, predict radio light curves, although @ter95 do remark on the relevance of SN1988Z, a radio supernova, to their models. Figure 3 shows the model light curves for SN1988Z at 1.4 and 4.9 GHz, based on three years of monitoring in the radio regime [@van93] but extrapolated to an age of 50 years. The powers of the VLA detections of NGC4395, measured on 1982 Feb 8 UT and 1990 Mar 3 UT by @mor99 with matched resolutions, and on 1999 Aug 29 and Oct 31 UT by @ho01a also with matched resolutions, are plotted at their minimum possible ages assuming a reference date of 1956 May 8 UT. (No point appears for the VLBA detection due to concern about the effects of source resolution between VLA and VLBA scales.) The power of the 91-m upper limit, obtained during 1963 between Feb 1 UT and Dec 31 UT, is also plotted at the minimum possible age of about 7.2 years relative to the same reference date. Figure 3 shows that NGC4395 is observed to be less powerful than SN1988Z, by factors ranging from at least 60 at an age of about 7 years to about 100–700 at an age of 43 years. This latter discrepancy could reflect the failure of the SN1988Z model after a few decades but model problems are unlikely through the first decade [@hym95]. The spectral powers of NGC4395 at 1.4 and 4.9 GHz thus appear to be too low since 1963 and too steady since 1982 to allow analogy with SN1988Z. The former trait is more constraining than the latter, as the degree of variability could be artificially suppressed if only a small fraction of the VLA emission arises from a cSNR. In summary, for the VLBA source in NGC4395, the constraints on $T_b$, $d\/$, and $P\/$ are consistent with an origin from a black hole but exclude an origin from a compact starburst or a supernova remnant like CassiopeiaA. Also, the spectral powers of NGC4395 at 1.4 and 4.9 GHz seem both too low and too stable for analogy with SN1988Z, a suggested prototype for models of compact supernova remnants. Future studies of the VLBA source in NGC4395 should focus on measuring its spectral index, reducing the upper limit to its diameter or seeking evidence for jet-like structures, assessing its astrometric stability (cf. Wrobel 2000), and imaging the missing VLA flux density. Like other Seyfert galaxies, NGC4395 exhibits copious evidence for thermal nuclear plasma. Estimates of the associated free-free optical depths $\tau_{\rm ff}$ at 1.4 GHz can constrain the relative geometries of the thermal and nonthermal plasmas. Equation 1 of @ulv99 reduces to $\tau_{\rm ff} = 0.038~T_{\rm e}^{-1.35}~E$ at 1.4 GHz for a thermal plamsa with electron temperature $T_{\rm e}$ in units of K and emission measure $E\/$ in units of cm$^{-6}$ pc. While $T_{\rm e}$ is either readily observed or plausibly estimated, neither condition generally applies to $E$. However, a notable exception is the thermal plasma responsible for the variable warm X-ray absorber in NGC4395 [@iwa00]. For this plasma, $T_{\rm e} = 10^6$ K is plausibly adopted, the column density is measured, and the variability time scale sets a lower limit to the density. The latter two quantities imply $E \gtrsim 2 \times 10^{13}$ cm$^{-6}$ pc, which, in combination with $T_{\rm e} = 10^6$ K, leads to $\tau_{\rm ff} \gtrsim 6000$. The variable warm X-ray absorber in NGC4395 has a free-free optical depth much larger than unity at 1.4 GHz and, therefore, cannot fully cover the VLBA source. Arguments such as these should be folded into discussions of unifying structures for the inner regions of quasars (eg, Elvis 2000). The authors thank Dr. J. Ulvestad for discussions. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, Caltech, under contract with the National Aeronautics and Space Administration. NRAO is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. Baars, J. W. M., Genzel, R., Pauliny-Toth, I. I. K., & Witzel, A. 1977, , 61, 99 Ball, J. A. 1975, in Methods in Computational Physics, Volume 14, eds. B. Alder, S. Fernbach, & M. Rotenberg (New York: Academic Press), 177 Bietenholz, M.F., Bartel, N., & Rupen, M.P. 2000, , 532, 895 Cecil, G., et al. 2000, , 536, 675 Condon, J. J., Huang, Z.-P., Yin, Q. F., & Thuan, T. X. 1991, , 378, 65 Cotton, W. D., Condon, J. J., & Arbizzani, E. 1999, , 125, 409 Elvis, M. 2000, , 545, 63 Falcke, H., Goss, W. M., Matsuo, H., Teuben, P., Zhao, J.-H., & Zylka, R. 1998, , 499, 731 Falcke, H., & Markoff, S. 2000, , 362, 113 Filippenko, A. V., & Sargent, W. L. W. 1989, , 342, L11 Filippenko, A. V., Ho, L. C., & Sargent, W. L. W. 1993, , 410, L75 Filippenko, A. V., & Ho, L. C. 2001, , submitted Green, D. A. 1984, , 209, 449 Heeschen, D. S., & Wade, C. M. 1964, , 69, 277 Ho, L. C., & Ulvestad, J. S. 2001, , 133, 77 Ho, L. C., et al. 2001, , 549, L51 Hyman, S. D., Van Dyk, S. D., Sramek, R. A., & Weiler, K. W. 1995, , 443, L77 Iwasawa, K., Fabian, A. C., Almaini, O., Lira, P., Lawrence, A., Hayashida, K., & Inoue, H. 2000, , 318, 879 Lira, P., Lawrence, A., O’Brien, P., Johnson, R. A., Terlevich, R., & Bannister, N. 1999, , 305, 109 Ma, C., et al. 1998, , 116, 516 Magorrian, J., et al. 1998, , 115, 2285 Moran, E. C., Filippenko, A. V., Ho, L. C., Shields, J. C., Belloni, T., Comastri, A., Snowden, S. L., & Sramek, R. A. 1999, , 111, 801 Napier, P. J., Bagri, D. S., Clark, B. G., Rogers, A. E. E., Romney, J. D., Thompson, A. R., & Walker, R. C. 1994, Proc. IEEE, 82, 658 Pedlar, A., Anatharmaiah, K. R., Ekers, R. D., Goss, W. M., van Gorkom, J. H., Schwarz, U. J., & Zhao, J.-H. 1989, , 342, 769 Reid, M. J., Readhead, A. C. S., Vermeulen, R. C., & Treuhaft, R. N. 1999, , 524, 816 Rowan-Robinson, M. 1985, The Cosmological Distance Ladder (New York: Freeman) Swaters, R. A., Schoenmakers, R. H. M., Sancisi, R., & van Albada, T. S. 1999, , 304, 330 Terlevich, R., Tenorio-Tagle, G., Rozyczka, M., Franco, J., & Melnick, J. 1995, , 272, 198 Ulvestad, J. S., Wrobel, J. M., Roy, A. L., Wilson, A. S., Falcke, H., & Krichbaum, T. P. 1999, , 517, L81 Ulvestad, J. S. 2000, VLBA Scientific Memorandum 25 Van Dyk, S. D., Weiler, K. W., Sramek, R. A., & Panagia, N. 1993, , 419, L69 van Zee, L., Salzer, J. J., Haynes, M. P., O’Donoghue, A. A., & Balonek, T. J. 1998, , 116, 2805 Wilkinson, P. N., Browne, I. W. A., Patnaik, A. R., Wrobel, J. M., & Sorathia, B. 1998, , 300, 790 Wrobel, J. M. 2000, , 531, 716 Wrobel, J. M., Walker, R. C., Benson, J. M, & Beasley, A. J., 2000, VLBA Scientific Memorandum 24 Wrobel, J. M., Machalski, J., & Condon, J. J. 2001, in preparation Figure 1 Figure 2 Figure 3

--- author: - Thomas Eiter - Axel Polleres date: January 2004 title: | Towards Automated Integration\ of Guess and Check Programs\ in Answer Set Programming:\ A Meta-Interpreter and Applications --- Introduction ============ Answer set programming (ASP) [@asp-2001; @gelf-lifs-91; @lifs-2002; @mare-trus-99; @niem-99], also called A-Prolog [@bald-gelf-2003; @bara-2002; @gelf-2002], is widely proposed as a useful tool for solving problems in a declarative manner, by encoding the solutions to a problem in the answer sets of a normal logic program. By well-known complexity results, in this way all problems with complexity in ${\mathrm{NP}}$ can be expressed and solved [@schl-95; @mare-remm-2003]; see also [@dant-etal-01]. A frequently considered example of an ${\mathrm{NP}}$-complete problem which can be elegantly solved in ASP is Graph-3-Colorability, i.e., deciding whether some given graph $G$ is 3-colorable. It is an easy exercise in ASP to write a program which determines whether a graph is 3-colorable. A straightforward encoding, following the “Guess and Check” [@eite-etal-2000c; @leon-etal-2002-dlv] respectively “Generate/Define/Test” approach [@lifs-2002], consists of two parts: - A “guessing” part, which assigns nondeterministically each node of the graph one of three colors: > $\mathtt{col(red,X)\ {\mbox{\texttt{v}\xspace}}\ col(green,X)\ {\mbox{\texttt{v}\xspace}}\ col(blue,X)\ {\mbox{\texttt{:\hspace{-0.15em}-}}\xspace}\ node(X).}$ - and a “checking” part, which tests whether no adjacent nodes have the same color: > $\mathtt{{\mbox{\texttt{:\hspace{-0.15em}-}}\xspace}edge(X,Y),\ col(C,X),\ col(C,Y).}$ Here, the graph $G$ is represented by a set of facts `node(x)` and `edge(x,y)`. Each legal 3-coloring of $G$ is a polynomial-size “proof” of its 3-colorability, and such a given proof can be validated in polynomial time. Furthermore, the answer sets of this program yield all legal 3-colorings of the graph $G$. However, we might encounter situations in which we want to express a problem which is complementary to some ${\mathrm{NP}}$ problem, and thus belongs to the class ${\textrm{co-}{\mathrm{NP}}}$. It is widely believed that in general, not all problems in ${\textrm{co-}{\mathrm{NP}}}$ are in ${\mathrm{NP}}$, and hence that it is not always the case that a polynomial-size “proof” of a ${\textrm{co-}{\mathrm{NP}}}$ property $P$ exists which can be verified in polynomial time. For such problems, we thus can not write a (polynomial-size propositional) normal logic program in ASP which guesses and verifies in its answer sets possible “proofs” of $P$. One such property, for instance, is the ${\textrm{co-}{\mathrm{NP}}}$-complete property that a given graph is [*not*]{} 3-colorable. However, this and similar properties $P$ can be dually expressed in ASP in terms of whether a normal logic program (equivalently, a head-cycle free disjunctive logic program [@bene-dech-94]) $\Pi_P$ has no answer set if and only if the property $p$ holds. Properties that are ${\textrm{co-}{\mathrm{NP}}}$-complete often occur within the context of problems that reside in the class ${{\Sigma}_{2}^{P}}$, which is above ${\mathrm{NP}}$ in the polynomial time hierarchy [@papa-94]. In particular, the solutions of a ${{\Sigma}_{2}^{P}}$-complete problem can be typically singled out from given candidate solutions by testing a ${\textrm{co-}{\mathrm{NP}}}$-complete property. Some well-known examples of such ${{\Sigma}_{2}^{P}}$-complete problems are the following ones, which will be further detailed in Section \[sec:applications\]: Quantified Boolean Formulas: : Evaluating a Quantified Boolean formula (QBF) of the form $\exists X\forall Y\Phi(X,Y)$, where $\Phi(X,Y)$ is a disjunctive normal form over propositional variables $X\cup Y$. Here, a solution is a truth value assignment $\sigma$ to the variables $X$ such that the formula $\forall Y\Phi(\sigma(X),Y)$ evaluates to true, i.e., $\Phi(\sigma(X),Y)$ is a tautology. Given a candidate solution $\sigma$, the ${\textrm{co-}{\mathrm{NP}}}$-complete property to check here is whether $\Phi(\sigma(X),Y)$ is a tautology. Strategic Companies: : Computing strategic companies sets [@eite-etal-2000c; @leon-etal-2002-dlv]. Roughly, here the problem is to compute, given a set of companies $C$ in a holding, a minimal subset $S \subseteq C$ which satisfies some constraints concerning the production of goods and control of companies. Any such set is called strategic; Given a candidate solution $S$ which satisfies the constraints, the ${\textrm{co-}{\mathrm{NP}}}$-complete property to check here is the minimality, i.e., that no set $S' \subset S$ exists which also satisfies the constraints. Conformant Planning: : Computing conformant plans under incomplete information and nondeterministic action effects. Here the problem is to generate from a description of the initial state $I$, the planning goal $G$, and the actions $\alpha$ and their effects a sequence of actions (a plan) $P = \alpha_1,\ldots,\alpha_n$ which carries the agent from the initial state to a goal-fulfilling state under all contingencies, i.e., regardless of the precise initial state and how non-deterministic actions work out. Given a candidate solution in terms of an [*optimistic*]{} plan $P$, which works under [ *some*]{} execution [@eite-etal-2001e], the property to check is whether it works under [*all* ]{} executions, i.e., whether it is conformant [@gold-bodd-96]. The latter problem is in ${\textrm{co-}{\mathrm{NP}}}$, provided that executability of actions is polynomially decidable, cf. [@eite-etal-2001e; @turn-2002]. This list can be extended, and further examples can be found, e.g., in [@eite-etal-97f; @eite-etal-2002-tplp; @grec-etal-2001; @saka-inou-2003]. The problems described above can be solved using ASP in a two-step approach as follows: 1. Generate a candidate solution $S$ by means of a logic program $\Pi_{guess}$. 2. Check the solution $S$ by “running” another logic program $\Pi_{check}$ (=$\Pi_p$) on $S$, such that $\Pi_{check}\cup S$ has no answer set if and only if $S$ is a valid solution. The respective programs $\Pi_{check}$ can be easily formulated (cf.Section \[sec:applications\]). On the other hand, ASP with disjunction, i.e. full extended disjunctive logic programming, allows one to formulate problems in ${{\Sigma}_{2}^{P}}$ in a [*single*]{} (disjunctive) program, since this formalism captures the complexity class ${{\Sigma}_{2}^{P}}$, cf.[@dant-etal-01; @eite-etal-97f]. Hence, efficient ASP engines such as [`DLV`]{} [@leon-etal-2002-dlv] or [[GnT]{}]{} [@janh-etal-2000] can be used to solve such programs directly in a one-step approach. A difficulty here is that sometimes, an encoding of a problem in a single logic program (e.g., for the conformant planning problem above) may not be easy to find. This raises the issue whether there exists an (effective) possibility to [*combine*]{} separate $\Pi_{guess}$ and $\Pi_{check}$ programs into a single program $\Pi_{solve}$, such that this unified program computes the same set of solutions as the two-step process outlined above. A potential benefit of such a combination is that the space of candidate solutions might be reduced in the evaluation due to its interaction with the checking part. Furthermore, automated program optimization techniques may be applied which consider both the guess and check part as well as the interactions between them. This is not possible for separate programs. The naive attempt of taking the union $\Pi_{guess} \cup \Pi_{check}$ unsurprisingly fails: indeed, each desired answer set of $\Pi_{guess}$ would be eliminated by $\Pi_{check}$ (assuming that, in a hierarchical fashion, $\Pi_{check}$ has no rules defining atoms from $\Pi_{guess}$). Therefore, some program transformation is necessary. A natural question here is whether it is possible to rewrite $\Pi_{check}$ to some other program $\Pi'_{check}$ such that an integrated logic program $\Pi_{solve} = \Pi_{guess} \cup \Pi'_{check}$ is feasible, and, moreover, whether this can be done automatically. From theoretical complexity results about disjunctive logic programs cf. [@dant-etal-01; @eite-etal-97f], one can infer that the program $\Pi'_{check}$ should be truly disjunctive in general, i.e., not rewritable to an equivalent non-disjunctive program in polynomial time. This and further considerations (see Section \[sec:trans\]) provide some evidence that a suitable rewriting of $\Pi_{check}$ to $\Pi'_{check}$ is not immediate. In this paper, we therefore address this issue and present a generic method for constructing the program $\Pi_{check}'$ by using a meta-interpreter approach. In particular, we make the following contributions: We provide a transformation ${\ensuremath{tr}\xspace}(\Pi)$ from propositional head-cycle-free [@bene-dech-94] (extended) disjunctive logic programs (HDLPs) $\Pi$ to disjunctive logic programs (DLPs), which enjoys the properties that the answer sets of ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ encode the answer sets of $\Pi$, if $\Pi$ has some answer set, and that ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ has a canonical answer set otherwise which is easy to recognize. The transformation ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ is polynomial and modular in the sense of [@janh-2000], and employs meta-interpretation of $\Pi$. Furthermore, we describe variants and modifications of ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ aiming at optimization of the transformation. In particular, we present a transformation to positive DLPs, and show that in a precise sense, modular transformations to such programs do not exist. We show how to use ${\ensuremath{tr}\xspace}(\cdot)$ for integrating separate guess and check programs $\Pi_{guess}$ and $\Pi_{check}$, respectively, into a single DLP $\Pi_{solve}$ such that the answer sets of $\Pi_{solve}$ yield the solutions of the overall problem. We demonstrate the method on the examples of QBFs, the Strategic Companies problem, and conformant planning [@gold-bodd-96] under fixed polynomial plan length (cf. [@eite-etal-2001e; @turn-2002]). Our method proves useful to loosen some restrictions of previous encodings, and to obtain disjunctive encodings for more general problem classes. We compare our approach on integrating separate guess and check programs experimentally against existing ad hoc encodings for QBFs and Strategic Companies and also applying it to conformant planning, where no such ad hoc encodings were known previously. For these experiments, we use [`DLV`]{} [@leon-etal-2002-dlv], a state-of-the-art Answer Set engine for solving DLPs. The results which we obtained reveal interesting aspects: While as intuitively expected, efficient ad hoc encodings have better performance than the synthesized integrated encodings in general, there are also cases where the performances scale similarly (i.e., the synthesized encoding is within a constant factor), or where even ad hoc encodings from the literature are outperformed. Our results contribute to further the “Guess and Check” resp.“Generate/Define/ Test” paradigms for ASP, and fill a gap by providing an automated construction for integrating guess and check programs. They relieve the user from the burden to use sophisticated techniques such as saturation, as employed e.g. in [@eite-etal-97f; @eite-etal-2000c; @leon-etal-2001], in order to overcome the technical intricacies in combining natural guess and check parts into a single program. Furthermore, our results complement recent results about meta-interpretation techniques in ASP, cf.[@mare-remm-2003; @delg-etal-01; @eite-etal-2002a]. The rest of this paper is organized a follows. In the next section, we very briefly recall the necessary concepts and fix notation. After that, we present in Section \[sec:trans\] our transformation ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ of a “checking” program $\Pi$ into a disjunctive logic program. We start there with making the informal desirable properties described above more precise, present the constituents of ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$, the factual program representation $F(\Pi)$ and a meta-interpreter $\Pi_{meta}$, and prove that our transformation satisfies the desirable properties. Section \[sec:optimizations\] thereafter is devoted to modifications towards optimization. In Section \[sec:integration\], we show how to synthesize separate guess and check programs to integrated encodings. Several applications are considered in Section \[sec:applications\], and experimental results for these are reported in Section \[sec:experiments\]. The final Section \[sec:conclusion\] gives a summary and presents issues for further research. Preliminaries {#sec:prelim} ============= We assume that the reader is familiar with logic programming and answer set semantics, see [@gelf-lifs-91; @asp-2001], and only briefly recall the necessary concepts. A *literal* is an atom $a(t_1, \ldots,t_n)$, or its negation ${\ensuremath{\neg}}{a(t_1, \ldots,t_n)}$, where “[$\neg$]{}” is the strong negation symbol, for which we also use the customary “–”, in a function-free first-order language (including at least one constant), which is customarily given by the programs considered. We write $|a| = |{\ensuremath{\neg}}{a}| = a$ to denote the atom of a literal. Extended disjunctive logic programs (EDLPs; or simply programs) are disjunctive logic programs with default (weak) and strong negation, i.e., finite sets $\Pi$ of rules $r$ $$\label{stmt:lprule} h_1 {\mbox{\texttt{v}\xspace}}\ \ldots\ {\mbox{\texttt{v}\xspace}}\ h_l\ {\mbox{\texttt{:\hspace{-0.15em}-}}\xspace}\ b_1,\ \ldots,\ b_m,\ {\ensuremath{\mathtt{not}\ }b_{m+1}},\ \ldots\ {\ensuremath{\mathtt{not}\ }b_n}.$$ $l,m,n \geq 0$, where each $h_i$ and $b_j$ is a literal and is weak negation (negation as failure). By ${\ensuremath{H(r)}} = \{h_1, \ldots, h_l\}$, ${\ensuremath{B^+(r)}} = \{b_1,\ldots, b_m\}$, ${\ensuremath{B^-(r)}} = \{b_{m+1},\ldots, b_n\}$, and ${\ensuremath{B(r)}} = {\ensuremath{B^+(r)}} \cup {\ensuremath{B^-(r)}}$ we denote the head and (positive, resp.negative) body of rule $r$. Rules with $|{\ensuremath{H(r)}}|{=}1$ and ${\ensuremath{B(r)}}{=}\emptyset$ are called *facts* and rules with ${\ensuremath{H(r)}}{=}\emptyset$ are called *constraints*. For convenience, we omit “extended” in what follows and refer to EDLPs as DLPs etc. Literals (resp. rules, programs) are *ground* if they are variable-free. Non-ground rules (resp. programs) amount to their *ground instantiation*, i.e., all rules obtained by substituting variables with constants from the (implicit) language. Rules (resp. programs) are *positive*, if “${\ensuremath{\mathtt{not}}\xspace}$” does not occur in them, and *normal*, if $|{\ensuremath{H(r)}}| \leq 1$. A ground program $\Pi$ is *head-cycle free* [@bene-dech-94], if no literals $l\neq l'$ occurring in the same rule head mutually depend on each other by positive recursion; $\Pi$ is stratified [@przy-89b; @przy-91], if no literal $l$ depends by recursion through negation on itself (counting disjunction as positive recursion). The [*answer set semantics*]{} [@gelf-lifs-91] for DLPs is as follows. Denote by $Lit(\Pi)$ the set of all ground literals for a program $\Pi$. Consider first positive (ground) programs $\Pi$. Let $S \subseteq Lit(\Pi)$ be a set of consistent literals. Such a set $S$ satisfies a positive rule $r$, if ${\ensuremath{H(r)}} \cap S \not= \emptyset$ whenever ${\ensuremath{B^+(r)}} \subseteq S$. An *answer set* for $\Pi$ then is a minimal (under $\subseteq$) set $S$ satisfying all rules.[^1] To extend this definition to programs with weak negation, the *reduct* $\Pi^S$ of a program $\Pi$ with respect to a set of literals $S$ is the set of rules $$h_1\ {\mbox{\texttt{v}\xspace}}\ \ldots\ {\mbox{\texttt{v}\xspace}}\ h_l\ {\mbox{\texttt{:\hspace{-0.15em}-}}\xspace}\ b_1,\ \ldots,\ b_m$$ for all rules (\[stmt:lprule\]) in $\Pi$ such that $S \cap {\ensuremath{B^-(r)}} = \emptyset$. Then $S$ is an [*answer set*]{} of $\Pi$, if $S$ is an answer set for $\Pi^S$. There is a rich literature on characterizations of answer sets of DLPs and restricted fragments; for our concerns, we recall here the following characterization of (consistent) answer sets for HDLPs, given by Ben-Eliyahu and Dechter [-@bene-dech-94]: \[theo:hedlp\] Given a ground HDLP $\Pi$, a consistent $S \subseteq Lit(\Pi)$ is an answer set iff 1. $S$ satisfies each rule in $\Pi$, and 2. there is a function $\phi: Lit(\Pi) \mapsto \Nat$ such that for each literal $l$ in $S$ there is a rule $r$ in $\Pi$ with 1. ${\ensuremath{B^+(r)}} \subseteq S$ 2. ${\ensuremath{B^-(r)}} \cap S = \emptyset$ 3. $l \in {\ensuremath{H(r)}}$ 4. $S \cap ({\ensuremath{H(r)}}\setminus \{l\}) = \emptyset$ 5. $\phi(l') < \phi(l)$ for each $l' \in {\ensuremath{B^+(r)}}$ We will use Theorem \[theo:hedlp\] as a basis for the transformation ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ in the next section. Meta-Interpreter Transformation {#sec:trans} =============================== As discussed in the Introduction, rewriting a given check program $\Pi_{check}$ to a program $\Pi_{check}'$ for integration with a separate guess program $\Pi_{guess}$ into a single program $\Pi_{solve} = \Pi_{guess}\cup \Pi'_{check}$ can be difficult in general. The problem is that the working of the answer set semantics, to be emulated in $\Pi_{check}'$, is not easy to express there. One difficulty is that for a given answer set $S$ of $\Pi_{guess}$, we have to test the [*non-existence*]{} of an answer set of $\Pi_{check}$ with respect to $S$, while $\Pi_{solve}$ [*should have an answer set*]{} extending $S$ to $\Pi'_{check}$ if the check succeeds. A possibility to work around this problem is to design $\Pi'_{check}$ in a way such that it has a dummy answer set with respect to $S$ if the check of $\Pi_{check}$ on $S$ succeeds, and no answer set if the check fails, i.e., if $\Pi_{check}$ has some answer set on $S$. While this may not look to be very difficult, the following observations suggest that this is not straightforward. Since $\Pi_{solve}$ may need to solve a ${{\Sigma}_{2}^{P}}$-complete problem, any suitable program $\Pi'_{check}$ must be truly disjunctive in general, i.e., contain disjunctions which are not head-cycle free (assuming that no head literal in $\Pi'_{check}$ occurs in $\Pi_{guess}$). Indeed, if both $\Pi_{guess}$ and $\Pi'_{check}$ are head-cycle free, then also $\Pi_{solve} = \Pi_{guess} \cup \Pi'_{check}$ is head-cycle free, and thus can only express a problem in ${\mathrm{NP}}$. Furthermore, we can make in $\Pi_{check}'$ only limited use of default negation on atoms which do not occur in $\Pi_{guess}$. The reason is that upon a “guess” $S$ for an answer set of $\Pi_{solve} = \Pi_{guess} \cup \Pi_{check}'$, the reduct $\Pi_{solve}^S$ is ${\ensuremath{\mathtt{not}}\xspace}$-free. Contrary to the case of $\Pi_{check}$ in the two-step approach, it is not possibile to explicitly consider for a guess $S_{guess}$ of an answer set of $\Pi_{guess}$ varying extensions $S = S_{guess}\cup S'_{check}$ to the whole program $\Pi_{solve}$ which activate different rules in $\Pi'_{check}$ (e.g., unstratified clauses $a {\mbox{\texttt{:\hspace{-0.15em}-}}\xspace}{\ensuremath{\mathtt{not}}\xspace}\,b$ and $b{\mbox{\texttt{:\hspace{-0.15em}-}}\xspace}{\ensuremath{\mathtt{not}}\xspace}\, a$ encoding a choice among $a$ and $b$). Therefore, default negation in rules of $\Pi_{check}$ must be handled with care and might cause major rewriting as well. These observations provide some evidence that a rule-rewriting approach for obtaining $\Pi'_{check}$ from $\Pi_{check}$ may be complicated. For this reason, we adopt at a generic level a Meta-interpreter approach, in which the ${\textrm{co-}{\mathrm{NP}}}$-check modeled by $\Pi_{check}$ is “emulated” by a minimality check for a positive DLP $\Pi_{check}'$. Basic approach -------------- The considerations above lead us to an approach in which the program $\Pi_{check}'$ is constructed by the use of meta-interpretation techniques [@mare-remm-2003; @delg-etal-01; @eite-etal-2002a]. The idea behind meta-interpretation is here that a program $\Pi$ is represented by a set of facts, $F(\Pi)$, which is input to a fixed program $\Pi_{meta}$, the meta-interpreter, such that the answer sets of $\Pi_{meta} \cup F(\Pi)$ correspond to the answer sets of $\Pi$. Note that the meta-interpreters available are normal logic programs (including arbitrary negation), and can not be used for our purposes for the reasons explained above. We thus have to construct a novel meta-interpreter which is essentially ${\ensuremath{\mathtt{not}}\xspace}$-free, i.e. uses negation as failure only in a restricted way, and contains disjunction. Basically, we present a general approach to translate normal LPs and HDLPs into stratified disjunctive logic programs. To this end, we exploit Theorem \[theo:hedlp\] as a basis for a transformation ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ from a given HDLP $\Pi$ to a DLP ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace} = F(\Pi) \cup \Pi_{meta}$ such that ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ fulfills the properties mentioned in the introduction. More precisely, it will satisfy the following properties: T0 : ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ is computable in time polynomial in the size of $\Pi$. T1 : Each answer set $S'$ of the transformed program ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ corresponds to an answer set $S$ of $\Pi$, such that $S = \{ l \mid \texttt{inS}(l) \in S'\}$ for some predicate `inS(\cdot)`, provided $\Pi$ is consistent, and conversely, each answer set $S$ of $\Pi$ corresponds to some answer set $S'$ of ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ such that $S = \{ l \mid \texttt{inS}(l) \in S'\}$. T2 : If the program $\Pi$ has no answer set, then ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ has exactly one designated answer set $\Omega$, which is easily recognizable. T3 : The transformation is of the form ${\ensuremath{tr}\xspace}(\Pi) = F(\Pi) \cup \Pi_{meta}$, where $F(\Pi)$ is a factual representation of $\Pi$ and $\Pi_{meta}$ is a fixed meta-interpreter. T4 : ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ is [*modular*]{} (at the syntactic level), i.e., ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace} = \bigcup_{r\in \Pi} {\ensuremath{{\ensuremath{tr}\xspace}(r)}\xspace}$ holds. Moreover, ${\ensuremath{tr}\xspace}(\Pi)$ returns a stratified DLP [@przy-89b; @przy-91] which uses negation only in its “deterministic” part. Note that properties [**T0**]{} – [**T4**]{} for ${\ensuremath{{\ensuremath{tr}\xspace}(\cdot)}\xspace}$ are similar yet different from the notion of polynomial faithful modular (PFM) transformation by Janhunen [-@janh-2000; -@janh-2001], which is a function $Tr$ mapping a class of logic programs $\cal C$ to another class $\cal C'$ of logic programs (where $\cal C'$ is assumed to be a subclass or superclass of $\cal C$), such that the following three conditions hold: (1) For each program $\Pi\in {\cal C}$, $Tr(\Pi)$ is computable in polynomial time in the size of $\Pi$ (called [*polynomiality*]{}), (2) the Herbrand base of $\Pi$, ${\ensuremath{\mathit H\!b}\xspace}(\Pi)$, is included in the Herbrand base of $Tr(\Pi)$, ${\ensuremath{\mathit H\!b}\xspace}(Tr(\Pi))$ and the models/interpretations of $\Pi$ and $Tr(\Pi)$, are in one-to-one correspondence and coincide up to ${\ensuremath{\mathit H\!b}\xspace}(\Pi)$ ([*faithfulness*]{}), and (3) $Tr(\Pi_1\cup \Pi_2)$ = $Tr(\Pi_1)\cup Tr(\Pi_2)$ for all programs $\Pi_1,\Pi_1$ in ${\cal C}$ and ${\cal C}'\subseteq {\cal C}$ implies $Tr(\Pi)=\Pi$ for all $\Pi$ in ${\cal C}'$ ([*modularity*]{}). Compared to PFM, also our transformation ${\ensuremath{{\ensuremath{tr}\xspace}(\cdot)}\xspace}$ is polynomially computable by [**T0**]{} and hence satisfies condition 1). Moreover, by [**T4**]{} and the fact that stratified disjunctive programs are not necessarily head-cycle free, it also satisfies condition 3). However, condition 2) fails. Its first part, that ${\ensuremath{\mathit H\!b}\xspace}(\Pi) \subseteq {\ensuremath{\mathit H\!b}\xspace}({\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace})$ and that answer sets coincide on $Lit(\Pi)$ could be fulfilled by adding rules $l\ {\mbox{\texttt{:\hspace{-0.15em}-}}\xspace}\ \texttt{inS}(l)$ for every $l \in Lit(\Pi)$); these polynomially many rules could be added during input generation. The second part of condition 2) is clearly in contradiction with [**T2**]{}, since for $\Omega$ never a corresponding answer set of $\Pi$ exists. Moreover, condition [**T1**]{} is a weaker condition than the one-to-one correspondence between the answer sets of $\Pi$ and ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ required for faithfulness: In fact, in case $\Pi$ has positive cycles, there might be several possible guesses for $\phi$ for an answer set $S$ of $\Pi$ in Theorem \[theo:hedlp\] reflected by different answer sets of ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$. We illustrate this by a short example: Let $\Pi$ be the program consisting of the following four rules:     a  b.  b  a.  a.  b. Then, $\Pi$ has a single answer set $S = {\ensuremath{\{{\ensuremath{\mathtt{a,b}}}\}}}$, while ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ has two answer sets such that $S_1 = {\ensuremath{\{{\ensuremath{\mathtt{inS(a),inS(b), phi(a,b), \ldots}}}\}}}$ and $S_2 = {\ensuremath{\{{\ensuremath{\mathtt{inS(a),inS(b), phi(b,a), \ldots}}}\}}}$, intuitively reflecting that here the order of applications of rules $r1$ and $r2$ does not matter, although they are cyclic. We remind that the different properties of our transformation ${\ensuremath{{\ensuremath{tr}\xspace}(\cdot)}\xspace}$ and PFM transformations is not an accident but a necessary feature, since we want to express nonexistence of certain answer sets via the transformation, and not merely preserve the exact semantics as targeted by PFM. Apart from this different objective, the other properties involved (polynomiality and modularity) are in effect the same. Input representation $F(\Pi)$ {#sec:transinput} ----------------------------- As input for our meta-interpreter $\Pi_{meta}$, which will be introduced in the next subsection, we choose the representation $F(\Pi)$ of the propositional program $\Pi$ defined below. We assume that each rule $r$ has a unique name $n(r)$; for convenience, we identify $r$ with $n(r)$. Let $\Pi$ be any ground (propositional) HDLP. The set $F(\Pi)$ consists of the facts XXX= `lit(h,l,r).  atom(l,|l|).` = for each literal $l$ $\in {\ensuremath{H(r)}}$,\ `lit(p,l,r).` for each literal $l \in {\ensuremath{B^+(r)}}$,\ `lit(n,l,r).` for each literal $l \in {\ensuremath{B^-(r)}}$, for every rule $r \in \Pi$. While the facts for predicate `lit` obviously encode the rules of $\Pi$, the facts for predicate `atom` indicate whether a literal is classically positive or negative. We only need this information for head literals; this will be further explained below. Meta-Interpreter $\Pi_{meta}$ {#sec:transmeta} ----------------------------- We construct our meta-interpreter program $\Pi_{meta}$, which in essence is a positive disjunctive program, in a sequence of several steps. They center around checking whether a guess for an answer set $S \subseteq Lit(\Pi)$, encoded by a predicate `inS(\cdot)`, is an answer set of $\Pi$ by testing the criteria of Theorem \[theo:hedlp\]. The steps of the transformation cast the various conditions there into rules of $\Pi_{meta}$, and also provide auxiliary machinery which is needed for this aim. #### Step 1 We add the following preprocessing rules: :    rule(L,R) lit(h,L,R), not lit(p,L,R), not lit(n,L,R).\ : ruleBefore(L,R) rule(L,R), rule(L,R1), R1 &lt; R.\ : ruleAfter(L,R) rule(L,R), rule(L,R1), R &lt; R1.\ : ruleBetween(L,R1,R2) rule(L,R1), rule(L,R2), rule(L,R3),\ R1 &lt; R3, R3 &lt; R2.\ : firstRule(L,R) rule(L,R), [$\mathtt{not}\ $ruleBefore(L,R)]{}.\ : lastRule(L,R) rule(L,R), [$\mathtt{not}\ $ruleAfter(L,R)]{}.\ : nextRule(L,R1,R2) rule(L,R1), rule(L,R2), R1 &lt; R2,\ [$\mathtt{not}\ $ruleBetween(L,R1,R2)]{}.\ : before(HPN,L,R) lit(HPN,L,R), lit(HPN,L1,R), L1 &lt; L.\ : after(HPN,L,R) lit(HPN,L,R), lit(HPN,L1,R), L &lt; L1.\ : between(HPN,L,L2,R) lit(HPN,L,R), lit(HPN,L1,R),\ lit(HPN,L2,R), L&lt;L1, L1&lt;L2.\ : next(HPN,L,L1,R) lit(HPN,L,R), lit(HPN,L1,R), L &lt; L1,\ [$\mathtt{not}\ $between(HPN,L,L1,R)]{}.\ : first(HPN,L,R) lit(HPN,L,R), [$\mathtt{not}\ $before(HPN,L,R)]{}.\ : last(HPN,L,R) lit(HPN,L,R), [$\mathtt{not}\ $after(HPN,L,R)]{}.\ : hlit(L) rule(L,R). Lines 1 to 7 fix an enumeration of the rules in $\Pi$ from which a literal $l$ may be derived, assuming a given order `<` on rule names (e.g. in [`DLV`]{}, built-in lexicographic order; `<` can also be easily generated using guessing rules). Note that under answer set semantics, we need only to consider rules where the literal $l$ to prove does not occur in the body. Lines 8 to 13 fix enumerations of ${\ensuremath{H(r)}}$, ${\ensuremath{B^+(r)}}$ and ${\ensuremath{B^-(r)}}$ for each rule. The final line 14 collects all literals that can be derived from rule heads. Note that the rules on lines 1-14 plus $F(\Pi)$ form a stratified program, which has a single answer set, cf. [@przy-89b; @przy-91]. #### Step 2 Next, we add rules which “guess” a candidate answer set $S \subseteq Lit(\Pi)$ and a total ordering `phi` on $S$ corresponding with the function $\phi$ in condition $2$ of Theorem \[theo:hedlp\]. We will explain this correspondence in more detail below (cf. proof of Theorem \[theo:corr\]). :    īnS(L)  ninS(L) hlit(L).\ : ninS(L) lit($pn$,L,R), [$\mathtt{not}\ $hlit(L)]{}.          \ :    notok inS(L), inS(NL), L[$\mathtt{\,!\!\!=}\,$]{}NL, atom(L,A), atom(NL,A).\ : phi(L,L1)  phi(L1,L) inS(L), inS(L1), L &lt; L1.\ : phi(L,L2) :- phi(L,L1),phi(L1,L2). Line 15 focuses the guess of $S$ to literals occurring in some relevant rule head in $\Pi$; only these can belong to an answer set $S$, but no others (line 16). Line 17 then checks whether $S$ is consistent, deriving a new distinct atom `notok` otherwise. Line 18 guesses a strict total order `phi` on `inS` where line 19 guarantees transitivity; note that minimality of answer sets prevents that `phi` is cyclic, i.e., that `phi(L,L)` holds. In the subsequent steps, we will check whether $S$ and `phi` violate the conditions of Theorem \[theo:hedlp\] by deriving the distinct atom `notok` (considered in Step 5 below) in case, indicating that $S$ is not an answer set or `phi` does not represent a proper function $\phi$. #### Step 3 Corresponding to condition $1$ in Theorem \[theo:hedlp\], `notok` is derived whenever there is an unsatisfied rule by the following program part: :   āllInSUpto(p,Min,R) inS(Min), first(p,Min,R).\ : allInSUpto(p,L1,R) īnS(L1), allInSUpto(p,L,R), next(p,L,L1,R).\ : allInS(p,R) allInSUpto(p,Max,R),last(p,Max,R). $\left.\mbox{\begin{minipage}{0.9\textwidth} \begin{tabbing} {\addtocounter{bctr}{1}\thebctr}:\ \ \ \=allNinSUpto($hn$,Min,R) {\mbox{\texttt{:\hspace{-0.15em}-}}\xspace}ninS(Min), first($hn$,Min,R).\\[0.33ex] {\addtocounter{bctr}{1}\thebctr}: \>allNinSUpto($hn$,L1,R) {\mbox{\texttt{:\hspace{-0.15em}-}}\xspace}\=ninS(L1), allNinSUpto($hn$,L,R),\\ \>\>next($hn$,L,L1,R).\\[0.33ex] {\addtocounter{bctr}{1}\thebctr}: \>allNinS($hn$,R) {\mbox{\texttt{:\hspace{-0.15em}-}}\xspace}allNinSUpto($hn$,Max,R), last($hn$,Max,R). \end{tabbing} \end{minipage} \hspace{-1ex}}\right\}$ :   hasHead(R) lit(h,L,R).\ : hasPBody(R) lit(p,L,R).\ : hasNBody(R) lit(n,L,R).\ : allNinS(h,R) lit(HPN,L,R), [$\mathtt{not}\ $hasHead(R)]{}.\ : allInS(p,R) lit(HPN,L,R), [$\mathtt{not}\ $hasPBody(R)]{}.\ : allNinS(n,R) lit(HPN,L,R), [$\mathtt{not}\ $hasNBody(R)]{}.\ : notok allNinS(h,R), allInS(p,R), allNinS(n,R), lit(HPN,L,R). These rules compute by iteration over ${\ensuremath{B^+(r)}}$ (resp. ${\ensuremath{H(r)}}$, ${\ensuremath{B^-(r)}}$) for each rule $r$, whether for all positive body (resp. head and default negated body) literals in rule $r$ `inS` holds (resp. `ninS` holds) (lines 20 to 25). Here, empty heads (resp. bodies) are interpreted as unsatisfied (resp. satisfied), cf. lines 26 to 31. The final rule 32 fires exactly if one of the original rules from $\Pi$ is unsatisfied. #### Step 4 We derive `notok` whenever there is a literal $l\in S$ which is not provable by any rule $r$ with respect to `phi`. This corresponds to checking condition $2$ from Theorem \[theo:hedlp\]. :   failsToProve(L,R) rule(L,R), lit(p,L1,R), ninS(L1).\ : failsToProve(L,R) rule(L,R), lit(n,L1,R), inS(L1).\ : failsToProve(L,R) rule(L,R), rule(L1,R), inS(L1), L1[$\mathtt{\,!\!\!=}\,$]{}L, inS(L).\ : failsToProve(L,R) rule(L,R), lit(p,L1,R), phi(L1,L).\ : allFailUpto(L,R) failsToProve(L,R), firstRule(L,R).\ : allFailUpto(L,R1) failsToProve(L,R1), allFailUpto(L,R),\ nextRule(L,R,R1).\ : notok allFailUpto(L,R), lastRule(L,R), inS(L). Lines 33 and 34 check whether condition $2.(a)$ or $(b)$ are violated, i.e. some rule can only prove a literal if its body is satisfied. Condition $2.(d)$ is checked in line 35, i.e. $r$ fails to prove $l$ if there is some $l'\neq l$ such that $l' \in {\ensuremath{H(r)}}\cap S$. Violations of condition $2.(e)$ are checked in line 36. Finally, lines 37 to 39 derive `notok` if all rules fail to prove some literal $l\in S$. This is checked by iterating over all rules with $l\in {\ensuremath{H(r)}}$ using the order from Step 1. Thus, condition $2.(c)$ is implicitly checked by this iteration. #### Step 5 Whenever `notok` is derived, indicating a wrong guess, then we apply a saturation technique as in [@eite-etal-97f; @eite-etal-2000c; @leon-etal-2001] to some other predicates, such that a canonical set $\Omega$ results. This set turns out to be an answer set iff no guess for $S$ and $\phi$ works out, i.e., $\Pi$ has no answer set. In particular, we saturate the predicates `inS`, `ninS`, and `phi` by the following rules: :   = phi(L,L1) =notok, hlit(L), hlit(L1).\ : inS(L) notok, hlit(L).\ : ninS(L) notok, hlit(L). Intuitively, by these rules, any answer set containing `notok` is “blown up” to an answer set $\Omega$ containing all possible guesses for `inS`, `ninS`, and `phi`. The program $\Pi_{meta}$ consists of the rule 1–42 from above. We then can formally define our transformation ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ as follows. Given any ground HDLP $\Pi$, its transformation ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ is given by the DLP ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}= F(\Pi) \cup \Pi_{meta}$. Examples of ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ will be provided in Section \[sec:applications\]. Properties of ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ ------------------------------------------------------------------- We now show that ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ satisfies indeed the properties [**T0**]{} – [**T4**]{} from the beginning of this section. As for [**T0**]{}, we note the following proposition, which is not difficult to establish. \[prop:poly\] Given $\Pi$, the transformation ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ and its ground instantiation are both computable in logarithmic workspace (and thus in polynomial time). The input representation $F(\Pi)$ is easily generated in a linear scan of $\Pi$, using the rule numbers as names, for which a counter (representable in logspace) is sufficient. The meta-interpreter part $\Pi_{meta}$ is fixed anyway. A naive grounding of ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ can be constructed by instantiating each rule $r$ from $\Pi_{meta}$ with constants from $\Pi$ and rule ids in all possible ways; for each variable `X` in $r$, all constants of $\Pi$ can be systematically considered, using counters to mark the start and end position in $\Pi$ (viewed as a string), and the rule ids by a rule number counter. A constant number of such counters is sufficient. Thus, the grounding of ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ is constructible in logarithmic work space. Notice that intelligent, efficient grounding methods such as those used in [`DLV`]{}[@leon-etal-2002-dlv] usually generate a smaller ground program than this naive ground instantiation. Clearly, ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ satisfies property [**T3**]{}, and as easily checked, ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ is modular. Moreover, strong negation does not occur in ${\ensuremath{tr}\xspace}(\Pi)$ and weak negation only stratified. The latter is not applied to literals depending on disjunction; it thus occurs only in the deterministic part of ${\ensuremath{tr}\xspace}(\Pi)$, which means [**T4**]{} holds. To establish [**T1**]{} and [**T2**]{}, we define the literal set $\Omega$ as follows: Let $\Pi_{meta}^i$ be the set of rules in $\Pi_{meta}$ established in Step $i \in \{1,\ldots,5\}$. For any program $\Pi$, let $\Pi_{\Omega} = F(\Pi) \cup \bigcup_{i\in \{1,3,4,5\}}\Pi_{meta}^i \cup {\ensuremath{\{\texttt{notok}.\}}}$. Then, $\Omega$ is defined as the answer set of $\Pi_{\Omega}$. $\Omega$ is well-defined and uniquely determined by $\Pi$. (Sketch) This follows immediately from the fact that $\Pi_{\Omega}$ is a (locally) stratified normal logic program without ${\ensuremath{\neg}}{}$ and constraints, which as well-known has a single answer set. \[theo:corr\] For a given HDLP $\Pi$ the following holds for ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$: 1. ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ always has some answer set, and $S' \subseteq \Omega$ for every answer set $S'$ of ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$. 2. $S$ is an answer set of $\Pi$ $\Leftrightarrow$ there exists an answer set $S'$ of ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ such that $S={\ensuremath{\{ l \mid \mathtt{inS(\mbox{$l$})} \in S'\}}}$ and $\texttt{notok} \not\in S'$. 3. $\Pi$ has no answer set $\Leftrightarrow$ ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ has the unique answer set $\Omega$. $1.$ The first part follows immediately from the fact that ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ has no constraints, no strong negation, and weak negation is stratified; this guarantees the existence of at least one answer set $S$ of ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ [@przy-91]. Moreover, $S'\subseteq\Omega$ must hold for every answer set: after removing $\{\texttt{notok.}\}$ from $\Pi_{\Omega}$ and adding $\Pi_{meta}^{2}$, we obtain ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$. Note that any rule in $\Pi_{meta}^{2}$ fires with respect to $S'$ only if all literals in its head are in $\Omega$, and `inS`, `ninS`, and `phi` are elsewhere not referenced recursively through negation or disjunction. Therefore, increasing $S'$ locally to the value of $\Omega$ on `inS`, `ninS`, `phi`, and `notok`, and closing off thus increases it globally to $\Omega$, which means $S' \subseteq \Omega$. $2.$ ($\Rightarrow$)  Assume that $S$ is an answer set of $\Pi$. Clearly, then $S$ is a consistent set of literals which has a corresponding set $S''= {\ensuremath{\{ \texttt{inS}(l) \mid l \in S\}}}$ $\cup$ ${\ensuremath{\{\texttt{ninS}(l) \mid l \in Lit(\Pi)\setminus S\}}}$ being one possible guess by the rules in lines 15 to 17 of $\Pi_{meta}$. Let now $\phi: Lit(\Pi) \rightarrow \Nat$ be the function from Theorem \[theo:hedlp\] for answer set $S$: Without loss of generality, we may assume two restrictions on this function $\phi$: - $\phi(l) = 0$ for all $l\in Lit(\Pi)\setminus S$ and $\phi(l) > 0$ for all $l\in S$. - $\phi(l)\neq\phi(l')$ for all $l,l' \in S$. Then, the function $\phi$ can be mapped to a total order over $S$ `phi` such that $$\mathtt{phi(\mbox{$l,l'$})} \Leftrightarrow \phi(l) > \phi(l') > 0.$$ This relation `phi` fixes exactly one possible guess by the lines 18 and 19 of $\Pi_{meta}$. Note that it is sufficient to define `phi` only over literals in $S$: Violations of condition $2.(e)$ have only to be checked for rules with ${\ensuremath{B^+(r)}} \subseteq S$, as otherwise condition $2.(a)$ already fails. Obviously, condition 2.$(e)$ of Theorem \[theo:hedlp\] is violated with respect to $\phi$ iff (a) `phi(Y,X)` holds for some `X` in the head of a rule with `Y` in its positive body or (b) if `X` itself occurs in its positive body. While (a) is checked in lines 36, (b) is implicit by definition of predicate `rule` (line 1) which says that a literal can not prove itself. Given $S''$ and `phi` from above, we can now verify by our assumption that $S$ is an answer set and by the conditions of Theorem \[theo:hedlp\] that (a) `notok` can never be derived in ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ and (b) $S''$ and `phi` uniquely determine an answer set $S'$ of ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ of the form we want to prove. This can be argued by construction of Steps 3 and 4 of ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$, where `notok` will only be derived if some rule is unsatisfied (Step 3) or there is a literal in $S$ (i.e. $S''$) which fails to be proved by all other rules (Step 4). ($\Leftarrow$)Assume that $S'$ is an answer set of [${\ensuremath{tr}\xspace}(\Pi)$]{} not containing `notok`. Then by the guess of `phi` in Step 5 a function $\phi: Lit(\Pi) \rightarrow \Nat$ can be constructed by the implied total order of `phi` as follows: We number all literals $l \in S={\ensuremath{\{l \mid \mathtt{inS(\mbox{$l$})} \in S'\}}}$ according to that order from $1$ to $|S|$ and fix $\phi(l)=0$ for all other literals. Again, by construction of Steps 3 to 5 and the assumption that $\texttt{notok} \not\in S'$, we can see that $S$ and the function $\phi$ constructed fulfill all the conditions of Theorem \[theo:hedlp\]; in particular, line 17 guarantees consistency. Hence $S$ is an answer set of $\Pi$. $3.$ ($\Leftarrow$)  Assume that $\Pi$ has an answer set. Then, by the already proved Part 2 of the Theorem, we know that there exists an answer set $S'$ of ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ such that `notok` $\not\in S'$. By minimality of answer sets, $\Omega$ can not be an answer set of ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$. ($\Rightarrow$)  By Part 1 of Theorem \[theo:corr\], we know that ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ always has an answer set $S' \subseteq \Omega$. Assume that there is an answer set $S'\subsetneqq\Omega$. We distinguish 2 cases: (a) `notok` $\not\in S'$ and (b) `notok` $\in S'$. In case (a), proving Part 2 of this proposition, we have already shown that $\Pi$ has an answer set; this is a contradiction. On the other hand, in case (b) the final “saturation” rules in Step 5 “blow up” any answer set containing `notok` to $\Omega$, which contradicts the assumption $S'\subsetneqq\Omega$. As noticed above, the transformation ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ uses weak negation only stratified and in a deterministic part of the program; we can easily eliminate it by computing in the transformation the complement of each predicate accessed through ${\ensuremath{\mathtt{not}}\xspace}$ and providing it in $F(\Pi)$ as facts; we then obtain a positive program. (The built-in predicates `<` and `!=` can be eliminated similarly if desired.) However, such a modified transformation is not modular. As shown next, this is not incidental. \[prop:non-modular\] There is no modular transformation ${\ensuremath{tr}\xspace}'(\Pi)$ from HDLPs to DLPs (i.e. such that ${\ensuremath{tr}\xspace}'(\Pi) = \bigcup_{r \in \Pi} {\ensuremath{tr}\xspace}'(r)$), satisfying [**T1**]{} such that ${\ensuremath{tr}\xspace}'(\Pi)$ is a positive program. Assuming such a transformation exists, we derive a contradiction. Let $\Pi_1 = \{ \texttt{ a {\mbox{\texttt{:\hspace{-0.15em}-}}\xspace}{\ensuremath{\mathtt{not}}\xspace}b.} \}$ and $\Pi_2 = \Pi_1 \cup \{ \texttt{b.} \}$. Then, ${\ensuremath{tr}\xspace}'(\Pi_2)$ has some answer set $S_2$. Since ${\ensuremath{tr}\xspace}'(\cdot)$ is modular, ${\ensuremath{tr}\xspace}'(\Pi_1) \subseteq {\ensuremath{tr}\xspace}'(\Pi_2)$ holds and thus $S_2$ satisfies each rule in ${\ensuremath{tr}\xspace}'(\Pi_1)$. Since ${\ensuremath{tr}\xspace}'(\Pi_1)$ is a positive program, $S_2$ contains some answer set $S_1$ of ${\ensuremath{tr}\xspace}'{\Pi_1}$. By [ **T1**]{}, we have that `inS(a)` $\in S_1$ must hold, and hence `inS(a)` $\in S_2$. By [**T1**]{} again, it follows that $\Pi_2$ has an answer set $S$ such that $\texttt{a} \in S$. But the single answer set of $\Pi_2$ is $\{ \texttt{b} \}$, which is a contradiction. We remark that Prop. \[prop:non-modular\] remains true if [**T1**]{} is generalized such that the answer set $S$ of $\Pi$ corresponding to $S'$ is given by $S=\{l \mid S'\models \Psi(l)\}$, where $\Psi(x)$ is a monotone query (e.g., computed by a normal positive program without constraints). Moreover, if a successor predicate `next(X,Y)` and predicates `first(X)` and `last(X)` for the constants are available, given that the universe is finite by the constants in $\Pi$ and rule names, then computing the negation of the non-input predicates accessed through ${\ensuremath{\mathtt{not}}\xspace}$ is feasible by a positive normal program, since such programs capture polynomial time computability by well-known results on the expressive power of Datalog [@papa-85]; thus, negation of input predicates in $F(\Pi)$ is sufficient in this case. Modifications towards Optimization {#sec:optimizations} ================================== The meta-interpreter $\Pi_{meta}$ from above can be modified in several respects. We discuss in this section some modifications which, though not necessarily reducing the size of the ground instantiation, intuitively prune the search of an answer set solver applied to ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$. Giving up modularity (OPT$_{mod}$) ---------------------------------- If we sacrifice modularity and allow that $\Pi_{meta}$ partly depends on the input, then we can circumvent the iterations in Step 3 and in part of Step 1. Intuitively, instead of iterating over the heads and bodies of all rules in order to determine whether these rules are satisfied, we add a single rule in ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ for each rule $r$ in $\Pi$ firing [$\mathtt{notok}$]{} whenever $r$ is unsatisfied. We therefore replace the rules from Step 3 by $$\label{stmt:rulesatisfied} \begin{split} \mathtt{notok\ {\mbox{\texttt{:\hspace{-0.15em}-}}\xspace}\ }&\mathtt{ninS(\mbox{$h_1$}),\ \ldots,\ ninS(\mbox{$h_l$}),\ inS(\mbox{$b_1$}),\ \ldots,\ inS(\mbox{$b_m$}),}\\ & \mathtt{ninS(\mbox{$b_{m+1}$}),\ \ldots\, ninS(\mbox{$b_n$}).} \end{split}$$ for each rule $r$ in $\Pi$ of form (\[stmt:lprule\]). These rules can be efficiently generated in parallel to $F(\Pi)$. Lines 8 to 13 of Step 1 then become unnecessary and can be dropped. We can even refine this further. For every normal rule $r \in \Pi$ with non-empty head, i.e.${\ensuremath{H(r)}}={\ensuremath{\{h\}}}$, which has a satisfied body, we can force the guess of $h$: we replace (\[stmt:rulesatisfied\]) by $$\label{stmt:forcenormal} \mathtt{inS(\mbox{$h$}) {\mbox{\texttt{:\hspace{-0.15em}-}}\xspace}\ inS(\mbox{$b_1$}),\ \ldots,\ inS(\mbox{$b_m$}), ninS(\mbox{$b_{m+1}$}),\ \ldots\ ninS(\mbox{$b_n$}).}$$ In this context, since constraints only serve to “discard” unwanted models but cannot prove any literal, we can ignore them during input generation $F(\Pi)$. Note that dropping input representation $\mathtt{lit(n,\mbox{$l$},\mbox{$c$}).}$ for literals only occurring in the negative body of constraints but nowhere else in $\Pi$ requires some care. Such $l$ can be removed by simple preprocessing, though, by removing all $l \in {\ensuremath{B^-(c)}}$ which do not occur in any rule head in $\Pi$. On the other hand, all literals $l\in {\ensuremath{B^-(c)}}$ which appear in some other (non-constraint) rule $r$ are not critical, since facts `lit(hpn,l,r).` ($hpn \in \{\texttt{h,p,n}\}$) from this other rule will ensure that either line 15 or line 16 in $\Pi_{meta}$ is applicable and therefore, either `inS(l)` or `ninS(l)` will be derived. Thus, after elimination of critical literals in constraints beforehand, we can safely drop the factual representation of constraints completely (including `lit(n,l,c).` for the remaining negative literals). Restricting to potentially applicable rules (OPT$_{pa}$) -------------------------------------------------------- We only need to consider literals in heads of [*potentially applicable*]{} rules. These are all rules with empty bodies, and rules where any positive body literal – recursively – is the head of another potentially applicable rule. This suggests the following definition: \[def:poss\_app\] A set $R$ of ground rules is *potentially applicable*, if there exists an enumeration ${\ensuremath{\langle}r_i\ensuremath{\rangle}}_{i \in I}$ of $R$, where $I$ is a prefix of $\Nat$ resp. $I{=}\Nat$, such that ${\ensuremath{B^+(r_i)}} \subseteq \bigcup_{j<i} {\ensuremath{H(r_j)}}$. The following proposition is then not difficult to establish. Let $\Pi$ be any ground HDLP. Then there exists a unique maximal set $R^* \subseteq \Pi$ of potentially applicable rules, denoted by $\mathrm{PA}(\Pi)$. Indeed, suppose ${\ensuremath{\langle}r_i\ensuremath{\rangle}}_{i \in I}$ and ${\ensuremath{\langle}r'_i\ensuremath{\rangle}}_{i \in I'}$ are enumerations witnessing that rule sets $R$ and $R'$ such that $R,R'\subseteq \Pi$ are potentially applicable. Then their union $R\cup R'$ is potentially applicable, witnessed by the enumeration obtained from the alternating enumeration $r_0,r'_0,r_1,r'_1$,…whose suffix are the rules from the larger set of $R$ and $R'$ if they have different cardinalities, from which duplicate rules are removed (i.e., remove any rule $r'_j$ if $r'_j=r_i$, for some $i\leq j$, and remove any rule $r_j$ if $r'_i=r_j$ and for some $i<j$). It follows that a unique largest set $R^* \subseteq \Pi$ of potentially applicable rules exists. The set $\mathrm{PA}(\Pi)$ can be computed by adding a rule:    pa($r$) :- lit(h,$b_1$,R$_1$), pa(R$_1$), …, lit(h,$b_m$,R$_m$), pa(R$_m$). for any rule $r$ of the form (\[stmt:lprule\]) in $\Pi$. In particular, if $m=0$ we simply add the fact `pa(r).` Finally, we change line 1 in $\Pi_{meta}$ to:    rule(L,R) lit(h,L,R), not lit(p,L,R), not lit(n,L,R), pa(R). such that only “interesting” rules are considered. We note, however, that computing `pa(\cdot)` incurs some cost: Informally, a profit of optimization **OPT$_{pa}$** might only be expected in domains where $\Pi_{check}$ contains a a reasonable number of rules which positively depend on each other and might on the other hand likely be “switched off” by particular guesses in $\Pi_{guess}$. Optimizing the order guess (OPT$_{dep}$) ---------------------------------------- We only need to guess and check the order $\phi$ for literals $L$, $L'$ if they allow for cyclic dependency, i.e., they appear in the heads of rules within the same strongly connected component of the program with respect to $S$.[^2] These dependencies with respect to $S$ are easily computed:    dep(L,L1) lit(h,L,R),lit(p,L1,R),inS(L),inS(L1).\ dep(L,L2) lit(h,L,R),lit(p,L1,R),dep(L1,L2),inS(L).\ cyclic dep(L,L1),dep(L1,L). The guessing rules for $\phi$ (line 18 and 19) are then be replaced by:    phi(L,L1)  phi(L,L1) dep(L,L1), dep(L1,L), L &lt; L1,cyclic.\ phi(L,L2) :- phi(L,L1),phi(L1,L2), cyclic. Moreover, we add the new atom `cyclic` also to the body of any other rule where `phi` appears (lines 36,40) to check `phi` only in case $\Pi$ has *any* cyclic dependencies with respect to $S$. In the following, we will denote the transformation obtained by the optimizations from this section as ${\ensuremath{{\ensuremath{tr}\xspace}_{Opt}(\Pi)}\xspace}$ while we refer to ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ for the original transformation. Integrating Guess and ${\textrm{co-}{\mathrm{NP}}}$ Check Programs {#sec:integration} ================================================================== In this section, we show how our transformation ${\ensuremath{tr}\xspace}$ (resp.${\ensuremath{tr}\xspace}_{Opt}$) from above can be used to automatically combine a HDLP $\Pi_{guess}$ which guesses in its answer sets solutions of a problem, and a HDLP $\Pi_{check}$ which encodes a ${\textrm{co-}{\mathrm{NP}}}$-check of the solution property, into a single DLP $\Pi_{solve}$ of the form $\Pi_{solve} = \Pi_{guess} \cup \Pi'_{check}$. We assume that the set $Lit(\Pi_{guess})$ is a Splitting Set [@lifs-turn-94] for $\Pi_{guess} \cup \Pi_{check}$, i.e. no head literal from $\Pi_{check}$ occurs in $\Pi_{guess}$. This can be easily achieved by introducing new predicate names, e.g., $\texttt{p}'$ for a predicate $\texttt{p}$, and adding a rule $\texttt{p}'(t) {\mbox{\texttt{:\hspace{-0.15em}-}}\xspace}\texttt{p}(t)$ in case there is an overlap. Each rule $r$ in $\Pi_{check}$ is of the form $$\label{eqn:int} \begin{split} h_1 {\mbox{\texttt{v}\xspace}}\ \cdots\ {\mbox{\texttt{v}\xspace}}\ h_l\ {\mbox{\texttt{:\hspace{-0.15em}-}}\xspace}\ & bc_1,\ \ldots,\ bc_m,\ {\ensuremath{\mathtt{not}\ }bc_{m+1}},\ \ldots,\ {\ensuremath{\mathtt{not}\ }bc_n}\\ & bg_1,\ \ldots,\ bg_p,\ {\ensuremath{\mathtt{not}\ }bg_{p+1}},\ \ldots,\ {\ensuremath{\mathtt{not}\ }bg_q}. \end{split}$$ where the $bg_i$ are the body literals defined in $\Pi_{guess}$. We write ${\sf body}_{guess}(r)$ for $bg_1,\ \ldots,\ bg_p,\ {\ensuremath{\mathtt{not}\ }bg_{p+1}},\ \ldots,\ {\ensuremath{\mathtt{not}\ }bg_q}$. We now define a new check program as follows. For any ground program $\Pi_{check}$ as above, the program $\Pi'_{check}$ contains the following rules and constraints: : \(i) The facts $F(\Pi_{check})$ in a conditional version: For each rule $r\in\Pi_{check}$ of form (\[eqn:int\]), the rules     `lit(p,bg_i,r)`= `lit(h,l,r)`$\!\!$ '$\!\!\!\!$[body]{}$_{guess}(r).$ `atom(l,|l|).` \` for each $l \in {\ensuremath{H(r)}}$;\ `lit(p,bc_i,r)`$\!\!$ '$\!\!\!\!$[body]{}$_{guess}(r).$ \` for each $i \in \{1,\ldots, m\}$;\ `lit(n,bc_j,r)`$\!\!$ '$\!\!\!\!$[body]{}$_{guess}(r).$ \` for each $j \in \{m+1,\ldots, n\}$; : \(ii) each rule in $\Pi_{meta}{\,=\,}{\ensuremath{{\ensuremath{tr}\xspace}(\Pi_{check})}\xspace}\!\setminus F(\Pi_{check})$ (resp. in ${\ensuremath{{\ensuremath{tr}\xspace}_{Opt}(\Pi_{check})}\xspace}\!\setminus\!F(\Pi_{check})$, where ${\sf body}_{guess}(r)$ must be added to the bodies of the rules (\[stmt:rulesatisfied\]) and (\[stmt:forcenormal\])); : \(iii) a constraint    = [$\mathtt{not}\ $notok]{}. It eliminates any answer set $S$ such that $\Pi_{check} \cup S$ has an answer set. The union of $\Pi_{guess}$ and $\Pi_{check}'$ then amounts to the desired integrated encoding $\Pi_{solve}$, which is expressed by the following result. \[theo:integrate\] Given separate guess and check programs $\Pi_{guess}$ and $\Pi_{check}$, the answer sets of $$\Pi_{solve}=\Pi_{guess}\cup\Pi_{check}',$$ denoted $S_{solve}$, are in 1-1 correspondence with the answer sets $S$ of $\Pi_{guess}$ such that $\Pi_{check}\!\cup\!S$ has no answer set. This result can be derived from Theorem \[theo:corr\] and the Splitting Set Theorem for logic programs under answer set semantics [@lifs-turn-94]. We consider the proof for the original transformation ${\ensuremath{{\ensuremath{tr}\xspace}(\cdot)}\xspace}$; the proof for the optimized transformation ${\ensuremath{{\ensuremath{tr}\xspace}_{Opt}(\cdot)}\xspace}$ is similar (with suitable extensions in places). In what follows, for any program $Q$ and any consistent literal set $S$, we let $Q[S]$ denote the program obtained from $Q$ by eliminating every rule $r$ such that ${\sf body}_{guess}(r)$ is false in $S$, and by removing ${\sf body}_{guess}(r)$ from the remaining rules. Notice that $\Pi_{check}\cup S$ and $\Pi_{check}[S]\cup S$ have the same answer sets. We can rewrite $\Pi_{solve}$ as $$\Pi_{solve} = \Pi_{guess} \cup F'(\Pi_{check}) \cup \Pi_{meta} \cup \{\,{\mbox{\texttt{:\hspace{-0.15em}-}}\xspace}\ {\ensuremath{\mathtt{not}\ }\texttt{notok}}.\,\}$$ where $F'(\Pi_{check})$ denotes the modified factual representation for $\Pi_{check}$, given in item $1.$ of the definition of $\Pi'_{check}$. By hypothesis on $\Pi_{guess}\cup \Pi_{check}$, the set $Lit(\Pi_{guess})$ is a splitting set for $\Pi_{solve}$. Hence, as easily seen also $Lit(\Pi_{guess} \cup F'(\Pi_{check}))$ is a splitting set for $\Pi_{solve}$, and $Lit(\Pi_{guess})$ is also a splitting set for $\Pi_{guess} \cup F'(\Pi_{check})$. Moreover, each answer set $S$ of $\Pi_{guess}$ is in 1-1 correspondence with an answer set $S'$ of $\Pi_{guess} \cup F'(\Pi_{check})$. Then $S' \setminus S = F(\Pi_{check}[S]) \cup A_S$, such that $F(\Pi_{check}[S])$ is the factual representation of $\Pi_{check}[S]$ in the transformation ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi_{check}[S])}\xspace}$ and $A_S = {\ensuremath{\{\texttt{atom}(l,|l|). \mid l \in {\ensuremath{H(\Pi_{check})}}\setminus {\ensuremath{H(\Pi_{check}[S])}}\}}}$[^3] is an additional set of facts emerging from $F'(\Pi_{check})$, since we added facts `atom`$(l,|l|)$. for all head literals of $r \in \Pi_{check}$, not only for those $r$ where ${\sf body}_{guess}(r)$ was satisfied. Now let $S_{solve}$ be any (consistent) answer set of $\Pi_{solve}$. From the Splitting Set Theorem [@lifs-turn-94], we can conclude that $S_{solve}$ can be written as $S_{solve} = S \cup S_{check} \cup A_S$ where $S$ and $S_{check} \cup A_S$ are disjoint, $S$ is an answer set of $\Pi_{guess}$, and $S_{check} \cup A_S$ is an answer set of the program $\Pi'_S = (\Pi_{solve} \setminus \Pi_{guess})[S]$. Since $F'(\Pi_{check})$ is the only part of $\Pi_{solve} \setminus \Pi_{guess}$ where literals from $Lit(\Pi_{guess})$ occur, we obtain $$\begin{aligned} \Pi'_S &=& F(\Pi_{check}[S]) \cup A_S \cup \Pi_{meta} \cup \{{\mbox{\texttt{:\hspace{-0.15em}-}}\xspace}\ {\ensuremath{\mathtt{not}\ }\texttt{notok}}.\}\\ &=& {\ensuremath{{\ensuremath{tr}\xspace}(\Pi_{check}[S])}\xspace} \cup A_S \cup \{{\mbox{\texttt{:\hspace{-0.15em}-}}\xspace}\ {\ensuremath{\mathtt{not}\ }\texttt{notok}}.\}.\end{aligned}$$ The additional facts $A_S$ can be viewed as independent part of any answer set of $\Pi'_S$, since the answer sets of $\Pi'_S$ are the sets $T \cup A_S$ where $T$ is any answer set of $\Pi'_S \setminus A_S$; note that $T\cap A_S = \emptyset$. Indeed, the only rule in $\Pi'_S$ where the facts of $A_S$ play a role, is line 17 of $\Pi_{meta}$. All ground instances of line 17 are of the following form: $$\texttt{\ \ \ notok {\mbox{\texttt{:\hspace{-0.15em}-}}\xspace}inS(l), inS(nl), l{\ensuremath{\mathtt{\,!\!\!=}\,}}nl, atom(l,|l|), atom(nl,|l|).}$$ We assume $r$ fires and $\texttt{atom(l,|l|)} \in A_S$ (resp. $\texttt{atom(nl,|l|} \in A_S$). Then, in order for the rule to fire, `inS(l)` (resp. `inS(nl)`) has to be true. However, this can only be the case for literals `l` (resp. `nl`) occurring in a rule head of $\Pi_{check}[S]$ (backwards, by the rules in line 15, 14 and 1 of $\Pi_{meta}$ and by definition of $\Pi'_{check}$), which contradicts our assumption that $\texttt{atom(l,|l|)} \in A_S$ (resp. $\texttt{atom(nl,|l|} \in A_S$). Therefore, the facts of $A_S$ do not affect the rule in line 17 and consequently $\Pi'_S$ has an answer set if and only if $\Pi'_S \setminus A_S$ has an answer set and these answer sets coincide on $Lit(\Pi'_S) \setminus A_S$. By Theorem \[theo:corr\], we know that (i) ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi_{check}[S])}\xspace}$ always has an answer set and (ii) ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi_{check}[S])}\xspace}$ has any answer set containing `notok` (which is unique) if and only if $\Pi_{check}[S]$ has no answer set. However, the constraint ${\mbox{\texttt{:\hspace{-0.15em}-}}\xspace}~{\ensuremath{\mathtt{not}\ }\texttt{notok}}.$ only allows for answer sets of $\Pi'_S$ containing `notok`. Hence, an answer set $S_{check}$ of $\Pi'_S \setminus A_S$ exists if and only if $\Pi_{check}[S]$ has no answer set, equivalently, $\Pi_{check}\cup S$ has no answer set. Conversely, suppose $S$ is an answer set of $\Pi_{guess}$ such that $\Pi_{check}\cup S$ has no answer set; equivalently, $\Pi_{check}[S]$ has no answer set. By Theorem \[theo:corr\], we know that ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi_{check}[S])}\xspace} = F(\Pi_{check}[S]) \cup \Pi_{meta}$ has a unique answer set $S_{check}$, and $S_{check}$ contains `notok`. Hence, also the program $$Q_S = F(\Pi_{check}[S]) \cup \Pi_{meta} \cup \{{\mbox{\texttt{:\hspace{-0.15em}-}}\xspace}\ {\ensuremath{\mathtt{not}\ }\texttt{notok}}.\}$$ has the unique answer set $S_{check}$. On the other hand, since $S$ is an answer set of $\Pi_{guess}$ and $Lit(\Pi_{guess})$ is a splitting set for $\Pi_{solve}$, for each answer set $S''$ of the program $\Pi'_S = (\Pi_{solve} \setminus \Pi_{guess})[S]$, we have that $S \cup S''$ is an answer set of $\Pi_{solve}$. However, $\Pi'_S = Q_S \cup A_S$; hence, $S''=S_{check} \cup A_S$ must hold and $S_{solve} = S \cup S_{check} \cup A_S$ is the unique answer set of $\Pi_{solve}$ which extends $S$. This proves the result. The optimizations **OPT$_{pa}$** and **OPT$_{dep}$** in Section \[sec:optimizations\] still apply. However, concerning **OPT$_{mod}$**, the following modifications are necessary: 1. Like the input representation, rules (\[stmt:rulesatisfied\]) and (\[stmt:forcenormal\]) have to be extended by adding [body]{}$_{guess}$`(r)`. 2. As for constraints $c$, we mentioned above that the factual representation of literals in ${\ensuremath{B(c)}}$ may be skipped. This now only applies to literals in ${\ensuremath{B^+(c)}}$; the rule `lit(n,l,c)  body_{guess}(c).` for $l\in {\ensuremath{B^-(c)}}$ may no longer be dropped in general, as shown by the following example. Let $\Pi_{guess} = \{\ \texttt{g {\mbox{\texttt{v}\xspace}}-g.}\ \}$ and $\Pi_{check}=\{\ r1:\texttt{~x {\mbox{\texttt{:\hspace{-0.15em}-}}\xspace}g.},\ r2:\texttt{~{\mbox{\texttt{:\hspace{-0.15em}-}}\xspace}{\ensuremath{\mathtt{not}\ }x}.}\ \}$ The “input” representation of $\Pi_{check}$ with respect to optimization **OPT$_{mod}$**, i.e., the variable part of $\Pi'_{check}$, now consists of:     lit(h,x,$r1$)$\,$$\,$g.  lit(n,x,$r2$).  inS(x)$\,$:-$\,$g.  notok$\,$:-$\,$ninS(x). where the latter correspond to rules (\[stmt:forcenormal\]) and (\[stmt:rulesatisfied\]). If we now assume that we want to check answer set $S={\ensuremath{\{\,\mbox{\tt-g} \,\}}}$ of $\Pi_{guess}$, it is easy to see that $\Pi_{check}$ has no answer set for $S$, and therefore $S$ should be represented by some answer set of our integrated encoding. Now assume that [lit(n,x,$r2$).]{} is dropped and we proceed in generating the integrated encoding as outlined above with respect to **OPT$_{mod}$**. Since $\texttt{g} \not\in S$ and we have dropped [lit(n,x,$r2$).]{}, the “input” representation of $\Pi_{check}$ for $S$ comprises only the final rule `notok :- ninS(x).`. However, this rule can never fire because neither line 15 nor line 16 of $\Pi_{meta}$ can ever derive [ninS(c)]{}. Therefore, also `notok` can not be derived and the integrated check fails. On the other hand, [lit(n,x,$r2$).]{} suffices to derive [ninS(x)]{} from line 16 of $\Pi_{meta}$, such that `notok` can be derived and the integrated check works as intended. In certain cases, we can still drop $l\in {\ensuremath{B^-(c)}}$. For example, if $l$ occurs in the head of a rule $r$ with [body]{}$_{guess}(r) = \emptyset$, since in this case [lit(h,l,$r$)]{} will always be added to the program (see also respective remarks in Section \[sec:applications\]). Integrating Guess and ${\mathrm{NP}}\!$ Check Programs ------------------------------------------------------ In contrast to the situation above, integrating a guess program $\Pi_{guess}$ and a check program $\Pi_{check}$ which succeeds iff $\Pi_{check}\cup S$ has *some* answer set, is easy. Given that $\Pi_{check}$ is a HDLP again, this amounts to integrating a check which is in ${\mathrm{NP}}$. After a rewriting to ensure the Splitting Set property (if needed), simply take $\Pi_{solve}=\Pi_{guess} \cup \Pi_{check}$; its answer sets correspond on the predicates in $\Pi_{guess}$ to the desired solutions. Applications {#sec:applications} ============ We now give examples of the use of our transformation for three well-known ${{\Sigma}_{2}^{P}}$-complete problems from the literature, which involve ${\textrm{co-}{\mathrm{NP}}}$-complete checking for a polynomial-time solution guess: the first is about quantified Boolean formulas (QBFs) with one quantifier alternation, which are well-studied in Answer Set Programming, the second about conformant planning [@eite-etal-2001e; @turn-2002; @leon-etal-2001], and the third is about strategic companies in the business domain [@leon-etal-2002-dlv]. Further examples and ad hoc encodings of such problems can be found e.g. in [@eite-etal-97f; @eite-etal-2002-tplp; @leon-etal-2002-dlv] (and solved similarly). However, note that our method is applicable to [*any*]{} checks encoded by inconsistency of a HDLP; ${\textrm{co-}{\mathrm{NP}}}$-hardness is not a prerequisite. Quantified Boolean formulas {#sec:QBF} --------------------------- Given a QBF $F = \exists x_1 \cdots \exists x_m \forall y_1 \cdots \forall y_n\,\Phi$, where $\Phi = c_1 \vee \cdots \vee c_k $ is a propositional formula over $x_1,\ldots,x_m, y_1,\ldots,y_n$ in disjunctive normal form, i.e. each $c_i = a_{i,1} \wedge \cdots \wedge a_{i,l_i}$ and $|a_{i,j}| \in \{x_1,\ldots,x_m,y_1\ldots,y_n\}$, the problem is to compute some resp. all assignments to the variable $x_1,\ldots,x_m$ which witness that $F$ evaluates to true. Intuitively, this problem can be solved by “guessing and checking” as follows: ($\mathit{QBF}_{guess}$) : Guess a truth assignment for the variables $x_1,\ldots, x_m$. ($\mathit{QBF}_{check}$) : Check whether this (fixed) assignment satisfies $\Phi$ for all assignments of variables $y_1, \ldots, y_n$. Both parts can be encoded by very simple HDLPs (or similarly by normal programs): =\ $x_1$  $-x_1.$ … $x_m$  $-x_m.$\ \[3ex\]\ $y_1$ =  $-y_1.$ … $y_n$  $-y_n.$\ $a_{1,1}, \ldots, a_{1,l_1}$.\ \ $a_{k,1}, \ldots, a_{k,l_1}$. Clearly, both programs are head-cycle free. Moreover, for every answer set $S$ of $\mathit{QBF}_{guess}$ –representing an assignment to $x_1,\ldots,x_m$– the program $\mathit{QBF}_{check} \cup S$ has no answer set thanks to the constraints, iff every assignment for $y_1, \ldots, y_n$ satisfies formula $\Phi$. `%%%% GUESS PART`\ ` x0 v -x0.  x1 v -x1.`\ \ `%%%% REWRITTEN CHECK PART`\ `%% 1. Create dynamically the facts for the check program:`\ \ `% y0 v -y0.                               % y1 v -y1.`\ `  lit(h,`“`y0`”`,1). lit(h,`“`-y0`”`,1).            lit(h,`“`y1`”`,2). lit(h,`“`-y1`”`,2).`\ `  atom(`“`y0`”`,`“`y0`”`). atom(`“`-y0`”`,`“`y0`”`).        atom(`“`y1`”`,`“`y1`”`). atom(`“`-y1`”`,`“`y1`”`).`\ \ ` % :- -y0, -x0.`\ ` % :-  y0, -x0.`\ ` % :- -y0, y1, x0.`\ ` % :- -y0, y0, -x1.`\ \ `%% 2. Optimized meta-interpreter`\ `%% 2.1 – program dependent part`\ \ `  notok :- ninS(`“`y0`”`),ninS(`“`-y0`”`).`\ `  notok :- ninS(`“`y1`”`),ninS(`“`-y1`”`).`\ `  notok :- inS(`“`-y0`”`),-x0.`\ `  notok :- inS(`“`y0`”`),-x0.`\ `  notok :- inS(`“`y1`”`),inS(`“`-y0`”`),x0.`\ `  notok :- inS(`“`y0`”`),inS(`“`-y0`”`),-x1.`\ \ `%% 2.2 – fixed rules`\ \ `% Iterate only over rules which contain L in the head:`\ `  rule(L,R) :- lit(h,L,R), not lit(p,L,R), not lit(n,L,R).`\ `  ruleBefore(L,R) :- rule(L,R), rule(L,R1), R1<R.`\ `  ruleAfter(L,R) :- rule(L,R), rule(L,R1), R<R1.`\ `  ruleBetween(L,R1,R2) :- rule(L,R1), rule(L,R2), rule(L,R3), R1<R3, R3<R2.`\ `  firstRule(L,R) :- rule(L,R), not ruleBefore(L,R).`\ `  lastRule(L,R) :- rule(L,R), not ruleAfter(L,R).`\ `  nextRule(L,R1,R2) :- rule(L,R1), rule(L,R2), R1<R2, not ruleBetween(L,R1,R2).`\ \ `% hlits are only those from active rules:`\ `  hlit(L) :- rule(L,R).`\ `  inS(L) v ninS(L) :- hlit(L).`\ `  ninS(L) :- lit(HPN,L,R), not hlit(L).`\ \ `% Consistency check could be skipped for programs without class. negation:`\ `  notok :- inS(L), inS(NL), L != NL, atom(L,A), atom(NL,A).`\ \ `  dep(L,L1) :- rule(L,R),lit(p,L1,R),inS(L1), inS(L).`\ `  dep(L,L2) :- rule(L,R),lit(p,L1,R),dep(L1,L2),inS(L).`\ `  cyclic :- dep(L,L1), dep(L1,L).`\ `  phi(L,L1) v phi(L1,L) :- dep(L,L1), dep(L1,L), L<L1, cyclic.`\ `  phi(L,L2) :- phi(L,L1),phi(L1,L2), cyclic.`\ `  failsToProve(L,R) :- rule(L,R), lit(p,L1,R), ninS(L1).`\ `  failsToProve(L,R) :- rule(L,R), lit(n,L1,R), inS(L1).`\ `  failsToProve(L,R) :- rule(L,R), rule(L1,R), inS(L1), L1`[$\mathtt{\,!\!\!=}\,$]{}`L.`\ `  failsToProve(L,R) :- lit(p,L1,R), rule(L,R), phi(L1,L), cyclic.`\ `  allFailUpto(L,R) :- failsToProve(L,R), firstRule(L,R).`\ `  allFailUpto(L,R1) :- failsToProve(L,R1), allFailUpto(L,R), nextRule(L,R,R1).`\ `  notok :- allFailUpto(L,R), lastRule(L,R), inS(L).`\ `  phi(L,L1) :- notok, hlit(L), hlit(L1), cyclic.`\ `  inS(L) :- notok, hlit(L).`\ `  ninS(L) :- notok, hlit(L).`\ \ `%%% 3. constraint`\ `   :- not notok.` By the method described in Section \[sec:integration\], we can automatically generate a single program $\Pi_{solve}$ integrating the guess and check programs. For illustration, we consider the following QBF: $$\exists x_0x_1\forall y_0y_1 ({\ensuremath{\neg}}{x_0} \wedge {\ensuremath{\neg}}{y_0}) \vee (y_0 \wedge {\ensuremath{\neg}}{x_0}) \vee (y_1 \wedge x_0 \wedge {\ensuremath{\neg}}{y_0}) \vee (y_0 \wedge {\ensuremath{\neg}}{x_1} \wedge {\ensuremath{\neg}}{y_0})$$ This QBF evaluates to true: for the assignments $x_0=0,x_1=0$ and $x_0=0,x_1=1$, the subformula $\forall y_0y_1(\cdots)$ is a tautology. The integrated program $\mathit{QBF}_{solve} = \mathit{QBF}_{guess} \cup \mathit{QBF}'_{check}$ under use of the optimized transformation ${\ensuremath{{\ensuremath{tr}\xspace}_{Opt}(\cdot)}\xspace}$ of ${\ensuremath{tr}\xspace}(\cdot)$ as discussed is shown in Figure \[fig:qbf\]. It has two answer sets of the form $S_1 = \{ x_0, -x_1, \ldots, \}$ and $S_2 = \{ x_0, x_1, \ldots, \}$, respectively. With respect to the variants of the transformation, we remark that for the QBF encoding considerations upon negative literals in constraints in **OPT$_{mod}$** do not play a role, because all literals in the constraints of $\mathit{QBF}_{check}$ are positive. Also **OPT$_{pa}$** does not play a role, since the only rules in $\mathit{QBF}_{check}$ with non-empty heads are always potentially applicable because their bodies are empty. Note that the customary (but tricky) saturation technique in disjunctive logic programming to solve this problem, as used e.g. in [@eite-etal-97f; @leon-etal-2002-dlv] and shown in \[app:adhoc-qbf\], is fully transparent to the non-expert, who might easily come up with the two programs above. Conformant planning {#sec:conformant} ------------------- Loosely speaking, planning is the problem of finding a sequence of actions $P=\alpha_1$, $\alpha_2$,…, $\alpha_n$, a [*plan*]{}, which takes a system from an initial state $s_0$ to a state $s_n$ in which a goal (often, given by an atom $g$) holds, where a state $s$ is described by values of fluents, i.e., predicates which might change over time. [*Conformant planning*]{} [@gold-bodd-96] is concerned with finding a plan $P$ which works under all contingencies which may arise because of incomplete information about the initial state and/or nondeterministic action effects. As well-known, conformant planning in a STRIPS-style formulation is a ${{\Sigma}_{2}^{P}}$-complete problem (precisely, deciding plan existence) in certain settings, e.g. if the plan length $n$ (of polynomial size) is given and executability of actions is guaranteed, cf.[@eite-etal-2001e; @turn-2002]. Hence, the problem can be solved with a guess and (${\textrm{co-}{\mathrm{NP}}}$) check strategy. As an example, we consider a simplified version of the well-known “[*Bomb in the Toilet*]{}” planning problem [@mcde-87] as in [@eite-etal-2001e]: We have been alarmed that a possibly armed bomb is in a lavatory which has a toilet bowl. Possible actions are dunking the bomb into the bowl and flushing the toilet. After just dunking, the bomb may be disarmed or not; only flushing the toilet guarantees that it is really disarmed. Using the following guess and check programs $\mathit{Bomb_{guess}}$ and $\mathit{Bomb_{check}}$, respectively, we can compute a plan for having the bomb disarmed by two actions: X=\ % Timestamps:\ time(0). time(1).\ % Guess a plan:\ dunk(T) v -dunk(T) :- time(T).\ flush(T) v -flush(T) :- time(T).\ % Forbid concurrent actions:\ :- flush(T), dunk(T).\ \ % Initial state:\ armed(0) v -armed(0).\ % Frame Axioms:\ armed(T1) :- armed(T), not -armed(T1), time(T), T1=T+1.\ dunked(T1) :- dunked(T), T1=T+1.\ % Effect of dunking:\ dunked(T1) :- dunk(T), T1=T+1.\ armed(T1) v -armed(T1) dunk(T), armed(T), T1=T+1.\ % Effect of flushing:\ -armed(T1) :- flush(T), dunked(T), T1=T+1.\ % Check whether goal holds in stage 2:\ :- not armed(2). $\mathit{Bomb_{guess}}$ guesses all candidate plans $P=\alpha_1,\alpha_2$, starting from possible time points for action execution, while $\mathit{Bomb_{check}}$ checks whether any such plan $P$ is conformant for the goal $g$ = `not armed(2)`. Here, the closed world assumption (CWA) on `armed` is used, i.e., absence of `armed(t)` is viewed as `-armed(t)`, which saves a negative frame axiom on `-armed`. The final constraint eliminates a plan execution iff it reaches the goal; thus, $\mathit{Bomb_{check}}$ has no answer set iff the plan $P$ is conformant. As can be checked, the answer set $S=\{ \texttt{time(0)}, \texttt{time(1)}, \texttt{dunk(0)}, \texttt{flush(1)}\}$ of $\mathit{Bomb_{guess}}$ corresponds to the (single) conformant plan $P$= `dunk`, `flush` for the goal `not armed(2)`. By using the method from Section \[sec:integration\], the programs $\mathit{Bomb_{guess}}$ and $\mathit{Bomb_{check}}$ can be integrated automatically into a single program $\mathit{Bomb_{plan}}= \mathit{Bomb_{guess}} \cup \mathit{Bomb_{check}}'$ (cf. \[app:planning\]). It has a single answer set, corresponding to the single conformant plan $P$ = `dunk`, `flush` as desired. We point out that our rewriting method is more generally applicable than the encoding for conformant planning proposed in [@leon-etal-2001]. It loosens some of the restrictions there: While [@leon-etal-2001] requires that the state transition function is specified by a positive constraint-free logic program, our method can still safely be used in presence of negation and constraints, provided action execution will always lead to a consistent successor state and not entail absurdity; see [@eite-etal-2001e; @turn-2002] for a discussion of this setting. Concerning **OPT$_{mod}$**, we point out that there is the interesting constraint $$\texttt{$c:$ :- not armed(2).}$$ in program $\mathit{Bomb_{check}}$. Here, we may drop $\texttt{lit(h,"armed(2)",c)}$ safely: For the frame axiom $$\texttt{$r:$ armed(2)\ {\mbox{\texttt{:\hspace{-0.15em}-}}\xspace}\ armed(1),\ not\ -armed(2),\ time(1).}$$ (cf. \[app:planning\]), we have [body]{}$_{guess}(r) = {\ensuremath{\{\texttt{time(1)}\}}}$. Therefore, we obtain: $$\texttt{lit(h,"armed(2)",r) :- time(1).}$$ However, this rule will always be added since $\texttt{time(1)}$ is a deterministic consequence of $\mathit{Bomb_{guess}}$. As for **OPT$_{pa}$** and considering the “Bomb in the Toilet” instances from [@eite-etal-2001e], there might be rules which are *not* possible applicable with respect to a guessed plan; however, in experiments, the additional overhead for computing unfounded sets did not pay off. A generalization of the method demonstrated here on a small planning problem expressed in Answer Set Programming to conformant planning in the [${\texttt{\small DLV}\xspace}^{{\ensuremath{\mathcal{K}}\xspace}}$]{} planning system [@eite-etal-2001e], is discussed in detail in [@poll-2003]. In this system, planning problems are encoded in a logical action language, and the encodings are mapped to logic programs. For conformant planning problems, separate guess and check programs have been devised [@eite-etal-2001e], which by our method can be automatically integrated into a single logic program. Such an encoding was previously unkown. Strategic Companies {#sec:strategic} ------------------- Another ${{\Sigma}_{2}^{P}}$-complete problem is the strategic companies problem from [@cado-etal-97]. Briefly, a holding owns companies, each of which produces some goods. Moreover, several companies may jointly have control over another company. Now, some companies should be sold, under the constraint that all goods can be still produced, and that no company is sold which would still be controlled by the holding after the transaction. A company is [*strategic*]{}, if it belongs to a [*strategic set*]{}, which is a minimal set of companies satisfying these constraints. Guessing a strategic set, and checking its minimality can be done by the following two programs, where we adopt the constraint in [@cado-etal-97] that each product is produced by at most two companies and each company is jointly controlled by at most three other companies. XX=\ strat(X) = v -strat(X) company(X).\ prod\_by(X,Y,Z), not strat(Y), not strat(Z).\ = contr\_by(W,X,Y,Z), not strat(W),\ strat(X), strat(Y), strat(Z).\ \ strat1(X) = v -strat1(X) strat(X).\ prod\_by(X,Y,Z), not strat1(Y), not strat1(Z).\ = contr\_by(W,X,Y,Z), not strat1(W),\ strat1(X), strat1(Y), strat1(Z).\ smaller '  -strat1(X).\ not smaller. Here, [strat$(C)$]{} means that $C$ is strategic, [prod\_by$(P,C1,C2)$]{} that product $P$ is produced by companies $C1$ and $C2$, and [contr\_by]{}$(C,C1,$ $C2,C3)$ that $C$ is jointly controlled by $C1,C2$ and $C3$. We assume facts `company(\cdot)., prod_by(\cdot,\cdot,\cdot).`, and `contr_by(\cdot,\cdot,\cdot,\cdot).` to be defined in a separate program which can be considered as part of $SC_{guess}$. The two programs above intuitively encode guessing a set `strat` of companies which fulfills the production and control preserving constraints, such that no real subset `strat1` fulfills these constraints. While the ad hoc encodings from [@eite-etal-2000c; @leon-etal-2002-dlv], which can also be found in \[app:adhoc-sc\], are not immediate (and require some thought), the above programs are very natural and easy to come up with. [|c|c|c|]{} PRODUCT & COMPANY \#1 & COMPANY \#2\ Pasta & Barilla & Saiwa\ Tomatoes & Frutto & Barilla\ Wine & Barilla & –\ Bread & Saiwa & Panino\ As an example, let us consider the following production and control relations from [@cado-etal-97] in a holding as shown in Tables \[tab:prod\] and \[tab:contr\]. The symbol “–” there means that the entry is void, which we simply represent by duplicating the single producer (or one of the controlling companies, respectively) in the factual representation; a possible representation is thus [ company(barilla). company(saiwa). company(frutto). company(panino). prod\_by(pasta,barilla,saiwa). prod\_by(tomatoes,frutto,barilla). prod\_by(wine,barilla,barilla). prod\_by(bread,saiwa,panino). contr\_by(frutto,barilla,saiwa,saiwa). ]{} If we would consider only the production relation, then Barilla and Saiwa together would form a strategic set, because they jointly produce all goods but neither of them alone. On the other hand, Frutto would not be strategic. However, given the company control as in Table \[tab:contr\] means that Barilla and Saiwa together have control over Frutto. Taking into account that therefore Frutto can be sold only if either Barilla or Saiwa is also sold, the minimal sets of companies that produce all goods change completely: $\{$Barilla, Saiwa$\}$ is no longer a strategic set, while $s_1=\{$Barilla, Saiwa, Frutto$\}$ is. Alternatively, $s_2=\{$Barilla, Panino$\}$ is another strategic set. [|c|c|c|c|]{} CONTROLLED & CONT \#1 & CONT \#2 & CONT \#3\ Frutto & Barilla & Saiwa & –\ Integration of the programs $SC_{guess}$ and $SC_{check}$ after grounding is again possible by the method from Section \[sec:integration\] in an automatic way. Here, the facts representing the example instance are to be added as part of $SC_{guess}$, yielding two answer sets corresponding to $s_1$ and $s_2$ (cf. \[app:stratcomp\]). With regard to **OPT$_{mod}$**, we remark that depending on the concrete problem instance, $SC_{check}$ contains critical constraints $c$, where `\mathtt{not}\ strat1(\cdot)` occurs, such that `lit(n,strat1(\cdot),c)` may not be dropped here (cf. \[app:stratcomp\]). Furthermore, as for **OPT$_{pa}$** all rules with non-empty heads are either possibly applicable or “switched off” by $SC_{guess}$. Since there are no positive dependencies among the rules, `pa(\cdot)` does not play a role there. As a final remark, we note that modifying the guess and check programs $SC_{guess}$ and $SC_{check}$ to allow for unbounded numbers of producers for each product and controllers for each company, respectively, is easy. Assume that production and control are represented instead of relations `prod_by` and `contr_by` by an arbitrary number of facts of the form `produces(c,p).` and `controls(c_1,g,c).`, which state that company $c$ produces $p$ and that company $c_1$ belongs to a group $g$ of companies which jointly control $c$, respectively. Then, we would simply have to change the constraints in $SC_{guess}$ to:     no\_control(G,C) :- controls(C1,G,C), not strat(C1).\ :- controls(C1,G,C), not no\_control(G,C), not strat(C).\ produced(P) :- produces(C,P), strat(C).\ :- produces(C,P), not produced(P). The constraints in $SC_{check}$ are changed similarly. Then, the synthesized integrated encoding according to our method gives us a DLP solving this problem. The ad hoc encodings in [@eite-etal-2000c; @leon-etal-2002-dlv] can not be adapted that easily, and in fact require substantial changes. Experiments {#sec:experiments} =========== As for evaluation of the proposed approach we have conducted a series of experiments for the problems outlined in the previous Section. Here, we were mainly interested in the following questions: *What is the performance impact of our automatically generated, integrated encoding compared with ad hoc encodings of ${{\Sigma}_{2}^{P}}$ problems?* We have therefore compared our automatically generated integrated encoding of QBFs and Strategic Companies against the following ad hoc encodings: 1. QBF against the ad hoc encoding for QBFs described in [@leon-etal-2002-dlv] (which assumes that the quantifier-free part is in 3DNF, i.e., contains three literals per disjunct); see \[app:adhoc-qbf\]. 2. Strategic companies against the two ad hoc encodings for the Strategic Companies problem from [@eite-etal-2000c]; see \[app:adhoc-sc\]. These two encodings significantly differ: The first encoding, $ad hoc_1$ is very concise, and integrates guessing and checking in only two rules; it is an illustrative example of the power of disjunctive rules and tailored for a DLP system under answer set semantics. The second encoding, $ad hoc_2$, has a more obvious separate structure of the guessing and checking parts of the problem at the cost of some extra rules. However, in our opinion, none of these ad hoc encodings is obvious at first sight compared with the separate guess and check programs shown above. Concerning (i) we have tested randomly generated QBF instances with $n$ existentially and $n$ universally quantified variables (QBF-$n$), and concerning (ii) we have chosen randomly generated instances involving $n$ companies (SC-$n$). *What is the performance impact of the automatically generated, integrated encoding compared with interleaved computation of guess and check programs?* To this end, we have tested the performance of solving some conformant planning problems with integrated encodings compared with the ASP based planning system [${\texttt{\small DLV}\xspace}^{{\ensuremath{\mathcal{K}}\xspace}}$]{} [@eite-etal-2001e] which solves conformant planning problems by interleaving the guess of a plan with checking plan security. For its interleaved computation, [${\texttt{\small DLV}\xspace}^{{\ensuremath{\mathcal{K}}\xspace}}$]{}hinges on translations of the planning problem to HDLPs, by computing “optimistic” plans as solutions of a HDLP $\Pi^{plan}_{guess}$ and interleaved checking of plan security by non-existence of solutions of a new program $\Pi^{plan}_{guess}$ which is dynamically generated with respect to the plan at hand. [${\texttt{\small DLV}\xspace}^{{\ensuremath{\mathcal{K}}\xspace}}$]{}generalizes in some sense solving the small planning example in Section \[sec:conformant\] for arbitrary planning problems specified in a declarative language, [$\mathcal{K}$]{}[@eite-etal-2001d]. For our experiments we have used elaborations of “Bomb in the Toilet” as described in [@eite-etal-2001d], namely “Bomb in the Toilet with clogging” BTC($i$), where the toilet is clogged after dunking a package, and “Bomb in the Toilet with Uncertain Clogging” BTUC($i$) where this clogging effect is non-deterministic and there are $i$ many possibly armed packages. Test Environment and General Setting ------------------------------------ All tests were performed on an AMD Athlon 1200MHz machine with 256MB of main memory running SuSE Linux 8.1. All our experiments have been conducted using the [`DLV`]{} system [@leon-etal-2002-dlv; @dlv-web], which is a state-of-the-art Answer Set Programming engine capable of solving DLPs. Another available system, [[GnT]{}]{} [@janh-etal-2000][^4] which is not reported here showed worse performance/higher memory consumption on the tested instances. Since our method works on ground programs, we had to ground all instances (i.e. the corresponding guess and check programs) beforehand whenever dealing with non-ground programs. Here, we have used [`DLV`]{}grounding with most optimizations turned off:[^5] Some optimizations during [`DLV`]{}grounding rewrite the program, adding new predicate symbols, etc. which we turned off in order to obtain correct input for the meta-interpreters. In order to assess the effect of various optimizations and improvements to the transformation [${\ensuremath{tr}\xspace}(\cdot)$]{}, we have also conducted the above experiments with the integrated encodings based on different optimized versions of [${\ensuremath{tr}\xspace}(\cdot)$]{}. Results ------- The results of our experiments are shown in Tables \[tab:QBF\]-\[tab:BIT\]. We report there the following tests on the various instances: - $meta$  indicates the unoptimized meta-interpreter $\Pi_{meta}$ - $mod$  indicates the non-modular optimization **OPT$_{mod}$** including the refinement for constraints. - $dep$indicates the optimization **OPT$_{dep}$** where [$\mathtt{phi}$]{} is only guessed for literals mutually depending on each other through positive recursion. - $opt$indicates both optimizations **OPT$_{mod}$** and **OPT$_{dep}$** turned on. We did not include optimization **OPT$_{pa}$** in our experiments, since the additional overhead for computing unfounded rules in the check programs which we have considered did not pay off (in fact, **OPT$_{pa}$** is irrelevant for QBF and Strategic Companies). All times reported in the tables represent the execution times for finding the first answer set under the following resource constraints. We set a time limit of 10 minutes (=600 seconds) for QBFs and Strategic Companies, and of 4.000 seconds for the “Bomb in the Toilet” instances. Furthermore, the limit on memory consumption was 256 MB (in order to avoid swapping). A dash ’-’ in the tables indicates that one or more instances exceeded these limits. [|@[ ]{}l@[ ]{}|\*[10]{}[c|]{}]{} & & & & &\ & AVG & MAX & AVG & MAX & AVG & MAX & AVG & MAX & AVG & MAX\ QBF-4 & 0.01s & 0.02s & 0.16s & 0.18s & 0.10s & 0.15s & 0.09s & 0.11s & 0.07s & 0.09s\ QBF-6 & 0.01s & 0.02s & 1.11s & 1.40s & 0.25s & 1.12s & 0.17s & 0.21s & 0.08s & 0.12s\ QBF-8 & 0.01s & 0.06s & 10.4s & 16.3s & 1.18s & 7.99s & 0.49s & 0.87s & 0.10s & 0.23s\ QBF-10 & 0.02s & 0.09s & 82.7s & 165s & 4.34s & 30.7s & 1.74s & 3.67s & 0.12s & 0.36s\ QBF-12 & 0.02s & 0.16s & - & - & - & - & - & - & 0.15s & 0.79s\ QBF-14 & 0.06s & 1.21s & - & - & - & - & - & - & 0.34s & 5.87s\ QBF-16 & 0.08s & 1.85s & - & - & - & - & - & - & 0.44s & 10.3s\ QBF-18 & 0.19s & 7.12s & - & - & - & - & - & - & 1.04s & 38.8s\ QBF-20 & 1.49s & 21.3s & - & - & - & - & - & - & 7.14s & 101s\ \ [|l|c|c|c|c|c|c|c|c|c|c|c|c|]{} & & & & & &\ & AVG & MAX & AVG & MAX & AVG & MAX & AVG & MAX & AVG & MAX & AVG & MAX\ SC-10 & 0.01s & 0.02s & 0.05s & 0.05s & 0.66s & 0.69s & 0.49s & 0.51s & 0.36s & 0.38s & 0.13s & 0.15s\ SC-15 & 0.01s & 0.02s & 0.11s & 0.13s & 1.82s & 3.23s & 1.50s & 3.12s & 0.64s & 0.68s & 0.20s & 0.22s\ SC-20 & 0.02s & 0.02s & 0.26 & 0.27s & 3.75s & 3.90s & 3.34s & 3.61s & 1.07s & 1.13s & 0.26s & 0.27s\ SC-25 & 0.02s & 0.02s & 0.51s & 0.54s & - & - & - & - & 1.63s & 1.68s & 0.33s & 0.35s\ SC-30 & 0.02s & 0.03s & 0.91s & 0.97s & - & - & - & - & 2.35s & 2.47s & 0.42s & 0.44s\ SC-35 & 0.02s & 0.03s & 1.50s & 1.60s & - & - & - & - & 3.17s & 3.27s & 0.54s & 0.56s\ SC-40 & 0.03s & 0.03s & 2.52s & 2.70s & - & - & - & - & 4.25s & 4.43s & 0.68s & 0.71s\ SC-45 & 0.03s & 0.04s & 4.503 & 4.97s & - & - & - & - & 5.46s & 5.77s & 0.84s & 0.90s\ SC-50 & 0.03s & 0.04s & 8.38s & 8.68s & - & - & - & - & 6.73s & 6.86s & 1.00s & 1.02s\ SC-60 & 0.04s & 0.05s & 22.6s & 24.3s & - & - & - & - & 10.2s & 10.6s & 1.47s & 1.53s\ SC-70 & 0.04s & 0.05s & 44.2s & 48.1s & - & - & - & - & 14.7s & 15.4s & 2.05s & 2.10s\ SC-80 & 0.04s & 0.05s & 75.9s & 82.5s & - & - & - & - & 19.7s & 21.0s & 2.78s & 3.05s\ SC-90 & 0.05s & 0.06s & 125s & 130s & - & - & - & - & 26.8s & 27.6s & 3.67s & 3.85s\ SC-100 & 0.06s & 0.08s & 196s & 208s & - & - & - & - & 34.8s & 36.3s & 4.70s & 4.80s\ \ [|@[ ]{}l@[ ]{}|c|c|c|c|c|]{} & [[`DLV`$^{\ensuremath{\mathcal{K}}\xspace}$]{}]{}[@eite-etal-2001e] & $meta$ & $mod$ & $dep$ & $opt$\ BTC(2) & 0.01s & 1.16s & 0.80s & 0.15s & 0.08s\ BTC(3) & 0.11s & 9.33s & 9.25s & 8.18s & 4.95s\ BTC(4) & 4.68s & 71.3s & 67.8s & 333s & 256s\ BTUC(2) & 0.01s & 6.38s & 6.26s & 0.22s & 0.17s\ BTUC(3) & 1.78s & - & - & 28.1s & 13.0s\ BTUC(4) & 577s & - & - & - & 2322s\ \ The results in Tables \[tab:SC\]-\[tab:BIT\] show that the “guess and saturate” strategy in our approach benefits a lot from optimizations for all problems considered. However, we emphasize that it might depend on the structure of $\Pi_{guess}$ and $\Pi_{check}$ which optimizations are beneficial. We strongly believe that there is room for further improvements both on the translation and for the underlying [`DLV`]{}engine. We note the following observations: - Interestingly, for the QBF problem, the performance of our optimized translation stays within reach of the ad hoc encoding in [@leon-etal-2002-dlv] for small instances. Overall, the performance shown in Table \[tab:QBF\] is within roughly a factor of 5-6 (with few exceptions for small instances), and thus scales similarly. - For the Strategic Companies problem, the picture in Table \[tab:SC\] is even more interesting. Unsurprisingly, the automatically generated encoding is inferior to the succinct ad hoc encoding $ad hoc_1$; it is more than an order of magnitude slower and scales worse. However, while it is slower by a small factor than the ad hoc encoding $ad hoc_2$ (which is more involved) on small instances, it scales much better and quickly outperforms this encoding. - For the planning problems, the integrated encodings tested still stay behind the interleaved calls of ${\ensuremath{{\texttt{\small DLV}\xspace}^{{\ensuremath{\mathcal{K}}\xspace}}}\xspace}$. - In all cases, the time limit was exceeded (for smaller instances) rather than the memory limit, but especially for bigger instances of “Bomb in the toilet” and “Strategic Companies,” in some cases the memory limit was exceeded before timeout (e.g. for BTUC(5), even with the optimized version of our transformation). Summary and Conclusion {#sec:conclusion} ====================== We have considered the problem of integrating separate “guess” and “check” programs for solving expressive problems in the Answer Set Programming paradigm with a 2-step approach, into a single logic program. To this end, we have first presented a polynomial-time transformation of a head-cycle free, disjunctive program $\Pi$ into a disjunctive program ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ which is stratified and constraint-free, such that in the case where $\Pi$ is inconsistent (i.e., has no answer set), ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ has a single designated answer set which is easy to recognize, and otherwise the answer sets of $\Pi$ are encoded in the answer sets of ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$. We then showed how to exploit ${\ensuremath{{\ensuremath{tr}\xspace}(\Pi)}\xspace}$ for combining a “guess” program $\Pi_{solve}$ and a “check” program $\Pi_{check}$ for solving a problem in Answer Set Programming automatically into a single disjunctive logic program, such that its answer sets encode the solutions of the problem. Experiments have shown that such a synthesized encoding has weaker performance than the two-step method or an optimal ad hoc encoding for a problem, but can also outperform (reasonably looking) ad hoc encodings. This is noticeable since in some cases, finding any arbitrary “natural” (not necessarily optimal) encoding of a problem in a single logic program appears to be very difficult, such as e.g.,for conformant planning [@leon-etal-2001] or determining minimal update answer sets [@eite-etal-2002-tplp], where such encodings were not known for the general case. Several issues remain for being tackled in future work. The first issue concerns extending the scope of programs which can be handled. The rewriting method which we have presented here applies to propositional programs only. Thus, before transformation, the program should be instantiated. In [@leon-etal-2002-dlv] instantiations of a logic program used in [`DLV`]{}have been described, which keep the grounding small and do not necessarily ground over the whole Herbrand universe. For wider applicability and better scalability of the approach, a more efficient lifting of our method to non-ground programs is needed. Furthermore, improvements to the current transformations might be researched. Some preliminary experimental results suggest that a structural analysis of the given guess and check program might be valuable for this purpose. A further issue are alternative transformations, which are possibly tailored for certain classes of programs. The work of Ben-Eliyahu and Dechter [-@bene-dech-94], on which we build, aimed at transforming head-cycle free disjunctive logic programs into SAT problems. It might be interesting to investigate whether related methods such as the one developed for ASSAT [@lin-zhao-2002], which was recently generalized by Lee and Lifschitz [-@lee-lifs-2003] to disjunctive programs, can be adapted for our approach. Acknowledgments {#acknowledgments .unnumbered} --------------- We thank Gerald Pfeifer for his help on experimental evaluation and fruitful discussions. We are also grateful to the reviewers of the paper as well as the reviewers of the preliminary conference versions for their comments and constructive suggestions for improvement. Integrated Program for Conformant Planning {#app:planning} ========================================== The integrated program for the planning problem in Section \[sec:conformant\], $\mathit{Bomb_{plan}} = \mathit{Bomb_{guess}} \cup \mathit{Bomb_{check}}'$, is given below. It has a single answer set $S = \{$ `dunk(0)`, `-flush(0)`, `flush(1)`, `-dunk(1)`, …$\}$ which corresponds to the single conformant plan $P$= `dunk`, `flush` as desired. `%%%% GUESS PART`\ \ `% Timestamps:`\ `  time(0). time(1).`\ \ `% Guess a plan:`\ `  dunk(T) v -dunk(T) :- time(T).`\ `  flush(T) v -flush(T) :- time(T).`\ `  :- flush(T), dunk(T).`\ \ `%%%% REWRITTEN CHECK PART (after grounding)`\ `%% 1. Create dynamically the facts for the program:`\ \ `% armed(0) v -armed(0).`\ `   lit(h,`“`armed(0)`”`,1). atom(`“`armed(0)`”`,`“`armed(0)`”`).`\ `   lit(h,`“`-armed(0)`”`,1). atom(`“`-armed(0)`”`,`“`armed(0)`”`).`\ \ `% armed(T1) :- armed(T), not -armed(T1), time(T), T1=T+1.`\ `   lit(h,`“`armed(1)`”`,2) :- time(0). atom(`“`armed(1)`”`,`“`armed(1)`”`).`\ `   lit(p,`“`armed(0)`”`,2) :- time(0).`\ `   lit(n,`“`-armed(1)`”`,2) :- time(0).`\ \ `   lit(h,`“`armed(2)`”`,3) :- time(1).  atom(`“`armed(2)`”`,`“`armed(2)`”`).`\ `   lit(p,`“`armed(1)`”`,3) :- time(1).`\ `   lit(n,`“`-armed(2)`”`,3) :- time(1).`\ \ `% dunked(T1) :- dunked(T), T1=T+1.`\ `   lit(h,`“`dunked(1)`”`,4).  atom(`“`dunked(1)`”`,`“`dunked(1)`”`).`\ `   lit(p,`“`dunked(0)`”`,4).`\ \ `   lit(h,`“`dunked(2)`”`,5). atom(`“`dunked(2)`”`,`“`dunked(2)`”`).`\ `   lit(p,`“`dunked(1)`”`,5).`\ \ `% dunked(T1) :- dunk(T), T1=T+1.`\ `   lit(h,`“`dunked(1)`”`,6) :- dunk(0).`\ \ `   lit(h,`“`dunked(2)`”`,7) :- dunk(1).`\ \ `% armed(T1) v -armed(T1) :- dunk(T), armed(T), T1=T+1.`\ `   lit(h,`“`armed(1)`”`,8) :- dunk(0).`\ `   lit(h,`“`-armed(1)`”`,8) :- dunk(0). atom(`“`-armed(1)`”`,`“`armed(1)`”`).`\ `   lit(p,`“`armed(0)`”`,8) :- dunk(0).`\ \ `   lit(h,`“`armed(2)`”`,9) :- dunk(1).`\ `   lit(h,`“`-armed(2)`”`,9) :- dunk(1). atom(`“`-armed(2)`”`,`“`armed(2)`”`).`\ `   lit(p,`“`armed(1)`”`,9) :- dunk(1).`\ \ `% -armed(T1) :- flush(1), dunked(T),T1=T+1.`\ `   lit(h,`“`-armed(1)`”`,10) :- flush(0). lit(p,`“`dunked(0)`”`,10) :- flush(0).`\ \ `   lit(h,`“`-armed(2)`”`,11) :- flush(1). lit(p,`“`dunked(1)`”`,11) :- flush(1).`\ \ `% :- not armed(2).`\ \ `%% 2. Optimized meta-interpreter`\ \ `%% 2.1 – program dependent part`\ \ `   notok :- ninS(`“`armed(0)`”`), ninS(`“`-armed(0)`”`).`\ `   inS(`“`armed(1)`”`) :- inS(`“`armed(0)`”`), ninS(`“`-armed(1)`”`), time(0).`\ `   inS(`“`armed(2)`”`) :- inS(`“`armed(1)`”`), ninS(`“`-armed(2)`”`), time(1).`\ `   inS(`“`dunked(1)`”`) :- inS(`“`dunked(0)`”`).`\ `   inS(`“`dunked(2)`”`) :- inS(`“`dunked(1)`”`).`\ `   inS(`“`dunked(1)`”`) :- dunk(0).`\ `   inS(`“`dunked(2)`”`) :- dunk(1).`\ `   notok :- ninS(`“`armed(1)`”`),ninS(`“`-armed(1)`”`), inS(`“`armed(0)`”`), dunk(0).`\ `   notok :- ninS(`“`armed(2)`”`),ninS(`“`-armed(2)`”`),inS(`“`armed(1)`”`), dunk(1).`\ `   inS(`“`-armed(1)`”`) :- inS(`“`dunked(0)`”`), flush(0).`\ `   inS(`“`-armed(2)`”`) :- inS(`“`dunked(1)`”`), flush(1).`\ `   notok :- ninS(`“`armed(2)`”`).`\ \ `%% 2.2 – fixed rules`\ \ `% Skipped, see QBF Encoding`\ \ `%%% 3. constraint`\ `   :- not notok.` Integrated Program for Strategic Companies {#app:stratcomp} ========================================== The integrated program for the strategic companies problem instance in Section \[sec:strategic\], $\mathit{SC_{strategic}}$ = $\mathit{SC_{guess}} \cup \mathit{SC_{check}}'$, is given below. It has two answer sets $S_1$ = {`strat(barilla)`, `strat(saiwa)`, `strat(frutto)`, …} and $S_2$ = {`strat(barilla)`, `strat(panino)`, …} which correspond to the strategic sets as identified above. `%%%% GUESS PART`\ `  company(barilla). company(saiwa). company(frutto). company(panino).`\ `  prod_by(pasta,barilla,saiwa).  prod_by(tomatoes,frutto,barilla).`\ `  prod_by(wine,barilla,barilla). prod_by(bread,saiwa,panino).`\ `  contr_by(frutto,barilla,saiwa,barilla).`\ \ `%% Guess Program: Not necessarily minimal`\ \ `  strat(X) v -strat(X) :- company(X).`\ `   :-  prod_by(X,Y,Z), not strat(Y), not strat(Z).`\ `   :-  contr_by(W,X,Y,Z), not strat(W),`\ `       strat(X), strat(Y), strat(Z).`\ \ `%%%% REWRITTEN CHECK PART (after grounding)`\ `%% 1. Create dynamically the facts for the program:`\ \ `% smaller :- -strat1(X).`\ `  lit(h,`“`smaller`”`,1). atom(`“`smaller`”`,`“`smaller`”`).`\ `  lit(p,`“`-strat1(saiwa)`”`,1).`\ `  lit(h,`“`smaller`”`,2). atom(`“`smaller`”`,`“`smaller`”`).`\ `  lit(p,`“`-strat1(panino)`”`,2).`\ `  lit(h,`“`smaller`”`,3). atom(`“`smaller`”`,`“`smaller`”`).`\ `  lit(p,`“`-strat1(frutto)`”`,3).`\ `  lit(h,`“`smaller`”`,4). atom(`“`smaller`”`,`“`smaller`”`).`\ `  lit(p,`“`-strat1(barilla)`”`,4).`\ \ `% strat1(X) v -strat1(X) :- strat(X).`\ `  lit(h,`“`strat1(saiwa)`”`,5) :- strat(saiwa).    atom(`“`strat1(saiwa)`”`,`“`strat1(saiwa)`”`).`\ `  lit(h,`“`-strat1(saiwa)`”`,5) :- strat(saiwa).   atom(`“`-strat1(saiwa)`”`,`“`strat1(saiwa)`”`).`\ `  lit(h,`“`strat1(panino)`”`,6) :- strat(panino).  atom(`“`strat1(panino)`”`,`“`strat1(panino)`”`).`\ `  lit(h,`“`-strat1(panino)`”`,6) :- strat(panino). atom(`“`-strat1(panino)`”`,`“`strat1(panino)`”`).`\ `lit(h,`“`strat1(frutto)`”`,7) :- strat(frutto).    atom(`“`strat1(frutto)`”`,`“`strat1(frutto)`”`).`\ `lit(h,`“`-strat1(frutto)`”`,7) :- strat(frutto).   atom(`“`-strat1(frutto)`”`,`“`strat1(frutto)`”`).`\ `lit(h,`“`strat1(barilla)`”`,8) :- strat(barilla).  atom(`“`strat1(barilla)`”`,`“`strat1(barilla)`”`).`\ `lit(h,`“`-strat1(barilla)`”`,8) :- strat(barilla). atom(`“`-strat1(barilla)`”`,`“`strat1(barilla)`”`).` `% For constraints, critical negative literals need to be represented (cf.` **OPT$_{mod}$**`)` `% :- prod_by(X,Y,Z), not strat1(Y), not strat1(Z).`\ `  lit(n,`“`strat1(saiwa)`”`,10) :- prod_by(bread,saiwa,panino).`\ `  lit(n,`“`strat1(panino)`”`,10) :- prod_by(bread,saiwa,panino).`\ `  lit(n,`“`strat1(frutto)`”`,11) :- prod_by(tomatoes,frutto, barilla).`\ `  lit(n,`“`strat1(barilla)`”`,11) :- prod_by(tomatoes,frutto, barilla).`\ `  lit(n,`“`strat1(barilla)`”`,12) :- prod_by(wine,barilla,barilla).`\ `  lit(n,`“`strat1(barilla)`”`,13) :- prod_by(pasta,barilla,saiwa).`\ `  lit(n,`“`strat1(saiwa)`”`,13) :- prod_by(pasta,barilla,saiwa).`\ \ `% :- contr_by(W,X,Y,Z), not strat1(W), strat1(X), strat1(Y), strat1(Z).`\ `  lit(n,`“`strat1(frutto)`”`,14) :- contr_by(frutto,barilla,saiwa,saiwa).`\ \ `%% 2. Optimized meta-interpreter`\ \ `%% 2.1 – program dependent part`\ `  inS(`“`smaller`”`) :- inS(`“`-strat1(saiwa)`”`).`\ `  inS(`“`smaller`”`) :- inS(`“`-strat1(panino)`”`).`\ `  inS(`“`smaller`”`) :- inS(`“`-strat1(frutto)`”`).`\ `  inS(`“`smaller`”`) :- inS(`“`-strat1(barilla)`”`).`\ `  notok :- ninS(`“`strat1(saiwa)`”`),ninS(`“`-strat1(saiwa)`”`),strat(saiwa).`\ `  notok :- ninS(`“`strat1(panino)`”`),ninS(`“`-strat1(panino)`”`),strat(panino).`\ `  notok :- ninS(`“`strat1(frutto)`”`),ninS(`“`-strat1(frutto)`”`),strat(frutto).`\ `  notok :- ninS(`“`strat1(barilla)`”`),ninS(`“`-strat1(barilla)`”`),strat(barilla).`\ `  notok :- ninS(`“`smaller`”`).`\ `  notok :- ninS(`“`strat1(saiwa)`”`),ninS(`“`strat1(panino)`”`).`\ `  notok :- ninS(`“`strat1(frutto)`”`),ninS(`“`strat1(barilla)`”`).`\ `  notok :- ninS(`“`strat1(barilla)`”`).`\ `  notok :- ninS(`“`strat1(barilla)`”`),ninS(`“`strat1(saiwa)`”`).`\ `  notok :- inS(`“`strat1(barilla)`”`),inS(`“`strat1(saiwa)`”`),ninS(`“`strat1(frutto)`”`).`\ \ `%% 2.2 – fixed rules`\ \ `% Skipped, see QBF Encoding`\ \ `%%% 3. constraint`\ `   :- not notok.` Ad Hoc Encoding for Quantified Boolean Formulas {#app:adhoc-qbf} =============================================== The ad hoc encoding in [@leon-etal-2002-dlv] for evaluating a QBF of form $F = \exists x_1 \cdots \exists x_m \forall y_1 \cdots \forall y_n\,\Phi$, where $\Phi = c_1 \vee \cdots \vee c_k $ is a propositional formula over $x_1,\ldots,x_m, y_1,\ldots,y_n$ in 3DNF, i.e. each $c_i = a_{i,1} \wedge \cdots \wedge a_{i,3}$ and $|a_{i,j}| \in \{x_1,\ldots,$ $x_m,$ $y_1\ldots,$ $y_n\}$, represents $F$ by the following facts: - `exists(x_i).` for each existential variable $x_i$; - `forall(y_j).` for each universal variable $y_j$; and - `term(p_1, p_2, p_3, q_1, q_2, q_3).` for each disjunct $c_j = l_{i,1} \land l_{i,2} \land l_{i,3}$ in $\Phi$, where (i) if $l_{i,j}$ is a positive atom $v_k$, then $p_j = v_k$, otherwise $p_j$= $\mathtt{true}$, and (ii) if $l_{i,j}$ is a negated atom $\neg v_k$, then $q_i=v_k$, otherwise $q_i$ = $\mathtt{false}$. For example, $term(x_1,\mathtt{true}, y_4, \mathtt{false}, y_2, \mathtt{false})$, encodes the term $x_1\land \neg y_2 \land y_4$. For instance, our sample instance from Section \[sec:QBF\] $$\exists x_0x_1\forall y_0y_1 ({\ensuremath{\neg}}{x_0} \wedge {\ensuremath{\neg}}{y_0}) \vee (y_0 \wedge {\ensuremath{\neg}}{x_0}) \vee (y_1 \wedge x_0 \wedge {\ensuremath{\neg}}{y_0}) \vee (y_0 \wedge {\ensuremath{\neg}}{x_1} \wedge {\ensuremath{\neg}}{y_0})$$ would be encoded by the following facts: `   exists(x0).  exists(x1).  forall(y1).  forall(y2).`\ `   term(true,true,true,x0,y0,false).`\ `   term(y0,true,true,x0,false,false).`\ `   term(y1,x0,true,y0,false,false).`\ `   term(y0,true,true,x1,y0,false).` These facts are conjoined with the following facts and rules: `   t(true). f(false).`\ `   t(X)  f(X) :- exists(X).`\ `   t(Y)  f(Y) :- forall(Y).`\ `             w :- term(X,Y,Z,Na,Nb,Nc),t(X),t(Y),t(Z),`\ `                  f(Na),f(Nb),f(Nc).`\ `          t(Y) :- w, forall(Y).`\ `          f(Y) :- w, forall(Y).`\ `               :- not w.` The guessing part “initializes” the logical constants $\mathtt{true}$ and $\mathtt{false}$ and chooses a witnessing assignment $\sigma$ to the variables in $X$, which leads to an answer set $M_G$ for this part. The more tricky checking part then tests whether $\phi[X/ \sigma(X)]$ is a tautology, using a saturation technique similar to our meta-interpreter. Ad Hoc Encodings for Strategic Companies {#app:adhoc-sc} ======================================== The first ad hoc encoding for Strategic Companies in [@eite-etal-2000c], $ad hoc_1$, solves the problem in a surprisingly elegant way by the following two rules conjoined to the facts representing the $\mathit{prod\_by}$ and $\mathit{contr\_by}$ relations: `   strat(Y)  strat(Z) :- prod_by(X,Y,Z).`\ `              strat(W) :- contr_by(W,X,Y,Z), strat(X),`\ `strat(Y),`\ `strat(Z).` Here, the minimality of answer sets plays together with the first rule generating candidate strategic sets and the second rule enforcing the constraint on the controls relation. It constitutes a sophisticated example of intermingled guess and check. Howewer, this succinct encoding relies very much on the fixed number of producing and controlling companies; an extension to arbitrarily many producers and controllers seems not to be as easy as in our separate guess and check programs from Section \[sec:strategic\]. The second ad hoc encoding from [@eite-etal-2000c], $ad hoc_2$, strictly separates the guess and checking parts, and uses the following rules and constraints: `   strat(X)  -strat(X) :- company(X).`\ `     :- prod_by(X,Y,Z), not strat(Y), not strat(Z).`\ `     :- contr_by(W,X,Y,Z), not strat(W), strat(X), strat(Y), strat(Z).`\ `     :- not min(X), strat(X).`\ `     :- strat’(X,Y), -strat(Y).`\ `     :- strat’(X,X).`\ `   min(X)  strat’(X,Y)  strat’(X,Z) :- prod_by(G,Y,Z),strat(X).`\ `   min(X)  strat’(X,C) :- contr_by(C,W,Y,Z), strat(X),`\ `                           strat’(X,W), strat’(X,Y), strat’(X,Z).`\ `            strat’(X,Y) :- min(X), strat(X), strat(Y), X`[[$\mathtt{\,!\!\!=}\,$]{}]{}`Y.` Informally, the first rule and the first two constraints generate a candidate strategic set, whose minimality is checked by the remainder of the program. For a detailed explanation, we refer to [@eite-etal-2000c]. [10]{} Marcello Balduccini and Michael Gelfond. . , 3:425–461, July/September 2003. Chitta Baral. . Cambridge University Press, 2002. R. Ben-Eliyahu and R. Dechter. . , 12:53–87, 1994. Marco Cadoli, Thomas Eiter, and Georg Gottlob. . , 9(3):448–463, May/June 1997. A. Cimatti and M. Roveri. .  [*13*]{}, 305–338, 2000. Evgeny Dantsin, Thomas Eiter, Georg Gottlob, and Andrei Voronkov. . , 33(3):374–425, 2001. Jim Delgrande, Torsten Schaub, and Hans Tompits. . In Thomas Eiter, Wolfgang Faber, and Miros[ł]{}aw Truszczy[ń]{}ski, editors, [*[Proceedings of the 6th International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR-01)]{}*]{}, number 2173 in LNCS, pages 411–415. Springer, 2001. Thomas Eiter, Wolfgang Faber, Nicola Leone, and Gerald Pfeifer. . In Jack Minker, editor, [*[Logic-Based Artificial Intelligence]{}*]{}, pages 79–103. Kluwer Academic Publishers, 2000. Thomas Eiter, Wolfgang Faber, Nicola Leone, and Gerald Pfeifer. . , 3:463–498, July/September 2003. Thomas Eiter, Wolfgang Faber, Nicola Leone, Gerald Pfeifer, and Axel Polleres. . , 144(1–2):157–211, March 2003. Thomas Eiter, Wolfgang Faber, Nicola Leone, Gerald Pfeifer, and Axel Polleres. . , 5(2), April 2004. Thomas Eiter, Michael Fink, Giuliana Sabbatini, and Hans Tompits. . , 2(6):721–777, 2002. Thomas Eiter, Georg Gottlob, and Heikki Mannila. . , 22(3):364–418, September 1997. Wolfgang Faber and Gerald Pfeifer. homepage, [since 1996]{}. <http://www.dlvsystem.com/>. M. Gelfond and V. Lifschitz. . , 9:365–385, 1991. Michael Gelfond. . In Antonis C. Kakas and Fariba Sadri, editors, [*[Computational Logic. Logic Programming and Beyond]{}*]{}, number 2408 in LNCS, pages 413–451. Springer, 2002. R. Goldman and M. Boddy. . In [*[Proceedings AIPS-96]{}*]{}, pages 110–117. AAAI Press, 1996. G. Greco, S. Greco, and E. Zumpano. A logic programming approach to the integration, repairing and querying of inconsistent databases. In [*Proceedings 17th International Conference on Logic Programming (ICLP 2001)*]{}, number 2237 in [Lecture Notes in AI (LNAI)]{}, pages 348–364. Springer Verlag, 2001. Tomi Janhunen. Comparing the expressive powers of some syntactically restricted classes of logic porgrams. In John Lloyd, Veronica Dahl, Ulrich Furbach, Manfred Kerber, Kung-Kiu Lau, Catuscia Palamidessi, Lu[í]{}s Moniz Pereira, Yehoshua Sagiv, and Peter J. Stuckey, editors, [*Computational Logic - CL 2000, First International Conference, Proceedings*]{}, number 1861 in [Lecture Notes in AI (LNAI)]{}, pages 852–866, London, UK, July 2000. Springer Verlag. Tomi Janhunen. On the effect of default negation on the expressiveness of disjunctive rules. In Thomas Eiter, Wolfgang Faber, and Miros[ł]{}aw Truszczyński, editors, [*[Logic Programming and Nonmonotonic Reasoning — 6th International Conference, LPNMR’01, Vienna, Austria, September 2001, Proceedings]{}*]{}, number 2173 in [Lecture Notes in AI (LNAI)]{}, pages 93–106. Springer Verlag, September 2001. Tomi Janhunen, Ilkka Niemel[ä]{}, Patrik Simons, and Jia-Huai You. . In Anthony G. Cohn, Fausto Giunchiglia, and Bart Selman, editors, [*[Proceedings of the Seventh International Conference on Principles of Knowledge Representation and Reasoning (KR 2000), April 12-15, Breckenridge, Colorado, USA]{}*]{}, pages 411–419. Morgan Kaufmann Publishers, Inc., 2000. Joohyung Lee and Vladimir Lifschitz. . In [*Proceedings of the Nineteenth International Conference on Logic Programming (ICLP-03)*]{}. Springer Verlag, December 2003. Nicola Leone, Gerald Pfeifer, Wolfgang Faber, Thomas Eiter, Georg Gottlob, Simona Perri, and Francesco Scarcello. , 2004. To appear. Available via <http://www.arxiv.org/ps/cs.AI/0211004>. Nicola Leone, Riccardo Rosati, and Francesco Scarcello. . In Alessandro Cimatti, H[é]{}ctor Geffner, Enrico Giunchiglia, and Jussi Rintanen, editors, [*[IJCAI-01 Workshop on Planning under Uncertainty and Incomplete Information]{}*]{}, pages 33–42, August 2001. V. Lifschitz and H. Turner. . In Pascal [Van Hentenryck]{}, editor, [*[Proceedings of the 11th International Conference on Logic Programming (ICLP’94)]{}*]{}, pages 23–37, Santa Margherita Ligure, Italy, June 1994. MIT Press. Vladimir Lifschitz. . , 138:39–54, 2002. Fangzhen Lin and Yuting Zhao. . In [*[Proceedings of the Eighteenth National Conference on Artificial Intelligence (AAAI-2002)]{}*]{}, Edmonton, Alberta, Canada, 2002. AAAI Press / MIT Press. Victor W. Marek and Jeffrey B. Remmel. . , 3:551–567, November 2003. Victor W. Marek and Miros[ł]{}aw Truszczy[ń]{}ski. . In K. Apt, V. W. Marek, M. Truszczy[ń]{}ski, and D. S. Warren, editors, [*[The Logic Programming Paradigm – A 25-Year Perspective]{}*]{}, pages 375–398. Springer Verlag, 1999. Drew McDermott. . , 3:151–237, 1987. Cited in [@cima-rove-2000]. Ilkka Niemel[ä]{}. . , 25(3–4):241–273, 1999. Christos H. Papadimitriou. A note on the expressive power of prolog. , 26:21–23, 1985. Christos H. Papadimitriou. . Addison-Wesley, 1994. Axel Polleres. . PhD thesis, [Institut f[ü]{}r Informationssysteme, Technische Universit[ä]{}t Wien]{}, [Wien, [Ö]{}sterreich]{}, 2003. Alessandro Provetti and Son Tran Cao, editors. , Stanford, CA, March 2001. AAAI Press. T. Przymusinski. . , 5(2):167–205, 1989. Teodor C. Przymusinski. . , 9:401–424, 1991. Chiaki Sakama and Katsumi Inoue. . , 3:671–713, November 2003. J.S. Schlipf. . , 51(1):64–86, 1995. Abstract in Proc. PODS 90, pp. 196–204. Hudson Turner. . In Sergio Flesca, Sergio Greco, Giovambattista Ianni, and Nicola Leone, editors, [*Proceedings of the 8th European Conference on Logics in Artificial Intelligence (JELIA)*]{}, number 2424 in [Lecture Notes in Computer Science]{}, pages 111–124, Cosenza, Italy, September 2002. Springer. [^1]: We disregard a possible inconsistent answer set, which is not of much interest for our concerns. [^2]: Similarly, in [@bene-dech-94] $\phi: Lit(\Pi) \rightarrow \{1, \ldots, r\}$ is only defined for a range $r$ bound by the longest acyclic path in any strongly connected component of the program. [^3]: Here, for any program $\Pi$, we write ${\ensuremath{H(\Pi)}}= \bigcup_{r \in \Pi} {\ensuremath{H(r)}}$. [^4]: [[GnT]{}]{}, available from <http://www.tcs.hut.fi/Software/gnt/>, is an extension of [[Smodels]{}]{}solving DLPs by interleaved calls of [[Smodels]{}]{}, which itself is only capable of solving normal LPs. [^5]: Respective ground instances have been produced with the command `dlv -OR- -instantiate`, (cf. the [`DLV`]{}-Manual [@dlv-web]), which turns off most of the grounding optimizations.

--- abstract: 'We investigate the minimization of the energy per point $E_f$ among $d$-dimensional Bravais lattices, depending on the choice of pairwise potential equal to a radially symmetric function $f(|x|^2)$. We formulate criteria for minimality and non-minimality of some lattices for $E_f$ at fixed scale based on the sign of the inverse Laplace transform of $f$ when $f$ is a superposition of exponentials, beyond the class of completely monotone functions. We also construct a family of non-completely monotone functions having the triangular lattice as the unique minimizer of $E_f$ at any scale. For Lennard-Jones type potentials, we reduce the minimization problem among all Bravais lattices to a minimization over the smaller space of unit-density lattices and we establish a link to the maximum kissing problem. New numerical evidence for the optimality of particular lattices for all the exponents are also given. We finally design one-well potentials $f$ such that the square lattice has lower energy $E_f$ than the triangular one. Many open questions are also presented.' address: - 'QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark' - 'Departamento de Matematicas, Pontificia Universidad Catolica de Chile, Av. Vicuna Mackenna 4860, Macul, Santiago, 6904441, Chile' author: - Laurent Bétermin - Mircea Petrache title: 'Optimal and non-optimal lattices for non-completely monotone interaction potentials' --- **AMS Classification:** Primary 74G65; Secondary 82B20, 11F27\ **Keywords:** Lattice energies, Theta functions, Lennard-Jones potentials, Triangular lattice, Completely monotone functions, Laplace transform Introduction and main results ============================= Efforts for rigorously proving crystallization phenomena, i.e. the fact that ground states of systems exhibit a periodic order, have recently been very active. This phenomenon is observed numerically and experimentally in several settings, but its rigorous mathematical justification appears to often be very challenging and the principles at work seem to be far from being completely understood (see the reviews [@radin1987low; @Blanc:2015yu]). In physics-inspired phenomenological models, results are known for one-dimensional models [@VN1; @vn1d1; @vn1d2; @Rad1; @BHS; @BlancLebris; @SandierSerfaty1d; @Leble1d; @BetKnupfNolte] and for some higher-dimensional cases [@Rad2; @Rad3; @Crystal; @ELi; @Stef1; @Stef2; @TheilFlatley; @Luca:2016aa]. In parallel to this direction, the study of the problem by number theory and related combinatorial techniques (see the book [@ConSloanPacking]) provided important results in dimensions $2$ and also $3$ [@Rankin; @Cassels; @Diananda; @Ennola; @Eno2; @Mont; @OPS; @hales2005proof; @Suto1; @Suto2], as well as in some particular higher dimensions [@CohnElkies; @banasz; @Coulangeon:2010uq; @CoulLazzarini; @SarStromb; @CoulSchurm2018], leading to the recent proof of optimality of best packings in dimensions $8$ and $24$ [@Viazovska; @CKMRV]. For $d\geq 1$, let $\mathcal L_d:=\{A\mathbb Z^d:\ A\in GL(d)\}$ be the space of all $d$-dimensional Bravais lattices. Our goal is to study here the main model problem for the crystallization question, namely the minimization of the potential energy $E_f$ defined, for any $L\in \mathcal{L}_d$, by $$\label{defintro-Ef} E_f[L]:=\sum_{p\in L\setminus\{0\}}{\vphantom{\sum}}f(|p|^2),$$ where $|\cdot |$ is the euclidean norm on ${\mathbb R}^d$ and $f$ is an admissible function in the sense of Definition \[def:admissible\] below, i.e. $f(|x|^2)$ is integrable away from the origin. Studying $E_f[L]$ globally amongst lattices gives an important information about the case of infinite systems of points in $\mathbb R^d$, if we answer the question of characterizing for which $f$ some special lattices (see the list at the end of this section) are, or are not, global minima amongst lattices. This type of question appeared before in several different contexts, which include Ginzburg-Landau vortices [@Sandier_Serfaty], Bose-Einstein Condensates [@AftBN; @Mueller:2002aa], Gaussian Core models [@cohnkumarsoft] and Thomas-Fermi models for solids [@MR1842045; @Betermin:2014fy] (see the review [@Blanc:2015yu]). There are two main sources of normalization which “fix the scale” of minimizers of $E_f$ and ensure that the minimum over $L$ (or more generally over all configurations) can be achieved: - either we *fix the density of our configurations*, as a constraint in our minimization - or *the shape of $f$ itself selects a preferred scale*, when we minimize amongst lattices of all possible scales. For the first situation the typical example is that of Gaussian kernels $f(r^2)=e^{-a r^2}$, and in the second case the typical case is that of $f$ equal to a one-well potential such as the Lennard-Jones case $f(r^2)=a_1r^{-12}- a_2 r^{-6}$. In fact historically, the main motivation in physics for introducing one-well potentials is precisely the above, namely to provide *good phenomenological models*, in which the scale-fixing is encoded directly via the potential itself, and needs not be artificially fixed (see e.g. [@Kaplan p. 7]). For both the above settings, the global optimality of a given lattice for $E_f$ amongst all lattices can be proved rigorously in very few examples, and the general treatment is based mainly on the following two principles: - **(cristallization at fixed density)** The minimization for $f(r^2)=e^{-a r^2}$ being a Gaussian, and amongst lattices of fixed density has been treated in $d=2$ (in which case $E_f$ is the so-called *lattice theta function* ), was studied in the fundamental work [@Mont] (see also higher-dimensional results [@Coulangeon:2010uq]), which prove that at fixed density the triangular lattice (defined by ) is the unique minimizer, *for all choices of the variance* $a>0$. By a change of variable, this means also that for any Gaussian kernel and amongst lattices of *any fixed density*, the triangular lattice is the unique minimizer. This result can be extended and transferred to all functions which are superposition of Gaussians with positive coefficients, which translates to requiring that $f$ is a *completely monotone function* (see Definition \[def:complmonot\] and the following discussion), a class which includes all inverse power admissible functions. Some conjectures from number theory are then naturally formulated for this class of $f$ (see e.g. [@CohnKumar]), due to the above basic principle. - **(minimization for one-well $f$)** In the absence of the complete monotone assumption on $f$, all known results on crystallization work under *strong localization* hypotheses, and in the setting in which the study of $f$ can effectively be reduced to a *finite-range situation*. The model-case to which proofs reduce is the so-called ”sticky disk potential“ in dimension $d=2$, with $f$ defined by $f(1)=-1$, $f(r)=0$ for $r>1$ and $f(r)=+\infty$ for $r<1$. In this case the general crystallization result (for the case in which we allow as competitors to $E_f$ general configurations too) can easily be proved by only discussing the nearest-neighbors of a given point, as first done by Heitmann and Radin [@Rad1]. The most general $f$ whose study is known to be reducible to the Heitmann-Radin situation is to our knowledge the one appearing in [@Crystal], to which we refer for further references. Note that there is a huge difference between the two above settings: any completely monotone function is in particular positive, decreasing and convex, whereas any one-well potential is negative at infinity and not monotone, and not convex. This means that *a wide class of potentials does not fit in either category*, and thus escapes treatment by the known techniques. In this work, we extend the scope and clarify the limitations of the abovementioned Gaussian superposition and localization principles. We concentrate on the minimization amongst Bravais lattices, and in dimension $d=2$, because we feel that the main principles at work for $d=2$ are at the moment the same ones that can work also in $d\ge 3$, and there is no gain of information in treating the higher dimensions in higher generality in this work. Some of our results generalize directly to other dimensions $d\ge 3$, and we will point this out whenever this is the case. Dimension $d=1$ is better understood, but presents some important open questions. Section \[sec:1d\] contains a survey of the main available $1$-dimensional results and examples, which are presented here as a source of inspiration for possible behaviours to test in dimension $d\ge 2$ in future work. As mentioned before, for the main physically relevant simple potentials, crystallization remains not rigorously proved. We mention here the following basic questions: \[q1\] If $f(r^2)$ is not a positive superposition of gaussians, can the triangular lattice still be a minimizer of $E_f$ among lattices at any fixed density? \[q2\] Does there exist a Lennard-Jones type potential $f(r^2)=a_1 r^{-x_1} - a_2 r^{-x_2}$ with $a_1,a_2>0, x_1>x_2>2$, such that we can prove crystallization amongst periodic configurations in dimension $d=2$? In other words, can we prove that $E_f$ has the triangular lattice as minimizer when considered on lattice configurations? \[q3\] Can we prove crystallization (i.e. that the minimum of $E_f$ amongst all configurations is achieved by a lattice) for some $f$ which has not extremely fast decay at infinity? For example can we prove it in dimension $d=2$ for some $f$ such that $|f(r^2)|\ge C r^{-6}$ for all large enough $r$? Our main results are as follows: 1. For the long-range case, we formulate criteria depending on the expression of $f$ as a superposition of exponentials, and we distinguish different behaviours based on the sign of the inverse Laplace transform of $f$ in Theorem \[MainTh1\]. In particular we provide an example in Section \[not\_cm\] which gives a positive answer to Question \[q1\]. 2. We prove, in Proposition \[Mainprop1\] and Theorem \[MainTh4\] that for a large class of one-well potentials *the square lattice has lower energy than the triangular one*, which proves for the first time in dimension $d>1$ that the two frameworks described above can have essentially distinct properties. 3. In case of one-well potentials, in Theorem \[MainTh2\] and \[MainTh3\], we reduce the number of parameters needed to understand the behavior of $f$ belonging to the class of Lennard-Jones type potentials (this larger class was originally introduced by Mie in 1903 [@mie1903kinetischen]), and we give new evidence for a positive answer to a *stronger version* of Question \[q2\], which however we do not have the tools to answer. The proofs of our results and counterexamples are implemented by extracting precise principles which could also be applied in dimension higher than $2$. We have already studied the above kind of questions for some more specific choices of $f$ and in some related problems, in [@Betermin:2014fy; @BetTheta15; @Beterloc; @BeterminPetrache; @Beterminlocal3d; @MorseLB], again with special emphasis on “physical” dimensions $d\in \{2,3\}$. The first author and Knüpfer have treated the case of two-dimensional interaction of masses located on lattice sites in [@BetKnupfdiffuse; @Softtheta], and the second author and Serfaty have treated the case of Jellium-type energies in $2$-dimensions for power-law interactions in [@petrache2017next]. Some noteworthy Bravais lattices, which play important roles in our energy minimization problem, are the square lattice $\mathbb Z^2$ and the triangular lattice $\mathsf{A}_2$ and its renormalized version $\Lambda_1$, defined by $$\label{def-triangularintro} \mathsf{A}_2:={\mathbb Z}(1,0)\oplus {\mathbb Z}\left(\frac{1}{2},\frac{\sqrt{3}}{2} \right),\quad \Lambda_1:=\sqrt{\frac{2}{\sqrt{3}}}\mathsf{A}_2.$$ In dimension $3$, by abuse of notation (because the lattices $\mathsf{D}_n$ are not usually with a unit density), special roles will be played by the Face-Centred-Cubic (FCC) lattice $\mathsf{D}_3$ and its dual, the Body-Centred-Cubic (BCC) lattice $\mathsf{D}_3^*$, which will be defined by $$\begin{aligned} &\mathsf{D}_3:=2^{-\frac{1}{3}}\left[ {\mathbb Z}(0,1,1)\oplus {\mathbb Z}(1,0,1) \oplus {\mathbb Z}(1,1,0) \right],\label{defFCC}\\ &\mathsf{D}_3^*:=2^{\frac{1}{3}}\left[{\mathbb Z}(1,0,0)\oplus {\mathbb Z}(0,1,0)\oplus {\mathbb Z}\left(\frac{1}{2},\frac{1}{2},\frac{1}{2} \right) \right].\label{defBCC}\end{aligned}$$ We have represented these lattices in Figure \[Lattices\]. ![Representation of the triangular and square lattices $\mathsf{A}_2,{\mathbb Z}^2$ (first line) and the simple cubic, FCC and BCC lattices ${\mathbb Z}^3,\mathsf{D}_3,\mathsf{D}_3^*$ (second line)[]{data-label="Lattices"}](2dlattices.png "fig:"){width="6cm"}\ ![Representation of the triangular and square lattices $\mathsf{A}_2,{\mathbb Z}^2$ (first line) and the simple cubic, FCC and BCC lattices ${\mathbb Z}^3,\mathsf{D}_3,\mathsf{D}_3^*$ (second line)[]{data-label="Lattices"}](3dlatticeZ3.png "fig:"){width="3cm"}![Representation of the triangular and square lattices $\mathsf{A}_2,{\mathbb Z}^2$ (first line) and the simple cubic, FCC and BCC lattices ${\mathbb Z}^3,\mathsf{D}_3,\mathsf{D}_3^*$ (second line)[]{data-label="Lattices"}](3dlatticeFCC.png "fig:"){width="3cm"}![Representation of the triangular and square lattices $\mathsf{A}_2,{\mathbb Z}^2$ (first line) and the simple cubic, FCC and BCC lattices ${\mathbb Z}^3,\mathsf{D}_3,\mathsf{D}_3^*$ (second line)[]{data-label="Lattices"}](3dlatticeBCC.png "fig:"){width="3cm"} In higher dimensions $d\in \{4,8,24\}$, we consider (unit-density versions of) the classical lattices $\mathsf{D}_4,\mathsf{E}_8,\Lambda_{24}$, which are respectively defined in [@ConSloanPacking Sect. 7.2, 8.1 and 11]. We now pass to introducing the precise statements of our results. Minimization at fixed density by writing $f(r^2)$ as a superposition of Gaussians {#sec:intro_gauss} --------------------------------------------------------------------------------- We recall the notion of completely monotone function: \[def:complmonot\] We say that $f:(0,+\infty)\to [0,+\infty)$ is *completely monotone* if for any $k\in {\mathbb N}_0$ and any $r>0$ $(-1)^{k}f^{(k)}(r)\geq 0$. We introduce the following normalizations and notations on spaces of lattices. Denote respectively by $\mathcal{L}_d^\circ\subset \mathcal L_d $ and $\mathcal{L}_d^1\subset \mathcal L_d$ the subsets of lattices with respectively unit density and unit shortest non-zero vector. Furthermore, we denote by $D_{\mathcal{L}_d}$ and $D_{\mathcal{L}_d^\circ}$ the fundamental domains of $\mathcal L_d$ and $\mathcal L_d^\circ$ (see Section \[sect-notations\] for a precise definition), where each Bravais lattice appears only once. Moreover, the shape $[L]$ of a Bravais lattice $L\in \mathcal{L}_d^\circ$ is its equivalence class modulo rotation and dilation among Bravais lattices (see Definition \[defn:shape\]). Our starting point is the well-known result (see e.g. [@CohnKumar p. 169] or [@BetTheta15 Prop 3.1]) which says that if $f$ is a completely monotone (admissible) function, then the triangular lattice $\Lambda_1$ is the unique minimizer of $L\mapsto E_f[\lambda L]$ on $D_{\mathcal L_d^\circ}$ for any fixed $\lambda>0$. Indeed, this follows by superposition, from the following two celebrated results. In dimension $d=2$, the triangular lattice $\Lambda_1$ is the unique minimum of $D_{\mathcal{L}_2^\circ}\ni L\mapsto \theta_L(\alpha)$, for any fixed $\alpha>0$. The lattice theta function $\theta_L$ mentioned above is defined for $L\in \mathcal{L}_d$ (or, more generally, for any configuration $L\subset \mathbb R^d$ for which the below sum is finite) and $\alpha>0$ by $$\label{def-thetalattice} \theta_L(\alpha):=\sum_{p\in L} e^{-\pi \alpha |p|^2}.$$ A function $f$ is completely monotone if and only if $f$ is the Laplace transform of a positive Borel measure $\rho_f$ on $(0,+\infty)$. A direct important consequence, also proved by Rankin [@Rankin], Ennola [@Eno2], Cassels [@Cassels] and Diananda [@Diananda], is the minimality of $\Lambda_1$ on $D_{\mathcal{L}_2^\circ}$, for any $s>2$, for the Epstein zeta function defined by $$\label{def-epsteinzeta} \zeta_L(s)=\sum_{p\in L\setminus\{0\}} \frac{1}{|p|^s}.$$ The next natural question is now to study the minimization of $L\mapsto E_f[\lambda L]$ on $D_{\mathcal{L}_d^\circ}$, for fixed $\lambda>0$, when instead of being positive as in the above theorem, the measure $\mu_f$ is negative on some open sets of $(0,+\infty)$, i.e. $f$ is not completely monotone. This question has been studied by the first author and Zhang in [@Betermin:2014fy; @BetTheta15] for the special case of the Lennard-Jones type potentials in dimension $d=2$, where the minimality of $\Lambda_1$ for $\lambda$ small enough was proved (with an explicit upper bound), as well as its non-minimality for $\lambda$ large enough (again with an explicit lower bound). Another natural question is the nature of lattices $\Lambda$ that are minimizers of $L\mapsto E_f[\lambda L]$ at fixed density $\lambda$ for any $\lambda>0$. Using the Laplace transform representation $$\label{laplacerepresent} \forall r>0,\quad f(r)=\int_0^{+\infty} e^{-rt}d\mu_f(t),$$ where $\mu_f=\mathcal{L}^{-1}[f]$ is the inverse Laplace transform of $f$, which we assume to be well defined and to be a Radon measure, we prove the following results in Proposition \[prop-asymptnonopt\] and Proposition \[prop-anyscale2\] below. \[MainTh1\] Let $d\geq 1$ and assume that $L_m$ is, for all $\alpha>0$, the unique minimizer on $D_{\mathcal{L}_d^\circ}$ of $L\mapsto \theta_L(\alpha)$, defined by . Let $f$ be an admissible potential with representation . Then the following holds for any $r_0>0$: 1. If $\mu_f$ is negative on $(0,r_0)$, then for any Bravais lattice $L\in \mathcal{L}_d^\circ\backslash\{L_m\}$, there exists $\lambda_0$ such that for any $\lambda>\lambda_0$, $E_f[\lambda L]<E_f[\lambda L_m]$; 2. If $\mu_f$ is negative on $(r_0,+\infty)$, then for any Bravais lattice $L\in \mathcal{L}_d^\circ\backslash\{L_m\}$, there exists $\lambda_1$ such that for any $0<\lambda<\lambda_1$, $E_f[\lambda L]<E_f[\lambda L_m]$. 3. If $\mu_f$ is positive on $(0,r_0)$ or on $(r_0,+\infty)$, and $\Lambda$ is a minimizer of $L\mapsto E_f[\lambda L]$ for any $\lambda >0$ on $D_{\mathcal{L}_d^\circ}$, then $\Lambda=L_m$. 4. If $\mu(r)$ is negative on $(0,r_0)$ or on $(r_0,+\infty)$, then the minimizer of $L\mapsto E_f[\lambda L]$ cannot be the same for all $\lambda>0$. In particular, these results hold in dimension $d=2$ for the triangular lattice $L_m=\Lambda_1$. The major issue is to identify $L_m$ for a given dimension $d$. This result is only known, so far, in dimension $d=2$ where $L_m=\Lambda_1$. We have previously worked on this problem in higher dimensions and we have given in [@BeterminPetrache] many results about minimizers of $L\mapsto \theta_L(\alpha)$ on different subclasses of lattices. It is actually conjectured by Cohn and Kumar in [@CohnKumar Conjecture 9.4] that $\Lambda_1,\mathsf{E}_8$ and $\Lambda_{24}$ are the unique minimizers of $L\mapsto \theta_L(\alpha)$ on $D_{\mathcal{L}_d^\circ}$, $d$ respectively equal to $\{2,8,24\}$ for any fixed $\alpha>0$. Their conjecture is even more general than that: they claim that these lattices, as well as $\Lambda_1$ in dimension $d=2$, are the unique minimizers of $\mathcal C\mapsto\theta_{\mathcal C}(\alpha)$, for all $\alpha>0$, among *all periodic configurations $\mathcal C$* of density $1$. For a space $X$ (in our case we always assume $X=\mathbb R^d$), configurations in $X$ with the property of minimizing the energy $E_f$, at fixed density, for all Gaussian kernels $f(r^2)=e^{-ar^2}$, are called *universally optimal*. The local minimality of $\mathsf{D}_4$ among four-dimensional periodic configurations of unit density for the lattice theta function, for all $\alpha>0$, proved in [@Coulangeon:2010uq], suggests that $\mathsf{D}_4$ should also be universally optimal in dimension $d=4$. Furthermore, recent results by Viazovska et al. [@Viazovska; @CKMRV] about the best packings in dimensions $8$ and $24$ have shown the efficiency of the Cohn-Elkies linear programming bounds for sphere packing [@CohnElkies] and could possibly lead to a proof of this conjecture in those dimensions (see [@CohnKumar] for the link between linear programming bounds and energy minimization problems). A natural conjecture would be that for an admissible function $f$, if for any $\lambda>0$, the minimizer $L_m$ of the theta function for any $\alpha>0$ is the unique minimizer of $L\mapsto E_f[\lambda L]$ on $D_{\mathcal L_d^\circ}$, then $f$ is completely monotone (see e.g. [@BetTheta15]), i.e. $\mu_f$ is a positive measure. Our example of Section \[not\_cm\] shows that this is not true in dimension $d=2$ for $L_m=\Lambda_1$. Indeed, considering, for $\varepsilon\geq 0$, the following potential $$\label{deffepsilon} f_\varepsilon(r):=\frac{6}{r^4}-\frac{2(2+\varepsilon)}{r^3}+\frac{1+\varepsilon}{r^2},$$ such that $\mu_{f_\varepsilon}$ is negative on $(1,1+\varepsilon)$, we numerically show that there exists $\varepsilon_0\approx 1.148$ such that for any $0< \varepsilon<\varepsilon_0$, the triangular lattice $\Lambda_1$ is the unique minimizer of $L\mapsto E_{f_\varepsilon}[\lambda L]$ on $D_{\mathcal{L}_d^\circ}$ for any fixed $\lambda>0$. We therefore observe that the fact that $\mathcal{L}^{-1}f$ is negative in a small interval, and if the measure of this negative part is not too big with respect to the measure of its positive part, the triangular lattice stays optimal at all scales for $E_f$. This results leads to another one that is more general, concerning the measure of the negative and positive parts of $f$: “How negative” can the inverse Laplace transform $\mu_f:=\mathcal L^{-1}f$ be, while preserving the property that the minimum $\min_{L\in D_{\mathcal L_2^\circ}}E_f[\lambda L]$ is achieved at all $\lambda>0$ by the triangular lattice? For example, if $\mu_f=\mu_f^+-\mu_f^-$ with $\mu_f^\pm$ positive finite measures, then we can ask more precisely: how large can the ratio of $R_f:=\int d\mu_f^-/\int d\mu_f^+$ be while preserving the above property? An answer to the above question would be of a great interest for understanding the following problem that we leave open: What is the largest class of functions $f$ such that for any $\lambda>0$, the triangular lattice $\Lambda_1$ is the unique minimizer of $L\mapsto E_f[\lambda L]$ on $D_{\mathcal L_2^\circ}$? Note that a similar question arises also in general dimension $d\ge 2$, and is also open in that case. Regarding the case $d=1$, we also do not know the complete answer, but more results are available. See Section \[sec:1d\] for a discussion. Concerning our example , it is also natural to ask whether the triangular lattice is a minimizer, for the same potential, also amongst general configurations. More generally, the following question is completely open: Is there any non-completely monotone $f$ for which the minimizer of $E_f[\lambda \mathcal{C}]$ is the triangular lattice for all $\lambda>0$, among periodic configurations $\mathcal{C}$ of unit density? We note again, by Theorem \[MainTh1\], that one-well potentials $f$ are not a good candidate in the above question, and we do not know of a good underlying principle which would allow to find a candidate. It is for instance not clear if $f_\varepsilon$ defined by could satisfy this property. However, we conjecture that the local minimality of $\Lambda_1$ for $E_{f_\varepsilon}$ should hold among periodic configurations of fixed density by a small modification of [@Coulangeon:2010uq Cor. 4.5]. Also note that, while no proof of any lattice-like configuration being *universally optimal* is available in the literature, our proof in Section \[not\_cm\] shows that the property of a triangular lattice to be a *minimizer of the energy amongst lattices at any fixed scale*, usually conjectured in universal optimality frameworks, *could hold beyond that setting*, i.e. for non-completely monotone functions. Therefore an answer to the above question would be important for understanding/testing the relevance of the concept of universal optimality, and we leave the following general question open: What is the largest class of functions $f$ such that the unique minimizer $L_m$ of $L\mapsto \theta_L(\alpha)$ on $D_{\mathcal{L}_d^\circ}$, for any $\alpha>0$ is the unique minimizer of $L\mapsto E_f[\lambda \mathcal{C}]$, for any $\lambda>0$, among periodic configurations $\mathcal{C}$ of unit density? Furthermore, about the minimality of some lattice at all scales, the following question can be asked: \[q3d\] For $d=3$ does there exist a continuous potential $f$ and a lattice $L_f$ for which $\min_{L\in D_{\mathcal L_3^\circ}}E_f[\lambda L]$ is achieved by $L_f$ at all scales $\lambda>0$? The reason why the above is not clear is that in $d=3$ the analogue of the Rankin-Montgomery theorem [@Rankin; @Mont], is false, as noted for example in [@SarStromb p. 117], i.e. no lattice can achieve the minimum for the theta functions at all scales. This implies that if a function $f$ as required in Question \[q3d\] exists, then it cannot have positive inverse Laplace transform. On the other hand, for the best packing problem, Hales’ result [@hales2005proof] implies that the unique minimizer amongst lattices is given by the FCC lattice at all scales, and it is well known that this minimization problem is the limit $s\to+\infty$ of the minimization for $f(r)=r^{-s}$. Minimization for one-well potentials $f$ without density constraint ------------------------------------------------------------------- The second problem we are setting in this paper is the global optimality of some lattices for $E_f$ on $D_{\mathcal{L}_d}$ (without a density restriction) where $f$ is a one-well potentials, i.e. $f:(0,+\infty)\to {\mathbb R}\cup \{+\infty\}$ that is decreasing on $(0,a)$ and increasing on $(a,+\infty)$ for some $a>0$. In [@BetTheta15; @Crystal], three examples of one-well potentials have been studied, where the global optimality of a triangular lattice was proved in dimension $2$ (on $D_{\mathcal{L}_2^\circ}$ for the two first and among all configurations, in the thermodynamic limit sense, for the third one). Those are: 1. Lennard-Jones type potentials $f_{\vec a,\vec x}^{LJ}$ with parameters $\vec a=(a_1,a_2)\in (0,+\infty)^2$ and $\vec x=(x_1,x_2)$, $x_2>x_1>2$, such that, for any $r>0$, $$f_{\vec a,\vec x}^{LJ}(r)=\frac{a_2}{r^{x_2/2}}-\frac{a_1}{r^{x_1/2}} \quad \textnormal{with}\quad \pi^{-\frac{x_2}{2}}\Gamma\left(\frac{x_2}{2}\right)\frac{x_2}{2}\leq \pi^{-\frac{x_1}{2}}\Gamma\left(\frac{x_1}{2}\right)\frac{x_1}{2}.$$ The classical Lennard-Jones potential is $r\mapsto f_{\vec a,\vec x}^{LJ}(r^2)$ with $x_1=6, x_2=12$ in our notation. The exponent $x_1=6$ is justified equals the long-range behavior of the Van der Waals interaction (see e.g. [@Kaplan p. 10]), whereas we do not know a good physical intuition behind the choice $x_2=12$. Note that $f_{\vec a,\vec x}^{LJ}$ is admissible in dimension $d$ according to our definition, if and only if $x_1>d$. 2. Differences of (three-dimensional) Yukawa potentials $f_{\vec a,\vec x}^Y$ with parameters $\vec a=(a_1,a_2)$, $0<a_1<a_2$ and $\vec x=(x_1,x_2)$, $0<x_1<x_2$, such that, for any $r>0$, $$f_{\vec a, \vec x}^Y(r)=\frac{a_2 e^{-x_2 r}-a_1 e^{-x_1 r}}{r} \quad \textnormal{with} \quad \frac{a_1\left( a_1x_2+x_1(a_2-a_1)\pi \right)}{a_2x_2\left(a_1+(a_2-a_1)\pi\right)}e^{\left(1-\frac{x_1}{x_2} \right)\left( \frac{a_2}{a_1}-1 \right)\pi}\geq 1.$$ This type of interacting potential arises in physics. For example, it turns out that Neumann [@Neumann] has shown that a linear combinations of Yukawa potentials are the most general laws ensuring the stability of electric charges. 3. Abstract potentials $V_\alpha$ described in [@Crystal], such that, for $\alpha$ small enough, there is a large repulsion at $0$ of order $1/\alpha$, a well with width of order $\alpha$ and behavior at infinity controlled by $r\mapsto \alpha r^{-7}$. We first focus on Lennard-Jones type potentials and we will write, in the following two results, $$\label{defLJ} f(r)=f_{\vec a,\vec x}^{LJ}(r)=\frac{a_2}{r^{x_2/2}}-\frac{a_1}{r^{x_1/2}}, \quad x_2>x_1>d,\quad (a_1,a_2)\in (0,+\infty).$$ In this case, the lattice energy of a Bravais lattice $L\in \mathcal{L}_d$ is given by $$E_f[L]=a_2\zeta_L(x_2)-a_1\zeta_L(x_1),$$ where the Epstein zeta function $\zeta_L$ is defined by . These potentials arise naturally in physics models of matter (see e.g. [@mie1903kinetischen; @LJonesPhasediagram; @Kaplan; @LJonesnoblegas]) in the case of the Born-Oppenheimer adiabatic approximation of the interaction energy where the electrons effect is neglected and the energy is reduced to the atomic interaction of the nuclei (see e.g. [@CondensMatter p. 33]). Thus the potential energy of the system is expressed in terms of many-body interactions and the simplest case – which is also relevant in many situations (see e.g. [@CondensMatter p. 945]) – is the one in which the energy is a sum of $2$-body interaction potentials. Lennard-Jones type potentials also appear in social aggregation model [@MEKBS]. In dimension $d=2$ (resp. $d=3$), the global minimizer of $E_f$ on $D_{\mathcal{L}_d}$ is expected to be a triangular lattice (resp. a FCC lattice) for all $x_2>x_1>d$, as conjectured in [@BetTheta15; @Beterloc; @Beterminlocal3d] from numerical evidences and local optimality results. In the following result, proved in Proposition \[prop-LJglobal\].(1) and Proposition \[prop:equiv\] below, we show that the problem of minimizing $E_f$ on the space of all Bravais lattices $D_{\mathcal{L}_d}$ can be reduced to a minimization problem on the space of lattices with unit density $D_{\mathcal{L}_d^\circ}$. In particular, the shape of the global minimizer of $E_f$ does not depend on $a_1,a_2$, which we have already observed in [@BetTheta15]. Furthermore, inspired by the $d\in \{2,3\}$ cases where the minimizer of $E_f$ on $D_{\mathcal{L}_d}$ seems to be the same for all $x_2>x_1>d$, we give different statements that are equivalent to this optimality for all the parameters $x_1,x_2$. \[MainTh2\] Let $d<x_1<x_2$, $(a_1,a_2)\in (0,+\infty)^2$ and $L_0=\lambda \Lambda$ where $\Lambda \in \mathcal{L}_d^\circ$. Then for $f$ a Lennard-Jones type potential as in , $L_0$ is a global minimizer of $E_f$ on $D_{\mathcal{L}_d}$ if and only if $\Lambda $ is a minimizer on $D_{\mathcal{L}_d^\circ}$ of $\tilde{E}_f$ defined by $$\tilde{E}_f[L]:=\frac{\zeta_L(x_2)^{x_1}}{\zeta_L(x_1)^{x_2}}.$$ Furthermore, if $L\in \mathcal L_d^\circ$ then we define functions $H_L,h_L:(d,+\infty)\to {\mathbb R}$ by $$H_L(x):=\frac{1}{x}\log\left( \frac{\zeta_L(x)}{\zeta_{\Lambda }(x)}\right), \quad h_L(x):=-\log \zeta_L(x) +x\frac{\partial_x\zeta_L(x)}{\zeta_L(x)}.$$ The following conditions are equivalent: 1. For any $x_2>x_1>d$, the lattice $L_0 $ is the unique minimizer of $E_f$ on $D_{\mathcal L_d}$. 2. For any $x_2>x_1>d$, the lattice $\Lambda $ is the unique minimizer of $\tilde{E}_f$ on $D_{\mathcal L_2^\circ}$. 3. For any Bravais lattice $L\in \mathcal L_d^\circ\setminus\{ \Lambda \}$, the function $H_L$ is strictly increasing on $(d,+\infty)$. 4. For any $x>d$, $\Lambda $ is the unique minimizer of $h_L(x)$ on $D_{\mathcal{L}_d^\circ}$. About the case $d=2$, (1) of Theorem \[MainTh2\] has been proved in [@BetTheta15 Thm. 1.2.B.2 and Lem. 6.17] on the small interval $I=(2,2\psi^{-1}(\log \pi)-2)$ where $\psi$ is the digamma function and $2\psi^{-1}(\log \pi)-2\approx 5.256$. We conjecture that the equivalent statements of Theorem \[MainTh2\] hold true in dimensions $d\in \{ 2,3,4,8,24\}$ for $\Lambda\in \{\Lambda_1,\mathsf{D}_3,\mathsf{D}_4,\mathsf{E}_8,\Lambda_{24}\}$. In dimensions $d\in \{2,3\}$, new numerical evidence for $\tilde{E}_f$ supporting our conjecture is included in Figures \[H2to10square\], \[fig:ratio\], \[Z3vsFCCBCC\] and \[fig-BCCFCC\]. Next, we prove the global optimality of these lattices for large values of the parameters and find their asymptotically optimal scaling as $x_1,x_2\to +\infty$. For a definition and some known results on the *kissing number* (also called *coordination number* in crystallography) of a Bravais lattice $L\in\mathcal L_d$, denoted $\tau(L)$, see Definition \[def:kissing\] and Remark \[rmk:kissing\] below. The following result is proved in Proposition \[prop-LJglobal\].(2) and Proposition \[prop:asymin\] below. \[MainTh3\] For $L,\Lambda\in\mathcal L_d$ and $f$ a Lennard-Jones type potential as in there holds $$\label{limit-kissing} \lim_{x_1\to +\infty}\lim_{x_2\to +\infty}\frac{\displaystyle \min_{\lambda>0}E_f[\lambda\Lambda]}{\displaystyle \min_{\lambda>0}E_f[\lambda L]}=\frac{\tau(\Lambda)}{\tau(L)}\ .$$ As the above minimum values are negative, if $\tau(\Lambda)$ uniquely realizes the optimal kissing number $k(d)$ amongst lattices, then there exists $x_0=x_0(d)$ such that for any $x_2>x_1>x_0$, the unique minimizer of $E_f$ on $D_{\mathcal{L}_d}$ has shape $[\Lambda]$. In particular, this holds for the cases $(d,\Lambda)\in\{(2,\mathsf{A}_2), (3,\mathsf{D}_3), (4,\mathsf{D}_4),(8,\mathsf{E}_8),(24,\Lambda_{24})\}$. If the minimum of $f(r)$ is achieved at $r_f=r_f(x_1,x_2,a_1,a_2)>0$, and $\lambda^{L_2}_0=\lambda^{L_2}_0(x_1,x_2,a_1,a_2)>0$ is the factor such that $\lambda^{L_2}_0 L_2$, $L_2\in \mathcal{L}_d^1$, realizes the minimum energy among lattices of the same shape as $L_2$, i.e. $\min_{\lambda>0} E_f[\lambda L_2]=E_f[\lambda^{L_2}_0 L_2]$, then the following limit exists and satisfies $$\label{limit_r0intro} \lim_{\substack{x_1,x_2\to +\infty\\x_1<x_2, r_f=r_0}}\lambda^{L_2}_0(x_1,x_2,a_1,a_2)=\sqrt{r_0}.$$ This result goes in a similar direction as Theil’s work [@Crystal], valid for general (not necessarily lattice-like) configurations: increasing the parameters makes at the same time the repulsion near the origin larger, the decay at infinity faster and the well of $f$ narrower. Thus, in the limit, the energy takes only into account the nearest-neighbours, that is why the lattices achieving the optimal kissing number $k(d)$ are globally optimal and the length of these lattices tend to the square root of the value of the minimizer of $f$ as in . In the three-dimensional case, the phase diagram of the classical Lennard-Jones energy $f(r^2)=r^{-12}-r^{-6}$ has been numerically studied in detail in [@LJonesPhasediagram; @StillingerLJ]. In particular, by [@StillingerLJ], the optimizer of $E_f$ amongst all configurations seems to be the hexagonal close-packing (HCP, see [@ConSloanPacking Sect. 6.5] for a precise definition). The minimization of $E_f$ corresponds to the minimization of the Helmholtz free energy $H=U-TS$ in the regime of small temperature and pressure $T,P\approx 0$, where $U$ is the internal energy and $S$ is the entropy of the system. On the other hand, for high pressures, numerically, it seems that the global minimizer is the FCC lattice. The HCP is not a Bravais lattice but is also known to be a best packing in dimension $d=3$ (see [@hales2005proof]) as well as the solid structure of many chemical components. As the number of closest neighbors of any point is the same for HCP and FCC, also gives the optimality of the HCP among all periodic configuration for sufficiently large parameters, as it is also true for the FCC lattice, but distinguishing the difference of energies of these two best packings. Note that if we try to fix a scale constraint while minimizing $E_f$ for one-well potentials, then in general *we will find different minimizers at different scales*. The *local minima* for this problem have been numerically studied in [@Beterloc], proving that, as $\lambda>0$ grows, it is expected that the minimizer changes from a triangular lattice to a rhombic one, then to a square one, then to a rectangular lattice and then to a degenerate rectangular one, and parts of this result are actually rigorously proved for high and low densities in [@BetTheta15; @Beterloc]. It is therefore interesting to understand numerically and then to prove rigorously what are possible behaviours of the global minimizers e.g. of simple/explicit one-well physically inspired potentials at fixed scale, and what are the principles that govern this behavior: What is the phase diagram with respect to $\lambda>0$ of $\min_{L\in D_{\mathcal L^\circ_3}} E_f[\lambda L]$, where $f$ is the classical Lennard-Jones potential $f(r^2)=r^{-12}-r^{-6}$? I.e., how are the minimizing lattices at fixed density varying, as $\lambda$ increases? It would be interesting to know whether the minimizer of $E_f[\lambda L]$ on $D_{\mathcal{L}_2^\circ}$ changes with $\lambda$, similarly to the classical Lennard-Jones case described in [@Beterloc], but for the more general case in which $f$ is the difference of two completely monotone functions (see also [@radinclassground] for a study of such $f$ under extra conditions in dimension $d=1$). More precisely, we ask the following question: Let $g_1,g_2$ be completely monotone functions and $f$ be defined by $f(r):=g_1(r)-g_2(r)$. Is the phase diagram, with respect to $\lambda>0$, of $\min_{L\in D_{\mathcal{L}_3^\circ}}E_f[\lambda L]$ the same as the one for the classical Lennard-Jones potential $f(r)=r^{-12}-r^{-6}$ described in [@Beterloc], i.e. triangular-rhombic-square-rectangular-degenerate as $\lambda$ increases? Constructions of $f$ which favor the square lattice over the triangular one --------------------------------------------------------------------------- Since the shape of the global minimizer of $E_f$ does not depend on $a_1,a_2$, it turns out that it is possible to construct potential with a well as wide or narrow as we want. Hence, the existence of a one-well potential $f$ such that the global minimizer is not triangular appears as an interesting question. Inspired by Ventevogel’s counter-example [@VN1 Sec. 5] given in the one-dimensional setting, we give in Section \[subsec-onewell\] an example of discontinuous potential (see Figure \[CE1\]) and an example of continuous potential such that $$\label{min-lambda} \displaystyle \min_{\lambda>0} E_f\left[\lambda {\mathbb Z}^2\right]=E_f\left[\lambda_0^{\mathbb Z^2} {\mathbb Z}^2\right]<E_f\left[\lambda_0^{\mathsf{A}_2} \mathsf{A}_2\right]=\min_{\lambda>0} E_f[\displaystyle \lambda \mathsf{A}_2],$$ where $f$ is defined by $f(r^2)=g(r)$ (change of notation justified by the fact that $g$ is the potential whose derivative is estimated over the lattices distances in the proof). More precisely, the proof of the following proposition is given in Appendix \[appendix\]. \[Mainprop1\] Let $g$ be the continuous potential defined by $$g(r):= \left\{ \begin{array}{ll} \displaystyle \frac{\left( \frac{2}{3} \right)\left(\frac{4}{9} \right)^{p}}{r^{p}} & \mbox{if $0<r<4/9$}\\ 2-3 r & \mbox{if $4/9\leq r\leq 1$}\\ -r^{-4} & \mbox{if $r>1$.}\\ \end{array}\right.$$ Then there exists $p_0$ such that for any $p>p_0$, holds for $f$ defined by $f(r^2):=g(r)$. ![Plot of function $f$ defined in Proposition \[Mainprop1\][]{data-label="CE1"}](CE1.png "fig:"){width="9cm"}\ It is also possible to construct a large class of $C^1$-functions $f$ having property . More precisely, we have the following result (see Proposition \[prop:theil\_ZA\] for a statement giving a more detailed descriptions of the functions $g$ below). \[MainTh4\] There exists an uncountable class of functions $g:[0,+\infty)\to{\mathbb R}\cup\{+\infty\}$ such that the function defined by $f(r^2)=g(r)$ is admissible, and holds. Here we have designed potentials that favour a square lattice instead of a triangular lattice. These two results are in the same spirit as the work of Torquato et al. [@TorquatoHCb; @Torquato09; @Torquato1] where a (truncated) potential is designed in such a way that a targeted lattice structure is the minimizer of the interaction energy. Our potential is however not truncated and the proof of Theorem \[MainTh4\] can be generalized to another $d$-dimensional lattices (see Remark \[rmk-generalprinciple\]). However, there is nothing in Theorem \[MainTh4\] that shows what is the global minimum of $E_f$ (it is not necessarily a square lattice). Finally, we mention the following open direction: Study and classify natural distances between (or other measures of the size of perturbations of) interaction kernels $f$, with respect to which small perturbations of $f$ can be ensured to preserve the crystallization properties of the kernels, such as the existence and shape of the global minimum amongst periodic configurations. In fact our counterexample in Section \[not\_cm\] of a non-completely monotone $f$ which is a good candidate for crystallization at all scales is based on a *small-perturbation method*, and the same can be said more generally for the many of the available proofs of crystallization. Thus it seems extremely important to understand what is the right notion of stability of potentials $f$, as hinted at above, even if the known cases are at the moment extremely episodic and prevent us from formulating any more precise question in a compelling way. We mention, as a simple possible starting point for such studies in dimension $d=1$, the fact that in [@hamrickradin], a potential which is a small (in $C^1$-norm but not in $C^2$-norm) perturbation of a one-well potential was produced, for which $N$-point minimizers converge to a *quasicrystal*. The principle in that case is to “add small wells” to $f$ an irregular pattern: this creates the possibility to find, next to the standard periodic configuration, a slightly better (in terms of energy), but non-periodic, ground state. Plan of the paper. {#plan-of-the-paper. .unnumbered} ------------------ In Section \[sect-notations\] we give the precise definitions of the objects we are working with. A survey of one-dimensional results is presented in Section \[sec:1d\]. Section \[sect-allscales\] is devoted to the problem of minimizing $E_f$ at fixed scale. In particular, we prove Theorem \[MainTh1\] in this section. The Lennard-Jones type case is investigated in Section \[sectLJ\], where Theorem \[MainTh2\] and \[MainTh3\] are proved. Finally, in Section \[sect-contrex\], several counter-examples are stated and proved, included Theorem \[MainTh4\]. Proposition \[Mainprop1\] is proved in Appendix \[appendix\]. Notations and definitions {#sect-notations} ========================= Let $\mathcal L_d:=\{A\mathbb Z^d:\ A\in GL(d)\}$ be the space of all $d$-dimensional Bravais lattices and $\mathcal{P}_d:=\{M\in \mathbb R^{d\times d}:\ M^t=M, M\mbox{ positive definite}\}$ the cone of positive definite matrices, in turn identified with the cone of positive definite quadratic forms in $d$ (real) variables. Recall that the link between the above settings is the following: to each $A\mathbb Z^d\in\mathcal L_d$ we can associate $M:=A^tA\in\mathcal{P}_d$ and the quadratic form $M[x]:=x^tMx$ in $d$ variables. Let $\mathcal L_d^\circ$ be the space of all $d$-dimensional lattices of density $1$: $\mathcal L_d^\circ:=\{A\mathbb Z^d:\ A\in SL(d)\}$, and $\mathcal{P}_d^\circ$ the cone of positive definite matrices of determinant $1$ in $d$ variables. Given a Bravais lattice $L\in \mathcal{L}_d$, its dual lattice $L^*$ is defined by $L^*:=\{x\in {\mathbb R}^2 : \forall p\in L, x\cdot p\in {\mathbb Z}\}$. For $L=A\mathbb Z^d\in \mathcal L_d$ or $L\in \mathcal L_d^\circ$, and $f:(0,+\infty)\to {\mathbb R}$, whose associated matrix and quadratic form are given by $M=A^tA\in\mathcal P_d$ or $\mathcal P_d^\circ$, we define $$\label{energy} E_f[L]:=\sum_{p\in L\setminus\{0\}} f(|p|^2)=\sum_{x\in\mathbb Z^d\setminus\{0\}}f(M[x]).$$ In order for the sum $E_f[L]$ to be equal to an absolutely convergent sum for $L\in\mathcal L^d$, we require that if $F:\mathbb R^d\setminus\{0\}\to(-\infty,+\infty]$ is given by $F(x):=f(|x|^2)$, then $F$ is integrable outside any neighborhood of the origin. For this, we consider the following class of $f$: \[def:admissible\] Let $k\in\mathbb N$. A function $\varphi_k:{\mathbb R}^d\to{\mathbb R}$ which is constant on each one of the cubes $\frac1k[0,1)^d+\frac1k \vec a$, with $\vec a\in \mathbb Z^d$, will be called a *$k$-coarse* function. A function $f:(0,+\infty)\to{\mathbb R}$ is called *admissible in dimension $d$* if for each $k\in\mathbb N$ there exists $k$-coarse functions $\varphi_k^+,\varphi_k^-\in L^1(\mathbb R^d)$ such that $\varphi_k^-(x)\le f(|x|^2)\le \varphi_k^+(x)$ for all $x\in\mathbb R^d\setminus[-1/k,1/k]^d$. A function $f:(0,+\infty)\to{\mathbb R}$ is called *weakly admissible in dimension $d$* if $$\label{admiss_eq} r^{d-1}f(r^2)\in\bigcap_{\epsilon>0}L^1([\epsilon,+\infty)).$$ Note that admissibility implies weak admissibility. Moreover, for $f:(0,+\infty)\to{\mathbb R}$ which is monotone (in particular for $f$ that has positive inverse Laplace transform) or which satisfies $C^1$-bounds away from the origin, the other implication holds: if such $f$ is weakly admissible in dimension $d$ then $f$ is admissible in dimension $d$. The weak admissibility condition in general does not guarantee that $E_f[L]$ is absolutely summable for $L\in\mathcal L^d$, as such $f$ could blow up at $r^2$ corresponding to the distances in lattice $L$, and this is why the more complicated condition in terms of $k$-coarse functions seems justified. On the other hand the simpler condition will suffice for the very regular class of potentials which are treated in this paper. The above functional $E_f$ is invariant under rotations $R\in O(d)$. Indeed, spaces $\mathcal L_d, \mathcal L_d^\circ$ (respectively, $\mathcal P_d,\mathcal P_d^\circ$) have natural actions by rotations, defined as follows. For $R\in O(d)$ and $A\mathbb Z\in \mathcal L_d$ (respectively, $M\in\mathcal P_d$), we define $R\cdot L:=RA\mathbb Z$ (respectively, $R\cdot M:=R^tMR=R^{-1}MR$, which is the induced action under the identification $M=A^tA$, because then $R\cdot M=(RA)^t(RA)$). As $E_f$ only depends on $M[x]$, which in turn is invariant under our $O(d)$-action, we find that $E_f[L]=E_f[R\cdot L]$ for all $R\in O(d)$, as claimed. We denote by $D_{\mathcal L_d},D_{\mathcal L_d^\circ},D_{\mathcal P_d}, D_{\mathcal P_d^\circ}$ fundamental domains for the above actions. Furthermore: - If $L$ is a lattice, let $r_0(L):=0<r_1(L)<r_2(L)<\cdots<r_n(L)<\cdots$ be an enumeration of the set of distances $\{|v|:v\in L\}$. In this case we call $L^{(j)}:=\{v\in L:\ |v|=r_j\}$ the $j$-th shell of $L$. We also denote $L^{(j,k)}:=L^{(j)}\cup L^{(k)}$ and $L^{(\ge j)}:=\bigcup_{k\ge j}L^{(k)}$. - Let $\mathcal L_d^1$ be the space of all $d$-dimensional lattices whose shortest nonzero vector has lenght $1$. In terms of the above notation, $\mathcal L_d^1:=\{L\in \mathcal L_d:\ r_1=1\}$. \[defn:onewell\]We call $f:(0,+\infty)\to \mathbb R\cup\{+\infty\}$ a *one-well potential* if there exists $a>0$ such that $f$ is nonincreasing on $(0,a)$ and nondecreasing on $(a,+\infty)$. \[defn:shape\] Define an equivalence relation $\sim$ on $\mathcal L_d$ by stating that for $L,L^\prime\in\mathcal L_d$ there holds $L\sim L^\prime$ if there exists $\lambda>0$ and $R\in O(d)$ such that $L=\lambda R\cdot L^\prime$. The intersection of the equivalence class $[L]$ of $L\in\mathcal L_d$ under $\sim$, with the fundamental domain $D_{\mathcal L_d^\circ}$ of $\mathcal L_d^\circ$ under $O(d)$ action is called the *shape* of $L$. Survey of known results in dimension $d=1$ {#sec:1d} ========================================== Note that in dimension $1$ it is common usage to define $$\label{energy1d} E^\mathrm{1D}_g[L]:=\sum_{p\in L\setminus\{0\}} g(|p|) \quad \text{and}\quad E^\mathrm{1D}_g[L_N]:=\frac{1}{N}\sum_{j=1}^N\sum_{i=-\infty\atop i\neq 0}^{+\infty} g(x_{i+j}-x_i),$$ where $L\in\mathcal L_1$ is a lattice and $L_N$ is an $N$-periodic configuration of density $\rho$, i.e. it satisfies $x_{i+N}-x_i=\rho^{-1}$ for some $\rho>0$. We usually suppose that $g(-x)=g(x)$ for all $x\neq 0$, and we note that then we just sum $g(|p|)$ (rather than $g(|p|^2)$, as done here in ). The asymptotics in periodic fixed-density situations ---------------------------------------------------- Recall that, corresponding to the setting in which the shape of $g$ does not automatically determine one scale for minimizers, as explained in Section \[sec:intro\_gauss\], for $d\ge 2$ we resorted to writing $g(r^2)$ as a superposition of Gaussians, i.e. to looking at the inverse Laplace transform of $g$, and discussing the ensuing coefficients. This setting turns out to be too restrictive in $d=1$. For example we have the following representative result, in a periodic case, and we refer to [@BHS] for several further theorems: Let $g:\mathbb R\to \mathbb R\cup \{+\infty\}$ be an even, lower semicontinuous function, invariant under translations by $N\mathbb Z$, so that the values of $g$ on $[0,N/2]$ completely determine $g$. If $g$ is convex decreasing on $[0,N/2]$, then the energy $E^\mathrm{1D}_g[L_N]$ attains a global minimum on the periodic configuration $\mathbb Z$. In the same work, it is proven that if $g$ as above is *concave and decreasing* on $[0,N/2]$, then the minimum is very degenerate and the points concentrate on a translated copy of $N/2\mathbb Z$, and it is a configuration with multiplicity $\lfloor N/2\rfloor$. Originally this last phenomenon was observed in [@VN1 Section 5]. A related result appears in [@georgakoul], where an equivalent proof of precisely the same statement is given depending on the *force* between points, giving a satisfactory answer to the question: “which configurations on the real line are in equilibrium with respect to repulsive $2$-point forces which are strictly decreasing with respect to the distance”, the answer being that only a “crystalline” periodic configuration can be stable. The following questions are at the moment still open: \[q:necsuf1d\] What is a necessary and sufficient condition on $g$ under which for any $N\in\mathbb{N}^*$ all minimizing $1$-dimensional $N$-periodic configurations of density $\rho$ are up to translation equal to the lattice $\rho^{-1}\mathbb Z$? \[q:necsuf\_force1d\] What is a necessary and sufficient condition on a repulsive force $G$ under which for any $N\in\mathbb{N}^*$ a $1$-dimensional $N$-periodic configuration of density $\rho$ in which forces balance at each point, are up to translation equal to the lattice $\rho^{-1}\mathbb Z$? One-well potentials $g$ without constraints on the density ---------------------------------------------------------- - In [@vn1d1] it was shown that there exist nonconvex potentials $g$ which have $\rho^{-1}\mathbb Z$ as unique minimizer for its average energy per particle among $N$-periodic configurations of density $\rho$ for any $N$ and any $\rho$. - In [@vn1d1 p. 284], for $g(x)=(1+|x|^4)^{-1}$, it has been proved that the average energy of the regular $2$-periodic configuration $1/2{\mathbb Z}$ is larger than the $2$-periodic most degenerate case, in which two points share each position $n\in {\mathbb Z}$. - In [@vn1d2] it was shown that in the class of $f$ such that $g(x)=g(-x)$, such that $g''$ exists and is well-behaved at infinity, and such that the Fourier transform $\hat g$ exists, a **necessary condition** for $\mathbb Z$ to satisfy the optimality condition among $N$-periodic configurations for any $N$ at high enough density $\rho\ge \rho_0$, is that $\hat g\ge 0$. This results has been generalized to two-component systems with three kind of interacting potentials in [@BetKnupfNolte]. - In [@gardnerradin] it was proved that for the classical Lennard-Jones potential $g(r)=r^{-12}-r^{-6}$ the unique minimizer amongst all configurations in dimension $d=1$ is a periodic crystal. This was later extended in [@radinclassground] to more general potentials of the form $g(r)=g_1(r)-g_2(r)$ with $g_1,g_2$ convex and with good decay properties, based on Sinai’s theorem on thermodynamic limits and on the previous work [@VN1], and necessary conditions for crystallization were given. These conditions seem relatively cumbersome, and as far as we could check, they were not further improved in later works. The following question seems to be still open: In dimension $d=1$, what is the largest class of potentials $g$ which have the property that the minimizers of $N$-point energies asymptotically as $N\to +\infty$ approximate (up to translation and rotation) a periodic configuration? Optimality and non-optimality at all scales {#sect-allscales} =========================================== We recall the following well-known result in dimension $d=2$, which follows from the Montgomery [@Mont Thm 1] and Bernstein-Hausdorff-Widder [@Bernstein] theorems (see the statements of these theorems in the Introduction): \[prop:completemonot\] If $f$ is completely monotone, then for any $\lambda>0$, the triangular lattice $\Lambda_1$ is the unique minimizer of $L\mapsto E_f[\lambda L]$ in $D_{\mathcal L_2^\circ}$. The above sufficient condition is not necessary, due to our new counterexample of Section \[not\_cm\]. On the other hand, too simple criteria do not furnish sufficient conditions strong enough to replace complete monotonicity. Indeed, by methods related to Montgomery’s approach, in [@BetTheta15 Prop 3.4], the first author also proved that the following positive, decreasing and convex function $$\label{Counterexample} V(r)=\frac{14}{r^2}-\frac{40}{r^3}+\frac{35}{r^4}$$ is such that the triangular lattice is not a minimizer of $E_V$ in $\lambda \mathcal L_2^\circ$ where $\lambda_1<\lambda<\lambda_2$ for values $\lambda_1\approx 1.522$ and $\lambda_2\approx 1.939$. In the two next subsections, we are prove Theorems \[MainTh1\] and \[MainTh2\] about the minimality of our special lattices at fixed scale. Non-optimality at high or low density for a subclass of functions ----------------------------------------------------------------- In this part, we consider $f$ admissible in $\mathbb R^d$ and such that $$\label{def-flaplace} f(r)=\int_0^{+\infty} e^{-rt} d\mu_f(t) :=\mathcal L[\mu_f](r),$$ where $\mu_f:=\mathcal{L}^{-1}[f]$ is the inverse Laplace transform of $f$, which we assume to be a well-defined Radon measure. We start by recalling Jacobi’s transformation formula for the lattice theta function defined by $$\label{def-theta} \theta_L(\alpha)=\sum_{p\in L} e^{-\pi \alpha |p|^2}.$$ which is in fact a simple application of Poisson Summation Formula. A proof of this identity can be found for instance in [@Bochnertheta]. \[lem:mont\] For any $d\geq 1$, any $\alpha>0$ and any Bravais lattice $L\in \mathcal{L}_d^\circ$, $$\label{identity-thetaA} \theta_L(\alpha)=\alpha^{-\frac{d}{2}}\theta_{L^*}\left(\frac{1}{\alpha}\right).$$ From and , it is possible to write $E_f[L]$ in terms of $\theta_L$ as follows: \[prop:tri2\] For any $\lambda>0$, any admissible function $f$ having the representation with absolutely continuous $d\mu_f(t)=\rho_f(t)dt$ and any Bravais lattice $L\in \mathcal{L}_d^\circ$, we have $$\begin{aligned} E_f[\lambda L]&=\frac{\pi}{\lambda^2}\int_0^{+\infty} \left(\theta_{L}(u)-1 \right)\rho_f\left( \frac{\pi u}{\lambda^2} \right)du \label{identity-Ef1}\\ &=\frac{\pi}{\lambda^2}\int_0^{+\infty} \left(u^{\frac{d}{2}}\theta_{L^*}(u)-1 \right)\rho_f\left( \frac{\pi}{\lambda^2u} \right)u^{-2}du.\label{identity-Ef2}\end{aligned}$$ The first equality is clear by definition of $f$, by the change of variables $u=\frac{\lambda^2 t}{\pi}$, $$\begin{aligned} E_f[\lambda L]=\int_0^{+\infty}\left( \theta_{\lambda L}\left( \frac{t}{\pi} \right)-1 \right)\rho_f(t)dt &=\int_0^{+\infty}\left( \theta_{L}\left( \frac{ \lambda^2 t}{\pi} \right)-1 \right)\rho_f(t)dt \\ &=\frac{\pi}{\lambda^2}\int_0^{+\infty} \left(\theta_{L}(u)-1 \right)\rho_f\left( \frac{\pi u}{\lambda^2} \right)du.\end{aligned}$$ The second equality is proved using for $\alpha=\frac{\lambda^2 t}{\pi}$ and by change of variables $u=\frac{\pi}{t \lambda^2}$: $$\begin{aligned} E_f[\lambda L] &=\int_0^{+\infty}\left( \theta_{L}\left( \frac{ \lambda^2 t}{\pi} \right)-1 \right)\rho_f(t)dt \\ &=\int_0^{+\infty} \left( \left( \frac{\pi}{t \lambda^2}\right)^{\frac{d}{2}} \theta_{L^*}\left(\frac{\pi}{t \lambda^2} \right)-1 \right)\rho_f(t)dt\\ &=\frac{\pi}{\lambda^2}\int_0^{+\infty} \left(u^{\frac{d}{2}}\theta_{L^*}(u)-1 \right)\rho_f\left( \frac{\pi}{\lambda^2 u} \right)u^{-2}du.\end{aligned}$$ We next assume that the dimension $d$ is such that there exists only one lattice $L_m\in \mathcal L_d^\circ$ which is the unique minimizer of $L\mapsto \theta_L(\alpha)$ on $D_{\mathcal L_d^\circ}$ for any fixed $\alpha>0$. This is known to be the case for $d=2$, where $L_m=\Lambda_1$ by Montgomery Theorem [@Mont Thm 1]. We now prove our first result about the non-optimality of $L_m$ for some admissible potentials $f$ that have the representation , when its inverse Laplace transform is negative in the neighbourhood of $0$ or $+\infty$. \[prop-asymptnonopt\] Assume that there exists a lattice $L_m\in \mathcal{L}_d^\circ$ which is the unique minimizer of $L\mapsto \theta_L(\alpha)$ on $D_{\mathcal L_d^\circ}$ for any fixed $\alpha>0$. Let $f$ be an admissible potential having the representation , then the following holds for any $r_0>0$: 1. If $\mu_f<0$ on $(0,r_0)$, then for any Bravais lattice $L\in \mathcal{L}_d^\circ\backslash\{L_m\}$, there exists $\lambda_0$ such that for any $\lambda>\lambda_0$, $E_f[\lambda L]<E_f[\lambda L_m]$; 2. If $\mu_f<0$ on $(r_0,+\infty)$, then for any Bravais lattice $L\in \mathcal{L}_d^\circ\backslash\{L_m\}$, there exists $\lambda_1$ such that for any $0<\lambda<\lambda_1$, $E_f[\lambda L]<E_f[\lambda L_m]$. In particular, these results hold in dimension $d=2$ for the triangular lattice $L_m=\Lambda_1$. \[rmk:824\] This result would hold in dimensions $8$ and $24$ for $\mathsf{E}_8$ or the Leech lattice $\Lambda_{24}$, once the universality of these lattice, i.e. their minimality for the theta function among periodic configurations of fixed unit density and for any $\alpha>0$, conjectured in [@CohnKumar Conjecture 9.4], will be proved. The same result in dimension $d=4$ could also be proved for $\mathsf{D}_4$, according to its local minimality for the lattice theta function proved in [@Coulangeon:2010uq]. Assume first that $\mu_f$ is absolutely continuous with respect to the Lebesgue measure and $\mu_f(t)=\rho_f(t)dt$. If the hypothesis of point (1) holds, for any Bravais lattice $L\in \mathcal{L}_d^\circ$, we write, using , $$E_f[\lambda L]-E_f[\lambda L_m]=\frac{\pi}{\lambda^2}\int_0^{+\infty} \left(\theta_{L}(u)-\theta_{L_m}(u) \right)\rho_f\left( \frac{\pi u}{\lambda^2} \right)du.$$ By assumption, $\theta_{L}(u)-\theta_{L_m}(u)>0$ for all $u>0$ and $\rho_f\left( \frac{\pi u}{\lambda^2} \right)<0$ if $u<\frac{r_0 \lambda^2}{\pi}$. By the exponential decay of $u\mapsto\theta_{L}(u)$ for any fixed $L\in \mathcal{L}_d^\circ$, we obtain that for any Bravais lattice $L\in \mathcal{L}_d^\circ$, there exists $\lambda_0$ such that for any $\lambda>\lambda_0$, $$\int_0^{+\infty} \left(\theta_{L}(u)-\theta_{L_m}(u) \right)\rho_f\left( \frac{\pi u}{\lambda^2} \right)du<0,$$ and the first part of the proposition is proved. For the second case, by , we get $$E_f[\lambda L]-E_f[\lambda L_m]=\frac{\pi}{\lambda^2}\int_0^{+\infty} \left(\theta_{L^*}(u)-\theta_{L_m^*}(u) \right)\rho_f\left( \frac{\pi}{\lambda^2 u} \right)u^{\frac{d}{2}-2}du.$$ It is clear from and the fact that $L_m$ is the unique minimizer of the lattice theta function for all $\alpha>0$ that we necessarily have $L_m^*=L_m$. By assumption, $\theta_{L^*}(u)-\theta_{L_m^*}(u)>0$ for all $u>0$ and $\rho_f\left( \frac{\pi }{\lambda^2 u} \right)<0$ if $u<\frac{\pi}{\lambda^2 r_0}$. As in the previous case, by the exponential decay of the lattice theta function, there exists $\lambda_1$ such that for any $0<\lambda<\lambda_1$, $$\int_0^{+\infty} \left(\theta_{L^*}(u)-\theta_{L_m^*}(u) \right)\rho_f\left( \frac{\pi}{\lambda^2 u} \right)u^{\frac{d}{2}-2}du<0$$ and the second case of the proposition is proved. If $\mu_f$ is a Radon measure but not absolutely continuous with respect to the Lebesgue measure, then we can repeat the proof of Proposition \[prop:tri2\] and prove $$E_f[\lambda L]=\int_0^{+\infty} \left(\theta_{L}(u)-1 \right)\bar\mu_f(u)=\int_0^{+\infty} \left(u^{\frac{d}{2}}\theta_{L^*}(u)-1 \right)\tilde \mu_f(u),$$ where $\bar \mu_f=(g_\lambda)_\#\mu_f$ with $g_\lambda(t)=\frac{\lambda^2t}{\pi}$ and $\tilde \mu_f=(h_\lambda)_\#\mu_f$ with $h_\lambda(t)=\frac{\pi}{t\lambda^2}$, and the proof continues as above. About lattices that are optimal for any density ----------------------------------------------- The next result shows that, if the inverse Laplace transform of $f$ has a sign in a neighbourhood of the origin or $+\infty$ (which is the case for all the classical example or functions constructed with inverse power laws, exponentials, Yukawa potentials, Gaussians...) and if $L_m$ is the unique minimizer of $L\mapsto \theta_L(\alpha)$ in $D_{\mathcal{L}_d^\circ}$ for all $\alpha>0$, then the only lattice that could be a minimizer for all the densities is $L_m$. As recalled in the previous section (see Proposition \[prop-asymptnonopt\] and Remark \[rmk:824\]), this is the case in dimension $d=2$ for $L_m=\Lambda_1$ is the triangular lattice and is conjectured to extend in dimensions $d\in\{4,8,24\}$. \[prop-anyscale2\] Let $d\geq 1$ and assume that $L_m$ is the unique minimizer of $L\mapsto \theta_L(\alpha)$ in $D_{\mathcal{L}_d^\circ}$ for any $\alpha>0$. Let $f$ be an admissible potential with representation such that $\mu_f>0$ on the interval $(0,r_0)$ or on on the interval $(r_0,+\infty)$. If $\Lambda$ is a minimizer of $L\mapsto E_f[\lambda L]$ for any $\lambda >0$ on $D_{\mathcal{L}_d^\circ}$, then $\Lambda=L_m$. Furthermore, if $\mu(r)<0$ on $(0,r_0)$ or on $(r_0,+\infty)$, then the minimizer of $L\mapsto E_f[\lambda L]$ cannot be the same for all $\lambda>0$. In particular, this result holds in dimension $d=2$ for the triangular lattice $L_m=\Lambda_1$. We perform the proof under the assumption that $\mu_f$ is absolutely continuous with respect to the Lebesgue measure, i.e. $d\mu_f(t)=\rho_f(t)dt$, and the general case follows by the same adaptation as in the proof of Proposition \[prop-asymptnonopt\]. We prove the first point. Let $\Lambda$ be a minimizer of $L\mapsto E_f[\lambda L]$ on $D_{\mathcal{L}_d^\circ}$ for any $\lambda>0$. We assume that $\Lambda\neq L_m$. Therefore, by strict minimality of $L_m$ and continuity of $L\mapsto \theta_L(u)$ for any fixed $u>0$, there exists $L_0\in D_{\mathcal{L}_d^\circ}$ such that $\theta_{L_0}(u)-\theta_{\Lambda}(u)<0$ for all $u>0$. We now use exactly the same approach as in Proposition \[prop-asymptnonopt\], using and . In the first case, we write by assumption $$\label{assumpt} \forall V>0, \quad 0<E_f[\lambda L_0]-E_f[\lambda\Lambda]=\frac{\pi}{\lambda^2}\int_0^{+\infty} \left(\theta_{L_0}(u)-\theta_{\Lambda}(u) \right)\rho_f\left( \frac{\pi u}{\lambda^2} \right)du.$$ Since $\rho_f\left( \frac{\pi u}{\lambda^2} \right)<0$ if $u<\frac{r_0 \lambda^2}{\pi}$, by the exponential decay of $u\mapsto \theta_L(u)$ for any Bravais lattice $L\in \mathcal{L}_d^\circ$, there exists $\lambda_0$ such that for any $\lambda>\lambda_0$, $$\frac{\lambda^2}{\pi}\left(E_f[\lambda L_0]-E_f[\lambda \Lambda]\right)=\int_0^{+\infty} \left(\theta_{L_0}(u)-\theta_{\Lambda}(u) \right)\rho_f\left( \frac{\pi u}{\lambda^2} \right)du<0,$$ that contradicts . The second case is proved similarly using the second equality in , as in the proof of Proposition \[prop-asymptnonopt\]. For the second point, i.e. $\mu$ is negative in the neighbourhood of $0$ or $+\infty$, we use the same arguments. By Proposition \[prop-asymptnonopt\], $L_m$ cannot be a minimizer of $L\mapsto E_f[\lambda L]$ for any $\lambda>0$. Let us assume that there exists another Bravais lattice $\Lambda\in D_{\mathcal{L}_d^\circ}$ that is a minimizer of the energy for all $\lambda>0$. Therefore, there exists $L_0$ such that, for any $u>0$, $\theta_{L_0}(u)-\theta_\Lambda(u)>0$. If $\mu(r)<0$ on $(0,r_0)$ the same argument as before shows that there exists $\lambda_0$ such that for any $\lambda>\lambda_0$, $E_f[\lambda L_0]-E_f[\lambda \Lambda]<0$ and that contradicts our assumption. The second case is proved similarly as explained before and shows the existence of $\lambda_1$ such that for any $0<\lambda<\lambda_1$, $E_f[\lambda L_0]-E_f[\lambda \Lambda]<0$, that again contradicts our assumption. One-well potentials and optimality of lattices {#sectLJ} ============================================== Potentials with one well ------------------------ This Section is devoted to the proof of Theorem \[MainTh3\] and \[MainTh4\]. We also give several numerical evidence for the minimality of certain lattices for the Lennard-Jones type energy. Lennard-Jones type potentials ----------------------------- In this section, we are discussing the minimization problem for the Lennard-Jones type potentials defined by $$\label{lj2} f(r)=\frac{a_2}{r^{x_2/2}}-\frac{a_1}{r^{x_1/2}}, \quad x_2>x_1>d\quad (a_1,a_2)\in (0,+\infty)^2,$$ and, for any Bravais lattice $L\in \mathcal L_d$, $$E_f[L]=a_2 \zeta_L(x_2)-a_1 \zeta_L(x_1).$$ We first prove that the shape of a global minimizer $L_0$ (which is defined as a canonical choice of a lattice equivalent to $L_0$ under rotation and dilation, see Definition \[defn:shape\]) of $E_f$ for fixed parameters $(a_1,a_2,x_1,x_2)$ does not depend on $(a_1,a_2)$. \[prop-LJglobal\] Let $d<x_1<x_2$ and $(a_1,a_2)\in (0,+\infty)^2$, and $L_0=\lambda_1 L_1=\lambda_2 L_2$, where the normalizations of $L_1, L_2$ are chosen so that $$\label{normaliz_L1} L_1\in\mathcal L_d^\circ,\quad L_2\in\mathcal L_d^1.$$ Then for $f$ a Lennard-Jones type potential as in , the following hold: 1. $L_0$ is a global minimizer of $E_f$ on $D_{\mathcal L_d}$ if and only if $L_1$ is a minimizer on $D_{\mathcal L_d^\circ}$ of the energy $$\label{def-Etilde} \tilde{E}_f[L]:=\frac{\zeta_L(x_2)^{x_1}}{\zeta_L(x_1)^{x_2}}.$$ 2. If the minimum of $f(r)$ is achieved at $r_f=r_f(x_1,x_2,a_1,a_2)>0$, and $\lambda^{L_2}_0=\lambda^{L_2}_0(x_1,x_2,a_1,a_2)>0$ is the factor such that $\lambda^{L_2}_0L_2$ realizes the minimum energy among lattices of the same shape as $L_2$, i.e. $\min_{\lambda>0} E_f[\lambda L_2]=E_f[\lambda^{L_2}_0 L_2]$, then the following limit exists and satisfies $$\label{limit_r0} \lim_{\substack{x_1,x_2\to+\infty\\x_1<x_2, r_f=r_0}}\lambda^{L_2}_0 (x_1,x_2,a_1,a_2)=\sqrt{r_0}.$$ For any Bravais lattice $L\in\mathcal L_d^\circ$ and any $\lambda>0$, we have $$E_f[\lambda L]=\frac{a_2}{\lambda^{x_2}}\zeta_L(x_2)-\frac{a_1}{\lambda^{x_1}}\zeta_L(x_1)\ .$$ Therefore, with the notation for $\lambda^L_0$ as in above point (2), we get $$\label{value_AL} \partial_\lambda E_f[\lambda L]\geq 0 \iff \lambda\geq \lambda^L_0,\quad \mbox{and}\quad \lambda^L_0=\left( \frac{a_2x_2\zeta_L(x_2)}{a_1x_1\zeta_L(x_1)} \right)^{\frac{1}{(x_2-x_1)}}\ .$$ It follows that, for any $L \in\mathcal L_d^\circ$, $$\label{min_A} \min_{\lambda>0} E_f[\lambda L]=E_f[\lambda^L_0 L]=\frac{a_1^{\frac{x_2}{x_2-x_1}}\zeta_L(x_1)^{\frac{x_2}{x_2-x_1}}}{a_2^{\frac{x_1}{x_2-x_1}}\zeta_L(x_2)^{\frac{x_1}{x_2-x_1}}}\left( \left(\frac{x_1}{x_2}\right)^{\frac{x_2}{x_2-x_1}} -\left(\frac{x_1}{x_2}\right)^{\frac{x_1}{x_2-x_1}} \right)<0.$$ Therefore, we have, for any Bravais lattices $L,L^\prime\in\mathcal L_d^\circ$, $$\min_{\lambda>0} E_f[\lambda L]\leq \min_{\lambda>0} E_f[\lambda L^\prime]\iff \tilde{E}_f[L]\leq \tilde{E}_f[L^\prime]\ ,$$ which proves point (1) of the Proposition. Then by computing the minimum of $f$ we may find $$\label{rel_rf_AL} r_f=\left(\frac{a_2x_2}{a_1x_1}\right)^{\frac2{(x_2-x_1)}}\quad\mbox{and}\quad \lambda^L_0\stackrel{\eqref{value_AL}}{=}\sqrt{r_f}\left(\frac{\zeta_L(x_2)}{\zeta_L(x_1)}\right)^{\frac1{(x_2-x_1)}}\ .$$ Now in order to prove point (2) we note that due to we have $\zeta_{L_2}(x)>1$ for all $x>0$ and for each $\epsilon>0$ there exists a finite bound $C_\epsilon>0$ such that $\zeta_{L_2}(x)\le C_\epsilon$ for all $x>d+\epsilon$. Moreover, we have $$\label{limit_L1} \lim_{x\to+\infty}\zeta_{L_2}(x)=\#L_2^{(1)}\ .$$ From and we find $$\lim_{\substack{x_1,x_2\to+\infty\\x_1<x_2, r_f=r_0}}\lambda^{L_2}_0(x_1,x_2,a_1,a_2)= \sqrt{r_0}\lim_{\substack{x_1,x_2\to+\infty\\x_1<x_2, r_f=r_0}}\left(\frac{\zeta_{L_2}(x_2)}{\zeta_{L_2}(x_1)}\right)^{\frac1{x_2-x_1}}= \sqrt{r_0}\ ,$$ which proves and concludes the proof. Next, extending point (1) of Proposition \[prop-LJglobal\], we formulate some simple equivalent condition for $\Lambda$ to minimize $E_f[L]$ among lattices of unit density: \[prop:equiv\] Let $L_0=\lambda \Lambda$ for $\Lambda\in \mathcal{L}_d^\circ$. If $L\in \mathcal L_d^\circ$ then we define functions $H_L,h_L:(d,+\infty)\to {\mathbb R}$ by $$H_L(x):=\frac{1}{x}\log\left( \frac{\zeta_L(x)}{\zeta_{\Lambda }(x)}\right), \quad h_L(x):=-\log \zeta_L(x) +x\frac{\partial_x\zeta_L(x)}{\zeta_L(x)}.$$ The following conditions are equivalent: 1. For any $x_2>x_1>d$, the lattice $L_0 $ is the unique minimizer of $E_f$ on $D_{\mathcal L_d}$. 2. For any $x_2>x_1>d$, the lattice $\Lambda $ is the unique minimizer of $\tilde{E}_f$, defined by , on $D_{\mathcal L_2^\circ}$. 3. For any Bravais lattice $L\in \mathcal L_d^\circ\setminus\{ \Lambda \}$, the function $H_L$ is strictly increasing on $(d,+\infty)$. 4. For any $x>d$, $\Lambda $ is the unique minimizer of $h_L(x)$ on $D_{\mathcal{L}_d^\circ}$. The equivalence between (1) and (2) is treated in point (1) of Proposition \[prop-LJglobal\]. To prove the equivalence between (2) and (3), note that $$\tilde{E}_f[\Lambda]\leq \tilde{E}_f[L]\iff \left( \frac{\zeta_L(x_1)}{\zeta_\Lambda(x_1)} \right)^{\frac{1}{x_1}}\leq \left( \frac{\zeta_L(x_2)}{\zeta_\Lambda(x_2)} \right)^{\frac{1}{x_2}}.$$ For proving the equivalence between (3) and (4), note that $H'(x)=\frac{1}{x^2}\left( h_L(x)-h_{\Lambda }(x) \right)$. We conjecture that the equivalent statements of Proposition \[prop:equiv\] hold true for the triangular lattice $\Lambda=\Lambda_1=\frac{2}{\sqrt 3}\mathsf{A}_2$ in dimension $d=2$ and for $\Lambda\sim \mathsf{D}_3, \sim\mathsf{D}_4, \sim\mathsf{E}_8,\sim\Lambda_{24}$ respectively in dimensions $3,4,8$ and $24$, where the equivalence relation is as in Definition \[defn:shape\] and indicates that $\Lambda$ has the shape of the corresponding lattices. About the case $d=2$, (3) of Lemma \[prop:equiv\] has been proved in [@BetTheta15 Thm. 1.2.B.2 and Lem. 6.17] on the small interval $I=(2,2\psi^{-1}(\log \pi)-2)$ where $\psi$ is the digamma function and $2\psi^{-1}(\log \pi)-2\approx 5.256$. Figure \[H2to10square\], where we have plotted $H_{{\mathbb Z}^2}$, shows the monotonicity of the function and is then a numerical evidence of the optimality of the triangular lattice against the square lattice. The same conclusion follows from Figure \[fig:ratio\] where we find that for all $x_1<x_2<2x_1$, that we tested (in particular for all half-integer values of $x_1$, $x_2$ in the range $1<x_1<x_2\le 25$), the triangular lattice has an energy $E_f$ lower than the square lattice. Moreover, numerically we find that the value $ \frac{\min_{A>0}E_f[\sqrt{A}\Lambda_1]}{\min_{A>0}E_f[\sqrt{A}\mathbb Z^2]}$ is increasing in $x_1$, $x_2$. In dimension $d=3$ we also have plotted in Figure \[Z3vsFCCBCC\] the function $H_{{\mathbb Z}^3}$ where $\Lambda\in \{\mathsf{D}_3,\mathsf{D}_3^*\}$ are the FCC lattice and the BCC lattice of unit density. Both functions seem to be increasing, showing the optimality of the FCC lattice (resp. BCC lattice) against the cubic lattice. The same occurs if $L$ is the BCC lattice and $\Lambda$ is the FCC lattice as we can see in Figure \[fig-BCCFCC\]. These numerics are consistent with [@Beterminlocal3d Conjecture 1.7] which states that optimality of $\mathsf{D}_3$ holds in dimension $3$ for any exponents $x_2>x_1>3$. ![Plot of function $H_{{\mathbb Z}^2}$ on $[2,50]$, for the triangular lattice $\Lambda=\Lambda_1$. The triangular lattice seems to have lower Lennard-Jones type energy than the square lattice for any values of the parameters.[]{data-label="H2to10square"}](figHZ2.png){width="8cm"} ![Plot in the $(x_1,x_2)$-plane of the quantity $\frac{\min_{\lambda>0}E_f[\lambda\Lambda_1]}{\min_{\lambda>0}E_f[\lambda\mathbb Z^2]}$, for $3\le x_1<x_2\le 50$ and $x_1,x_2\in\mathbb Z$. Note that the smallest value is taken at $x_1=3, x_2=4$ and equals $\approx 1.061$, i.e. is already larger than $1$, and then the value increases in both coordinate directions, so that for $x_1=49$, $x_2=50$ it equals $\approx 1.499$, very close to the asymptotic value $3/2$.[]{data-label="fig:ratio"}](tri-sq-50x50.pdf "fig:"){width="9cm"}\ ![Plot of function $H_{{\mathbb Z}^3}$ on $[4,50]$ where $\Lambda=\mathsf{D}_3$ is the FCC lattice (on the left) and $\Lambda=\mathsf{D}_3^*$ the BCC lattice (on the right). The FCC and BCC lattices seem to have lower Lennard-Jones type energy than the cubic lattice for any values of the parameters.[]{data-label="Z3vsFCCBCC"}](HFCCZ3.png "fig:"){width="6cm"} ![Plot of function $H_{{\mathbb Z}^3}$ on $[4,50]$ where $\Lambda=\mathsf{D}_3$ is the FCC lattice (on the left) and $\Lambda=\mathsf{D}_3^*$ the BCC lattice (on the right). The FCC and BCC lattices seem to have lower Lennard-Jones type energy than the cubic lattice for any values of the parameters.[]{data-label="Z3vsFCCBCC"}](HZ3BCC.png "fig:"){width="6cm"} ![Plot of function $H_{\mathsf{D}_3^*}$ on $[4,50]$ where $\Lambda=\mathsf{D}_3$ is the FCC lattice. The FCC lattice seems to have lower Lennard-Jones type energy than the BCC lattice for any values of the parameters.[]{data-label="fig-BCCFCC"}](HBCCFCC.png){width="8cm"} \[def:kissing\] The *kissing number* (also called *coordination number* in crystallography) of a Bravais lattice $L\in\mathcal L_d$, denoted $\tau(L)$, is defined as the number of nearest-neighbors in $L$ of the origin, $\tau(L):=\#L^{(1)}$ in our notation. We also define the *optimal kissing number in dimension $d$* by $k(d):=\max_{L\in\mathcal L_d}\tau(L)$. \[rmk:kissing\] It is known (see [@ConSloanPacking Table 1.1], [@musinkiss] and the references therein) that $k(2)=\tau(\mathsf{A}_2)=6$, $k(3)=\tau(\mathsf{D}_3)=12$, $k(4)=\tau(\mathsf{D}_4)=24$, $k(8)=\tau(\mathsf{E}_8)=240$, $k(24)=\tau(\Lambda_{24})=196560$, while all other cases are not known. The above lattices are known to be unique optimizers among lattices, and in $2,8,24$ dimensions also among general packing configurations, while in dimension $3$ this is not the case, see [@kusnerkusner], and in dimension $4$ it is not known, see [@musinkiss]. \[prop:asymin\] For $L,\Lambda\in\mathcal L_d$ and $f$ a Lennard-Jones type potential as in there holds $$\label{limit_ratioenergy} \lim_{x_1\to +\infty}\lim_{x_2\to +\infty}\frac{\displaystyle \min_{\lambda>0}E_f[\lambda\Lambda]}{\displaystyle \min_{\lambda>0}E_f[\lambda L]}=\frac{\tau(\Lambda)}{\tau(L)}\ .$$ As the above minimum values are negative, if $\tau(\Lambda)$ uniquely realizes the optimal kissing number $k(d)$ amongst lattices, then there exists $x_0=x_0(d)$ such that for any $x_2>x_1>x_0$, the unique minimizer of $E_f$ in $D_{\mathcal{L}_d}$ has shape $[\Lambda]$. In particular, this holds for the cases $(d,\Lambda)\in\{(2,\mathsf{A}_2), (3,\mathsf{D}_3), (4,\mathsf{D}_4),(8,\mathsf{E}_8),(24,\Lambda_{24})$. Due to we have, for any Bravais lattices $L,\Lambda\in\mathcal L_d^1$, $$\frac{\displaystyle \min_{\lambda>0}E_f[\lambda \Lambda]}{\displaystyle \min_{\lambda>0}E_f[\lambda L]}=\frac{\zeta_{\Lambda}(x_1)^{\frac{x_2}{x_2-x_1}} \zeta_{L}(x_2)^{\frac{x_1}{x_2-x_1}} }{\zeta_{\Lambda}(x_2)^{\frac{x_1}{x_2-x_1}} \zeta_{L}(x_1)^{\frac{x_2}{x_2-x_1}} }.$$ We have, for any $L\in \mathcal{L}_d^1$, $$\begin{aligned} \displaystyle \lim_{x_2\to +\infty} \zeta_L(x_2)^{\frac{x_1}{x_2-x_1}}&=\lim_{x_2\to +\infty}\left( \tau(L) +\sum_{p\in L^{(>1)}}|p|^{-x_2} \right)^{\frac{x_1}{x_2-x_1}}\\ &=\lim_{x_2\to +\infty} \tau(L)^{\frac{x_1}{x_2-x_1}}\left( 1+\tau(L)^{-1}\sum_{p\in L^{(>1)}}|p|^{-x_2} \right)^{\frac{x_1}{x_2-x_1}}=1\ .\end{aligned}$$ It follows that, using the fact that we have the normalization $L,\Lambda\in \mathcal L_d^1$, $$\begin{aligned} \lim_{x_1\to +\infty}\lim_{x_2\to +\infty}\frac{\displaystyle \min_{\lambda>0}E_f[\lambda\Lambda]}{\displaystyle \min_{\lambda>0}E_f[\lambda L]} &=\lim_{x_1\to +\infty} \frac{\zeta_{\Lambda}(x_1)}{\zeta_L(x_1)}\\ &=\lim_{x_1\to +\infty} \frac{\tau(\Lambda) + \sum_{p\in\Lambda^{(> 1)}} |p|^{-x_1}}{\tau(L) + \sum_{p\in L^{(>1)}} |p|^{-x_1}}=\frac{\tau(\Lambda)}{\tau(L)}.\end{aligned}$$ The final statement in the proposition holds by noting that $\min_{\lambda>0}E_f[\lambda\Lambda]<0$ as follows by taking $\lambda$ large enough, and by then using Remark \[rmk:kissing\]. The optimality of the FCC lattice for large exponent proved above is consistent with [@Beterminlocal3d Conjecture 1.7] which in the absolutely summable case states that optimality of the $FCC$ lattice should hold in dimension $3$ for any exponents $x_2>x_1>3$. We note that in [@Crystal Th. 1.1] there were given conditions for a potential $V$ with one well, and needed to produce crystallization to a triangular lattice. Although these conditions are robust enough to accomodate a potential which is of power-law type near $r=0$ and near $r=+\infty$, these conditions on the other hand are not weak enough to include the case of Lennard-Jones type potentials of the form considered here. Indeed, if we normalize our potentials as indicated in [@Crystal], so that $f(1)=-1$ is the minimum of $f$, namely we look at $$f_{x_1,x_2}(r):=\frac{x_1}{x_2-x_1}\frac{1}{r^{x_2}} - \frac{x_2}{x_2-x_1}\frac{1}{r^{x_1}}\ ,$$ then we find numerically the conditions $f_{x_1,x_2}''(1+\alpha)\ge 1$ and $f_{x_1,x_2}(1+\alpha)\ge -\alpha$, required as sufficient conditions in [@Crystal (3), (4)] for suitable $\alpha\in(0,\alpha_0)$ (where the upper bound $\alpha_0<1/3$ is implicitly found during the proof of [@Crystal Thm. 1.1], and needs in fact to be fixed as a positive number much smaller than $1/3$). What we found is that the above conditions on $f_{x_1,x_2}$ are never simultaneously achieved for any choice of $\alpha<1/3$ for powers $2<x_1< x_2$. Some counterexamples {#sect-contrex} ==================== The triangular lattice can be minimizer at all scales for non-completely monotone $f$ {#not_cm} ------------------------------------------------------------------------------------- In the following example, we have numerically checked that there exist functions which are not completely monotone and such that the triangular lattice is the minimizer of $E_f$ at all scales. For any $\varepsilon>0$, define $f_\varepsilon:(0,+\infty)\to {\mathbb R}$ by $$f_\varepsilon(r):=\frac{6}{r^4}-\frac{2(2+\varepsilon)}{r^3}+\frac{1+\varepsilon}{r^2}.$$ The inverse Laplace transform of $f_\varepsilon$ is $\mathcal{L}^{-1}[f_\varepsilon](x)=x(x-1)(x-1-\varepsilon)$, and in particular $\mathcal{L}^{-1}[f_\varepsilon](x)\leq 0$ on $[1,1+\varepsilon]$. In Figure \[fig-lapinvepsilon\] and Figure \[fig-graphepsilon\], we have respectively plotted $\mathcal{L}^{-1}[f_\varepsilon]$ and $f_\varepsilon$ for different values of $\varepsilon$. ![Plot of function $\mathcal{L}^{-1}[f_\varepsilon]$ for $\varepsilon\in \{0,0.5,1,1.148\}$[]{data-label="fig-lapinvepsilon"}](InvLapepsilon.png "fig:"){width="11cm"}\ ![Plot of function $f_\varepsilon$ for $\varepsilon\in \{0,0.5,1,1.148\}$[]{data-label="fig-graphepsilon"}](graphfepsilon.png "fig:"){width="11cm"}\ We now justify the following claim, based on numerical evidence: There exists $\varepsilon_0\approx 1.148>0$ such that for any $0\leq \varepsilon<\varepsilon_0$ and any $\lambda>0$, $\Lambda_1$ is the unique minimizer of $L\mapsto E_{f_\varepsilon}[\lambda L]$ on $D_{\mathcal{L}_2^\circ}$. We have, using the scaling property of the Epstein zeta function, for any Bravais lattice $L\in D_{\mathcal{L}_2^\circ}$ $$\label{difference} E_{f_\varepsilon}[\lambda L]-E_{f_\varepsilon}[\lambda \Lambda_1]=\frac{P_{\varepsilon,L}(\lambda^2)}{\lambda^8}$$ where, using the notation $d_s(L):=\zeta_{L}(s)-\zeta_{\Lambda_1}(s)$, we have $$P_{\varepsilon,L}(X):=(1+\varepsilon)d_4(L)X^2 -2(2+\varepsilon)d_6(L)X+6d_8(L).$$ The discriminant of polynomial $P_{\varepsilon, L}$ is $$\label{delta_eps_theta} \Delta(\varepsilon,L)=4(2+\varepsilon)^2d_6(L)^2-24(1+\varepsilon)d_4(L)d_8(L).$$ Note that, for all any $s>2$, $d_s(L)\ge 0$ with equality only for $L=\Lambda_1$ on $D_{\mathcal{L}_2^\circ}$. Furthermore, we have that $\Delta(\varepsilon,L)< 0$ for $\varepsilon\geq 0$ and $L\neq \Lambda_1$ (on $D_{\mathcal{L}_2^\circ}$) if and only if $$h(\varepsilon):=\frac{(2+\varepsilon)^2}{6(1+\varepsilon)}< \frac{d_4(L)d_8(L)}{d_6(L)^2}=:c(L).$$ Note that $h$ is an increasing function on $[0,+\infty)$ and $h(0)=2/3$. Furthermore, we can prove that $\lim_{L\mapsto \Lambda_1} c(L)=c\in {\mathbb R}$. Thus, if we find $\varepsilon_0$ such that $$h(\varepsilon_0)< \inf_{L, |L|=1} c(L),$$ then $\Delta(\varepsilon,L)< 0$ for any $0\leq \varepsilon \leq \varepsilon_0$ and any $L\in D_{\mathcal{L}_2^\circ}$. As in [@Coulangeon:kx Sect. 3], we recall that for $L\in\mathcal L^\circ_2$, we can write $$\zeta_L(s)=\sum_{p\in L\setminus\{0\}}\frac{1}{|p|^s}=\sum_{x\in{\mathbb Z}^2\setminus\{0\}}\frac{1}{(x^tAx)^{s/2}}\ ,$$ for suitable $A\in\mathcal{P}_2^\circ$. We can view $\mathcal P_2^\circ$ as a differential submanifold of the vector space of $2\times2$ symmetric matrices with real entries $\mathcal{S}_2({\mathbb R})$. The tangent space to $\mathcal P_2^\circ$ at any point $A\in\mathcal S_2({\mathbb R})$ then identifies with the set $\{H\in \mathcal{S}_2({\mathbb R}):\ Tr(A^{-1}H)=0\}$. Moreover, the exponential map $H\mapsto e_A(H):=A \exp(A^{-1}H)$ induces a local diffeomorphism from the tangent space to $\mathcal{P}_2^\circ$. Noting that for $\Lambda_1$ all layers hold a $4$-design (see [@venkov Def. 6.3, Thm. 6.12]), we therefore obtain, by [@Coulangeon:kx Eq. (3.5), with dimension $n=2$ and exponent $s/2$ in our case], that for $A$ corresponding to the lattice $\Lambda_1$ and for $L$ another lattice corresponding to the quadratic form with matrix $e_A(H)$ there holds $$\label{TaylorEpstein} \zeta_L(s)-\zeta_{\Lambda_1}(s)= \frac{s(s-2)}{32}{{\rm{Tr}}}(A^{-1}H)^2\zeta_{\Lambda_1}(s) + o\left(\|H^2\|\right)\ .$$ Thus we have $$c(L)=\frac{(8 Tr(A^{-1}H)^2\zeta_{\Lambda_1}(4)+o(\|H^2\|))(48\ Tr(A^{-1}H)^2\zeta_{\Lambda_1}(8)+o(\|H^2\|))}{(24\ Tr(A^{-1}H)^2\zeta_{\Lambda_1}(6)+o(\|H^2\|))^2}.$$ And we then get, taking $H\to 0$, $H\in\mathcal S_2({\mathbb R})\setminus\{0\}$, $$\begin{aligned} \lim_{L\to \Lambda_1}c(L)=\frac{2}{3}\frac{\zeta_{\Lambda_1}(4)\zeta_{\Lambda_1}(8)}{\zeta_{\Lambda_1}(6)^2}\approx 0.7719234>\frac{2}{3}.\end{aligned}$$ Numerically, we find that $$\inf_{L, |L|=1} c(L) > 0.769.$$ To obtain the above, we have proceeded as follows. We parametrized $L\in D_{\mathcal L_2^\circ}$ as $L(x,y)$ with $(x,y)$ a point of the half elliptic fundamental domain $$\tilde D=\{(x,y)\in {\mathbb R}^2 : 0\leq x\leq 1/2, x^2+y^2\geq 1 \},$$ and denote by slight abuse of notation $c(L(x,y))=c(x,y)$. We have first computed the limit of $c(x,y)$ for fixed $x\in [0,1/2]$ as $y\to +\infty$. Using the fact that $\zeta_L(s)=2y^{\frac{s}{2}}\zeta(s)+h(s,y)$ where $h(s,y)=o(y^{\frac{s}{2}})$ (see [@katsurada Thm 1]) and the well-known exact values of $\zeta(2k)$ for $k\in \{2,3,4\}$ of the Riemann zeta function $\zeta(s)=\sum_{m>0}m^{-s}$, which are $$\zeta(4)=\frac{\pi^4}{90},\quad \zeta(6)=\frac{\pi^6}{945},\quad \zeta(8)=\frac{\pi^8}{9450},$$ we obtain, for any fixed $x\in [0,1/2]$, $$\lim_{y\to +\infty} c(x,y)=\lim_{y\to +\infty} \frac{\zeta_L(4)\zeta_L(8)}{\zeta_L(6)^2}=\frac{\zeta(4)\zeta(8)}{\zeta(6)^2}=\frac{21}{20}=1.05.$$ It is therefore possible to reduce the research of the minimizer of $c$ to a compact subset of $\tilde{D}$. A numerical study of $c$ shows that, for $y>3$, $c(x,y)>0.9$. Therefore, we have computed the values of $c(x,y)$ for $(x,y)$ on a grid and on the set $\{(x,y)\in \tilde{D} : x^2+y^2=1\}$, i.e. among rhombic lattices. The results can be seen in Figure \[fig-c3\]. ![Plot of $c(x,y)$ for $\sqrt{3}/2<y<3$ (on the left) and $c(\cos(t),\sin(t))$, $t\in \left[\frac{\pi}{3},\frac{\pi}{2} \right]$ (on the right).[]{data-label="fig-c3"}](graphCy3.png "fig:"){width="6cm"} ![Plot of $c(x,y)$ for $\sqrt{3}/2<y<3$ (on the left) and $c(\cos(t),\sin(t))$, $t\in \left[\frac{\pi}{3},\frac{\pi}{2} \right]$ (on the right).[]{data-label="fig-c3"}](graphCrhombic.png "fig:"){width="6cm"} We finally get, from an enough refined mesh of the values $(x,y)\in \tilde{D}$ such that $y\leq 3$, that $\min_{(x,y)\in \tilde{D}} c(x,y)>0.769$ and this minimum is approximatively equal to $0.7699393$, achieved very close to the point $(x,y)=(0.2607474,0.9654071)$. Hence, solving $h(\varepsilon)\leq 0.769$, we get $\varepsilon_0\approx 1.1485753<1.148$. Therefore, we have numerically checked that, for any $0\leq \varepsilon \leq 1.148$ and any Bravais lattice $L\in \mathcal{L}_2^\circ$, $\Delta(\varepsilon,L)< 0$, i.e., for any $\lambda>0$ and any $\varepsilon,L$ as above, $$E_{f_\varepsilon}[\lambda L]-E_{f_\varepsilon}[\lambda \Lambda_1]\geq 0,$$ with equality if and only if $L=\Lambda_1$. We have checked that, for $\varepsilon=\varepsilon_0\approx 1.148$, $f_\varepsilon$ satisfies $$\forall r>0,\quad \forall k\leq 4,\quad (-1)^k f_\varepsilon^{(k)}(r)\geq 0,$$ and $f_\varepsilon^{(5)}$ is not non-positive on $(0,+\infty)$. Potentials with one (strict) well such that the global minimum is not triangular {#subsec-onewell} -------------------------------------------------------------------------------- In this section we provide an abstract result (in Proposition \[prop:theil\_ZA\]) in which the existence of a whole class of such potentials (in particular, as is easy to verify by Remark \[rmk:existza\], we can construct such potentials with arbitrary smoothness). We present first an explicit example of a Lipschitz potential for which the square lattice is favored over the triangular one. The ease to create examples, even under the constraint of having precisely one well, and even at a high level of smoothness, suggests that even beyond the case of the square lattice, the possibilities to construct potentials that favor a given lattice over the triangular one seem to be relatively vast. At the end of this section, we present a remark on how to do this for more general lattices (see Remark \[rmk-generalprinciple\]). Inspired by a counter-example by Ventevogel [@VN1 Sec. 5], we define the following potential, which gives another more explicit example: $$g(r):= \left\{ \begin{array}{ll} \displaystyle 30000 & \mbox{if $0<r<4/9$}\\ 2-3r & \mbox{if $4/9\leq r\leq 1$}\\ -r^{-4} & \mbox{if $r>1$,}\\ \end{array}\right.$$ This function is clearly strictly decreasing on $(0,1)$ and strictly increasing on $(1,+\infty)$. Defining $f$ by $f(r^2)=g(r)$, we have numerically computed the following quantities: $$\min_{\lambda>0} E_f[\lambda {\mathbb Z}^2]\approx -8.5915114 , \quad \min_{\lambda >0} E_f[\lambda\mathsf{A}_2]\approx -7.7107743.$$ We therefore can see that $\min_{\lambda >0} E_f[\lambda {\mathbb Z}^2]< \min_{\lambda>0} E_f[\lambda \mathsf{A}_2]$, and that (numerically) shows that a global minimizer of $E_f$ cannot be a triangular lattice $\lambda \mathsf{A}_2$. We then can construct the following continuous potential $g$, equal to the above one for $r\geq 4/9$ and with strong repulsion near the origin, which satisfies the same property of non-minimality for triangular lattices. The proof of this proposition is given in Appendix \[appendix\]. \[prop:example\_vn\] Let $g$ be the continuous potential defined by $$g(r):= \left\{ \begin{array}{ll} \displaystyle \frac{\left( \frac{2}{3} \right)\left(\frac{4}{9} \right)^{p}}{r^{p}} & \mbox{if $0<r<4/9$}\\ 2-3 r & \mbox{if $4/9\leq r\leq 1$}\\ -r^{-4} & \mbox{if $r>1$,}\\ \end{array}\right.$$ Then there exists $p_0$ such that for any $p>p_0$ we have for $f(r^2)=g(r)$, $$\label{min_za} \displaystyle \min_{\lambda>0} E_f\left[\lambda {\mathbb Z}^2\right]<\min_{\lambda>0} E_f[\displaystyle \lambda \mathsf{A}_2].$$ Furthermore, we finally prove the existence of a non-countable family of $C^1$-functions $f(r^2)=g(r)$ for which the global minimizer of $E_f$ cannot be triangular. \[prop:theil\_ZA\] For any choice of parameters $0<\alpha_0<\alpha_1<1<\sqrt 3 \alpha_0$, there exist $C_0,C_1>0$ such that if a $C^1$-function $g:[0,+\infty)\to{\mathbb R}\cup\{+\infty\}$ is such that $g(|x|), g^\prime(|x|)$ are integrable on $\mathbb R^2\setminus B_\epsilon$ for $0<\epsilon\le\min\{\alpha_1/2,\sqrt 3\alpha_0/2\}$, and satisfies the conditions \[imposed\] $$g(r)\ge 0\quad\mathrm{for}\quad r\in(0,\alpha_1],\quad\mathrm{and}\quad g(r)\le 0\quad\mathrm{for}\quad r\in[1,+\infty),\label{pos}$$ $$g'(r)\ge 0\quad\mathrm{for}\quad r\in(\sqrt 3\alpha_0,+\infty),\label{incr}$$ $$\min_{r>0}g(r)=g(1)=-1,\label{normalize}$$ $$\forall r\in(0,\alpha_0],\quad g(r)\ge -\frac{C_0}{r^2}\int_{\alpha_1}^{+\infty} g(\rho)\rho\mathrm{d}\rho+ \frac{C_0}{r}\int_{\alpha_1/2}^{+\infty}|g^\prime(\rho)|\rho \mathrm{d}\rho,\label{hardwall}$$ $$\min_{\alpha_0\le r\le1}\left(g(r)+g\left(\sqrt{2} r\right)\right)\le -\frac32 +\frac14 E_f\left[\alpha_0 \mathsf{A}_2^{(\ge 2)}\right] ,\label{zbettera}$$ then, for $f(r^2)=g(r)$, and $\lambda_0^{\mathbb Z^2},\lambda_0^{\mathsf{A}_2}\in[\alpha_0,1]$, we have $$\displaystyle \min_{\lambda>0} E_f\left[\lambda {\mathbb Z}^2\right]= E_f\left[\lambda_0^{{\mathbb Z}^2}{\mathbb Z}^2\right]<E_f\left[\lambda_0^{\mathsf{A}_2}\mathsf{A}_2\right]=\min_{\lambda>0} E_f[\displaystyle \lambda \mathsf{A}_2].$$ \[rmk:existza\] The sign conditions and could possibly be relaxed, at the cost of complicating the proof, and in the present form they are already true for a large class of one-well potentials, while is just a normalization condition. Condition seems to be the most restrictive one, but we can construct a $g$ that satisfies it by the following rough procedure. First, we ensure that there exists $\alpha \in(\alpha_0,1]$ such that $g(\alpha)+g(\sqrt 2\alpha)\le -3/2-\epsilon$. Then, we make $|g(r)|$ very small compared to $\epsilon$ in the range $r>\sqrt 3\alpha_0$. If $g(|x|), g^\prime(|x|)$ are integrable on $\mathbb R^2$ away from the origin, then condition is true if $g(r)\ge \tilde C/r^2$ in a neighborhood of the origin. The role of this condition is to form an “effective hard core” for our interactions: as we will see in the proof of the proposition, it implies that the optimal value from is larger than $\alpha_0$. Then $\alpha_1$ can be defined to be the only value where $g(\alpha_1)=0$, therefore does not represent a constraint on the possible choices of $g$, and we merely introduced the notation for it in the statement for the sake of explicitly representing the behavior of $g$. Note that both lattices $\Lambda=\mathbb Z^2, \mathsf{A}_2$ have as first shell a set of points ($4$ points for $\Lambda=\mathbb Z^2$ and $6$ points for $\Lambda=\mathsf{A}_2$) at distance $1$ from the origin. The proof consists in comparing the contributions from this first shell to $E_f[\lambda \Lambda]$ to the contributions from all the remaining shells. While $\mathbb Z^2$ has fewer points in the first shell, the second shell of $\mathbb Z^2$ is closer to the origin, at distance $\sqrt 2$ from the origin for $\Lambda=\mathbb Z^2$ and it is farther from the origin, at distance $\sqrt 3$ from the origin, for $\Lambda=\mathsf{A}_2$. The principle of the proof is the following: the above properties of $g$ are such that the effect of the disparity of the contributions from the second shell (which is advantageous for $\mathbb Z^2$) “wins” over the effect of the disparity of the first shell (which itself would be more advantageous to $\mathsf{A}_2$). We first show that $\lambda_0^{\mathbb Z^2}, \lambda_0^{\mathsf{A}_2}\in[\alpha_0,1]$, in steps 1 and 2, after which in step 3 we use the above specific discussion on shells to conclude. **Step 1.** We show that for $\Lambda\in\{\mathbb Z^2,\mathsf{A}_2\}$ and $\lambda\in ]0,\alpha_0]$ there holds $E_f[\lambda\Lambda]\ge 0$. This will follow by and from the bound $$\label{energypositive} E_f[\lambda\Lambda\cap B_{\alpha_0}]\ge - E_f[\lambda\Lambda\setminus B_{\alpha_1}],$$ which we are proving now. We use a rough quadrature estimate in order to bound the left hand side of as follows. By Poincaré inequality (which holds for bounded convex domains such as $\mathcal V_{\lambda\Lambda}$) $$\label{taylor} \frac{1}{|\mathcal V_{\lambda \Lambda}|}\int_{\mathcal V_{\lambda \Lambda}(p)}\left|g(|p|)-g(|x|)\right|\mathrm{d}x\le C\frac{\mathrm{diam}\mathcal V_{\lambda\Lambda}}{|\mathcal V_{\lambda \Lambda}|} \int_{\mathcal V_{\lambda\Lambda}(p)}|g'(|x|)|\mathrm{d}x,$$ where for the Voronoi cells of $\lambda\Lambda$ we use the notation $\mathcal V_{\lambda\Lambda}(p)=\{x\in\mathbb R^2: |x-p| \leq\mathrm{dist}(x, \lambda\Lambda)\}$ and $\mathcal V_{\lambda \Lambda}=\mathcal V_{\lambda \Lambda}(0)$. By summing formula over $p\in\lambda\Lambda\setminus B_{\alpha_1}$, we find the bound $$\begin{gathered} \label{quadbound} \left|\frac{1}{|\mathcal V_{\lambda\Lambda}|}\int_{\bigcup_{p\in\lambda\Lambda\setminus B_{\alpha_1}}\mathcal V_{\lambda\Lambda}(p)}g(|x|)\mathrm{d}x - \sum_{p\in\lambda\Lambda\setminus B_{\alpha_1}}g(|p|)\right| \\ \le C\frac{\mathrm{diam}\mathcal V_{\lambda\Lambda}}{|\mathcal V_{\lambda\Lambda}|}\int_{\mathbb R^2\setminus B_{\alpha_1/2}}|g^\prime(r)|r \mathrm{d}r,\end{gathered}$$ where in the last integral we used the fact that for $\lambda<\alpha_1$, Voronoi cells of $\lambda\Lambda$ corresponding to points outside $B_{\alpha_1}$ do not intersect $B_{\alpha_1/2}$. The bound gives the following control on the right-hand side of , in which the constants $C_\Lambda>0$ only depend on the dimension and on the shape of $\mathcal V_{\lambda\Lambda}$: $$\label{controlright} -E_f[\lambda\Lambda\setminus B_{\alpha_1}]\le -\frac{C_\Lambda}{\lambda^2}\int_{\alpha_1}^{+\infty} g(r)r\mathrm{d} r +\frac{C_\Lambda}{\lambda}\int_{\alpha_1/2}^{+\infty}|g^\prime(r)|r\mathrm{d}r.$$ Now in order to pass from to it suffices to note that for $\lambda<\alpha_0$ and $\Lambda\in\{{\mathbb Z}^2,\mathsf{A}_2\}$ the first shell of $\Lambda$ is at distance $\lambda<\alpha_0$ from the origin. Thus we may use the version of the bound , together with : we obtain that is true, provided that the constant $C_0>C_\Lambda/\#(\Lambda^{(1)})$, where $\Lambda^{(1)}\subset\Lambda$ is the first shell in $\Lambda$ and $C_\Lambda$ is the constant from . Now due to we have that $ E_f[\lambda\Lambda]\ge 0$ whenever $\lambda<\alpha_0$, whereas due to we have that $\min_{\lambda>0} E_f[\lambda\Lambda]\le E_f[\Lambda]<0$. This shows that the value $\lambda_0^\Lambda$ at which $\min_{\lambda>0} E_f[\lambda\Lambda]$ is achieved, does not lie in the interval $]0,\alpha_0[$. **Step 2.** Now we use condition on $g^\prime$, which implies that for $\lambda\in[1,+\infty)$, the sum over the shells different than the first is increasing in $\lambda$, whereas shows that the sum over first shell is minimized at $\lambda = 1$, showing that the case $\lambda_0^\Lambda>1$ is also not possible. This shows that the minimum of $\lambda\mapsto E_f[\lambda\Lambda]$ occurs at $\lambda_0^\Lambda\in[\alpha_0,1]$ for our lattices $\Lambda\in \{\mathbb Z^2,\mathsf{A}_2\}$. **Step 3.** Now that we know that the minimum of $\lambda\mapsto E_f[\lambda\Lambda]$ is achieved for $\lambda_0^\Lambda$ belonging to the interval $[\alpha_0,1]$ for both choices $\Lambda\in\{{\mathbb Z}^2,\mathsf{A}_2\}$, we need only to check the validity of the inequality $$\label{boundza_rough} E_f\left[\lambda_0^{\mathsf{A}_2} \mathsf{A}_2\right]> E_f\left[\lambda_0^{\mathbb Z^2} \mathbb Z^2\right].$$ For simplicity of notation, we simply denote $\lambda_1=\lambda_0^{\mathsf A_2}, \lambda_2:=\lambda_0^{\mathbb Z^2}$ for the rest of the proof. We note that due to the constraints $\lambda_1,\lambda_2\in[\alpha_0,1]$ we have that - $\left(\lambda_1 \mathsf{A}_2\right)^{(\ge 2)}\subset \mathbb R^2\setminus B_{\sqrt 3 \alpha_0}$, - $\left(\lambda_2 \mathbb Z^2\right)^{(\ge 3)}\subset \mathbb R^2\setminus B_{2 \alpha_0}$. This has the following two consequences: Firstly, due to the fact that $1<\sqrt 3\alpha_0<2\alpha_0$ and the second sign condition in , we have $$\label{negative_tail} E_f\left[\left(\lambda_1 \mathsf{A}_2\right)^{(\ge 2)}\right]<0, \quad E_f\left[\left(\lambda_2 \mathbb Z^2\right)^{(\ge 3)}\right]<0.$$ Secondly, due to the condition , we have that $E_f\left[\left(\lambda \mathsf{A}_2\right)^{(\ge 2)}\right]$ is increasing in $\lambda$ for $\lambda>\sqrt 3\alpha_0$, therefore in particular $$\label{min_tail_A} \min_{\lambda\in[\alpha_0,1]}E_f\left[\left(\lambda \mathsf{A}_2\right)^{(\ge 2)}\right]=E_f\left[\left(\alpha_0 \mathsf{A}_2\right)^{(\ge 2)}\right].$$ Thus, e can use , , and to prove the following inequalities: $$\begin{aligned} \lefteqn{\sum_{p\in (\lambda_1\mathsf{A}_2)^{(1)}}g(|p|) - \sum_{p\in(\lambda_2\mathbb Z^2)^{(1,2)}}g(|p|)=6g(\lambda_1)-4g(\lambda_2)-4g\left(\sqrt 2 \lambda_2\right)}\nonumber\\ &\stackrel{\eqref{normalize}}{\ge}&-4g(\lambda_2)-4g\left(\sqrt 2 \lambda_2\right)-6\nonumber\\ &\stackrel{\eqref{zbettera}}{>}& - E_f\left[\left(\alpha_0\mathsf{A}_2\right)^{(\ge 2)}\right]\nonumber\\ &\stackrel{\eqref{negative_tail}, \eqref{min_tail_A}}{>}& -E_f\left[\left(\lambda_1\mathsf{A}_2\right)^{(\ge 2)}\right] + E_f\left[\left(\lambda_2\mathbb Z^2\right)^{(\ge 3)}\right],\label{boundza}\end{aligned}$$ Now the last line in by reordering terms gives , and completes the proof of . In [@Crystal Fig. 1], Theil numerically noticed that $\min_{\lambda>0} E_f[\lambda {\mathbb Z}^2]<\min_{\lambda>0} E_f[\lambda \mathsf{A}_2]$ for $g(r)=\frac{1}{r^{12}}+\tanh\left(4r-\frac{13}{2} \right)-1$, $f(r^2)=g(r)$, based on a similar principle as in our above theorem. This potential is not a one-well potentinal, as it is decreasing at infinity. Potentials with several wells, and possibly with the property of being decreasing at infinity (property which could be interpreted as having “a well at infinity”), are not discussed in the present paper. \[rmk-generalprinciple\] Note that the proof of Proposition \[prop:theil\_ZA\] is relatively robust, and allows to introduce also perturbations of the square lattice, and to favor them over the triangular lattice. In a setting of arbitrary dimension and for arbitrary lattices, we formulate the following principle: If for two lattices $\Lambda_1,\Lambda_2$ there holds $$|\Lambda_1|=|\Lambda_2|\quad\mbox{and}\quad\exists\ r>0,\quad\#\left(\Lambda_1\cap B(0,r)\right)>\#\left(\Lambda_2\cap B(0,r)\right),$$ then it is possible to construct a potential $f$ with one single well, such that holds with the lattices $\mathsf{A}_2,\mathbb Z^2$ replaced by $\Lambda_1,\Lambda_2$, respectively. This can be done, amongst other methods, by imitating the proof of Proposition \[prop:theil\_ZA\]. #### **Acknowledgements.** The authors wish to thank the anonymous referee for helping clarify the paper. LB is grateful for the support of the Mathematics Center Heidelberg (MATCH) during his stay in Heidelberg. He also acknowledges support from ERC advanced grant Mathematics of the Structure of Matter (project No. 321029) and from VILLUM FONDEN via the QMATH Centre of Excellence (grant No. 10059). MP is grateful for the stimulating work environment provided by ICERM (Brown University), during the Semester Program on “Point Configurations in Geometry, Physics and Computer Science” in spring 2018 supported by the National Science Foundation under Grant No. DMS-1439786, and acknowledges support from the FONDECYT *Iniciacion en Investigacion 2017* grant N. 11170264. Proof of Proposition \[prop:example\_vn\] {#appendix} ========================================= In the following, we will write $c=\left( \frac{2}{3} \right)\left(\frac{4}{9} \right)^{p}$ and $I_\lambda=\left[\frac{4}{9\lambda},\frac{1}{\lambda} \right]$. Let $L\in \mathcal{L}_d$ be a Bravais lattice, then we have, for any $\lambda>0$, $$\label{formula_egl} E_f[\lambda L]=\frac{c}{\lambda^{p}}\sum_{x\in L \atop |x|<\frac{4}{9\lambda}}\frac{1}{|x|^{p}}+\sum_{x\in L \atop |x|\in I_\lambda}(2-3\lambda |x|)-\frac{1}{\lambda^4}\sum_{x\in L \atop |x|>\frac{1}{\lambda}} \frac{1}{|x|^4}:=S_1+S_2+S_3.$$ **Step 1.** We remark that, for any $L$ with minimal distance $1$ (like ${\mathbb Z}^2$ and $\mathsf{A}_2$), therefore if $\lambda>1$, then $S_1=S_2=0$ and we get $E_f[\lambda L]=S_3=-\lambda^{-4} \zeta_{L}(4)$. Since $\lambda\mapsto \lambda^{-4}$ is decreasing, it follows that $E_f[\lambda L]$ is increasing in $\lambda$ for $\lambda\in(1,+\infty)$. **Step 2.** We now treat the case $L={\mathbb Z}^2$, for $\lambda\in[4/9,1]$, a range in which $S_1=0$ but $S_2,S_3\neq 0$. Note that the distances to the origin for the square lattice are $1$ (achieved $4$ times), $\sqrt{2}$ (achieved $4$ times), $2$ (achieved $4$ times) and $\sqrt{5}$ (achieved $8$ times). We base our case subdivision on these values. 1. *Values $4/9\leq \lambda \leq 1/\sqrt{5}$ for $L={\mathbb Z}^2$.* Then $$E_f[\lambda {\mathbb Z}^2]=40-3(12+4\sqrt{2}+8\sqrt{5})\lambda -\frac{(\zeta_{{\mathbb Z}^2}(4)-5-1/4-8/25)}{\lambda^4}.$$ Therefore, $\frac{d}{d\lambda}E_f[\lambda {\mathbb Z}^2]\geq 0$ if and only if $\lambda\leq \left(\frac{4(\zeta_{{\mathbb Z}^2}(4)-5-1/4-8/25)}{3(12+4\sqrt{2}+8\sqrt{5})} \right)^{1/5}=:\lambda_1\approx 0.4433$. Thus, $\lambda\mapsto E_f[\lambda {\mathbb Z}^2]$ is decreasing on $[4/9,1/\sqrt{5}]$. 2. *Values $1/\sqrt{5}< \lambda\leq 1/2$ for $L={\mathbb Z}^2$.* In this case $$\label{case2} E_f[\lambda {\mathbb Z}^2]=24-3(12+4\sqrt{2})\lambda -\frac{(\zeta_{{\mathbb Z}^2}(4)-5-1/4)}{\lambda^4}.$$ The $\tfrac{d}{d\lambda}$-derivative of is positive for $ \lambda<\left(\frac{4(\zeta_{{\mathbb Z}^2}(4)-5-1/4)}{3(12+4\sqrt{2})} \right)^{1/5}\approx 0.56$ and thus $\lambda\mapsto E_f[\lambda {\mathbb Z}^2]$ is increasing for $\lambda\in(1/\sqrt{5},1/2]$. 3. *Values $1/2<\lambda\leq 1/\sqrt{2}$ for $L={\mathbb Z}^2$.* In this case $$E_f[\lambda {\mathbb Z}^2]=16-3(4+4\sqrt{2})\lambda -\frac{(\zeta_{{\mathbb Z}^2}(4)-5)}{\lambda^4}.$$ Now for the critical value $\left(\frac{4(\zeta_{{\mathbb Z}^2}(4)-5)}{3(4+4\sqrt{2})} \right)^{1/5}=:\lambda_2\approx 0.6765$ we find that $\lambda\mapsto E_f[\lambda {\mathbb Z}^2]$ is increasing on $(1/2,\lambda_2]$ and decreasing on $[\lambda_2,1/\sqrt{2}]$. 4. *Values $1/\sqrt{2}<\lambda\leq 1$ for $L={\mathbb Z}^2$.* In this case $$E_f[\lambda {\mathbb Z}^2]=8-12\lambda -\frac{(\zeta_{{\mathbb Z}^2}(4)-4)}{\lambda^4}.$$ Therefore, defining $\left( \frac{4(\zeta_{{\mathbb Z}^2}(4)-4)}{12} \right)^{1/5}=:\lambda_3\approx 0.9245$ the map $\lambda\mapsto E_f[\lambda {\mathbb Z}^2]$ is increasing on $[1/\sqrt{2},\lambda_3]$ and decreasing on $[\lambda_3,1]$. Now, comparing the values of $E_f[\lambda L]$ for $L={\mathbb Z}^2$ for $\lambda\in \{1/\sqrt{5},1/\sqrt{2},1\}$, we find that, based on the above discussion and on Step 1, $$\label{minsquare} \displaystyle \min_{\lambda\ge 4/9}E_f[\lambda {\mathbb Z}^2]=E_f\left[\frac{1}{\sqrt{5}}{\mathbb Z}^2 \right]\approx -19.108745$$ **Step 3.** By performing a similar discussion as in Steps 1 and 2, based on the distances to the origin for points in $L=\mathsf{A}_2$ lower than $9/4$, that are $1,\sqrt{3}$ and $2$ (all achieved $6$ times), we obtain $$\label{mintri} \displaystyle \min_{\lambda\ge 4/9}E_f[\lambda \mathsf{A}_2]=E_f\left[\frac{4}{9}\mathsf{A}_2 \right]\approx -19.013358.$$ **Step 4.** We now assume that $\lambda<\frac{4}{9}$ and we compute a lower bound for the energy . We bound $S_1$ by the first term in the sum and $S_3$ by the sum over the whole lattice without the constraint $|x|>\tfrac1\lambda$, and we obtain $$S_1> \frac{4\left(\frac{2}{3} \right)\left(\frac{4}{9} \right)^p}{\lambda^p},\quad S_3>-\frac{\zeta_{{\mathbb Z}^2}(4)}{\lambda^4}.$$ For $S_2$, we use the fact that $\#\{ x\in {\mathbb Z}^2; |x|\leq r \}=\pi r^2 + R(r)$ where $|R(r)|\leq 2\sqrt{2}\pi r$. We therefore get $$\begin{aligned} S_2=\sum_{x\in {\mathbb Z}^2 \atop |x|\in I_\lambda}(2-3\lambda |x|)&> \left(2-\frac{4}{3\lambda}\right)\#\{x\in {\mathbb Z}^2; |x|\in I_\lambda \}\\ &> \left(2-\frac{4}{3\lambda}\right)\left(\frac{\pi}{\lambda^2}-\frac{16\pi}{81\lambda^2}+\frac{2\sqrt{2}\pi}{\lambda}+\frac{8\sqrt{2}\pi}{9\lambda} \right)\\ &=-\frac{260\pi}{243\lambda^3}-\left(\frac{104\sqrt{2}\pi}{27\lambda^2}-\frac{130\pi}{81\lambda^2} \right)+\frac{52\sqrt{2}\pi}{9\lambda}. \end{aligned}$$ Thus, we have obtained $$E_f[\lambda {\mathbb Z}^2]>\frac{4\left(\frac{2}{3} \right)\left(\frac{4}{9} \right)^p}{\lambda^p}-\frac{\zeta_{{\mathbb Z}^2}(4)}{\lambda^4}-\frac{260\pi}{243\lambda^3}-\left(\frac{104\sqrt{2}\pi}{27\lambda^2}-\frac{130\pi}{81\lambda^2} \right)+\frac{52\sqrt{2}\pi}{9\lambda} .$$ We now want to determine a value $\lambda<4/9$ such that $E_f[\lambda {\mathbb Z}^2]>E[\frac{1}{\sqrt{5}}{\mathbb Z}^2]$. A sufficient condition is, setting $X=\lambda^{-1}$, to know all the $X>\frac{9}{4}$ satisfy $$\begin{gathered} \label{ineq_qp} 4\left(\frac{2}{3} \right)\left(\frac{4}{9} \right)^p X^p+\frac{52\sqrt{2}\pi}{9}X \\ \ge\zeta_{{\mathbb Z}^2}(4) X^4 +\frac{260\pi}{243}X^3+\left(\frac{104\sqrt{2}\pi}{27}+\frac{130\pi}{81} \right)X^2+E\left[\frac{1}{\sqrt{5}}{\mathbb Z}^2\right]. \end{gathered}$$ Defining the following coefficient $$\alpha_4:=\zeta_{{\mathbb Z}^2}(4),\quad \alpha_3:=\frac{260\pi}{243}, \quad \alpha_2:=\left(\frac{104\sqrt{2}\pi}{27}-\frac{130\pi}{81} \right),$$ it follows by a direct estimate that holds if $$X\geq U_p:=\max\left\{\left(\frac{9}{8}\left( \frac{9}{4} \right)^p \alpha_4 \right)^{\frac{1}{p-4}}, \left(\frac{9}{8}\left( \frac{9}{4} \right)^p \alpha_3 \right)^{\frac{1}{p-3}},\left(\frac{9}{8}\left( \frac{9}{4} \right)^p \alpha_2 \right)^{\frac{1}{p-2}}\right\}.$$ We observe that $U_p\to \frac{9}{4}$ as $p\to +\infty$ and is $U_p$ decreasing in $p$ for large $p$. Therefore, by continuity of $\lambda\mapsto E_f[\lambda {\mathbb Z}^2]$, for any $\varepsilon>0$, there exists $p_0$ such that $$\left| \min_{\lambda<4/9} E_f[\lambda {\mathbb Z}^2]- E_f\left[\frac{4}{9}{\mathbb Z}^2\right]\right|<\varepsilon.$$ Since $ E_f[\frac{4}{9}{\mathbb Z}^2]>E_f[\frac{1}{\sqrt{5}}{\mathbb Z}^2]$, it follows that, for enough large $p$, $$\min_{\lambda>0}E_f[\lambda {\mathbb Z}^2]=E_f[\frac{1}{\sqrt{5}}{\mathbb Z}^2]\approx -19.108745.$$ The same argument can be repeated for $L=\mathsf{A}_2$, obtaining that for any $\varepsilon>0$, there exists $p_0$ such that for any $p>p_0$ there holds $$\left| \min_{\lambda<4/9} E_f[\lambda \mathsf{A}_2]- E_f\left[\frac{4}{9}\mathsf{A}_2\right]\right|<\varepsilon.$$ Therefore, by , $\min_{\lambda>0} E_f[\lambda \mathsf{A}_2]> E_f[\frac{4}{9} \mathsf{A}_2]-\varepsilon>E_f[\frac{1}{\sqrt{5}}{\mathbb Z}^2]$ for $\varepsilon>0$ sufficiently small, which in turn is achievable for $p_0$ sufficiently large. These choices allow to complete the proof. [10]{} A. Aftalion, X. Blanc, and F. Nier. . , 241:661–702, 2006. W. Banaszczyk. . , 296(1):625–635, 1993. S. Bernstein. . , 52:1–66, 1929. L. B[é]{}termin. . , 48(5):3236–3269, 2016. L. B[é]{}termin. . , 9(1):403–426, 2019. L. B[é]{}termin. . , 31(9):3973–4005, 2018. L. B[é]{}termin. . *Preprint. arXiv:1809.00473*, 2019. L. B[é]{}termin. . *Preprint. arXiv:1901.08957*, 2018. L. B[é]{}termin and H. Kn[ü]{}pfer. . , 108(10):2213–2228, 2018. L. B[é]{}termin, H. Kn[ü]{}pfer, and F. Nolte. . *Preprint. arXiv:1804.05743*, 2018. L. B[é]{}termin and M. Petrache. . , 58:071902, 2017. L. B[é]{}termin and P. Zhang. . , 17(6):1450049, 2015. X. Blanc. . , 26(3-4):651–696, 2001. X. Blanc and C. Le Bris. . , 48(6):791–803, 2002. X. Blanc and M. Lewin. . , 2:255–306, 2015. S. Bochner. Theta relations with spherical harmonics. , 37(12):804–808, 1951. J. B. Brauchart, D. P. Hardin, and E. B. Saff. . , 7(2):77–108, 2012. J.W.S. Cassels. . , 4:73–80, 1959. Y. Choi and T. Ree. . , 99:9917, 1993. H. Cohn and N. Elkies. . , 157:689–714, 2003. H. Cohn and A. Kumar. . , 20(1):99–148, 2007. H. Cohn and A. Kumar. . , 78(6):061113, 7, 2008. H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, and M. Viazovska. . , 185(3):1017–1033, 2017. J. H. Conway and N. J. A. Sloane. , volume 290. Springer, 1999. R. Coulangeon. . , ID 49620(16), 2006. R. Coulangeon and G. Lazzarini. . , 141:288–315, 2014. R. Coulangeon and A. Sch[ü]{}rmann. . , pages 829–848, 2012. R. Coulangeon and A. Sch[ü]{}rmann. . *Preprint. arXiv:1802.02072*, 2018. P. H. Diananda. . , 6:202–204, 1964. V. Ennola. . , 6:198–201, 1964. V. Ennola. . , 60:855–875, 1964. L. Flatley and F. Theil. . , 219(1):363–416, 2015. C. S. Gardner and C. Radin. . , 20:719–724, 1979. C. S. Gardner and C. Radin. . , 20(6):719–724, 1979. A. Georgakopoulos and M. Kolountzakis. . , 145(8):3501–3511, 2017. T. C. Hales. . , 162(3):1065–1185, 2005. G. C. Hamrick and C. Radin. . , 21(5):601–607, 1979. R. C. Heitmann and C. Radin. . , 22:281–287, 1980. I. G. Kaplan. . John Wiley and Sons Ltd, 2006. M. Katsurada. . , 14:249–272, 2007. R. Kusner, W. Kusner, J. C. Lagarias, and S. Shlosman. . , 2016. T. Lebl[é]{}. . , 268(7):1649–1677, 2015. L. De Luca and G. Friesecke. . , 28(1):69–90, 2018. E. Mainini, P. Piovano, and U. Stefanelli. . , 27:717–737, 2014. E. Mainini and U. Stefanelli. . , 328:545–571, 2014. E. Marcotte, F. H. Stillinger, and S. Torquato. . , 134:164105, 2011. G. Mie. . , 316(8):657–697, 1903. A. Mogilner, L. Edelstein-Keshet, L. Bent, and A. Spiros. . , 47:353–389, 2003. H. L. Montgomery. . , 30(1):75–85, 1988. E. J. Mueller and T.-L. Ho. . , 88(18), 2002. O. R. Musin. . , 168:1–32, 2008. K. Neumann. . Teubner (Leipzig), 1896. B. Osgood, R. Phillips, and P. Sarnak. . , 80:148–211, 1988. M. Petrache and S. Serfaty. . , 16(3):501–569, 2017. C. Poole. . Elsevier, 1st edition edition, 2004. C. Radin. . , 26(2):365–373, 1981. C. Radin. . , 35(1):109–117, 1984. C. Radin. . , 1(05n06):1157–1191, 1987. R. A. Rankin. . , 1:149–158, 1953. M. C. Rechtsman, F. H. Stillinger, and S. Torquato. . , 95, 2005. G. Rutkai, M. Thol, R. Span, and J. Vrabec. , 115(9-12):1104–1121, 2017. E. Sandier and S. Serfaty. . , 313(3):635–743, 2012. E. Sandier and S. Serfaty. . , 162(3–4):795–846, 2015. P. Sarnak and A. Str[ö]{}mbergsson. . , 165:115–151, 2006. F. H. Stillinger. . , 115(11):5208–5212, 2001. A. S[ü]{}to. . , 95(26):265501, 2005. A. S[ü]{}to. . , 305:657–710, 2011. F. Theil. . , 262(1):209–236, 2006. S. Torquato. . , 5:1157, 2009. B. Venkov. . , 37:10–86, 2001. W. J. Ventevogel. . , 92A(3-4):343–361, 1978. W. J. Ventevogel and B. R. A. Nijboer. . , 98(1-2):274–288, 1979. W. J. Ventevogel and B. R. A. Nijboer. . , 99(3):569–580, 1979. M. Viazovska. . , 185(3):991–1015, 2017. E W. and D. Li. . , 286:1099–1140, 2009.

--- abstract: 'Being able to automatically and quickly understand the user context during a session is a main issue for recommender systems. As a first step toward achieving that goal, we propose a model that observes in real time the diversity brought by each item relatively to a short sequence of consultations, corresponding to the recent user history. Our model has a complexity in constant time, and is generic since it can apply to any type of items within an online service (*e.g.* profiles, products, music tracks) and any application domain (e-commerce, social network, music streaming), as long as we have partial item descriptions. The observation of the diversity level over time allows us to detect implicit changes. In the long term, we plan to characterize the context, *i.e.* to find common features among a contiguous sub-sequence of items between two changes of context determined by our model. This will allow us to make context-aware and privacy-preserving recommendations, to explain them to users. As this is an on-going research, the first step consists here in studying the robustness of our model while detecting changes of context. In order to do so, we use a music corpus of 100 users and more than 210,000 consultations (number of songs played in the global history). We validate the relevancy of our detections by finding connections between changes of context and events, such as ends of session. Of course, these events are a subset of the possible changes of context, since there might be several contexts within a session. We altered the quality of our corpus in several manners, so as to test the performances of our model when confronted with sparsity and different types of items. The results show that our model is robust and constitutes a promising approach.' author: - - - title: 'Toward a Robust Diversity-Based Model to Detect Changes of Context' --- User Modeling; Diversity; Context; Real-Time Analysis of Navigation Path; Recommender Systems Introduction ============ Despite their growing success in industry (e-commerce, social networks, VOD, music streaming platforms) and their impressive predictive performances [@Simpson:2014], two major user concerns frequently show up about recommender systems in online services. First, people are more and more preoccupied by privacy issues. To maintain a good trust level, we should thus provide models and algorithms that offer the best compromise between quality of recommendations, ethics as regards data collection [@Cranor:2005], and users’ policy [@Knijnenburg:2013]. Second, recommendations are still too often made out of context. Recommending is not only a question of maximizing the accuracy, but also providing relevant items at the right time in the good manner [@Jones:2010]. This is the reason why the literature about context-aware recommender systems is increasing fast [@Hariri:2014]. Starting from these observations, we wondered what could possibly be the necessary and sufficient data to understand as quickly as possible the user context, and then to adapt recommendations. As regards privacy, Cranor suggests to favor methods where personal data are transient (*i.e.* deleted after the task or the session) [@Cranor:2005]. The system should also rely on item profiles, rather than user profiles. Thus, it is reasonable to study the short history of recently consulted items, and see what are the common features or differences that could explain or characterize the current user context. This line of reasoning implies that we have a precise description of each item available in the online service, or at least an exhaustive set of description attributes like those we have in product catalogs, but for every type of items (music tracks, social network profiles of users and companies, …). Besides these considerations, Castagnos *et al.* took an interest in the role of diversity within the user decision-making process [@Castagnos:2010]. They provide two interesting conclusions within the frame of e-commerce applications. On one hand, the diversity in recommender systems seems to significantly improve user satisfaction, and is correlated to the intention to buy. On the other hand, the user need for diversity evolves over time, and should be carefully controlled to provide the correct amount of diversity and novelty. Bringing too much diversity risks to transform recommendations into novelty. Recent works confirmed that satisfaction is negatively dependent on novelty [@Ekstrand:2014], and badly-used diversity can lead users to mistrust the system [@Castagnos:2013]. Finally, in [@Castagnos:2010], we showed that recommender systems should increase the diversity level at the end of a session to make users more confident in their buying decisions. Yet, predicting when the session will end is not an easy task. This conclusion led us to ask if we could take the opposite view: would it be possible to monitor the diversity level within user sequences of consultations over time, and find connections between variations of diversity and changes of context? Through an exploratory research, we proposed the first model that measure the diversity brought by each consulted item, relatively to a short user history [@LHuillier:2014]. We showed that variations of diversity often match with ends of session. However, these conclusions were made *a posteriori*, *i.e.* by analyzing the whole sequence of consultations for each user, and then knowing how each session ended and how the next session started. Furthermore, our model was built by considering that all consulted items were of the same type. As an example, if the active user is listening to music, it should be possible to measure the diversity between each pair of items. In this paper, we want to bring this model a step further. First, we aim at investigating if it allows us to predict ends of session in real time, without knowing what happens next. Then, we will test the robustness of our model, by reconsidering our strong hypothesis according to which we always have a complete description of items. We will thus evaluate the performances of our model when we have sparse data about items. At last, we will extend our model to a situation where the active user consults different types of items (*e.g.* music tracks, social network profiles, ...). In this case, it is not always possible to measure the diversity between items, since their attributes may be different. Thank to a corpus of more than 210,000 consultations, we show that the performances of our system remain stable up to 60% of missing diversity measures. The rest of this paper is organized as follows: Section \[related-work\] offers an overview of the literature as regards diversity and context in recommender systems. Section \[model\] is dedicated to the presentation of our model and our hypotheses about its robustness to sparsity and diversification of types of items. Section \[experiment\] presents and discusses its performances. Related Work ============ Diversity in Recommender Systems {#diversity} -------------------------------- Diversity has long been proven to improve the interactions between users and recommender systems [@McGinty:2003]. This dimension is considered in two different ways in the literature. Some analyze the impact of diversity on users’ behavior, while others integrate diversity in machine learning algorithms of recommender systems. Diversity has first been defined by Smyth and McClave [@Smyth:2001] as the opposite dimension to similarity. More precisely, this measure quantifies the dissimilarity within a set of items. Thus, diversifying recommendations consists in determining the best set of items that are highly similar to the users’ known preferences while reducing the similarity between those recommendations. A classification of diversity has been proposed by Adomavicius and Kwon [@Adomavicius:2012]. It distinguishes individual diversity and aggregated diversity, depending on if we are interested in generating recommendations to individuals, or to groups of users. Here, we focus on individual diversity. Many works focus on controlling the diversity level brought by recommender systems. Diversity was initially dedicated to content-based algorithms, especially in the case we have attribute values for each item. We distinguish 3 practices: we can compute the diversity between two items [@Smyth:2001], the diversity within a set of items [@Ziegler:2005], or the relative diversity brought by a single item relatively to a set of items [@Smyth:2001] (see Equation \[eq:reldiv\]). These metrics have then been used in content-based filtering to reorder the recommendation list, according to a diversity criterion [@Bradley:2001; @Zhang:2008]. In addition to these content-based algorithms, some works have focused on a way to integrate diversity in collaborative filtering [@Ziegler:2005; @Said:2012]. In parallel to the integration of diversity in recommender systems, many user studies took interest in the role and perception of diversity. McGinty and Smyth showed that diversity improves the efficiency of recommendations [@McGinty:2003]. Many works showed that diversity is perceived by users [@Zhang:2008; @Lathia:2010; @Jones:2010], and positively correlated to user satisfaction [@Castagnos:2013; @Ekstrand:2014]. Nevertheless, it came out that the user need for diversity evolves over time and diversity should not be integrated in the same way at each recommendation stage [@McGinty:2003; @Castagnos:2010]. At last, recent works focus on how the amount of diversity should be provided by recommender systems [@Hasan:2014]. Contrary to this literature, we do not want to adapt the amount of diversity in recommendations. We aim at observing the natural diversity level within users’ navigation path to infer their context. Thus, the following subsection will be dedicated to this notion of context. Context-Aware Recommender Systems {#context} --------------------------------- Integrating the context into the recommendation process is an increasing research field known as `CARS`, acronym for Context Aware Recommender Systems. In their state-of-the-art, Adomavicius *et al.* present several approaches like contextual modeling, pre/post filtering method for using contextual factors in order to adapt recommendation to the users’ context [@Adomavicius:2011b]. Contextual factors are all the information which can be gathered and used by a system to determine and characterize the current context of the user. For example, a system can use the location of the user to adapt the recommendation [@Kaminskas:2013]. The most important drawbacks of these kinds of systems lies in the fact that they are invasive, by using personal informations and most often require a complex representational model. For example, such systems can use ontologies to determine user context [@Chen:2014]. Yet, such an ontology cannot be transferred from one domain to another. As Adomavicius and Tuzhilin explain in their conclusion, “most of the work on CARS has focused on the representation view of the context and the alternative methods have been underexplored” [@Adomavicius:2011b]. This fact has also been highlighted by Hariri *et al.* who have developed a `CARS` based on users’ feedback on items presented in a interactive recommender system [@Hariri:2014]. Even if this approach dynamically adapts to changes of context, it requires user effort to obtain user’ feedback on which the system is based. We thus aim at proposing a similar method having the same objectives, but more transparent for users by relying on item profiles and users’ navigation path. In the following, we propose to distinguish two different types of context: explicit context and implicit context. Explicit context is close to the definition of contextual factors, that is to say physical context, social context, interaction media context and modal context are different kinds of explicit context [@Adomavicius:2011b]. Conversely, implicit context will refer to the common characteristics shared by the consulted items during a certain time lapse. The motivation behind this notion is that detecting implicit context does not increase user involvement, enhances the privacy and can be used in any application domain without heavy modifications. Model and Hypotheses {#model} ==================== Overview -------- As explained above, the role of our model is to monitor the diversity level within users’ navigation path over time, and then derive their implicit context. Concretely, each time a user consults a new item, we compute the added value of this item – called `target` – relatively to a short history (*i.e.* the $k$ previously consulted items) as regards to diversity. To provide a better understanding of our model, we will rely on an example shown in Figure \[fig:dance\]. Let us imagine an online service that allows users to listen to music, and to browse different kinds of profiles like we can do on social networks (profiles of other users, profiles of artists, information about record companies and so on). For each user, we can then pay attention to his/her sequence of consultations. In this example, we understand that there might be several contexts within a session, and several ways to classify them. ![Overview of our model.[]{data-label="fig:dance"}](images/dance.eps){width="50.00000%"} One strength of our model is that it allows us to measure in real time the diversity brought by each item, for each attribute independently, and for the whole set of attributes. Thus, it can be configured to detect and characterize various kinds of implicit contexts, or to cut the navigation path at some points where diversity reaches the highest levels (*i.e.* what we called the changes of implicit context). In the rest of this article, we will give meaning to these changes of implicit context, by verifying that they match with some events such as ends of session in many cases. But, of course, there can be several successive implicit contexts, and several changes of context, within a session. Let us notice that, in the case where we want to force the detection of events and to optimize the characterization of the implicit context according to user’s expectations, all we have to do is to complete a learning phase to find the optimal weight of each attribute within our computation of the diversity over time. The quality of our model has been demonstrated in [@LHuillier:2014]. However, the purpose of this paper is to test the robustness of our model in the case where we have sparse data within item descriptions, that is to say detecting the same changes of implicit context with less data. We see two different scenarios which can explain sparse data. Either we have a single type of items (for example music tracks), but an incomplete description of each item, which is often the case in real applications. Or the users’ navigation path are made of different types of items, and there may be a partial overlap of attributes between items. In Figure \[fig:dance\], common attributes between items are displayed on the same line. Formalism --------- Before evaluating the robustness of our model, we will present it more formally and will introduce some notations. We call $U$ ={*$u_{1}$, $u_{2}$,..., $u_{n}$*} the set of users. $u$ refers to the active user. $I$ ={*$i_{1}$, $i_{2}$,..., $i_{m}$*} is the whole set of consulted items. The recent user history of size $k$ at time $t$, called $C_{k,t}^{u}$, can be written under the form of a sequence of items $<c_{t-k}^{u}$, ..., $c_{t-2}^{u}$, $c_{t-1}^{u}$, $c_{t-1}^{u}>$. At last, $A_i$ = *$a_{1}$, $a_{2}$,..., $a_{h} $* is the set of attributes of an item $i$. Let us note that each consulted item, such as $c_{t}^{u}$, refers to an item $i$ of the set $I$. Our model is a Markov model. At each time-step (*i.e.* each time the active user consults a new item), our model computes the relative diversity brought by the new consulted item $c_{t}^{u}$ relatively to $C_{k,t}^{u}$. In order to do so, we strongly took inspiration from the formula proposed by Smyth and McClave [@Smyth:2001] (see Equation \[eq:reldiv\]). The only difference here is that we count the number of times $s$ when we can compute the similarity between the target item $c_{t}^{u}$ and one of the items in the history $C_{k,t}^{u}$. As the active user can browse different types of items, there may be situations where there is no common attributes between two items, and no way to compute the similarity between this pair of items (*i.e.* it returns NaN). Consequently, $s$ is included in $[0;k]$. [$$\begin{gathered} \label{eq:reldiv} \raggedright{ \scriptstyle RD(c_{t}^{u},C_{k,t}^{u})~= \begin{cases} &\scriptstyle ~\text{NaN~if $C_{k,t}^{u}$~=~$\emptyset$ or if $s~=~0$,}\\ &\scriptstyle ~\frac{\sum_{j=1..k}(1-sim(c_{t}^{u},c_{t-j}^{u}))}{s}~\text{otherwise.} \end{cases} }\end{gathered}$$ ]{} Measuring RD (Equation \[eq:reldiv\]) involves to compute the similarity between each pair of items, using Equation \[eq:sim\]. In this equation, the function $sim_{a}$ computes the similarity between two items relatively to a specific attribute $a$. $\alpha_{a}$ is the weight of this attribute $a$ in the computation of the similarity. In this paper, since we want mainly want to test the robustness of our model as regards sparse data, we will use a naive approach where each weight $\alpha_{a}$ is equal to 1. But we could parameter these weights to adapt our model, according to the kind of changes of implicit context and/or the kind of events we want to detect. [$$\begin{gathered} \label{eq:sim} \raggedright{ \scriptstyle sim(c_{t}^{u},c_{t-j}^{u})~= \begin{cases} &\scriptstyle ~\text{NaN~if $(A_{c_{t}^{u}}\cap{}A_{c_{t-j}^{u}})$ or $c_{t}^{u}.a$ or $c_{t-j}^{u}.a~=~\emptyset$,}\\ &\scriptstyle ~\frac{\sum_{a\in{}A_{c_{t}^{u}}\cap{}A_{c_{t-j}^{u}}} (\alpha_{a}~*~sim_{a}(c_{t}^{u},c_{t-j}^{u}))}{\sum_{a\in{}A_{c_{t}^{u}}\cap{}A_{c_{t-j}^{u}}} \alpha_{a}}~\text{otherwise.} \end{cases} }\end{gathered}$$ ]{} In Equation \[eq:sim\], $i.a$ refers to the values (or set of values) of an attribute $a$ for a given item $i$. Starting from here, we developed 5 generic formulas to compute similarities per attribute, according to the type of attribute we have. If the values $i.a$ are expressed under the form of a list (*e.g.* the attribute “similar artists” for a song), we will use Equation \[eq:sima1\]. $$\label{eq:sima1} sim_{a}(c_{t}^{u},c_{t-j}^{u})=\frac{card(c_{t}^{u}.a\cap c_{t-j}^{u}.a)}{min(card(c_{t}^{u}.a), card(c_{t-j}^{u}.a))} %sim_{attribut=a}(c_{t}^{u},c_{t-1}^{u})=1-\left(\frac{\sum_{i=1}^{nb_term(i)} terme(j).i=term(j+1).i}{\frac{nbterme(j)+nbterme(j+1)}{2}}\right)$$ If the values $i.a$ correspond to intervals (*e.g.* the attribute “period of activity of a singer”), we will use Equation \[eq:sima2\]. $$\label{eq:sima2} sim_{a}(c_{t}^{u},c_{t-j}^{u})=\frac{card(c_{t}^{u}.a\cap c_{t-j}^{u}.a)}{max(card(c_{t}^{u}.a), card(c_{t-j}^{u}.a))}$$ If $i.a$ have binary values (*e.g.* the mode of a song), we will use Equation \[eq:sima3\]. [$$\begin{gathered} \raggedright{ \scriptstyle sim_{a}(c_{t}^{u},c_{t-j}^{u})~= \begin{cases} &\scriptstyle ~1~\text{if}~c_{t-j}^{u}.a~=~c_{t}^{u}.a\text{,} \\ &\scriptstyle ~0~\text{otherwise.}\hspace*{5em} \end{cases} \label{eq:sima3}}\end{gathered}$$ ]{} If $i.a$ take numerical values (*e.g.* user ratings), we will use Equation \[eq:sima4\]. $$\label{eq:sima4} sim_{a}(c_{t}^{u},c_{t-j}^{u})=e^{-10*\left(\frac{c_{t}^{u}.a-c_{t-j}^{u}.a}{max_{a} - min{a}}\right)^2}$$ At last, if $i.a$ express coordinates (*e.g.* the localization of two artists), we will use the Equation \[eq:sima5\]. $$\label{eq:sima5} sim_{a}(c_{t}^{u},c_{t-j}^{u})=1~-~\frac{distance(c_{t}^{u},c_{t-j}^{u})}{max_{distance}}$$ Finally, we are considering that there is a change of implicit context if the 4 conditions of Equation \[eq:detection\] are met. $\tau$ allows us to focus on relative diversity measures $RD(c_{t}^{u},C_{k,t}^{u})$ that exceed a given threshold. $$\begin{gathered} \label{eq:detection} RD(c_{t-1}^{u},C_{k,t-1}^{u})<>\text{NaN}~\text{and}~RD(c_{t}^{u},C_{k,t}^{u})<>\text{NaN}\\ \text{and}~RD(c_{t-1}^{u},C_{k,t-1}^{u}) < RD(c_{t}^{u},C_{k,t}^{u})~\text{and}~RD(c_{t}^{u},C_{k,t}^{u}) > \tau\end{gathered}$$ Hypotheses ---------- The scientific question is now to test if our model is robust to a realistic situation where: (1) we do not know what will happen after the current time $t$, (2) we have sparse data as regards item descriptions. For these reasons, we will make 3 assumptions that will be discussed in Section \[experiment\]. This assumption has not been considered in our preliminary work in [@LHuillier:2014], since we were analyzing variations of diversity *a posteriori* on the whole user’s navigation path, knowing consultations at each time. We will thus check how many ends of session we can retrieve by only using data at time $t$, even if this does not lower the interests and relevancy of our other detections, as explained above (see Subsection \[overview\]). Considering that we have a single type of items, we expect to retrieve the same amount of events and changes of implicit context. In this scenario, the attributes may be different from one type of items to another, leading to another form of sparsity. Experiments {#experiment} =========== In this section, we present 3 experiments we developed to validate these assumptions. In the first experiment (**H1**), we test the ability of our model to detect changes of implicit context in real time, and check if the detected contexts could be correlated with some particular events like ends of sessions. However, unlike our exploratory research [@LHuillier:2014], our new model only uses data available at the current time $t$ (that is to say, we do not look at how diversity evolves beyond the current time). Indeed, our previous model was looking for local maxima on the curve of relative diversity and used thereby information unavailable at time $t$ to detect changes of context. In real situations, only present and past information are available. That is one of the reason which motivated us to extend our model (the other one is the consultations of different types of items). The principle of our model remains quite similar to [@LHuillier:2014]. However, the inputs used to detect changes of context are different. For each consulted item, we compute the corresponding values of relative diversity. As relative diversity can be computed for each attribute, there are as many relative diversity values as attributes. In this paper, we set the relative diversity of the current item to the average of all relative diversities per attribute. From now on, when we will talk about a relative diversity value according to an item, we will refer to the average relative diversity calculated from all the attributes for this item relatively to the history (Equation \[eq:sim\]). Inside a given context, we assume that the relative diversity of each item is quite constant and low, but that the relative diversity suddenly increases when changes of implicit context occur. This increase is due to the fact that different contexts do not share the same characteristics (*i.e.* the same attribute values). Our model aims to detect these peaks of relative diversity over time. To achieve this, our model checks at each time-step if the conditions of Equation \[eq:detection\] are satisfied. In this case, we assume that $c_{t}^{u}$ is the first item of a new implicit context. For each new implicit context detected, we check if $c_{t}^{u}$ corresponds to the beginning of a new session. In the second experiment (**H2**), we put to the proof our model by deleting information within our corpus. Indeed, data sparsity is a well-known problem in the field of recommender systems, and we want to know how our model can face this problem. In [@LHuillier:2014], we were using a complete dataset (*i.e.* with no missing information about items), but that is rarely the case in real situations. For instance, in a musical corpus, we could have the song title and artist name for each track but some information like the release date, the popularity or the keywords may be missing. Thus, we want to test if: - our model is able to compute a relative diversity value, even if some pieces of information about attributes are not known; - our model is robust to missing information and still performs well for detecting changes of context. To answer these questions, we randomly delete values of attributes in our dataset, until we reach an intended rate of sparsity. We test the performances of our model for rates of sparsity between 1 and 99%. Because of that random deletion, some similarity measures between two items, or even some relative diversity measures could not be computed. As soon as we can compute the similarity on at least one attribute for at least one pair of items (the target item and one of the items within the history), a value of relative diversity can be set for the target. Otherwise, if we cannot compute any similarity per attribute on any pair of items, we set the relative diversity of the target to *NaN*. Let us notice that we set the diversity to *NaN*, because a value of 0 would indicate that there is no diversity brought by the current item, not that the diversity cannot be calculated. Of course, we do not consider NaN values as changes of context (see Equation \[eq:detection\]). In the last experiment (**H3**), the purpose is to examine the consequences of having several types of items in our dataset on context detection performances. Indeed, the previous experiments were tested with a single type of items but in practice, this may not be always the case. When the target item and the history items are of the same type (*i.e.* music), the relative diversity can be computed on all attributes for all items (except when there are missing data). However, when these types may change from a consultation to another, the relative diversity can only be computed for common attributes (see Figure \[fig:dance\]). Considering that our initial dataset contained a single type of items (songs), we modified it in order to test our third hypothesis. Criteria for simulating the different types of items were as follows: First, a number of types of items is determined, and all items are randomly assigned to a type of items. Afterward, for each type of items, we randomly select a subset of $x$ attributes (from the whole set of attributes) that will characterize these items. Another parameter, called $y$, corresponds to the minimum number of attributes in common with all the other types of items. Let us notice that the common attributes between pairs of types of items are not necessarily the same (*i.e.* ($A_{type1}\cap{}A_{type2})<>(A_{type2}\cap{}A_{type3})$). In this way, we can artificially obtain a dataset composed of different types of items, with only a few attributes in common. For instance, if the initial dataset contains 7 attributes ($A=\{a_{1},a_{2},a_{3},a_{4},a_{5},a_{6},a_{7}\}$) and we want to create 3 types with $x=4$ and $y=2$, we randomly get this kind of situation: $A_{type1}=\{a_{1},a_{4},a_{6},a_{7}\}$, $A_{type2}=\{a_{1},a_{2},a_{3},a_{4}\}$, and $A_{type3}=\{a_{2},a_{3},a_{4},a_{6}\}$. In that case, $A^{type1}\cap{}A^{type2}=\{a_{1},a_{4}\}$, $A^{type1}\cap{}A^{type3}=\{a_{4},a_{6}\}$, and $A^{type2}\cap{}A^{type3}=\{a_{2},a_{3},a_{3}\}$. Material -------- In order to test our different hypotheses, we decided to base our evaluation on a musical dataset. This choice was made because musical items offer many advantages. First, musical items have their own consultation time, that is to say the time spent to consult a song cannot vary from a user to another. Second, meta data on songs can be easily retrieved using some specialized services like Echnonest[^1] or Musicbrainz[^2]. At last, users frequently listen to several songs consecutively, contrary to a movie corpus for example. Our dataset contains 212,233 plays which were listened by 100 users. We obtained these consultations by using the Last.fm[^3] API to collect listening events from 28 June 2005 to 18 December 2014. Our dataset is made of 41,742 single tracks, performed by 5,370 single artists. In order to create the sessions for all the users, we assumed that a session is composed by a sequence of consultations without any interruption bigger than 15 minutes. When this threshold is reached, we consider that the user started a new session. According to this standard, we computed 22,212 sessions with an average of 9.6 consultations per session (42.71 min per session). Then, using the Echonest API, we gathered meta data on these songs. For each song, we retrieved 13 attributes: 7 of these attributes are specific to songs, and 6 of them are related to artists. - song attributes: duration, tempo, mode, hotttness, danceability, energy and loudness; - artist attributes: hotttness, familiarity, similar artists (10 artists names), terms, years of activity, and location of the artist (geographical coordinates). Table \[tab:corpus\] summarizes the values of the attributes. Results and Discussion {#results} ---------------------- **Results as regards the first experiment (H1).** Previously, we presented Equation \[eq:detection\] which allow our model to determine if the current consultation is the start of a new implicit context. In order to fix the threshold $\tau$, we calculated the mean and the standard deviation of all values of relative diversity for all users within our corpus. In Table \[tab:statistique\_rd\], we can notice that the standard deviation is pretty high compared to the mean of the relative diversity. This result means that users’ relative diversity over time takes a large range of values. We cannot know *a priori* the best value for $\tau$, since we do not know how many implicit contexts are present in our dataset. However, we previously assumed that diversity is pretty low within a given context and increases when a change of context occurs. This assessment can easily be confirmed *a posteriori*, by noticing that the average level of relative diversity for consultations that correspond to a session opening ($average=0.36, standard deviation=0.13$) is much higher than those of other consultations ($average=0.21, standard deviation=0.16$). We finally decided to set $\tau$ to the global average of relative diversity within our dataset ($0.23$), so as to favor the detection of consultations above an average rate, but without fixing this threshold too high since there might be significant increase of diversity after a long period of decreasing (leading to values near the global average). When relative diversity exceeds this threshold and all the conditions of Equation \[eq:detection\] are satisfied, we consider that there is a change of implicit context. The results are reported in Table \[tab:detection\_naive\]. In total, our model detects 51,795 changes of implicit context. Among those changes of context, the number of sessions detected is important, since our model is able to detect more than 63% of the sessions. This significant overlap between changes of context and events indicates that our model remains efficient when we only use information available at the current time (*i.e.* without considering consultations at time $t+1$ and beyond), since we can easily justify/explain these changes of context by a end of session. This means that, when the explicit context changes (at least as regards the time dimension[^4] since there is a temporal gap between two sessions), the songs listened in those two explicit contexts do usually not share common characteristics (since they are in different implicit contexts). We can also note that there are 37,743 changes of implicit context which do not match with changes of session. This is not a surprising result and can be explained in a simply manner. There can exist more than one implicit context within a session. We can easily imagine the case where a user starts listening to calm and down tempo songs, and suddenly changes to energetic and rapid tempo songs within the same session. As a conclusion of these results, we can say that our model seems to perform well by detecting possibly interesting points with the navigation path, that corresponds to changes of implicit context according to our definition, and can often by confirmed by changes of explicit context (events). But, as a perspective, we need to confront these results to real users, in order to study how they perceive and accept these implicit contexts, before using them as a support for recommender systems. Also, let us remind that we can easily change every parameter of our model (weights of attributes, size of history, value of the threshold $\tau$, ...) after a learning phase, to match users’ expectations and maximize the acceptance and adoption rates. **Results as regards the second experiment (H2).** In order to understand how our model performs with a lack of data, we operated a controlled deterioration of our corpus. By controlled, we mean that the amount of missing data (that is to say missing values of attributes for the songs) was fixed for each execution. We monitored the number of sessions and implicit contexts detected, while progressively deteriorating the corpus percent after percent (see Figure \[fig:degradation\_session\]). ![Performance of our model against sparcity[]{data-label="fig:degradation_session"}](images/degradation.eps){width="50.00000%"} From Figure \[fig:degradation\_session\], we can see that the performances of our model are pretty stable until up to 60% of missing data. These results highlight the fact that our model can perform well, even with a large and realistic amount of missing data. **Results as regards the third experiment (H3).** Derived from some popular social networks like Facebook[^5], LinkedIn[^6], or Yupeek[^7], we observed that the number of different types of items was usually around 4. That is why we decided to create 4 types of items from our initial corpus. On this basis, we tested different combinations as regards the number of attributes per item $x$ and the number of common attributes $y$. For each combination, we compute the number of sessions and implicit contexts detected. The results are presented in Table \[tab:types\_differents\]. These values result from 10 executions, with the intent to limit bias due to the random selection of attributes. Indeed, according to the attributes which are selected for each type of items, the performance could vary as some attributes may be more representative than others in the detection of implicit contexts. From Table \[tab:types\_differents\], we can observe that performances are quite good even if the number of attributes per type of items $x$ is low. Moreover, the highest the number of common attributes between types of items $y$ is, the more we detect changes of session and implicit contexts. We see that the standard deviation has high values when both the number of attributes $x$ and the number of common attributes $y$ are low. This confirms that all attributes have not the same impact in detecting changes of implicit context. It can be supposed that a difference between the value of the energy of two songs is more characteristic of a change of context than a variation of the artist location. Adapting the weight of each attribute in the calculation of the relative diversity for a given item is a perspective. Conclusions and Future Work {#conclusion} =========================== Our model allows to monitor the natural diversity contained in users’ navigation path over time and, although part of an on-going research, already presents many strengths to characterize user context. First, it has a complexity in constant time since, at each time-step, we only compute relative diversity on a fixed and small history size. This makes our model highly scalable. In addition, it preserves privacy, since it does not require personal information about the active user (even if it can make use of information that other users accept to share, as shown in Figure \[fig:dance\]) and allows to forget the navigation path beyond the recent history. At last, it is generic since our equations fit any kind of attributes, and does not require an ontology to put words on the context. One of the questions addressed in this paper was to check our ability to predict changes of implicit context at time $t$, without knowing what will happen next. So as to give meaning to these implicit contexts detected by our model, we tried to find a matching with explicit factors and events such as ends of session. Our results showed that we got a significant overlap between changes of implicit contexts and ends of session. Thus, this reinforce our conviction that this model highlights interesting points within users’ navigation path. First, it allows us to anticipate ends of session, and will then be useful to adapt recommendations when users are near to reach a decision. Second, the changes of implicit context detected by our model that do not match with events are very promising results to be, on the long-term, able to formally characterize the user context and provide context-aware recommendations that fit privacy issues. Another purpose of this paper was to test the robustness of our model when confronted to sparse data. We distinguished two different scenarios where we have a single type of items with incomplete descriptions, or several types of items with small intersections of attributes. In both cases, the performances of our model remained stable in tough conditions, with about 60% of missing data. Among our perspectives, we aim at confronting our model to real users, so as to measure their perception and acceptance rate of implicit contexts. We expect to map implicit and explicit contexts so as to reach the same performances as systems based on explicit contexts, but with a deeper consideration of privacy issues. Finally, by characterizing implicit contexts, we will be able to explain recommendations based on implicit contexts and provide new interaction modes to make user decisions easier. Acknowledgements {#acknowledgements .unnumbered} ================ This work was financed by the region of Lorraine and the Urban Community of Greater Nancy, in collaboration with the Yupeek company. [1]{} G. Adomavicius and Y. Kwon. Improving aggregate recommendation diversity using ranking-based techniques. , 24(5):896–911, 2012. G. Adomavicius, B. Mobasher, F. Ricci, and A. Tuzhilin. Context-aware recommender systems. , pages 67–80, 2011. A. L’Huillier, S. Castagnos, and A. Boyer. Understanding Usages by Modeling Diversity over Time. In [*ACM Conference on User Modelling, Adaptation and Personalization (UMAP)*]{}, 2014. K. Bradley and B. Smith. Improving recommendation diversity. In [*irish Conference on Artificial Intelligence and Cognitive Science*]{}, AICS’01, pages 85–94, San Francisco, USA, 2001. S. Castagnos, A. Brun, and A. Boyer. When diversity is needed... but not expected! In [*International Conference on Advances in Information Mining and Management (IMMM)*]{}, pages 44–50, 2013. S. Castagnos, N. Jones, and P. Pu. Eye–tracking product recommenders’ usage. In [*RecSys*]{}, pages 29–36, 2010. G. Chen and L. Chen. Recommendation based on contextual opinions. In [*User Modeling, Adaptation, and Personalization, UMAP ’14*]{}, pages 61–73. Springer, 2014. L. F. Cranor. Hey, that‘s personal! In L. Ardissono, P. Bruna, and A. Mitrovic, editors, [*User Modeling 2005*]{}, volume 3538 of [*Lecture Notes in Computer Science*]{}, pages 4–4. Springer Berlin Heidelberg, 2005. M. D. Ekstrand, F. M. Harper, M. C. Willemsen, and J. A. Konstan. User perception of differences in recommender algorithms. In [*Proceedings of the 8th ACM Conference on Recommender Systems*]{}, RecSys ’14, pages 161–168, New York, USA, 2014. N. Hariri, B. Mobasher, and R. Burke. Context adaptation in interactive recommender systems. In [*Proceedings of the 8th ACM Conference on Recommender Systems*]{}, RecSys ’14, pages 41–48, New York, NY, USA, 2014. ACM. M. Hasan, A. Kashyap, V. Hristidis, and V. Tsotras. User effort minimization through adaptive diversification. In [*Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining*]{}, KDD ’14, pages 203–212, New York, NY, USA, 2014. ACM. N. Jones. . These, Ecole polytechnique de Lausanne, July 2010. M. Kaminskas, F. Ricci, and M. Schedl. Location-aware music recommendation using auto-tagging and hybrid matching. In [*Proceedings of the 7th ACM Conference on Recommender Systems*]{}, RecSys ’13, pages 17–24, New York, USA, 2013. B. P. Knijnenburg, A. Kobsa, and H. Jin. Dimensionality of information disclosure behavior. , 71(12):1144 – 1162, 2013. N. Lathia, S. Hailes, L. Capra, and X. Amatriain. Temporal diversity in recommender systems. In [*Proceedings of the 33rd International ACM SIGIR Conference on Research and Development in Information Retrieval*]{}, SIGIR ’10, pages 210–217, New York, USA, 2010. L. McGinty and B. Smyth. On the role of diversity in conversational recommender systems. In [*Proceedings of the Fifth International Conference on Case–Based Reasoning*]{}, pages 276–290. Springer, 2003. A. Said, B. Kille, J. Brijnesh, and S. Albayrak. Increasing diversity throught furhest neighbor–based recommandation. In [*Proceedings of the Workshop on Diversity in Document Retrieval*]{}, WSDM’12, Seattle, USA, 2012. C. Simpson. Amazon will sell you things before you know you want to buy them. The Wire, 2014. B. Smyth and P. McClave. Similarity vs. diversity. In [*Proceedings of the 4th International Conference on Case–Based Reasoning: Case–Based Reasoning Research and Development*]{}, ICCBR ’01, pages 347–361, London, UK, 2001. M. Zhang and N. Hurley. Avoiding monotony: Improving the diversity of recommendation lists. In [*Proceedings of the 2008 ACM Conference on Recommender Systems*]{}, RecSys ’08, pages 123–130, New York, NY, USA, 2008. ACM. C.-N. Ziegler, S. M. McNee, J. A. Konstan, and G. Lausen. Improving recommendation lists through topic diversification. In [*Proceedings of the 14th International Conference on World Wide Web*]{}, pages 22–32, New York, NY, USA, 2005. ACM. [^1]: http://developer.echonest.com/ [^2]: https://musicbrainz.org/ [^3]: http://www.lastfm.fr/ [^4]: Among other common explicit context factors such as localization, mood, people nearby and so on. [^5]: https://www.facebook.com/ [^6]: https://www.linkedin.com/ [^7]: http://yupeek.com/

--- abstract: 'We show that the technique known as concatenated continuous dynamical decoupling (CCD) can be applied to a trapped-ion setup for a robust implementation of the quantum Rabi model in a variety of parameter regimes. These include the case where the Dirac equation emerges, and the limit in which a quantum phase transition takes place. We discuss the applicability of the CCD scheme in terms of the fidelity between different initial states evolving under an ideal quantum Rabi model and their corresponding trapped-ion realization, and demonstrate the effectiveness of noise suppression of our method.' address: 'Institut für Theoretische Physik and IQST, Albert-Einstein Allee 11, Universität Ulm, 89069 Ulm, Germany' author: - 'Ricardo Puebla, Jorge Casanova and Martin B. Plenio' title: A robust scheme for the implementation of the quantum Rabi model in trapped ions --- [*Keywords*]{}: Dynamical decoupling, trapped ions, Rabi model\ Introduction ============ Quantum coherence is an essential prerequisite to observe and exploit the intriguing phenomena in the quantum realm [@Nielsen00]. Indeed, technologies relying on those quantum properties are expected to surpass their classical counterparts in efficiency and performance. This new generation of quantum technologies encompasses a large diversity of possible applications which inlcude quantum simulation [@Feynman82], quantum metrology [@Giovannetti04], quantum communication [@Gisin07] and quantum sensing [@Wu16], all of them requiring the preservation of quantum coherence for their correct functioning. In this respect, the loss of quantum coherence, or simply decoherence, is a crucial limitation as it occurs due to the unavoidable interaction of the quantum system with an uncontrolled environment as well as to the presence of experimental imperfections. Hence, the long-time maintenance of the quantum coherence of an evolving system is highly desired although its realization constitutes a formidable task. During the past decades considerable efforts have been invested in the development of theoretical schemes to circumvent, as much as possible, the effect of the noise in the system with the goal of prolonging coherence times. Among them we find techniques such as decoherence-free subspaces [@Lidar13], quantum error correction [@Lidarbook13], or dynamical decoupling [@Souza12]. These are methods designed to handle specific noise scenarios, and present different benefits concerning noise supression. In particular, dynamical decoupling constitutes a promising tool to handle non-Markovian noise, and it is the central object of study in this article. In its continuous wave configuration, the effect of dynamical decoupling corresponds to the creation of a dressed basis with an energy gap such that, under certain circumstances that will be later developed, the effect of noise is suppressed. In addition, this technique allows for a [*concatenated*]{} configuration known as *concatenated continuous decoupling* (CCD) [@Cai:12] that consists in applying concurrently different driving fields to eliminate further sources of noise, including those from imperfect driving fields themselves. Standard dynamical decoupling has been theoretically proposed in its continuous [@Bermudez:12; @Lemmer:13njp; @Cohen15; @Mikelsons15] and pulsed [@Souza12; @Carr54; @Meiboom58; @Casanova15] configurations. Furthermore, these techniques have already been used in both radio frequency and Penning traps in  [@Timonei11; @Tan13] (continuous case) and in [@UyS09; @Biercuk09; @Biercuk09bis; @Biercuk09bisbis] (pulsed case) as a method to suppress noises on the registers and to drive robust single- and two-qubit gates. Furthermore, dynamical decoupling has been used to explore different models involving spin-spin interactions [@Cohen15bis]. On the other hand, the CCD scheme has experimentally demonstrated its feasibility to preserve the coherence of an isolated nitrogen-vacancy center in diamond [@Cai:12]. However, the convenience and possible benefits of the CCD method in an ion trap platform for quantum simulation purposes has not been proven yet. In the present article we show how to apply the CCD scheme in a trapped-ion setting for a robust implementation of the paradigmatic quantum Rabi model that describes the interaction between a two-level system and one bosonic field mode. Despite of its apparent simplicity, this model exhibits a rich variety of physics, ranging from the relativistic Dirac equation [@Lamata07; @Gerritsma09; @Casanova10r; @Gerritsma11] to critical phenomena as it can undergo a second-order quantum phase transition [@Hwang:15; @Puebla:16]. We demonstrate that, within the CCD scheme, high fidelities can be achieved and maintained during long evolution times in an ion trap setup in the presence of different noise sources and realistic conditions. While an experimental verification of such scheme in an ion trap is still required, the present theoretical results are promising and open the door to the study of robust and noise-resilient trapped-ion quantum simulations. We exemplify and support by means of detailed numerics the applicability of the CCD scheme realizing the quantum Rabi model in three different parameter regimes. First, the case where the energy splitting of the two-level system matches the motional frequency and the rotating-wave approximation can be applied. In this situation the Jaynes-Cummings model [@Jaynes63] emerges and we can observe Rabi oscillations. Second, the realization of the Dirac equation [@Lamata07; @Gerritsma09; @Casanova10r; @Gerritsma11] whose main hallmark is the Zitterbewegung, and finally, the extreme parameter regime [@Casanova10] required to witness critical dynamics as a consequence of the emergence of a second-order quantum phase transition in the limit of strong coupling [@Hwang:15; @Puebla:16]. Additionally, we discuss possible drawbacks in the CCD scheme and identify particular situations where the method does not lead to an improved performance. The present article is organized as follows. In Sec. \[sec:OU\] we introduce the Orstein-Uhlenbeck stochastic process [@Orstein:30; @Wang:45], which we will use to model fluctuations in the trapped-ion setting as well as of the externally applied control fields. In Sec. \[sec:ccd\] the CCD scheme is presented and explained. Furthermore, we show how CCD adapts to trapped-ion Hamiltonians giving rise to a noise protected quantum Rabi model in Sec. \[sec:TI\], while specific examples and their numerical simulations are shown in Sec. \[sec:num\]. Finally, we summarize the main conclusions in Sec. \[sec:conc\]. Stochastic fluctuations: Orstein-Uhlenbeck process {#sec:OU} ================================================== A quantum system looses its quantum coherence due to an uncontrolled interaction with the environment. Such interaction introduces a stochastic noise or fluctuation in the system that we will model as an Orstein-Uhlenbeck (OU) stochastic process [@Orstein:30; @Wang:45; @Gillespie:96]. This effective description successfully reproduces the exponential decay of the quantum coherence due to dephasing noise as measured by Ramsey interferometry [@Wineland:98], as well as the behavior of a quantum system under fluctuations on the intensity of the applied radiation [@Cai:12]. Moreover, as we will see later on, it also allows to vary the width of the spectral density, which quantifies the amount of power per unit of frequency. In this manner the OU process can describe different noise scenarios, and thus, it has been extensively used in the literature [@Bermudez:12; @Lemmer:13njp; @Bermudez:13; @Lemmer:13]. An OU process is characterized by two parameters, namely, $\tau$ and $c$, relaxation or correlation time and diffusion constant, respectively. While the former fixes the time in which the noise is correlated, the latter is proportional to the noise amplitude. A stochastic variable $X(t)$ that obeys an OU process has an exact update formula [@Gillespie:96], $$\label{eq:OU} X(t+\Delta t)=X(t)e^{-\Delta t/\tau}+\left[\frac{c\tau}{2}\left(1-e^{-2\Delta t/\tau}\right)\right]^{1/2}N(t),$$ for an arbitrary value of $\Delta t$. The term $N(t)$ stands for a temporally uncorrelated normally distributed random variable, i.e., $\overline{N(t)}=0$ and $\overline{N(t)N(t')}=\delta(t-t')$, where the overline denotes the stochastic average. The OU process is Gaussian, and hence, fully determined by its first and second moments, $$\begin{aligned} \overline{X(t)}&=0 \\ \sigma^2[X(t)]&=\frac{c\tau}{2}\left(1-e^{-2t/\tau}\right),\end{aligned}$$ where $\sigma^2[X]$ denotes the variance of $X$, and thus, $\sigma[X]$ its standard deviation. The power spectrum or spectral density, $S_X(f)$, characterizes the nature of the noise, since it measures the amount of power per unit of frequency of $X(t)$ at a frequency $f$. The stochastic variable $X(t)$ can be written in Fourier series as $X(t)=\sum_n P_n e^{2\pi i f_n t}$ for $t\in[0,T]$ where $P_n$ are the corresponding Fourier coefficients at frequency $f_n$. Then, the spectral density can be defined in the $T\rightarrow \infty$ limit, as shown in [@Wang:45], as $S_X(f_n)=\lim_{T\rightarrow \infty} \frac{1}{T}\left|P_n \right|^2$. The spectral density will be of importance in the next section, Sec. \[sec:ccd\], for the understanding of the noise decoupling efficiency of the CCD method. Indeed, for the particular case of an OU process, $S_X(f)$ can be analytically calculated giving rise to [@Wang:45] $$\label{eq:Sxf} S_X(f)=\frac{c\tau^2}{1+4\pi^2\tau^2f^2}.$$ Therefore, the relaxation time $\tau$ sets a boundary in the frequency domain between [*white*]{} noise, i.e. $S_X(f)\propto f^0$, and [*Brownian*]{} or [*red*]{} noise, i.e. $S_X(f)\propto f^{-2}$. This *crossover* frequency $f_{cr}$ can be estimated as $S_X(f_{cr})/S_X(0)=1/2$, that is, $f_{cr}=1/(2\pi\tau)$. In Fig. \[fig:FFT\] we show a typical trajectory of an OU process for a fluctuating variable $\delta_m(t)$ and its Fourier transform. Note that $S_X(f_n)\propto \left|P_n\right|^2$. Here we are interested in magnetic-field fluctuations or simply dephasing noise, which can be written as $H=\delta_m(t)/2 \ \sigma_z$ where $\delta_m(t)$ follows Eq. (\[eq:OU\]). The coherence time of the system depends then on the properties of $\delta_m(t)$. For example, consider an initial state ${{\left|\textstyle{\uparrow}\right\rangle}_x}$ at $t=0$, i.e. $\sigma_x{{\left|\textstyle{\uparrow}\right\rangle}_x}=+{{\left|\textstyle{\uparrow}\right\rangle}_x}$, evolving under $H=\delta_m(t)/2 \ \sigma_z$, then it is easy to prove that $$\label{eq:sx} {\left\langle\textstyle{\sigma_x(t)}\right\rangle}=e^{-\frac{1}{2}\overline{\varphi^2(t)}},$$ where $\varphi(t)=\int_0^t ds \ \delta_m(s)$ is the time integral of the stochastic variable $\delta_m(t)$ and $\overline{\varphi^2(t)}$ its autocorrelation function that can be written as [@Gillespie:96] $$\label{eq:phi2} \overline{\varphi^2(t)}=c\tau^2\left[t-\tau\left(\frac{3}{2}-2e^{-t/\tau}+\frac{1}{2}e^{-2t/\tau} \right)\right].$$ The coherence time $T_2$ is defined as the time instant at which ${\left\langle\textstyle{\sigma_x(T_2)}\right\rangle}=e^{-1}$. Hence, from Eq. (\[eq:phi2\]) and (\[eq:sx\]) it follows that $$\label{eq:c} c=\frac{4e^{2T_2/\tau}}{\tau^2\left(4e^{T_2/\tau}\tau-\tau+e^{2T_2/\tau}(2T_2-3\tau) \right)}$$ that is, for a given $\tau$ and a coherence time $T_2$, the diffusion constant can be determined. Nevertheless, depending on whether the noise is fast, i.e. with short memory, meaning $\tau\ll T_2$, or slow, i.e. with long memory, which corresponds to $\tau\gtrsim T_2$, the coherence decays differently. Indeed, exponential decay is achieved when $\tau\ll T_2$ which is the typical scenario in ion traps [@Wineland:98]. In this case Eq. (\[eq:c\]) acquires a simpler form: $T_2\approx2/c\tau^2$. In contrast, for slow noise a Gaussian decay is observed. In Fig. \[fig:coherences\] we plot ${\left\langle\textstyle{\sigma_x(t)}\right\rangle}$ as a function of the evolution time $t$ for an initial state ${{\left|\textstyle{\uparrow}\right\rangle}_x}$ evolved under fast and slow noise, considering $T_2=3$ ms, $\tau=50 \ \mu$s and $\tau=5$ ms, and $c$ obtained according to Eq. (\[eq:c\]). We can observe how the numerical stochastic average ${\left\langle\textstyle{\sigma_x(t)}\right\rangle}$ agrees with the exact expression in Eq. (\[eq:sx\]). Concatenated Continuous Decoupling (CCD) {#sec:ccd} ======================================== In this section we explain the technique known as dynamical decoupling in a concatenated scheme (CCD) [@Cai:12] that corresponds to the addition of several continuous decoupling fields. Note that the use of continuous fields, not pulsed, will be maintained throughout the article. Consider a situation where the Hamiltonian is $H=\omega_0(t)/2\ \sigma_z$ where $\omega_0(t)=\omega_0+\delta_m(t)$ with $\delta_m(t)$ the stochastic fluctuation of $\omega_0$, which strongly affects the quantum coherence of the system. Then, in order to eliminate its effects a continuous driving field with Rabi frequency $\Omega$ is introduced. This situation is described by the Hamiltonian $$\label{eq:ccd1} H=\frac{\omega_0}{2}\sigma_z+\frac{\delta_m(t)}{2}\sigma_z+\Omega\cos(\omega t)\sigma_x.$$ In an interaction picture w.r.t. $\omega_0/2\sigma_z$ we have $$H^I=\frac{\delta_m(t)}{2}\sigma_z +\frac{\Omega}{2}\left[\sigma^+\left(e^{i(\omega_0+\omega)t}+e^{i(\omega_0-\omega)t}\right)+\textrm{H.c.}\right],$$ thus, selecting $\omega=\omega_0$ and invoking the rotating-wave approximation (RWA), the previous Hamiltonian (in the case $\Omega\ll\omega_0$) reads $$H^I\approx \frac{\delta_m(t)}{2}\sigma_z+\frac{\Omega}{2}\sigma_x.$$ The first term on the r.h.s of the above equation produces no transition in the basis $\left\{{{\left|\textstyle{\uparrow}\right\rangle}_x},{{\left|\textstyle{\downarrow}\right\rangle}_x}\right\}$ as long as the fluctuating term, $\delta_m(t)$, has vanishing Fourier coefficients, $|P_n|\ll 1$, in the vicinity of frequencies $f_n\approx \Omega$. In other words, to protect the system against the noise, the Rabi frequency $\Omega$ must lie in the region in which the noise spectrum is negligible. In this manner, transitions in the dressed basis $\left\{{{\left|\textstyle{\uparrow}\right\rangle}_x},{{\left|\textstyle{\downarrow}\right\rangle}_x}\right\}$ as a consequence of the stochastic term $\delta_m(t)/2 \ \sigma_z$ have an energy penalty and can be neglected. We will denote this first step as the [*first layer*]{} of protection, since only one additional driving has been introduced. From a more rigorous point of view, that noise elimination is achieved after the application of a RWA on each of the noise components as a consequence of the presence of the term $\frac{\Omega}{2} \sigma_x$. In addition, and because the RWA presents a slightly different behavior depending on the initial state of the system, the proposed method inherits its dependence. Note however the existence of certain states for which its evolution under noise and the Hamiltonian just gives rise to a global phase. For such dark states, introducing a first layer deteriorates the coherent evolution since, in the rotated basis, noisy terms are able to produce transitions. In \[subsec:QRM\] we will comment more about this scenario and show an example. Now one should also consider that the Rabi frequency $\Omega$ is not completely stable and represents another source of fluctuations, that is, $\Omega \equiv \Omega [1 + \delta_\Omega(t) ]$ with $\delta_\Omega(t)$ another stochastic fluctuation with a small amplitude. However, the CCD scheme offers the possibility to further protect the system against $\delta_\Omega(t)$ with a [*second layer*]{} by introducing one additional driving to cancel $\delta_\Omega(t)$ [@Cai:12]. In Fig. \[fig:schemeCCD\] we sketch the main idea behind the effectiveness of dynamical decoupling to cancel interfering stochastic processes. In Fig. \[fig:schemeCCD\] (b) the evolution of the coherences as a function of the evolution time is plotted for three different drivings. The success depends on the properties of the noise (a): when the Rabi frequency of the driving does not exceed the crossover frequency of the noise ($\Omega_1<f_{cr}$) no protection is achieved. On the contrary, as the Rabi frequency gets larger, $\Omega_{2,3}\gtrsim f_{cr}$, the quantum coherence is preserved during longer times since transitions due to the original noise occur with a smaller probability in the new dressed basis. This shows the crucial interplay between noise properties and driving frequencies in a dynamical decoupling scheme. Then, one can apply the same criteria to cancel further fluctuations of additional drivings fields in the CCD scheme. Note that the same techniques can be applied to other noise models that present a similar behavior, i.e. models exhibiting a spectral density that vanishes for asymptotically large frequencies. ![[Schematic representation of the CCD scheme. In (a) the normalized power spectrum of the noise is plotted. Depending on the Rabi frequency $\Omega_i$ of the additional pulse, different evolution of the coherences is observed (b). As sketched in (c), the original basis suffers dephasing. Then, if the introduced $\Omega_i$ is small compared to the characteristic frequency of the noise, there is essentially no protection, while $\Omega_i\gtrsim f_{cr}$ coherence times are enhanced significantly as the noise term $\delta(t)\sigma_z$ is not enough to produce transitions in the new dressed basis. Noise parameters are $\tau=50 \ \mu$s and $T_2=3$ ms, while the Rabi frequencies $\Omega_1=2\pi\times0.5 \ {\textrm{kHz}}$, $\Omega_2=2\pi\times 5 \ {\textrm{kHz}}$ and $\Omega_3=2\pi\times50 \ {\textrm{kHz}}$, and $\omega_0\gg\Omega_{3}$ such that a RWA can be safely applied.]{}[]{data-label="fig:schemeCCD"}](fig3){width="1\linewidth"} Trapped-ion Hamiltonian and CCD {#sec:TI} =============================== Consider a trapped-ion with internal electronic structure described by $\omega_I/2 \ \sigma_z$ and $\nu{a^{\dagger}a}$ representing the motional mode energy with $\nu$ the trap frequency. The interaction created by a laser irradiation is captured in the term $\Omega_j/2\sigma_x\left[e^{i(k\hat{x}-\omega_jt-\phi_j)}+\textrm{H.c.}\right]$. Hence, under the influence of applied radiation the trapped-ion Hamiltonian reads [@Leibfried:03] $$\label{eq:TIH} H=\frac{\omega_I}{2}\sigma_z+\nu {a^{\dagger}a}+\sum_j \frac{\Omega_j}{2}\sigma_x\left[e^{i(k_j\hat{x}-\omega_jt-\phi_j)} +\textrm{H.c.} \right].$$ where $k_j$ is the wave vector of each laser field, $\omega_j$ its frequency, $\phi_j$ an initial phase, $\Omega_j$ the Rabi frequency of the $j$th laser, and $\hat{x}$ the ion position operator. Before starting with further developments, let us introduce some typical values of the parameters in the previous equation according to the state-of-the-art in experiments with $^{40}\rm{Ca}^{+}$ [@Gerritsma09; @Gerritsma11]. Here, the axial trap frequency is $\nu = 2 \pi \times 1.36$ MHz, $\omega_I$ is on the optical regime at $729$ nm, i.e. $\omega_I =2\pi \times 4\cdot 10^{14}$ Hz, and the Rabi frequency is typically on the order of several kHz [@Gerritsma09; @Gerritsma11]. Additionally, we should consider the coherence time of the internal levels of the ions as the main limiting factor that affects to the quality of the experiments with $^{40}\rm{Ca}^+$ [@Gerritsma09; @Gerritsma11]. As we already commented this is caused by magnetic-field fluctuations which give rise to a coherence time $T_2 \approx 3$ ms, see [@Gerritsma11]. We will consider this value throughout the present article. Note however that, by using a cryogenic setup [@Brandl:16], a longer coherence time of $T_2\approx18$ ms has already been achieved. Additionally, laser-intensity fluctuations are present in any realistic ion trap experiment, while its frequency $\omega_j$ and phase $\phi_j$ can be very accurate. Although these magnetic and intensity fluctuations are the main limiting factor for the coherence time of the system, there are still another sources of noise which will be not considered here as they will produce significant effects only on time scales significantly longer than $T_2=3$ ms. In this respect, phonon dephasing has been measured with an incidence of few ${\textrm{Hz}}$ [@Kaler03]. This provides a limit of the time scale across which the dynamics can be observed, which is, approximately, two orders of magnitude larger than the one we could consider if the magnetic noise is not eliminated. Concerning the heating rate it can be estimated that, on average, one phonon is gathered in $\sim 100$ ms [@Kaler03], or in $\sim 500$ ms for a cryogenic setup [@Brandl:16]. Furthermore, the lifetime of the qubit for the D$_{5/2}$ state of $^{40}$Ca$^+$ is $\sim 1$s [@Kaler03]. Regarding the trapped-ion Hamiltonian, in the interaction picture w.r.t. $H_0=\frac{\omega_0}{2}\sigma_z+\nu {a^{\dagger}a}$, it reads $$\begin{aligned} \label{eq:Hi_orwa} H^{I}&=e^{i(\frac{\omega_I}{2}\sigma_z+\nu {a^{\dagger}a})t}H_1e^{-i(\frac{\omega_I}{2}\sigma_z+\nu {a^{\dagger}a})t}\nonumber \\ &\approx \sum_j \frac{\Omega_j}{2}\left[\sigma^+e^{i\eta_j(ae^{-i\nu t}+ae^{i\nu t})}e^{i(\omega_I-\omega_j)t-i\phi_j}+\textrm{H.c.} \right],\end{aligned}$$ where we have already performed the optical RWA, i.e., we neglect the terms that rotate at frequency $\omega_I+\omega_j$ (counter rotating terms). Since $\omega_j$ will be chosen such that $\omega_j\approx \omega_I$ and because $\Omega_j\ll \omega_I+\omega_j$, this approximation can be safely carried out. We denote $\Delta_j=\omega_I-\omega_j$, thus, choosing $\Delta=0$, $\nu$ or $-\nu$ one arrives to a [*carrier*]{}, [*red sideband*]{} or [*blue sideband*]{} interaction, respectively, when the system is adjusted to lie within the Lamb-Dicke regime ($\eta_j\sqrt{ {\left\langle\textstyle{(a+{a^{\dagger}})^2}\right\rangle} }\ll 1$). Here, the Lamb-Dicke parameter $\eta_j$ is $\eta_j=k_j x_0$ where $x_0=(2m\nu)^{-1/2}$, $m$ the mass of the ion and $\hbar=1$ throughout the whole article; thus, $\hat{x}=x_0\left(a+{a^{\dagger}}\right)$. Finally, we would like to remark that all the numerical simulations of trapped-ion Hamiltonians presented in this article have been performed after the optical RWA and without further assumptions. CCD for a single trapped-ion setup {#subsec:CCDRabi} ---------------------------------- We discuss now how to employ a CCD scheme in a single trapped-ion setup. In [@Pedernales:15] it is demonstrated that, by using two traveling waves to excite the red- and blue sideband transitions, and by setting properly the parameters $\Omega_{1,2}$, $\phi_{1,2}$ and $\omega_{1,2}$, the Rabi model can be simulated in a variety of parameter regimes which includes the Dirac equation as a particular case. However, the presence of different noise sources could significantly deteriorate its realization. Therefore, a noise-resilient implementation is desired to enhance coherence control and fidelity. For that reason, in the following we apply a CCD scheme to a single trapped-ion setup. We use the [*first layer*]{} (\[subsubsec:1layer\]) to tackle the dephasing noise as it is the main limiting factor for the coherence time of the system, while the [*second layer*]{} is introduced to handle laser-intensity fluctuations (\[subsubsec:2layer\]). ### First layer {#subsubsec:1layer} In order to achieve the Rabi model within the CCD scheme, we apply an extra laser, denoted by the subscript $a$, with the objective to introduce a term $\Omega_a \cos(\omega_I t)\sigma_x$ into the dynamics. This is accomplished by setting $\omega_a=\omega_I$ (resonant with the frequency splitting of the ion), $\phi_a=0$ and a Rabi frequency $\Omega_a\ll \omega_I$. Then, the trapped-ion Hamiltonian in a rotating frame w.r.t. $H_0=\omega_I/2 \sigma_z +\nu{a^{\dagger}a}$ and after the optical RWA reads $$\begin{aligned} \label{eq:hifirst} H^I_1=\frac{\delta_m(t)}{2}\sigma_z&+\frac{\Omega_a}{2}\left[\sigma^+e^{i\eta_a(ae^{-i\nu t}+{a^{\dagger}}e^{i\nu t})}+\textrm{H.c.} \right]+ \nonumber \\ &+\sum_{j}\frac{\Omega_j}{2}\left[\sigma^+e^{i\eta_j(ae^{-i\nu t}+{a^{\dagger}}e^{i\nu t})}e^{i(\Delta_j t-\phi_j)}+\textrm{H.c.}\right],\end{aligned}$$ where $\delta_m(t)$ follows an OU process and is responsible of the dephasing noise, $\Delta_j=\omega_I-\omega_j$ is the detuning and $\eta_j$ the Lamb-Dicke of the $j$th laser. Note that the additional laser $a$ has zero detuning, $\Delta_a=0$ which ensures a carrier interaction (i.e. a $\sigma_x$ proportional term) within the Lamb-Dicke regime, and when other terms, i.e. the ones with a linear dependence in the Lamb-Dicke parameter, can be averaged out because of the condition $\Omega_a\eta \ll \nu$. Hence, only the first term of the following expansion is considered, $$\begin{aligned} \label{eq:exp_LD} e^{i\eta(ae^{-i\nu t}+{a^{\dagger}}e^{i\nu t})}=I&+i\eta\left(ae^{-i\nu t}+{a^{\dagger}}e^{i\nu t} \right)-\nonumber\\ &-\frac{\eta^2}{2}\left(2{a^{\dagger}a}+1+a^2e^{-2i\nu t}+({a^{\dagger}})^2 e^{2i\nu t}\right)+\mathcal{O}(\eta^3).\end{aligned}$$ In this way the additional continuous driving $a$, provides a dressed spin-basis, $\left\{{{\left|\textstyle{\uparrow}\right\rangle}_x},{{\left|\textstyle{\downarrow}\right\rangle}_x}\right\}$, in which the system is protected against the magnetic-field fluctuation or dephasing noise, $\delta_m(t)/2 \sigma_z$, as long as $\Omega_a$ fulfills the criteria given in Sec. \[sec:ccd\]. Then, the magnetic-field fluctuation can be eliminated and the Hamiltonian (\[eq:hifirst\]) is $$\begin{aligned} \label{eq:H1Ieff} H^{I}_1\approx \frac{\Omega_a}{2}\sigma_x +\sum_{j}\frac{\Omega_j}{2}\left[\sigma^+e^{i\eta_j(ae^{-i\nu t}+{a^{\dagger}}e^{i\nu t})}e^{i(\Delta_j t-\phi_j)}+\textrm{H.c.}\right].\end{aligned}$$ Furthermore, choosing properly the detunings and phases, $\Delta_j$ and $\phi_j$, a tunable Rabi model can be obtained from the previous effective Hamiltonian. This can be accomplished by setting two lasers $j=1,2$ with $\Delta_1=\nu-\xi$ and $\Delta_2=-\nu+\xi$ (detuned red and blue sideband), for which only the terms at first order in $\eta$ ($\eta_{1,2}=\eta$) of the expansion in Eq. (\[eq:exp\_LD\]) survive, provided by $\xi\ll \nu$ and $\Omega_j\ll \nu$; that is, we are applying the vibrational RWA. Finally, the Rabi model is achieved when an interaction term is orthogonal to the free energy term of the two-level system, which in this case is $\sigma_x$. Therefore, it suffices to set the phases as $\phi_1=\phi_2=0$ and the Rabi frequencies $\Omega_{1,2}=\Omega$, $$\begin{aligned} H^{I}_1\approx \frac{\Omega_a}{2}\sigma_x -\frac{\Omega\eta}{2}\sigma_y\left(ae^{-i\xi t}+{a^{\dagger}}e^{i\xi t} \right).\end{aligned}$$ The previous Hamiltonian corresponds to a Rabi model in a rotating frame w.r.t. $\xi{a^{\dagger}a}$, i.e. $$\begin{aligned} \label{eq:Hi_final1} H_R= \frac{\Omega_a}{2}\sigma_x +\xi {a^{\dagger}a}-\frac{\Omega\eta}{2}\sigma_y\left(a+{a^{\dagger}}\right).\end{aligned}$$ We remark that the previous effective Hamiltonian is only valid under both optical and vibrational RWA, within the Lamb-Dicke regime and when $\Omega_a$ is such that the noise $\delta_m(t)$ has vanishing small component at that frequency. Under the same approximations, the Dirac equation can be obtained. The corresponding Hamiltonian of the $(1+1)$ Dirac equation [@Lamata07; @Casanova10r] reads $H_D=c_D \hat{p} \sigma_x +m_D c^2 \sigma_z$, where $c_D$ is the speed of light, $m_D$ the mass of the $\frac{1}{2}$-spin particle, and $\hat{p}$ the momentum operator. To realize such a Hamiltonian from Eq. (\[eq:H1Ieff\]), we select $\Delta_1=\nu$, $\Delta_2=-\nu$ (red and blue sideband), $\phi_1=3\pi/2$, $\phi_2=\pi/2$ considering $\eta_{1,2}=\eta$ and $\Omega_{1,2}=\Omega$ (together with $\Delta_a=0$ and $\phi_a=0$). Then, Eq. (\[eq:H1Ieff\]) reads $$\begin{aligned} H_1^I\approx \frac{\Omega_a}{2}\sigma_x +\eta\Omega\sigma_y\hat{p},\end{aligned}$$ where $\hat{p}=i({a^{\dagger}}-a)/2$. This is equivalent to the Dirac equation with the following parameters $c_D=\eta\Omega$ and $m_D=\Omega_a/(2\eta^2\Omega^2)$. ### Second layer {#subsubsec:2layer} Once the main source of noise, magnetic field fluctuations, is overcome by means of the first layer, the following step consists in facing laser-intensity fluctuations which can still spoil quantum coherence. The intensity of a $j$th laser is now modeled as $\Omega_j(t)=\Omega_j\left( 1+\delta_{\Omega_j}(t) \right)$, where $\Omega_j$ is the desired Rabi frequency and $\delta_{\Omega_j}(t)$ describes a small stochastic fluctuation. Such fluctuation will be present for all the lasers used in the setup. That is, the laser intensities are not completely stable, but fluctuate around its mean value $\Omega_j$. We characterize these fluctuations as an OU process with $\tau_{\Omega}=1$ ms following [@Haffner08], and an amplitude of $0.1\%$ ($p=0.001$) of the laser intensity $\Omega_j$. Thus, one can characterize this as $\sigma[\delta_\Omega]=p$, which leads to $c_{\Omega}=2p^2/\tau_{\Omega}$. Note that the laser-amplitude noise is chosen to be slow, compared to $\delta_m(t)$. This fact can be seen as a technological requirement as otherwise the noise might not be easily handled within the CCD scheme as we will discuss later on. In this way, once $\delta_m(t)/2 \ \sigma_z$ is overcome, the main fluctuation in Eq. (\[eq:H1Ieff\]) appears in the free energy term of the two-level system (i.e. as dephasing noise). Note that the rest of the Rabi frequencies, $\Omega_j$, are multiplied by a Lamb-Dicke parameter which reduces the influence of the errors introduced into the system by their fluctuating character. Therefore, we can proceed as for the first layer to deal with the term $\Omega_a\delta_{\Omega_a}(t)/2 \ \sigma_x$. To eliminate its contribution an additional continuous driving, denoted by the subscript $b$, is introduced, but with a time-dependent Rabi frequency $\Omega_b2\cos(\Omega_a t)$. The Hamiltonian describing this situation in a rotating frame w.r.t. $H_0=\omega_I/2 \sigma_z +\nu{a^{\dagger}a}$ reads $$\begin{aligned} H^I_2\approx \frac{\delta_m(t)}{2}\sigma_z&+\frac{\Omega_a}{2}\sigma_x +\frac{\Omega_a\delta_{\Omega_a}(t)}{2}\sigma_x+\nonumber \\ &+\sum_{j}\frac{\Omega_j}{2}\left[\sigma^+e^{i\eta_j(ae^{-i\nu t}+{a^{\dagger}}e^{i\nu t})}e^{i(\Delta_j t-\phi_j)}+\textrm{H.c.}\right]\nonumber \\ &+\frac{2\Omega_b\cos(\Omega_at)}{2}\left[\sigma^+e^{i\eta_b(ae^{-i\nu t}+{a^{\dagger}}e^{i\nu t})}e^{-i\phi_b}\right],\end{aligned}$$ where we have already fixed $\Delta_b=0$. By simplicity, we only write down explicitly the fluctuation $\delta_m(t)$ and $\delta_{\Omega_a}(t)$, although all the functions $\delta_{\Omega_j}(t)$ have been taken into account in our numerical simulations, see next Section. As we need an orthogonal carrier with respect to $\sigma_x$ for $\Omega_b$, we select $\phi_b=\pi/2$ which leads to $\Omega_b\cos(\Omega_at)\sigma_y$. Now we move to a rotating frame w.r.t. $\Omega_a/2 \sigma_x$ obtaining $$\begin{aligned} H^{II}_2&=e^{i\frac{\Omega_a}{2}\sigma_xt}H^I_2e^{-i\frac{\Omega_a}{2}\sigma_xt}\nonumber \\&\approx\frac{\delta_m(t)}{2}\left[\cos(\Omega_at)\sigma_z+\sin(\Omega_at)\sigma_y\right]+\frac{\Omega_a\delta_{\Omega_a}(t)}{2}\sigma_x+\nonumber \\ &+\frac{\Omega_b}{2}\left[\cos^2(\Omega_at)\sigma_y-\cos(\Omega_at)\sin(\Omega_at)\sigma_z\right]+\nonumber\\&+\sum_j\frac{\Omega_j}{2}\left[e^{i\frac{\Omega_a}{2}\sigma_xt}\sigma^+e^{-i\frac{\Omega_a}{2}\sigma_xt} e^{i\eta_j(ae^{-i\nu t}+{a^{\dagger}}e^{i\nu t})}e^{i(\Delta_j t-\phi_j)}+\textrm{H.c}\right].\end{aligned}$$ The spin raising and lowering operators have contributions of $\sigma_x$ and $\sigma_y$, i.e. $\sigma^{\pm}=\frac{1}{2}(\sigma_x\pm i\sigma_y)$, which in a rotating frame with respect to $\Omega_a/2 \ \sigma_x$ makes $\sigma_y$ to rotate at frequencies $\pm\Omega_a$ while it does not affect $\sigma_x$. We then invoke the RWA to average out those rotating terms. Note that this is valid under the assumption $\Omega_b\ll\Omega_a$. The free energy term of the effective two-level system is given now by $\sigma_y$, and hence, the new dressed spin-basis is $\left\{{\left|\textstyle{\uparrow}\right\rangle}_y,{\left|\textstyle{\downarrow}\right\rangle}_y \right\}$. In this basis the fluctuating term $\Omega_a\delta_{\Omega_a}(t)/2 \ \sigma_x$ can be depreciated following the same arguments given in Sec. \[sec:ccd\], as well as $\delta_m(t)$. Hence, the Hamiltonian can be approximated by $$\begin{aligned} \label{eq:H2IIeff} H_2^{II}\approx \frac{\Omega_b}{2}\sigma_y+\sum_j\frac{\Omega_j}{2}\left[\frac{\sigma_x}{2}e^{i\eta_j(ae^{-i\nu t}+{a^{\dagger}}e^{i\nu t})}e^{i(\Delta_j t-\phi_j)}+\textrm{H.c}\right].\end{aligned}$$ We can summarize the operating regime on the second layer as $\Omega_b\ll\Omega_a\ll\omega_I$. Additionally, $\Omega_a$ has to be large enough to ensure decoupling with respect to $\delta_m(t)$, this condition is $\Omega_a\gtrsim 1/(2\pi\tau_m)$ or, in different words, $\Omega_a$ has to be larger than the crossover frequency, see Sec. \[sec:OU\]. At the same time, and following the same arguments, $\Omega_b$ needs to handle the fluctuation $\Omega_a\delta_{\Omega_a}(t)/2 \ \sigma_x$, and hence, $\Omega_b\gtrsim 1/(2\pi\tau_{\Omega})$ which implies the relation $\tau_{\Omega_a} \gg \tau_m$. Yet, both the intensity of the noise and the RWA ($\Omega_b\ll\Omega_a$) play a decisive role to successfully apply a second layer of protection in the CCD scheme. We note that now we may use only one traveling wave to produce the Rabi-like interaction. Setting $\Delta_1=+\nu-\xi$, $\phi_1=3\pi/2$ , we arrive to $$\begin{aligned} \label{eq:H2R} H_2^{II}\approx \frac{\Omega_b}{2}\sigma_y-\frac{\Omega_1 \eta_1}{4}\sigma_x\left(ae^{-i\xi t}+{a^{\dagger}}e^{i\xi t} \right),\end{aligned}$$ after using the vibrational RWA. The previous equation is equivalent to the Rabi model in a rotating frame w.r.t. $\xi{a^{\dagger}a}$, $$\begin{aligned} H_R=\frac{\Omega_b}{2}\sigma_y+\xi {a^{\dagger}a}-\frac{\Omega \eta}{4}\sigma_x\left(a+{a^{\dagger}}\right).\end{aligned}$$ As in the case of the first layer, the Dirac equation can be realized in a straightforward manner. Choosing $\Omega_1=\Omega$, $\eta_1=\eta$, $\Delta_1=\nu$ and $\phi_1=\pi$ the Eq. (\[eq:H2IIeff\]) reduces to $$\label{eq:H2D} H_2^{II}\approx \frac{\Omega_b}{2}\sigma_y+\frac{\eta \Omega}{2}\sigma_x\hat{p},$$ which is equivalent to the Dirac Hamiltonian with $c_D=\eta\Omega/2$ and $m_D=2\Omega_b/(\eta^2\Omega^2)$. Note that the effective Hamiltonians given in Eqs. (\[eq:H2R\]) and (\[eq:H2D\]) are valid under a number of approximations, as for the first layer. Additionally, we now require $\Omega_b\ll\Omega_a$ due to a RWA, but at the same time $\Omega_b$ must be still large enough to decouple with respect to the noisy term $\Omega_a\delta_{\Omega_a}(t)\sigma_x$. Numerical results {#sec:num} ================= Here we present numerical simulations of the previous derived effective Hamiltonians. We compare the usefulness of CCD scheme in contrast to the bare realization, denoted here as [*zeroth layer*]{} (see for example [@Pedernales:15] and \[ap:1\] for a derivation), i.e., when no protection against noise is provided. We explore two physical regimes in the realized quantum Rabi model, namely, the paradigmatic resonant case to observe Rabi oscillations, and the limiting case where a quantum phase transition takes place [@Hwang:15; @Puebla:16]. Then, we present the case of the evolution of a Dirac particle. We emphasize that all the numerical simulations involving trapped-ion Hamiltonians have been carried out after the optical RWA without performing further approximations. The bare realization or [*zeroth layer*]{} is accomplished by two lasers $$\begin{aligned} \label{eq:H0sim} H_0^I=\frac{\delta_m(t)}{2}\sigma_z&+\frac{\Omega_1(1+\delta_{\Omega_1}(t))}{2}\left[\sigma^+e^{i\eta_1\left(ae^{-i\nu t}+{a^{\dagger}}e^{i\nu t} \right)}e^{i(\Delta_1t-\phi_1)}+ \textrm{H.c.}\right]+\nonumber \\&+\frac{\Omega_2(1+\delta_{\Omega_2}(t))}{2}\left[\sigma^+e^{i\eta_2\left(ae^{-i\nu t}+{a^{\dagger}}e^{i\nu t} \right)}e^{i(\Delta_2t-\phi_2)}+ \textrm{H.c.}\right],\nonumber\\\end{aligned}$$ while the first layer involves and additional laser for protection purposes, $$\begin{aligned} \label{eq:H1sim} H_1^I=\frac{\delta_m(t)}{2}\sigma_z&+\frac{\Omega_1(1+\delta_{\Omega_1}(t))}{2}\left[\sigma^+e^{i\eta_1\left(ae^{-i\nu t}+{a^{\dagger}}e^{i\nu t} \right)}e^{i(\Delta_1t-\phi_1)}+ \textrm{H.c.}\right]\nonumber \\ &+\frac{\Omega_2(1+\delta_{\Omega_2}(t))}{2}\left[\sigma^+e^{i\eta_2\left(ae^{-i\nu t}+{a^{\dagger}}e^{i\nu t} \right)}e^{i(\Delta_2t-\phi_2)}+ \textrm{H.c.}\right]\nonumber\\ &+\frac{\Omega_a(1+\delta_{\Omega_a}(t))}{2}\left[\sigma^+e^{i\eta_a\left(ae^{-i\nu t}+{a^{\dagger}}e^{i\nu t} \right)}e^{i(\Delta_at-\phi_a)}+ \textrm{H.c.}\right].\nonumber\\\end{aligned}$$ Finally, the second layer adds a time-dependent Rabi frequency, $$\begin{aligned} \label{eq:H2sim} H_2^{I}=\frac{\delta_m(t)}{2}\sigma_z&+\frac{\Omega_1(1+\delta_{\Omega_1}(t))}{2}\left[\sigma^+e^{i\eta_1\left(ae^{-i\nu t}+{a^{\dagger}}e^{i\nu t} \right)}e^{i(\Delta_1t-\phi_1)}+ \textrm{H.c.}\right]\nonumber \\ &+\frac{\Omega_a(1+\delta_{\Omega_a}(t))}{2}\left[\sigma^+e^{i\eta_a\left(ae^{-i\nu t}+{a^{\dagger}}e^{i\nu t} \right)}e^{i(\Delta_at-\phi_a)}+ \textrm{H.c.}\right]\nonumber\\ &+\frac{2\Omega_b\cos(\Omega_at)(1+\delta_{\Omega_b}(t))}{2}\left[\sigma^+e^{i\eta_b\left(ae^{-i\nu t}+{a^{\dagger}}e^{i\nu t} \right)}e^{i(\Delta_bt-\phi_b)}+ \textrm{H.c.}\right].\nonumber\\\end{aligned}$$ The effective magnetic-field fluctuation is described by $\delta_m(t)$, as shown in Sec. \[sec:OU\] and \[sec:ccd\], with parameters $\tau_m=50 \ \mu$s and $T_2=3$ ms. Note that distinct experimental setups may suffer different magnetic-field fluctuation, and thus $\tau_m$ may differ. In this respect, depending on the correlation time $\tau_m$, our scheme can be adapted to suppress magnetic-field fluctuations by setting properly the Rabi frequencies $\Omega_j$, as discussed in Sec. \[sec:ccd\]. However, for a too short noise correlation time, i.e. in the limit of Markovian noise $\tau_m/T_2\rightarrow 0$, the tunability of the simulated Rabi models using CCD scheme is reduced as the Rabi frequency must fulfill $\Omega_a>1/(2\pi\tau_m)$ to ensure decoupling. We recall that the characteristic frequency from which the spectral density starts to decay as $1/f^2$ corresponds to $f_{cr}=1/(2\pi\tau_m)$, and therefore $\Omega_a>f_{cr}$, as explained in Sec. \[sec:ccd\]. In addition, the fluctuation of the $j$th laser’s amplitude, denoted as $\delta_{\Omega_j}(t)$, is parametrized with $\tau_\Omega=1$ ms and $c_\Omega=2p^2/\tau_\Omega$ as it describes a relative amplitude fluctuation, with $p=0.1\%$. We have considered an equal noise for the lasers with intensities $\Omega_1$ and $\Omega_2$, i.e. $\delta_{\Omega_1}(t)=\delta_{\Omega_2}(t)$, while the fluctuations of the rest are completely independent. However, we also performed simulations with uncorrelated noise between $\Omega_1$ and $\Omega_2$ and no significant differences have been observed. In all the simulations, the trap frequency has been chosen as $\nu=2\pi\times1.36 \ {\textrm{MHz}}$, the Lamb-Dicke parameter as $\eta_{1,2}=0.06$ and $\eta_{a,b}=0.01$ [@Gerritsma09; @Gerritsma11]. Quantum Rabi model realization {#subsec:QRM} ------------------------------ Here we present the numerical simulations of the trapped-ion Hamiltonian realizing the quantum Rabi model to observe the paradigmatic Rabi oscillations. The simulated quantum Rabi model in the $i$th layer can be written as $$\label{eq:simR} H_{R,i}=\frac{\tilde{\Omega}_i}{2}\sigma^i_{\tiny{\textrm{TLS}}}+\tilde{\omega}_i{a^{\dagger}a}-\tilde{\lambda}_i\sigma_{\perp}^i\left(a+{a^{\dagger}}\right),$$ where $\sigma_{\tiny{\textrm{TLS}}}^i$ and $\sigma_{\perp}^i$ stand for the Pauli matrices of the free energy term of the two-level system and the orthogonal direction of the interaction, respectively. The parameters used to simulate this model using Eqs. (\[eq:H0sim\]), (\[eq:H1sim\]) and (\[eq:H2sim\]) are gathered in Table \[tab:1\], as well as their relation with the effective frequencies given in Eq. (\[eq:simR\]), $\tilde{\Omega}_i$, $\tilde{\omega}_i$ and $\tilde{\lambda}_i$. Note that $\Omega_{1,2}=\Omega$ and $\eta_{1,2}=\eta$ for zeroth and first layer. [@llll]{} ${}$&Zeroth layer&First layer&Second layer\ $\Delta_1$ & $\nu+\delta_1$& $\nu-\omega_1$&$\nu-\omega_2$\ $\Delta_2$ & $-\nu+\delta_2$& $-\nu+\omega_1$& —\ $\Delta_a$ & — & $0$ & 0\ $\Delta_b$ & — & — & 0\ $\phi_{1,2}$& $3\pi/2$ & $3\pi/2$ & $3\pi/2$\ $\phi_a$ & — & $0$ & 0\ $\phi_b$ & — & — & $\pi/2$\ $\sigma_{\tiny{\textrm{TLS}}}^i$ & $\sigma_z$ & $\sigma_x$ & $\sigma_y$\ $\sigma_{\perp}^i$ & $\sigma_x$ & $\sigma_y$ & $\sigma_x$\ $\tilde{\Omega}_i$ &$\frac{1}{2}(\delta_2+\delta_1)$ & $\Omega_a$ & $\Omega_b$\ $\tilde{\omega}_i$ & $\frac{1}{2}(\delta_2-\delta_1)$ & $\omega_1$ & $\omega_2$\ $\tilde{\lambda}_i$ & $\frac{\eta\Omega}{2}$ & $\frac{\eta\Omega}{2}$ & $\frac{\eta_1\Omega_1}{4}$\ In order to achieve the same effective model, regardless of the layer, we will introduce dimensionless constants to define a target Hamiltonian. These are $R\equiv\tilde{\Omega}_i/\tilde{\omega}_i$ and $g\equiv2\tilde{\lambda}_i/(\tilde{\omega}_i\sqrt{R})$. Hence, fixing $R$ and $g$, $H_{R,i}/\tilde{\omega}_i$ represents the same effective quantum Rabi model. We set $\tilde{\omega}_{0,1,2}=\tilde{\Omega}_{0,1,2}=2\pi\times 5 \ {\textrm{kHz}}$ to simulate a resonant case $R=1$, and a dimensionless coupling constant $g=1/4$. This implies that: (i) for $H_0^I$, i.e. for the bare realization, $\delta_2=2\pi\times10 \ {\textrm{kHz}}$, $\delta_1=0$ and $\Omega_{1,2}=2\pi\times20.83 \ {\textrm{kHz}}$; (ii) for $H_1^I$ (first layer) $\omega_1=2\pi\times 5 \ {\textrm{kHz}}$, $\Omega_a=2\pi\times 5 \ {\textrm{kHz}}$ and $\Omega_{1,2}=2\pi\times20.83 \ {\textrm{kHz}}$; (iii) for $H_2^I$ (second layer) $\omega_2=2\pi\times 5 \ {\textrm{kHz}}$, $\Omega_b=2\pi\times 5 \ {\textrm{kHz}}$ and $\Omega_a=40\Omega_b=2\pi\times200 \ {\textrm{kHz}}$, $\Omega_1=2\pi\times41.67 \ {\textrm{kHz}}$. We illustrate how CCD improves the realization of the Rabi model by means of the fidelity among the wavefunction of the ideal Rabi model, ${\left|\textstyle{\psi_{R,i}(t)}\right\rangle}$, and its noisy trapped-ion realization ${\left|\textstyle{\psi_i(t)}\right\rangle}$ for the $i$th layer of protection, which reads $$\begin{aligned} F_i(t)=\left| \left< \psi_{R,i}(t)\right| \left.\psi_i(t) \right> \right|.\end{aligned}$$ We will also compare the oscillations of the population on the excited state of the qubit which is given by ${\left\langle\textstyle{\sigma_{\tiny{\textrm{TLS}}}^i+1}\right\rangle}/2$ in both cases, ideal and the trapped ion realization with different noisy contributions. In Figs. \[fig:TIRabiCCD\] the improvement achieved by applying the CCD scheme is clearly demonstrated for two different initial states, ${\left|\textstyle{\psi(0)}\right\rangle}={\left|\textstyle{0}\right\rangle}{\left|\textstyle{\uparrow}\right\rangle}_{\tiny{\textrm{TLS}}}$ and ${\left|\textstyle{\psi(0)}\right\rangle}={\left|\textstyle{0}\right\rangle}{\left|\textstyle{\uparrow}\right\rangle}_{\perp}$, where $\sigma_{\tiny{\textrm{TLS}}}{\left|\textstyle{\uparrow}\right\rangle}_{\tiny{\textrm{TLS}}}=+{\left|\textstyle{\uparrow}\right\rangle}_{\tiny{\textrm{TLS}}}$ and $\sigma_{\perp}{\left|\textstyle{\uparrow}\right\rangle}_{\perp}=+{\left|\textstyle{\uparrow}\right\rangle}_{\perp}$. To the contrary, there are specific situations in which CCD scheme could deteriorate the desired realization. In particular, if the considered initial state is parallel to both magnetic noise, $\delta_m\sigma_z$ and Hamiltonian (i.e. when we deal with the dark state), to apply CCD scheme is counterproductive since it changes a source of noise, that originally just gives rise to a global phase, to an orthogonal noise producing transitions and distorting the dynamics. This is the case for ${\left|\textstyle{\psi(0)}\right\rangle}={\left|\textstyle{0}\right\rangle}{\left|\textstyle{\downarrow}\right\rangle}_{\tiny{\textrm{TLS}}}$ in the Rabi model when $g\ll 1$ i.e. when the Jaynes-Cummings model arises. As we see in Fig. \[fig:TIRabiCCD\_SdownZ\_F\], for $R=1$ and $g=1/4$ the fidelity of the first layer is noticeably worse that an unprotected realization, while the second layer is just as good as the original. This reveals that CCD scheme does not necessarily lead to an improved realization; it depends on several factors which have to be taken into account beforehand. Critical dynamics of the superradiant quantum phase transition in the Rabi model -------------------------------------------------------------------------------- In order to illustrate the versatility of the CCD scheme, we analyze the realization of a time-dependent Rabi Hamiltonian in the ultra-strong coupling regime. In this respect, it has been recently shown that the Rabi model (Eq. (\[eq:simR\])) undergoes a quantum phase transition in the $R=\Omega/\omega_0\rightarrow\infty$ limit at the critical point $g_c=2\lambda_c/\sqrt{\Omega\omega_0}=1$ despite of consisting only of a single two-level system and a single-mode bosonic field [@Hwang:15]. For finite $R$, critical behavior is revealed in the form of *finite-frequency* scaling functions, in an approach that is equivalent to finite-size scaling in traditional phase transitions [@Fisher:72; @Botet:82]. As shown in [@Puebla:16], the presence of the quantum phase transition can be observed with a single trapped-ion that interacts with one of its vibrational modes. This can be achieved resorting to non-equilibrium universal scaling functions [@Acevedo:14; @Puebla:16] in terms of the expectation value $\left<\sigma^i_{\tiny{\textrm{TLS}}}\right>$ of Eq. (\[eq:simR\]), which can be measured with high-fidelity in a trapped-ion system [@Myerson08; @Burrell10]. To obtain such non-equilibrium universal scaling functions one can proceed as follows. Prepare an initial state ${\left|\textstyle{\psi(0)}\right\rangle}={\left|\textstyle{0}\right\rangle}{\left|\textstyle{\downarrow}\right\rangle}_{\tiny{\textrm{TLS}}}$ at $g=0$ for a fixed $R$, such that $\sigma_{\tiny{\textrm{TLS}}}^i{\left|\textstyle{\downarrow}\right\rangle}_{\tiny{\textrm{TLS}}}=-{\left|\textstyle{\downarrow}\right\rangle}_{\tiny{\textrm{TLS}}}$, and then quench continuously in a time $\tau_Q$ the coupling constant $g$ until $g=g_c=1$ is reached. Then, at $g(\tau_Q)=1$ for a frequency ratio $R$ we calculate the quantity ${\left\langle\textstyle{\sigma_{\tiny{\textrm{TLS}}}^i}\right\rangle}_R(\tau_Q,R)=\left|\left<\psi(\tau_Q)\left|\sigma_{\tiny{\textrm{TLS}}}^i \right|\psi(\tau_Q)\right>-{\left\langle\textstyle{\sigma{\tiny{\textrm{TLS}}}^i}\right\rangle}_{GS}(R) \right|$, where ${\left\langle\textstyle{\sigma{\tiny{\textrm{TLS}}}^i}\right\rangle}_{GS}(R)$ is the ground-state expectation value of $\sigma_{\tiny{\textrm{TLS}}}^i$ at $g=1$ and $R$. The non-equilibrium universal function is found as $S(T)=R^{\mu}{\left\langle\textstyle{\sigma_{\tiny{\textrm{TLS}}}^i}\right\rangle}_R$ where $T\equiv R^{-\gamma/(\mu(1+\zeta))}\tau_Q$. The critical exponents are $\mu=2/3$, $\gamma=1$ and $\zeta=1/2$ [@Hwang:15; @Puebla:16]. Note however that the driving time $\tau_Q$ cannot be arbitrarily short since $S(T)$ is obtained assuming adiabatic dynamics away from the critical point. On the other hand, in an ion-trap realization, the duration of the dynamics to reconstruct $S(T)$ is severely restricted due to the presence of various sources of noise [@Puebla:16]. Here, by applying the CCD scheme, we offer a way to overcome these noises, which facilitates the observation of universal scaling functions, and illustrate that the CCD scheme is valid in an extreme parameter regime and even when quench dynamics is considered. Note however that, due to the large desired value of $R$, the second layer is expected to fail as $R\propto \Omega_b$ but $\Omega_b\ll\Omega_a$ is required to fulfill the RWA. Hence, for this specific case the approximations leading to the quantum Rabi model will break down. The Fig. \[fig:TIQPTuniv\] shows the universal non-equilibrium function $S(T)$ as a function of the rescaled driving time $T$. The solid black line corresponds to the ideal quantum Rabi model, while the points to the trapped-ion realization using a first layer protection with $R=50$ (circles) and $R=100$ (squares) for $0.02\leq\tau_Q\leq 8.6$ in units of $2\pi/\tilde{\omega}_i$. In the inset the results using zeroth and second layer are plotted. Observe the remarkable improvement compared to the zeroth layer, and the failure of the second layer as $\Omega_b$ becomes comparable to $\Omega_a$. The simulation parameters are $\tilde{\omega}_{0,1}=2\pi\times 1 \ {\textrm{kHz}}$, $\tilde{\omega}_2=2\pi\times400{\textrm{Hz}}$, while $\tilde{\Omega}_i=R\tilde{\omega}_i$. For the second layer $\Omega_{a}$ is set to $2\pi\times200 \ {\textrm{kHz}}$, and hence $\Omega_a/\Omega_b=10$ and $5$ for $R=50$ and $100$, respectively, which already provides evidence of the expected failure of the RWA. Additionally, the quench is attained by tuning linearly in time the laser intensities from $0$ to $\Omega_f$. For the zeroth and first layer, $\Omega_f$ results in $2\pi\times 117.8 \ {\textrm{kHz}}$ and $2\pi\times 166.7 \ {\textrm{kHz}}$ for $R=50$ and $R=100$, respectively. For the second layer $\Omega_f$ amounts to $2\pi\times 94.3 \ {\textrm{kHz}}$ and $2\pi\times 133.3 \ {\textrm{kHz}}$ for $R=50$ and $R=100$, respectively. Dirac equation realization in a trapped-ion setting --------------------------------------------------- The parameters to realize the Dirac equation, $H_{D,i}/c_D=r\sigma_{\tiny{\textrm{TLS}}}^i+\hat{p}\sigma_{\perp}^i$ with $r\equiv m_Dc_D$, using Eqs. (\[eq:H0sim\]), (\[eq:H1sim\]) and (\[eq:H2sim\]) are gathered in the Table \[tab:2\]. [@llll]{} ${}$&Zeroth layer&First layer&Second layer\ $\Delta_1$ & $\nu+\delta$& $\nu$&$\nu$\ $\Delta_2$ & $-\nu+\delta$& $-\nu$ & —\ $\Delta_a$ & — & $0$ & $0$\ $\Delta_b$ & — & — & $0$\ $\phi_{1}$& $\pi$ & $3\pi/2$ & $\pi$\ $\phi_2$ & $0$ & $\pi/2$ & —\ $\phi_a$ & — & $0$ & $0$\ $\phi_b$ & — & — & $\pi/2$\ $\sigma_{\tiny{\textrm{TLS}}}^i$ & $\sigma_z$ & $\sigma_x$ & $\sigma_y$\ $\sigma_{\perp}^i$ & $\sigma_x$ & $\sigma_y$ & $\sigma_x$\ $m_Dc_D^2$ &$\frac{\delta}{2}$ & $\frac{\Omega_a}{2}$ & $\frac{\Omega_b}{2}$\ $c_D$ & $\eta\Omega$ & $\eta\Omega$ & $\frac{\eta_1\Omega_1}{2}$\ In order to observe the paradigmatic Zitterbewegung [@Gerritsma09] we calculate the expectation value of the position operator $\hat{x}=(a+{a^{\dagger}})$ as a function of time for an initial state ${\left|\textstyle{\psi(0)}\right\rangle}$, eigenstate of $\sigma_{\perp}^i$ (in particular we consider ${\left|\textstyle{\uparrow}\right\rangle}_{\perp}$). We then set a value $m_D$ and $c_D$, or equivalently, $r$. Note that the presented scheme for first and second layer does not allow for a realization of the strict massless limit, $r=0$, since $r$ is proportional to $\Omega_{a}$ or $\Omega_b$ and $\Omega_{a,b}=0$ does not provide a protected Hamiltonian against fluctuations, while in the zeroth layer, $r$ is just proportional to the detuning $\delta$. Nevertheless, for $r>0$, CCD scheme still improves the simulated Dirac equation, as we illustrate in the following. We set $r=2$, (i) $\delta=2\pi\times5 \ {\textrm{kHz}}$, (ii) $\Omega_a=2\pi\times5 \ {\textrm{kHz}}$, (iii) $\Omega_b=2\pi\times5 \ {\textrm{kHz}}$ and $\Omega_a=2\pi\times200 \ {\textrm{kHz}}$. This implies (i) for Eq. (\[eq:H0sim\]) $\Omega_{1,2}=2\pi\times20.8 \ {\textrm{kHz}}$ and $\Delta_{1,2}=\pm\nu+\delta$, (ii) for Eq. (\[eq:H1sim\]) $\Omega_{1,2}=2\pi\times 20.8 \ {\textrm{kHz}}$ and (iii) for Eq. (\[eq:H2sim\]) $\Omega_1=2\pi\times41.7 \ {\textrm{kHz}}$. In Fig. \[fig:TIDiracCCD\] we plot the fidelity $F_{0,1,2}(t)$ (a) and position expectation value ${\left\langle\textstyle{x(t)}\right\rangle}$ (b) as a function of time. The fidelity corresponds to $F_i(t)=\left| \left<\psi_{D,i}(t)\right| \left.\psi_{i}(t) \right>\right|$, where ${\left|\textstyle{\psi_i(t)}\right\rangle}$ and ${\left|\textstyle{\psi_{D,i}(t)}\right\rangle}$ are the wave-function of the trapped-ion and ideal Dirac equation of the $i$th layer, respectively. Note that the final time corresponds to $t=3(2\pi/c_D)=2.4$ ms. The improvement is clearly shown in Fig. \[fig:TIDiracCCD\]. The second layer works worse at longer times than the first one, which is mainly due to laser-amplitude fluctuations and breakdown of RWA (note that $\Omega_a=40\Omega_b$). Nevertheless, for shorter times, the simulation of Dirac equation in the second layer is considerably enhanced. Finally we want to comment that the access to motional variables is achieved by, for example, adding a second ion to the trap and computing the time derivative of the qubit expectation value [@Gerritsma09; @Gerritsma11], see \[ap:2\] for more details. In principle, this protocol requires to prepare the ancillary ion in a certain quantum state that we will select as parallel to the magnetic noise $\delta_m(t)$. Hence, during the realization of the dynamics, this ion is not affected by external fluctuations, while, for the reconstruction of the time derivatives, a fast evolution is required. In this manner the noise will have an small incidence in the reconstruction of $\langle x(t)\rangle$. Summary {#sec:conc} ======= In the present article we demonstrate that concatenated continuous dynamical decoupling (CCD) can be applied to a trapped-ion setup for a robust realization of the quantum Rabi model. We show that the use of the CCD scheme can significantly improve the coherence times and fidelities of quantum simulations in ion-trap experiments. We exemplify this by means of numerical simulations exploiting the rich physics of the quantum Rabi model in three completely different parameter regimes. This work is supported by an Alexander von Humboldt Professorship, the EU STREP project EQUAM, the ERC Synergy grant BioQ and the CRC TRR21. The authors acknowledge support by the state of Baden-Württemberg through bwHPC and the Germany Research Foundation (DFG) through grant no INST 40/467-1 FUGG. J. C. acknowledges support to the Alexander von Humboldt foundation. Zeroth layer realization of the quantum Rabi model {#ap:1} ================================================== Here we recall briefly the procedure to realize the Rabi model and the Dirac equation without resorting to CCD scheme, as shown in [@Pedernales:15]. A tunable quantum Rabi model can be realized as follows. The trapped-ion Hamiltonian, in the rotating frame with respect to $\omega_I/2\sigma_z+\nu{a^{\dagger}a}$ and after the optical RWA, reads $$\begin{aligned} \label{eq:H0I} H_0^I=&\frac{\delta_m(t)}{2}\sigma_z+\frac{\Omega_1}{2}\left[\sigma^+e^{i\eta_1\left(ae^{-i\nu t}+{a^{\dagger}}e^{i\nu t} \right)}e^{i(\Delta_1t-\phi_1)}+ \textrm{H.c.}\right]\nonumber \\ &+\frac{\Omega_2}{2}\left[\sigma^+e^{i\eta_2\left(ae^{-i\nu t}+{a^{\dagger}}e^{i\nu t} \right)}e^{i(\Delta_2t-\phi_2)}+ \textrm{H.c.}\right].\end{aligned}$$ Now, choosing frequency detunings such that $\Delta_1=\nu+\delta_1$ $\Delta_2=-\nu+\delta_2$, together with $\Omega_{1,2}=\Omega$, $\eta_{1,2}=\eta$ and $\phi_{1,2}=3\pi/2$ we obtain $$\begin{aligned} H_0^I&=\frac{\delta_m(t)}{2}\sigma_z-\frac{\eta\Omega}{2}\left[\sigma^+\left(ae^{i\delta_1t}+{a^{\dagger}}e^{i\delta_2t}\right) +\textrm{H.c.}\right]\\ &=\frac{\delta_m(t)}{2}\sigma_z-\frac{\eta\Omega}{2}\left[(\sigma^+e^{i\tilde{\Omega}_0t}+\sigma^-e^{-i\tilde{\Omega}_0t})(ae^{-i\tilde{\omega}_0t}+{a^{\dagger}}e^{i\tilde{\omega}_0t}) \right],\end{aligned}$$ which corresponds to a Rabi model in a rotating frame with respect to $\tilde{\Omega}_0/2\sigma_z+\tilde{\omega}_0{a^{\dagger}a}$, being $\tilde{\Omega}_0=(\delta_1+\delta_2)/2$ and $\tilde{\omega}_0=(\delta_2-\delta_1)/2$. In a straightforward manner, the Dirac equation is realized when choosing $\delta_{1,2}=\delta$, $\phi_1=\pi$, $\phi_2=0$, $\eta_{1,2}=\eta$ and $\Omega_{1,2}=\Omega$. Then, the Eq. (\[eq:H0I\]) adopts the following form $$\begin{aligned} H_0^I\approx \frac{\delta_m(t)}{2}\sigma_z+\eta\Omega\left[\sigma^+e^{i\delta t}+\sigma^-e^{-i\delta t}\right]\hat{p},\end{aligned}$$ where $\hat{p}=i({a^{\dagger}}-a)/2$. The previous Hamiltonian is then equivalent to the Dirac Hamiltonian $H_D=\frac{\delta}{2}\sigma_z+\eta\Omega\sigma_x\hat{p}$ in a rotating frame with respect to $\delta/2\sigma_z$ (omitting fluctuations). Thus, $c_D=\eta\Omega$ and $m_Dc^2=\delta/2$. Measurement of vibrational operators {#ap:2} ==================================== After the system evolution within the CCD scheme we have that the final state is $|\psi(t')\rangle$. Then, we can use another ion which is initialized into the state ${{\left|\textstyle{\uparrow}\right\rangle}}$, and therefore does not suffer from the action of a noisy term $\delta_m(t)/2 \sigma_z^A$, where $\sigma_i^A$ are the Pauli operators of the ancillary ion. Hence, it does not require CCD protection. After the final time $t'$, a short evolution of time $t$ of the form $U = e^{-i\Omega t \sigma_x^A \hat{x}}$ is applied to the state ${\left|\textstyle{\psi(t')}\right\rangle}{{\left|\textstyle{\uparrow}\right\rangle}}$. Then, it is easy to demonstrate that $$\partial_t \langle \sigma_ y^A \rangle\bigg|_{t=0} = 2\Omega \langle \psi(t') | \hat{x} | \psi(t') \rangle.$$ References {#references .unnumbered} ========== [99]{} Nielsen M A and Chuang I L [*Quantum Computation and Quantum Information*]{} (Cambridge University Press, Cambridge, England, 2000) Feynman R P 1982 *Int. J. Theor. Phys.* [**21**]{} 467 Giovannetti V, Lloyd S and Maccone L 2004 *Science* [**306**]{} 1330 Gisin N and Thew R 2007 *Nature Photonics* [**1**]{} 165 Wu Y, Jelezko F, Plenio M B and Weil T 2016 *Angew. Chem. Int. Ed.* [**55**]{} 6586 Lidar D A 2012 *Adv. Chem. Phys.* [**154**]{} 295 Lidar D A and Brun T A *Quantum Error Correction* (Cambridge University Press 2013) Souza A M, Álvarez G A and Suter D 2012 *Phil. Trans. R. Soc. A* [**370**]{} 4748 Cai J M, Naydenov B, Pfeiffer R, McGuinness L P, Jahnke K D, Jelezko F, Plenio M B and Retzker A 2012 *New J. Phys.* [**14**]{} 113023 Bermudez A, Schmidt P O, Plenio M B and Retzker A 2012 *Phys. Rev. A* [**85**]{}(4) 040302 Lemmer A, Bermudez A and Plenio M B 2013 *New J. Phys.* [**15**]{} 083001 Cohen I, Weidt S, Hensinger W K and Retzker A 2015 *New J. Phys.* [**17**]{} 043008 Mikelsons G, Cohen I, Retzker A and Plenio M B 2015 *New. J. Phys.* [**17**]{} 053032 Carr H Y and Purcell E M 1954 *Phys. Rev.* [**94**]{} 630 Meiboom S and Gill D 1958 *Rev. Sci. Instrum.* [**29**]{} 688 Casanova J, Wang Z Y, Haase J F and Plenio MB 2015 *Phys. Rev. A* [**92**]{} 042304 Timoney N, Baumgart I, Johanning M, Varón A F, Plenio M B, Retzker A and Wunderlich Ch 2011 *Nature* [**476**]{} 185 Tan T R, Gaebler J P, Bowler R, Lin Y, Jost J D, Leibfried D and Wineland D J 2013 *Phys. Rev. Lett.* [**110**]{} 263002 Uys H, Biercuk M J and Bollinger J J 2009 arxiv:0904.0036 Biercuk M J, Uys H, VanDevender A P, Shiga N, Itano W M and Bollinger J J 2009 arxiv:0906.0398 Biercuk M J, Uys H, VanDevender A P, Shiga N, Itano W M and Bollinger J J 2009 *Phys. Rev. A* [**79**]{} 062324 Biercuk M J, Uys H, VanDevender A P, Shiga N, Itano W M and Bollinger J J 2009 *Nature* [**458**]{} 996 Cohen I, Richerme P, Gong Z -X, Monroe C and Retzker A 2015 *Phys. Rev. A* [**92**]{} 012334 Lamata L, León J, Schätz T and Solano E 2007 *Phys. Rev. Lett.* [**98**]{} 253005 Gerritsma R, Kirchmair G, Zähringer F, Solano E, Blatt R and Roos C F 2010 *Nature* [**463**]{} 68 Casanova J, García-Ripoll J J, Gerritsma R, Roos C F and Solano E 2010 *Phys. Rev. A* [**82**]{} 020101(R) Gerritsma R, Lanyon B P, Kirchmair G, Zähringer F, Hempel C, Casanova J, García-Ripoll J J, Solano E, Blatt R and Roos C F 2011 *Phys. Rev. Lett.* [**106**]{} 060503 Hwang M J, Puebla R and Plenio M B 2015 *Phys. Rev. Lett.* [**115**]{} 180404 Puebla R, Hwang M J, Casanova J and Plenio M B 2016 arxiv:1607.03781 Jaynes E T and Cummings F W 1963 *Proc. IEEE* [**51**]{} 89 Casanova J, Romero G, Lizuain I, García-Ripoll J J and Solano E 2010 *Phys. Rev. Lett.* [**155**]{} 263603 Uhlenbeck G E and Orstein L, 1930 *Phys. Rev.* [**36**]{} 823 Wang M C and Uhlenbeck G E, 1945 *Rev. Mod. Phys.* [**17**]{} 323 Gillespie D T 1996 *Phys. Rev. E* [**54**]{}(2) 2084 Wineland D J, Monroe C, Itano W M, Leibfried D, King B E and Meekhof D M 1998 *J. Res. Natl. Inst. Stand. Technol* [**103**]{} 259 Bermudez A, Bruderer M and Plenio M B 2013 *Phys. Rev. Lett.* [**111**]{}(4) 040601 Lemmer A, Bermudez A and Plenio M B 2015 *Proceedings of the International School of Physics “Enrico Fermi”, Course 189,* edited by M. Knoop, I. Marzoli and G. Morigi. Leibfried D, Blatt R, Monroe C and Wineland D J 2003 *Rev. Mod. Phys.* [**75**]{} 281 Brandl M F *et al* 2016 arXiv:1607.04980 Schmidt-Kaler F, Gulde S, Riebe M, Deuschle T, Kreuter A, Lancaster G, Becher C, Eschner J, Häffner H and Blatt R 2003 *J. Phys. B: At. Mol. Opt. Phys.* [**36**]{} 623 Pedernales J S, Lizuain I, Felicetti S, Romero G, Lamata L and Solano E 2015 *Scientific Reports* [**5**]{} 15472 Häffner H, Roos C F and Blatt 2008 *Physics Reports* [**469**]{} 155 Fisher M E and Barber M N, 1972 *Phys. Rev. Lett.* [**28**]{} 1516 Botet R, Jullien R and Pfeuty P, 1982 *Phys. Rev. Lett.* [**49**]{} 478 Acevedo O L, Quiroga L, Rodríguez F J and Johnson N F, 2014 *Phys. Rev. Lett.* [**112**]{} 030403 Myerson A H, Szwer D J, Webster S C, Allcock D T C, Curtis M J, Imreh G, Sherman J A, Stacey D N, Steane A M and Lucas D M, 2008 *Phys. Rev. Lett.* [**100**]{} 200502 Burrell A H, Szwer D J, Webster S C and Lucas D M, 2010 *Phys. Rev. A* [**81**]{} 040302

=15truecm =22truecm =-0.5cm [**[Two-Dimensional Seven-State Potts Model\ \ Under External Magnetic Field]{}**]{} xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx\ \ \ \ xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx The two-dimensional Potts Model with seven states under external field\ is studied using a cluster algorithm. Cluster size distribution and the fluctuations in\ the average cluster size provide helpful information on the order of phase transitions.\ Potts model [@Potts:1952] is known to have a very rich critical behaviour and considered as a testing ground for both analytical and numerical methods. Recently, the problem is mainly focused on determining the nature of the phase transition, especially the weak first-order transition occuring in this model [@Billoire:1995; @Lee:1990]. In two dimensions, the Potts model displays a second-order phase transition for the number of states $q = 2, 3$ and $4$, and first-order transition for $q \geq 5$, where transitions become stronger as $q$ increases [@Baxter:1973; @Wu:1982]. One of the most reliable methods to study first-order transitions is to observe the doubly-peaked probability distribution for energy [@Binder:1984; @Binder:1986]. At the critical point, the order parameter and the cluster size distribution should also exhibit the same behaviour. A first-order phase transition however is expected to become weaker under an increasing external field and it becomes a difficult task to recognize a double-peak in a weak first-order phase transition. Further increase in external field reduce the strength of the transition and the phase transition disappears for large external field strengths. This process is the result of two competing interactions: spins form sizable clusters in some regions because of the tendency to align in the field direction, while the thermal fluctuations result in disintegration of the clusters. The external field yields the ordering of the system at higher temperatures, shifting the critical temperature to higher values as well as reducing fluctuations in the system. On the other hand, the dynamical changes associated with a phase transition in the system are reflected in the fluctuations and variations in the cluster size. The aim in this work is to observe, by using a cluster algorithm, the dynamical behaviour of cluster size variations with respect to temperature and the external field. $\;$ In the present work, the critical behaviour of the two-dimensional Potts model with seven states under external magnetic field is studied using cluster algorithm. Equilibrium averages, fluctuations in the average cluster size and the histograms for energy, order parameter and the average cluster size are obtained as a function of the temperature for different field strengths. $\;$ The Hamiltonian of the two-dimensional Potts model is given by $${\cal H} = K \sum_{<i,j>} \delta_{\sigma_{i},\sigma_{j}} + H \sum_i \delta_{\sigma_{i},o}.$$ Here $K=J/kT$ ; where $k$ and $T$ are the Boltzmann constant and the temperature respectively, and $J$ is the magnetic interaction between spins $\sigma_{i}$ and $\sigma_{j}$, which can take values $0,1,2, ..., q-1$ for the $q$-state Potts model and $H=h/kT$ with $h$ is the external field along the orientation 0. Reader can refer to the review article by Wu [@Wu:1982] for detailed information about the model. Order of the transitions can be studied by calculating specific heat $$C=\frac {1}{kT^{2}} (<E^{2}>-<E>^{2})$$ and the Binder cumulant [@Binder:1981] $$B=1-\frac {<E^{4}>}{3<E^{2}>}$$ on finite lattices, where $E$ is the energy of the system. $\;$ The dynamical evolution of clusters and fluctuations in the observables are strongly dependent on the correlations in the system. When the temperature is high the clusters are small, and they start growing as the critical point is approached. If the correlation length in the system is finite (especially when it is shorter than the lattice size), existing large clusters may break down to smaller ones with thermal fluctuations in the system. Hence in such a system, very large and very small clusters may coexist, resulting in large fluctuations in the cluster size. If the correlation length is very large as in the case of weak first-order phase transition or infinite as in the case of a second-order transition, the existing large clusters can not easily disintegrate with thermal fluctuations. A similar effect can be seen when an external field is turned on in a system with first-order phase transition. The effect of the external magnetic field aligns the spins parallel to the field direction creating large clusters and preserve these clusters from the effects of thermal fluctuations. These considerations led us to the usage of the average cluster size, fluctuations in the cluster size, cluster size distribution as the operators to investigate the changes in the phase transition with respect to variations of temperature and external field strength. The cluster algorithm which introduces a global update by means of selecting clusters and updating all spins in the cluster proved to be very successful in eliminating critical slowing down and super critical slowing down as in the first-order transitions in systems such as Potts [@Swendsen:1987] and O(N) [@Wolf:1989] models. Studying any operator which uses information related to the clusters is extremely straightforward since this information is inherited in the cluster algorithm. The algorithm used in this work is similar to Wolf’s algorithm, with the exception that, $H$ is incorporated following Dotsenko et al [@Dotsenko:1991] and before calculating the observables, searching the clusters is continued until the total number of sites in all searched clusters is equal to or exceeds the total number of sites in the lattice. $\;$ xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxFig. 1 . [Energy histograms at the critical ]{} Fig. 2 . [Specific heat peaks vs. $H$.]{}\ [point $K_c$ for several values of $H$.]{}\ In this work, the two-dimensional, $q = 7$ Potts model with external magnetic field has been simulated on $64 \times 64$ square lattice. After thermalization with $10^{5} - 2 \times 10^{5}$ sweeps, $5 \times 10^{5} - 2 \times 10^6$ iterations are performed at different values of the coupling $K$ and the field $H$. Longer runs with up to $3 \times 10^{6}$ iterations are done near the finite-size critical value $K_{c}$ at each field strength $H$. For $H = 0.00, 0.01, 0.015, 0.016, 0.02, 0.03$ and $0.05$ the long runs are performed at $K_{c} = 1.2909, 1.2774, 1.2707, 1.2696, 1.2652, 1.2531$ and $1.2299$ respectively, where these points are chosen as the estimated peak positions of the specific heat. From these long runs we have evaluated by appropriate reweighting [@Ferrenberg:1988] the energy as a function of the temperature for each value of $H$. The continuous curves of energy, specific heat and the Binder cumulant obtained through extrapolation from one long run are in very good agreement with the data points over rather wide ranges of $K$ for each $H$ value . The point should be stressed here is that the success of an extrapolation depends on number of statistically independent configurations used, which is provided by the cluster algorithm employed here. As a first criterion to distinguish phase transition, we have checked for each value of $H$, the energy distribution at the critical temperature where the specific heat possess a peak. At $H = 0$, where seven-state Potts model exhibits first-order character, a distinct double-peak is observed. As one can see from figure 1, with increasing $H$ the double peak converges to a single peak while $H$ reaches to the value of 0.016. Our simulations indicate that $H = 0.016$ is almost at the verge of the first-order phase transition region and for larger values of $H$ we have seen no double peak behaviour but from the observation of a single gaussian energy distribution, it is hard to conclude whether the transition becomes second-order or it is totaly wiped out. $\;$ xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxFig. 3 . [Cluster size distributions for seve-]{} Fig. 4 . [Fluctuations in cluster size]{}\ [ral values of $H$.]{}\ From specific heat peaks and the Binder cumulant minima, relevant information for the phase transition may also be obtained. For a first-order phase transition, the specific heat exhibits Dirac-delta function like shape. While the system moves towards the region of softer phase transitions with increasing $H$, the peak widens as well as the height is reduced. In the case of no phase transition, the system may still be expected to possess some fluctuations in the energy. For the set of values of the increasing external field, one obtaines a series of specific heat curves which become lower and wider as one moves towards the lower valus of $K$. Instead of a polynomial fit, here we have simply joined the tips of the successive specific heat curves by straight lines and displayed it in figure 2. This most naive presentation already prevails the existence of two distinct regimes and the turnover point is at $H=0.016$ where the line of first order transitions comes to an end. Our data on the Binder cumulant minima led to the same conclusion. $\;$ In order to have more microscopic insight and demonstrate what possibly the cluster search may add to the knowledge, we studied the time evolution of average cluster size (CS), its fluctuations (FSC) and the cluster size distributions (CSD). While the average energy fluctuates within a limited range about a mean value, the size of then existing clusters may easily fluctuate between the two extremes, namely from a single spin to a cluster of lattice size. Hence it seems to us that one may draw more information from cluster formation about the dynamical changes occuring in the system. Although the clusters we consider are of the Swendsen-Wang type, since the changes in cluster size are due to the update algorithm used, the observables not dealing with the exact cluster size (like $FCS$ and $CSD$) should give correct information on the system. We have studied the time evolutions over 2 - 3 million iterations of average cluster sizes about $K_c$ for each value of the external field. These data are very much the same of the corresponding time-series for energy but more amplified and displays more insight about the grouping of spins leading to the concerned energy value. Two-state structures of first-order transitions and the long range fluctuations of second order phase transitions are more easily detectable comparing to the energy-time series. In figure 3, we show the corresponding cluster size distributions in terms of histograms over the cluster size (normalized to the volume) in the range of 0 to 1. From figure 3, one can easily follow the appearence of the two-state of small and large size clusters as for the $H=0.00$ case, almost equal weight for all sizes as for $H=0.016$ and loosing the small and large clusters in favor of medium sizes for $H > 0.016$. The point to bring into the readers attention here is that the distribution of small clusters (bins $<$ 0.1) are very different for the three cases considered. For a first-order phase transition a substantial amount of small clusters are present and they gradually disappear while the phase transition weakens. $\;$ The fluctuations $FCS$ ($FCS=<(CS)^{2}>-<CS>^{2}$) in the average cluster size are calculated for different values of $H$ , as a function of $K$. For the case of no external field, at small $K$ values (high temperatures) the cluster sizes are small, hence $FCS$ is small. As $K$ increases, the cluster sizes are getting larger, but $FCS$ increases due to the existence of the small-size clusters as well as the large ones. At the critical point, formation of the largest clusters leads to the largest value of $FCS$. One can make a similar discussion for $K > K_{c}$ (when the critical point is approached from above). When the external field is turned on, spin alignments in the field direction reduces the fluctuations and the probability of finding large and small clusters in coexistence decreases with the increasing magnetic field. $FCS$ for $H=0.00, 0.01, 0.016, 0.02$ and $0.05$ are plotted in figure 4. Joining the tips of the successive $FCS$ curves in figure 4 yields nothing but exactly what is displayed in figure 2 apart from a small shift in $K$ due to the finite size effects. As can be seen from these plots, increasing magnetic field reduces the critical temperature, as well as the peak heights. As $H$ increases, the $FCS$ curves widen and after some $H$ value, it is hard to see a distinct maximum point. At $H=0.05$ the $FCS$ curve is step function like where the phase transition is already wiped out. Besides the appearence of two different regimes on either side of the field value $H=0.016$, figure 4 also displays differences in the distinct shapes of the $FCS$ curves belonging to two regimes. We observed that the $FCS$ curves for $H > 0.016$ are continuous and a polynomial fit is possible which may be denoted as the sign of a smooth change in the average cluster size in the system. The curves for $H < 0.016$ have the shape of a spike, for which the only possible polynomial fit is to have two distinct polynomials joining at the tip. This appearent discontinuity in the rate of fluctuations in cluster size may be attributed to the two state structure of first-order phase transitions. The low temperature asymptotic values for all $H$ seem to be almost the same regardless of the order of transition. Because of small thermal fluctuations at low temperatures, the clusters freeze after they are formed, resulting in a small $FCS$ value. $\;$ In conclusion, one can see from $CSD$ and $FCS$ plots and from the energy histograms obtained near $K_{c}$ for each considered value of $H$ and from the extrapolations performed using the fitted values of the maximums of the specific heat, the first-order transition in 2D $q = 7$ Potts model seems to disappear at $H=0.016$. What we would like to bring into attention here is that the observation of the temperature dependence of fluctuations in the average cluster size and the cluster size distributions gives valuable information about the nature of the transition and this kind of investigation may enrich the physical insight when employed in studying phase transitions. $\;$ Further work is planned to study random-field and random-bond Potts models using the same algorithm. $\;$ $\star \star \star$ The support from TÜBİTAK through project TBAG-1299 is acknowledged. [20]{} POTTS R. B., [*Proc. Camb. Phil. Soc.*]{}, [**48**]{} (1952) 106. BILLOIRE A., [*Nuc. Phys.*]{} (Proc. Suppl.), [**B42**]{} (1995) 21. LEE J. and KOSTERLITZ J. M., [*Phys. Rev. Lett.*]{}, [**65**]{} (1990) 137. BAXTER R. J., [*J. Phys.*]{}, [**C6**]{} (1973) L445. WU F. Y., [*Rev. Mod. Phys.*]{}, [**54**]{} (1982) 235. BINDER K. and LANDAU D. P., [*Phys. Rev.*]{}, [**B30**]{} (1984) 1477. CHALLA M. S. S., LANDAU D. P. and BINDER K., [*Phys. Rev.*]{}, [**B34**]{} (1986) 1841. BINDER K., [*Phys. Rev. Lett.*]{}, [**47**]{} (1981) 693. SWENDSEN R. H. and WANG J. S., [*Phys. Rev. Lett.*]{}, [**58**]{} (1987) 86. WOLF U., [*Phys. Rev. Lett.*]{}, [**62**]{} (1989) 361. DOTSENKO V. S., SELKE W. and TALAPOV A. L., [*Physica*]{}, [**A170**]{} (1991) 278. FERRENBERG A.M. and SWENDSEN R. H., [*Phys. Rev. Lett.*]{}, [**61**]{} (1988) 2635. FIGURE CAPTIONS [Figure 1.]{} Energy histograms at the critical point $K_c$ for several values of $H$. [Figure 2.]{} Specific heat peaks vs. $H$. Dashline is drawn to guide the eye. [Figure 3.]{} Cluster size distributions for several values of $H$. [Figure 4.]{} Fluctuations in cluster size vs. $K$.

--- author: - 'D. Kaledin[^1]' date: '*To Yu. I. Manin, the founder, on the occasion of his 70-th birthday*' title: Cyclic homology with coefficients --- Introduction {#introduction .unnumbered} ============ Ever since it was discovered in 1982 by A. Connes [@C1] and B. Tsygan [@tsy], cyclic homology occupies a strange place in the realm of homological algebra. Normally in homological algebra problems, one expects to start from some data, such e.g. a topological space $X$, then construct some abelian category, such as the category of sheaves on $X$, and then define the cohomology of $X$ by computing the derived functors of some natural functor, such as e.g. the global sections functor $\Gamma(X,-)$. Admittedly, this is a modern formulation, but it had certainly been current already in 1982. Cyclic homology starts with an associative algebra $A$, and defines its homology groups $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A)$, but there are absolutely no derived functors in sight. Originally, $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A)$ were defined as the homology of an explicit complex, – which anyone trained to use triangulated categories cannot help but take as an insult. Later A. Connes [@C] improved on the definition by introducing the abelian category of so-called cyclic vector spaces. However, the passage from $A$ to its associated cyclic vector space $A_\#$ is still done by an explicit [*ad hoc*]{} formula. It is as if we were to know the bar-complex which computes the homology of a group, without knowing the definition of the homology of a group. This situation undoubtedly irked many people over the years, but to the best of my knowledge, no satisfactory solution has been proposed, and it may not exist – indeed, many relations to the de Rham homology notwithstanding, it is not clear whether cyclic homology properly forms a part of homological algebra at all (to the point that e.g. in [@FT] the word “homology” is not used at all for $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A)$, and it is called instead [*additive $K$-theory*]{} of $A$). In the great codification of homological algebra done in [@GM1], cyclic homology only appears in the exercises. This is not surprising, since the main unifying idea of [@GM1] is the ideology of “linearization”: homological algebra linearizes geometry, just as functional analysis used to do 50 years ago; triangulated categories and adjoint functors are modern-day versions of Banach spaces and adjoint linear operators. This has been an immensely successful and clarifying point of view, in general, but $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A)$ sticks out on a complete tangent – there is simply no natural place for it in this framework. This paper arouse as one more attempt to propose a solution to the difficulty – to find a natural triangulated category where $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(-)$ would be able to live with a certain level of comfort (and with all the standard corollaries such as the notion of cyclic homology with coefficients, the ability to compute cyclic homology by whatever resolution is convenient, not just the bar resolution, and so on). In a sense, our attempt has been successful: we define a triangulated category which can serve as the natural “category of coefficients” for cyclic homology of an algebra $A$, and we prove the comparison theorem that shows that when the coefficients are trivial, the new definition of cyclic homology is equivalent to the old one. In fact, the algebra $A$ enters into the construction only through the category $A{\operatorname{\!-\sf bimod}}$ of $A$-bimodules; we also show how to generalize the construction so that $A{\operatorname{\!-\sf bimod}}$ is replaced with a more general tensor abelian category ${{\mathcal C}}$. From a different point of view, though, out attempt failed miserably: the correspondence $A \mapsto A_\#$, being thrown out of the window, immediately returns through the door in a new and “higher-level disguise”: it is now applied not to the algebra $A$, but to the tensor category ${{\mathcal C}}= A{\operatorname{\!-\sf bimod}}$. Then in practice, the freedom to choose an arbitrary resolution to compute the derived functors leads, in our approach to $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(-)$, to complexes which are even larger than the original complex, and at some point the whole exercise starts to look pointless. Still, we believe that, all said and done, some point can be found, and some things are clarified in our approach; one such thing is, for instance, the version of Gauss-Manin connection for cyclic homology discovered by E. Getzler [@getz]. Beside, we do propose a definition of cyclic homology which makes sense for a general tensor category; and in some particular questions, even the computations can be simplified. As for the presence of the $A_\#$-construction, this might be in the nature of things, after all – not a bug of the theory, but a necessary feature. However, we best leave it to the reader to be the judge. The paper is organized as follows. In Section 1 we recall A. Connes’ second definition of cyclic homology which uses the cyclic category $\Lambda$; we also recall some facts about homology of small categories that we will need. We have tried to give only the absolute minimum – the reader not familiar with the material will have to consult the references. In Section 2 we introduce our main object: the notion of a [*cyclic bimodule*]{} over an associative algebra $A$, and the derived category of such bimodules. We also introduce cyclic homology $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M)$ with coefficients in a cyclic bimodule $M$. In Section 3 we give a very short derivation of the Gauss-Manin connection; strictly speaking, the language of cyclic bimodules is not needed for this, but we believe that it shows more clearly what is really going on. In Section 4, we show how to replace the category $A{\operatorname{\!-\sf bimod}}$ everywhere with a more general tensor abelian category ${{\mathcal C}}$. Section 5 is a postface, or a “discussion” (as they do in medical journals) – we discuss some of the further things one might (and should) do with cyclic bimodules, and how to correct some deficiencies of the theory developed in Sections 2 and 4. Acknowledgements. {#acknowledgements. .unnumbered} ----------------- In the course of this work, I have benefited greatly from discussions with A. Beilinson, E. Getzler, V. Ginzburg, A. Kuznetsov, N. Markarian, D. Tamarkin, and B. Tsygan. I am grateful to Northwestern Univeristy, where part of this work was done, and where some of the results were presented in seminars, with great indulgence from the audience towards the unfinished state they were in. And, last but not least, it is a great pleasure and a great opportunity to dedicate the paper to Yuri Ivanovich Manin on his birthday. Besides all the usual things, I would like to stress that it is the book [@GM1], – and [@GM2], to a lesser extent – which shaped the way we look at homological algebra today, at least “we” of my generation and of Moscow school. Without Manin’s decisive influence, this paper certainly would not have appeared (as in fact at least a half of the papers I ever wrote). Recollection on cyclic homology. ================================ We start by recalling, extremely briefly, A. Connes’ approach to cyclic homology, which was originally introduced in [@C] (for detailed overviews, see e.g. [@L Section 6] or [@FT Appendix]; a brief but complete exposition using the same language and notation as in this paper can be found in [@Ka Section 1]). Connes’ approach relies on the technique of homology of small categories. Fix a base field $k$. Recall that for every small category $\Gamma$, the category ${\operatorname{Fun}}(\Gamma,k)$ of functors from $\Gamma$ to the category $k{\operatorname{\it\!-Vect}}$ of $k$-vector spaces is an abelian category with enough projectives and enough injectives, with derived category ${{\mathcal D}}(\Gamma,k)$. For any object $E \in {\operatorname{Fun}}(\Gamma,k)$, the homology $H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Gamma,E)$ of the category $\Gamma$ with coefficients in $E$ is by definition the derived functor of the direct limit functor $$\displaystyle\lim_{\overset{\to}{\Gamma}}:{\operatorname{Fun}}(\Gamma,k) \to k{\operatorname{\it\!-Vect}}.$$ Analogously, the cohomology $H^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(\Lambda,E)$ is the derived functor of the inverse limit $\displaystyle\lim_{\overset{\gets}{\Gamma}}$. Equivalently, $$H^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(\Gamma,E) = {\operatorname{Ext}}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(k,E),$$ where $k \in {\operatorname{Fun}}(\Gamma,k)$ is the constant functor (all objects in $\Gamma$ go to $k$, all maps go to identity). In particular, $H^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(\Gamma,k)$ is an algebra. For any $E \in {\operatorname{Fun}}(\Gamma,k)$, the cohomology $H^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(\Gamma,E)$ and the homology $H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Gamma,E)$ are modules over $H^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(\Gamma,k)$. We also note, although it is not needed for the definition of cyclic homology, that for any functor $\gamma:\Gamma' \to \Gamma$ between two small categories, we have the pullback functor $\gamma^*:{\operatorname{Fun}}(\Gamma,k) \to {\operatorname{Fun}}(\Gamma',k)$, and for any $E \in {\operatorname{Fun}}(\Gamma,k)$, we have natural maps $$\label{dir.im} H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Gamma',\gamma^*E) \to H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Gamma,E),\qquad H^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(\Gamma,E) \to H^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(\Gamma',\gamma^*E).$$ Moreover, the pullback functor $\gamma^*$ has a left adjoint $\gamma_!:{\operatorname{Fun}}(\Gamma',k) \to {\operatorname{Fun}}(\Gamma,k)$ and a right-adjoint $f_*:{\operatorname{Fun}}(\Gamma',k) \to {\operatorname{Fun}}(\Gamma,k)$, known as the left and right Kan extensions. In general, $f_!$ is right-exact but it need not be left-exact. We will need one particular case where it is exact. Assume given a covariant functor $V:\Gamma \to {\operatorname{Sets}}$ from a small category $\Gamma$ to the category of sets, and consider the category $\Gamma'$ of pairs $\langle [a],v \rangle$ of an object $[a] \in \Gamma$ and an element $v \in V([a])$ (maps in $\Gamma'$ are those maps $\gamma:[a] \to [a']$ which send $v \in V([a])$ to $v' \in V([a'])$. Such a category is known as a [*discrete cofibration*]{} over $\Gamma$ associated to $V$, see [@SGA]. Then the Kan extension $f_!$ associated to the forgetful functor $f:\Gamma' \to \Gamma$, $\langle [a],v \rangle \mapsto [a]$ is exact, and is easy to compute: for any $E \in {\operatorname{Fun}}(\Gamma',k)$ and $[a] \in \Gamma$, we have $$\label{discr} f_!E([a]) = \bigoplus_{v \in V([a])}E(\langle [a],v \rangle).$$ Moreover, for any $E \in {\operatorname{Fun}}(\Gamma,k)$, this imediately gives the projection formula: $$\label{projj} f_!f^*E \cong E \otimes F_!k,$$ where, as before, $k \in {\operatorname{Fun}}(\Gamma',k)$ stands for the constant functor. For applications to cyclic homology, one starts with introducing the [*cyclic category*]{} $\Lambda$. This is a small category whose objects $[n]$ are numbered by positive integers $n \geq 1$. One thinks of an object $[n]$ as a circle $S^1$ with $n$ distinct marked points; we denote the set of these points by $V([n])$. The set of maps $\Lambda([n'],[n])$ from $[n']$ to $[n]$ is then the set of homotopy classes of continuous maps $f:S^1 \to S^1$ such that - $f$ has degree $1$, sends marked points to marked points, and is non-decreasing with respect to the natural cyclic order on $S^1$ (that is, if a point $a \in S^1$ lies between points $b$ and $c$ when counting clockwise, then the same is true for $f(a)$, $f(b)$ and $f(c)$). In particular, we have $\Lambda([1],[n]) = V([n])$. This topological description of the cyclic category $\Lambda$ is easy to visualize, but there are also alternative combinatorial descriptions (e.g. [@GM1 Exercize II.1.6], [@L Section 6], or [@FT A.2], retold in [@Ka Section 1.4]). All the descriptions are equivalent. Objects in ${\operatorname{Fun}}(\Lambda,k)$ are usually called [*cyclic vector spaces*]{}. The cyclic category $\Lambda$ is related to the more familiar [*simplicial category*]{} $\Delta^{opp}$, the opposite to the category $\Delta$ of finite non-empty linearly ordered sets. To understand the relation, consider the discrete cofibration $\Lambda_{[1]}/\Lambda$ associated to the functor $V:\Lambda \to {\operatorname{Sets}}$ – equivalently, $\Lambda_{[1]}$ is the category of objects $[n]$ in $\Lambda$ eqipped with a map $[1] \to [n]$. Then it is easy to check that $\Lambda_{[1]}$ is equivalent to the $\Delta^{opp}$. From now on, we will abuse the notation and identify $\Lambda_{[1]}$ and $\Delta^{opp}$. We then have a natural projection $\Delta^{opp} = \Lambda_{[1]} \to \Lambda$, $\langle [n],v \rangle \mapsto [n]$, which we denote by $j:\Delta^{opp} \to \Lambda$. For any cyclic $k$-vector space $E \in {\operatorname{Fun}}(\Lambda,k)$, we have its restriction $j^*E \in {\operatorname{Fun}}(\Delta^{opp},E)$, a simplicial vector space. One defines the cyclic homology $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(E)$ and the Hochschild homology $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}$ of $E$ by $$HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(E) {\overset{\text{\sf\tiny def}}{=}}H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Lambda,E), \qquad HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(E) {\overset{\text{\sf\tiny def}}{=}}H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Delta^{opp},j^*E).$$ By , we have a natural map $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(E) \to HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(E)$ (moreover, since $j:\Delta^{opp} \to \Lambda$ is a discrete cofibration, the Kan extension $j_!$ is exact, so that we have $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(E) \cong HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(j_!j^*E)$, and the natural map is induced by the adjunction map $j_!j^*E \to E$). It has been shown by A. Connes that this map fits into a long exact sequence $$\label{connes} \begin{CD} HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(E) @>>> HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(E) @>{u}>> HC_{{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}-2}(E) @>>>. \end{CD}$$ Here the map $u$ is the so-called [*periodicity map*]{} on $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(E)$: one shows that the algebra $H^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(\Lambda,k)$ is isomorphic to the polynomial algebra $k[u]$ in one generator $u$ of degree $2$, and the periodicity map on homology is simply the action of this generator. This allows to define a third homological invariant, the [*periodic cyclic homology*]{} $HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(E)$ – to do it, one inverts the periodicity map. For any cyclic $k$-vector space $E \in {\operatorname{Fun}}(\Lambda,k)$, the [*periodic cyclic homology*]{} of $E$ is defined by $$HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(E) = \dlim_{\overset{u}{\gets}}HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(E),$$ where $\dlim_{\gets}$ denotes the derived functor of the inverse limit $\displaystyle\lim_{\gets}$. Assume now given an associative unital algebra $A$ over $k$. To define its cyclic homology, we associate to $A$ a canonical cyclic vector space $A_\#$ in the following way. We set $A_{\#}([n]) = A^{\otimes V([n])}$, the tensor product of $n$ copies of the vector space $A$ numbered by marked points $v \in V([n])$. Then for any map $f \in \Lambda([n'],[n])$, we define $$\label{hash.def} A_\#(f) = \bigotimes_{v \in V([n])}m_{f^{-1}(v)}:A^{\otimes V([n'])} = \bigotimes_{v \in V([n])}A^{\otimes f^{-1}(v)} \to A^{\otimes V([n])},$$ where for any linearly ordered finite set $S$, $m_S:A^{\otimes S} \to A$ is the canonical multiplication map induced by the associative algebra structure on $A$ (and if $S$ is empty, we set $A^{\otimes S} = k$, and $m_S$ is the embedding of the unity). This is obviously compatible with compositions, and it is well-defined since for any $v \in V([n])$, its preimage $f^{-1} \subset V([m])$ carries a natural linear order induced by the orientation of the circle $S^1$. \[alg.def\] For any associative unital algebra $A$ over $k$, its Hochschild, cyclic and periodic cyclic homology $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A)$, $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A)$, $HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A)$ is defined as the corresponding homology of the cyclic $k$-vector space $A_\#$: $$HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A) {\overset{\text{\sf\tiny def}}{=}}HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A_\#),\quad HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A) {\overset{\text{\sf\tiny def}}{=}}HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A_\#),\quad HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(P) {\overset{\text{\sf\tiny def}}{=}}HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A_\#).$$ Cyclic bimodules. {#naive} ================= Among all the homology functors introduced in Definition \[alg.def\], Hochschild homology is the most accesible, and this is because it has another definition: for any associative unital algebra $A$ over $k$, we have $$\label{hh.def} HH_{{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}} = {\operatorname{Tor}}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}_{A^{opp} \otimes A}(A,A),$$ where ${\operatorname{Tor}}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ is taken over the algebra $A^{opp} \otimes A$ (here $A^{opp}$ denotes $A$ with the multiplication taken in the opposite direction). This has a version with coefficients: if $M$ is a left module over $A^{opp} \otimes A$, – in other words, an $A$-bimodule, – one defines Hochschild homology of $A$ with coefficients in $M$ by $$\label{hoch.coeff} HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M) = {\operatorname{Tor}}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}_{A^{opp} \otimes A}(M,A).$$ The category $A{\operatorname{\!-\sf bimod}}$ of $A$-bimodules is a unital (non-symmetric) tensor category, with tensor product $- \otimes_A -$ and the unit object $A$. Hochschild homology is a homological functor from $A{\operatorname{\!-\sf bimod}}$ to $k{\operatorname{\it\!-Vect}}$. To obtain a small category interpretation of $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M)$, one notes that for any $n,n' \geq 0$, the $A$-bimodule structure on $M$ induces a multiplication map $$A^{\otimes n} \otimes M \otimes A^{\otimes n'} \to M.$$ Therefore, if to any $\langle [n],v \rangle \in \Delta^{opp}$ we associate the $k$-vector space $$\label{M.Delta} M^\Delta_\#([n]) = M \otimes A^{\otimes (V([n]) \setminus \{v\})},$$ with $M$ filling the place corresponding to $v \in V([n])$, then make perfect sense for those maps $f:[n'] \to [n]$ which preserve the distinguished points. Thus to any $M \in A{\operatorname{\!-\sf bimod}}$, we can associate a simplicial $k$-vector space $M^\Delta_\# \in {\operatorname{Fun}}(\Delta^{opp},k)$. In the particular case $M=A$, we have $A_\#^\Delta = j^*A_\#$. \[hoch\] For any $M \in A{\operatorname{\!-\sf bimod}}$, we have a canonical isomorphism $$\label{hh.iso} HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M) \cong H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Delta^{opp},M^\Delta_\#).$$ [[*Proof.*]{}]{} It is well-known that for any simplicial $k$-vector space $E$, the homology $H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Delta^{opp},E)$ can be computed by the standard complex of $E$ (that is, the complex with terms $E([n])$ and the differential $d = \sum_i(-1)^id_i$, where $d_i$ are the face maps). In particular, $H_0(\Delta^{opp},M^\Delta_\#)$ is the cokernel of the map $d:A \otimes M \to M$ given by $d(a \otimes m) = am-ma$. The natural projection $M \to M \otimes_{A^{opp} \otimes A} A$ obviously factors through this cokernel, so that we have a natural map $$\rho_0:H_0(\Delta^{opp},M^\Delta_\#) \to HH_0(A,M).$$ Both sides of are homological functors in $M$, and $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M)$ is a universal homological functor (=the derived functor of $HH_0(A,M)$); therefore the map $\rho_0$ extends to a map $\rho_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}:H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Delta^{opp},M^\Delta_\#) \to HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M)$. To prove that $\rho_{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ is an isomorphism for any $M$, it suffices to prove it when $M$ is free over $A^{opp} \otimes A$, or in fact, when $M = A^{opp} \otimes A$. Then on one hand, $HH_0(A,M) = A$, and $HH_i(A,M) = 0$ for $i \geq 1$. And on the other hand, the standard complex associated to the simplicial $k$-vector space $(A^{opp} \otimes A)_\#^\Delta$ is just the usual bar resolution of the diagonal $A$-bimodule $A$. It is more or less obvious that for an arbitrary $M \in A{\operatorname{\!-\sf bimod}}$, $M_\#^\Delta$ does not extend to a cyclic vector space – in order to be able to define $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M)$, we have to equip the bimodule $M$ with some additional structure. To do this, we want to use the tensor structure on $A{\operatorname{\!-\sf bimod}}$. The slogan is the following: - To find a suitable category of coefficients for cyclic homology, we have to repeat the definition of the cyclic vector space $A_\# \in {\operatorname{Fun}}(\Lambda,k)$, but replace the associative algebra $A$ in this definition with the tensor category $A{\operatorname{\!-\sf bimod}}$. Let us explain what this means. First, consider an arbitrary associative unital monoidal category ${{\mathcal C}}$ with unit object $I$ (at this point, not necessarily abelian). For any integer $n$, we have the Cartesian product ${{\mathcal C}}^n = {{\mathcal C}}\times {{\mathcal C}}\times \dots \times {{\mathcal C}}$. Moreover, the product on ${{\mathcal C}}$ induces a product functor $$m:{{\mathcal C}}^n \to {{\mathcal C}},$$ where if $n=0$, we let ${{\mathcal C}}^n={\operatorname{{\sf pt}}}$, the category with one object and one morphism, and let $m:{\operatorname{{\sf pt}}}\to {{\mathcal C}}$ be the embedding of the unit object. More generally, for any finite linearly ordered set $S$ with $n$ elements, we have a product functor $m_S:{{\mathcal C}}^S \to {{\mathcal C}}$, where ${{\mathcal C}}^S = {{\mathcal C}}^n$ with multiples in the product labeled by elements of $S$. Then for any $[n],[n'] \in \Lambda$, and any $f:[n'] \to [n]$, we can define a functor $f_!:{{\mathcal C}}^{V([n'])} \to {{\mathcal C}}^{V([n])}$ by the same formula as in : $$\label{trans.f} f_! = \prod_{v \in V([n])}m_{f^{-1}(v)}:{{\mathcal C}}^{V([n'])} = \prod_{v \in V([n])}{{\mathcal C}}^{f^{-1}(v)} \to {{\mathcal C}}^{V([n])}.$$ The natural associativity isomorphism for the product on ${{\mathcal C}}$ induces natural isomorphisms $(f \circ f')_! \cong f_! \circ f'_!$, and one checks easily that they satisfy natural compatibility conditions. All in all, setting $[n] \mapsto {{\mathcal C}}^{V([n])}$, $f \mapsto f_!$ defines a weak functor (a.k.a. lax functor, a.k.a.$2$-functor, a.k.a. pseudofunctor in the original terminology of Grothendieck) from $\Lambda$ to the category of categories. Informally, we have a “cyclic category”. To work with weak functors, it is convenient to follow Grothendieck’s approach in [@SGA]. Namely, instead of considering a weak functor directly, we define a [*category*]{} ${{\mathcal C}}_\#$ in the following way: its objects are pairs $\langle [n],M_n \rangle$ of an object $[n]$ of $\Lambda$ and an object $M_n \in {{\mathcal C}}^n$, and morphisms from $\langle [n'],M_{n'} \rangle$ to $\langle [n],M_n\rangle$ are pairs $\langle f,\iota_f\rangle$ of a map $f:[n'] \to [n]$ and a bimodule map $\iota_f:f_!(M_{n'}) \to M_n$. A map $\langle f, \iota_f \rangle$ is called [*cocartesian*]{} if $\iota_f$ is an isomorphism. For the details of this construction, – in particular, for the definition of the composition of morphisms, – we refer the reader to [@SGA]. The category ${{\mathcal C}}_\#$ comes equipped with a natural forgetful projection $\tau:{{\mathcal C}}_\# \to \Lambda$, and this projection is a [*cofibration*]{} in the sense of [@SGA]. A [*section*]{} of this projection is a functor $\sigma:\Lambda \to {{\mathcal C}}_\#$ such that $\tau \circ \sigma = {\operatorname{\sf id}}$ (since $\Lambda$ is small, there is no harm in requiring that two functors from $\Lambda$ to itself are equal, not just isomorphic). These sections obviously form a category which we denote by ${\operatorname{\sf Sec}}({{\mathcal C}}_\#)$. Explicitly, an object $M_\# \in {\operatorname{\sf Sec}}({{\mathcal C}}_\#)$ is given by the following: 1. a collection of objects $M_n = M_\#([n]) \in {{\mathcal C}}^n$, and 2. a collection of transition maps $\iota_f:f_!M_{n'} \to M_n$ for any $n$, $n'$, and $f \in \Lambda([n'],[n])$, subject to natural compatibility conditions. A section $\sigma:\Lambda \to {{\mathcal C}}_\#$ is called cocartesian if $\sigma(f)$ is a cocartesian map for any $[n],[n'] \in \Lambda$ and $f:[n'] \to [n]$ – equivalently, a section is cocartesian if all the transition maps $\iota_f$ are isomorphisms. Cocartesian sections form a full subcategory ${\operatorname{\sf Sec}}_{cart}({{\mathcal C}}_\#)$ \[cycl.str\] The category ${\operatorname{\sf Sec}}_{cart}({{\mathcal C}}_\#)$ of cocartesian objects $M_\# \in {\operatorname{\sf Sec}}({{\mathcal C}}_\#)$ is equivalent to the category of the following data: 1. an object $M = M_\#([1]) \in {{\mathcal C}}$, and 2. an isomorphism $\tau: I \times M \to M \times I$ in the category ${{\mathcal C}}^2 = {{\mathcal C}}\times {{\mathcal C}}$, such that, if we denote by $\tau_{ij}$ the endomorphism of $I \times I \times M \in {{\mathcal C}}^3$ obtained by applying $\tau$ to the $i$-th and $j$-th multiple, we have $\tau_{31} \circ \tau_{12} \circ \tau_{23} = {\operatorname{\sf id}}$. [[*Proof.*]{}]{} Straghtforward and left to the reader. Thus the natural forgetfull functor ${\operatorname{\sf Sec}}_{cart}({{\mathcal C}}_\#) \to {{\mathcal C}}$, $M_\# \mapsto M_\#([1])$ is faithful: an object in ${\operatorname{\sf Sec}}_{cart}({{\mathcal C}}_\#)$ is given by $M_\#([1])$ plus some extra structure on it, and all the higher components $M_\#([n])$, $n \geq 2$, together with the transition maps $\iota_f$, can be recovered from $M_\#([1])$ and this extra structure. Return now to the abelian situation: we are given an associative unital algebra $A$ over a field $k$, and our monoidal category is ${{\mathcal C}}= A{\operatorname{\!-\sf bimod}}$, with the natural tensor product. Then for every $n$, the product $A{\operatorname{\!-\sf bimod}}^n$ has a fully faithful embedding $A{\operatorname{\!-\sf bimod}}^n \to A^{\otimes n}{\operatorname{\!-\sf bimod}}$, $M_1 \times M_2 \times \dots \times M_n \mapsto M_1 \boxtimes M_2 \boxtimes \dots \boxtimes M_n$, and one checks easily that the multiplication functors $m_S$ actually extend to right-exact functors $$m_S:A^{\otimes S}{\operatorname{\!-\sf bimod}}\to A{\operatorname{\!-\sf bimod}};$$ for instance, one can define $m_S$ as $$m_S(M) = M/\{ a_{v'}m - ma_v \mid v \in S, a \in A, m \in M \},$$ where $a_v = 1 \otimes \dots \otimes a \otimes \dots \otimes 1 \in A^{\otimes S}$ with $a$ at the $v$-th position, and $v' \in S$ is the next element after $v$. We can therefore define the cofibered category $A{\operatorname{\!-\sf bimod}}_\#/\Lambda$ with fiber $A^{\otimes V([n])}{\operatorname{\!-\sf bimod}}$ over $[n] \in \Lambda$, and transition functors $f_!$ as in . We also have the category of sections ${\operatorname{\sf Sec}}(A{\operatorname{\!-\sf bimod}}_\#)$ and the subcategory of cocartesian sections ${\operatorname{\sf Sec}}_{cart}(A{\operatorname{\!-\sf bimod}}_\#) \subset {\operatorname{\sf Sec}}(A{\operatorname{\!-\sf bimod}}_\#)$. \[sec.ab\] The category ${\operatorname{\sf Sec}}(A{\operatorname{\!-\sf bimod}}_\#)$ is a $k$-linear abelian category. [[*Sketch of a proof.*]{}]{} This is a general fact about cofibered categories; the proof is straightforward. The kernel ${\operatorname{{\sf Ker}}}\phi$ and cokernel ${\operatorname{{\sf Coker}}}\phi$ of a map $\phi:M_\# \to M'_\#$ between objects $M_\#,M'_\# \in {\operatorname{\sf Sec}}(A{\operatorname{\!-\sf bimod}}_\#)$ are taken pointwise: for every $n$, we have an exact sequence $$0 \to ({\operatorname{{\sf Ker}}}\phi)([n]) \to M_\#([n]) \overset{\phi}{\to} M'_\#([n]) \to ({\operatorname{{\sf Coker}}}\phi)([n]) \to 0.$$ The transtition maps $\iota_f$ for ${\operatorname{{\sf Ker}}}\phi$ are obtained by restriction from those for $M_\#$; for ${\operatorname{{\sf Coker}}}\phi$, one uses the fact that the functors $f_!$ are right-exact. A [*cyclic bimodule $M$*]{} over a unital associative algebra $A$ is a cocartesian section $M_\# \in {\operatorname{\sf Sec}}_{cart}(A{\operatorname{\!-\sf bimod}}_\#)$. A [*complex of cyclic bimodules $M_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}$*]{} over $A$ is an object in the derived category ${{\mathcal D}}({\operatorname{\sf Sec}}(A{\operatorname{\!-\sf bimod}}_\#))$ whose homology objects are cocartsian. Complexes of cyclic bimodules obviously form a full triangulated subcategory in ${{\mathcal D}}({\operatorname{\sf Sec}}(A{\operatorname{\!-\sf bimod}}_\#))$; consistent notation for this category would be ${{\mathcal D}}_{cart}({\operatorname{\sf Sec}}(A{\operatorname{\!-\sf bimod}}_\#))$, but for simplicity we will denote it ${{\mathcal D}}\Lambda(A{\operatorname{\!-\sf bimod}})$. We have to define complexes separately for the following reasons: 1. The category ${\operatorname{\sf Sec}}_{cart}(A{\operatorname{\!-\sf bimod}}_\#) \subset {\operatorname{\sf Sec}}(A{\operatorname{\!-\sf bimod}}_\#)$ need not be abelian – since the transition functors $f_!$ are only right-exact, the condition of being cocartesian need not be preserved when passing to kernels. 2. Even if ${\operatorname{\sf Sec}}_{cart}(A{\operatorname{\!-\sf bimod}}_\#)$ is abelian, its derived category might be much smaller than ${{\mathcal D}}\Lambda(A{\operatorname{\!-\sf bimod}})$. \[const.exa\] An extreme example of is the case $A = k$: in this case ${\operatorname{\sf Sec}}(A{\operatorname{\!-\sf bimod}}_\#)$ is just the category of cyclic vector spaces, ${\operatorname{Fun}}(\Lambda,k)$, and $E \in {\operatorname{Fun}}(\Lambda,k)$ is cocartesian if and only if $E(f)$ is invertible for any map $f:[n'] \to [n]$. One deduces easily that $E$ must be a constant functor, so that ${\operatorname{\sf Sec}}_{cart}(k{\operatorname{\!-\sf bimod}}_\#) = k{\operatorname{\it\!-Vect}}$. Then ${{\mathcal D}}\Lambda(k{\operatorname{\!-\sf bimod}})$ is the full subcategory ${{\mathcal D}}_{const}(\Lambda,k) \subset {{\mathcal D}}(\Lambda,k)$ of complexes whose homology is constant. If we were to consider $\Delta^{opp}$ instead of $\Lambda$, we would have ${{\mathcal D}}_{const}(\Delta^{opp},k) \cong {{\mathcal D}}(k{\operatorname{\it\!-Vect}})$ – since $H^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(\Delta^{opp},k) = k$, the embedding ${{\mathcal D}}(k{\operatorname{\it\!-Vect}}) \to {{\mathcal D}}(\Delta^{opp},k)$ is fully faithful, and ${{\mathcal D}}_{const}(\Delta^{opp},k)$ is its essential image. However, $H^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(\Lambda,k)$ is $k[u]$, not $k$. Therefore there are maps between constant functors in ${{\mathcal D}}(\Lambda,k)$ which do not come from maps in ${{\mathcal D}}(k{\operatorname{\it\!-Vect}})$, and the cones of these maps give objects in ${{\mathcal D}}_{const}(\Lambda,k)$ which do not come from ${{\mathcal D}}(k{\operatorname{\it\!-Vect}})$. This phenomenon is quite common in homological algebra – examples are, for instance, the triangulated category of complexes of étale sheaves with constructible homology, the category of complex of ${{\mathcal D}}$-modules with holonomic homology, or the so-called “equivariant derived category” of sheaves on a topological space $X$ acted upon by a topological group $G$ (which is not in fact the derived category of anything useful). The upshot is that it is the triangulated category ${{\mathcal D}}\Lambda(A{\operatorname{\!-\sf bimod}})$ which should be treated as the basic object, wherever categories are discussed. \[const.rem\] We note one interesting property of the category ${{\mathcal D}}_{const}(\Lambda,k)$. Fix an integer $n \geq 1$, and consider the full subcategory $\Lambda_{\leq n} \subset \Lambda$ of objects $[n'] \in \Lambda$ with $n' \leq n$. Then one can show that $H^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(\Lambda_{\leq n},k) = k[u]/u^n$, so that we have a natural exact triangle $$\label{conn.2} \begin{CD} H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Lambda_{\leq n},E^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}) @>>> HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(E^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}) @>{u^n}>> HC_{{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}+2n}(E) @>>>, \end{CD}$$ for every $E^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}\in {{\mathcal D}}_{const}(\Lambda,k)$. We note that for any $E^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}\in {{\mathcal D}}(\Lambda,k)$, extends to a spectral sequence $$\label{conn.sp} HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(E^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}})[u^{-1}] \Rightarrow HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(E),$$ where the expression on the left-hand side reads as “polynomials in one formal variable $u^{-1}$ of homological degree $2$ with coefficients in $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(E^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}})$”. Then shows that for $E^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}\in {{\mathcal D}}_{const}(\Lambda,k)$, the first $n$ differentials in depend only on the restriction of $E^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ to $\Lambda_{\leq (n+1)} \subset \Lambda$. This is useful because in practice, one is often interested only in the first differential in the spectral sequence. As in Lemma \[cycl.str\], a cyclic $A$-bimodule $M_\#$ essentially consists of an $A$-bimodule $M = M_\#([1])$ equipped with an extra structure. Explicitly, this structure is a map $\tau:A \otimes_k M \to M \otimes_k A$ which respects the $A^{\otimes 2}$-bimodule structure on both sides, and satisfies the condition $\tau_{31} \circ \tau_{12} \circ \tau_{23} = {\operatorname{\sf id}}$, as in Lemma \[cycl.str\]. Another way to view this structure is the following. One checks easily that for any cyclic $A$-bimodule $M_\#$, the restriction $j^*M_\# \in {\operatorname{Fun}}(\Delta^{opp},k)$ is canonically isomorphic to the simplicial $k$-vector space $M^\Delta_\#$ associated to the underlying $A$-bimodule $M$ as in . By adjunction, we have a natural map $$\tau_\#:j_!M^\Delta_\# \to M_\#.$$ Then $j_!M^\Delta_\#$ in this formula only depends on $M \in A{\operatorname{\!-\sf bimod}}$, and all the structure maps which turn $M$ into the cyclic bimodule $M_\#$ are collected in the map $\tau_\#$. We can now define cyclic homology with coefficients. The definition is rather tautological. We note that for any cyclic $A$-bimodule $M_\#$ – or in fact, for any $M_\# \in {\operatorname{\sf Sec}}(A{\operatorname{\!-\sf bimod}}_\#)$ – we can treat $M_\#$ as a cyclic vector space by forgetting the bimodule structure on its components $M_n$. \[cycl.def\] The [*cyclic homology $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M_\#)$ with coefficients*]{} in a cyclic $A$-bimodule $M$ is equal to $H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Lambda,M_\#)$. Of course, , being valid for any cyclic $k$-vector space, also applies to $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M_\#)$, so that we automatically get the whole package – the Connes’ exact sequence, the periodicity endomorphism, and the periodic cyclic homology $HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M)$. By Lemma \[hoch\], $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(M_\#)$ coincides with $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M)$ as defined in . Gauss-Manin connection. {#getz} ======================= To illustate the usefulness of the notion of a cyclic bimodule, let us study the behavior of cyclic homology under deformations. There are two types of deformation theory objects that one can study for an associative algebra $A$. The first is the notion of a [*square-zero extension*]{} of the algebra $A$ by a $A$-bimodule $M$. This is an associative algebra ${\widetilde}{A}$ which fits into a short exact sequence $$\begin{CD} 0 @>>> M @>{i}>> {\widetilde}{A} @>{p}>> A @>>> 0, \end{CD}$$ where $p$ is an algebra map, and $i$ is an ${\widetilde}{A}$-bimodule map, under the ${\widetilde}{A}$-bimodule structure on $M$ induced from the given $A$-bimodule structure by means of the map $p$. In other words, the multiplication on the ideal ${\operatorname{{\sf Ker}}}p \subset {\widetilde}{A}$ is trivial, so that the ${\widetilde}{A}$-bimodule structure on ${\operatorname{{\sf Ker}}}p$ is induced by an $A$-bimodule structure, and $i$ identifies the $A$-bimodule ${\operatorname{{\sf Ker}}}p$ with $M$. Square-zero extensions are classified up to an isomorphism by elements in the second Hochschild cohomology group $HH^2(A,M)$, defined as $$HH^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(A,M) = {\operatorname{Ext}}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}_{A^{opp} \otimes A}(A,M).$$ In this setting, we can consider the cyclic homology of the algebra ${\widetilde}{A}$ and compare with the cyclic homology of $A$. Th. Goodwillie’s theorem [@go] claims that if the base field $k$ has characteristic $0$, the natural map $$HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}({\widetilde}{A}) \to HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A)$$ is an isomorphism, and there is also some information on the behaviour of $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A)$. A second type of deformation theory data includes a commutative $k$-algebra $R$ with a maximal ideal ${{\mathfrak m}}\subset R$. A [*deformation*]{} $A_R$ of the algebra $A$ over $R$ is a flat associative unital algebra $A_R$ over $R$ equipped with an isomorphism $A_R/{{\mathfrak m}}\cong A$. In this case, one can form the [*relative*]{} cyclic $R$-module $A_{R\#}$ by taking the tensor products over $R$; thus we have relative homology $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A_R/R)$, $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A_R/R)$, $HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A_R/R)$. The fundamental fact discovered by E. Getzler [@getz] is that we have an analog of the Gauss-Manin connection: if ${\operatorname{Spec}}R$ is smooth, the $R$-module $HP_i(A_R/R)$ carries a canonical flat connection for every $i$. Consider now the case when $R$ is not smooth but, on the contrary, local Artin. Moreover, assume that ${{\mathfrak m}}^2=0$, so that $R$ is itself a (commutative) square-zero extension of $k$. Then a deformation $A_R$ of $A$ over $R$ is also a square-zero extension of $A$, by the bimodule $A \otimes {{\mathfrak m}}$ (${{\mathfrak m}}$ here is taken as a $k$-vector space). But this square-zero extension is special – for a general square-zero extension ${\widetilde}{A}$ of $A$ by some $M \in A{\operatorname{\!-\sf bimod}}$, there does not exist any analog of the relative cyclic $R$-module $A_{R\#} \in {\operatorname{Fun}}(\Lambda,R)$. We observe the following: the data needed to define such an analog is precisely a cyclic bimodule structure on the bimodule $M$. Namely, assume given a square-zero extension ${\widetilde}{A}$ of the algebra $A$ by some $A$-bimodule $M$, and consider the cyclic $k$-vector space ${\widetilde}{A}_\# \in {\operatorname{Fun}}(\Lambda,k)$. Let us equip ${\widetilde}{A}$ with a descreasing two-step filtration $F^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ by setting $F^1{\widetilde}{A} = M$. Then this induces a decreasing filtration $F^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ on tensor powers ${\widetilde}{A}^{\otimes n}$. Since ${\widetilde}{A}$ is square-zero, $F^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ is compatible with the multiplication maps; therefore we also have a filtration $F^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ on ${\widetilde}{A}_\#$. Consider the quotient $$\overline{A_\#} = {\widetilde}{A}_\#/F^2{\widetilde}{A}_\#.$$ One checks easily that ${\operatorname{\sf gr}}^0_F{\widetilde}{A}_\# \cong A_\#$ and ${\operatorname{\sf gr}}^1_F{\widetilde}{A}_\# \cong j_!M^\Delta_\#$ in a canonical way, so that $\overline{A_\#}$ fits into a canonical short exact sequence $$\label{A.bar} \begin{CD} 0 @>>> j_!M^\Delta_\# @>>> \overline{A_\#} @>>> A_\# @>>> 0 \end{CD}$$ of cyclic $k$-vector spaces. Now assume in addition that $M$ is equipped with a structure of a cyclic $A$-bimodule $M_\#$, so that $M^\Delta_\# \cong j^*M_\#$, and we have the structure map $\tau_\#:j_!M^\Delta_\# \to M_\#$. Then we can compose the extension with the map $\tau_\#$, to obtain a commutative diagram $$\label{A.hat} \begin{CD} 0 @>>> j_!M^\Delta_\# @>>> \overline{A_\#} @>>> A_\# @>>> 0\\ @. @V{\tau_\#}VV @VVV @|\\ 0 @>>> M_\# @>>> {\widehat}{A_\#} @>>> A_\# @>>> 0 \end{CD}$$ of short exact sequences in ${\operatorname{Fun}}(\Lambda,k)$, with cartesian left square. It is easy to check that when ${\widetilde}{A} = A_R$ for some square-zero $R$, so that $M = A \otimes {{\mathfrak m}}$, and we take the cyclic $A$-bimodule structure on $M$ induced by the tautological structure on $A$, then ${\widehat}{A_\#}$ coincides precisely with the relative cyclic object $A_{R\#}$ (which we consider as a $k$-vector space, forgetting the $R$-module structure). We believe that this is the proper generality for the Getzler connection; in this setting, the main result reads as follows. \[spl\] Assume given a square-zero extension ${\widetilde}{A}$ of an associative algebra $A$ by an $A$-bimodule $M$, and assume that $M$ is equipped with a structure of a cyclic $A$-bimodule. Then the long exact sequence $$\begin{CD} HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M) @>>> HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}({\widehat}{A_\#}) @>>> HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A) @>>> \end{CD}$$ of periodic cyclic homology induced by the second row in admits a canonical splitting $HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A) \to HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}({\widehat}{A_\#})$. [[*Proof.*]{}]{} By definition, we have two natural maps $$\label{comp} \begin{aligned} HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\overline{A_\#}) &\to HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A_\#) = HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A),\\ HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\overline{A_\#}) &\to HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}({\widehat}{A_\#}), \end{aligned}$$ and the cone of the first map is isomorphic to $HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(j_!M^\Delta_\#)$. Since $j_!$ is exact, we have $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(j_!M^\Delta_\#) \cong HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(M_\#)$, and the periodicity map $u:HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(j_!M^\Delta_\#) \to HC_{{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}-2}(j_!M^\Delta_\#)$ is equal to $0$, so that $HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(j_!M^\Delta_\#) = 0$. Thus the first map in is an isomorphism, and the second map is then the required splitting. Assume given a commutative $k$-algebra $R$ with a maximal ideal ${{\mathfrak m}}\subset R$, and a deformation $A_R$ of the algebra $A$ over $R$. Then if ${\operatorname{Spec}}R$ is smooth, the $R$-modules $HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A_R/R)$ carry a natural connection. [[*Sketch of a proof.*]{}]{} Consider the $R \otimes R$-algebras $A_R \otimes R$ and $R \otimes A_R$, and their restrictions to the first infinitesemal neighborhood of the diagonal in ${\operatorname{Spec}}(R \otimes R) = {\operatorname{Spec}}R \times {\operatorname{Spec}}R$. Then Proposition \[spl\], suitably generalized, shows that $HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(-)$ of these two restrictions are canonically isomorphic. It is well-known that giving such an isomorphism is equivalent to giving a connection on $HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A_R/R)$. We note that we do not claim that the connection is [*flat*]{}. It certainly is, at least in characteristic $0$; but our present method does not allow one to go beyond square-zero extensions. Thus we cannot analyse the second infinitesemal neighborhood of the diagonal in ${\operatorname{Spec}}(R \otimes R)$, and we cannot prove flatness. Unfortunately, at present, we do not understand what is the proper cyclic bimodule context for higher-level infinitesemal extensions. Of course, if one is only interested in an $R$-deformation ${\widetilde}{A} = A_R$ over an Artin local base $R$, not in its cyclic bimodule generalizations, one can use Goodwillie’s Theorem: using the full cyclic object ${\widetilde}{A}_\#$ instead of its quotient $\overline{A_\#}$ in Proposition \[spl\] immediately gives a splitting $HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A) \to HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A_R/R)$ of the augmentation map $HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A_R/R) \to HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A)$, and this extends by $R$-linearity to an isomorphism $HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A_R/R) \cong HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A) \otimes R$. However, this is not quite satisfactory from the conceptual point of view, and it does not work in positive characteristic (where Goodwillie’s Theorem is simply not true). If ${\operatorname{\sf char}}k \neq 2$, the latter can be cured by using ${\widetilde}{A}_\#/F^3{\widetilde}{A}_\#$, but the former remains. We plan to return to this elsewhere. Categorical approach. {#cat} ===================== Let us now try to define cyclic homology in a more general setting – we will attempt to replace $A{\operatorname{\!-\sf bimod}}$ with an arbitrary associative unital $k$-linear tensor category ${{\mathcal C}}$ with a unit object ${\operatorname{{\sf I}}}\in {{\mathcal C}}$. We do not assume that ${{\mathcal C}}$ is symmetric in any way. However, we will assume that the tensor product $- \otimes -$ is right-exact in each variable, and we will need to impose additional technical assumptions later on. The first thing to do is to try to define Hochschild homology; so, let us look more closely at . The formula in the right-hand side looks symmetric, but this is an optical illusion – the two copies of $A$ are completely different objects: one is a left module over $A^{opp} \otimes A$, and the other is a right module ($A$ just happens to have both structures at the same time). It is better to separate them and introduce the functor $${\operatorname{\sf tr}}:A{\operatorname{\!-\sf bimod}}\to k{\operatorname{\it\!-Vect}}$$ by ${\operatorname{\sf tr}}(M) = M \otimes_{A^{opp} \otimes A} A$ – or, equivalently, by $$\label{tr.A} {\operatorname{\sf tr}}(M) = M/\{ am-ma \mid a \in A, m \in M \}.$$ Then ${\operatorname{\sf tr}}$ is a right-exact functor, and we have $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M) = L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}(M)$. We want to emphasize that the functor ${\operatorname{\sf tr}}$ can not be recovered from the tensor structure on $A{\operatorname{\!-\sf bimod}}$ – this really is an extra piece of data. For a general tensor category ${{\mathcal C}}$, it does not exist a priori; we have to impose it as an additional structure. Let us axiomatize the situation. First, forget for the moment about the $k$-linear and abelian structure on ${{\mathcal C}}$ – let us treat it simply as a monoidal category. Assume given some other category ${{\mathcal B}}$ and a functor $T:{{\mathcal C}}\to {{\mathcal B}}$. \[trace.defn\] The functor $T:{{\mathcal C}}\to {{\mathcal B}}$ is a [*trace functor*]{} if it is extended to a functor ${{\mathcal C}}_\# \to {{\mathcal B}}$ which sends any cocartesian map $f:M \to M'$ in ${{\mathcal C}}_\#$ to an invertible map. Another way to say the same thing is the following: the categories ${\operatorname{Fun}}({{\mathcal C}}^n,{{\mathcal B}})$ of functors from ${{\mathcal C}}^n$ to ${{\mathcal B}}$ form a fibered category over $\Lambda$, and a trace functor is a cartesian section of this fibration. Explicitly, a trace functor is defined by $T:{{\mathcal C}}\to {{\mathcal B}}$ and a collection of isomorphisms $$T(M \otimes M') \to T(M' \otimes M)$$ for any $M,M' \in {{\mathcal C}}$ which are functorial in $M$ and $M'$ and satisfy some compatibility conditions analogous to those in Lemma \[cycl.str\]; we leave it to the reader to write down these conditions precisely. Thus $T$ has a trace-like property with respect to the product in ${{\mathcal C}}$, and this motivates our terminology. Recall now that ${{\mathcal C}}$ is a $k$-linear abelian category. To define Hochschild homology, we have to assume that it is equipped with a right-exact trace functor ${\operatorname{\sf tr}}:{{\mathcal C}}\to k{\operatorname{\it\!-Vect}}$; then for any $M \in {{\mathcal C}}$, we set $$\label{hh.def.gen} HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(M) = L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}(M).$$ The functor ${\operatorname{\sf tr}}:A{\operatorname{\!-\sf bimod}}\to k{\operatorname{\it\!-Vect}}$ canonically extends to a right-exact trace functor in the sense of Definition \[trace.defn\]. [[*Proof.*]{}]{} For any object $\langle [n],M_n \rangle \in A{\operatorname{\!-\sf bimod}}_\#$, $[n] \in \Lambda$, $M_n \in A^{\otimes n}{\operatorname{\!-\sf bimod}}$, let $${\operatorname{\sf tr}}(\langle [n],M_n \rangle) = M_n/\{ a_{v'}m - ma_v \mid v \in V([n]), m \in M_n, a \in A \},$$ where $a_v = 1 \otimes 1 \otimes \dots \otimes a \otimes \dots \otimes 1 \in A^{\otimes V([n])}$ has $a$ in the multiple corresponding to $v \in V([n])$, and $v' \in V([n])$ is the next marked point after $v$ counting clockwise. The compatibility with maps in the category $A{\operatorname{\!-\sf bimod}}_\#$ is obvious. We note that here, in the case ${{\mathcal C}}=A{\operatorname{\!-\sf bimod}}$, the category $A{\operatorname{\!-\sf bimod}}_\#$ is actually larger than what we would have had purely from the monoidal structure on ${{\mathcal C}}$: $M_n$ is allowed to be an arbitrary $A^{\otimes n}$-bimodule, not a collection of $n$ $A$-bimodules. To do the same for general $k$-linear ${{\mathcal C}}$, we need to replace $A^{\otimes n}{\operatorname{\!-\sf bimod}}$ with some version of the tensor product ${{\mathcal C}}^{\otimes n}$. Here we have a difficulty: for various technical reasons, it is not clear how to define tensors products for sufficiently general abelian categories. One way around it is the following. For any (small) $k$-linear abelian category ${{\mathcal B}}$, a $k$-linear functor ${{\mathcal B}}^{opp} \to k{\operatorname{\it\!-Vect}}$ is left-exact if and only if it is a sheaf for for the canonical Grothendieck topology on ${{\mathcal B}}$ ([@BD 5, §10]); the category ${\operatorname{Shv}}({{\mathcal B}})$ of such functors is abelian and $k$-linear, and ${{\mathcal B}}$ itself is naturally embedded into ${\operatorname{Shv}}({{\mathcal B}})$ by Yoneda. The embedding is a fully faithfull exact functor. Every functor in ${\operatorname{Shv}}({{\mathcal B}})$ is in fact a direct limit of representable functors, so that ${\operatorname{Shv}}({{\mathcal B}})$ is an inductive completion of the abelian category ${{\mathcal B}}$. Now, if are given two (small) $k$-linear abelian categories ${{\mathcal B}}_1$, ${{\mathcal B}}_2$, then their product ${{\mathcal B}}_1 \times {{\mathcal B}}_2$ is no longer abelian. However, we still have the abelian category ${\operatorname{Shv}}({{\mathcal B}}_1 \times {{\mathcal B}}_2)$ of bilinear functors ${{\mathcal B}}_1^{opp} \times {{\mathcal B}}_2^{opp} \to k{\operatorname{\it\!-Vect}}$ which are left-exact in each variable, and the same goes for polylinear functors. Moreover, for any right-exact functor $F:{{\mathcal B}}_1 \to {{\mathcal B}}_2$ between small abelian categories, we have the restriction functor $F^*:{\operatorname{Shv}}({{\mathcal B}}_2) \to {\operatorname{Shv}}({{\mathcal B}}_1)$, which is left-exact, and its left-adjoint $F_!:{\operatorname{Shv}}({{\mathcal B}}_1) \to {\operatorname{Shv}}({{\mathcal B}}_2)$, which is right-exact. The functor $F_!$ is an extension of the functor $F$: on Yoneda images ${{\mathcal B}}_i \subset {\operatorname{Shv}}({{\mathcal B}}_i)$, we have $F_! = F$. And, again, the same works for polylinear functors. In particular, given our $k$-linear abelian tensor category ${{\mathcal C}}$, we can form the category ${\operatorname{Shv}}({{\mathcal C}})_\#$ of pairs $\langle E,[n] \rangle$, $[n] \in \Lambda$, $E \in {\operatorname{Shv}}({{\mathcal C}}^n)$, with a map from $\langle E',[n'] \rangle$ to $\langle E,[n] \rangle$ given by a pair of a map $f:[n'] \to [n]$ and either a map $E' \to (f_!)^*E$, or map $(f_!)_!E' \to E$ – this is equivalent by adjunction. Then ${\operatorname{Shv}}({{\mathcal C}})_\#$ is bifibered category over $\Lambda$ in the sense of [@SGA]. The category of sections $\Lambda \to {\operatorname{Shv}}({{\mathcal C}})_\#$ of this bifibration can also be described as the full subcategory ${\operatorname{Shv}}({{\mathcal C}}_\#) \subset {\operatorname{Fun}}({{\mathcal C}}^{opp}_\#,k)$ spanned by those functors $E_\#:{{\mathcal C}}^{opp}_\# \to k{\operatorname{\it\!-Vect}}$ whose restriction to $({{\mathcal C}}^{opp})^n \subset {{\mathcal C}}^{opp}_\#$ is a sheaf – that is, an object in ${\operatorname{Shv}}({{\mathcal C}}^n) \subset {\operatorname{Fun}}(({{\mathcal C}}^{opp})^n,k)$. Since the transition functors $(f_!)_!$ are right-exact, ${\operatorname{Shv}}({{\mathcal C}}_\#)$ is an abelian category (this is proved in exactly the same way as Lemma \[sec.ab\]). We denote by ${\operatorname{Shv}}_{cart}({{\mathcal C}}_\#) \subset {\operatorname{Shv}}({{\mathcal C}}_\#)$ the full subcategory of sections $E:\Lambda \to {\operatorname{Shv}}({{\mathcal C}})_\#$ which are cocartesian, and moreover, are such that $E([1]) \in {\operatorname{Shv}}({{\mathcal C}})$ actually lies in the Yoneda image ${{\mathcal C}}\subset {\operatorname{Shv}}({{\mathcal C}})$. We also denote by ${{\mathcal D}}\Lambda({{\mathcal C}}) \subset {{\mathcal D}}({\operatorname{Shv}}({{\mathcal C}}_\#))$ the full triangulated subcategory of complexes $E^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}_\# \in {{\mathcal D}}({\operatorname{Shv}}({{\mathcal C}}_\#))$ with homology in ${\operatorname{Shv}}_{cart}({{\mathcal C}}_\#)$. If ${{\mathcal C}}$ is the category of $A$-bimodules for some algebra $A$ – or better yet, of $A$-bimodules of cardinality not more than that of $A \times {{\mathbb N}}$, so that ${{\mathcal C}}$ is small – then ${\operatorname{Shv}}({{\mathcal C}})$ is equivalent to $A{\operatorname{\!-\sf bimod}}$ (one shows easily that every sheaf $E \in {\operatorname{Shv}}({{\mathcal C}})$ is completely determined by its value at $A^{opp} \otimes A \in {{\mathcal C}}$). In this case, ${{\mathcal D}}\Lambda({{\mathcal C}})$ is our old category ${{\mathcal D}}\Lambda(A{\operatorname{\!-\sf bimod}})$. Now, we assume that ${{\mathcal C}}$ is equipped with a right-exact trace functor ${\operatorname{\sf tr}}:{{\mathcal C}}\to k{\operatorname{\it\!-Vect}}$, we would like to define cyclic homology $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(M_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}})$ for any $M_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\in {{\mathcal D}}\Lambda({{\mathcal C}})$, and we immediately notice a problem: for a general ${{\mathcal C}}$, we do not have a forgetful functor to vector spaces. However, it turns out that the forgetful functor [*is not needed*]{} for the definition – it can be replaced with the trace functor ${\operatorname{\sf tr}}$. We proceed as follows. By definition, ${\operatorname{\sf tr}}$ is extended to a functor ${{\mathcal C}}_\# \to k{\operatorname{\it\!-Vect}}$; we extend it canonically to a functor ${\operatorname{Shv}}({{\mathcal C}})_\# \to k{\operatorname{\it\!-Vect}}$, and consider the product $${\operatorname{\sf tr}}\times \tau: {\operatorname{Shv}}({{\mathcal C}})_\# \to k{\operatorname{\it\!-Vect}}\times \Lambda,$$ where $\tau:{\operatorname{Shv}}({{\mathcal C}})_\# \to \Lambda$ is the projection. This is a functor compatible with the projections to $\Lambda$, and therefore, it induces a functor of the categories of sections. The category of sections of the projection $k{\operatorname{\it\!-Vect}}\times \Lambda \to \Lambda$ is tautologically the same as ${\operatorname{Fun}}(\Lambda,k{\operatorname{\it\!-Vect}})$, so that we have a functor $${\operatorname{\sf tr}}_\#:{\operatorname{Shv}}({{\mathcal C}}_\#) \to {\operatorname{Fun}}(\Lambda,k).$$ One checks easily that this functor is right-exact. \[cycl.def.gen\] For any $M_\# \in {\operatorname{\sf Sec}}({{\mathcal C}}_\#)$, its cyclic homology $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(M_\#)$ is defined by $$HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(M_\#) {\overset{\text{\sf\tiny def}}{=}}HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}_\#(M_\#)) = H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Lambda,L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}_\#(M_\#)).$$ \[clean\] The pair $\langle {{\mathcal C}},{\operatorname{\sf tr}}\rangle$ is called [*homologically clean*]{} if for any $n$, the category ${\operatorname{Shv}}({{\mathcal C}}^n)$ has enough objects $E$ such that 1. $E$ is acyclic both for functors $(f_!)_!:{\operatorname{Shv}}({{\mathcal C}}^n) \to {\operatorname{Shv}}({{\mathcal C}}^{n'})$, for any $f:[n] \to [n']$, and for the trace functor ${\operatorname{\sf tr}}:{\operatorname{Shv}}({{\mathcal C}}^n) \to k{\operatorname{\it\!-Vect}}$, and 2. for any $f:[n] \to [n']$, $(f_!)_!E \in {\operatorname{Shv}}({{\mathcal C}}^{n'})$ is acyclic for ${\operatorname{\sf tr}}:{\operatorname{Shv}}({{\mathcal C}}^{n'}) \to k{\operatorname{\it\!-Vect}}$. \[cln.exa\] Assume that the category ${{\mathcal C}}$ has enough projectives, and moreover, $P_1 \otimes P_2$ is projective for any projective $P_1,P_2 \in {{\mathcal C}}$ (this is satisfied, for instance, for ${{\mathcal C}}=A{\operatorname{\!-\sf bimod}}$). Then the pair $\langle {{\mathcal C}},{\operatorname{\sf tr}}\rangle$ is homologically clean, for any trace functor ${\operatorname{\sf tr}}$. Indeed, ${\operatorname{Shv}}({{\mathcal C}}^n)$ then also has enough projectives, say sums of objects $$\label{box} P = P_1 \boxtimes P_2 \boxtimes \dots \boxtimes P_n \in {\operatorname{Shv}}({{\mathcal C}}^n)$$ for projective $P_1,\dots,P_n \in {{\mathcal C}}\subset {\operatorname{Shv}}({{\mathcal C}})$, and these projectives automatically satisfy the condition . To check , one decomposes $f:[n] \to [n']$ into a surjection $p:[n] \to [n'']$ and an injection $i:[n''] \to [n']$. Since the tensor product of projective objects is projective, $(p_!)_!(P) \in {\operatorname{Shv}}({{\mathcal C}}^{n''})$ is also an object of the type , so we may as well assume that $f$ is injective. Then one can find a left-inverse map $f':[n'] \to [n]$, $f' \circ f = {\operatorname{\sf id}}$; since $P'=(f_!)_!(P)$ is obviously acyclic for $(f'_!)_!$, and $(f'_!)_!(P') = ((f' \circ f)_!)_!(P) = P$ is acyclic for ${\operatorname{\sf tr}}$, $P'$ itself is acyclic for ${\operatorname{\sf tr}}= {\operatorname{\sf tr}}\circ (f'_!)_!$. \[b.ch\] Assume that $\langle {{\mathcal C}},{\operatorname{\sf tr}}\rangle$ is homologically clean. Then for any object $[n] \in \Lambda$ and any $M_\# \in {\operatorname{Shv}}({{\mathcal C}}_\#)$, we have $$\label{equa} L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}_\#(M_\#)([n]) \cong L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}(M_\#([n])).$$ For any $M_\#^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}\in {{\mathcal D}}\Lambda({{\mathcal C}})$, we have $L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}_\#(M_\#) \in {{\mathcal D}}_{const}(\Lambda,k) \subset {{\mathcal D}}(\Lambda,k)$. [[*Proof.*]{}]{} The natural restriction functor ${\operatorname{Shv}}({{\mathcal C}}_\#) \to {\operatorname{Shv}}({{\mathcal C}}^n)$, $M_\# \mapsto M_\#([m])$ has a left-adjoint functor $I_{n!}:{\operatorname{Shv}}({{\mathcal C}}^n) \to {\operatorname{Shv}}({{\mathcal C}}_\#)$; explicitly, it is given by $$\label{ind} I_{n!}(E)([n']) = \bigoplus_{f:[n] \to [n']}(f_!)_!(E).$$ Let us say that an object $E \in {\operatorname{Shv}}({{\mathcal C}}^n)$ is admissible if it satisfies the conditions , of Definition \[clean\]. By assumption, ${\operatorname{Shv}}({{\mathcal C}}^n)$ has enough admissible objects for any $n$. Then ${\operatorname{Shv}}({{\mathcal C}}_\#)$ has enough objects of the form $I_{n!}E$, $[n] \in \Lambda$, $E \in {\operatorname{Shv}}({{\mathcal C}}^n)$ admissible, and to prove the first claim, it suffices to consider $M_\#=I_{n!}E$ of this form. In degree $0$, is the definition of the functor ${\operatorname{\sf tr}}_\#$, and the higher degree terms in the right-hand side vanish by Definition \[clean\] . Therefore it suffices to prove that $M_\#=I_{n!}E$ is acyclic for the functor ${\operatorname{\sf tr}}_\#$. This is obvious: applying ${\operatorname{\sf tr}}_\#$ to any short exact sequence $$\begin{CD} 0 @>>> M'_\# @>>> M''_\# @>>> M_\# @>>> 0 \end{CD}$$ in ${\operatorname{Shv}}({{\mathcal C}}_\#)$, we see that, since $M_\#([n'])$ is acyclic for any $[n'] \in \Lambda$, the sequence $$\begin{CD} 0 @>>> {\operatorname{\sf tr}}M'_\#([n']) @>>> {\operatorname{\sf tr}}M''_\# ([n']) @>>> {\operatorname{\sf tr}}M_\# ([n']) @>>> 0 \end{CD}$$ is exact; this means that $$\begin{CD} 0 @>>> {\operatorname{\sf tr}}M'_\# @>>> {\operatorname{\sf tr}}M''_\# @>>> {\operatorname{\sf tr}}M_\# @>>> 0 \end{CD}$$ is an exact sequence in ${\operatorname{Fun}}(\Lambda,k)$, and this means that $M_\#$ is indeed acyclic for ${\operatorname{\sf tr}}_\#$. With the first claim proved, the second amounts to showing that the natural map $$L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}\circ L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(f_!)_!(E) \to L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}(E)$$ is a quasiismorphism for any $f:[n] \to [n']$ and any $E \in {\operatorname{Shv}}({{\mathcal C}}^n)$. It suffices to prove it for admissible $M$; then the higher derived functors vanish, and the isomorphism ${\operatorname{\sf tr}}\circ (f_!)_! \cong {\operatorname{\sf tr}}$ is Definition \[trace.defn\]. In the assumptions of Lemma \[b.ch\], for any complex $M^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}_\# \in {{\mathcal D}}\Lambda({{\mathcal C}})$ with the first component $M^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}= M^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}_\#([1])$ we have $$HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(M^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}) \cong HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}_\#(M^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}_\#)).$$ [[*Proof.*]{}]{} By Lemma \[b.ch\], the left-hand side, $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(M^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}})$, is canonically isomorphic to the complex $L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}_\#(M^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}_\#) \in {{\mathcal D}}(\Lambda,k)$ evaluated at $[1] \in \Lambda$, and moreover, $L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}_\#(M^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}_\#)$ lies in the subcategory ${{\mathcal D}}_{const}(\Lambda,k) \subset {{\mathcal D}}(\Lambda,k)$. It remains to apply the general fact: for any $E^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}\in {{\mathcal D}}_{const}(\Lambda,k)$, we have a natural isomorphism $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(E^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}) \cong E^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}([1])$. Indeed, by definition we have $$HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(E^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}) = H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Delta^{opp},j^*E^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}),$$ and $j^*E^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ lies in the category ${{\mathcal D}}_{const}(\Delta^{opp},k)$ which is equivalent to ${{\mathcal D}}(k{\operatorname{\it\!-Vect}})$ (see Example \[const.exa\], and also Remark \[const.rem\]: the isomorphism we constructed here is a special case of for $n=1$). The Lemma shows that if the pair $\langle {{\mathcal C}},{\operatorname{\sf tr}}\rangle$ is homologically clean, Definition \[cycl.def.gen\] is consistent with , and we get the whole periodicity package of – the periodicity map $u$, the Connes’ exact sequence $$\begin{CD} HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(M^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}) @>>> HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(M^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}) @>{u}>> HC_{{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}-2}(M^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}) @>>>, \end{CD}$$ and the periodic cyclic homology $HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(M^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}})$. In general, objects in ${{\mathcal D}}\Lambda({{\mathcal C}})$ may be hard to construct, but we always have at least one – the identity section ${\operatorname{{\sf I}}}_\#:\Lambda \to {\operatorname{Shv}}({{\mathcal C}})_\#$, given by $${\operatorname{{\sf I}}}_\#([n]) = {\operatorname{{\sf I}}}^{\boxtimes n} \in {{\mathcal C}}^{\otimes n},$$ where ${\operatorname{{\sf I}}}\in {{\mathcal C}}$ is the unit object. Thus we can define cyclic homology of a tensor category equipped with a trace functor. \[cycl.cat\] For any $k$-linear abelian unital tensor category ${{\mathcal C}}$ equipped with a trace functor ${\operatorname{\sf tr}}:{{\mathcal C}}\to k{\operatorname{\it\!-Vect}}$, its Hochschild and cyclic homology is given by $$HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}({{\mathcal C}},{\operatorname{\sf tr}}) {\overset{\text{\sf\tiny def}}{=}}HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}({\operatorname{{\sf I}}}), \qquad HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}({{\mathcal C}},{\operatorname{\sf tr}}) {\overset{\text{\sf\tiny def}}{=}}HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}({\operatorname{{\sf I}}}_\#),$$ where ${\operatorname{{\sf I}}}\in {{\mathcal C}}$ is the unit object, and ${\operatorname{{\sf I}}}_\# \in {{\mathcal D}}\Lambda({{\mathcal C}})$ is the identity section. We now have to check that in the case ${{\mathcal C}}= A{\operatorname{\!-\sf bimod}}$, Definition \[cycl.def.gen\] is compatible with our earlier Definition \[cycl.def\] – in other words, that the cyclic homology computed by means of the forgetfull functor is the same as the cyclic homology computed by means of the trace. This is not at all trivial. Indeed, if for instance $M_\# \in {\operatorname{Shv}}({{\mathcal C}}_\#)$ is cocartesian, then, while $L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}^\#M_\#$ lies in the subcategory ${{\mathcal D}}_{const}(\Lambda,k) \subset {{\mathcal D}}(\Lambda,k)$, the same is certainly not true for the object $M_\# \in {\operatorname{Fun}}(\Lambda,k)$ obtained by forgetting the bimodule structure on $M_n$. Thus these two objects are different. However, they do become equal after taking cyclic (or Hochschild, or periodic cyclic) homology. Namely, for any $M_\# \in {\operatorname{\sf Sec}}(A{\operatorname{\!-\sf bimod}}_\#)$ we have a natural map $$\label{natu} M_\# \to L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}^\#M_\#$$ in the derived category ${{\mathcal D}}(\Lambda,k)$, and we have the following result. \[main\] For every $M_\# \in {\operatorname{\sf Sec}}(A{\operatorname{\!-\sf bimod}}_\#)$, the natural map induces isomorphisms $$\begin{aligned} HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(M_\#) &\cong HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}M_\#),\\ HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(M_\#) &\cong HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}M_\#),\\ HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(M_\#) &\cong HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}M_\#).\end{aligned}$$ [[*Proof.*]{}]{} By , it suffices to consider $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(-)$; as in the proof of Lemma \[b.ch\], it suffices to consider $M_\# = I_{n!}E$ given in , with $E$ being the free bimodule $$E = (A^{opp} \otimes A)^{\otimes n} \in {\operatorname{Shv}}({{\mathcal C}}^n) = A^{\otimes n}{\operatorname{\!-\sf bimod}}$$ for some fixed $n$. Explicitly, we have $$\label{ind.2} I_{n!}E([n']) = \bigoplus_{f:[n] \to [n']}\bigotimes_{v' \in V([n'])} A^{opp} \otimes A^{\otimes f^{-1}(v')}$$ for any $[n'] \in \Lambda$. Then $L^p{\operatorname{\sf tr}}_\#I_{n!}E = 0$ for $p \geq 1$, and one checks easily that $${\operatorname{\sf tr}}_\# I_{n!}E = i_{n!}{\operatorname{\sf tr}}E = i_{n!}A^{\otimes n} \in {\operatorname{Fun}}(\Lambda,k),$$ where $i_n:{\operatorname{{\sf pt}}}\to \Lambda$ is the embedding of the object $[n] \in \Lambda$ (${\operatorname{{\sf pt}}}$ is the category with one object and one morphism). Therefore $$HC_0(L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}_\#I_{n!}E) = H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Lambda,i_{n!}A^{\otimes n}) = A^{\otimes n},$$ and $HC_p(L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}_\#i_{n!}E) = 0$ for $p \geq 1$. We have to compare it with $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(i_{n!}E)$. To do this, consider the category $\Lambda_{[n]}$ of objects $[n'] \in \Lambda$ equipped with a map $[n] \to [n']$, and let $j_n:\Lambda_{[n]} \to \Lambda$ be the forgetful functor. Then $j_n$ is obviously a discrete cofibration. Comparing and , we see that $$I_{n!}E = j_{n!}E_\#^{[n]}$$ for some $E_\#^{[n]} \in {\operatorname{Fun}}(\Lambda_{[n]})$. Moreover, fix once and for all a map $[1] \to [n]$. Then we see that the discrete cofibration $j_n:\Lambda_{[n]} \to \Lambda$ factors through the discrete cofibration $j:\Lambda_{[1]} = \Delta^{opp} \to \Lambda$ by means of a discrete cobifbration $\gamma_n:\Lambda_{[n]} \to \Lambda_{[1]}$, and we observe that $$E^{[n]}_\#([n']) = (A^{opp})^{\otimes n'} \otimes A^{\otimes n}$$ only depends on $\gamma_n([n']) \in \Delta^{opp}$. More precisely, we have $E^{[n]}_\# = \gamma^*_nE_n^{\Delta}$, where $E_n^{\Delta} \in {\operatorname{Fun}}(\Delta^{opp},k)$ is as in , and $E_n$ is the free $A$-bimodule $$E_n = A^{opp} \otimes A^{\otimes (n-1)} \otimes A.$$ The conclusion: we have $$HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(I_{n!}E) = H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Lambda_{[n]},E^{[n]}_\#) = H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Delta^{opp},\gamma_{n!}\gamma^*_nE_n^{\Delta}) = H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Delta^{opp},E_n^{\Delta} \otimes \gamma_{n!}k),$$ where we have used the projection formula in the right-hand side. The homology of the category $\Delta^{opp}$ can be computed by the standard complex; then by the Künneth formula, the right-hand side is isomorphic to $$H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Delta^{opp},E_n^{\Delta}) \otimes H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Delta^{opp},\gamma_{n!}k) \cong H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Delta^{opp},E_n^{\Delta}) \otimes H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Lambda_{[n]},k).$$ By Lemma \[hoch\], $$H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Delta^{opp},E_n^{\Delta}) \cong HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,E_n) \cong A^{\otimes n}.$$ Since the category $\Lambda_{[n]}$ has an initial object $[n] \in \Lambda_{[n]}$, we have $k = i_{n!k}$, so that the second multiple $H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\Lambda_{[n]},k)$ is just $k$ in degree $0$. The essential point of Proposition \[main\] is the following: the cyclic object $A_\#$ associated to an algebra $A$ inconveniently contains two things at the same time – the cyclic structure, which seems to be essential to the problem, and the bar resolution, which is needed only to compute the Hochschild homology $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A)$. Replacing $A_\#$ with the cyclic complex $L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}_\#A_\# \in {{\mathcal D}}(\Lambda,k)$ disentagles these two. We note that while one still has to prove that this does not change the final answer, the construction itself looks pretty straightforward – if one wants to remove the non-essential bar resolution from the definition of the cyclic homology, Definition \[cycl.cat\] seems to be the obvious thing to try. However, it was actually arrived at by a sort of a reverse engeneering process. To finish the section, perhaps it would be useful to show the reader the first stage of this process. Assume given an associative algebra $A$, and fix a projective resolution $P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}$ of the diagonal $A$-module $A$. Then $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M)$ can be computed by the complex $${\operatorname{\sf tr}}(P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}) = P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\otimes_{A^{opp} \otimes A} A.$$ How can one see the cyclic homology in terms of this complex? Or even simpler – what is the first differential in the spectral sequence , the Connes’ differential $B:HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A) \to HH_{{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}+1}(A)$? There is the following recepy which gives the answer. Let $\tau:P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\to A$ be the augmentation map. Consider the tensor product $P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\otimes_A P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}$. This is also a projective resotuion of $A$, and we actually have [*two*]{} natural quasiisomorphisms $$\tau_1,\tau_2:P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\otimes_A P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\to P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}},$$ given by $\tau_1 = \tau \otimes {\operatorname{\sf id}}$, $\tau_2 = {\operatorname{\sf id}}\otimes \tau$. These quasiisomorphisms are different. However, since both are maps between projective resolutions of the same object, there should be a chain homotopy between them. Fix such a homotopy $\iota:P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\otimes_A P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\to P_{{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}+1}$. Now we apply the trace functor ${\operatorname{\sf tr}}$, and obtain two maps $\tau_1,\tau_2:{\operatorname{\sf tr}}(P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\otimes P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}) \to {\operatorname{\sf tr}}(P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}})$, and a homotopy $\iota:{\operatorname{\sf tr}}(P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\otimes P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}) \to {\operatorname{\sf tr}}(P_{{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}+1})$ between them. However, by the trace property of $\tau$, we also have an involution $\sigma:{\operatorname{\sf tr}}(P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\otimes_A P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}})$ which interchanges the two multiples. This involution obviously also interchages $\tau_1$ and $\tau_2$, but there is no reason why it should fix the homotopy $\iota$ – in fact, it sends $\iota$ to a second homotopy $\iota':{\operatorname{\sf tr}}(P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\otimes_A P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}) \to {\operatorname{\sf tr}}(P_{{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}+1})$ between $\tau_1$ and $\tau_2$. The difference $\iota' - \iota$ is then a well-defined map of complexes $$\label{homo} \iota'-\iota:{\operatorname{\sf tr}}(P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\otimes_A P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}) \to {\operatorname{\sf tr}}(P_{{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}+1}).$$ On the level of homology, both sides are $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A)$; the map $\iota'-\iota$ then induces exactly the Connes’ differential $B:HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A) \to HH_{{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}+1}(A)$. To justify this recepy, we use Proposition \[main\] and identify $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A)$ with $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}_\#(A_\#))$ rather than $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A_\#)$. Then $L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}_\#(A_\#)$ is an object in ${{\mathcal D}}_{const}(\Lambda,k)$. Therefore, as noted in Remark \[const.rem\], the Connes’ differential $B$ only depends on the restriction of $L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}_\#(A_\#)$ to $\Lambda_{\leq 2} \subset \Lambda$. In other words, we do not need to compute the full $L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}_\#(A_\#)$ and to construct a full resolution $P^\#_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}$ of the cyclic $A$-bimodule $A_\#$; it suffices to construct $P^i_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}= P^\#_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}([i])$ for $i=1,2$ (and then apply the functor ${\operatorname{\sf tr}}$). With the choices made above, we set $P^1_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}= P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}$, and we let $P^2_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}$ be the cone of the map $$\begin{CD} P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\boxtimes P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}@>{(\tau \boxtimes {\operatorname{\sf id}})\oplus({\operatorname{\sf id}}\boxtimes \tau)}>> (A \boxtimes P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}})\oplus(P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\boxtimes A). \end{CD}$$ The involution $\sigma:[2] \to [2]$ acts on $P^2_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}$ in the obvious way. We also need to define the transition maps $\iota_f$ for the two injections $d,d':[1] \to [2]$ and the two surjections $s,s':[2] \to [1]$. For $d_1$, the transition map $\iota_d:A \boxtimes P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\to P^2_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}$ is the obvious embedding, and so is the transition map $\iota_{d'}$. For the surjection $s$, we need a map $\iota_s$ from the cone of the map $$\begin{CD} P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\otimes_A P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}@>{(\tau \otimes {\operatorname{\sf id}})\oplus({\operatorname{\sf id}}\otimes \tau)}>> P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\oplus P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\end{CD}$$ to $P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}$. On $P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\oplus P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}$, the map $\iota_s$ is just the difference map $a \oplus b \mapsto a - b$; on $P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\otimes_AP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}$, $\iota_s$ is our fixed homotopy $\iota:P_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\otimes_AP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}\to P_{{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}+1}$. And similarly for the other surjection $s'$. We leave it to the reader to check that if one computes $L^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}{\operatorname{\sf tr}}_\#(A_\#)\mid_{\Lambda_{\leq 2}}$ using this resolution $P^\#_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}$, then one obtains exactly for the Connes’ differential $B$. Discussion ========== One of the most unpleasant features of the construction presented in Section \[cat\] is the strong assumptions we need to impose on the tensor category ${{\mathcal C}}$. In fact, the category one would really like to apply the construction to is the category ${\operatorname{End}}{{\mathcal B}}$ of endofunctors – whatever that means – of the category ${{\mathcal B}}$ of coherent sheaves on an algebraic variety $X$. But if $X$ is not affine, ${\operatorname{End}}{{\mathcal B}}$ certainly does not have enough projectives, so that Example \[cln.exa\] does not apply, and it is unlikely that ${\operatorname{End}}{{\mathcal B}}$ can be made homologically clean in the sense of Definition \[clean\]. We note that Definition \[clean\] has been arranged so as not impose anything more than strictly necessary for the proofs; but in practice, we do not know any examples which are not covered by Example \[cln.exa\]. As for the category ${\operatorname{End}}{{\mathcal B}}$, there is an even bigger problem with it: while there are ways to define endofunctors so that ${\operatorname{End}}{{\mathcal B}}$ is an abelian category with a right-exact tensor product, it cannot be equipped with a right-exact trace functor ${\operatorname{\sf tr}}$. Indeed, it immediately follows from Definition \[cycl.cat\] that the Hochschild homology groups $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}({{\mathcal C}})$ of a tensor category ${{\mathcal C}}$ are trivial in negative homological degrees. If ${{\mathcal C}}= {\operatorname{End}}{{\mathcal B}}$, one of course expects $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}({{\mathcal C}}) = HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(X)$, the Hochschild homology $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(X)$ of the variety $X$, which by now is well-understood (see e.g. [@w]). And if $X$ is not affine, $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(X)$ typically is non-trivial both in positive and in negative degrees. If $X$ is smooth and proper, $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(X)$ in fact carries a non-degenerate pairing, so that it is just as non-trivial in degrees $>0$ as in degrees $<0$. Thus the case of a non-affine algebraic variety is far beyond the methods developed in this paper. The real reason for these difficulties is that we are dealing with abelian categories, while the theory emphatically wants to live in the triangulated world; as we explained in Example \[const.exa\], even our main topic, cyclic bimodules, are best understood as objects of a triangulated category ${{\mathcal D}}\Lambda({{\mathcal C}})$. Unfortunately, we cannot develop the theory from scratch in the triangulated context, since we do not have a strong and natural enough notion of an enhanced triangulated category (and working with the usual triangulated categories is out of the question because, for instance, the category of triangulated functors between triangulated categories is usually not a triangulated category itself). A well-developed theory would probably require a certain compromise between the abelian and the triangulated approach. We will return to it elsewhere. Another thing which is very conspicously not done in the present paper is the combination of Section \[cat\] and Section \[getz\]. Indeed, in Section \[getz\], we are dealing with cyclic homology in the straightforward naive way of Section \[naive\], and while we define the cyclic object ${\widehat}{A_\#}$ associated to a square-zero extension ${\widetilde}{A}$, we make no attempt to find an appropriate category ${\widehat}{{\operatorname{\sf Sec}}(A{\operatorname{\!-\sf bimod}}_\#)}$ where it should live. This is essentially the reason why we cannot go further than square-zero extensions. At present, sadly, we do not really understand this hypothetical category ${\widehat}{{\operatorname{\sf Sec}}(A{\operatorname{\!-\sf bimod}}_\#)}$. One suspects that treating this properly would require studying deformations in a much more general context – instead of considering square-zero extensions of an algebra, we should look at the deformations of the abelian category of its modules, or at the deformations of the tensor category of its bimodules. This brings us to another topic completely untouched in the paper: the Hochschild cohomology $HH^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(A)$. Merely [*defining*]{} Hochschild cohomology for an arbitrary tensor category ${{\mathcal C}}$ is in fact much simpler than the definition of $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}({{\mathcal C}})$, and one does not need a trace functor for this – we just set $HH^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}({{\mathcal C}})={\operatorname{Ext}}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}({\operatorname{{\sf I}}},{\operatorname{{\sf I}}})$, where ${\operatorname{{\sf I}}}\in {{\mathcal C}}$ is the unit object. However, it is well understood by now that just as Hochschild homology always comes equipped with the Connes’ differential, the spectral sequence , and the whole cyclic homology package, Hochschild cohomology should be considered not as an algebra but as the so-called [*Gerstenhaber*]{} algebra; in fact, the pair $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(-),HH^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(-)$ should form a version of “non-commutative calculus”, as proposed for instance in [@TT]. Deformations of the tensor category ${{\mathcal C}}$ should be controlled by $HH^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}({{\mathcal C}})$, and the behaviour of $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}({{\mathcal C}})$ and $HC_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}({{\mathcal C}})$ under these deformations reflects various natural actions of $HH^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(-)$ on $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(-)$. We believe that a convenient development of the “non-commutative calculus” for a tensor category ${{\mathcal C}}$ might be possible along the same lines as our Section \[cat\]. Just as our category ${{\mathcal D}}\Lambda({{\mathcal C}})$ is defined as the category of sections of the cofibration ${{\mathcal C}}_\#/\Lambda$, whose definition imitates the usual cyclic object $A_\#$, one can construct a cofibration ${{\mathcal C}}^\#/\Delta$ which imitates the standard cosimplicial object computing $HH^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(A)$ – for any $[n] \in \Delta$, ${{\mathcal C}}^\#([n])$ is the category of polylinear right-exact functors from ${{\mathcal C}}^{n-1}$ to ${{\mathcal C}}$, and the transition functors between various ${{\mathcal C}}^\#([n])$ are induced by the tensor product on ${{\mathcal C}}$. Then one can define a triangulated category ${{\mathcal D}}\Delta({{\mathcal C}})$, the subcategory in ${{\mathcal D}}({\operatorname{\sf Sec}}({{\mathcal C}}^\#))$ of complexes with cocartesian homology; the higher structures on $HH^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}({{\mathcal C}})$ should be encoded in the structure of the category ${{\mathcal D}}\Delta({{\mathcal C}})$, and relations between $HH_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}({{\mathcal C}})$ and $HH^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}({{\mathcal C}})$ should be reflected in a relation between ${{\mathcal D}}\Lambda({{\mathcal C}})$ and ${{\mathcal D}}\Delta({{\mathcal C}})$. We will proceed in this direction elsewhere. At present, the best we can do is to make the following hopeful observation: - the category ${\operatorname{\sf Sec}}_{cart}({{\mathcal C}}^\#)$ is naturally a [*braided*]{} tensor category over $k$. The reason for this is very simple: if one writes out explicitly the definition of ${\operatorname{\sf Sec}}_{cart}({{\mathcal C}}^{\#})$ along the lines of Lemma \[cycl.str\], one finds out that it coincides on the nose with the Drinfeld double of the tensor category ${{\mathcal C}}$. [GM2]{} I. Bucur and A. Deleanu, [*Introduction to the theory of categories and functors*]{}, Interscience Publication John Wiley & Sons, Ltd., London-New York-Sydney 1968. A. Connes, [*Non-commutative differential geometry, I, II*]{}, preprint IHES oct. 1982, may 1983. A. Connes, [*Cohomologie cyclique et foncteur ${\operatorname{Ext}}^n$*]{}, Comptes Rendues Ac. Sci. Paris Sér. A-B, [**296**]{} (1983), 953–958. B. Feigin and B. Tsygan, [*Additive $K$-Theory*]{}, in Lecture Notes in Math. [**1289**]{} (1987), 97–209. S. Gelfand and Yu. Manin, [*Methods of homological algebra*]{}, Nauka Publishers, Moscow, 1988 (in Russian). S. Gelfand and Yu. Manin, [*Homological algebra*]{}, Itogi vol. 38 (Algebra V), VINITI, Moscow, 1989 (in Russian). E. Getzler, [*Cartan homotopy formulas and the Gauss-Manin connection in cyclic homology*]{}, in [*Quantum deformations of algebras and their representations (Ramat-Gan, 1991/1992; Rehovot, 1991/1992)*]{}, Israel Math. Conf. Proc. [**7**]{}, Bar-Ilan Univ., Ramat Gan, 1993, 65–78. Th. Goodwillie, [*Cyclic homology, derivations, and the free loopspace*]{}, Topology [**24**]{} (1985), 187–215. A. Grothendieck, [*Expose VI: Catégories fibré et descente*]{}, in [*SGAI: Revétements étales et groupe fondamental*]{}, Lecture Notes in Math., [**224**]{}, Springer, Berlin; 145–194. D. Kaledin, [*Non-commutative Hodge-to-de Rham degeneration via the method of Deligne-Illusie*]{}, math.KT/0611623. J.-L. Loday, [*Cyclic Homology*]{}, second ed., Springer, 1998. D. Tamarkin and B. Tsygan, [*The ring of differential operators on forms in noncommutative calculus*]{}, in [*Graphs and patterns in mathematics and theoretical physics*]{}. Proc. Sympos. Pure Math. [**73**]{}, AMS, Providence, RI, 2005; 105–131. B. Tsygan, [*Homology of Lie algebras over rings and Hochshild homology*]{}, Uspekhi Mat. Nauk, [**38**]{} (1983), 217–218. C. Weibel, [*Cyclic homology for schemes*]{}, Proc. AMS [**124**]{} (1996), 1655-1662. [Steklov Math Institute\ Moscow, USSR]{} [*E-mail address*]{}: [[email protected]]{} [^1]: Partially supported by CRDF grant RUM1-2694-MO05.

--- abstract: | We present a brief summary of observations of the transiting extrasolar planet, HD209458b, designed to detect the secondary eclipse. We employ the method of ‘occultation spectroscopy’, which searches in combined light (star and planet) for the disappearance and reappearance of weak infrared spectral features due to the planet as it passes behind the star and reappears. We have searched for a continuum peak near 2.2  (defined by CO and H$_2$O absorption bands), as predicted by some models of the planetary atmosphere to be $\sim 6 \times 10^{-4}$ of the stellar flux, but no such peak is detected at a level of $\sim 3 \times 10^{-4}$ of the stellar flux. Our results represent the strongest limits on the infrared spectrum of the planet to date and carry significant implications for understanding the planetary atmosphere. author: - 'L. Jeremy Richardson *and* Drake Deming' - Sara Seager title: 'Strong Limits on the Infrared Spectrum of HD209458b Near 2.2 ' --- Background ========== Our observations cover two predicted secondary eclipse events, and we obtained 1036 individual spectra of the HD209458 system using the SpeX instrument at the NASA IRTF in September 2001. Our spectra extend from 1.9 to 4.2  with a resolution ($\lambda/\Delta \lambda$) of 1500. A summary of the method of occultation spectrosocpy, as well as the details of the data analysis, can be found in Richardson, Deming, & Seager (2003). Results and Discussion ====================== The final difference spectrum, shown in Figure 1, represents the average candidate planetary spectrum, as calculated from the ‘in-eclipse’ minus the ‘out-of-eclipse’ spectra from the two nights during which a secondary eclipse was predicted to occur. The average spectrum represents data from 550 individual spectra of HD209458, as well as an equal number of spectra of the comparison star HD210483. The comparison star was used in our analysis to remove variability due to changes in the terrestrial atmsophere and to normalize the data to the stellar flux density. Also shown in Figure 1 is the baseline model for HD204958b calculated by Sudarsky, Burrows, & Hubeny (2003), which exhibits a peak near 2.2 . A least-squares analysis indicates this peak is not present in the candidate planetary spectrum as derived from the data. We believe this result has significant implications for the structure of the planetary atmosphere. In particular, some models that assume the stellar irradiation is re-radiated entirely on the sub-stellar hemisphere predict this flux peak, which is inconsistent with our observations. Several physical mechanisms can improve agreement with our observations, including the re-distribution of heat by global circulation, a nearly isothermal atmosphere, and/or the presence of a high cloud. =3.7in Richardson, L. J., Deming, D., & Seager, S. 2003, , 597, 581 Sudarksy, S., Burrows, A., & Hubeny, I. 2003, , 588, 1121

--- bibliography: - '/home/osegu/LyxDocs/osegu.bib' --- *Int. J. Bifurcation and Chaos* **13**[, 3147-3233, (2003). Tutorial and Review paper.]{} **TOWARD A THEORY OF CHAOS** A. Sengupta\ Department of Mechanical Engineering\ Indian Institute of Technology. Kanpur\ E-Mail: [email protected] **ABSTRACT** [This paper formulates a new approach to the study of chaos in discrete dynamical systems based on the notions of inverse ill-posed problems, set-valued mappings, generalized and multivalued inverses, graphical convergence of a net of functions in an extended multifunction space [@Sengupta2000], and the topological theory of convergence. Order, chaos, and complexity are described as distinct components of this unified mathematical structure that can be viewed as an application of the theory of convergence in topological spaces to increasingly nonlinear mappings, with the boundary between order and complexity in the topology of graphical convergence being the region in $\textrm{Multi}(X)$ that is susceptible to chaos. The paper uses results from the discretized spectral approximation in neutron transport theory [@Sengupta1988; @Sengupta1995] and concludes that the numerically exact results obtained by this approximation of the Case singular eigenfunction solution is due to the graphical convergence of the Poisson and conjugate Poisson kernels to the Dirac delta and the principal value multifunctions respectively. In $\textrm{Multi}(X)$, the continuous spectrum is shown to reduce to a point spectrum, and we introduce a notion of]{} *latent chaotic states* [to interpret superposition over generalized eigenfunctions. Along with these latent states, spectral theory of nonlinear operators is used to conclude that nature supports complexity to attain efficiently a multiplicity of states that otherwise would remain unavailable to it. ]{} *Keywords:* [chaos, complexity, ill-posed problems, graphical convergence, topology, multifunctions.]{} **Prologue** **1.** @Peitgen1992 **2.** Mitchell Feigenbaum’s *Foreword* (pp 1-7) in @Peitgen1992 **3.** [^1] Opening address of Heitor Gurgulino de Souza, Rector United Nations University, Tokyo @Grebogi1997 **4.** [^2] @Gallagher1999 **5.** @Goldenfeld1999 **6.** @Gleick1987 **7.** @Waldrop1992 **8.** @Brown1996 **9.** @Falconer1990 **10.** @Robinson1999 **1. Introduction** The purpose of this paper is to present an unified, self-contained mathematical structure and physical understanding of the nature of chaos in a discrete dynamical system and to suggest a plausible explanation of *why* natural systems tend to be chaotic. The somewhat extensive quotations with which we begin above, bear testimony to both the increasingly significant — and perhaps all-pervasive — role of nonlinearity in the world today as also our imperfect state of understanding of its manifestations. The list of papers at both the UN Conference [@Grebogi1997] and in *Science* [@Gallagher1999] is noteworthy if only to justify the observation of @Gleick1987 that “chaos seems to be everywhere”. Even as everybody appears to be finding chaos and complexity in all likely and unlikely places, and possibly because of it, it is necessary that we have a clear mathematically-physical understanding of these notions that are supposedly reshaping our view of nature. This paper is an attempt to contribute to this goal. To make this account essentially self-contained we include here, as far as this is practicable, the basics of the background material needed to understand the paper in the form of *Tutorials* and an extended *Appendix.* The paradigm of chaos of the kneading of the dough is considered to provide an intuitive basis of the mathematics of chaos [@Peitgen1992], and one of our fundamental objectives here is to recount the mathematical framework of this process in terms of the theory of ill-posed problems arising from non-injectivity [@Sengupta1997], *maximal ill-posedness,* and *graphical convergence* of functions [@Sengupta2000]. A natural mathematical formulation of the kneading of the dough in the form of *stretch-cut-and-paste* and *stretch-cut-and-fold* operations is in the ill-posed problem arising from the increasing non-injectivity of the function $f$ modeling the kneading operation. ***Begin Tutorial1: Functions and Multifunctions*** A *relation,* or *correspondence,* between two sets $X$ and $Y$, written $\mathscr{M}\!:X\qquad Y$, is basically a rule that associates subsets of $X$ to subsets of $Y$; this is often expressed as $(A,B)\in\mathscr{M}$ where $A\subset X$ and $B\subset Y$ and $(A,B)$ is an ordered pair of sets. The domain $$\mathcal{D}(\mathscr{M})\overset{\textrm{def}}=\{ A\subset X\!:(\!\exists Z\in\mathscr{M})(\pi_{X}(Z)=A)\}$$ and range $$\mathcal{R}(\mathscr{M})\overset{\textrm{def}}=\{ B\subset Y\!:(\!\exists Z\in\mathscr{M})(\pi_{Y}(Z)=B)\}$$ of $\mathscr{M}$ are respectively the sets of $X$ which under $\mathscr{M}$ corresponds to sets in $Y$; here $\pi_{X}$ and $\pi_{Y}$ are the projections of $Z$ on $X$ and $Y$ respectively. Equivalently, $\mathcal{D}(\mathcal{M})=\{ x\in X\!:\mathscr{M}(x)\neq\emptyset\}$ and $\mathcal{R}(\mathscr{M})=\bigcup_{x\in\mathcal{D}(\mathcal{M})}\mathscr{M}(x)$. The *inverse* $\mathscr M^{-}$ of $\mathscr{M}$ is the relation $$\mathscr M^{-}=\{(B,A)\!:(A,B)\!\in\mathscr{M}\}$$ so that $\mathscr M^{-}$ assigns $A$ to $B$ iff $\mathscr{M}$ assigns $B$ to $A$. In general, a relation may assign many elements in its range to a single element from its domain; of especial significance are *functional relations* $f$[^3] that can assigns only a unique element in $\mathcal{R}(f)$ to any element in $\mathcal{D}(f)$. Fig. \[Fig: functions\] illustrates the distinction between arbitrary and functional relations $\mathscr{M}$ and $f$. This difference between functions (or maps) and multifunctions is basic to our developments and should be fully understood. Functions can again be classified as injections (or $1:1$) and surjections (or onto). $f\!:X\rightarrow Y$ is said to be *injective* (or *one-to-one*) if $x_{1}\neq x_{2}\Rightarrow f(x_{1})\neq f(x_{2})$ for all $x_{1},x_{2}\in X$, while it is *surjective* (or *onto*) if $Y=f(X)$. $f$ is *bijective* if it is both $1:1$ and onto. Associated with a function $f\!:X\rightarrow Y$ is its inverse $f^{-1}\!:Y\supseteq\mathcal{R}(f)\rightarrow X$ that exists on $\mathcal{R}(f)$ iff $f$ is injective. Thus when $f$ is bijective, $f^{-1}(y):=\{ x\in X\!:y=f(x)\}$ exists for every $y\in Y$; infact $f$ is bijective iff $f^{-1}(\{ y\})$ is a singleton for each $y\in Y$. Non-injective functions are not at all rare; if anything, they are very common even for linear maps and it would be perhaps safe to conjecture that they are overwhelmingly predominant in the nonlinear world of nature. Thus for example, the simple linear homogeneous differential equation with constant coefficients of order $n>1$ has $n$ linearly independent solutions so that the operator $D^{n}$ of $D^{n}(y)=0$ has a $n$-dimensional null space. Inverses of non-injective, and in general non-bijective, functions will be denoted by $f^{-}$. If $f$ is not injective then $$A\subset f^{-}f(A)\overset{\textrm{def}}=\textrm{sat}(A)$$ where $\textrm{sat}(A)$ is the *saturation of* $A\subseteq X$ *induced by* $f$; if $f$ is not surjective then $${\textstyle ff^{-}(B):=B\bigcap f(X)\subseteq B.}$$ If $A=\textrm{sat}(A)$, then $A$ is said to be *saturated,* and $B\subseteq\mathcal{R}(f)$ whenever $ff^{-}(B)=B$. Thus for non-injective $f$, $f^{-}f$ is not an identity on $X$ just as $ff^{-}$ is not **$\mathbf{1}_{Y}$** if $f$ is not surjective. However the set of relations $$ff^{-}f=f,\qquad f^{-}ff^{-}=f^{-}\label{Eqn: f_inv_f}$$ that is always true will be of basic significance in this work. Following are some equivalent statements on the injectivity and surjectivity of functions $f\!:X\rightarrow Y$. (Injec) $f$ is $1:1$ $\Leftrightarrow$ there is a function $f_{\textrm{L }}\!:Y\rightarrow X$ called the left inverse of $f$, such that $f_{\textrm{L}}f=\mathbf{1}_{X}$ $\Leftrightarrow$ $A=f^{-}f(A)$ for all subsets $A$ of $X$$\Leftrightarrow$$f(\bigcap A_{i})=\bigcap f(A_{i})$. (Surjec) $f$ is onto $\Leftrightarrow$ there is a function $f_{\textrm{R }}\!:Y\rightarrow X$ called the right inverse of $f$, such that $ff_{\textrm{R}}=\mathbf{1}_{Y}$ $\Leftrightarrow$ $B=ff^{-}(B)$ for all subsets $B$ of $Y$. As we are primarily concerned with non-injectivity of functions, saturated sets generated by equivalence classes of $f$ will play a significant role in our discussions. A relation $\mathscr{E}\mathcal{E}$ on a set $X$ is said to be an *equivalence relation* if it is[^4] (ER1) Reflexive: $(\forall x\in X)(x\mathcal{E}x)$. (ER2) Symmetric: $(\forall x,y\in X)(x\mathcal{E}y\Longrightarrow y\mathcal{E}x)$. (ER3) Transitive: $(\forall x,y,z\in X)(x\mathcal{E}y\wedge y\mathcal{E}z\Longrightarrow x\mathcal{E}z)$. Equivalence relations group together unequal elements $x_{1}\neq x_{2}$ of a set as equivalent according to the requirements of the relation. This is expressed as $x_{1}\sim x_{2}\textrm{ }(\textrm{mod }\mathcal{E})$ and will be represented here by the shorthand notation $x_{1}\sim_{\mathcal{E}}x_{2}$, or even simply as $x_{1}\sim x_{2}$ if the specification of $\mathcal{E}$ is not essential. Thus for a noninjective map if $f(x_{1})=f(x_{2})$ for $x_{1}\neq x_{2}$, then $x_{1}$ and $x_{2}$ can be considered to be equivalent to each other since they map onto the same point under $f$; thus $x_{1}\sim_{f}x_{2}\Leftrightarrow f(x_{1})=f(x_{2})$ defines the equivalence relation $\sim_{f}$ induced by the map $f$. Given an equivalence relation $\sim$ on a set $X$ and an element $x\in X$ the subset $$[x]\overset{\textrm{def}}=\{ y\in X\!:y\sim x\}$$ is called the *equivalence class of $x$;* thus $x\sim y\Leftrightarrow[x]=[y]$*.* In particular, equivalence classes generated by $f\!:X\rightarrow Y$, $[x]_{f}=\{ x_{\alpha}\in X\!:f(x_{\alpha})=f(x)\}$, will be a cornerstone of our analysis of chaos generated by the iterates of non-injective maps, and the equivalence relation $\sim_{f}:=\{(x,y)\!:f(x)=f(y)\}$ generated by $f$ is uniquely defined by the partition that $f$ induces on $X$. Of course as $x\sim x$, $x\in[x]$. It is a simple matter to see that any two equivalence classes are either disjoint or equal so that the equivalence classes generated by an equivalence relation on $X$ form a disjoint cover of $X.$ The *quotient set of $X$ under $\sim$,* denoted by $X/\sim\;:=\{[x]\!:x\in X\}$, has the equivalence classes $[x]$ as its elements; thus $[x]$ plays a dual role either as subsets of $X$ or as elements of $X/\sim$. The rule $x\mapsto[x]$ defines a surjective function $Q\!:X\rightarrow X/\sim$ known as the *quotient map.* **Example 1.1.** Let $$S^{1}=\{(x,y)\in\mathbb{R}^{2})\!:x^{2}+y^{2}=1\}$$ be the unit circle in $\mathbb{R}^{2}$. Consider $X=[0,1]$ as a subspace of $\mathbb{R}$, define a map $$q\!:X\rightarrow S^{1},\qquad s\longmapsto(\cos2\pi s,\sin2\pi s),\,\, s\in X,$$ from $\mathbb{R}$ to $\mathbb{R}^{2}$, and let $\sim$ be the equivalence relation on $X$ $$s\sim t\Longleftrightarrow(s=t)\vee(s=0,t=1)\vee(s=1,t=0).$$ If we bend $X$ around till its ends touch, the resulting circle represents the quotient set $Y=X/\sim$ whose points are equivalent under $\sim$ as follows $$[0]=\{0,1\}=[1],\qquad[s]=\{ s\}\,\textrm{for all }s\in(0,1).$$ Thus $q$ is bijective for $s\in(0,1)$ but two-to-one for the special values $s=0\textrm{ and }1$, so that for $s,t\in X$,$$s\sim t\Longleftrightarrow q(s)=q(t).$$ This yields a bijection $h\!:X/\sim\:\rightarrow S^{1}$ such that $$q=h\circ Q$$ defines the quotient map $Q\!:X\rightarrow X/\sim$ by $h([s])=q(s)$ for all $s\in[0,1]$. The situation is illustrated by the commutative diagram of Fig. \[Fig: quotient\] that appears as an integral component in a different and more general context in Sec. 2. It is to be noted that commutativity of the diagram implies that if a given equivalence relation $\sim$ on $X$ is completely determined by $q$ that associates the partitioning equivalence classes in $X$ to unique points in $S^{1}$, then $\sim$ is identical to the equivalence relation that is induced by $Q$ on $X$. Note that a larger size of the equivalence classes can be obtained by considering $X=\mathbb{R}_{+}$ for which $s\sim t\Leftrightarrow|s-t|\in\mathbb{Z}_{+}$.$\qquad\blacksquare$ ***End Tutorial1*** One of the central concepts that we consider and employ in this work is the inverse $f^{-}$ of a nonlinear, non-injective, function $f$; here the equivalence classes $[x]_{f}=f^{-}f(x)$ of $x\in X$ are the saturated subsets of $X$ that partition $X$. While a detailed treatment of this question in the form of the non-linear ill-posed problem and its solution is given in Sec. 2 [@Sengupta1997], it is sufficient to point out here from Figs. \[Fig: functions\](c) and \[Fig: functions\](d), that the inverse of a noninjective function is not a function but a multifunction while the inverse of a multifunction is a noninjective function. Hence one has the general result that$$\begin{aligned} f\textrm{ is a non injective function} & \Longleftrightarrow & f^{-}\textrm{ is a multifunction}.\label{Eqn: func-multi}\\ f\textrm{ is a multifunction} & \Longleftrightarrow & f^{-}\textrm{ is a non injective function}\nonumber \end{aligned}$$ The inverse of a multifunction $\mathscr{M}\!:X\qquad Y$ is a generalization of the corresponding notion for a function $f\!:X\rightarrow Y$ such that $$\mathscr M^{-}(y)\overset{\textrm{def}}=\{ x\in X\!:y\in\mathscr{M}(x)\}$$ leads to $${\textstyle \mathscr M^{-}(B)=\{ x\in X\!:\mathscr{M}(x)\bigcap B\neq\emptyset\}}$$ for any $B\subseteq Y$, while a more restricted inverse that we shall not be concerned with is given as $\mathscr M^{+}(B)=\{ x\in X\!:\mathscr{M}(x)\subseteq B\}$. Obviously, $\mathscr M^{+}(B)\subseteq\mathscr M^{-}(B)$. A multifunction is injective if $x_{1}\neq x_{2}\Rightarrow\mathscr{M}(x_{1})\bigcap\mathscr{M}(x_{2})=\emptyset$, and in common with functions it is true that $$\begin{aligned} \mathscr{M}\left(\bigcup_{\alpha\in\mathbb{{D}}}A_{\alpha}\right)= & \bigcup_{\alpha\in\mathbb{{D}}}\mathscr{M}(A_{\alpha})\\ \mathscr{M}\left(\bigcap_{\alpha\in\mathbb{{D}}}A_{\alpha}\right)\subseteq & \bigcap_{\alpha\in\mathbb{{D}}}\mathscr{M}(A_{\alpha})\end{aligned}$$ and where $\mathbb{D}$ is an index set. The following illustrates the difference between the two inverses of $\mathscr{M}$. Let $X$ be a set that is partitioned into two disjoint $\mathscr{M}$-invariant subsets $X_{1}$ and $X_{2}$. If $x\in X_{1}$ (or $x\in X_{2}$) then $\mathscr{M}(x)$ represents that part of $X_{1}$ (or of $X_{2}$ ) that is realized immediately after one application of $\mathscr{M}$, while $\mathscr M^{-}(x)$ denotes the possible precursors of $x$ in $X_{1}$ (or of $X_{2}$) and $\mathscr M^{+}(B)$ is that subset of $X$ whose image lies in $B$ for any subset $B\subset X$. In this work the multifunctions we are explicitly concerned with arise as the inverses of non-injective functions. The second major component of our theory is the *graphical convergence of a net of functions to a multifunction.* In Tutorial2 below, we replace for the sake of simplicity and without loss of generality, the net (which is basically a sequence where the index set is not necessarily the positive integers; thus every sequence is a net but the family[^5] indexed, for example, by $\mathbb{Z}$, the set of *all* integers, is a net and not a sequence) with a sequence and provide the necessary background and motivation for the concept of graphical convergence. ***Begin Tutorial2: Convergence of Functions*** This Tutorial reviews the inadequacy of the usual notions of convergence of functions either to limit functions or to distributions and suggests the motivation and need for introduction of the notion of graphical convergence of functions to multifunctions. Here, we follow closely the exposition of @Korevaar1968, and use the notation $(f_{k})_{k=1}^{\infty}$ to denote real or complex valued functions on a bounded or unbounded interval $J$. A sequence of piecewise continuous functions $(f_{k})_{k=1}^{\infty}$ is said to converge to the function $f$, notation $f_{k}\rightarrow f$, on a bounded or unbounded interval $J$[^6] \(1) *Pointwise* if$$f_{k}(x)\longrightarrow f(x)\qquad\textrm{for all }x\in J,$$ that is: Given any arbitrary real number $\varepsilon>0$ there exists a $K\in\mathbb{N}$ that may depend on $x$, such that $|f_{k}(x)-f(x)|<\varepsilon$ for all $k\geq K$. \(2) *Uniformly* if $$\sup_{x\in J}|f(x)-f_{k}(x)|\longrightarrow0\qquad\textrm{as }k\longrightarrow\infty,$$ that is: Given any arbitrary real number $\varepsilon>0$ there exists a $K\in\mathbb{N}$, such that $\sup_{x\in J}|f_{k}(x)-f(x)|<\varepsilon$ for all $k\geq K$. \(3) *In the mean of order $p\geq1$* if $|f(x)-f_{k}(x)|^{p}$ is integrable over $J$ for each $k$ $$\int_{J}|f(x)-f_{k}(x)|^{p}\longrightarrow0\qquad\textrm{as }k\rightarrow\infty.$$ For $p=1$, this is the simple case of *convergence in the mean.* \(4) *In the mean $m$-integrally* if it is possible to select indefinite integrals $$f_{k}^{(-m)}(x)=\pi_{k}(x)+\int_{c}^{x}dx_{1}\int_{c}^{x_{1}}dx_{2}\cdots\int_{c}^{x_{m-1}}dx_{m}f_{k}(x_{m})$$ and $$f^{(-m)}(x)=\pi(x)+\int_{c}^{x}dx_{1}\int_{c}^{x_{1}}dx_{2}\cdots\int_{c}^{x_{m-1}}dx_{m}f(x_{m})$$ such that for some arbitrary real $p\geq1$, $$\int_{J}|f^{(-m)}-f_{k}^{(-m)}|^{p}\longrightarrow0\qquad\textrm{as }k\rightarrow\infty.$$ where the polynomials $\pi_{k}(x)$ and $\pi(x)$ are of degree $<m$, and $c$ is a constant to be chosen appropriately. \(5) *Relative to test functions $\varphi$* if $f\varphi$ and $f_{k}\varphi$ are integrable over $J$ and $$\int_{J}(f_{k}-f)\varphi\longrightarrow0\qquad\textrm{for every }\varphi\in\mathcal{C}_{0}^{\infty}(J)\textrm{ as }k\longrightarrow\infty,$$ where $\mathcal{C}_{0}^{\infty}(J)$ is the class of infinitely differentiable continuous functions that vanish throughout some neighbourhood of each of the end points of $J$. For an unbounded $J$, a function is said to vanish in some neighbourhood of $+\infty$ if it vanishes on some ray $(r,\infty)$. While pointwise convergence does not imply any other type of convergence, uniform convergence on a bounded interval implies all the other convergences. It is to be observed that apart from pointwise and uniform convergences, all the other modes listed above represent some sort of an averaged contribution of the entire interval $J$ and are therefore not of much use when pointwise behaviour of the limit $f$ is necessary. Thus while limits in the mean are not unique, oscillating functions are tamed by $m$-integral convergence for adequately large values of $m$, and convergence relative to test functions, as we see below, can be essentially reduced to $m$-integral convergence. On the contrary, our graphical convergence — which may be considered as a pointwise biconvergence with respect to both the direct and inverse images of $f$ just as usual pointwise convergence is with respect to its direct image only — allows a sequence (in fact, a net) of functions to converge to an arbitrary relation, unhindered by external influences such as the effects of integrations and test functions. To see how this can indeed matter, consider the following **Example 1.2.** Let $f_{k}(x)=\sin kx,\, k=1,2,\cdots$ and let $J$ be any bounded interval of the real line. Then $1$-integrally we have$$f_{k}^{(-1)}(x)=-\frac{1}{k}\cos kx=-\frac{1}{k}+\int_{0}^{x}\sin kx_{1}dx_{1},$$ which obviously converges to $0$ uniformly (and therefore in the mean) as $k\rightarrow\infty$. And herein lies the point: even though we cannot conclude about the exact nature of $\sin kx$ as $k$ increases indefinitely (except that its oscillations become more and more pronounced), we may very definitely state that $\lim_{k\rightarrow\infty}(\cos kx)/k=0$ uniformly. Hence from$$f_{k}^{(-1)}(x)\longrightarrow0=0+\int_{0}^{x}\lim_{k\rightarrow\infty}\sin kx_{1}dx_{1}$$ it follows that $$\lim_{k\rightarrow\infty}\sin kx=0\label{Eqn: intsin}$$ $1$-integrally. Continuing with the same sequence of functions, we now examine its test-functional convergence with respect to $\varphi\in\mathcal{C}_{0}^{1}(-\infty,\infty)$ that vanishes for all $x\notin(\alpha,\beta)$. Integrating by parts, $$\begin{aligned} {\displaystyle {\displaystyle \int_{-\infty}^{\infty}f_{k}\varphi}}= & {\displaystyle \int_{\alpha}^{\beta}\varphi(x_{1})\sin kx_{1}dx_{1}}\\ = & -\frac{1}{k}\left[\varphi(x_{1})\cos kx_{1}\right]_{\alpha}^{\beta}-\frac{1}{k}\int_{\alpha}^{\beta}\varphi^{\prime}(x_{1})\cos kx_{1}dx_{1}\end{aligned}$$ The first integrated term is $0$ due to the conditions on $\varphi$ while the second also vanishes because $\varphi\in\mathcal{C}_{0}^{1}(-\infty,\infty)$. Hence $$\int_{-\infty}^{\infty}f_{k}\varphi\longrightarrow0=\int_{\alpha}^{\beta}\lim_{k\rightarrow\infty}\varphi(x_{1})\sin ksdx_{1}$$ for all $\varphi$, and leading to the conclusion that $$\lim_{k\rightarrow\infty}\sin kx=0\label{Eqn: testsin}$$ test-functionally.$\qquad\blacksquare$ This example illustrates the fact that if $\textrm{Supp}(\varphi)=[\alpha,\beta]\subseteq J$[^7], integrating by parts sufficiently large number of times so as to wipe out the pathological behaviour of $(f_{k})$ gives $$\begin{aligned} \int_{J}f_{k}\varphi= & \int_{\alpha}^{\beta}f_{k}\varphi\\ = & \int_{\alpha}^{\beta}f_{k}^{(-1)}\varphi^{\prime}=\cdots=(-1)^{m}\int_{\alpha}^{\beta}f_{k}^{(-m)}\varphi^{m}\end{aligned}$$ where $f_{k}^{(-m)}(x)=\pi_{k}(x)+\int_{c}^{x}dx_{1}\int_{c}^{x_{1}}dx_{2}\cdots\int_{c}^{x_{m-1}}dx_{m}f_{k}(x_{m})$ is an $m$-times arbitrary indefinite integral of $f_{k}$. If now it is true that $\int_{\alpha}^{\beta}f_{k}^{(-m)}\rightarrow\int_{\alpha}^{\beta}f^{(-m)}$, then it must also be true that $f_{k}^{(-m)}\varphi^{(m)}$ converges in the mean to $f^{(-m)}\varphi^{(m)}$ so that $$\int_{\alpha}^{\beta}f_{k}\varphi=(-1)^{m}\int_{\alpha}^{\beta}f_{k}^{(-m)}\varphi^{(m)}\longrightarrow(-1)^{m}\int_{\alpha}^{\beta}f^{(-m)}\varphi^{(m)}=\int_{\alpha}^{\beta}f\varphi.$$ In fact the converse also holds leading to the following Equivalences between $m$-convergence in the mean and convergence with respect to test-functions, [@Korevaar1968]. **Type 1 Equivalence.** If $f$ and $(f_{k})$ are functions on $J$ that are integrable on every interior subinterval, then the following are equivalent statements. \(a) For every interior subinterval $I$ of $J$ there is an integer $m_{I}\geq0$, and hence a smallest integer $m\geq0$, such that certain indefinite integrals $f_{k}^{(-m)}$ of the functions $f_{k}$ converge in the mean on $I$ to an indefinite integral $f^{(-m)}$; thus $\int_{I}|f_{k}^{(-m)}-f^{(-m)}|\rightarrow0.$ \(b) $\int_{J}(f_{k}-f)\varphi\rightarrow0$ for every $\varphi\in\mathcal{C}_{0}^{\infty}(J)$. A significant generalization of this Equivalence is obtained by dropping the restriction that the limit object $f$ be a function. The need for this generalization arises because metric function spaces are known not to be complete: Consider the sequence of functions (Fig. \[Fig: FuncSpace\](a)) $$\begin{aligned} f_{k}(x)= & \left\{ \begin{array}{lcl} 0 & \textrm{} & \textrm{if }a\leq x\leq0\\ kx & \textrm{} & \textrm{if }0\leq x\leq1/k\\ 1 & \textrm{} & \textrm{if }1/k\leq x\leq b\end{array}\right.\label{Eqn: Lp[a,b]}\end{aligned}$$ which is not Cauchy in the uniform metric $\rho(f_{j},f_{k})=\sup_{a\leq x\leq b}|f_{j}(x)-f_{k}(x)|$ but is Cauchy in the mean $\rho(f_{j},f_{k})=\int_{a}^{b}|f_{j}(x)-f_{k}(x)|dx$, or even pointwise. However in either case, $(f_{k})$ cannot converge in the respective metrics to a *continuous function* and the limit is a discontinuous unit step function $$\Theta(x)=\left\{ \begin{array}{lcl} 0 & & \textrm{if }a\leq x\leq0\\ 1 & & \textrm{if }0<x\leq b\end{array}\right.$$ with graph $([a,0],0)\bigcup((0,b],1)$, which is also integrable on $[a,b]$. Thus even if the limit of the sequence of continuous functions is not continuous, both the limit and the members of the sequence are integrable functions. This Riemann integration is not sufficiently general, however, and this type of integrability needs to be replaced by a much weaker condition resulting in the larger class of the Lebesgue integrable complete space of functions $L[a,b]$.[^8] The functions in Fig \[Fig: FuncSpace\](b1), $$\delta_{k}(x)=\left\{ \begin{array}{ccl} k & & \textrm{if }0<x<1/k\\ 0 & & x\in[a,b]-(0,1/k),\end{array}\right.$$ can be associated with the arbitrary indefinite integrals $$\Theta_{k}(x)\overset{\textrm{def}}=\delta_{k}^{(-1)}(x)=\left\{ \begin{array}{lcl} 0 & & a\leq x\leq0\\ kx & & 0<x<1/k\\ 1 & & 1/k\leq x\leq b\end{array}\right.$$ of Fig. \[Fig: FuncSpace\](b2), which, as noted above, converge in the mean to the unit step function $\Theta(x)$; hence $\int_{-\infty}^{\infty}\delta_{k}\varphi\equiv\int_{\alpha}^{\beta}\delta_{k}\varphi=-\int_{\alpha}^{\beta}\delta_{k}^{(-1)}\varphi^{\prime}\rightarrow-\int_{0}^{\beta}\varphi^{\prime}(x)dx=\varphi(0)$. But there can be no *functional relation $\delta(x)$* for which $\int_{\alpha}^{\beta}\delta(x)\varphi(x)dx=\varphi(0)$ for *all* $\varphi\in C_{0}^{1}[\alpha,\beta]$, so that unlike in the case in Type 1 Equivalence, the limit in the mean $\Theta(x)$ of the indefinite integrals $\delta_{k}^{(-1)}(x)$ *cannot be expressed as the indefinite integral $\delta^{(-1)}(x)$ of some function $\delta(x)$ on any interval containing the origin.* This leads to the second more general type of equivalence **Type 2 Equivalence.** If $(f_{k})$ are functions on $J$ that are integrable on every interior subinterval, then the following are equivalent statements. \(a) For every interior subinterval $I$ of $J$ there is an integer $m_{I}\geq0$, and hence a smallest integer $m\geq0$, such that certain indefinite integrals $f_{k}^{(-m)}$ of the functions $f_{k}$ converge in the mean on $I$ to an integrable function $\Theta$ which, unlike in Type 1 Equivalence, need not itself be an indefinite integral of some function $f$. \(b) $c_{k}(\varphi)=\int_{J}f_{k}\varphi\rightarrow c(\varphi)$ for every $\varphi\in\mathcal{C}_{0}^{\infty}(J)$. Since we are now given that $\int_{I}f_{k}^{(-m)}(x)dx\rightarrow\int_{I}\Psi(x)dx$, it must also be true that $f_{k}^{(-m)}\varphi^{(m)}$ converges in the mean to $\Psi\varphi^{(m)}$ whence $$\int_{J}f_{k}\varphi=(-1)^{m}\int_{I}f_{k}^{(-m)}\varphi^{(m)}\longrightarrow(-1)^{m}\int_{I}\Psi\varphi^{(m)}\left(\neq(-1)^{m}\int_{I}f^{(-m)}\varphi^{(m)}\right).$$ The natural question that arises at this stage is then: What is the nature of the relation (not function any more) $\Psi(x)$? For this it is now stipulated, despite the non-equality in the equation above, that as in the mean $m$-integral convergence of $(f_{k})$ to a *function* $f$, $$\Theta(x):=\lim_{k\rightarrow\infty}\delta_{k}^{(-1)}(x)\overset{\textrm{def}}=\int_{-\infty}^{x}\delta(x^{\prime})dx^{\prime}\label{Eqn: delta1}$$ *defines* the non-functional relation (“generalized function”) $\delta(x)$ integrally as a solution of the integral equation (\[Eqn: delta1\]) of the first kind; hence formally[^9] $$\delta(x)=\frac{d\Theta}{dx}\label{Eqn: delta2}$$ ***End Tutorial2*** The above tells us that the “delta function” is not a function but its indefinite integral is the piecewise continuous *function* $\Theta$ obtained as the mean (or pointwise) limit of a sequence of non-differentiable functions with the integral of $d\Theta_{k}(x)/dx$ being preserved for all $k\in\mathbb{Z}_{+}$. What then is the delta (and not its integral)? The answer to this question is contained in our multifunctional extension $\textrm{Multi}(X,Y)$ of the function space $\textrm{Map}(X,Y)$ considered in Sec. 3. Our treatment of ill-posed problems is used to obtain an understanding and interpretation of the numerical results of the discretized spectral approximation in neutron transport theory [@Sengupta1988; @Sengupta1995]. The main conclusions are the following: In a one-dimensional discrete system that is governed by the iterates of a nonlinear map, the dynamics is chaotic if and only if the system evolves to a state of *maximal ill-posedness.* The analysis is based on the non-injectivity, and hence ill-posedness, of the map; this may be viewed as a mathematical formulation of the *stretch-and-fold* and *stretch-cut-and-paste* kneading operations of the dough that are well-established artifacts in the theory of chaos and the concept of maximal ill-posedness helps in obtaining a *physical understanding* of the nature of chaos. We do this through the fundamental concept of the *graphical convergence* of a sequence (generally a net) of functions [@Sengupta2000] that is allowed to converge graphically, when the conditions are right, to a set-valued map or multifunction. Since ill-posed problems naturally lead to multifunctional inverses through functional generalized inverses [@Sengupta1997], it is natural to seek solutions of ill-posed problems in multifunctional space $\textrm{Multi}(X,Y)$ rather than in spaces of functions $\textrm{Map}(X,Y)$; here $\textrm{Multi}(X,Y)$ is an extension of $\textrm{Map}(X,Y)$ that is generally larger than the smallest dense extension $\textrm{Multi}_{\mid}(X,Y)$. Feedback and iteration are natural processes by which nature evolves itself. Thus almost every process of evolution is a self-correction process by which the system proceeds from the present to the future through a controlled mechanism of input and evaluation of the past. Evolution laws are inherently nonlinear and complex; here *complexity* is to be understood as the natural manifestation of the nonlinear laws that govern the evolution of the system. This work presents a mathematical description of complexity based on [@Sengupta1997] and [@Sengupta2000] and is organized as follows. In Sec. 1, we follow [@Sengupta1997] to give an overview of ill-posed problems and their solution that forms the foundation of our approach. Secs. 2 to 4 apply these ideas by defining a chaotic dynamical system as a *maximally ill-posed problem;* by doing this we are able to overcome the limitations of the three Devaney characterizations of chaos [@Devaney1989] that apply to the specific case of iteration of transformations in a metric space, and the resulting graphical convergence of functions to multifunctions is the basic tool of our approach. Sec. 5 analyzes graphical convergence in $\textrm{Multi}(X)$ for the discretized spectral approximation of neutron transport theory, which suggests a natural link between ill-posed problems and spectral theory of non-linear operators. This seems to offer an answer to the question of *why* a natural system should increase its complexity, and eventually tend toward chaoticity, by becoming increasingly nonlinear. **2. Ill-Posed Problem and its solution** This section based on @Sengupta1997 presents a formulation and solution of ill-posed problems arising out of the non-injectivity of a function $f\!:X\rightarrow Y$ between topological spaces $X$ and $Y$. A workable knowledge of this approach is necessary as our theory of chaos leading to the characterization of chaotic systems as being a *maximally ill-posed* state of a dynamical system is a direct application of these ideas and can be taken to constitute a mathematical representation of the familiar *stretch-cut-and paste* and *stretch-and-fold* paradigms of chaos. The problem of finding an $x\in X$ for a given $y\in Y$ from the functional relation $f(x)=y$ is an inverse problem that is *ill-posed* (or, the equation $f(x)=y$ is ill-posed) if any one or more of the following conditions are satisfied. (IP1) $f$ *is not injective.* This *non-uniqueness* problem of the solution for a given $y$ is the single most significant criterion of ill-posedness used in this work. (IP2) *$f$ is not surjective.* For a $y\in Y$, this is the *existence* problem of the given equation. (IP3) When $f$ *is bijective,* the inverse *$f^{-1}$* is not continuous, which means that small changes in $y$ may lead to large changes in $x$. A problem $f(x)=y$ for which a solution exists, is unique, and small changes in data $y$ lead to only small changes in the solution $x$ is said to be *well-posed* or *properly posed.* This means that $f(x)=y$ is well-posed if $f$ is bijective and the inverse $f^{-1}\!:Y\rightarrow X$ is continuous; otherwise the equation is *ill-posed* or *improperly posed.* It is to be noted that the three criteria are not, in general, independent of each other. Thus if $f$ represents a bijective, bounded linear operator between Banach spaces $X$ and $Y$, then the inverse mapping theorem guarantees that the inverse $f^{-1}$ is continuous. Hence ill-posedness depends not only on the algebraic structures of $X$, $Y$, $f$ but also on the topologies of $X$ and $Y$. **Example 2.1.** As a non-trivial example of an inverse problem, consider the heat equation$$\frac{\partial\theta(x,t)}{\partial t}=c^{2}\frac{\partial^{2}\theta(x,t)}{\partial x^{2}}$$ for the temperature distribution $\theta(x,t)$ of a one-dimensional homogeneous rod of length $L$ satisfying the initial condition $\theta(x,0)=\theta_{0}(x),\textrm{ }0\leq x\leq L$, and boundary conditions $\theta(0,t)=0=\theta(L,t),\,0\leq t\leq T$, having the Fourier sine-series solution $$\theta(x,t)=\sum_{n=1}^{\infty}A_{n}\sin\left(\frac{n\pi}{L}x\right)e^{-\lambda_{n}^{2}t}\label{Eqn: heat1}$$ where $\lambda_{n}=(c\pi/a)n$ and $$A_{n}=\frac{2}{L}\int_{0}^{a}\theta_{0}(x^{\prime})\sin\left(\frac{n\pi}{L}x^{\prime}\right)dx^{\prime}$$ are the Fourier expansion coefficients. While the direct problem evaluates $\theta(x,t)$ from the differential equation and initial temperature distribution $\theta_{0}(x)$, the inverse problem calculates $\theta_{0}(x)$ from the integral equation $$\theta_{T}(x)=\frac{2}{L}\int_{0}^{a}k(x,x^{\prime})\theta_{0}(x^{\prime})dx^{\prime},\qquad0\leq x\leq L,$$ when this final temperature $\theta_{T}$ is known, and $$k(x,x^{\prime})=\sum_{n=1}^{\infty}\sin\left(\frac{n\pi}{L}x\right)\sin\left(\frac{n\pi}{L}x^{\prime}\right)e^{-\lambda_{n}^{2}T}$$ is the kernel of the integral equation. In terms of the final temperature the distribution becomes $$\theta_{T}(x)=\sum_{n=1}^{\infty}B_{n}\sin\left(\frac{n\pi}{L}x\right)e^{-\lambda_{n}^{2}(t-T)}\label{Eqn: heat2}$$ with Fourier coefficients $$B_{n}=\frac{2}{L}\int_{0}^{a}\theta_{T}(x^{\prime})\sin\left(\frac{n\pi}{L}x^{\prime}\right)dx^{\prime}.$$ In $L^{2}[0,a]$, Eqs. (\[Eqn: heat1\]) and (\[Eqn: heat2\]) at $t=T$ and $t=0$ yield respectively $$\Vert\theta_{T}(x)\Vert^{2}=\frac{L}{2}\sum_{n=1}^{\infty}A_{n}^{2}e^{-2\lambda_{n}^{2}T}\leq e^{-2\lambda_{1}^{2}T}\Vert\theta_{0}\Vert^{2}\label{Eqn: heat3}$$ $$\Vert\theta_{0}\Vert^{2}=\frac{L}{2}\sum_{n=1}^{\infty}B_{n}^{2}e^{2\lambda_{n}^{2}T}.\label{Eqn: heat4}$$ The last two equations differ from each other in the significant respect that whereas Eq. (\[Eqn: heat3\]) shows that the direct problem is well-posed according to (IP3), Eq. (\[Eqn: heat4\]) means that in the absence of similar bounds the inverse problem is ill-posed.[^10]$\qquad\blacksquare$ **Example 2.2.** Consider the ****Volterra integral equation of the first kind $$y(x)=\int_{a}^{x}r(x^{\prime})dx^{\prime}=Kr$$ where $y,r\in C[a,b]$ and $K\!:C[0,1]\rightarrow C[0,1]$ is the corresponding integral operator. Since the differential operator $D=d/dx$ under the sup-norm $\Vert r\Vert=\sup_{0\leq x\leq1}|r(x)|$ is unbounded, the inverse problem $r=Dy$ for a differentiable function $y$ on $[a,b]$ is ill-posed, see Example 6.1. However, $y=Kr$ becomes well-posed if $y$ is considered to be in $C^{1}[0,1]$ with norm $\Vert y\Vert=\sup_{0\leq x\leq1}|Dy|$. This illustrates the importance of the topologies of $X$ and $Y$ in determining the ill-posed nature of the problem when this is due to (IP3).$\qquad\blacksquare$ Ill-posed problems in nonlinear mathematics of type (IP1) arising from the non-injectivity of $f$ can be considered to be a generalization of non-uniqueness of solutions of linear equations as, for example, in eigenvalue problems or in the solution of a system of linear algebraic equations with a larger number of unknowns than the number of equations. In both cases, for a given $y\in Y$, the solution set of the equation $f(x)=y$ is given by $$f^{-}(y)=[x]_{f}=\{ x^{\prime}\in X:f(x^{\prime})=f(x)=y\}.$$ A significant point of difference between linear and nonlinear problems is that unlike the special importance of 0 in linear mathematics, there are no preferred elements in nonlinear problems; this leads to a shift of emphasis from the null space of linear problems to equivalence classes for nonlinear equations. To motivate the role of equivalence classes, let us consider the null spaces in the following linear problems. \(a) Let $f:\mathbb{R}^{2}\rightarrow\mathbb{R}$ be defined by $f(x,y)=x+y$, $(x,y)\in\mathbb{R}^{2}$. The null space of $f$ is generated by the equation $y=-x$ on the $x$-$y$ plane, and the graph of $f$ is the plane passing through the lines $\rho=x$ and $\rho=y.$ For each $\rho\in\textrm{R}$ the equivalence classes $f^{-}(\rho)=\{(x,y)\in\textrm{R}^{2}\!:x+y=\rho\}$ are lines on the graph parallel to the null set. \(b) For a linear operator $A\!:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$, $m<n$, satisfying (1) and (2), the problem $Ax=y$ reduces $A$ to echelon form with rank $r$ less than $\min\{ m,n\}$, when the given equations are consistent. The solution however, produces a generalized inverse leading to a set-valued inverse $A^{-}$ of $A$ for which the inverse images of $y\in\mathcal{R}(A)$ are multivalued because of the non-trivial null space of $A$ introduced by assumption (1). Specifically, a null-space of dimension $n-r$ is generated by the free variables $\{ x_{j}\}_{j=r+1}^{n}$ which are arbitrary: this is illposedness of type (1). In addition, $m-r$ rows of the row reduced echelon form of $A$ have all 0 entries that introduces restrictions on $m-r$ coordinates $\{ y_{i}\}_{i=r+1}^{m}$ of $y$ which are now related to $\{ y_{i}\}_{i=1}^{r}$: this illustrates illposedness of type (2). Inverse ill-posed problems therefore generate multivalued solutions through a generalized inverse of the mapping. \(c) The eigenvalue problem $$\left(\frac{d^{2}}{dx^{2}}+\lambda^{2}\right)y=0\qquad y(0)=0=y(1)$$ has the following equivalence class of 0 $$[0]_{D^{2}}=\{\sin(\pi mx)\}_{m=0}^{\infty},\qquad D^{2}=\left(d^{2}/dx^{2}+\lambda^{2}\right),$$ as its eigenfunctions corresponding to the eigenvalues $\lambda_{m}=\pi m$. Ill-posed problems are primarily of interest to us explicitly as noninjective maps $f$, that is under the condition of (IP1). The two other conditions (IP2) and (IP3) are not as significant and play only an implicit role in the theory. In its application to iterative systems, the degree of non-injectivity of $f$ defined as the number of its injective branches, increases with iteration of the map. A necessary (but not sufficient) condition for chaos to occur is the increasing non-injectivity of $f$ that is expressed descriptively in the chaos literature as *stretch-and-fold* or *stretch-cut-and-paste* operations. This increasing noninjectivity that we discuss in the following sections, is what causes a dynamical system to tend toward chaoticity. Ill-posedness arising from non-surjectivity of (injective) $f$ in the form of *regularization* [@Tikhonov1977] **has received wide attention in the literature of ill-posed problems; this however is not of much significance in our work. ***Begin Tutorial3: Generalized Inverse*** In this Tutorial, we take a quick look at the equation $a(x)=y$, where $a\!:X\rightarrow Y$ is a linear map that need not be either one-one or onto. Specifically, we will take $X$ and $Y$ to be the Euclidean spaces $\mathbb{R}^{n}$ and $\mathbb{R}^{m}$ so that $a$ has a matrix representation $A\in\mathbb{R}^{m\times n}$ where $\mathbb{R}^{m\times n}$ is the collection of $m\times n$ matrices with real entries. The inverse $A^{-1}$ exists and is unique iff $m=n$ and $\textrm{rank}(A)=n$; this is the situation depicted in Fig. \[Fig: functions\](a). If $A$ is neither one-one or onto, then we need to consider the multifunction $A^{-}$, a functional choice of which is known as the *generalized inverse* $G$ of $A$. A good introductory text for generalized inverses is @Campbell1979Figure \[Fig: MP\_Inverse\](a) introduces the following definition of the *Moore-Penrose* generalized inverse $G_{\textrm{MP}}$. **Definition 2.1.** ***Moore-Penrose Inverse.*** *If $a\!:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ is a linear transformation with matrix representation $A\in\mathbb{R}^{m\times n}$ then the* Moore-Penrose inverse $G_{\textrm{MP}}\in\mathbb{R}^{n\times m}$ of $A$ *(we will use the same notation* $G_{\textrm{MP}}\!:\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$ *for the inverse of the map $a$) is the noninjective map defined in terms of the row and column spaces of $A$,* $\textrm{row}(A)=\mathcal{R}(A^{\textrm{T}})$, $\textrm{col}(A)=\mathcal{R}(A)$*, as* $$G_{\textrm{MP}}(y)\overset{\textrm{def}}=\left\{ \begin{array}{lcl} (a|_{\textrm{row}(A)})^{-1}(y), & & \textrm{if }y\in\textrm{col}(A)\\ 0 & & \textrm{if }y\in\mathcal{N}(A^{\textrm{T}}).\end{array}\right.\qquad\square\label{Eqn: Def: Moore-Penrose}$$ Note that the restriction $a|_{\textrm{row}(A)}$ of $a$ to $\mathcal{R}(A^{\textrm{T}})$ is bijective so that the inverse $(a|_{\textrm{row}(A)})^{-1}$ is well-defined. The role of the transpose matrix appears naturally, and the $G_{\textrm{MP}}$ of Eq. (\[Eqn: Def: Moore-Penrose\]) is the unique matrix that satisfies the conditions $$\begin{array}{c} AG_{\textrm{MP}}A=A,\quad G_{\textrm{MP}}AG_{\textrm{MP}}=G_{\textrm{MP}},\\ (G_{\textrm{MP}}A)^{\textrm{T}}=G_{\textrm{MP}}A,\quad(AG_{\textrm{MP}})^{\textrm{T}}=AG_{\textrm{MP}}\end{array}\label{Eqn: MPInverse}$$ that follow immediately from the definition (\[Eqn: Def: Moore-Penrose\]); hence $G_{\textrm{MP}}A$ and $AG_{\textrm{MP}}$ are orthogonal projections[^11] onto the subspaces $\mathcal{R}(A^{\textrm{T}})=\mathcal{R}(G_{\textrm{MP}})$ and $\mathcal{R}(A)$ respectively. Recall that the range space $\mathcal{R}(A^{\textrm{T}})$ of $A^{\textrm{T}}$ is the same as the *row space* $\textrm{row}(A)$ of $A$, and $\mathcal{R}(A)$ is also known as the *column space* of $A$, $\textrm{col}(A)$. **Example 2.3.** For $a\!:\mathbb{R}^{5}\rightarrow\mathbb{R}^{4}$, let $$A=\left(\begin{array}{rrrrr} 1 & -3 & 2 & 1 & 2\\ 3 & -9 & 10 & 2 & 9\\ 2 & -6 & 4 & 2 & 4\\ 2 & -6 & 8 & 1 & 7\end{array}\right)$$ By reducing the augmented matrix $\left(A|y\right)$ to the row-reduced echelon form, it can be verified that the null and range spaces of $A$ are $3$- and $2$-dimensional respectively. A basis for the null space of $A^{\textrm{T}}$ and of the row and column space of $A$ obtained from the echelon form are respectively $$\left(\begin{array}{r} -2\\ 0\\ 1\\ 0\end{array}\right),\textrm{ }\left(\begin{array}{r} 1\\ -1\\ 0\\ 1\end{array}\right);\quad\textrm{and }\left(\begin{array}{r} 1\\ -3\\ 0\\ 3/2\\ 1/2\end{array}\right),\textrm{ }\left(\begin{array}{r} 0\\ 0\\ 1\\ -1/4\\ 3/4\end{array}\right);\textrm{ }\left(\begin{array}{r} 1\\ 0\\ 2\\ -1\end{array}\right),\textrm{ }\left(\begin{array}{r} 0\\ 1\\ 0\\ 1\end{array}\right).$$ According to its definition Eq. (\[Eqn: Def: Moore-Penrose\]), the Moore-Penrose inverse maps the middle two of the above set to $(0,0,0,0,0)^{\textrm{T}}$, and the $A$-image of the first two (which are respectively $(19,70,38,51)^{\textrm{T}}$ and $(70,275,140,205)^{\textrm{T}}$ lying, as they must, in the span of the last two), to the span of $(1,-3,2,1,2)^{\textrm{T}}$ and $(3,-9,10,2,9)^{\textrm{T}}$ because $a$ restricted to this subspace of $\mathbb{R}^{5}$ is bijective. Hence $$G_{\textrm{MP}}\left(A\left(\begin{array}{r} 1\\ -3\\ 0\\ 3/2\\ 1/2\end{array}\right)\textrm{ }A\left(\begin{array}{r} 0\\ 0\\ 1\\ -1/4\\ 3/4\end{array}\right)\begin{array}{rr} -2 & 1\\ 0 & -1\\ 1 & 0\\ 0 & 1\end{array}\right)=\left(\begin{array}{rrrr} 1 & 0 & 0 & 0\\ -3 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 3/2 & -1/4 & 0 & 0\\ 1/2 & 3/4 & 0 & 0\end{array}\right).$$ The second matrix on the left is invertible as its rank is $4$. This gives $${\displaystyle G_{\textrm{MP}}=\left(\begin{array}{rrrr} 9/275 & -1/275 & 18/275 & -2/55\\ -27/275 & 3/275 & -54/275 & 6/55\\ -10/143 & 6/143 & -20/143 & 16/143\\ 238/3575 & -57/3575 & 476/3575 & -59/715\\ -129/3575 & 106/3575 & -258/3575 & 47/715\end{array}\right)}\label{Eqn: MPEx5}$$ as the Moore-Penrose inverse of $A$ that readily verifies all the four conditions of Eqs. (\[Eqn: MPInverse\]). The basic point here is that, as in the case of a bijective map, $G_{\textrm{MP}}A$ and $AG_{\textrm{MP}}$ are identities on the row and column spaces of $A$ that define its rank. For later use — when we return to this example for a simpler inverse $G$ — given below are the orthonormal bases of the four fundamental subspaces with respect to which $G_{\textrm{MP}}$ is a representation of the generalized inverse of $A$; these calculations were done by MATLAB. The basis for \(a) the column space of $A$ consists of the first $2$ columns of the eigenvectors of $AA^{\textrm{T}}$: $$\begin{array}{c} (-1633/2585,-363/892,\textrm{ }3317/6387,\textrm{ }363/892)^{\textrm{T}}\\ (-929/1435,\textrm{ }709/1319,\textrm{ }346/6299,-709/1319)^{\textrm{T}}\end{array}$$ \(b) the null space of $A^{\textrm{T}}$ consists of the last $2$ columns of the eigenvectors of $AA^{\textrm{T}}$:$$\begin{array}{c} (-3185/8306,\textrm{ }293/2493,-3185/4153,\textrm{ }1777/3547)^{\textrm{T}}\\ (323/1732,\textrm{ }533/731,\textrm{ }323/866,\textrm{ }1037/1911)^{\textrm{T}}\end{array}$$ \(c) the row space of $A$ consists of the first $2$ columns of the eigenvectors of $A^{\textrm{T}}A$: $$\begin{array}{c} (421/13823,\textrm{ }44/14895,-569/918,-659/2526,\textrm{ }1036/1401)\\ (661/690,\textrm{ }412/1775,\textrm{ }59/2960,-1523/10221,-303/3974)\end{array}$$ \(d) the null space of $A$ consists of the last $3$ columns of the of $A^{\textrm{T}}A$:$$\begin{array}{c} (-571/15469,-369/776,\textrm{ }149/25344,-291/350,-389/1365)\\ (-281/1313,\textrm{ }956/1489,\textrm{ }875/1706,-1279/2847,\textrm{ }409/1473)\\ (292/1579,-876/1579,\textrm{ }203/342,\textrm{ }621/4814,\textrm{ }1157/2152)\end{array}$$ The matrices $Q_{1}$ and $Q_{2}$ with these eigenvectors $(x_{i})$ satisfying $\Vert x_{i}\Vert=1$ and $(x_{i},x_{j})=0$ for $i\neq j$ as their columns are *orthogonal matrices* with the simple inverse criterion $Q^{-1}=Q^{\textrm{T}}$.$\qquad\blacksquare$ ***End Tutorial3*** The basic issue in the solution of the inverse ill-posed problem is its reduction to an well-posed one when restricted to suitable subspaces of the domain and range of $A$. Considerations of geometry leading to their decomposition into orthogonal subspaces is only an additional feature that is not central to the problem: recall from Eq. (\[Eqn: f\_inv\_f\]) that any function $f$ must necessarily satisfy the more general set-theoretic relations $ff^{-}f=f$ and $f^{-}ff^{-}=f^{-}$ of Eq. (\[Eqn: MPInverse\]) for the multiinverse $f^{-}$ of $f\!:X\rightarrow Y$. The second distinguishing feature of the MP-inverse is that it is defined, by a suitable extension, on all of $Y$ and not just on $f(X)$ which is perhaps more natural. The availability of orthogonality in inner-product spaces allows this extension to be made in an almost normal fashion. As we shall see below the additional geometric restriction of Eq. (\[Eqn: MPInverse\]) is not essential to the solution process, and infact, only results in a less canonical form of the inverse. ***Begin Tutorial4: Topological Spaces*** This Tutorial is meant to familiarize the reader with the basic principles of a topological space. A topological space $(X,\mathcal{U})$ is a set $X$ with a class[^12] $\mathcal{U}$ of distinguished subsets, called *open sets of $X$,* that satisfy (T1) The empty set $\emptyset$ and the whole $X$ belong to $\mathcal{U}$ (T2) Finite intersections of members of $\mathcal{U}$ belong to $\mathcal{U}$ (T3) Arbitrary unions of members of $\mathcal{U}$ belong to $\mathcal{U}$. **Example 2.4.** (1) The smallest topology possible on a set $X$ is its *indiscrete topology* when the only open sets are $\emptyset$ and $X$; the largest is the *discrete topology* where every subset of $X$ is open (and hence also closed). \(2) In a metric space $(X,d)$, let $B_{\varepsilon}(x,d)=\{ y\in X\!:d(x,y)<\varepsilon\}$ be an open ball at $x$. Any subset $U$ of $X$ such that for each $x\in U$ there is a $d$-ball $B_{\varepsilon}(x,d)\subseteq U$ in $U$, is said to be an open set of $(X,d)$. The collection of all these sets is the topology induced by $d$. The topological space $(X,\mathcal{U})$ is then said to be *associated with (induced by)* $(X,d)$. \(3) If $\sim$ is an equivalence relation on a set $X$, the set of all saturated sets $[x]_{\sim}=\{ y\in X\!:y\sim x\}$ is a topology on $X;$ this topology is called the *topology of saturated sets.* We argue in Sec. 4.2 that this constitutes the defining topology of a chaotic system. \(4) For any subset $A$ of the set $X$, the $A$-*inclusion topology on $X$* consists of $\emptyset$ and every superset of $A$, while the $A$-*exclusion topology on* $X$ consists of all subsets of $X-A$. Thus $A$ is open in the inclusion topology and closed in the exclusion, and in general every open set of one is closed in the other. The special cases of the *$a$-inclusion* and *$a$-exclusion* topologies for $A=\{ a\}$ are defined in a similar fashion. \(5) The *cofinite* and *cocountable topologies* in which the open sets of an infinite (resp. uncountable) set $X$ are respectively the complements of finite and countable subsets, are examples of topologies with some unusual properties that are covered in Appendix A1. If $X$ is itself finite (respectively, countable), then its cofinite (respectively, cocountable) topology is the discrete topology consisting of all its subsets. It is therefore useful to adopt the convention, unless stated to the contrary, that cofinite and cocountable spaces are respectively infinite and uncountable.$\qquad\blacksquare$ In the space $(X,\mathcal{U})$, a *neighbourhood of a point* $x\in X$ is a nonempty subset $N$ of $X$ that contains an open set $U$ containing $x$; thus $N\subseteq X$ is a neighbourhood of $x$ iff $$x\in U\subseteq N\label{Eqn: Def: nbd1}$$ for some $U\in\mathcal{U}$. The largest open set that can be used here is $\textrm{Int}(N)$ (where, by definition, $\textrm{Int}(A)$ is the largest open set that is contained in $A$) so that the above neighbourhood criterion for a subset $N$ of $X$ can be expressed in the equivalent form $$N\subseteq X\textrm{ is a }\mathcal{U}-\textrm{neighbourhood of }x\textrm{ iff }x\in\textrm{Int}_{\mathcal{U}}(N)\label{Eqn: Def: nbd2}$$ implying that a subset of $(X,\mathcal{U})$ is a neighbourhood of all its interior points, so that $N\in\mathcal{N}_{x}\Rightarrow N\in\mathcal{N}_{y}$ for all $y\in\textrm{Int}(N)$. The collection of all neighbourhoods of $x$ $$\mathcal{N}_{x}\overset{\textrm{def}}=\{ N\subseteq X\!:x\in U\subseteq N\textrm{ for some }U\in\mathcal{U}\}\label{Eqn: Def: nbd system}$$ **is the *neighbourhood system* at $x$, and the subcollection $U$ of the topology used in this equation constitutes a *neighbourhood* (*local*) *base* or *basic neighbourhood system, at* $x$, see Def. A1.1 of Appendix A1. The properties (N1) $x$ belongs to every member $N$ of *$\mathcal{N}_{x}$,* (N2) The intersection of any two neighbourhoods of *$x$* is another neighbourhood of $x$: $N,M\in\mathcal{N}_{x}\Rightarrow N\bigcap M\in\mathcal{N}_{x}$, (N3) Every superset of **any neighbourhood of $x$ is a neighbourhood of $x$: $(M\in\mathcal{N}_{x})\wedge(M\subseteq N)\Rightarrow N\in\mathcal{N}_{x}$. that characterize *$\mathcal{N}_{x}$* completely are a direct consequence of the definition (\[Eqn: Def: nbd1\]), (\[Eqn: Def: nbd2\]) that may also be stated as (N0) Any neighbourhood $N\in\mathcal{N}_{x}$ contains another neighbourhood $U$ of $x$ that is a *neighbourhood of each of its point*s: $((\forall N\in\mathcal{N}_{x})(\exists U\in\mathcal{N}_{x})(U\subseteq N))\!:(\forall y\in U\Rightarrow U\in\mathcal{N}_{y})$. Property (N0) infact serves as the defining characteristic of an open set, and *$U$* can be identified with the largest open set $\textrm{Int}(N)$ contained in $N$; hence *a set $G$ in a topological space is open iff it is a neighbourhood of each of its points.* Accordingly if *$\mathcal{N}_{x}$* is a given class of subsets of $X$ associated with each $x\in X$ satisfying $(\textrm{N}1)-(\textrm{N}3)$, then (N0) defines the special class of neighbourhoods $G$ $$\mathcal{U}=\{ G\in\mathcal{N}_{x}\!:x\in B\subseteq G\textrm{ for all }x\in G\textrm{ and a basic nbd }B\in\mathcal{N}_{x}\}\label{Eqn: nbd-topology}$$ as the unique topology on $X$ that contains a basic neighbourhood of each of its points, for which the neighbourhood system at $x$ **coincides exactly with the assigned collection** *$\mathcal{N}_{x}$*; compare Def A1.1.** Neighbourhoods in topological spaces are a generalization of the familiar notion of distances of metric spaces that quantifies “closeness” of points of $X$. A *neighbourhood of a nonempty subset* $A$ of $X$ that will be needed later on is defined in a similar manner: $N$ is a neighbourhood of $A$ iff $A\subseteq\textrm{Int}(N)$, that is $A\subseteq U\subseteq N$; thus the neighbourhood system at $A$ is given by $\mathcal{N}_{A}=\bigcap_{a\in A}\mathcal{N}_{a}:=\{ G\subseteq X\!:G\in\mathcal{N}_{a}\textrm{ for every }a\in A\}$ is the class of common neighbourhoods of each point of $A$. Some examples of neighbourhood systems at a point $x$ in $X$ are the following: \(1) In an indiscrete space $(X,\mathcal{U})$, $X$ is the only neighbourhood of every point of the space; in a discrete space any set containing $x$ is a neighbourhood of the point. \(2) In an infinite cofinite (or uncountable cocountable) space, every neighbourhood of a point is an open neighbourhood of that point. \(3) In the topology of saturated sets under the equivalence relation $\sim$, the neighbourhood system at $x$ consists of all supersets of the equivalence class $[x]_{\sim}$. \(4) Let $x\in X$. In the $x$-inclusion topology, $\mathcal{N}_{x}$ consists of all the non-empty open sets of $X$ which are the supersets of $\{ x\}$. For a point $y\neq x$ of $X$, $\mathcal{N}_{y}$ are the supersets of $\{ x,y\}$. For any given class $_{\textrm{T}}\mathcal{S}$ of subsets of $X$, a unique topology $\mathcal{U}(_{\textrm{T}}\mathcal{S})$ can always be constructed on $X$ by taking all *finite* *intersections* $_{\textrm{T}}\mathcal{S}_{\wedge}$ of members of $\mathcal{S}$ followed by *arbitrary* *unions* $_{\textrm{T}}\mathcal{S}_{\wedge\vee}$ of these finite intersections. $\mathcal{U}(_{\textrm{T}}\mathcal{S}):=\,_{\textrm{T}}\mathcal{S}_{\wedge\vee}$ is the smallest topology on $X$ that contains $_{\textrm{T}}\mathcal{S}$ and is said to be *generated by* $_{\textrm{T}}\mathcal{S}$. For a given topology $\mathcal{U}$ on $X$ satisfying $\mathcal{U}=\mathcal{U}(_{\textrm{T}}\mathcal{S})$, $_{\textrm{T}}\mathcal{S}$ is a *subbasis,* and $_{\textrm{T}}\mathcal{S}_{\wedge}:=\,_{\textrm{T}}\mathcal{B}$ a *basis, for the topology* $\mathcal{U}$; for more on topological basis, see Appendix A1. The topology generated by a subbase essentially builds not from the collection $_{\textrm{T}}\mathcal{S}$ itself but from the finite intersections $_{\textrm{T}}\mathcal{S}_{\wedge}$ of its subsets; in comparison the base generates a topology directly from a collection $_{\textrm{T}}\mathcal{S}$ of subsets by forming their unions. Thus whereas *any* class of subsets can be used as a subbasis, a given collection must meet certain qualifications to pass the test of a base for a topology: these and related topics are covered in Appendix A1. Different subbases, therefore, can be used to generate different topologies on the same set $X$ as the following examples for the case of $X=\mathbb{R}$ demonstrates; here $(a,b)$, $[a,b)$, $(a,b]$ and $[a,b]$, for $a\leq b\in\mathbb{R}$, are the usual open-closed intervals in $\mathbb{R}$[^13]. The subbases $_{\textrm{T}}\mathcal{S}_{1}=\{(a,\infty),(-\infty,b)\}$, $_{\textrm{T}}\mathcal{S}_{2}=\{[a,\infty),(-\infty,b)\}$, $_{\textrm{T}}\mathcal{S}_{3}=\{(a,\infty),(-\infty,b]\}$ and $_{\textrm{T}}\mathcal{S}_{4}=\{[a,\infty),(-\infty,b]\}$ give the respective bases $_{\textrm{T}}\mathcal{B}_{1}=\{(a,b)\}$, $_{\textrm{T}}\mathcal{B}_{2}=\{[a,b)\}$, $_{\textrm{T}}\mathcal{B}_{3}=\{(a,b]\}$ and $_{\textrm{T}}\mathcal{B}_{4}=\{[a,b]\}$, $a\leq b\in\mathbb{R}$, leading to the *standard (usual*)*, lower limit (Sorgenfrey*)*, upper limit,* and *discrete* (take $a=b$) topologies on $\mathbb{R}$. Bases of the type $(a,\infty)$ and $(-\infty,b)$ provide the *right* and *left ray* topologies on $\mathbb{R}$. *This feasibility of generating different topologies on a set can be of great practical significance because open sets determine convergence characteristics of nets and continuity characteristics of functions, thereby making it possible for nature to play around with the structure of its working space in its kitchen to its best possible advantage.*[^14] ** Here are a few essential concepts and terminology for topological spaces. **Definition 2.2.** ***Boundary, Closure, Interior*.** *The* *boundary of $A$ in $X$* *is the set of points $x\in X$ such that every neighbourhood $N$ of $x$ intersects both $A$ and $X-A$:* $${\textstyle \textrm{Bdy}(A)\overset{\textrm{def}}=\{ x\in X\!:(\forall N\in\mathcal{N}_{x})((N\bigcap A\neq\emptyset)\wedge(N\bigcap(X-A)\neq\emptyset))\}}\label{Eqn: Def: Boundary}$$ *where $\mathcal{N}_{x}$ is the neighbourhood system of Eq. (\[Eqn: Def: nbd system\]) at $x$.* *The* *closure of $A$* *is the set of all points $x\in X$ such that each neighbourhood of $x$ contains at least one point of $A$* ***that may be $\boldmath{x}$ itself****. Thus the set* $${\textstyle \textrm{Cl}(A)\overset{\textrm{def}}=\{ x\in X\!:(\forall N\in\mathcal{N}_{x})\textrm{ }(N\bigcap A\neq\emptyset)\}}\label{Eqn: Def: Closure}$$ *of all points in $X$ adherent* *to* *$A$ is given by* *is the union* ***of $A$ with its boundary.* *The* *interior of $A$* $$\textrm{Int}(A)\overset{\textrm{def}}=\{ x\in X\!:(\exists N\in\mathcal{N}_{x})\textrm{ }(N\subseteq A)\}\label{Eqn: Def: Interior}$$ *consisting of those points of $X$ that are in $A$ but not in its boundary,* $\textrm{Int}(A)=A-\textrm{Bdy}(A)$*, is the largest open subset of $X$ that is contained in $A$. Hence it follows that* $\textrm{Int}(\textrm{Bdy}(A))=\emptyset$, *the boundary of $A$ is the intersection of the closures of $A$ and $X-A$,* *and a subset $N$ of $X$ is a neighbourhood of $x$ iff* $x\in\textrm{Int}(N)$*.$\qquad\square$* The three subsets $\textrm{Int}(A)$, $\textrm{Bdy}(A)$ and *exterior* of $A$ defined as $\textrm{Ext}(A):=\textrm{Int}(X-A)=X-\textrm{Cl}(A)$, are pairwise disjoint and have the full space $X$ as their union. **Definition 2.3.** ***Derived and Isolated sets.*** *Let $A$ be a subset of $X$. A point $x\in X$ (which may or may not be a point of $A$) is a* *cluster point of* $A$ *if every neighbourhood $N\in\mathcal{N}_{x}$ contains atleast one point of $A$* ***different from*** *$\mathbf{x}$. The* *derived set of $A$* $${\textstyle \textrm{Der}(A)\overset{\textrm{def}}=\{ x\in X\!:(\forall N\in\mathcal{N}_{x})\textrm{ }(N\bigcap(A-\{ x\})\neq\emptyset)\}}\label{Eqn: Def: Derived}$$ *is the set of all cluster points of $A$. The complement of* $\textrm{Der}(A)$ in $A$ $$\textrm{Iso}(A)\overset{\textrm{def}}=A-\textrm{Der}(A)=\textrm{Cl}(A)-\textrm{Der}(A)\label{Eqn: Def: Isolated}$$ *are the* *isolated* *points* *of* $A$ *to which no proper sequence in $A$ converges, that is there exists a neighbourhood of any such point that contains no other point of $A$ so that* *the only sequence that converges to* $a\in\textrm{Iso}(A)$ *is the constant sequence $(a,a,a,\cdots)$.$\qquad\square$* Clearly, $$\begin{array}{ccl} {\textstyle \textrm{Cl}(A)} & = & A\bigcup\textrm{Der}(A)=A\bigcup\textrm{Bdy}(A)\\ & = & \textrm{Iso}(A)\bigcup\textrm{Der}(A)=\textrm{Int}(A)\bigcup\textrm{Bdy}(A)\end{array}$$ with the last two being disjoint unions, and $A$ is closed iff $A$ contains all its cluster points, $\textrm{Der}(A)\subseteq A$, iff $A$ contains its closure. Hence $$\begin{gathered} A=\textrm{Cl}(A)\Longleftrightarrow\textrm{Cl}(A)=\{ x\in A\!:((\exists N\in\mathcal{N}_{x})(N\subseteq A))\vee((\forall N\in\mathcal{N}_{x})(N\bigcap(X-A)\neq\emptyset))\}\end{gathered}$$ Comparison of Eqs. (\[Eqn: Def: Boundary\]) and (\[Eqn: Def: Derived\]) also makes it clear that $\textrm{Bdy}(A)\subseteq\textrm{Der}(A)$. The special case of $A=\textrm{Iso}(A)$ with $\textrm{Der}(A)\subseteq X-A$ is important enough to deserve a special mention: **Definition 2.4.** ***Donor set.*** *A proper, nonempty subset $A$ of $X$ such that* $\textrm{Iso}(A)=A$ *with* $\textrm{Der}(A)\subseteq X-A$ *will be called* *self-isolated* *or* *donor.* *Thus sequences eventually in a donor set converges only in its complement; this is the opposite of the characteristic of a closed set where all converging sequences eventually in the set must necessarily converge in it. A closed-donor set with a closed neighbour has no derived or boundary sets, and will be said to be* *isolated in $X$.*$\qquad\square$ **Example 2.5.** In an isolated set sequences converge, if they have to, simultaneously in the complement (because it is donor) and in it (because it is closed). Convergent sequences in such a set can only be constant sequences. Physically, if we consider adherents to be contributions made by the dynamics of the corresponding sequences, then an isolated set is secluded from its neighbour in the sense that it neither receives any contributions from its surroundings, nor does it give away any. In this light and terminology, a closed set is a *selfish* set (recall that a set $A$ is closed in $X$ iff every convergent net of $X$ that is eventually in $A$ converges in $A$; conversely a set is open in $X$ iff the only nets that converge in $A$ are eventually in it), **whereas a set with a derived set that intersects itself and its complement may be considered to be *neutral.* Appendix A3 shows the various possibilities for the derived set and boundary of a subset $A$ of $X$.$\qquad\blacksquare$ Some useful properties of these concepts for a subset $A$ of a topological space $X$ are the following. \(a) $\textrm{Bdy}_{X}(X)=\emptyset$, \(b) $\textrm{Bdy}(A)=\textrm{Cl}(A)\bigcap\textrm{Cl}(X-A)$, \(c) $\textrm{Int}(A)=X-\textrm{Cl}(X-A)=A-\textrm{Bdy}(A)=\textrm{Cl}(A)-\textrm{Bdy}(A)$, \(d) $\textrm{Int}(A)\bigcap\textrm{Bdy}(A)=\emptyset$, \(e) $X=\textrm{Int}(A)\bigcup\textrm{Bdy}(A)\bigcup\textrm{Int}(X-A)$, \(f) $${\textstyle \textrm{Int}(A)=\bigcup\{ G\subseteq X\!:G\textrm{ is an open set of }X\textrm{ contained in }A\}}\label{Eqn: interior}$$ \(g) $${\textstyle \textrm{Cl}(A)=\bigcap\{ F\subseteq X\!:F\textrm{ is a closed set of }X\textrm{ containing }A\}}\label{Eqn: closure}$$ A straightforward consequence of property (b) is that the boundary of any subset $A$ of a topological space $X$ is closed in $X$; this significant result may also be demonstrated as follows. If $x\in X$ is not in the boundary of $A$ there is some neighbourhood $N$ of $x$ that does not intersect both $A$ and $X-A$. For each point $y\in N$, $N$ is a neighbourhood of that point that does not meet $A$ and $X-A$ simultaneously so that $N$ is contained wholly in $X-\textrm{Bdy}(A)$. We may now take $N$ to be open without any loss of generality implying thereby that $X-\textrm{Bdy}(A)$ is an open set of $X$ from which it follows that $\textrm{Bdy}(A)$ is closed in $X$. Further material on topological spaces relevant to our work can be found in Appendix A3. ***End Tutorial4*** Working in a general topological space, we now recall the solution of an ill-posed problem $f(x)=y$ [@Sengupta1997] that leads to a multifunctional inverse $f^{-}$ through the generalized inverse $G$. Let $f\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$ be a (nonlinear) function between two topological space $(X,\mathcal{U})$ and $(Y,\mathcal{V})$ that is neither one-one or onto. Since $f$ is not one-one, $X$ can be partitioned into disjoint equivalence classes with respect to the equivalence relation $x_{1}\sim x_{2}\Leftrightarrow f(x_{1})=f(x_{2})$. Picking a representative member from each of the classes (this is possible by the Axiom of Choice; see the following Tutorial) produces a *basic set* $X_{\textrm{B}}$ of $X$; it is basic as it corresponds to the row space in the linear matrix example which is all that is needed for taking an inverse. $X_{\textrm{B}}$ is the counterpart of the quotient set $X/\sim$ of Sec. 1, with the important difference that whereas the points of the quotient set are the equivalence classes of $X$, $X_{\textrm{B}}$ *is a subset of* $X$ with each of the classes contributing a point to $X_{\textrm{B}}$. It then follows that $f_{\textrm{B}}\!:X_{\textrm{B}}\rightarrow f(X)$ is the bijective restriction $a|_{\textrm{row}(A)}$ that reduces the original ill-posed problem to a well-posed one with $X_{\textrm{B}}$ and $f(X)$ corresponding respectively to the row and column spaces of $A$, and $f_{\textrm{B}}^{-1}\!:f(X)\rightarrow X_{\textrm{B}}$ is the basic inverse from which the multiinverse $f^{-}$ is obtained through $G$, which in turn corresponds to the Moore-Penrose inverse $G_{\textrm{MP}}$. The topological considerations (obviously not for inner product spaces that applies to the Moore-Penrose inverse) needed to complete the solution are discussed below and in Appendix A1. ***Begin Tutorial5: Axiom of Choice and Zorn’s Lemma*** Since some of our basic arguments depend on it, this Tutorial contains a short description of the Axiom of Choice that has been described as “one of the most important, and at the same time one of the most controversial, principles of mathematics”. What this axiom states is this: For any set $X$ there exists a function $f_{\textrm{C}}\!:\mathcal{P}_{0}(X)\rightarrow X$ such that $f_{\textrm{C}}(A_{\alpha})\in A_{\alpha}$ for every non-empty subset $A_{\alpha}$ of $X$; here $\mathcal{P}_{0}(X)$ is the class of all subsets of $X$ except $\emptyset$. Thus, if $X=\{ x_{1},x_{2},x_{3}\}$ is a three element set, a possible choice function is given by $$\begin{array}{c} f_{\textrm{C}}(\{ x_{1},x_{2},x_{3}\})=x_{3},\quad f_{\textrm{C}}(\{ x_{1},x_{2}\})=x_{1},\quad f_{\textrm{C}}(\{ x_{2},x_{3}\})=x_{3},\quad f_{\textrm{C}}(\{ x_{3},x_{1}\})=x_{3},\\ f_{\textrm{C}}(\{ x_{1}\})=x_{1},\quad f_{\textrm{C}}(\{ x_{2}\})=x_{2},\quad f_{\textrm{C}}(\{ x_{3}\})=x_{3}.\end{array}$$ It must be appreciated that the axiom is only an existence result that asserts *every set* to have a choice function, even when nobody knows how to construct one in a specific case. Thus, for example, how does one pick out the isolated irrationals $\sqrt{2}$ or $\pi$ from the uncountable reals? There is no doubt that they do exist, for we can construct a right angled triangle with sides of length $1$ or a circle of radius $1$. The axiom tells us that these choices are possible even though we do not know how exactly to do it; all that can be stated with confidence is that we can actually pick up rationals arbitrarily close to these irrationals. The axiom of choice is essentially meaningful when $X$ is infinite as illustrated in the last two examples. This is so because even when $X$ is denumerable, it would be physically impossible to make an infinite number of selections either all at a time or sequentially: the Axiom of Choice nevertheless tells us that this is possible. The real strength and utility of the Axiom however is when $X$ and some or all of its subsets are uncountable as in the case of the choice of the *single element* $\pi$ from the reals. To see this more closely in the context of maps that we are concerned with, let $f\!:X\rightarrow Y$ be a non-injective, onto map. To construct a functional right inverse $f_{r}\!:Y\rightarrow X$ of $f$, we must choose, for each $y\in Y$ one *representative* element $x_{\textrm{rep}}$ from the set $f^{-}(y)$ and define $f_{r}(y)$ to be that element according to $f\circ f_{r}(y)=f(x_{\textrm{rep}})=y$. If there is no preferred or natural way to make this choice, the axiom of choice allows us to make an arbitrary selection from the infinitely many that may be possible from $f^{-}(y)$. When a natural choice is indeed available, as for example in the case of the initial value problem $y^{\prime}(x)=x;\, y(0)=\alpha_{0}$ on $[0,a]$, the definite solution $\alpha_{0}+x^{2}/2$ may be selected from the infinitely many $\int_{0}^{x}x^{\prime}dx^{\prime}=\alpha+x^{2}/2,\textrm{ }0\leq x\leq a$ that are permissible, and the axiom of choice sanctions this selection. In addition, each $y\in Y$ gives rise to the family of solution sets $A_{y}=\{ f^{-}(y)\!:y\in Y\}$ and the real power of the axiom is its assertion that it is possible to make a choice $f_{\textrm{C}}(A_{y})\in A_{y}$ on every $A_{y}$ simultaneously; this permits the choice **on every $A_{y}$ of the collection to be made at the same time. ***Pause Tutorial5*** Figure \[Fig: GenInv\] shows our [@Sengupta1997] formulation and solution of the inverse ill-posed problem $f(x)=y$. In sub-diagram $X-X_{\textrm{B}}-f(X)$, the surjection $p\!:X\rightarrow X_{\textrm{B}}$ is the counterpart of the quotient map $Q$ of Fig. \[Fig: quotient\] that is known in the present context as the *identification* of $X$ with $X_{\mathrm{B}}$ (as it *identifies* each saturated subset of $X$ with its representative point in $X_{\textrm{B}}$), with the space $(X_{\textrm{B}},\textrm{FT}\{\mathcal{U};p\})$ carrying the *identification topology* $\textrm{FT}\{\mathcal{U};p\}$ being known as an *identification space.* By sub-diagram $Y-X_{\textrm{B}}-f(X)$, the image $f(X)$ of $f$ gets the *subspace topology*[^15] $\textrm{IT}\{ j;\mathcal{V}\}$ from $(Y,\mathcal{V})$ by the inclusion $j\!:f(X)\rightarrow Y$ when its open sets are generated as, and only as, $j^{-1}(V)=V\bigcap f(X)$ for $V\in\mathcal{V}$. Furthermore if the bijection $f_{\textrm{B}}$ connecting $X_{\textrm{B}}$ and $f(X)$ (which therefore acts as a $1:1$ correspondence between their points, implying that these sets are set-theoretically identical except for their names) is image continuous, then by Theorem A2.1 of Appendix 2, so is the *association* $q=f_{\textrm{B}}\circ p\!:X\rightarrow f(X)$ that associates saturated sets of $X$ with elements of $f(X)$; this makes $f(X)$ look like an identification space of $X$ by assigning to it the topology $\textrm{FT}\{\mathcal{U};q\}$. On the other hand if $f_{\textrm{B}}$ happens to be preimage continuous, then $X_{\textrm{B}}$ acquires, by Theorem A2.2, the initial topology $\textrm{IT}\{ e;\mathcal{V}\}$ by the *embedding* $e\!:X_{\textrm{B}}\rightarrow Y$ that embeds $X_{\textrm{B}}$ into $Y$ through $j\circ f_{\textrm{B}}$, making it look like a subspace of $Y$[^16]. In this dual situation, $f_{\textrm{B}}$ has the highly interesting topological property of being simultaneously image and preimage continuous when the open sets of $X_{\textrm{B}}$ and $f(X)$ — which are simply the $f_{\textrm{B}}^{-1}$-images of the open sets of $f(X)$ which, in turn, are the $f_{\textrm{B}}$-images of these saturated open sets — can be considered to have been generated by $f_{\textrm{B}}$, and are respectively the smallest and largest collection of subsets of $X$ and $Y$ that makes $f_{\textrm{B}}$ *ini*(tial-fi)*nal continuous* [@Sengupta1997]*.* A bijective ininal function such as $f_{\textrm{B}}$ is known as a *homeomorphism* and ininality for functions that are neither $1:1$ **nor onto is a generalization of homeomorphism for bijections; refer Eqs. (\[Eqn: INI\]) and (\[Eqn: HOM\]) for a set-theoretic formulation of this distinction. A homeomorphism $f\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$ renders the homeomorphic spaces $(X,\mathcal{U})$ and $(Y,\mathcal{V})$ topologically indistinguishable which may be considered to be identical in as far as their topological properties are concerned. **Remark.** It may be of some interest here to speculate on the significance of *ininality* in our work. Physically, a map $f\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$ between two spaces can be taken to represent an interaction between them and the algebraic and topological characters of $f$ determine the nature of this interaction. A simple bijection merely sets up a correspondence, that is an interaction, between every member of $X$ with some member $Y$, whereas a continuous map establishes the correspondence among the special category of “open” sets. Open sets, as we see in Appendix A1, are the basic ingredients in the theory of convergence of sequences, nets and filters, and the characterization of open sets in terms of convergence, namely that *a set $G$ in $X$ is open in it if every net or sequence that converges in $X$ to a point in $G$ is eventually in $G$*, see Appendix A1, may be interpreted to mean that such sets represent groupings of elements that require membership of the group before permitting an element to belong it; an open set unlike its complement the closed or *selfish* set, however, does not forbid a net that has been eventually in it to settle down in its selfish neighbour, who nonetheless will never allow such a situation to develop in its own territory. An ininal map forces these well-defined and definite groups in $(X,\mathcal{U})$ and $(Y,\mathcal{V})$ to interact with each other through $f$; this is not possible with simple continuity as there may be open sets in $X$ that are not derived from those of $Y$ and non-open sets in $Y$ whose inverse images are open in $X$. *It is our hypothesis that the driving force behind the evolution of a system represented by the input-output relation $f(x)=y$ is the attainment of the ininal triple state $(X,f,Y)$ for the system.* A preliminary analysis of this hypothesis is to be found in Sec. 4.2. For ininality of the interaction, it is therefore necessary to have$$\begin{aligned} \textrm{FT}\{\mathcal{U};f_{<}\} & = & \textrm{IT}\{ j;\mathcal{V}\}\label{Eqn: ininal}\\ \textrm{IT}\{\,_{<}f;\mathcal{V}\} & = & \textrm{FT}\{\mathcal{U};p\}\};\nonumber \end{aligned}$$ in what follows we will refer to the injective and surjective restrictions of $f$ by their generic topological symbols of embedding $e$ and association $q$ respectively. What are the topological characteristics of $f$ in order that the requirements of Eq. (\[Eqn: ininal\]) be met? From Appendix A1, it should be clear by superposing the two parts of Fig. \[Fig: Initial-Final\] over each other that given $q\!:(X,\mathcal{U})\rightarrow(f(X),\textrm{FT}\{\mathcal{U};q\})$ in the first of these equations, $\textrm{IT}\{ j;\mathcal{V}\}$ will equal $\textrm{FT}\{\mathcal{U};q\}$ iff $j$ is an ininal open inclusion and $Y$ receives $\textrm{FT}\{\mathcal{U};f\}$. In a similar manner, preimage continuity of $e$ requires $p$ to be open ininal and $f$ to be preimage continuous if the second of Eq. (\[Eqn: ininal\]) is to be satisfied. Thus under the restrictions imposed by Eq. (\[Eqn: ininal\]), the interaction $f$ between $X$ and $Y$ must be such as to give $X$ the smallest possible topology of $f$-saturated sets and $Y$ the largest possible topology of images of all these sets: $f$, under these conditions, is an ininal transformation. Observe that a direct application of parts (b) of Theorems A2.1 and A2.2 to Fig. \[Fig: GenInv\] implies that Eq. (\[Eqn: ininal\]) is satisfied iff $f_{\textrm{B}}$ is ininal, that is iff it is a homeomorphism. Ininality of $f$ is simply a reflection of this as it is neither $1:1$ nor onto. The $f$- and $p$-images of each saturated set of $X$ are singletons in $Y$ (these saturated sets in $X$ arose, in the first place, as $f^{-}(\{ y\})$ for $y\in Y$) and in $X_{\textrm{B}}$ respectively. This permits the embedding $e=j\circ f_{\textrm{B}}$ to give $X_{\textrm{B}}$ the character of a virtual subspace of $Y$ just as $i$ makes $f(X)$ a real subspace. Hence the inverse images $p^{-}(x_{r})=f^{-}(e(x_{r}))$ with $x_{r}\in X_{\textrm{B}}$, and $q^{-}(y)=f^{-}(i(y))$ with $y=f_{\textrm{B}}(x_{r})\in f(X)$ are the same, and are just the corresponding $f^{-}$ images via the injections $e$ and $i$ respectively. $G$, a left inverse of $e$, is a generalized inverse of $f$. $G$ is a generalized inverse because the two set-theoretic defining requirements of $fGf=f$ and $GfG=G$ for the generalized inverse are satisfied, as Fig. \[Fig: GenInv\] shows, in the following forms $$jf_{\textrm{B}}Gf=f\qquad Gjf_{\textrm{B}}G=G.$$ In fact the commutativity embodied in these equalities is self evident from the fact that $e=if_{\textrm{B}}$ is a left inverse of $G$, that is $eG=\bold1_{Y}$. On putting back $X_{\textrm{B}}$ into $X$ by identifying each point of $X_{\textrm{B }}$ with the set it came from yields the required set-valued inverse $f^{-}$, and $G$ may be viewed as a functional selection of the multiinverse $f^{-}$. An *injective branch* of a function $f$ in this work refers to the restrictions $f_{\mathrm{B}}$ and its associated inverse $f_{\mathrm{B}}^{-1}$. The following example of an inverse ill-posed problem will be useful in fixing the notations introduced above. Let $f$ on $[0,1]$ be the function of \[Fig: gen-inv\]. Then $f(x)=y$ is well-posed for $[0,1/4)$, and ill-posed in [\[]{}1/4,1[\]]{}. There are two injective branches of $f$ in $\{[1/4,3/8)$$\bigcup$ $(5/8,1]\}$, and $f$ is constant ill-posed in $[3/8,5/8]$. Hence the basic component $f_{\textrm{B}}$ of $f$ can be taken to be $f_{\textrm{B}}(x)=2x$ for $x\in[0,3/8)$ having the inverse $f_{\textrm{B}}^{-1}(y)=x/2$ with $y\in[0,3/4]$. The generalized inverse is obtained by taking $[0,3/4]$ as a subspace of $[0,1]$, while the multiinverse $f^{-}$ follows by associating with every point of the basic domain $[0,1]_{\textrm{B}}=[0,3/8]$, the respective equivalent points $[3/8]_{f}=[3/8,5/8]$ and $[x]_{f}=\{ x,7/4-3x\}\textrm{ for }x\in[1/4,3/8)$. Thus the inverses $G$ and $f^{-}$ of $f$ are[^17] $$G(y)=\left\{ \begin{array}{ccl} y/2, & & y\in[0,3/4]\\ 0, & & y\in(3/4,1]\end{array}\right.,\quad f^{-}(y)=\left\{ \begin{array}{ccl} y/2, & & y\in[0,1/2)\\ \{ y/2,7/4-3y/2\}, & & y\in[1/2,3/4)\\ {}[3/8,5/8], & & y=3/4\\ 0, & & y\in(3/4,1],\end{array}\right.$$ which shows that $f^{-}$ is multivalued. In order to avoid cumbersome notations, an injective branch of $f$ will always refer to a representative basic branch $f_{\textrm{B}}$, and its “inverse” will mean either $f_{\textrm{B}}^{-1}$ or $G$. **Example 2.3, Revisited.** The row reduced echelon form of the augmented matrix $(A|b)$ of Example 2.3 is $${\displaystyle (A|b)\longrightarrow\left(\begin{array}{rrrrrcl} 1 & -3 & 0 & 3/2 & 1/2 & & 5b_{1}/2-b_{2}/2\\ 0 & 0 & 1 & -1/4 & 3/4 & & -3b_{1}/4+b_{2}/4\\ 0 & 0 & 0 & 0 & 0 & & -2b_{1}+b_{3}\\ 0 & 0 & 0 & 0 & 0 & & b_{1}-b_{2}+b_{4}\end{array}\right)}\label{Eqn: RowReduce}$$ The multifunctional solution $x=A^{-}b$, with $b$ any element of $Y=\mathbb{R}^{4}$ not necessarily in the of image of $a$, is$$x=A^{-}b=Gb+x_{2}\left(\begin{array}{c} 3\\ 1\\ 0\\ 0\\ 0\end{array}\right)+x_{4}\left(\begin{array}{r} -3/2\\ 0\\ 1/4\\ 1\\ 0\end{array}\right)+x_{5}\left(\begin{array}{r} -1/2\\ 0\\ -3/4\\ 0\\ 1\end{array}\right),$$ with its multifunctional character arising from the arbitrariness of the coefficients $x_{2}$, $x_{4},$ and $x_{5}$. The generalized inverse $$G=\left(\begin{array}{rrrr} 5/2 & -1/2 & 0 & 0\\ 0 & 0 & 0 & 0\\ -3/4 & 1/4 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\end{array}\right)\!:Y\rightarrow X_{\textrm{B}}\label{Eqn: GenInvEx5}$$ is the unique matrix representation of the functional inverse $a_{\textrm{B}}^{-1}\!:a(\mathbb{R}^{5})\rightarrow X_{\textrm{B}}$ extended to $Y$ defined according to[^18] $$g(b)=\left\{ \begin{array}{ccl} a_{\textrm{B}}^{-1}(b), & & \textrm{ if }b\in\mathcal{R}(a)\\ 0, & & \textrm{ if }b\in Y-\mathcal{R}(a),\end{array}\right.\label{Eqn: Def: GenInv}$$ that bears comparison with the basic inverse $$A_{\textrm{B}}^{-1}(b^{*})=\left(\begin{array}{rrrr} 5/2 & -1/2 & 0 & 0\\ 0 & 0 & 0 & 0\\ -3/4 & 1/4 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\end{array}\right)\left(\begin{array}{c} b_{1}\\ b_{2}\\ 2b_{1}\\ b_{2}-b_{1}\end{array}\right)\!:a(\mathbb{R}^{5})\rightarrow X_{\textrm{B}}$$ between the $2$-dimensional column and row spaces of $A$ which is responsible for the particular solution of $Ax=b$. Thus $G$ is simply $A_{\textrm{B}}^{-1}$ acting on its domain $a(X)$ considered a subspace of $Y$, suitably extended to the whole of $Y$. That it is indeed a generalized inverse is readily seen through the matrix multiplications $GAG$ and $AGA$ that can be verified to reproduce $G$ and $A$ respectively. Comparison of Eqs. (\[Eqn: Def: Moore-Penrose\]) and (\[Eqn: Def: GenInv\]) shows that the Moore-Penrose inverse differs from ours through the geometrical constraints imposed in its definition, Eqs. (\[Eqn: MPInverse\]). Of course, this results in a more complex inverse (\[Eqn: MPEx5\]) as compared to our very simple (\[Eqn: GenInvEx5\]); nevertheless it is true that both the inverses satisfy $$\begin{aligned} E((E(G_{\textrm{MP}}))^{\textrm{T}}) & = & \left(\begin{array}{ccccc} 1 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\end{array}\right)\\ \\ & = & E((E(G))^{\textrm{T}})\end{aligned}$$ where $E(A)$ is the row-reduced echelon form of $A$. The canonical simplicity of Eq. (\[Eqn: GenInvEx5\]) as compared to Eq. (\[Eqn: MPEx5\]) is a general feature that suggests a more natural choice of bases by the map $a$ than the orthogonal set imposed by Moore and Penrose. This is to be expected since the MP inverse, governed by Eq. (\[Eqn: MPInverse\]), is a subset of our less restricted inverse described by only the first two of (\[Eqn: MPInverse\]); more specifically the difference is made clear in Fig. \[Fig: MP\_Inverse\](a) which shows that for any $b\notin\mathcal{R}(A)$, only $G_{\textrm{MP}}(b_{\bot})=0$ as compared to $G(b)=0$. This seems to imply that introducing extraneous topological considerations into the purely set theoretic inversion process may not be a recommended way of inverting, and the simple bases comprising the row and null spaces of $A$ and $A^{\textrm{T}}$ — that are mutually orthogonal just as those of the Moore-Penrose — are a better choice for the particular problem $Ax=b$ than the general orthonormal bases that the MP inverse introduces. These “good” bases, with respect to which the generalized inverse $G$ has a considerably simpler representation, are obtained in a straight forward manner from the row-reduced forms of $A$ and $A^{\textrm{T}}$. These bases are \(a) The column space of $A$ is spanned by the columns $(1,\textrm{ }3,\textrm{ }2,\textrm{ }2)^{\textrm{T}}$ and $(1,\textrm{ }5,\textrm{ }2,\textrm{ }4)^{\textrm{T}}$ of $A$ that correspond to the basic columns containing the leading $1$’s in the row-reduced form of $A$, \(b) The null space of $A^{\textrm{T}}$ is spanned by the solutions $(-2,\textrm{ }0,\textrm{ }1,\textrm{ }0)^{\textrm{T}}$ and $(1,-1,\textrm{ }0,\textrm{ }1)^{\textrm{T}}$ of the equation $A^{\textrm{T}}b=0$, \(c) The row space of $A$ is spanned by the rows $(1,-3,\textrm{ }2,\textrm{ }1,\textrm{ }2)$ and $(3,-9,\textrm{ }10,\textrm{ }2,\textrm{ }9)$ of $A$ corresponding to the non-zero rows in the row-reduced form of $A$, \(d) The null space of $A$ is spanned by the solutions $(3,\textrm{ }1,\textrm{ }0,\textrm{ }0,\textrm{ }0)$, $(-6,\textrm{ }0,\textrm{ }1,\textrm{ }4,\textrm{ }0)$, and $(-2,\textrm{ }0,-3,\textrm{ }0,\textrm{ }4)$ of the equation $Ax=0$.$\qquad\blacksquare$ The main differences between the natural “good” bases and the MP-bases that are responsible for the difference in form of the inverses, is that the later have the additional restrictions of being orthogonal to each other (recall the orthogonality property of the $Q$-matrices), and the more severe one of basis vectors mapping onto basis vectors according to $Ax_{i}=\sigma_{i}b_{i}$, $i=1,\cdots,r$, where the $\{ x_{i}\}_{i=1}^{n}$ and $\{ b_{j}\}_{j=1}^{m}$ are the eigenvectors of $A^{\textrm{T}}A$ and $AA^{\textrm{T}}$ respectively and $(\sigma_{i})_{i=1}^{r}$ are the positive square roots of the non-zero eigenvalues of $A^{\textrm{T}}A$ (or of $AA^{\textrm{T}}$), with $r$ denoting the dimension of the row or column space. This is considered as a serious restriction as the linear combination of the basis $\{ b_{j}\}$ that $Ax_{i}$ should otherwise have been equal to, allows a greater flexibility in the matrix representation of the inverse that shows up in the structure of $G$. These are, in fact, quite general considerations in the matrix representation of linear operators; thus the basis that diagonalizes an $n\times n$ matrix (when this is possible) is not the standard “diagonal” orthonormal basis of $\mathbb{R}^{n}$, but a problem-dependent, less canonical, basis consisting of the $n$ eigenvectors of the matrix. The $0$-rows of the inverse of Eq. (\[Eqn: GenInvEx5\]) result from the $3$-dimensional null-space variables $x_{2}$, $x_{4}$, and $x_{5}$, while the $0$-columns come from the $2$-dimensional image-space dependency of $b_{3}$, $b_{4}$ on $b_{1}$ and $b_{2}$, that is from the last two zero rows of the reduced echelon form (\[Eqn: RowReduce\]) of the augmented matrix. We will return to this theme of the generation of a most appropriate problem-dependent topology for a given space in the more general context of chaos in Sec. 4.2. In concluding this introduction to generalized inverses we note that the inverse $G$ of $f$ comes very close to being a right inverse: thus even though $AG\not\neq\bold1_{2}$ its row-reduced form $$\left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\end{array}\right)$$ is to be compared with the corresponding less satisfactory $$\left(\begin{array}{cccr} 1 & 0 & 2 & -1\\ 0 & 1 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\end{array}\right)$$ representation of $AG_{\textrm{MP}}$. **3. Multifunctional extension of function spaces** The previous section has considered the solution of ill-posed problems as multifunctions and has shown how this solution may be constructed. Here we introduce the multifunction space $\textrm{Multi}_{\mid}(X)$ as the first step toward obtaining a smallest dense extension $\textrm{Multi}(X)$ of the function space $\textrm{Map}(X)$. $\textrm{Multi}_{\mid}(X)$ is basic to our theory of chaos [@Sengupta2000] in the sense that a chaotic state of a system can be fully described by such an indeterminate multifunctional state. In fact, multifunctions also enter in a natural way in describing the spectrum of nonlinear functions that we consider in Section 6; this is required to complete the construction of the smallest extension $\textrm{Multi}(X)$ of the function space $\textrm{Map}(X)$. The main tool in obtaining the space $\textrm{Multi}_{\mid}(X)$ from $\textrm{Map}(X)$ is a generalization of the technique of pointwise convergence of continuous functions to (discontinuous) functions. In the analysis below, we consider nets instead of sequences as the spaces concerned, like the topology of pointwise convergence, may not be first countable, Appendix A1. ***3.1. Graphical convergence of a net of functions*** Let $(X,\mathcal{U})$ and $(Y,\mathcal{V})$ be Hausdorff spaces and $(f_{\alpha})_{\alpha\in\mathbb{D}}:X\rightarrow Y$ be a net of piecewise continuous functions, not necessarily with the same domain or range, and suppose that for each $\alpha\in\mathbb{D}$ there is a finite set $I_{\alpha}=\{1,2,\cdots P_{\alpha}\}$ such that $f_{\alpha}^{-}$ has $P_{\alpha}$ functional branches possibly with different domains; obviously $I_{\alpha}$ is a singleton iff $f$ is a injective. For each $\alpha\in\mathbb{D}$, define functions $(g_{\alpha i})_{i\in I_{\alpha}}\!:Y\rightarrow X$ such that $$f_{\alpha}g_{\alpha i}f_{\alpha}=f_{\alpha i}^{I}\qquad i=1,2,\cdots P_{\alpha,}$$ where $f_{\alpha i}^{I}$ is a basic injective branch of $f_{\alpha}$ on some subset of its domain: $g_{\alpha i}f_{\alpha i}^{I}=1_{X}$ on $\mathcal{D}(f_{\alpha i}^{I})$, $f_{\alpha i}^{I}g_{\alpha i}=1_{Y}$ on $\mathcal{D}(g_{\alpha i})$ for each $i\in I_{\alpha}$. The use of nets and filters is dictated by the fact that we do not assume $X$ and $Y$ to be first countable. In the application to the theory of dynamical systems that follows, $X$ and $Y$ are compact subsets of $\mathbb{R}$ when the use of sequences suffice. In terms of the residual and cofinal subsets $\textrm{Res}(\mathbb{D})$ and $\textrm{Cof}(\mathbb{D})$ of a directed set $\mathbb{D}$ (Def. A1.7), with $x$ and $y$ in the equations below being taken to belong to the required domains, define subsets $\mathcal{D}_{-}$ of $X$ and $\mathcal{R}_{-}$ of $Y$ as $$\mathcal{D}_{-}=\{ x\in X\!:((f_{\nu}(x))_{\nu\in\mathbb{D}}\textrm{ converges in }(Y,\mathcal{V}))\}\label{Eqn: D-}$$ $$\mathcal{R}_{-}=\{ y\in Y\!:\textrm{ }(\exists i\in I_{\nu})((g_{\nu i}(y))_{\nu\in\mathbb{D}}\textrm{ converges in }(X,\mathcal{U}))\}\label{Eqn: R-}$$ Thus: $\mathcal{D}_{-}$ is the set of points of $X$ on which the values of a given net of functions $(f_{\alpha})_{\alpha\in\mathbb{D}}$ converge pointwise in $Y$. Explicitly, this is the subset of $X$ on which subnets[^19] in $\textrm{Map}(X,Y)$ combine to form a net of functions that converge pointwise to a limit function $F:\mathcal{D}_{-}\rightarrow Y$. $\mathcal{R}_{-}$ is the set of points of $Y$ on which the values of the nets in $X$ generated by the injective branches of $(f_{\alpha})_{\alpha\in\mathbb{D}}$ converge pointwise in $Y$. Explicitly, this is the subset of $Y$ on which subnets of injective branches of $(f_{\alpha})_{\alpha\in\mathbb{D}}$ in $\textrm{Map}(Y,X)$ combine to form a net of functions that converge pointwise to a family of limit functions $G:\mathcal{R}_{-}\rightarrow X$. Depending on the nature of $(f_{\alpha})_{\alpha\in\mathbb{D}}$, there may be more than one $\mathcal{R}_{-}$ with a corresponding family of limit functions on each of them. To simplify the notation, we will usually let $G:\mathcal{R}_{-}\rightarrow X$ denote all the limit functions on all the sets $\mathcal{R}_{-}$. If we consider cofinal rather than residual subsets of $\mathbb{D}$ then corresponding $\mathbb{D}_{+}$ and $\mathbb{R}_{+}$ can be expressed as $$\mathcal{D}_{+}=\{ x\in X\!:((f_{\nu}(x))_{\nu\in\textrm{Cof}(\mathbb{D})}\textrm{ converges in }(Y,\mathcal{V}))\}\label{Eqn: D+}$$ $$\mathcal{R}_{+}=\{ y\in Y\!:(\exists i\in I_{\nu})((g_{\nu i}(y))_{\nu\in\textrm{Cof}(\mathbb{D})}\textrm{ converges in }(X,\mathcal{U}))\}.\label{Eqn: R+}$$ It is to be noted that the conditions $\mathcal{D}_{+}=\mathcal{D}_{-}$ and $\mathcal{R}_{+}=\mathcal{R}_{-}$ are necessary and sufficient for the Kuratowski convergence to exist. Since $\mathcal{D}_{+}$ and $\mathcal{R}_{+}$ differ from $\mathcal{D}_{-}$ and $\mathcal{R}_{-}$ only in having cofinal subsets of $D$ replaced by residual ones, and since residual sets are also cofinal, it follows that $\mathcal{D}_{-}\subseteq\mathcal{D}_{+}$ and $\mathcal{R}_{-}\subseteq\mathcal{R}_{+}$. The sets $\mathcal{D}_{-}$ and $\mathcal{R}_{-}$ serve for the convergence of a net of functions just as $\mathcal{D}_{+}$ and $\mathcal{R}_{+}$ are for the convergence of subnets of the nets (*adherence*). The later sets are needed when subsequences are to be considered as sequences in their own right as, for example, in dynamical systems theory in the case of $\omega$-limit sets. As an illustration of these definitions, consider the sequence of injective functions on the interval $[0,1]$ $f_{n}(x)=2^{n}x$, for $x\in\left[0,1/2^{n}\right],\textrm{ }n=0,1,2\cdots$. Then $\mathbb{D}_{0.2}$ is the set $\{0,1,2\}$ and only $\mathbb{D}_{0}$ is eventual in $\mathbb{D}$. Hence $\mathcal{D}_{-}$ is the single point set {0}. On the other hand $\mathbb{D}_{y}$ is eventual in $\mathbb{D}$ for all $y$ and $\mathcal{R}_{-}$ is $[0,1]$. **Definition** **[<span style="font-variant:small-caps;">3.1</span>]{}***[<span style="font-variant:small-caps;">.</span>]{}* ***Graphical Convergence of a net of functions.*** *A net of functions $(f_{\alpha})_{\alpha\in D}\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$ is said to* *converge graphically* *if either $\mathcal{D}_{-}\neq\emptyset$ or $\mathcal{R}_{-}\neq\emptyset$; in this case let $F\!:\mathcal{D}_{-}\rightarrow Y$ and $G:\mathcal{R}_{-}\rightarrow X$ be the entire collection of limit functions. Because of the assumed Hausdorffness of $X$ and $Y$, these limits are well defined.* *The graph of the* *graphical limit* $\mathscr{M}$ *of the net* $(f_{\alpha})\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$ ***denoted by* ***$f_{\alpha}\overset{\mathbf{G}}\longrightarrow\mathscr{M}$, is the subset of $\mathcal{D}_{-}\times\mathcal{R}_{-}$that is the union of the graphs of the function $F$ and the multifunction $G^{-}$ $$\mathbf{G}_{\mathscr{M}}=\mathbf{G}_{F}\bigcup\mathbf{G}_{G^{-}}$$* *where $$\mathbf{G}_{G^{-}}=\{(x,y)\in X\times Y\!:(y,x)\in\mathbf{G}_{G}\subseteq Y\times X\}.\qquad\square$$* ***Begin Tutorial6: Graphical Convergence*** The following two examples are basic to the understanding of the graphical convergence of functions to multifunctions and were the examples that motivated our search of an acceptable technique that did not require vertical portions of limit relations to disappear simply because they were non-functions: the disturbing question that needed an answer was how not to mathematically sacrifice these extremely significant physical components of the limiting correspondences. Furthermore, it appears to be quite plausible to expect a physical interaction between two spaces $X$ and $Y$ to be a consequence of both the direct interaction represented by $f\!:X\rightarrow Y$ and also the inverse interaction $f^{-}\!:Y\rightarrow X$, and our formulation of pointwise biconvergence is a formalization of this idea. Thus the basic examples (1) and (2) below produce multifunctions instead of discontinuous functions that would be obtained by the usual pointwise limit. **Example 3.1.** (1) $$f_{n}(x)=\left\{ \begin{array}{lc} 0 & -1\leq x\leq0\\ nx & 0\leq x\leq1/n\\ 1 & 1/n\leq x\leq1\end{array}\right.:\quad[-1,1]\rightarrow[0,1]$$ $$g_{n}(y)=y/n:\quad[0,1]\rightarrow[0,1/n]$$ Then$$F(x)=\left\{ \begin{array}{cc} 0 & -1\leq x\leq0\\ 1 & 0<x\leq1\end{array}\right.\qquad\mathrm{on}\qquad\mathcal{D}_{-}=\mathcal{D}_{+}=[-1,0]\bigcup(0,1]$$ $$G(y)=0\quad\mathrm{on}\quad\mathcal{R}_{-}=[0,1]=\mathcal{R}_{+}.$$ The graphical limit is $([-1,0],0)\bigcup(0,[0,1])\bigcup((0,1],1)$. \(2) $f_{n}(x)=nx$ for $x\in[0,1/n]$ gives $g_{n}(y)=y/n:[0,1]\rightarrow[0,1/n].$ Then $$F(x)=0\quad\mathrm{on}\quad\mathcal{D}_{-}=\{0\}=\mathcal{D}_{+},\qquad G(y)=0\quad\mathrm{on}\quad\mathcal{R}_{-}=[0,1]=\mathcal{R}_{+}.$$ The graphical limit is $(0,[0,1])$.$\qquad\blacksquare$ [1.4]{} In these examples that we consider to be the prototypes of graphical convergence of functions to multifunctions, $G(y)=0$ on $\mathcal{R}_{-}$ because $g_{n}(y)\rightarrow0$ for all $y\in\mathcal{R}_{-}$. Compare the graphical multifunctional limits with the corresponding usual pointwise functional limits characterized by discontinuity at $x=0$. Two more examples from @Sengupta2000 that illustrate this new convergence principle tailored specifically to capture one-to-many relations are shown in Fig. \[Fig: Example2\_1\] which also provides an example in Fig. \[Fig: Example2\_1\](c) of a function whose iterates do not converge graphically because in this case both the sets $\mathcal{D}_{-}$ and $\mathcal{R}_{-}$are empty. The power of graphical convergence in capturing multifunctional limits is further demonstrated by the example of the sequence $(\sin n\pi x)_{n=1}^{\infty}$ that converges to $0$ both $1$-integrally and test-functionally, Eqs. (\[Eqn: intsin\]) and (\[Eqn: testsin\]). [$\quad$(b) $F(x)=1$ on $\mathcal{D}_{-}=\{0\}$ and $G(y)=0$ on $\mathcal{R}_{-}=\{1\}$. Also $F(x)=-1/2,\textrm{ }0,\textrm{ }1,\textrm{ }3/2$ respectively on $\mathcal{D}_{+}=(0,3],\textrm{ }\{2\},\textrm{ }\{0\},\textrm{ }(0,2)$ and $G(y)=0,\textrm{ }0,\textrm{ }2,\textrm{ }3$ respectively on $\mathcal{R}_{+}=(-1/2,1],\textrm{ }[1,3/2),\textrm{ }[0,3/2),\textrm{ }[-1/2,0)$. ]{} [$\quad$(c) For $f(x)=-0.05+x-x^{2}$, no graphical limit as $\mathcal{D}_{-}=\emptyset=\mathcal{R}_{-}$.]{} [$\quad$(d) For $f(x)=0.7+x-x^{2}$, $F(x)=\alpha$ on $\mathcal{D}_{-}=[a,c]$, $G_{1}(y)=a$ and $G_{2}(y)=c$ on $\mathcal{R}_{-}=(-\infty,\alpha]$. Notice how the two fixed points and their equivalent images define the converged limit rectangular multi. As in example (1) one has $\mathcal{D}_{-}=\mathcal{D}_{+}$; also $\mathcal{R}_{-}=\mathcal{R}_{+}$.]{} It is necessary to understand how the concepts of *eventually in* and *frequently in* of Appendix A2 apply in examples (a) and (b) of Fig. [\[Fig: Example2\_1\].]{} In these two examples we have two subsequences one each for the even indices and the other for the odd. For a point-to-point functional relation, this would mean that the sequence frequents the adherence set $\textrm{adh}(x)$ of the sequence $(x_{n})$ but does not converge anywhere as it is not eventually in every neighbourhood of any point. For a multifunctional limit however it is possible, as demonstrated by these examples, for the subsequences to be eventually in every neighbourhood of certain *subsets* common to the eventual limiting sets of the subsequences; this intersection of the subsequential limits is now *defined to be the limit of the original sequence.* A similar situation obtains, for example, in the solution of simultaneous equations: The solution of the equation $a_{11}x_{1}+a_{12}x_{2}=b_{1}$ for one of the variables $x_{2}$ say with $a_{12}\neq0$, is the set represented by the straight line $x_{2}=m_{1}x_{1}+c_{1}$ for all $x_{1}$ in its domain, while for a different set of constants $a_{21}$, $a_{22}$ and $b_{2}$ the solution is the entirely different set $x_{2}=m_{2}x_{1}+c_{2}$, under the assumption that $m_{1}\neq m_{2}$ and $c_{1}\neq c_{2}$. Thus even though the individual equations (subsequences) of the simultaneous set of equations (sequence) may have distinct solutions (limits), the solution of the equations is their common point of intersection. Considered as sets in $X\times Y$, the discussion of convergence of a sequence of graphs $f_{n}\!:X\rightarrow Y$ would be incomplete without a mention of the convergence of a sequence of sets under the Hausdorff metric that is so basic in the study of fractals. In this case, one talks about the convergence of a sequence of compact subsets of the metric space $\mathbb{R}^{n}$ so that the sequences, as also the limit points that are the fractals, are compact subsets of $\mathbb{R}^{n}$. Let $\mathcal{K}$ denote the collection of all nonempty compact subsets of $\mathbb{R}^{n}$. Then the *Hausdorff metric* $d_{\textrm{H}}$ between two sets on $\mathcal{K}$ is defined to be $$d_{\textrm{H}}(E,F)=\max\{\delta(E,F),\delta(F,E)\}\qquad E,F\in\mathcal{K},$$ where $$\delta(E,F)=\max_{x\in E}\textrm{ }\min_{y\in F}\Vert\mathbf{x-y}\Vert_{2}$$ is $\delta(E,F)$ is the non-symmetric $2$-norm in $\mathbb{R}^{n}$. The power and utility of the Hausdorff distance is best understood in terms of the dilations $E+\varepsilon:=\bigcup_{x\in E}D_{\varepsilon}(x)$ of a subset $E$ of $\mathbb{R}^{n}$ by $\varepsilon$ where $D_{\varepsilon}(x)$ is a closed ball of radius $\varepsilon$ at $x$; physically a dilation of $E$ by $\varepsilon$ is a closed $\varepsilon$-neighbourhood of $E$. Then a fundamental property of $d_{\textrm{H}}$ is that $d_{\textrm{H}}(E,F)\leq\varepsilon$ iff both $E\subseteq F+\varepsilon$ and $F\subseteq E+\varepsilon$ hold simultaneously which leads [@Falconer1990] to the interesting consequence that *If $(F_{n})_{n=1}^{\infty}$ and $F$ are nonempty compact sets, then $\lim_{n\rightarrow\infty}F_{n}=F$ in the Hausdorff metric iff $F_{n}\subseteq F+\varepsilon$ and $F\subseteq F_{n}+\varepsilon$ eventually. Furthermore if $(F_{n})_{n=1}^{\infty}$ is a decreasing sequence of elements of a filter-base in $\mathbb{R}^{n}$, then the nonempty and compact limit set $F$ is given by $$\lim_{n\rightarrow\infty}F_{n}=F=\bigcap_{n=1}^{\infty}F_{n}.$$* Note that since $\mathbb{R}^{n}$ is Hausdorff, the assumed compactness of $F_{n}$ ensures that they are also closed in **$\mathbb{R}^{n}$; $F$, therefore, is just the adherent set of the filter-base. In the deterministic algorithm for the generation of fractals by the so-called iterated function system (IFS) approach, $F_{n}$ is the inverse image by the $n^{\textrm{th}}\textrm{ }$ iterate of a non-injective function $f$ having a finite number of injective branches and converging graphically to a multifunction. Under the conditions stated above, the Hausdorff metric ensures convergence of any class of compact subsets in $\mathbb{R}^{n}$. It appears eminently plausible that our multifunctional graphical convergence on $\textrm{Map}(\mathbb{R}^{n})$ implies Hausdorff convergence on $\mathbb{R}^{n}$: in fact pointwise biconvergence involves simultaneous convergence of image and preimage nets on $Y$ and $X$ respectively. Thus confining ourselves to the simpler case of pointwise convergence, if $(f_{\alpha})_{\alpha\in\mathbb{D}}$ is a net of functions in $\textrm{Map}(X,Y)$, then the following theorem expresses the link between convergence in $\textrm{Map}(X,Y)$ and in $Y$. **Theorem 3.1.** *A net of functions* $(f_{\alpha})_{\alpha\in\mathbb{D}}$ *converges to a function* $f$ *in* $(\textrm{Map}(X,Y),\mathcal{T})$ *in the topology of pointwise convergence iff* $(f_{\alpha})$ *converges pointwise to $f$ in the sense that $f_{\alpha}(x)\rightarrow f(x)$ in $Y$ for every $x$ in $X$.$\qquad\square$* **Proof.** *Necessity.* First consider $f_{\alpha}\rightarrow f$ in $(\textrm{Map}(X,Y),\mathcal{T})$. For an open neighbourhood $V$ of $f(x)$ in $Y$ with $x\in X$, let $B(x;V)$ be a local neighbourhood of $f$ in $(\textrm{Map}(X,Y),\mathcal{T})$, see Eq. (\[Eqn: point\]) in Appendix A1. By assumption of convergence, $(f_{\alpha})$ must eventually be in $B(x;V)$ implying that $f_{\alpha}(x)$ is eventually in $V$. Hence $f_{\alpha}(x)\rightarrow f(x)$ in $Y$. *Sufficiency.* Conversely, if $f_{\alpha}(x)\rightarrow f(x)$ in $Y$ for every $x\in X$, then for a *finite* collection of points $(x_{i})_{i=1}^{I}$ of $X$ ($X$ may itself be uncountable) and corresponding open sets $(V_{i})_{i=1}^{I}$ in $Y$ with $f(x_{i})\in V_{i}$, let $B((x_{i})_{i=1}^{I};(V_{i})_{i=1}^{I})$ be an open neighbourhood of $f$. From the assumed pointwise convergence $f_{\alpha}(x_{i})\rightarrow f(x_{i})$ in $Y$ for $i=1.2.\cdots.I$, it follows that $(f_{\alpha}(x_{i}))$ is eventually in $V_{i}$ for every $(x_{i})_{i=1}^{I}$. Because $\mathbb{D}$ is a directed set, the existence of a residual applicable globally for all $i=1,2,\cdots,I$ is assured leading to the conclusion that $f_{\alpha}(x_{i})\in V_{i}$ eventually for every $i=1,2,\cdots,I$. Hence $f_{\alpha}\in B((x_{i})_{i=1}^{I};(V_{i})_{i=1}^{I})$ eventually; this completes the demonstration that $f_{\alpha}\rightarrow f$ in $(\textrm{Map}(X,Y),\mathcal{T})$, and thus of the proof.$\qquad\blacksquare$ ***End Tutorial6*** ***3.2. The Extension*** **Multi$_{\mid}$(*X,Y*)** ***of*** **Map(*X,Y*)** In this Section we show how the topological treatment of pointwise convergence of functions to functions given in Example A1.1 of Appendix 1 can be generalized to generate the boundary $\textrm{Multi}_{\mid}(X,Y)$ between $\textrm{Map}(X,Y)$ and $\textrm{Multi}(X,Y)$; here $X$ and $Y$ are Hausdorff spaces and $\textrm{Map}(X,Y)$ and $\textrm{Multi}(X,Y)$ are respectively the sets of all functional and non-functional relations between $X$ and $Y$. The generalization we seek defines neighbourhoods of $f\in\textrm{Map}(X,Y)$ to consist of those functional relations in $\textrm{Multi}(X,Y)$ whose images at any point $x\in X$ lies not only arbitrarily close to $f(x)$ (this generates the usual topology of pointwise convergence $\mathcal{T}_{Y}$ of Example A1.1) but whose inverse images at $y=f(x)\in Y$ contain points arbitrarily close to $x$. Thus the graph of $f$ must not only lie close enough to $f(x)$ at $x$ in $V$, but must additionally be such that $f^{-}(y)$ has at least branch in $U$ about $x$; thus $f$ is constrained to cling to $f$ as the number of points on the graph of $f$ increases with convergence and, unlike in the situation of simple pointwise convergence, no gaps in the graph of the limit object is permitted not only, as in Example A1.1 on the domain of $f$, but simultaneously on it range too. We call the resulting generated topology the *topology of pointwise biconvergence on* $\textrm{Map}(X,Y)$, to be denoted by $\mathcal{T}$. Thus for any given integer $I\geq1$, the generalization of Eq. (\[Eqn: point\]) gives for $i=1,2,\cdots,I$, the open sets of $(\textrm{Map}(X,Y),\mathcal{T})$ to be $$\begin{gathered} B((x_{i}),(V_{i});(y_{i}),(U_{i}))=\{ g\in\mathrm{Map}(X,Y)\!:\\ (g(x_{i})\in V_{i})\wedge(g^{-}(y_{i})\bigcap U_{i}\neq\emptyset)\textrm{ },i=1,2,\cdots,I\},\label{Eqn: func_bi}\end{gathered}$$ where $(x_{i})_{i=1}^{I},(V_{i})_{i=1}^{I}$ are as in that example, $(y_{i})_{i=1}^{I}\in Y$, and the corresponding open sets $(U_{i})_{i=1}^{I}$ in $X$ are chosen arbitrarily[^20]. A local base at $f$, for $(x_{i},y_{i})\in\mathbf{G}_{f}$, is the set of functions of (\[Eqn: func\_bi\]) with $y_{i}=f(x_{i})$ and the collection of all local bases $$B_{\alpha}=B((x_{i})_{i=1}^{I_{\alpha}},(V_{i})_{i=1}^{I_{\alpha}};(y_{i})_{i=1}^{I_{\alpha}},(U_{i})_{i=1}^{I_{\alpha}}),\label{Eqn: local_base}$$ for every choice of $\alpha\in\mathbb{D}$, is a base $_{\textrm{T}}\mathcal{B}$ of $(\textrm{Map}(X,Y),\mathcal{T})$. Here the directed set $\mathbb{D}$ is used as an indexing tool because, as pointed out in Example A1.1, the topology of pointwise convergence is not first countable. In a manner similar to Eq. (\[Eqn: func\_bi\]), the open sets of $(\mathrm{Multi}(X,Y),\widehat{\mathcal{T}})$, where $\textrm{Multi}(X,Y)$ are multifunctions with only countably many values in $Y$ for every point of $X$ (so that we exclude continuous regions from our discussion except for the “vertical lines” of $\textrm{Multi}_{\mid}(X,Y)$), can be defined by $$\begin{gathered} \widehat{B}((x_{i}),(V_{i});(y_{i}),(U_{i}))=\{\mathscr{G}\in\mathrm{Multi}(X,Y)\!:(\mathscr{G}(x_{i})\bigcap V_{i}\neq\emptyset)\wedge(\mathscr G^{-}(y_{i})\bigcap U_{i}\neq\emptyset)\}\label{Eqn: multi_bi}\end{gathered}$$ where $$\mathscr G^{-}(y)=\{ x\in X\!:y\in\mathscr{G}(x)\}.$$ and $(x_{i})_{i=1}^{I}\in\mathcal{D}(\mathscr{G}),(V_{i})_{i=1}^{I};(y_{i})_{i=1}^{I}\in\mathcal{R}(\mathscr{G}),(U_{i})_{i=1}^{I}$ are chosen as in the above. The topology $\widehat{\mathcal{T}}$ of $\textrm{Multi}(X,Y)$ is generated by the collection of all local bases $\widehat{B_{\alpha}}$ for every choice of $\alpha\in\mathbb{D}$, and it is not difficult to see from Eqs. (\[Eqn: func\_bi\]) and (\[Eqn: multi\_bi\]), that the restriction **** of $\widehat{\mathcal{T}}$ to $\textrm{Map}(X,Y)$ is just $\mathcal{T}$. Henceforth $\widehat{\mathcal{T}}$ and $\mathcal{T}$ will be denoted by the same symbol $\mathcal{T}$, and convergence in the topology of pointwise biconvergence in $(\textrm{Multi}(X,Y),\mathcal{T})$ will be denoted by $\rightrightarrows$, with the notation being derived from Theorem 3.1. **Definition 3.2.** ***Functionization of a multifunction.*** *A net of functions* $(f_{\alpha})_{\alpha\in\mathbb{D}}$ *in* $\textrm{Map}(X,Y)$ *converges in* $(\textrm{Multi}(X,Y),\mathcal{T})$, $f_{\alpha}\rightrightarrows\mathscr{M}$, *if it biconverges pointwise in* $(\textrm{Map}(X,Y),\mathcal{T}^{*})$. *Such a net of functions will be said to be a* *functionization of* $\mathscr{M}$*.$\qquad\square$* **Theorem 3.2.** *Let $(f_{\alpha})_{\alpha\in\mathbb{D}}$ be a net of functions in $\textrm{Map}(X,Y)$. Then $$f_{\alpha}\overset{\mathbf{G}}\longrightarrow\mathscr{M}\Longleftrightarrow f_{\alpha}\rightrightarrows\mathcal{M}.\qquad\square$$* **Proof.** If $(f_{\alpha})$ converges graphically to $\mathscr{M}$ then either $\mathcal{D}_{-}$ or $\mathcal{R}_{-}$ is nonempty; let us assume both of them to be so. Then the sequence of functions $(f_{\alpha})$ converges pointwise to a function $F$ on $\mathcal{D}_{-}$ and to functions $G$ on $\mathcal{R}_{-}$, and the local basic neighbourhoods of $F$ and $G$ generate the topology of pointwise biconvergence. Conversely, for pointwise biconvergence on $X$ and $Y$, $\mathcal{R}_{-}$ and $\mathcal{D}_{-}$ must be non-empty.$\qquad\blacksquare$ Observe that the boundary of $\textrm{Map}(X,Y)$ in the topology of pointwise biconvergence is a “line parallel to the $Y$-axis”. We denote this closure of $\textrm{Map}(X,Y)$ as **Definition 3.3.** $\textrm{Multi}_{\mid}((X,Y),\mathcal{T})=\mathrm{Cl}(\mathrm{Map}((X,Y),\mathcal{T})).$$\qquad\square$ The sense in which $\textrm{Multi}_{\mid}(X,Y)$ is the smallest closed topological extension of $M=\textrm{Map}(X,Y)$ is the following, refer Thm. A1.4 and its proof. Let $(M,\mathcal{T}_{0})$ be a topological space and suppose that$${\textstyle \widehat{M}=M\bigcup\{\widehat{m}\}}$$ is obtained by adjoining an extra point to $M$; here $M=\textrm{Map}(X,Y)$ and $\widehat{m}\in\textrm{Cl}(M)$ is the multifunctional limit in $\widehat{M}=\textrm{Multi}_{\mid}(X,Y)$. Treat all open sets of $M$ generated by local bases of the type (\[Eqn: local\_base\]) with finite intersection property as a filter-base $_{\textrm{F}}\mathcal{B}$ on $X$ that induces a filter $\mathcal{F}$ on $M$ (by forming supersets of all elements of $_{\textrm{F}}\mathcal{B}$; see Appendix A1) and thereby the filter-base $${\textstyle \widehat{_{\textrm{F}}\mathcal{B}}=\{\widehat{B}=B\bigcup\{\widehat{m}\}\!:B\in\,_{\textrm{F}}\mathcal{B}\}}$$ on $\widehat{M}$; this filter-base at $m$ can also be obtained independently from Eq. (\[Eqn: multi\_bi\]). Obviously $\widehat{_{\textrm{F}}\mathcal{B}}$ is an extension of $_{\textrm{F}}\mathcal{B}$ on $\widehat{M}$ and $_{\textrm{F}}\mathcal{B}$ is the filter induced on $M$ by $\widehat{_{\textrm{F}}\mathcal{B}}$. We may also consider the filter-base to be a topological base on $M$ that defines a coarser topology $\mathcal{T}$ on $M$ (through all unions of members of $_{\textrm{F}}\mathcal{B}$) and hence the topology$${\textstyle \widehat{\mathcal{T}}=\{\widehat{G}=G\bigcup\{\widehat{m}\}\!:G\in\mathcal{T}\}}$$ on $\widehat{M}$ to be the topology associated with $\widehat{\mathcal{F}}$. A finer topology on $\widehat{M}$ may be obtained by adding to $\widehat{\mathcal{T}}$ all the discarded elements of $\mathcal{T}_{0}$ that do not satisfy FIP. It is clear that $\widehat{m}$ is on the boundary of $M$ because every neighbourhood of $\widehat{m}$ intersects $M$ by construction; thus $(M,\mathcal{T})$ is dense in $(\widehat{M,}\widehat{\mathcal{T}})$ which is the required topological extension of $(M,\mathcal{T}).$ In the present case, a filter-base at $f\in\mathrm{Map}(X,Y)$ is the neighbourhood system $_{\textrm{F}}\mathcal{B}_{f}$ at $f$ given by decreasing sequences of neighbourhoods $(V_{k})$ and $(U_{k})$ of $f(x)$ and $x$ respectively, and the filter $\widehat{\mathcal{F}}$ is the neighbourhood filter $\mathcal{N}_{f}\bigcup G$ where $G\in$$\textrm{Multi}_{\mid}(X,Y)$. We shall present an alternate, and perhaps more intuitively appealing, description of graphical convergence based on the adherence set of a filter on Sec. 4.1. As more serious examples of the graphical convergence of a net of functions to multifunction than those considered above, Fig. \[Fig: tent4\] shows the first four iterates of the tent map $$t(x)=\left\{ \begin{array}{lc} 2x & 0\leq x<1/2\\ 2(1-x) & 1/2\leq x\leq1\end{array}\right.\qquad\begin{array}{c} (t^{1}=t).\end{array}$$ defined on $[0.1]$ and the sine map $f_{n}=|\sin(2^{n-1}\pi x)|,\; n=1,\cdots,4$ with domain $[0,1]$. These examples illustrate the important generalization that *periodic points may be replaced by the more general equivalence classes* where a sequence of functions converges graphically; this generalization based on the ill-posed interpretation of dynamical systems is significant for non-iterative systems as in second example above. The equivalence classes of the tent map for its two fixed points $0$ and $2/3$ generated by the first 4 iterates are $$[0]_{4}=\left\{ 0,\frac{1}{8},\frac{1}{4},\frac{3}{8},\frac{1}{2},\frac{5}{8},\frac{3}{4},\frac{7}{8},1\right\}$$ $$\left[\frac{2}{3}\right]_{4}=\left\{ c,\frac{1}{8}\mp c,\frac{1}{4}\mp c,\frac{3}{8}\mp c,\frac{1}{2}\mp c,\frac{5}{8}\mp c,\frac{3}{4}\mp c,\frac{7}{8}\mp c,1-c\right\}$$ where $c=1/24$. If the moduli of the slopes of the graphs passing through these equivalent fixed points are greater than 1 then the graphs converge to multifunctions and when these slopes are less than 1 the corresponding graphs converge to constant functions. It is to be noted that the number of equivalent fixed points in a class increases with the number of iterations $k$ as $2^{k-1}+1;$ this *increase in the degree of ill-posedness is typical of discrete chaotic systems and can be regarded as a paradigm of chaos generated by* *the convergence of a family of functions.* The $m^{\textrm{th}}$ iterate $t^{m}$ of the tent map has $2^{m}$ fixed points corresponding to the $2^{m}$ injective branches of $t^{m}$ $$x_{mj}=\left\{ \begin{array}{ll} {\displaystyle \frac{j-1}{2^{m}-1}}, & j=1,3,\cdots,(2^{m}-1),\\ {\displaystyle \frac{j}{2^{m}+1}}, & j=2,4,\cdots,2^{m},\end{array}\right.t^{m}(x_{mj})=x_{mj},\textrm{ }j=1,2,\cdots,2^{m}.$$ Let $X_{m}$ be the collection of these $2^{m}$ fixed points (thus $X_{1}=\{0,2/3\}$), and denote by $[X_{m}]$ the set of the equivalent points, one coming from each of the injective branches, for each of the fixed points: thus $$\begin{array}{crcl} \mathcal{D}_{-}= & [X_{1}] & = & \{[0],[2/3]\}\\ & [X_{2}] & = & \{[0],[2/5],[2/3],[4/5]\}\end{array}$$ and $\mathcal{D}_{+}=\bigcap_{m=1}^{\infty}[X_{m}]$ is a nonempty countable set dense in $X$ at each of which the graphs of the sequence $(t^{m})$ converge to a multifunction. New sets $[X_{n}]$ will be formed by subsequences of the higher iterates $t^{n}$ for $m=in$ with $i=1,2,\cdots$ where these subsequences remain fixed. For example, the fixed points $2/5$ and $4/5$ produced respectively by the second and fourth injective branches of $t^{2}$, are also fixed for the seventh and thirteenth branches of $t^{4}.$ For the shift map $2x\;\textrm{mod}(1)$ on $[0,1]$, $\mathcal{D}_{-}=\{[0],[1]\}$ where $[0]=\bigcap_{m=1}^{\infty}\{(i-1)/2^{m}\!:i=1,2,\cdots,2^{m}\}$ and $[1]=\bigcap_{m=1}^{\infty}\{ i/2^{m}\!:i=1,2,\cdots,2^{m}\}$. It is useful to compare the graphical convergence of $(\sin(\pi nx))_{n=1}^{\infty}$ to $[0,1]$ at $0$ and to $0$ at $1$ with the usual integral and test-functional convergences to $0$; note that the point $1/2$, for example, belongs to $\mathcal{D}_{+}$and not to $\mathcal{D}_{-}=\{0,1\}$ because it is frequented by even $n$ only. However for the subsequence $(f_{2^{m-1}})_{m\in\mathbb{Z}_{+}}$, $1/2$ is in $\mathcal{D}_{-}$ because if the graph of $f_{2^{m-1}}$ passes through $(1/2,0)$ for some $m$, then so do the graphs for all higher values . Therefore $[0]=\bigcap_{m=1}^{\infty}\{ i/2^{m-1}\!:i=0,1,\cdots,2^{m-1}\}$ is the equivalence class of $(f_{2^{m-1}})_{m=1}^{\infty}$ and this sequence converges to $[-1,1]$ on this set. Thus our extension $\textrm{Multi}(X)$ is distinct from the distributional extension of function spaces with respect to test functions, and is able to correctly generate the pathological behaviour of the limits that are so crucially vital in producing chaos. **4. Discrete chaotic systems are maximally ill-posed** The above ideas apply to the development of a criterion for chaos in discrete dynamical systems that is based on the limiting behaviour of the graphs of a sequence of functions $(f_{n})$ on $X,$ rather than on the values that the sequence generates as is customary. For the development of the maximality of ill-posedness criterion of chaos, we need to refresh ourselves with the following preliminaries. ***Resume Tutorial5: Axiom of Choice and Zorn’s Lemma*** Let us recall from the first part of this Tutorial that for nonempty subsets $(A_{\alpha})_{\alpha\in\mathbb{D}}$ of a nonempty set $X$, the Axiom of Choice ensures the existence of a set $A$ such that $A\bigcap A_{\alpha}$ consists of a single element for every $\alpha$. The choice axiom has far reaching consequences and a few equivalent statements, one of which the Zorn’s lemma that will be used immediately in the following, is the topic of this resumed Tutorial. The beauty of the Axiom, and of its equivalents, is that they assert the existence of mathematical objects that, in general, cannot be demonstrated and it is often believed that Zorn’s lemma is one of the most powerful tools that a mathematician has available to him that is “almost indispensable in many parts of modern pure mathematics” with significant applications in nearly all branches of contemporary mathematics. This “lemma” talks about maximal (as distinct from “maximum”) elements of a partially ordered set, a set in which some notion of $x_{1}$ “preceding” $x_{2}$ for two elements of the set has been defined. A relation $\preceq$ on a set $X$ is said to be a *partial order* (or simply an *order*) if it is (compare with the properties (ER1)–(ER3) of an equivalence relation, Tutorial1) (OR1) Reflexive, that is $(\forall x\in X)(x\preceq x)$. (OR2) Antisymmetric: $(\forall x,y\in X)(x\preceq y\wedge y\preceq x\Longrightarrow x=y)$. (OR3) Transitive, that is $(\forall x,y,z\in X)(x\preceq y\wedge y\preceq z\Longrightarrow x\preceq z)$. Any notion of order on a set $X$ in the sense of one element of $X$ preceding another should possess at least this property. The **relation is a *preorder* $\precsim$ if it is only reflexive and transitive, that is if only (OR1) and (OR3) are true. If the hypothesis of (OR2) is also satisfied by a preorder, then this $\precsim$ induces an equivalence relation $\sim$ on $X$ according to $(x\precsim y)\wedge(y\precsim x)\Leftrightarrow x\sim y$ that evidently is actually a partial order iff $x\sim y\Leftrightarrow x=y$. For any element $[x]\in X/\sim$ of the induced quotient space, let $\leq$ denote the generated order in $X/\sim$ so that $$x\precsim y\Longleftrightarrow[x]\leq[y];$$ then $\leq$ is a partial order on $X/\sim$. If every two element of $X$ are *comparable*, in the sense that either $x_{1}\preceq x_{2}$ or $x_{2}\preceq x_{1}$ for all $x_{1},x_{2}\in X$, then $X$ is said to be a *totally ordered set* or a *chain.* A totally ordered subset $(C,\preceq)$ of a partially ordered set $(X,\preceq)$ with the ordering induced from $X$, is known as a *chain in $X$* if $$C=\{ x\in X\!:(\forall c\in X)(c\preceq x\vee x\preceq c)\}.\label{Eqn: chain}$$ The most important class of chains that we are concerned with in this work is that on the subsets $\mathcal{P}(X)$ of a set $(X,\subseteq)$ under the inclusion order; Eq. (\[Eqn: chain\]), as we shall see in what follows, defines a family of chains of nested subsets in $\mathcal{P}(X)$. Thus while the relation $\precsim$ in $\mathbb{Z}$ defined by $n_{1}\precsim n_{2}\Leftrightarrow\mid n_{1}\mid\,\leq\,\mid n_{2}\mid$ with $n_{1},n_{2}\in\mathbb{Z}$ preorders $\mathbb{Z}$, it is not a partial order because although $-n\precsim n\textrm{ and }n\precsim-n$ for any $n\in\mathbb{Z}$, it is does not follow that $-n=n$. A common example of partial order on a set of sets, for example on the power set $\mathcal{P}(X)$ of a set $X$ (see footnote \[Foot: notation\]), is the inclusion relation $\subseteq$: the ordered set $\mathcal{X}=(\mathcal{P}(\{ x,y,z\}),\subseteq)$ is partially ordered but not totally ordered because, for example, $\{ x,y\}\not\subseteq\{ y,x\}$, or $\{ x\}$ is not comparable to $\{ y\}$ unless $x=y$; however $C=\{\{\emptyset,\{ x\},\{ x,y\}\}$ does represent one of the many possible chains of $\mathcal{X}$. Another useful example of partial order is the following: Let $X$ and $(Y,\leq)$ be sets with $\leq$ ordering $Y$, and consider $f,g\in\textrm{Map}(X,Y)$ with $\mathcal{D}(f),\mathcal{D}(g)\subseteq X$. Then $$\begin{aligned} (\mathcal{D}(f)\subseteq\mathcal{D}(g))(f=g|_{\mathcal{D}(f)}) & \Longleftrightarrow & f\preceq g\nonumber \\ (\mathcal{D}(f)=\mathcal{D}(g))(\mathcal{R}(f)\subseteq\mathcal{R}(g)) & \Longleftrightarrow & f\preceq g\label{Eqn: FunctionOrder}\\ (\forall x\in\mathcal{D}(f)=\mathcal{D}(g))\textrm{ }(f(x)\leq g(x)) & \Longleftrightarrow & f\preceq g\nonumber \end{aligned}$$ define partial orders on $\textrm{Map}(X,Y)$. In the last case, the order is not total because any two functions whose graphs cross at some point in their common domain cannot be ordered by the given relation, while in the first any $f$ whose graph does not coincide with that of $g$ on the common domain is not comparable to it by this relation. Let $(X,\preceq)$ be a partially ordered set and let $A$ be a subset of $X$. An element $a_{+}\in(A,\preceq)$ is said to be a *maximal* element of $A$ with respect to $\preceq$ if $$(\forall a\in(A,\preceq))(a_{+}\preceq a)\Longrightarrow\textrm{ }a=a_{+},\label{Eqn: maximal}$$ that is iff there is no $a\in A$ with $a\neq a_{+}$ and $a\succ a_{+}$[^21]. Expressed otherwise, this implies that an element $a_{+}$ of a subset $A\subseteq(X,\preceq)$ is maximal in $(A,\preceq)$ iff it is true that $$(a\preceq a_{+}\in A)\textrm{ }(\textrm{for every }a\in(A,\preceq)\textrm{ comparable to }a_{+});\label{Eqn: maximal1}$$ thus $a_{+}$ in $A$ is a maximal element of $A$ iff it is strictly greater than every *other comparable* element of $A$. This of course does not mean that each element $a$ of $A$ satisfies $a\preceq a_{+}$ because every pair of elements of a partially ordered set need not be comparable: in a totally ordered set there can be at most one maximal element. In comparison, an element $a_{\infty}$ of a subset $A\subseteq(X,\preceq)$ is *the* unique *maximum* (*largest, greatest, last*) element of $A$ iff $$(a\preceq a_{\infty}\in A)\textrm{ }(\textrm{for every }a\in(A,\preceq)),\label{Eqn: maximum}$$ implying that $a_{\infty}$ is *the* element of $A$ that is strictly larger than every other element of $A$. As in the case of the maximal, although this also does not require all elements of $A$ to be comparable to each other, it does require $a_{\infty}$ to be larger than every element of $A$. The dual concepts of minimal and minimum can be similarly defined by essentially reversing the roles of $a$ and $b$ in relational expressions like $a\preceq b$. The last concept needed to formalize Zorn’s lemma is that of an upper bound: For a subset $(A,\preceq)$ of a partially ordered set $(X,\preceq)$, an element $u$ of $X$ is an *upper bound of* $A$ *in* $X$ iff $$(a\preceq u\in(X,\preceq))\textrm{ }(\textrm{for every }a\in(A,\preceq))\label{Eqn: upper bound}$$ which requires the upper bound $u$ to be larger than all members of $A$, with the corresponding lower bounds of $A$ being defined in a similar manner. Of course, it is again not necessary that the elements of $A$ be comparable to each other, and it should be clear from Eqs. (\[Eqn: maximum\]) and (\[Eqn: upper bound\]) that when an upper bound of a set is in the set itself, then it is the maximum element of the set. If the upper (lower) bounds of a subset $(A,\preceq)$ of a set $(X,\preceq)$ has a least (greatest) element, then this smallest upper bound (largest lower bound) is called *the* *least upper bound* (*greatest lower* *bound*) or *supremum* (*infimum*) *of $A$ in $X$*. Combining Eqs. (\[Eqn: maximum\]) and (\[Eqn: upper bound\]) then yields $$\begin{array}{rcl} {\displaystyle \sup_{X}A} & = & \{ a_{\leftarrow}\in\Omega_{A}\!:a_{\leftarrow}\preceq u\textrm{ }\forall\textrm{ }u\in(\Omega_{A},\preceq)\}\\ {\displaystyle \inf_{X}A} & = & \{_{\rightarrow}a\in\Lambda_{A}\!:l\preceq\,_{\rightarrow}a\textrm{ }\forall\textrm{ }l\in(\Lambda_{A},\preceq)\}\end{array}\label{Eqn: supinf1}$$ where **$\Omega_{A}=\{\textrm{ }u\in X\!:(\forall\textrm{ }a\in A)(a\preceq u)\}$ **and **$\Lambda_{A}=\{ l\in X\!:(\forall\textrm{ }a\in A)(l\preceq a)\}$ **are the sets of all upper and lower bounds of $A$ in $X$*.* Equation (\[Eqn: supinf1\]) may be expressed in the equivalent but more transparent form as $$\begin{array}{c} {\displaystyle a_{\leftarrow}={\displaystyle \sup_{X}A}\Longleftrightarrow(a\in A\Rightarrow a\preceq a_{\leftarrow})\wedge(a_{0}\prec a_{\leftarrow}\Rightarrow a_{0}\prec b\preceq a_{\leftarrow}\textrm{ for some }b\in A)}\\ _{\rightarrow}a={\displaystyle \inf_{X}A}\Longleftrightarrow(a\in A\Rightarrow\,_{\rightarrow}a\preceq a)\wedge(_{\rightarrow}a\prec a_{1}\Rightarrow\,_{\rightarrow}a\preceq b\prec a_{1}\textrm{ for some }b\in A)\end{array}\label{Eqn: supinf2}$$ to imply that *$a_{\leftarrow}$* ($_{\rightarrow}a$) is *the* upper (lower) bound of $A$ in $X$ which precedes (succeeds) every other upper (lower) bound of $A$ **in $X$. Notice that uniqueness in the definitions above is a direct consequence of the uniqueness of greatest and least elements of a set. **It must be noted that whereas maximal and maximum are properties of the particular subset and have nothing to do with anything outside it, upper and lower bounds of a set are defined only with respect to a superset that may contain it. The following example, beside being useful in Zorn’s lemma, is also of great significance in fixing some of the basic ideas needed in our future arguments involving classes of sets ordered by the inclusion relation. **Example 4.1.** Let $\mathcal{X}=\mathcal{P}(\{ a,b,c\})$ be ordered by the inclusion relation $\subseteq$. The subset $\mathcal{A}=\mathcal{P}(\{ a,b,c\})-\{ a,b,c\}$ has three maximals $\{ a,b\}$, $\{ b,c\}$ and $\{ c,a\}$ but no maximum as there is no $A_{\infty}\in\mathcal{A}$ satisfying $A\preceq A_{\infty}$ for every $A\in\mathcal{A}$, while $\mathcal{P}(\{ a,b,c\})-\emptyset$ the three minimals $\{ a\}$, $\{ b\}$, and $\{ c\}$ but no minimum. This shows that a subset of a partially ordered set may have many maximals (minimals) without possessing a maximum (minimum), but a subset has a maximum (minimum) iff this is its unique maximal (minimal). If $\mathcal{A}=\{\{ a,b\},\{ a,c\}\}$, then every subset of the intersection of the elements of $\mathcal{A}$, namely $\{ a\}$ and $\emptyset$, are lower bounds of $\mathcal{A}$, and all supersets in $\mathcal{X}$ of the union of its elements — which in this case is just $\{ a,b,c\}$ — are its upper bounds. Notice that while the maximal (minimal) and maximum (minimum) are elements of $\mathcal{A}$, upper and lower bounds need not be contained in their sets. In this class $(\mathcal{X},\subseteq)$ of subsets of a set $X$, $X_{+}$ is a maximal element of $\mathcal{X}$ iff $X_{+}$ is not contained in any other subset of $X$, while $X_{\infty}$ is a maximum of $\mathcal{X}$ iff $X_{\infty}$ contains every other subset of $X$. Let $\mathcal{A}:=\{ A_{\alpha}\in\mathcal{X}\}_{\alpha\in\mathbb{D}}$ be a nonempty subclass of $(\mathcal{X},\subseteq)$, and suppose that both $\bigcup A_{\alpha}$ and $\bigcap A_{\alpha}$ are elements of $\mathcal{X}$. Since each $A_{\alpha}$ is $\subseteq$-less than $\bigcup A_{\alpha}$, it follows that $\bigcup A_{\alpha}$ is an upper bound of $\mathcal{A}$; this is also be the smallest of all such bounds because if $U$ is any other upper bound then every $A_{\alpha}$ must precede $U$ by Eq. (\[Eqn: upper bound\]) and therefore so must $\bigcup A_{\alpha}$ (because the union of a class of subsets of a set is the smallest that contain each member of the class: $A_{\alpha}\subseteq U\Rightarrow\bigcup A_{\alpha}\subseteq U$ for subsets $(A_{\alpha})$ and $U$ of $X$). Analogously, since $\bigcap A_{\alpha}$ is $\subseteq$-less than each $A_{\alpha}$ it is a lower bound of $\mathcal{A}$; that it is the greatest of all the lower bounds $L$ in $\mathcal{X}$ follows because the intersection of a class of subsets is the largest that is contained in each of the subsets: $L\subseteq A_{\alpha}\Rightarrow L\subseteq\bigcap A_{\alpha}$ for subsets $L$ and $(A_{\alpha})$ of $X$. Hence the supremum and infimum of $\mathcal{A}$ in $(\mathcal{X},\subseteq)$ given by $$A_{\leftarrow}=\sup_{(\mathcal{X},\subseteq)}\mathcal{A}=\bigcup_{A\in\mathcal{A}}A\qquad\textrm{and}\qquad_{\rightarrow}A=\inf_{(\mathcal{X},\subseteq)}\mathcal{A}=\bigcap_{A\in\mathcal{A}}A\label{Eqn: supinf3}$$ are both elements of $(\mathcal{X},\subseteq)$. Intuitively, an upper (respectively, lower) bound of $\mathcal{A}$ in $\mathcal{X}$ is any subset of $\mathcal{X}$ that contains (respectively, is contained in) every member of $\mathcal{A}$.$\qquad\blacksquare$ The statement of Zorn’s lemma and its proof can now be completed in three stages as follows. For Theorem 4.1 below that constitutes the most significant technical first stage, let $g$ be a function on $(X,\preceq)$ that assigns to every $x\in X$ an *immediate successor* $y\in X$ such that $${\textstyle \mathscr{M}(x)=\{\textrm{ }y\succ x\!:\not\exists\textrm{ }x_{*}\in X\textrm{ satisfying }x\prec x_{*}\prec y\}}$$ are all the successors of $x$ in $X$ with no element of $X$ lying strictly between $x$ and $y$. Select a representative of $\mathscr{M}(x)$ by a choice function $f_{\textrm{C}}$ such that $$g(x)=f_{\textrm{C}}(\mathscr{M}(x))\in\mathscr{M}(x)$$ is an immediate successor of $x$ chosen from the many possible in the set $\mathscr{M}(x)$. The basic idea in the proof of the first of the three-parts is to express the existence of a maximal element of a partially ordered set $X$ in terms of the existence of a fixed point in the set, which follows as a contradiction of the assumed hypothesis that every point in $X$ has an immediate successor. Our basic application of immediate successors in the following will be to classes $\mathcal{X}\subseteq(\mathcal{P}(X),\subseteq)$ of subsets of a set $X$ ordered by inclusion. In this case for any $A\in\mathcal{X}$, the function $g$ can be taken to be the superset $${\textstyle g(A)=A\bigcup f_{\textrm{C}}(\mathscr{G}(A)),\quad\textrm{where }\mathscr{G}(A)=\{ x\in X-A\!:A\bigcup\{ x\}\in\mathcal{X}\}}\label{Eqn: FilterTower}$$ of $A$. Repeated application of $g$ to $A$ then generates a principal filter, and hence an associated sequence, based at $A$. **Theorem 4.1.** *Let $(X,\preceq)$ be a partially ordered set that satisfies* (ST1) *There is a smallest element $x_{0}$ of $X$ which has no immediate predecessor in $X$.* (ST2) *If $C\subseteq X$ is a totally ordered subset in $X$, then $c_{*}=\sup_{X}C$ is in $X$.* *Then there exists a maximal element $x_{+}$ of $X$ which has no immediate successor in $X$.*$\qquad\square$ **Proof.** Let $T\subseteq(X,\preceq)$ be a subset of $X$. If the conclusion of the theorem is false then the alternative (ST3) *Every element $x\in T$ has an immediate successor $g(x)$ in $T$*[^22] leads, as shown below, to a contradiction that can be resolved only by the conclusion of the theorem. A subset $T$ of $(X,\preceq)$ satisfying conditions (ST1)$-$(ST3) is sometimes known as an $g$*-tower* or an $g$*-sequence:* an obvious example of a tower is $(X,\preceq)$ itself. If $${\textstyle _{\rightarrow}T=\bigcap\{ T\in\mathcal{T}\!:T\textrm{ is an }x_{0}-\textrm{tower}\}}$$ is the $(\mathcal{P}(X),\subseteq)$-infimum of the class $\mathcal{T}$ of all sequential towers of $(X,\preceq)$, we show that this smallest sequential **tower is infact a *sequential totally ordered chain* in $(X,\preceq)$ built from $x_{0}$ by the $g$-function. Let the subset $$C_{\textrm{T}}=\{ c\in X\!:(\forall t\in\,_{\rightarrow}T)(t\preceq c\vee c\preceq t)\}\subseteq X\label{Eqn: tower-chain}$$ of $X$ be an $g$-chain in $_{\rightarrow}T$ in the sense that (cf. Eq. (\[Eqn: chain\])) it is that subset of $X$ each of whose elements is comparable with some element of $_{\rightarrow}T$. The conditions (ST1)$-$(ST3) for $C_{\textrm{T}}$ can be verified as follows to demonstrate that $C_{\textrm{T}}$ is an $g$-tower. \(1) $x_{0}\in C_{\textrm{T}}$, because it is less than each $x\in\,_{\rightarrow}T$. \(2) Let $c_{\leftarrow}=\sup_{X}C_{\textrm{T}}$ be the supremum of the chain $C_{\textrm{T}}$ in $X$ so that by (ST2), $c_{\leftarrow}\in X$. Let $t\in\,_{\rightarrow}T$. If there is *some* $c\in C_{\textrm{T}}$ such that $t\preceq c$, then surely $t\preceq c_{\leftarrow}$. Else, $c\preceq t$ for *every* $c\in C_{\textrm{T}}$ shows that $c_{\leftarrow}\preceq t$ because $c_{\leftarrow}$ is the smallest of all the upper bounds $t$ of $C_{\textrm{T}}$. Therefore $c_{\leftarrow}\in C_{\textrm{T}}$. \(3) In order to show that $g(c)\in C$ whenever $c\in C$ it needs to verified that for all $t\in\,_{\rightarrow}T$, either $t\preceq c\Rightarrow t\preceq g(c)$ or $c\preceq t\Rightarrow g(c)\preceq t$. As the former is clearly obvious, we investigate the later as follows; note that $g(t)\in\,_{\rightarrow}T$ by (ST3). The first step is to show that the subset $$C_{g}=\{ t\in\,_{\rightarrow}T\!:(\forall c\in C_{\textrm{T}})(t\preceq c\vee g(c)\preceq t)\}\label{Eqn: chain_g}$$ of $_{\rightarrow}T$, which is a chain in $X$ (observe the inverse roles of $t$ and $c$ here as compared to that in Eq. (\[Eqn: tower-chain\])), is a tower: Let $t_{\leftarrow}$ be the supremum of $C_{g}$ and take $c\in C$. If there is *some* $t\in C_{g}$ for which $g(c)\preceq t$, then clearly $g(c)\preceq t_{\leftarrow}$. Else, $t\preceq x$ for *each* $t\in C_{g}$ shows that $t_{\leftarrow}\preceq c$ because $t_{\leftarrow}$ is the smallest of all the upper bounds $c$ of $C_{g}$. Hence $t_{\leftarrow}\in C_{g}$. Property (ST3) for $C_{g}$ follows from a small yet significant modification of the above arguments in which the immediate successors $g(t)$ of $t\in C_{g}$ formally replaces the supremum $t_{\leftarrow}$ of $C_{g}$. Thus given a $c\in C$, if there is *some* $t\in C_{g}$ for which $g(c)\preceq t$ then $g(c)\prec g(t)$; this combined with $(c=t)\Rightarrow(g(c)=g(t))$ yields $g(c)\preceq g(t)$. On the other hand, $t\prec c$ for *every* $t\in C_{g}$ requires $g(t)\preceq c$ as otherwise $(t\prec c)\Rightarrow(c\prec g(t))$ would, from the resulting consequence $t\prec c\prec g(t)$, contradict the assumed hypothesis that $g(t)$ is the immediate successor of $t$. Hence, $C_{g}$ is a $g$-tower in $X$. To complete the proof that $g(c)\in C_{\textrm{T}}$, and thereby the argument that $C_{\textrm{T}}$ is a tower, we first note that as $_{\rightarrow}T$ is the smallest tower and $C_{g}$ is built from it, $C_{g}=\,_{\rightarrow}T$ must infact be $_{\rightarrow}T$ itself. From Eq. (\[Eqn: chain\_g\]) therefore, for every $t\in\,_{\rightarrow}T$ either $t\preceq g(c)$ or $g(c)\preceq t$, so that $g(c)\in C_{\textrm{T}}$ whenever $c\in C_{\textrm{T}}$. This concludes the proof that $C_{\textrm{T}}$ is actually the tower $_{\rightarrow}T$ in $X$. From (ST2), the implication of the chain $C_{\textrm{T}}$ $$C_{\textrm{T}}=\,_{\rightarrow}T=C_{g}\label{Eqn: ChainedTower}$$ being the minimal tower $_{\rightarrow}T$ is that the supremum $t_{\leftarrow}$ of the totally ordered $_{\rightarrow}T$ *in its own tower* (as distinct from in the tower $X$: recall that $_{\rightarrow}T$ is a subset of $X$) must be contained in itself, that is $$\sup_{C_{\textrm{T}}}(C_{\textrm{T}})=t_{\leftarrow}\in\,_{\rightarrow}T\subseteq X.\label{Eqn: sup chain}$$ This however leads to the contradiction from (ST3) that $g(t_{\leftarrow})$ be an element of $_{\rightarrow}T$, unless of course $$g(t_{\leftarrow})=t_{\leftarrow},\label{Eqn: fixed point}$$ which because of (\[Eqn: ChainedTower\]) may also be expressed equivalently as $g(c_{\leftarrow})=c_{\leftarrow}\in C_{\textrm{T}}$. As the sequential totally ordered set $_{\rightarrow}T$ is a subset of $X$, Eq. (48) implies that $t_{\leftarrow}$ is a maximal element of $X$ which allows (ST3) to be replaced by the remarkable inverse criterion that $(\textrm{ST}3^{\prime})$ If $x\in X$ and $w$ precedes $x,$ $w\prec x$, then $w\in X$ that is obviously false for a general tower $T$. In fact, it follows directly from Eq. (\[Eqn: maximal\]) that under $(\textrm{ST}3^{\prime})$ *any $x_{+}\in X$ is a maximal element of $X$ iff it is a fixed point of $g$* as given by Eq. (\[Eqn: fixed point\]). This proves the theorem and also demonstrates how, starting from a minimum element of a partially ordered set $X$, (ST3) can be used to generate inductively a totally ordered sequential subset of $X$ *leading to a maximal $x_{+}=c_{\leftarrow}\in(X,\preceq)$ that is a fixed point of the generating function $g$* *whenever the supremum* $t_{\leftarrow}$ *of the chain $_{\rightarrow}T$ is in* $X$.$\qquad\blacksquare$ **Remarks.** The proof of this theorem, despite its apparent length and technically involved character, carries the highly significant underlying message that [0.1in]{} *Any inductive sequential $g$-construction of an infinite chained tower* $C_{\textrm{T}}$ *starting with a smallest element $x_{0}\in(X,\preceq)$ such that a supremum $c_{\leftarrow}$ of the $g$-generated sequential chain* $C_{\textrm{T}}$ *in its own tower is contained in itself, must necessarily terminate with a fixed point relation of the type* (\[Eqn: fixed point\]) *with respect to the supremum. Note from Eqs. (\[Eqn: sup chain\]) and (\[Eqn: fixed point\]) that the role of* (ST2) *applied to a fully ordered tower is the identification of the maximal of the tower — which depends only the tower and has nothing to do with anything outside it — with its supremum that depends both on the tower and its complement.* Thus although purely set-theoretic in nature, the filter-base associated with a sequentially totally ordered set may be interpreted to lead to the usual notions of adherence and convergence of filters and thereby of a generated topology for $(X,\preceq)$, see Appendix A1 and Example A1.3. This very significant apparent inter-relation between topologies, filters and orderings will form the basis of our approach to the condition of maximal ill-posedness for chaos. In the second stage of the three stage programme leading to Zorn’s lemma, the tower Theorem 4.1 and the comments of the preceding paragraph are applied at one higher level to a very special class of the power set of a set, the class of all the chains of a partially ordered set, to directly lead to the physically significant **Theorem 4.2.** **Hausdorff Maximal Principle.** *Every partially ordered set $(X,\preceq)$ has a maximal totally ordered subset*.[^23]$\qquad\square$ **Proof.** Here the base level is $$\mathcal{X}=\{ C\in\mathcal{P}(X)\!:C\textrm{ is a chain in }(X,\preceq)\}\subseteq\mathcal{P}(X)\label{Eqn: Hausdorff}$$ be the set of all the totally ordered subsets of $(X,\preceq)$. Since $\mathcal{X}$ is a collection of (sub)sets of $X$, we order it by the inclusion relation on $\mathcal{X}$ and use the tower Theorem to demonstrate that $(\mathcal{X},\subseteq)$ has a maximal element $C_{\leftarrow}$, which by the definition of $\mathcal{X}$, is the required maximal chain in $(X,\preceq)$. Let $\mathcal{C}$ be a chain in $\mathcal{X}$ of the chains in $(X,\preceq)$. In order to apply the tower Theorem to $(\mathcal{X},\subseteq)$ we need to verify hypothesis (ST2) that the smallest $$C_{*}=\sup_{\mathcal{X}}\mathcal{C}=\bigcup_{C\in\mathcal{C}}C\label{Eqn: HausdorffChain}$$ of the possible upper bounds of $\mathcal{C}$ (see Eq. (\[Eqn: supinf3\])) is a chain of $(X,\preceq)$. Indeed, if $x_{1},x_{2}\in X$ are two points of $C_{\textrm{sup}}$ with $x_{1}\in C_{1}$ and $x_{2}\in C_{2}$, then from the $\subseteq$-comparability of $C_{1}$ and $C_{2}$ we may choose $x_{1},x_{2}\in C_{1}\supseteq C_{2}$, say. Thus $x_{1}$ and $x_{2}$ are $\preceq$-comparable as $C_{1}$ is a chain in $(X,\preceq)$; $C_{*}\in\mathcal{X}$ is therefore a chain in $(X,\preceq)$ which establishes that the supremum of a chain of $(\mathcal{X},\subseteq)$ is a chain in $(X,\preceq)$. The tower Theorem 4.1 can now applied to $(\mathcal{X},\subseteq)$ with $C_{0}$ as its smallest element to construct a $g$-sequentially towered fully ordered subset of $\mathcal{X}$ consisting of chains in $X$ $$\mathcal{C}_{\textrm{T}}=\{ C_{i}\in\mathcal{P}(X)\!:C_{i}\subseteq C_{j}\textrm{ for }i\leq j\in\mathbb{N}\}=\,_{\rightarrow}\mathcal{T}\subseteq\mathcal{P}(X)$$ of $(\mathcal{X},\subseteq)$ — consisting of the common elements of all $g$-sequential towers $\mathcal{T}\in\mathfrak{T}$ of $(\mathcal{X},\subseteq)$ — that infact is a principal filter base of chained subsets of $(X,\preceq)$ at $C_{0}$. The supremum (chain in $X$) $C_{\leftarrow}$ of $\mathcal{C}_{\textrm{T}}$ in $\mathcal{C}_{\textrm{T}}$ must now satisfy, by Thm. 4.1, the fixed point $g$-chain of $X$ $$\sup_{\mathcal{C}_{\textrm{T}}}(\mathcal{C}_{\textrm{T}})=C_{\leftarrow}=g(C_{\leftarrow})\in\mathcal{C}_{\textrm{T}}\subseteq\mathcal{P}(X),$$ where the chain $g(C)=C\bigcup f_{\textrm{C}}(\mathscr{G}(C)-C)$ with $\mathscr{G}(C)=\{ x\in X-C\!:C\bigcup\{ x\}\in\mathcal{X}\}$, is an immediate successor of $C$ obtained by choosing one point $x=f_{\textrm{C}}(\mathscr{G}(C)-C)$ from the many possible in $\mathscr{G}(C)-C$ such that the resulting $g(C)=C\bigcup\{ x\}$ is a strict successor of the chain $C$ with no others lying between it and $C$. Note that $C_{\leftarrow}\in(\mathcal{X},\subseteq)$ is only one of the many maximal fully ordered subsets possible in $(X,\preceq)$.$\qquad\blacksquare$ With the assurance of the existence of a maximal chain $C_{\leftarrow}$ among all fully ordered subsets of a partially ordered set $(X,\preceq)$, the arguments are completed by returning to the basic level of $X$. **Theorem 4.3. Zorn’s Lemma.** *Let $(X,\preceq)$ be a partially ordered set such that every totally ordered subset of $X$ has an upper bound in $X$. Then $X$ has at least one maximal element with respect to its order.$\qquad\square$* **Proof.** The proof of this final part is a mere application of the Hausdorff Maximal Principle on the existence of a maximal chain $C_{\leftarrow}$ in $X$ to the hypothesis of this theorem that $C_{\leftarrow}$ has an upper bound $u$ in $X$ that quickly leads to the identification of this bound as a maximal element $x_{+}$ of $X$. Indeed, if there is an element $v\in X$ that is comparable to $u$ and $v\not\preceq u$, then $v$ cannot be in $C_{\leftarrow}$ as it is necessary for every $x\in C_{\leftarrow}$ to satisfy $x\preceq u$. Clearly then $C_{\leftarrow}\bigcup\{ v\}$ is a chain in $(X,\preceq)$ bigger than $C_{\leftarrow}$ which contradicts the assumed maximality of $C_{\leftarrow}$ among the chains of $X$.$\qquad\blacksquare$ The sequence of steps leading to Zorn’s Lemma, and thence to the maximal of a partially ordered set, is summarised in Fig. \[Fig: Zorn\]. [(a) The one-level higher subset $\mathcal{X}=\{ C\in\mathcal{P}(X)\!:C\textrm{ is a chain in }(X,\preceq)\}$ of $\mathcal{P}(X)$ consisting of all the totally ordered subsets of $(X,\preceq)$, ]{} [(b) The smallest common $g$-sequential totally ordered towered chain $\mathcal{C}_{\textrm{T}}=\{ C_{i}\in\mathcal{P}(X)\!:C_{i}\subseteq C_{j}\textrm{ for }i\leq j\}\subseteq\mathcal{P}(X)$ of all sequential $g$-towers of $\mathcal{X}$ by Thm. 4.1, which infact is a principal filter base of totally ordered subsets of $(X,\preceq)$ at the smallest element $C_{0}$. ]{} [(c) Apply Hausdorff Maximal Principle to $(\mathcal{X},\subseteq)$ to get the subset $\sup_{\mathcal{C}_{\textrm{T}}}(\mathcal{C}_{\textrm{T}})=C_{\leftarrow}=g(C_{\leftarrow})\in\mathcal{C}_{\textrm{T}}\subseteq\mathcal{P}(X)$ of $(X,\preceq)$ as the supremum of $(\mathcal{X},\subseteq)$ in $\mathcal{C}_{\textrm{T}}$. The identification of this supremum as a maximal element of $(\mathcal{X},\subseteq)$ is a consequence of (ST2) and Eqs. (\[Eqn: sup chain\]), (\[Eqn: fixed point\]) that actually puts the supremum into $\mathcal{X}$ itself. ]{} [By returning to the original level $(X,\preceq)$ ]{} [(d) Zorn’s Lemma finally yields the required maximal element $u\in X$ as an upper bound of the maximal totally ordered subset $(C_{\leftarrow},\preceq)$ of $(X,\preceq)$. ]{} [The dashed segment denotes the higher Hausdorff $(\mathcal{X},\subseteq)$ level leading to the base $(X,\preceq)$ Zorn level. ]{} The three examples below of the application of Zorn’s Lemma clearly reflect the increasing complexity of the problem considered, with the maximals a point, a subset, and a set of subsets of $X$, so that these are elements of $X$, $\mathcal{P}(X)$, and $\mathcal{P}^{2}(X)$ respectively. **Example 4.2.** (1) Let $X=(\{ a,b,c\},\preceq)$ be a three-point base-level ground set ordered lexicographically, that is $a\prec b\prec c$. A chain $\mathcal{C}$ of the partially ordered Hausdorff-level set $\mathcal{X}$ consisting of subsets of $X$ given by Eq. (\[Eqn: Hausdorff\]) is, for example, $\{\{ a\},\{ a,b\}\}$ and the six $g$-sequential chained towers $$\begin{array}{c} \mathcal{C}_{1}=\{\emptyset,\{ a\},\{ a,b\},\{ a,b,c\}\},\qquad\mathcal{C}_{2}=\{\emptyset,\{ a\},\{ a,c\},\{ a,b,c\}\}\\ \mathcal{C}_{3}=\{\emptyset,\{ b\},\{ a,b\},\{ a,b,c\}\},\qquad\mathcal{C}_{4}=\{\emptyset,\{ b\},\{ b,c\},\{ a,b,c\}\}\\ \mathcal{C}_{5}=\{\emptyset,\{ c\},\{ a,c\},\{ a,b,c\}\},\qquad\mathcal{C}_{6}=\{\emptyset,\{ c\},\{ b,c\},\{ a,b,c\}\}\end{array}$$ built from the smallest element $\emptyset$ corresponding to the six distinct ways of reaching $\{ a,b,c\}$ from $\emptyset$ along the sides of the cube marked on the figure with solid lines, all belong to $\mathcal{X}$; see Fig. \[Fig: order\](b). An example of a tower in $(\mathcal{X},\subseteq)$ which is not a chain is $$\mathcal{T}=\{\emptyset,\{ a\},\{ b\},\{ c\},\{ a,b\},\{ a,c\},\{ b,c\},\{ a,b,c\}\}.$$ Hence the common infimum towered chained subset $$\mathcal{C}_{\textrm{T}}=\{\emptyset,\{ a,b,c\}\}=\,_{\rightarrow}\mathcal{T}\subseteq\mathcal{P}(X)$$ of $\mathcal{X}$, with $$\sup_{\mathcal{C}_{\textrm{T}}}(\mathcal{C}_{\textrm{T}})=C_{\leftarrow}=\{ a,b,c\}=g(C_{\leftarrow})\in\mathcal{C}_{\textrm{T}}\subseteq\mathcal{P}(X)$$ the only maximal element of $\mathcal{P}(X)$. Zorn’s Lemma now assures the existence of a maximal element of $c\in X$. Observe how the maximal element of $(X,\preceq)$ is obtained by going one level higher to $\mathcal{X}$ at the Hausdorff stage and returning to the base level $X$ at Zorn, see Fig. \[Fig: Zorn\] for a schematic summary of this sequence of steps. \(2) *Basis of a vector space.* A linearly independent set of vectors in a vector space $X$ that spans the space is known as the Hamel basis of $X$. To prove the existence of a Hamel basis in a vector space, Zorn’s lemma is invoked as follows. The ground base level of the linearly independent subsets of $X$ $$\mathcal{X}=\{\{ x_{i_{j}}\}_{j=1}^{J}\in\mathcal{P}(X)\!:\textrm{Span}(\{ x_{i_{j}}\}_{j=1}^{J})=0\Rightarrow(\alpha_{j})_{j=1}^{J}=0\,\forall J\geq1\}\subseteq\mathcal{P}(X)),$$ with $\textrm{Span}(\{ x_{i_{j}}\}_{j=1}^{J}):=\sum_{j=1}^{J}\alpha_{j}x_{i_{j}}$, is such that no $x\in\mathcal{X}$ can be expressed as a linear combination of the elements of $\mathcal{X}-\{ x\}$. $\mathcal{X}$ clearly has a smallest element, say $\{ x_{i_{1}}\}$, for some non-zero $x_{i_{1}}\in X$. Let the higher Hausdorff level $$\mathfrak{X}=\{\mathcal{C}\in\mathcal{P}^{2}(X)\!:\mathcal{C}\textrm{ is a chain in }(\mathcal{X},\subseteq)\}\subseteq\mathcal{P}^{2}(X)$$ collection of the chains $$\mathcal{C}_{i_{K}}=\{\{ x_{i_{1}}\},\{ x_{i_{1}},x_{i_{2}}\},\cdots,\{ x_{i_{1}},x_{i_{2}},\cdots,x_{i_{K}}\}\}\textrm{ }\in\mathcal{P}^{2}(X)$$ of $\mathcal{X}$ comprising linearly independent subsets of $X$ be $g$-built from the smallest $\{ x_{i_{1}}\}$. Any chain $\mathfrak{C}$ of $\mathfrak{X}$ is bounded above by the union $\mathcal{C}_{*}=\sup_{\mathfrak{X}}\mathfrak{C}=\bigcup_{\mathcal{C}\in\mathfrak{C}}\mathcal{C}$ which is a chain in $\mathcal{X}$ containing $\{ x_{i_{1}}\}$, thereby verifying (ST2) for $\mathfrak{X}$. Application of the tower theorem to $\mathfrak{X}$ implies that the chain $$\mathfrak{C}_{\textrm{T}}=\{\mathcal{C}_{i_{1}},\mathcal{C}_{i_{2}},\cdots,\mathcal{C}_{i_{n}},\cdots\}=\,_{\rightarrow}\mathfrak{T}\subseteq\mathcal{P}^{2}(X)$$ in $\mathfrak{X}$ of chains of $\mathcal{X}$ is a $g$-sequential fully ordered towered subset of $(\mathfrak{X},\subseteq)$ consisting of the common elements of all $g$-sequential towers of $(\mathfrak{X},\subseteq)$, that infact is a *chained* *principal ultrafilter on $(\mathcal{P}(X),\subseteq)$ generated by the filter-base $\{\{\{ x_{i_{1}}\}\}\}$* *at $\{ x_{i_{1}}\}$*, where $$\mathfrak{T}=\{\mathcal{C}_{i_{1}},\mathcal{C}_{i_{2}},\cdots,\mathcal{C}_{j_{n}},\mathcal{C}_{j_{n+1}},\cdots\}$$ for some $n\in\mathbb{N}$ is an example of non-chained $g$-tower whenever $(\mathcal{C}_{j_{k}})_{k=n}^{\infty}$ is neither contained in nor contains any member of the $(\mathcal{C}_{i_{k}})_{k=1}^{\infty}$ chain. Hausdorff’s chain theorem now yields the fixed-point $g$-chain $\mathcal{C}_{\leftarrow}\,\in\mathfrak{X}$ of $\mathcal{X}$ $$\sup_{\mathfrak{C}_{\textrm{T}}}(\mathfrak{C}_{\textrm{T}})=\mathcal{C}_{\leftarrow}=\{\{ x_{i_{1}}\},\{ x_{i_{1}},x_{i_{2}}\},\{ x_{i_{1}},x_{i_{2}},x_{i_{3}}\},\cdots\}=g(\mathcal{C}_{\leftarrow})\in\mathfrak{C}_{\textrm{T}}\subseteq\mathcal{P}^{2}(X)$$ as a maximal *totally ordered* *principal filter on $X$ that is generated by the filter-base $\{\{ x_{i_{1}}\}\}$* *at $x_{i_{1}}$*, whose supremum $B=\{ x_{i_{1}},x_{i_{2}},\cdots\}\in\mathcal{P}(X)$ is, by Zorn’s lemma, a maximal element of the base level $\mathcal{X}$. This maximal linearly independent subset of $X$ is the required Hamel basis for $X$: Indeed, if the span of $B$ is not the whole of $X$, then $\textrm{Span}(B)\bigcup x$, with $x\notin\textrm{Span}(B)$ would, by definition, be a linearly independent set of $X$ strictly larger than $B$, contradicting the assumed maximality of the later. It needs to be understood that since the infinite basis cannot be classified as being linearly independent, we have here an important example of the supremum of the maximal chained set not belonging to the set even though this criterion was explicitly used in the construction process according to (ST2) and (ST3). Compared to this purely algebraic concept of basis in a vector space, is the Schauder basis in a normed space which combines topological structure with the linear in the form of convergence: If a normed vector space contains a sequence $(e_{i})_{i\in\mathbb{Z}_{+}}$ with the property that for every $x\in X$ there is an unique sequence of scalars $(\alpha_{i})_{i\in\mathbb{Z}_{+}}$ such that the remainder $\parallel x-(\alpha_{1}e_{1}+\alpha_{2}e_{2}+\cdots+\alpha_{I}e_{I})\parallel$ approaches $0$ as $I\rightarrow\infty$, then the collection $(e_{i})$ is known as a Schauder basis for $X$. \(3) *Ultrafilter.* Let $X$ be a set. The set $${\textstyle _{\textrm{F}}\mathcal{S}=\{ S_{\alpha}\in\mathcal{P}(X)\!:S_{\alpha}\bigcap S_{\beta}\neq\emptyset,\textrm{ }\forall\alpha\neq\beta\}\subseteq\mathcal{P}(X)}$$ of all nonempty subsets of $X$ with finite intersection property is known as a *filter subbase on* $X$ and $_{\textrm{F}}\mathcal{B}=\{ B\subseteq X\!:B=\bigcap_{i\in I\subset\mathbb{D}}S_{i}\}$, for $I\subset\mathbb{D}$ a finite subset of a directed set $\mathbb{D}$, is a *filter-base on $X$* *associated with the subbase* $_{\textrm{F}}\mathcal{S}$; cf. Appendix A1. Then the *filter generated by* $_{\textrm{F}}\mathcal{S}$ consisting of every superset of the finite intersections $B\in\,_{\textrm{F}}\mathcal{B}$ of sets of $_{\textrm{F}}\mathcal{S}$ is the smallest filter that contain the subbase $_{\textrm{F}}\mathcal{S}$ and base $_{\textrm{F}}\mathcal{B}$. For notational simplicity, we will denote the subbase $_{\textrm{F}}\mathcal{S}$ in the rest of this example simply by $\mathcal{S}$. Consider the base-level ground set of all filter subbases on $X$ $$\mathfrak{S}=\{\mathcal{S}\in\mathcal{P}^{2}(X)\!:\bigcap_{\emptyset\neq\mathcal{R}\subseteq\mathcal{S}}\mathcal{R}\neq\emptyset\textrm{ for every finite subset of }\mathcal{S}\}\subseteq\mathcal{P}^{2}(X),$$ ordered by inclusion in the sense that $\mathcal{S}_{\alpha}\subseteq\mathcal{S}_{\beta}\textrm{ for all }\alpha\preceq\beta\in\mathbb{D}$, and let the higher Hausdorff-level $$\widetilde{\mathfrak{X}}=\{\mathfrak{C}\in\mathcal{P}^{3}(X)\!:\mathfrak{C}\textrm{ is a chain in }(\mathfrak{S},\subseteq)\}\subseteq\mathcal{P}^{3}(X)$$ comprising the collection of the totally ordered chains $$\mathfrak{C}_{\kappa}=\{\{ S_{\alpha}\},\{ S_{\alpha},S_{\beta}\},\cdots,\{ S_{\alpha},S_{\beta},\cdots,S_{\kappa}\}\}\in\mathcal{P}^{3}(X)$$ of $\mathfrak{S}$ be $g$-built from the smallest $\{ S_{\alpha}\}$ then an *ultrafilter* on $X$ is a maximal member $\mathcal{S}_{+}$ of $(\mathfrak{S},\subseteq)$ in the usual sense that any subbase $\mathcal{S}$ on $X$ must necessarily be contained in $\mathcal{S}_{+}$ so that $\mathcal{S}_{+}\subseteq\mathcal{S}\Rightarrow\mathcal{S}=\mathcal{S}_{+}$ for any $\mathcal{S}\subseteq\mathcal{P}(X)$ with FIP. The tower theorem now implies that the element $$\widetilde{\mathfrak{C}_{\textrm{T}}}=\{\mathfrak{C}_{\alpha},\mathfrak{C}_{\beta},\cdots,\mathfrak{C}_{\nu},\cdots\}=\,\widetilde{_{\rightarrow}\mathfrak{T}}\subseteq\mathcal{P}^{3}(X)$$ of $\mathcal{P}^{4}(X)$, which is a chain in $\widetilde{\mathfrak{X}}$ of the chains of $\mathfrak{S}$, is a $g$-sequential fully ordered towered subset of the common elements of all sequential towers of $(\widetilde{\mathfrak{X}},\subseteq)$ and a *chained* *principal ultrafilter on $(\mathcal{P}^{2}(X),\subseteq)$ generated by the filter-base $\{\{\{ S_{\alpha}\}\}\}$* *at* $\{ S_{\alpha}\}$; here $$\widetilde{\mathfrak{T}}=\{\mathfrak{C}_{\alpha},\mathfrak{C}_{\beta},\cdots,\mathfrak{C}_{\sigma},\mathfrak{C}_{\varsigma},\cdots\},$$ is an obvious example of non-chained $g$-tower whenever $(\mathfrak{C}_{\sigma})$ is neither contained in, nor contains, any member of the $\mathfrak{C}_{\alpha}$-chain. Hausdorff’s chain theorem now yields the fixed-point $\widetilde{\mathfrak{C}_{\leftarrow}}\,\in\widetilde{\mathfrak{X}}$ $$\sup_{\widetilde{\mathfrak{C}_{\textrm{T}}}}(\widetilde{\mathfrak{C}_{\textrm{T}}})=\widetilde{\mathfrak{C}_{\leftarrow}}=\{\{ S_{\alpha}\},\{ S_{\alpha},S_{\beta}\},\{ S_{\alpha},S_{\beta},S_{\gamma}\},\cdots\}=g(\widetilde{\mathfrak{C}_{\leftarrow}})\in\widetilde{\mathfrak{C}_{\textrm{T}}}\subseteq\mathcal{P}^{3}(X)$$ as a maximal *totally ordered* $g$-chained towered subset of $X$ that is, by Zorn’s lemma, a maximal element of the base level subset $\mathfrak{S}$ of $\mathcal{P}^{2}(X)$. $\widetilde{\mathfrak{C}_{\leftarrow}}$ is a *chained principal ultrafilter on* $(\mathcal{P}(X),\subseteq)$ *generated by the filter-base $\{\{ S_{\alpha}\}\}$* *at $S_{\alpha}$*, while $\mathcal{S}_{+}=\{ S_{\alpha},S_{\beta},S_{\gamma},\cdots\}\in\mathcal{P}^{2}(X)$ is an (non-principal) *ultrafilter on* $X$ — characterized by the property that any collection of subsets on $X$ with FIP (that is any filter subbase on $X$) must be contained the maximal set $\mathcal{S}_{+}$ having FIP — that is not a principal filter unless $\mathcal{S}_{\alpha}$ is a singleton set $\{ x_{\alpha}\}$. $\qquad\blacksquare$ [1.4]{} What emerges from these application of Zorn’s Lemma is the remarkable fact that *infinities (the dot-dot-dots) can be formally introduced as “limiting cases” of finite systems in a purely set-theoretic context* *without the need for topologies, metrics or convergences.* The significance of this observation will become clear from our discussions on filters and topology leading to Sec. 4.2 below. Also, the observation on the successive iterates of the power sets $\mathcal{P}(X)$ in the examples above was to suggest their anticipated role in the complex evolution of a dynamical system that is expected to play a significant part in our future interpretation and understanding of this adaptive and self-organizing phenomenon of nature. ***End Tutorial5*** From the examples in Tutorial5, it should be clear that the sequential steps summarized in Fig. \[Fig: Zorn\] are involved in an application of Zorn’s lemma to show that a partially ordered set has a maximal element with respect to its order. Thus for a partially ordered set $(X,\preceq)$, form the set $\mathcal{X}$ of all chains $C$ in $X$. If $C_{+}$ is a maximal chain of $X$ obtained by the Hausdorff Maximal Principle from the chain $\mathcal{C}$ of all chains of $X$, then its supremum $u$ is a maximal element of $(X,\preceq)$. This sequence is now applied, paralleling Example 4.2(1), to the set of arbitrary relations $\textrm{Multi}(X)$ on an infinite set $X$ in order to formulate our definition of chaos that follows. Let $f$ be a *noninjective map* in $\textrm{Multi}(X)$ and $P(f)$ the number of injective branches of $f$. Denote by $$F=\{ f\in\textrm{Multi}(X)\!:f\textrm{ is a noninjective function on }X\}\subseteq\textrm{Multi}(X)$$ the resulting basic collection of noninjective functions in $\textrm{Multi}(X)$. \(i) For every $\alpha$ in some directed set $\mathbb{D}$, let $F$ have the extension property $$(\forall f_{\alpha}\in F)(\exists f_{\beta}\in F)\!:P(f_{\alpha})\leq P(f_{\beta})$$ \(ii) Let a partial order $\preceq$ on $\textrm{Multi}(X)$ be defined, for $f_{\alpha},f_{\beta}\in\textrm{Map}(X)\subseteq\textrm{Multi}(X)$ by $$P(f_{\alpha})\leq P(f_{\beta})\Longleftrightarrow f_{\alpha}\preceq f_{\beta},\label{Eqn: chaos1}$$ with $P(f):=1$ for the smallest $f$, define a partially ordered subset $(F,\preceq)$ of $\textrm{Multi}(X)$. This is actually a preorder on $\textrm{Multi}(X)$ in which functions with the same number of injective branches are equivalent to each other. \(iii) Let $$C_{\nu}=\{ f_{\alpha}\in\textrm{Multi}(X)\!:f_{\alpha}\preceq f_{\nu}\}\in\mathcal{P}(F),\qquad\nu\in\mathbb{D},$$ be the $g$-chains of non-injective functions of $\textrm{Multi}(X)$ and $$\mathcal{X}=\{ C\in\mathcal{P}(F)\!:C\textrm{ is a chain in }(F,\preceq)\}\subseteq\mathcal{P}(F)$$ denote the corresponding Hausdorff level of the chains of $F$, with $$\mathcal{C}_{\textrm{T}}=\{ C_{\alpha},C_{\beta},\cdots,C_{\nu},\cdots\}=\,_{\rightarrow}\mathcal{T}\subseteq\mathcal{P}(F)$$ being a $g$-sequential chain in $\mathcal{X}$ . **By Hausdorff Maximal Principle, there is a maximal fixed-point $g$-towered chain $C_{\leftarrow}\in\mathcal{X}$ of $F$ $$\sup_{\mathcal{C}_{\textrm{T}}}(\mathcal{C}_{\textrm{T}})=C_{\leftarrow}=\{ f_{\alpha},f_{\beta},f_{\gamma},\cdots\}=g(C_{\leftarrow})\in\mathcal{C}_{\textrm{T}}\subseteq\mathcal{P}(F).$$ Zorn’s Lemma now applied to this maximal chain yields its supremum as the maximal element of $C_{\leftarrow}$, and thereby of $F$. It needs to be appreciated, as in the case of the algebraic Hamel basis, that the existence of this maximal non-functional element was obtained purely set theoretically as the “limit” of a net of functions with increasing non-linearity, without resorting to any topological arguments. Because it is not a function, this supremum does not belong to the functional $g$-towered chain having it as a fixed point, and this maximal chain does not possess a largest, or even a maximal, element, although it does have a supremum.[^24] The supremum is a contribution of the inverse functional relations $(f_{\alpha}^{-})$ in the following sense. From Eq. (\[Eqn: func-multi\]), the net of increasingly non-injective functions of Eq. (\[Eqn: chaos1\]) implies a corresponding net of increasingly multivalued functions ordered inversely by the inverse relation $f_{\alpha}\preceq f_{\beta}\Leftrightarrow f_{\beta}^{-}\preceq f_{\alpha}^{-}$. Thus the inverse relations which are as much an integral part of graphical convergence as are the direct relations, have a smallest element belonging to the multifunctional class. Clearly, this smallest element as the required supremum of the increasingly non-injective tower of functions defined by Eq. (\[Eqn: chaos1\]), serves to complete the significance of the tower by capping it with a “boundary” element that can be taken to bridge the classes of functional and non-functional relations on $X$. We are now ready to define a *maximally ill-posed problem $f(x)=y$* for *$x,y\in X$* in terms of a *maximally non-injective map $f$* as follows. **Definition 4.1.** ***Chaotic map.*** *Let $A$ be a non-empty closed set of a compact Hausdorff space $X.$ A function* $f\in\textrm{Multi}(X)$ **(*equivalently the sequence of functions $(f_{i})$*)** *is* *maximally non-injective* *or* *chaotic on* **$A$** *with respect to the order relation* **(\[Eqn: chaos1\])** *if* *(a) for any $f_{i}$ on $A$ there exists an $f_{j}$ on $A$ satisfying $f_{i}\preceq f_{j}$ for every $j>i\in\mathbb{N}$.* *(b) the set $\mathcal{D}_{+}$ consists of a countable collection of isolated singletons.$\qquad\square$* **Definition 4.2.** ***Maximally ill-posed problem.*** *[<span style="font-variant:small-caps;">L</span>]{}et $A$ be a non-empty closed set of a compact Hausdorff space $X$ and let $f$ be a functional relation in* $\textrm{Multi}(X)$*. The problem $f(x)=y$ is* *maximally ill-posed at* **$y$** *if $f$ is chaotic on $A$*.$\qquad\square$ As an example of the application of these definitions, on the dense set $\mathcal{D}_{+}$, the tent map satisfies both the conditions of sensitive dependence on initial conditions and topological transitivity [@Devaney1989] and is also maximally non-injective; the tent map is therefore chaotic on $\mathcal{D}_{+}.$ In contrast, the examples of Secs. 1 and 2 are not chaotic as the maps are not topologically transitive, although the Liapunov exponents, as in the case of the tent map, are positive. Here the $(f_{n})$ are identified with the iterates of $f,$ and the “fixed point” as one through which graphs of all the functions on residual index subsets pass. When the set of points $\mathcal{D}_{+}$ is dense in $[0,1]$ and both $\mathcal{D}_{+}$ and $[0,1]-\mathcal{D}_{+}=[0,1]-\bigcup_{i=0}^{\infty}f^{-i}(\textrm{Per}(f))$ (where $\textrm{Per}(f)$ denotes the set of periodic points of $f$) are totally disconnected, it is expected that at any point on this complement the behaviour of the limit will be similar to that on $\mathcal{D}_{+}$: these points are special as they tie up the iterates on $\textrm{Per}(f)$ to yield the multifunctions. Therefore in any neighbourhood $U$ of a $\mathcal{D}_{+}$-point, there is an $x_{0}$ at which the *forward orbit $\{ f^{i}(x_{0})\}_{i\geq0}$ is chaotic* in the sense that \(a) the sequence neither diverges nor does it converge in the image space of $f$ to a periodic orbit of any period, and \(b) the Liapunov exponent given by $$\begin{aligned} \lambda(x_{0}) & = & \lim_{n\rightarrow\infty}\ln\left|\frac{df^{n}(x_{0})}{dx}\right|^{1/n}\\ & = & {\displaystyle \lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=0}^{n-1}\ln\left|\frac{df(x_{i})}{dx}\right|,\, x_{i}=f^{i}(x_{0}),}\end{aligned}$$ which is a measure of the average slope of an orbit at $x_{0}$ or equivalently of the average loss of information of the position of a point after one iteration, is positive. Thus *an orbit with positive Liapunov exponent is chaotic if it is not* *asymptotic* (that is neither convergent nor adherent, having no convergent suborbit in the sense of Appendix A1) *to an unstable periodic orbit* *or to any other limit set on which the dynamics is simple.* A basic example of a chaotic orbit is that of an irrational in $[0,1]$ under the shift map and that of the chaotic set its closure, the full unit interval. Let $f\in\textrm{Map}((X,\mathcal{U}))$ and suppose that $A=\{ f^{j}(x_{0})\}_{j\in\mathbb{N}}$ is a sequential set corresponding to the orbit $\textrm{Orb}(x_{0})=(f^{j}(x_{0}))_{j\in\mathbb{N}}$, and let $f_{\mathbb{R}_{i}}(x_{0})=\bigcup_{j\geq i}f^{j}(x_{0})$ be the $i$-residual of the sequence $(f^{j}(x_{0}))_{j\in\mathbb{N}}$, with $_{\textrm{F}}\mathcal{B}_{x_{0}}=\{ f_{\mathbb{R}_{i}}(x_{0})\!:\textrm{Res}(\mathbb{N})\rightarrow X\textrm{ for all }i\in\mathbb{N}\}$ being the decreasingly nested filter-base associated with $\textrm{Orb}(x_{0})$. The so-called *$\omega$-limit set of* $x_{0}$ given by $$\begin{array}{ccl} \omega(x_{0}) & \overset{\textrm{def}}= & \{ x\in X\!:(\exists n_{k}\in\mathbb{N})(n_{k}\rightarrow\infty)\textrm{ }(f^{n_{k}}(x_{0})\rightarrow x)\}\\ & = & \{ x\in X\!:(\forall N\in\mathcal{N}_{x})(\forall f_{\mathbb{R}_{i}}\in\,_{\textrm{F}}\mathcal{B}_{x_{0}})\textrm{ }(f_{\mathbb{R}_{i}}(x_{0})\bigcap N\neq\emptyset)\}\end{array}\label{Eqn: Def: omega(x)}$$ is simply the adherence set $\textrm{adh}(f^{j}(x_{0}))$ of the sequence $(f^{j}(x_{0}))_{j\in\mathbb{N}}$, see Eq. (\[Eqn: net adh\]); hence Def. A1.11 of the filter-base associated with a sequence and Eqs. (\[Eqn: adh net2\]), (\[Eqn: adh filter\]), (\[Eqn: filter adh\*\]) and (\[Eqn: net-fil\]) allow us to express $\omega(x_{0})$ more meaningfully as $$\omega(x_{0})=\bigcap_{i\in\mathbb{N}}\textrm{Cl}(f_{\mathbb{R}_{i}}(x_{0})).\label{Eqn: adh_omega_x}$$ It is clear from the second of Eqs. (\[Eqn: Def: omega(x)\]) that for a continuous $f$ and any $x\in X$, $x\in\omega(x_{0})$ implies $f(x)\in\omega(x_{0})$ so that the entire orbit of $x$ lies in $\omega(x_{0})$ whenever $x$ does implying that the $\omega$-limit set is positively invariant; it is also closed because the adherent set is a closed set according to Theorem A1.3. Hence $x_{0}\in\omega(x_{0})\Rightarrow A\subseteq\omega(x_{0})$ reduces the $\omega$-limit set to the closure of $A$ without any isolated points, $A\subseteq\textrm{Der}(A)$. In terms of Eq. (\[Eqn: PrinFil\_Cl(A)\]) involving principal filters, Eq. (\[Eqn: adh\_omega\_x\]) in this case may be expressed in the more transparent form $\omega(x_{0})=\bigcap\textrm{Cl}(\,_{\textrm{F}}\mathcal{P}(\{ f^{j}(x_{0})\}_{j=0}^{\infty}))$ where the principal filter $_{\textrm{F}}\mathcal{P}(\{ f^{j}(x_{0})\}_{j=0}^{\infty})$ at $A$ consists of all supersets of $A=\{ f^{j}(x_{0})\}_{j=0}^{\infty}$, and $\omega(x_{0})$ represents the adherence set of the principal filter at $A$, see the discussion following Theorem A1.3. If $A$ represents a chaotic orbit under this condition, then $\omega(x_{0})$ is sometimes known as a *chaotic set* [@Alligood1997]; thus the chaotic orbit infinitely often visits every member of its chaotic set[^25] which is simply the $\omega$-limit set of a chaotic orbit that is itself contained in its own limit set. Clearly the chaotic set if positive invariant, and from Thm. A1.3 and its corollary it is also compact. Furthermore, if all (sub)sequences emanating from points $x_{0}$ in some neighbourhood of the set converge to it, then $\omega(x_{0})$ is called a *chaotic attractor,* see @Alligood1997. As common examples of chaotic sets that are not attractors mention may be made of the tent map with a peak value larger than $1$ at $0.5$, and the logistic map with $\lambda\geq4$ again with a peak value at $0.5$ exceeding $1$. [Figure \[Fig: logcob357\],]{} [contd: Multifunctional and cobweb plots of $\lambda_{*}x(1-x)$ where $\lambda_{*}=3.5699456$]{} It is important that the difference in the dynamical behaviour of the system on $\mathcal{D}_{+}$ and its complement be appreciated. At any fixed point $x$ of $f^{i}$ in $\mathcal{D}_{+}$ (or at its equivalent images in $[x]$) the dynamics eventually gets attached to the (equivalent) fixed point, and the sequence of iterates converges graphically in $\textrm{Multi}(X)$ to $x$ (or its equivalent points). [Figure \[Fig: logcob357\],]{} [contd: Mulnctional and cobweb plots of $3.57x(1-x)$. ]{} When $x\notin\mathcal{D}_{+}$, however, the orbit $A=\{ f^{i}(x)\}_{i\in\mathbb{N}}$ is chaotic in the sense that $(f^{i}(x))$ is not asymptotically periodic and not being attached to any particular point they wander about in the closed chaotic set $\omega(x)=\textrm{Der}(A)$ containing $A$ such that for any given point in the set, some subsequence of the chaotic orbit gets arbitrarily close to it. Such sequences do not converge anywhere but only frequent every point of $\textrm{Der}(A)$. Thus although in the limit of progressively larger iterations there is complete uncertainty of the outcome of an experiment conducted at either of these two categories of initial points, whereas on $\mathcal{D}_{+}$ this is due to a random choice from a multifunctional set of equally probable outputs as dictated by the specific conditions under which the experiment was conducted at that instant, on its complement the uncertainty is due to the chaotic behaviour of the functional iterates themselves. Nevertheless it must be clearly understood *that this later behaviour is* *entirely due to the multifunctional limits at the $\mathcal{D}_{+}$ points which completely determine the behaviour of the system on its complement.* As an explicit illustration of this situation, recall that for the shift map $2x\textrm{ mod}(1)$ the $\mathcal{D}_{+}$ points are the rationals on $[0,1]$, and any irrational is represented by a non-terminating and non-repeating decimal so that almost all decimals in $[0,1]$ in any base contain all possible sequences of any number of digits. For the logistic map, the situation is more complex, however. Here the onset of chaos marking the end of the period doubling sequence at $\lambda_{*}=3.5699456$ is signaled by the disappearance of all stable fixed points, Fig. \[Fig: logcob357\](c), with Fig. \[Fig: logcob357\](a) being a demonstration of the stable limits for $\lambda=3.569$ that show up as convergence of the iterates to constant valued functions (rather than as constant valued inverse functions) at stable fixed points, shown more emphatically in Fig\[Fig: log357\](a). What actually happens at $\lambda_{*}$ is shown in Fig. \[Fig: attractor\](a) in the next subsection: the almost vertical lines produced at a large, but finite, iterations $i$ (the multifunctions are generated only in the limiting sense of $i\rightarrow\infty$ and represent a boundary between functional and non-functional relations on a set), decrease in magnitude with increasing iterations until they reduce to points. This gives rise to a (totally disconnected) Cantor set on the $y$-axis in contrast with the connected intervals that the multifunctional limits at $\lambda>\lambda_{*}$ of Figs. \[Fig: attractor\](b)–(d) produce. By our characterization Definition 4.1 of chaos therefore, $\lambda x(1-x)$ is chaotic for the values of $\lambda>\lambda_{*}$ that are shown in Fig. \[Fig: attractor\]. We return to this case in the following subsection. ** [Figure]{} [\[Fig: log357\], contd: Isolated fixed points of logistic map. The sequence of points generated by the iterates of the map are marked on the $y$-axis of (a)–(c) in]{} *italics*[. The singletons $\{ x\}$ are $\omega$-limit sets of the respective fixed point $x$ and is generated by the constant sequence $(x,x,\cdots)$. Whereas in (a) this is the limit of every point in $(0,1)$, in the other cases these fixed points are isolated in the sense of Def. 2.3. The isolated points, however, give rise to sequences that converge to more than one point in the form of limit cycles as shown in figures (b)–(d). ]{} As an example of chaos *in a noniterative system*, we investigate the following question: While maximality of non-injectiveness produced by an increasing number of injective branches is necessary for a family of functions to be chaotic, is this also sufficient for the system to be chaotic? This is an important question especially in the context of a non-iterative family of functions where fixed points are of no longer relevant. Consider the sequence of functions $|\sin(\pi nx)|_{n=1}^{\infty}.$ The graphs of the subsequence $|\sin(2^{n-1}\pi x)|$ and of the sequence $(t^{n}(x))$ on [\[]{}0,1[\]]{} are qualitatively similar in that they both contain $2^{n-1}$ of their functional graphs each on a base of $1/2^{n-1}.$ Thus both $|\sin(2^{n-1}\pi x)|_{n=1}^{\infty}$ and $(t^{n}(x))_{n=1}^{\infty}$ converge graphically to the multifunction [\[]{}0,1[\]]{} on the same set of points equivalent to 0. This is sufficient for us to conclude that $|\sin(2^{n-1}\pi x)|_{n=1}^{\infty}$, and hence $|\sin(\pi nx)|_{n=1}^{\infty}$, is chaotic on the infinite equivalent set [\[]{}0[\]]{}. While Fig. \[Fig: tent4\] was a comparison of the first four iterates of the tent and absolute sine maps, Fig. [\[Fig: tent17\]]{} following shows the “converged” graphical limits for after 17 iterations. ***4.1. The chaotic attractor*** One of the most fascinating characteristics of chaos in dynamical systems is the appearance of attractors the dynamics on which are chaotic. **For a subset $A$ of a topological space $(X,\mathcal{U})$ such that $\mathcal{R}(f(A))$ is contained in $A$ — in this section, unless otherwise stated to the contrary, $f(A)$ *will* *denote the* *graph and not the range (image)* *of* $f$ — which ensures that the iteration process can be carried out in $A$, let $$\begin{array}{ccl} {\displaystyle f_{\mathbb{R}_{i}}(A)} & = & {\displaystyle \bigcup_{j\geq i\in\mathbb{N}}f^{j}(A)}\\ & = & {\displaystyle \bigcup_{j\geq i\in\mathbb{N}}\left(\bigcup_{x\in A}f^{j}(x)\right)}\end{array}\label{Eqn: absorbing set}$$ generate the filter-base $_{\textrm{F}}\mathcal{B}$ with $A_{i}:=f_{\mathbb{R}_{i}}(A)\in\,_{\textrm{F}}\mathcal{B}$ being decreasingly nested, $A_{i+1}\subseteq A_{i}$ for all $i\in\mathbb{N}$, in accordance with Def. A1.1. The existence of a maximal chain with a corresponding maximal element as asssured by the Hausdorff Maximal Principle and Zorn’s Lemma respectively implies a nonempty core of $_{\textrm{F}}\mathcal{B}$. As in Sec. 3 following Def. 3.3, we now identify the filterbase with the neighbourhood base at $f^{\infty}$ which allows us to define $$\begin{array}{ccl} {\displaystyle \textrm{Atr}(A_{1})} & \overset{\textrm{def}}= & \textrm{adh}(\,_{\textrm{F}}\mathcal{B})\\ & = & {\displaystyle \bigcap_{A_{i}\in\,_{\textrm{F}}\mathcal{B}}\textrm{Cl}(A_{i})}\end{array}\label{Eqn: attractor_adherence}$$ as the attractor of the set $A_{1}$, where the last equality follows from Eqs.(\[Eqn: Def: omega(A)\]) and (\[Eqn: Def: Closure\]) and the closure is with respect to the topology induced by the neighbourhood filter base $_{\textrm{F}}\mathcal{B}$. Clearly the attractor as defined here is the graphical limit of the sequence of functions $(f^{i})_{i\in\mathbb{N}}$ which may be verified by reference to Def. A1.8, Thm. A1.3 and the proofs of Thms. A1.4 and A1.5, together with the directed set Eq. (\[Eqn: DirectedIndexed\]) with direction (\[Eqn: DirectionIndexed\]). The *basin of attraction* of the attractor is $A_{1}$ because the graphical limit $(\mathcal{D}_{+},F(\mathcal{D}_{+}))\bigcup(G(\mathcal{R}_{+}),\mathcal{R}_{+})$ of Def. 3.1 may be obtained, as indicated above, by a proper choice of sequences associated with $\mathcal{A}$. Note that in the context of iterations of functions, the graphical limit $(\mathcal{D}_{+},y_{0})$ of the sequence $(f^{n}(x))$ denotes a stable fixed point $x_{*}$ with image $x_{*}=f(x_{*})=y_{0}$ to which iterations starting at any point $x\in\mathcal{D}_{+}$ converge. The graphical limits $(x_{i0},\mathcal{R}_{+})$ are generated with respect to the class $\{ x_{i*}\}$ of points satisfying $f(x_{i0})=x_{i*}$, $i=0,1,2,\cdots$ equivalent to unstable fixed point $x_{*}:=x_{0*}$ to which inverse iterations starting at any initial point in $\mathcal{R}_{+}$ must converge. Even though only $x_{*}$ is inverse stable, an equivalent class of graphically converged limit multis is produced at every member of the class $x_{i*}\in[x_{*}]$, resulting in the far-reaching consequence *that every member of the class is as significant as the parent fixed point $x_{*}$ from which they were born in determining the dynamics of the evolving system.* The point to remember about infinite intersections of a collection of sets having finite intersection property, as in Eq. (\[Eqn: attractor\_adherence\]), is that this may very well be empty; recall, however, that in a compact space this is guaranteed not to be so. In the general case, if $\textrm{core}(\mathcal{A})\neq\emptyset$ then $\mathcal{A}$ is the principal filter at this core, and $\textrm{Atr}(A_{1})$ by Eqs. (\[Eqn: attractor\_adherence\]) and (\[Eqn: PrinFil\_Cl(A)\]) is the closure of this core, which in this case of the topology being induced by the filterbase, is just the core itself. $A_{1}$ by its very definition, is a positively invariant set as any sequence of graphs converging to **$\textrm{Atr}(A_{1})$ must be eventually in $A_{1}$: the entire sequence therefore lies in $A_{1}$. Clearly, from Thm. A3.1 and its corollary, the attractor is a positively invariant compact set. A typical attractor is illustrated by the derived sets in the second column of Fig. \[Fig: DerSets\] which also illustrates that the set of functional relations are open in $\textrm{Multi}(X)$; specifically functional-nonfunctional correspondences are neutral-selfish related as in Fig. \[Fig: DerSets\], 3-2, with the attracting graphical limit of Eq. (\[Eqn: attractor\_adherence\]) forming the boundary of (finitely)many-to-one functions and the one-to-(finitely)many multifunctions. Equation (\[Eqn: attractor\_adherence\]) is to be compared with the *image definition of an attractor* [@Stuart1996] where $f(A)$ denotes the range and not the graph of $f$. Then Eq. (\[Eqn: attractor\_adherence\]) can be used to define a sequence of points $x_{k}\in A_{n_{k}}$ and hence the subset $$\begin{aligned} \omega(A) & \overset{\textrm{def}}= & \{ x\in X\!:(\exists n_{k}\in\mathbb{N})(n_{k}\rightarrow\infty)(\exists x_{k}\in A_{n_{k}})\textrm{ }(f^{n_{k}}(x_{k})\rightarrow x)\}\nonumber \\ & = & \{ x\in X\!:(\forall N\in\mathcal{N}_{x})(\forall A_{i}\in\mathcal{A})(N\bigcap A_{i}\neq\emptyset)\}\label{Eqn: Def: omega(A)}\end{aligned}$$ as the corresponding attractor of $A$ that satisfies an equation formally similar to (\[Eqn: attractor\_adherence\]) with the difference that the filter-base $\mathcal{A}$ is now in terms of the image $f(A)$ of $A$, which allows the adherence expression to take the particularly simple form $$\omega(A)=\bigcap_{i\in\mathbb{N}}\textrm{Cl}(f^{i}(A)).\label{Eqn: omega(A)_intersect}$$ The complimentary subset excluded from this definition of $\omega(A)$, as compared to $\textrm{Atr}(A_{1})$, that is required to complete the formalism is given by Eq. (\[Eqn: basin\]) below. Observe that the equation for $\omega(A)$ is essentially Eq. (\[Eqn: adh net1\]), even though we prefer to use the alternate form of Eq. (\[Eqn: adh net2\]) as this brings out more clearly the frequenting nature of the sequence. The basin of attraction $$\begin{array}{ccl} B_{f}(A) & = & \{ x\in A\!:\omega(x)\subseteq\textrm{Atr}(A)\}\\ & = & \{ x\in A\!:(\exists n_{k}\in\mathbb{N})(n_{k}\rightarrow\infty)\textrm{ }(f^{n_{k}}(x)\rightarrow x^{*}\in\omega(A)\textrm{ })\end{array}\label{Eqn: basin}$$ of the attractor is the smallest subset of $X$ in which sequences generated by $f$ must eventually lie in order to adhere at $\omega(A)$. Comparison of Eqs. (\[Eqn: Attractor\_R+\]) with (\[Eqn: R+\]) and (\[Eqn: basin\]) with (\[Eqn: D+\]) show that $\omega(A)$ can be identified with the subset $\mathcal{R}_{+}$ on the $y$-axis on which the multifunctional limits $G\!:\mathcal{R}_{+}\rightarrow X$ of graphical convergence are generated, with its basin of attraction being contained in the $\mathcal{D}_{+}$ associated with the injective branch of $f$ that generates $\mathcal{R}_{+}$. In summary it may be concluded that since definitions (\[Eqn: Def: omega(A)\]) and (\[Eqn: basin\]) involve both the domain and range of $f$, a description of the attractor in terms of the graph of $f$, like that of Eq. (\[Eqn: attractor\_adherence\]), is more pertinent and meaningful as it combines the requirements of both these equations. Thus, for example, as $\omega(A)$ is not the function $G(\mathcal{R}_{+})$, this attractor does not include the equivalence class of inverse stable points that may be associated with $x_{*}$, see for example Fig. \[Fig: omega\]. From Eq. (\[Eqn: Def: omega(A)\]), we may make the particularly simple choice of $(x_{k})$ to satisfy $f^{n_{k}}(x_{-k})=x$ so that $x_{-k}=f_{\textrm{B}}^{-n_{k}}(x)$, where $x_{-k}\in[x_{-k}]:=f^{-n_{k}}(x)$ is the element of the equivalence class of the inverse image of $x$ corresponding to the injective branch $f_{\textrm{B}}$. This choice is of special interest to us as it is the class that generates the $G$-function on $\mathcal{R}_{+}$ in graphical convergence. This allows us to express $\omega(A)$ as $$\omega(A)=\{ x\in X\!:(\exists n_{k}\in\mathbb{N})(n_{k}\rightarrow\infty)(f_{\textrm{B}}^{-n_{k}}(x)=x_{-k}\textrm{ converges in }(X,\mathcal{U}))\};\label{Eqn: Attractor_R+}$$ note that the $x_{-k}$ of this equation and the $x_{k}$ of Eq. (\[Eqn: Def: omega(A)\]) are, in general, quite different points. A simple illustrative example of the construction of $\omega(A)$ for the positive injective branch of the homeomorphism $(4x^{2}-1)/3$, $-1\leq x\leq1$, is shown in Fig. \[Fig: omega\], where the arrow-heads denote the converging sequences $f^{n_{i}}(x_{i})\rightarrow x$ and $f^{n_{i}-m}(x_{i})\rightarrow x_{-m}$ which proves invariance of $\omega(A)$ for a homeomorphic $f$; here continuity of the function and its inverse is explicitly required for invariance. Positive invariance of a subset $A$ of $X$ implies that for any $n\in\mathbb{N}$ and $x\in A$, $f^{n}(x)=y_{n}\in A$, while negative invariance assures that for any $y\in A$, $f^{-n}(y)=x_{-n}\in A$. Invariance of $A$ in both the forward and backward directions therefore means that for any $y\in A$ and $n\in\mathbb{N}$, there exists a $x\in A$ such that $f^{n}(x)=y$. In interpreting this figure, it may be useful to recall from Def. 4.1 that an increasing number of injective branches of $f$ is a necessary, but not sufficient, condition for the occurrence of chaos; thus in Figs. \[Fig: log357\](a) and \[Fig: omega\], increasing noninjectivity of $f$ leads to constant valued limit functions over a connected $\mathcal{D}_{+}$ in a manner similar to that associated with the classical Gibb’s phenomenon in the theory of Fourier series. Graphical convergence of an increasingly nonlinear family of functions implied by its increasing non-injectivity may now be combined with the requirements of an attractor to lead to the concept of a chaotic attractor to be that on which the dynamics is chaotic in the sense of Defs. 4.1. and 4.2. Hence **Definition 4.3.** ***Chaotic Attractor.*** *Let $A$ be a positively invariant subset of $X$. The attractor* $\textrm{Atr}(A)$ *is chaotic on $A$ if there is sensitive dependence on initial conditions for* all *$x\in A$. The sensitive dependence manifests itself as multifunctional graphical limits for all $x\in\mathcal{D}_{+}$ and as chaotic orbits when* $x\not\in\mathcal{D}_{+}$*.*$\qquad\square$ [$$f_{\textrm{f}}(x)=\left\{ \begin{array}{ccl} 2(1+x)/3, & & 0\leq x<1/2\\ 2(1-x), & & 1/2\leq x\leq1\end{array}\right.$$ ]{} The picture of chaotic attractors that emerge from the foregoing discussions and our characterization of chaos of Def. 4.1 is that it it is a subset of $X$ that is simultaneously “spiked” multifunctional on the $y$-axis and consists of a dense collection of singleton domains of attraction on the $x$-axis. This is illustrated in Figure \[Fig: attractor\] which shows some typical chaotic attractors. The first four diagrams (a)$-$(d) are for the logistic map with (b)$-$(d) showing the 4-, 2- and 1-piece attractors for $\lambda=3.575,\textrm{ }3.66,\textrm{ and }3.8$ respectively that are in qualitative agreement with the standard bifurcation diagram reproduced in (e). Figs. (b)$-$(d) have the advantage of clearly demonstrating how the attractors are formed by considering the graphically converged limit as the object of study unlike in Fig. (e) which shows the values of the 501-1001th iterates of $x_{0}=1/2$ as a function of $\lambda$. The difference in Figs. (a) and (b) for a change of $\lambda$ from [$\lambda>\lambda_{*}=3.5699456$]{} to 3.575 is significant as $\lambda=\lambda_{*}$ marks the boundary between the nonchaotic region for $\lambda<\lambda_{*}$ and the chaotic for $\lambda>\lambda_{*}$ (this is to be understood as being suitably modified by the appearance of the nonchaotic windows for some specific intervals in $\lambda>\lambda_{*}$). At $\lambda_{*}$ the generated fractal Cantor set $\Lambda$ is an attractor as it attracts almost every initial point $x_{0}$ so that the successive images $x^{n}=f^{n}(x_{0})$ converge toward the Cantor set $\Lambda$. In Fig. (f) the chaotic attractors for the piecewise continuous function on $[0,1]$ $$f_{\textrm{f}}(x)=\left\{ \begin{array}{ccc} 2(1+x)/3, & & 0\leq x<1/2\\ 2(1-x), & & 1/2\leq x\leq1,\end{array}\right.$$ is $[0,1]$ where the dotted lines represent odd iterates and the full lines even iterates of $f$; here the attractor disappears if the function is reflected about the $x$-axis. [Figure]{} [\[Fig: attractor\]]{}[, contd.]{} [Chaotic attractors for $\lambda=3.66$ and $\lambda=3.8$.]{} ***4.2. Why Chaos? A Preliminary Inquiry*** The question as to why a natural system should evolve chaotically is both interesting and relevant, and this section attempts to advance a plausible answer to this inquiry that is based on the connection between topology and convergence contained in the Corollary to Theorem A1.5. Open sets are groupings of elements that govern convergence of nets and filters, because the required property of being either eventually of frequently in (open) neighbourhoods of a point determines the eventual behaviour of the net; recall in this connection the unusual convergence characteristics in cofinite and cocountable spaces. Conversely for a given convergence characteristic of a class of nets, it is possible to infer the topology of the space that is responsible for this convergence, and it is this point of view that we adopt here to investigate the question of this subsection: recall that our Definitions 4.1 and 4.2 were based on purely algebraic set-theoretic arguments on ordered sets, just as the role of the choice of an appropriate problem-dependent basis was highlighted at the end of Sec. 2. [Figure]{} [\[Fig: attractor\], contd. Bifurcation diagram and attractors for $f_{\textrm{f}}(x)$.]{} Chaos as manifest in its attractors is a direct consequence of the increasing nonlinearity of the map with increasing iteration; we reemphasize that this is only a necessary condition so that the increasing nonlinearities of Figs. \[Fig: log357\] and \[Fig: omega\] eventually lead to stable states and not to chaotic instability. Under the right conditions as enunciated following Fig. \[Fig: Zorn\], chaos appears to be the natural outcome of the difference in the behaviour of a function $f$ and its inverse $f^{-}$ under their successive applications. Thus $f=ff^{-}f$ allows $f$ to take advantage of its multi inverse to generate all possible equivalence classes that is available to it, a feature not accessible to $f^{-}=f^{-}ff^{-}$. As we have seen in the foregoing, equivalence classes of fixed points, stable and unstable, are of defining significance in determining the ultimate behaviour of an evolving dynamical system and as the eventual (as also frequent) character of a filter or net in a set is dictated by open neighbourhoods of points of the set, *it is postulated that chaoticity on a set $X$ leads to a reformulation of the open sets of $X$ to equivalence classes generated by the evolving map $f$,* see Example 2.4(3). Such a redefinition of open sets of equivalence classes allow the evolving system to temporally access an ever increasing number of states even though the equivalent fixed points are not fixed under iterations of $f$ except for the parent of the class, and can be considered to be the governing criterion for the cooperative or collective behaviour of the system. The predominance of the role of $f^{-}$ in $f=ff^{-}f$ in generating the equivalence classes (that is exploiting the many-to-one character) of $f$ is reflected as limit multis for $f$ (that is constant $f^{-}$ on $\mathcal{R}_{+}$) in $f^{-}=f^{-}ff^{-}$; this interpretation of the dynamics of chaos is meaningful as graphical convergence leading to chaos is a result of pointwise biconvergence of the sequence of iterates of the functions generated by $f$. But as $f$ is a noninjective function *on* $X$ *possessing the property of increasing nonlinearity in the form of increasing noninjectivity with iteration,* various cycles of disjoint equivalence classes are generated under iteration, see for example Fig. \[Fig: tent4\](a) for the tent map. A reference to Fig. \[Fig: GenInv\] shows that the basic set $X_{\textrm{B}}$, for a finite number $n$ of iterations of $f$, contains the parent of each of these open equivalent sets in the domain of $f$, with the topology on $X_{\textrm{B}}$ being the corresponding $p$-images of these disjoint saturated open sets of the domain. In the limit of infinite iterations of $f$ leading to the multifunction $\mathcal{M}$ (this is the $f^{\infty}$ of Sec. 4.1), the generated open sets constitute a basis for a topology on $\mathcal{D}(f)$ and the basis for the topology of $\mathcal{R}(f)$ are the corresponding $\mathcal{M}$-images of these equivalent classes. *It is our contention that the motive force behind evolution toward a chaos, as defined by Def. 4.1, is the drive toward a state of the dynamical system that supports ininality of the limit multi* $\mathcal{M}$*;* see Appendix A2 with the discussions on Fig. \[Fig: GenInv\] and Eq. (\[Eqn: ininal\]) in Sec. 2. In the limit of infinite iterations therefore, the open sets of the range $\mathcal{R}(f)\subseteq X$ are the multi images that graphical convergence generates at each of these inverse-stable fixed points. $X$ therefore has two topologies imposed on it by the dynamics of $f$: the first of equivalence classes generated by the limit multi $\mathcal{M}$ in the domain of $f$ and the second as $\mathcal{M}$-images of these classes in the range of $f$. Quite clearly these two topologies need not be the same; their intersection therefore can be defined to be the *chaotic topology* *on* $X$ *associated with the chaotic map* $f$ on $X$. Neighbourhoods of points in this topology cannot be arbitrarily small as they consist of all members of the equivalence class to which any element belongs; hence a sequence converging to any of these elements necessarily converges to all of them, and the eventual objective of chaotic dynamics is to generate a topology in $X$ with respect to which elements of the set can be grouped together in as large equivalence classes as possible in the sense that if a net converges simultaneously to points $x\neq y\in X$ then $x\sim y$: $x$ is of course equivalent to itself while $x,y,z$ are equivalent to each other iff they are simultaneously in every open set in which the net may eventually belong. This hall-mark of chaos can be appreciated in terms of a necessary obliteration of any separation property that the space might have originally possessed, see property (H3) in Appendix A3. We reemphasize that a set in this chaotic context is required to act in a dual capacity depending on whether it carries the initial or final topology under $\mathcal{M}$. This preliminary inquiry into the nature of chaos is concluded in the final section of this work. **5. Graphical convergence works** We present in this section some real evidence in support of our hypothesis of graphical convergence of functions in $\textrm{Multi}(X,Y)$. The example is taken from neutron transport theory, and concerns the discretized spectral approximation [@Sengupta1988; @Sengupta1995] of Case’s singular eigenfunction solution of the monoenergetic neutron transport equation, [@Case1967]. The neutron transport equation is a linear form of the Boltzmann equation that is obtained as follows. Consider the neutron-moderator system as a mixture of two species of gases each of which satisfies a Boltzmann equation of the type$$\begin{gathered} \left(\frac{\partial}{\partial t}+v_{i}.\nabla\right)f_{i}(r,v,t)=\\ =\int dv^{\prime}\int dv_{1}\int dv_{1}^{\prime}\sum_{j}W_{ij}(v_{i}\rightarrow v^{\prime};v_{1}\rightarrow v_{1}^{\prime})\{ f_{i}(r,v^{\prime},t)f_{j}(r,v_{1}^{\prime},t)--f_{i}(r,v,t)f_{j}(r,v_{1},t)\})\end{gathered}$$ where$$W_{ij}(v_{i}\rightarrow v^{\prime};v_{1}\rightarrow v_{1}^{\prime})=\mid v-v_{1}\mid\sigma_{ij}(v-v^{\prime},v_{1}-v_{1}^{\prime})$$ $\sigma_{ij}$ being the cross-section of interaction between species $i$ and $j$. Denote neutrons by subscript 1 and the background moderator with which the neutrons interact by 2, and make the assumptions that \(i) The neutron density $f_{1}$ is much less compared with that of the moderator $f_{2}$ so that the terms $f_{1}f_{1}$ and $f_{1}f_{2}$ may be neglected in the neutron and moderator equations respectively. \(ii) The moderator distribution $f_{2}$ is not affected by the neutrons. This decouples the neutron and moderator equations and leads to an equilibrium Maxwellian $f_{\textrm{M}}$ for the moderator while the neutrons are described by the linear equation $$\begin{gathered} \left(\frac{\partial}{\partial t}+v.\nabla\right)f(r,v,t)=\\ =\int dv^{\prime}\int dv_{1}\int dv_{1}^{\prime}W_{12}(v\rightarrow v^{\prime};v_{1}\rightarrow v_{1}^{\prime})\{ f(r,v^{\prime},t)f_{\textrm{M}}(v_{1}^{\prime})--f(r,v,t)f_{\textrm{M}}(v_{1})\})\end{gathered}$$ This is now put in the standard form of the neutron transport equation [@Williams1967]$$\begin{gathered} \left(\frac{1}{v}\frac{\partial}{\partial t}+\Omega.v+\mathcal{S}(E)\right)\Phi(r,E,\widehat{\Omega},t)=\int d\Omega^{\prime}\int dE^{\prime}\mathcal{S}(r,E^{\prime}\rightarrow E;\widehat{\Omega}^{\prime}\cdot\widehat{\Omega})\textrm{ }\Phi(r,E^{\prime},\widehat{\Omega}^{\prime},t).\end{gathered}$$ where $E=mv^{2}/2$ is the energy and $\widehat{\Omega}$ the direction of motion of the neutrons. The steady state, monoenergetic form of this equation is Eq. (\[Eqn: NeutronTransport\]) $$\mu\frac{\partial\Phi(x,\mu)}{\partial x}+\Phi(x,\mu)=\frac{c}{2}\int_{-1}^{1}\Phi(x,\mu^{\prime})d\mu^{\prime},\qquad0<c<1,\,-1\leq\mu\leq1$$ and its singular eigenfunction solution for $x\in(-\infty,\infty)$ is given by Eq. (\[Eqn: CaseSolution\_FR\]) $$\begin{gathered} \Phi(x,\mu)=a(\nu_{0})e^{-x/\nu_{0}}\phi(\mu,\nu_{0})+a(-\nu_{0})e^{x/\nu_{0}}\phi(-\nu_{0},\mu)+\int_{-1}^{1}a(\nu)e^{-x/\nu}\phi(\mu,\nu)d\nu;\end{gathered}$$ see Appendix A4 for an introductory review of Case’s solution of the one-speed neutron transport equation. --------------- --------------- ------------------------ ------------------------------ ---------------------------------- $\mathcal{R}=X$ $\textrm{Cl}(\mathcal{R})=X$ $\textrm{Cl}(\mathcal{R})\neq X$ Not injective $\cdots$ $P\sigma(\mathscr{L})$ $P\sigma(\mathscr{L})$ $P\sigma(\mathscr{L})$ Not contiuous $C\sigma(\mathscr{L})$ $C\sigma(\mathscr{L})$ $R\sigma(\mathscr{L})$ Continuous $\rho(\mathscr{L})$ $\rho(\mathscr{L})$ $R\sigma(\mathscr{L})$ --------------- --------------- ------------------------ ------------------------------ ---------------------------------- : [\[Table: spectrum\]Spectrum of linear operator $\mathscr{L}\in\textrm{Map}(X)$. Here $\mathscr L_{\lambda}:=\mathscr{L}-\lambda$ satisfies the equation $\mathscr L_{\lambda}(x)=0$, with the resolvent set $\rho(\mathscr{L})$ of $\mathscr{L}$ consisting of all those complex numbers $\lambda$ for which $\mathscr L_{\lambda}^{-1}$]{} [exists as a continuous operator with dense domain. Any value of $\lambda$ for which this is not true is in the spectrum $\sigma(\mathscr{L})$ of $\mathscr{L}$, that is further subdivided into three disjoint components of the point, continuous, and residual spectra according to the criteria shown in the table. ]{} The term “eigenfunction” is motivated by the following considerations. Consider the eigenvalue equation $$(\mu-\nu)\mathscr F_{\nu}(\mu)=0,\qquad\mu\in V(\mu),\textrm{ }\nu\in\mathbb{R}\label{Eqn: eigen}$$ in the space of multifunctions $\textrm{Multi}(V(\mu),(-\infty,\infty))$, where $\mu$ is in either of the intervals $[-1,1]$ or $[0,1]$ depending on whether the given boundary conditions for Eq. (\[Eqn: NeutronTransport\]) is full-range or half range. If we are looking only for functional solutions of Eq. (\[Eqn: eigen\]), then the unique function $\mathcal{F}$ that satisfies this equation for all possible $\mu\in V(\mu)$ and $\nu\in\mathbb{R}-V(\mu)$ is $\mathcal{F}_{\nu}(\mu)=0$ which means, according to Table \[Table: spectrum\], that the point spectrum of $\mu$ is empty and $(\mu-\nu)^{-1}$ exists for all $\nu$. When $\nu\in V(\mu)$, however, this inverse is not continuous and we show below that in $\textrm{Map}(V(\mu),0)$, $\nu\in V(\mu)$ belongs to the continuous spectrum of $\mu$. This distinction between the nature of the inverses depending on the relative values of $\mu$ and $\nu$ suggests a wider “non-function” space in which to look for the solutions of operator equations, and in keeping with the philosophy embodied in Fig. \[Fig: GenInv\] of treating inverse problems in the space of multifunctions, we consider all $\mathscr F_{\nu}\in\textrm{Multi}(V(\mu),\mathbb{R}))$ satisfying Eq. (\[Eqn: eigen\]) to be eigenfunctions of $\mu$ for the corresponding eigenvalue $\nu$, leading to the following multifunctional solution of (\[Eqn: eigen\])$$\begin{aligned} \mathscr F_{\nu}(\mu) & = & \left\{ \begin{array}{ccl} (V(\mu),0), & & \textrm{if }\nu\notin V(\mu)\\ (V(\mu)-\nu,0)\bigcup(\nu,\mathbb{R})), & & \textrm{if }\nu\in V(\mu),\end{array}\right.\end{aligned}$$ where $V(\mu)-\nu$ is used as a shorthand for the interval $V(\mu)$ with $\nu$ deleted. Rewriting the eigenvalue equation (\[Eqn: eigen\]) as $\mu_{\nu}(\mathscr F_{\nu}(\mu))=0$ and comparing this with Fig. \[Fig: GenInv\], allows us to draw the correspondences $$\begin{aligned} f & \Longleftrightarrow & \mu_{\nu}\nonumber \\ X\textrm{ and }Y & \Longleftrightarrow & \{\mathscr F_{\nu}\in\textrm{Multi}(V(\mu),\mathbb{R})\!:\mathscr F_{\nu}\in\mathcal{D}(\mu_{\nu})\}\nonumber \\ f(X) & \Longleftrightarrow & \{0\!:0\in Y\}\label{Eqn: GenInv_Spectrum}\\ X_{\textrm{B}} & \Longleftrightarrow & \{0\!:0\in X\}\nonumber \\ f^{-} & \Longleftrightarrow & \mu_{\nu}^{-}.\nonumber \end{aligned}$$ Thus a multifunction in $X$ is equivalent to $0$ in $X_{\textrm{B}}$ under the linear map $\mu_{\nu}$, and we show below that this multifunction is infact the Dirac delta “function” $\delta_{\nu}(\mu)$, usually written as $\delta(\mu-\nu)$. This suggests that in $\textrm{Multi}(V(\mu),\mathbb{R})$*, every $\nu\in V(\mu)$ is in the point spectrum of $\mu$*, so that *discontinuous functions that are pointwise limits of functions in function space can be replaced by graphically converged multifunctions in the space of multifunctions*. Completing the equivalence class of $0$ in Fig. \[Fig: GenInv\], gives the multifunctional solution of Eq. (\[Eqn: eigen\]). From a comparison of the definition of ill-posedness (Sec. 2) and the spectrum (Table \[Table: spectrum\]), it is clear that $\mathscr L_{\lambda}(x)=y$ is ill-posed iff \(1) $\mathscr L_{\lambda}$ not injective $\Leftrightarrow$ $\lambda\in P\sigma(\mathscr L_{\lambda})$, which corresponds to the first row of Table \[Table: spectrum\]. \(2) $\mathscr L_{\lambda}$ not surjective $\Leftrightarrow$ the values of $\lambda$ correspond to the second and third columns of Table \[Table: spectrum\]. \(3) $\mathscr L_{\lambda}$ is bijective but not open $\Leftrightarrow$ $\lambda\textrm{ is either in }C\sigma(\mathscr L_{\lambda})\textrm{ or }R\sigma(\mathscr L_{\lambda})$ corresponding to the second row of Table \[Table: spectrum\]. We verify in the three steps below that $X=L_{1}[-1,1]$ of integrable functions, $\nu\in V(\mu)=[-1,1]$ belongs to the continuous spectrum of $\mu$. \(a) *$\mathcal{R}(\mu_{\nu})$ is dense, but not equal to $L_{1}$*. The set of functions $g(\mu)\in L_{1}$ such that $\mu_{\nu}^{-1}g\in L_{1}$ cannot be the whole of $L_{1}$. Thus, for example, the piecewise constant function $g=\textrm{const}\neq0$ on $\mid\mu-\nu\mid\leq\delta>0$ and $0$ otherwise is in $L_{1}$ but not in *$\mathcal{R}(\mu_{\nu})$* as $\mu_{\nu}^{-1}g\not\in L_{1}$. Nevertheless for any $g\in L_{1}$, we may choose the sequence of functions $$g_{n}(\mu)=\left\{ \begin{array}{ccl} 0, & & \textrm{if }\mid\mu-\nu\mid\leq1/n\\ g(\mu), & & \textrm{otherwise}\end{array}\right.$$ in $\mathcal{R}(\mu_{\nu})$ to be eventually in every neighbourhood of $g$ in the sense that $\lim_{n\rightarrow\infty}\int_{-1}^{1}\mid g-g_{n}\mid=0$. \(b) *The inverse $(\mu-\nu)^{-1}$ exists but is not continuous.* The inverse exists because, as noted earlier, $0$ is the only functional solution of Eq. (\[Eqn: eigen\]). Nevertheless although the net of functions $$\delta_{\nu\varepsilon}(\mu)=\frac{1}{\tan^{-1}(1+\nu)/\varepsilon+\tan^{-1}(1-\nu)/\varepsilon}\left(\frac{\varepsilon}{(\mu-\nu)^{2}+\varepsilon^{2}}\right),\qquad\varepsilon>0$$ is in the domain of $\mu_{\nu}$ because $\int_{-1}^{1}\delta_{\nu\varepsilon}(\mu)d\mu=1$ for all $\varepsilon>0$, $$\lim_{\varepsilon\rightarrow0}\int_{-1}^{1}\mid\mu-\nu\mid\delta_{\nu\varepsilon}(\mu)d\mu=0$$ implying that $(\mu-\nu)^{-1}$ is unbounded. Taken together, (a) and (b) show that functional solutions of Eq. (\[Eqn: eigen\]) lead to state 2-2 in Table \[Table: spectrum\]; hence $\nu\in[-1,1]=C\sigma(\mu)$. \(c) The two integral constraints in (b) also mean that $\nu\in C\sigma(\mu)$ is a *generalized eigenvalue* of $\mu$ which justifies calling the graphical limit $\delta_{\nu\varepsilon}(\mu)\overset{\mathbf{G}}\rightarrow\delta_{\nu}(\mu)$ a *generalized,* or singular, *eigenfunction*, see Fig. \[Fig: Poison\] which clearly indicates the convergence of the net of functions[^26]. From the fact that the solution Eq. (\[Eqn: CaseSolution\_FR\]) of the transport equation contains an integral involving the multifunction $\phi(\mu,\nu)$, we may draw an interesting physical interpretation. As the multi appears *every where* on $V(\mu)$ (that is there are no chaotic orbits but only the multifunctions that produce them), we have here a situation typical of *maximal ill-posedness* characteristic of chaos: note that both the functions comprising $\phi_{\varepsilon}(\mu,\nu)$ are non-injective. As the solution (\[Eqn: CaseSolution\_FR\]) involves an integral over all $\nu\in V(\mu)$, the singular eigenfunctions — that collectively may be regarded as representing a *chaotic substate of* the system represented by the solution of the neutron transport equation — combine with the functional components $\phi(\pm\nu_{0},\mu)$ to produce the well-defined, non-chaotic, experimental end result of the neutron flux $\Phi(x,\mu)$. The solution (\[Eqn: CaseSolution\_FR\]) is obtained by assuming $\Phi(x,\mu)=e^{-x/\nu}\phi(\mu,\nu)$ to get the equation for $\phi(\mu,\nu)$ to be $(\mu-\nu)\phi(\mu,\nu)=-c\nu/2$ with the normalization $\int_{-1}^{1}\phi(\mu,\nu)=1$. As $\mu_{\nu}^{-1}$ is not invertible in $\textrm{Multi}(V(\mu),\mathbb{R})$ and $\mu_{\nu\textrm{B}}\!:X_{\textrm{B}}\rightarrow f(X)$ does not exist, the alternate approach of regularization was adopted in [@Sengupta1988; @Sengupta1995] to rewrite $\mu_{\nu}\phi(\mu,\nu)=-c\nu/2$ as $\mu_{\nu\varepsilon}\phi_{\varepsilon}(\mu,\nu)=-c\nu/2$ with $\mu_{\nu\varepsilon}:=\mu-(\nu+i\varepsilon)$ being a net of bijective functions for $\varepsilon>0$; this is a consequence of the fact that for the multiplication operator every nonreal $\lambda$ belongs to the resolvent set of the operator. The family of solutions of the later equation is given by [@Sengupta1988; @Sengupta1995] $$\phi_{\varepsilon}(\nu,\mu)=\frac{c\nu}{2}\frac{\nu-\mu}{(\mu-\nu)^{2}+\varepsilon^{2}}+\frac{\lambda_{\varepsilon}(\nu)}{\pi_{\varepsilon}}\frac{\varepsilon}{(\mu-\nu)^{2}+\varepsilon^{2}}\label{Eqn: phieps}$$ where the required normalization $\int_{-1}^{1}\phi_{\varepsilon}(\nu,\mu)=1$ gives $$\begin{array}{ccl} {\displaystyle \lambda_{\varepsilon}(\nu)} & = & {\displaystyle \frac{\pi_{\varepsilon}}{\tan^{-1}(1+\nu)/\varepsilon+\tan^{-1}(1-\nu)/\varepsilon}\left(1-\frac{c\nu}{4}\ln\frac{(1+\nu)^{2}+\varepsilon^{2}}{(1-\nu)^{2}+\varepsilon^{2}}\right)}\\ & \overset{\varepsilon\rightarrow0}\longrightarrow & \pi\lambda(\nu)\end{array}$$ with $$\pi_{\varepsilon}=\varepsilon\int_{-1}^{1}\frac{d\mu}{\mu^{2}+\varepsilon^{2}}=2\tan^{-1}\left(\frac{1}{\varepsilon}\right)\overset{\varepsilon\rightarrow0}\longrightarrow\pi.$$ These discretized equations should be compared with the corresponding exact ones of Appendix A4. We shall see that the net of functions (\[Eqn: phieps\]) converges graphically to the multifunction Eq. (\[Eqn: singular\_eigen\]) as $\varepsilon\rightarrow0$. In the discretized spectral approximation., the singular eigenfunction $\phi(\mu,\nu)$ is replaced by $\phi_{\varepsilon}(\mu,\nu)$, $\varepsilon\rightarrow0$, with the integral in $\nu$ being replaced by an appropriate sum. The solution Eq. (\[Eqn: CaseSolution\_HR\]) of the physically interesting half-space $x\geq0$ problem then reduces to [@Sengupta1988; @Sengupta1995] $$\Phi_{\varepsilon}(x,\mu)=a(\nu_{0})e^{-x/\nu_{0}}\phi(\mu,\nu_{0})+\sum_{i=1}^{N}a(\nu_{i})e^{-x/\nu_{i}}\phi_{\varepsilon}(\mu,\nu_{i})\qquad\mu\in[0,1]\label{Eqn: DiscSpect_HR}$$ where the nodes $\{\nu_{i}\}_{i=1}^{N}$ are chosen suitably. This discretized spectral approximation to Case’s solution has given surprisingly accurate numerical results for a set of properly chosen nodes when compared with exact calculations. Because of its involved nature [@Case1967], the exact calculations are basically numerical which leads to nonlinear integral equations as part of the solution procedure. To appreciate the enormous complexity of the exact treatment of the half-space problem, we recall that the complete set of eigenfunctions $\{\phi(\mu,\nu_{0}),\{\phi(\mu,\nu)\}_{\nu\in[0,1]}\}$ are orthogonal with respect to the half-range weight function $W(\mu)$ of half-range theory, Eq. (\[Eqn: W(mu)\]), that is expressed only in terms of solution of the nonlinear integral equation Eq. (\[Eqn: Omega(-mu)\]). The solution of a half-space problem then evaluates the coefficients $\{ a(\nu_{0}),a(\nu)_{\nu\in[0,1]}\}$ from the appropriate half range (that is $0\leq\mu\leq1$) orthogonality integrals satisfied by the eigenfunctions $\{\phi(\mu,\nu_{0}),\{\phi(\mu,\nu)\}_{\nu\in[0,1]}\}$ with respect to the weight $W(\mu)$, see Appendix A4 for the necessary details of the half-space problem in neutron transport theory. As may be appreciated from this brief introduction, solutions to half-space problems are not simple and actual numerical computations must rely a great deal on tabulated values of the $X$-function. Self-consistent calculations of sample benchmark problems performed by the discretized spectral approximation in a full-range adaption of the half-range problem described below that generate all necessary data, independent of numerical tables, with the quadrature nodes $\{\nu_{i}\}_{i=1}^{N}$ taken at the zeros Legendre polynomials show that the full range formulation of this approximation [@Sengupta1988; @Sengupta1995] can give very accurate results not only of integrated quantities like the flux $\Phi$ and leakage of particles out of the half space, but of also basic ‘"raw‘" data like the extrapolated end point $$z_{0}=\frac{c\nu_{0}}{4}\int_{0}^{1}\frac{\nu}{N(\nu)}\left(1+\frac{c\nu^{2}}{1-\nu^{2}}\right)\ln\left(\frac{\nu_{0}+\nu}{\nu_{0}-\nu}\right)d\nu\label{Eqn: extrapolated}$$ and of the $X$-function itself. Given the involved nature of the exact theory, it is our contention that the remarkable accuracy of these basic data, some of which is reproduced in Table \[Table: extrapolated\], is due to the graphical convergence of the net of functions $$\phi_{\varepsilon}(\mu,\nu)\overset{\mathbf{G}}\longrightarrow\phi(\mu,\nu)$$ shown in Fig. \[Fig: Case\]; here $\varepsilon=1/\pi N$ so that $\varepsilon\rightarrow0$ as $N\rightarrow\infty$. By this convergence, the delta function and principal values in $[-1,1]$ are the multifunctions $([-1,0),0)\bigcup(0,[0,\infty)\bigcup((0,1],0)$ and $\{1/x\}_{x\in[-1,0)}\bigcup(0,(-\infty,\infty))\bigcup\{1/x\}_{x\in(0,1]}$ respectively. Tables \[Table: extrapolated\] and \[Table: X-function\], taken from @Sengupta1988 and @Sengupta1995, show respectively the extrapolated end point and $X$-function by the full-range adaption of the discretized spectral approximation for two different half range problems denoted as Problems A and B defined as $$\begin{aligned} Problem\textrm{ }A\quad & \textrm{Equation}\!:\textrm{ }{\textstyle {\mu\Phi_{x}+\Phi=(c/2)\int_{-1}^{1}\Phi(x,\mu^{\prime})d\mu^{\prime},\; x\geq0}}\\ & \textrm{Boundary condition}:\textrm{ }\Phi(0,\mu)=0,\;\mu\geq0\\ & \textrm{Asymptotic condition}:\textrm{ }\Phi\rightarrow e^{-x/\nu_{0}}\phi(\mu,\nu_{0}),\; x\rightarrow\infty.\\ Problem\textrm{ }B\quad & \textrm{Equation}\!:\textrm{ }{\textstyle {\mu\Phi_{x}+\Phi=(c/2)\int_{-1}^{1}\Phi(x,\mu^{\prime})d\mu^{\prime},\; x\geq0}}\\ & \textrm{Boundary condition}:\textrm{ }\Phi(0,\mu)=1,\;\mu\geq0\\ & \textrm{Asymptotic condition}:\textrm{ }\Phi\rightarrow0,\; x\rightarrow\infty.\end{aligned}$$ The full $-1\leq\mu\leq1$ range form of the half $0\leq\mu\leq1$ range discretized spectral approximation replaces the exact integral boundary condition at $x=0$ by a suitable quadrature sum over the values of $\nu$ taken at the zeros of Legendre polynomials; thus the condition at $x=0$ can be expressed as $$\psi(\mu)=a(\nu_{0})\phi(\mu,\nu_{0})+\sum_{i=1}^{N}a(\nu_{i})\phi_{\varepsilon}(\mu,\nu_{i}),\qquad\mu\in[0,1],\label{Eqn: BC}$$ where $\psi(\mu)=\Phi(0,\mu)$ is the specified incoming radiation incident on the boundary from the left, and the half-range coefficients $a(\nu_{0})$, $\{ a(\nu)\}_{\nu\in[0,1]}$ are to be evaluated using the $W$-function of Appendix 4. We now exploit the relative simplicity of the full-range calculations by replacing Eq. (\[Eqn: BC\]) by Eq. (\[Eqn: HRFR\_Discrete\]) following, where the coefficients $\{ b(\nu_{i})\}_{i=0}^{N}$ are used to distinguish the full-range coefficients from the half-range ones. The significance of this change lies in the overwhelming simplicity of the full-range weight function $\mu$ as compared to the half-range function $W(\mu)$, and the resulting simplicity of the orthogonality relations that follow, see Appendix A4. The basic data of $z_{0}$ and $X(-\nu)$ are then completely generated self-consistently [@Sengupta1988; @Sengupta1995] by the discretized spectral approximation from the full-range adaption $$\sum_{i=0}^{N}b_{i}\phi_{\varepsilon}(\mu,\nu_{i})=\psi_{+}(\mu)+\psi_{-}(\mu),\qquad\mu\in[-1,1],\textrm{ }\nu_{i}\geq0\label{Eqn: HRFR_Discrete}$$ of the discretized boundary condition Eq. (\[Eqn: BC\]), where $\psi_{+}(\mu)$ is by definition the incident flux $\psi(\mu)$ for $\mu\in[0,1]$ and $0$ if $\mu\in[-1,0]$, while $$\psi_{-}(\mu)=\left\{ \begin{array}{ccl} {\displaystyle \sum_{i=0}^{N}b_{i}^{-}\phi_{\varepsilon}(\mu,\nu_{i})} & & \textrm{if }\mu\in[-1,0],\textrm{ }\nu_{i}\geq0\textrm{ }\\ 0 & & \textrm{if }\mu\in[0,1]\end{array}\right.$$ is the the emergent angular distribution out of the medium. Equation (\[Eqn: HRFR\_Discrete\]) corresponds to the full-range $\mu\in[-1,1],\textrm{ }\nu_{i}\geq0$ form $$b(\nu_{0})\phi(\mu,\nu_{0})+\int_{0}^{1}b(\nu)\phi(\mu,\nu)d\nu=\psi_{+}(\mu)+\left(b^{-}(\nu_{0})\phi(\mu,\nu_{0})+\int_{0}^{1}b^{-}(\nu)\phi(\mu,\nu)d\nu\right)\label{Eqn: HRFR}$$ of boundary condition (\[Eqn: BC\_HR\]) with the first and second terms on the right having the same interpretation as for Eq. (\[Eqn: HRFR\_Discrete\]). This full-range simulation merely states that the solution (\[Eqn: CaseSolution\_HR\]) of Eq. (\[Eqn: NeutronTransport\]) holds for all $\mu\in[-1,1]$, $x\geq0$, although it was obtained, unlike in the regular full-range case, from the given radiation $\psi(\mu)$ incident on the boundary at $x=0$ over only half the interval $\mu\in[0,1]$. To obtain the simulated full-range coefficients $\{ b_{i}\}$ and $\{ b_{i}^{-}\}$ of the half-range problem, we observe that there are effectively only half the number of coefficients as compared to a normal full-range problem because $\nu$ is now only over half the full interval. This allows us to generate two sets of equations from (\[Eqn: HRFR\]) by integrating with respect to $\mu\in[-1,1]$ with $\nu$ in the half intervals $[-1,0]$ and $[0,1]$ to obtain the two sets of coefficients $b^{-}$ and $b$ respectively. Accordingly we get from Eq. (\[Eqn: HRFR\_Discrete\]) with $\textrm{ }j=0,1,\cdots,N$ the sets of equations $$\begin{array}{c} {\displaystyle {\displaystyle (\psi,\phi_{j-})_{\mu}^{(+)}=-\sum_{i=0}^{N}b_{i}^{-}(\phi_{i+},\phi_{j-})_{\mu}^{(-)}}}\\ b_{j}={\displaystyle \left((\psi,\phi_{j+})_{\mu}^{(+)}+\sum_{i=0}^{N}b_{i}^{-}(\phi_{i+},\phi_{j+})_{\mu}^{(-)}\right)}\end{array}\label{Eqn: FRBC1}$$ where $(\phi_{j\pm})_{j=1}^{N}$ represents $(\phi_{\varepsilon}(\mu,\pm\nu_{j}))_{j=1}^{N}$, $\phi_{0\pm}=\phi(\mu,\pm\nu_{0})$, the $(+)$ $(-)$ superscripts are used to denote the integrations with respect to $\mu\in[0,1]$ and $\mu\in[-1,0]$ respectively, and $(f,g)_{\mu}$ denotes the usual inner product in $[-1,1]$ with respect to the full range weight $\mu$. While the first set of $N+1$ equations give $b_{i}^{-}$, the second set produces the required $b_{j}$ from these ‘"negative‘" coefficients. By equating these calculated $b_{i}$ with the exact half-range expressions for $a(\nu)$ with respect to $W(\mu)$ as outlined in Appendix A4, it is possible to find numerical values of $z_{0}$ and $X(-\nu)$. Thus from the second of Eq. (\[Eqn: Constant\_Coeff\]), $\{ X(-\nu_{i})\}_{i=1}^{N}$ is obtained with $b_{i\textrm{B}}\textrm{ }=a_{i\textrm{B}}$, $i=1,\cdots,N$, which is then substituted in the second of Eq. (\[Eqn: Milne\_Coeff\]) with $X(-\nu_{0})$ obtained from $a_{\textrm{A}}(\nu_{0})$ according to Appendix A4, to compare the respective $a_{i\textrm{A}}$ with the calculated $b_{i\textrm{A}}$ from (\[Eqn: FRBC1\]). Finally the full-range coefficients of Problem A can be used to obtain the $X(-\nu)$ values from the second of Eqs. (\[Eqn: Milne\_Coeff\]) and compared with the exact tabulated values as in Table \[Table: X-function\]. The tabulated values of $cz_{0}$ from Eq. (\[Eqn: extrapolated\]) show a consistent deviation from our calculations of Problem A according to $a_{\textrm{A}}(\nu_{0})=-\exp(-2z_{0}/\nu_{0})$. Since the $X(-\nu)$ values of Problem A in Table \[Table: X-function\] also need the same $b_{0\textrm{A}}$ as input that was used in obtaining $z_{0}$, it is reasonable to conclude that the ‘"exact‘" numerical integration of $z_{0}$ is inaccurate to the extent displayed in Table \[Table: extrapolated\]. ----- --------- --------- --------- -------- $N=2$ $N=6$ $N=10$ Exact 0.2 0.78478 0.78478 0.78478 0.7851 0.4 0.72996 0.72996 0.72996 0.7305 0.6 0.71535 0.71536 0.71536 0.7155 0.8 0.71124 0.71124 0.71124 0.7113 0.9 0.71060 0.71060 0.71061 0.7106 ----- --------- --------- --------- -------- : \[Table: extrapolated\][Extrapolated end-point $z_{0}$.]{} -- -- ----------- ----------- ----------- ---------- $\nu_{i}$ Problem A Problem B Exact 0.2133 0.8873091 0.8873091 0.887308 0.7887 0.5826001 0.5826001 0.582500 0.0338 1.3370163 1.3370163 1.337015 0.1694 1.0999831 1.0999831 1.099983 0.3807 0.8792321 0.8792321 0.879232 0.6193 0.7215240 0.7215240 0.721524 0.8306 0.6239109 0.6239109 0.623911 0.9662 0.5743556 0.5743556 0.574355 0.0130 1.5971784 1.5971784 1.597163 0.0674 1.4245314 1.4245314 1.424532 0.1603 1.2289940 1.2289940 1.228995 0.2833 1.0513750 1.0513750 1.051376 0.4255 0.9058140 0.9058410 0.905842 0.5744 0.7934295 0.7934295 0.793430 0.7167 0.7102823 0.7102823 0.710283 0.8397 0.6516836 0.6516836 0.651683 0.9325 0.6136514 0.6136514 0.613653 0.9870 0.5933988 0.5933988 0.593399 -- -- ----------- ----------- ----------- ---------- : [\[Table: X-function\]$X(-\nu)$ by the full range method.]{} From these numerical experiments and Fig. \[Fig: Case\] we may conclude that the continuous spectrum $[-1,1]$ of the position operator $\mu$ acts as the $\mathcal{D}_{+}$ points in generating the multifunctional Case singular eigenfunction $\phi(\mu,\nu)$. Its rational approximation $\phi_{\varepsilon}(\mu,\nu)$ in the context of the simple simulated full-range computations of the complex half-range exact theory of Appendix A4, clearly demonstrates the utility of graphical convergence of sequence of functions to multifunction. The totality of the multifunctions $\phi(\mu,\nu)$ for all $\nu$ in Fig. \[Fig: Case\](c) and (d) endows the problem with the character of maximal ill-posedness that is characteristic of chaos. This chaotic signature of the transport equation is however latent as the experimental output $\Phi(x,\mu)$ is well-behaved and regular. This important example shows how nature can use hidden and complex chaotic substates to generate order through a process of superposition. **6. Does Nature support complexity?** The question of this section is basic in the light of the theory of chaos presented above as it may be reformulated to the inquiry of what makes nature support chaoticity in the form of increasing non-injectivity of an input-output system. It is the purpose of this Section to exploit the connection between spectral theory and the dynamics of chaos that has been presented in the previous section. Since linear operators on finite dimensional spaces do not possess continuous or residual spectra, spectral theory on infinite dimensional spaces essentially involves limiting behaviour to infinite dimensions of the familiar matrix eigenvalue-eigenvector problem. As always this means extensions, dense embeddings and completions of the finite dimensional problem that show up as generalized eigenvalues and eigenvectors. In its usual form, the goal of nonlinear spectral theory consists [@Appel2000] in the study of $T_{\lambda}^{-1}$ for nonlinear operators $T_{\lambda}$ that satisfy more general continuity conditions, like differentiability and Lipschitz continuity, than simple boundedness that is enough for linear operators. The following generalization of the concept of the spectrum of a linear operator to the nonlinear case is suggestive. For a nonlinear map, $\lambda$ need not appear only in a multiplying role, so that an eigenvalue equation can be written more generally as a fixed-point equation $$f(\lambda;x)=x$$ with a fixed point corresponding to the eigenfunction of a linear operator and an “eigenvalue” being the value of $\lambda$ for which this fixed point appears. The correspondence of the residual and continuous parts of the spectrum are, however, less trivial than for the point spectrum. This is seen from the following two examples, [@Roman1975]. Let $Ae_{k}=\lambda_{k}e_{k},\textrm{ }k=1,2,\cdots$ be an eigenvalue equation with $e_{j}$ being the $j^{\textrm{th}}$ unit vector. Then $(A-\lambda)e_{k}:=(\lambda_{k}-\lambda)e_{k}=0$ iff $\lambda=\lambda_{k}$ so that $\{\lambda_{k}\}_{k=1}^{\infty}\in P\sigma(A)$ are the only eigenvalues of $A$. Consider now $(\lambda_{k})_{k=1}^{\infty}$ to be a sequence of real numbers that tends to a finite $\lambda^{*}$; for example let $A$ be a diagonal matrix having $1/k$ as its diagonal entries. Then $\lambda^{*}$ belongs to the continuous spectrum of $A$ because $(A-\lambda^{*})e_{k}=(\lambda_{k}-\lambda^{*})e_{k}$ with $\lambda_{k}\rightarrow\lambda^{*}$ implies that $(A-\lambda^{*})^{-1}$ is an unbounded linear operator and $\lambda^{*}$ a generalized eigenvalue of $A$. In the second example $Ae_{k}=e_{k+1}/(k+1)$, it is not difficult to verify that: (a) The point spectrum of $A$ is empty, (b) The range of $A$ is not dense because it does not contain $e_{1}$, and (c) $A^{-1}$ is unbounded because $Ae_{k}\rightarrow0$. Thus the generalized eigenvalue $\lambda^{*}=0$ in this case belongs to the residual spectrum of $A$. In either case, $\lim_{j\rightarrow\infty}e_{j}$ is the corresponding generalized eigenvector that enlarges the trivial null space $\mathcal{N}(\mathscr L_{\lambda^{*}})$ of the generalized eigenvalue $\lambda^{*}$. In fact in these two and the Dirac delta example of Sec. 5 of continuous and residual spectra, the generalized eigenfunctions arise as the limits of a sequence of functions whose images under the respective $\mathcal{L}_{\lambda}$ converge to $0$; recall the definition of footnote \[Foot: gen\_eigen\]. This observation generalizes to the dense extension $\textrm{Multi}_{|}(X,Y)$ of $\textrm{Map}(X,Y)$ as follows. If $x\in\mathcal{D}_{+}$ is not a fixed point of $f(\lambda;x)=x$, but there is some $n\in\mathbb{N}$ such that $f^{n}(\lambda;x)=x$, then the limit $n\rightarrow\infty$ generates a multifunction at $x$ as was the case with the delta function in the previous section and the various other examples that we have seen so far in the earlier sections. One of the main goals of investigations on the spectrum of nonlinear operators is to find a set in the complex plane that has the usual desirable properties of the spectrum of a linear operator, @Appel2000. In this case, the focus has been to find a suitable class of operators $\mathcal{C}(X)$ with $T\in\mathcal{C}(X)$, such that the resolvent set is expressed as$$\rho(T)=\{\lambda\in\mathbb{C}\!:(T_{\lambda}\textrm{ is }1:1)(\textrm{Cl}(\mathcal{R}(T_{\lambda})=X)\textrm{ and }(T_{\lambda}^{-1}\in\mathcal{C}(X)\textrm{ on }\mathcal{R}(T_{\lambda}))\}$$ with the spectrum $\sigma(T)$ being defined as the complement of this set. Among the classes $\mathcal{C}(X)$ that have been considered, beside spaces of continuous functions $C(X)$, are linear boundedness $B(X)$, Frechet differentiability $C^{1}(X)$, Lipschitz continuity $\textrm{Lip}(X)$, and Granas quasiboundedness $Q(x)$, where $\textrm{Lip}(X)$ specifically takes into account the nonlinearity of $T$ to define $$\Vert T\Vert_{\textrm{Lip}}=\sup_{x\neq y}\frac{\Vert T(x)-T(y)\Vert}{\Vert x-y\Vert},\qquad|T|_{\textrm{lip}}=\inf_{x\neq y}\frac{\Vert T(x)-T(y)\Vert}{\Vert x-y\Vert}\label{Eqn: LipNorm}$$ that are plainly generalizations of the corresponding norms of linear operators. Plots of $f_{\lambda}^{-}(y)=\{ x\in\mathcal{D}(f-\lambda)\!:(f-\lambda)x=y\}$ for the functions $f\!:\mathbb{R}\rightarrow\mathbb{R}$$$\begin{array}{rcl} f_{\lambda\textrm{a}}(x) & = & \left\{ \begin{array}{cll} -1-\lambda x, & & x<-1\\ (1-\lambda)x, & & -1\leq x\leq1\\ 1-\lambda x, & & 1<x,\end{array}\right.\\ \\f_{\lambda\textrm{b}}(x) & = & \left\{ \begin{array}{cll} -\lambda x, & & x<1\\ (1-\lambda x)-1, & & 1\leq x\leq2\\ 1-\lambda x, & & 2<x\end{array}\right.\\ \\f_{\lambda\textrm{c}}(x) & = & \left\{ \begin{array}{cll} -\lambda x & & x<1\\ \sqrt{x-1}-\lambda x & & 1\leq x,\end{array}\right.\\ \\f_{\lambda\textrm{d}}(x) & = & \left\{ \begin{array}{cll} (x-1)^{2}+1-\lambda x & & 1\leq x\leq1\\ (1-\lambda)x & & \textrm{otherwise}\end{array}\right.\\ \\f_{\lambda\textrm{e}}(x) & = & \tan^{-1}(x)-\lambda x,\\ \\f_{\lambda\textrm{f}}(x) & = & \left\{ \begin{array}{cll} 1-2\sqrt{-x}-\lambda x, & & x<-1\\ (1-\lambda)x, & & -1\leq x\leq1\\ 2\sqrt{x}-1-\lambda x, & & 1<x\end{array}\right.\end{array}$$ taken from @Appel2000 are shown in Fig. \[Fig: Appel\]. It is easy to verify that the Lipschitz and linear upper and lower bounds of these maps are as in Table \[Table: Appel\_bnds\]. The point spectrum defined by $$P\sigma(f)=\{\lambda\in\mathbb{C}\!:(f-\lambda)x=0\textrm{ for some }x\neq0\}$$ is the simplest to calculate. Because of the special role played by the zero element $0$ in generating the point spectrum in the linear case, the bounds $m\Vert x\Vert\leq\Vert\mathscr{L}x\Vert\leq M\Vert x\Vert$ together with $\mathscr{L}x=\lambda x$ imply $\textrm{Cl}(P\sigma(\mathscr{L}))=[\Vert\mathscr{L}\Vert_{\textrm{b}},\Vert\mathscr{L}\Vert_{\textrm{B}}]$ — where the subscripts denote the lower and upper bounds in Eq. (\[Eqn: LipNorm\]) and which is sometimes taken to be a descriptor of the point spectrum of a nonlinear operator — as can be seen in Table \[Table: Appel\_spectra\] and verified from Fig. \[Fig: Appel\]. The remainder of the spectrum, as the complement of the resolvent set, is more difficult to find. Here the convenient characterization of the resolvent of a continuous linear operator as the set of all sufficiently large $\lambda$ that satisfy $|\lambda|>M$ is of little significance as, unlike for a linear operator, the non-existence of an inverse is not just due the set $\{ f^{-1}(0)\}$ which happens to be the only way a linear map can fail to be injective. Thus the map defined piecewise as $\alpha+2(1-\alpha)x$ for $0\leq x<1/2$ and $2(1-x)$ for $1/2\leq x\leq1$, with $0<\alpha<1$, is not invertible on its range although $\{ f^{-}(0)\}=1$. Comparing Fig. \[Fig: Appel\] and Table \[Table: Appel\_bnds\], it is seen that in cases (b), (c) and (d), the intervals $[|f|_{\textrm{b}},\Vert f\Vert_{\textrm{B}}]$ are subsets of the $\lambda$-values for which the respective maps are not injective; this is to be compared with (a), (e) and (f) where the two sets are the same. Thus the linear bounds are not good indicators of the uniqueness properties of solution of nonlinear equations for which the Lipschitzian bounds are seen to be more appropriate. [|c||c|c|c|c|]{} Function& $|f|_{\textrm{b}}$& $\Vert f\Vert_{\textrm{B}}$& $|f|_{\textrm{lip}}$& $\Vert f\Vert_{\textrm{Lip}}$[\ ]{} $f_{\textrm{a}}$& $0$& $1$& $0$& $1$[\ ]{} $f_{\textrm{b}}$& $0$& $1/2$& $0$& $1$[\ ]{} $f_{\textrm{c}}$& $0$& $1/2$& $0$& $\infty$[\ ]{} $f_{\textrm{d}}$& $2(\sqrt{2}-1)$& $\infty$& $0$& $2$[\ ]{}$f_{\textrm{e}}$& $0$& $1$& $0$& $1$[\ ]{} $f_{\textrm{f}}$& $0$& $1$& $0$& $1$[\ ]{} [|c||c|c|]{} Functions& $\sigma_{\textrm{Lip}}(f)$& $P\sigma(f)$[\ ]{} $f_{\textrm{a}}$& $[0,1]$& $(0,1]$[\ ]{} $f_{\textrm{b}}$& $[0,1]$& $[0,1/2]$[\ ]{} $f_{\textrm{c}}$& $[0,\infty)$& $[0,1/2]$[\ ]{} $f_{\textrm{d}}$& $[0,2]$& $[2(\sqrt{2}-1),1]$[\ ]{} $f_{\textrm{e}}$& $[0,1]$& $(0,1)$[\ ]{} $f_{\textrm{f}}$& $[0,1]$& $(0,1)$[\ ]{} In view of the above, we may draw the following conclusions. If we choose to work in the space of multifunctions $\textrm{Multi}(X,\mathcal{T})$, with $\mathcal{T}$ the topology of pointwise biconvergence, when all functional relations are (multi) invertible on their ranges, we may make the following definition for the net of functions $f(\lambda;x)$ satisfying $f(\lambda;x)=x$. **Definition 6.1.** *Let* $f(\lambda;\cdot)\in\textrm{Multi}(X,\mathcal{T})$ *be a function. The resolvent set of $f$ is given by* $$\rho(f)=\{\lambda\!:(f(\lambda;\cdot)^{-1}\in\textrm{Map}(X,\mathcal{T}))\wedge(\textrm{Cl}(\mathcal{R}(f(\lambda;\cdot))=X)\},$$ *and any $\lambda$ not in $\rho$ is in the spectrum of $f$.$\qquad\square$* Thus apart from multifunctions, $\lambda\in\sigma(f)$ also generates functions on the boundary of functional and non-functional relations in $\textrm{Multi}(X,\mathcal{T})$. While it is possible to classify the spectrum into point, continuous and residual subsets, as in the linear case, it is more meaningful for nonlinear operators to consider $\lambda$ as being either in the *boundary spectrum* $\textrm{Bdy}(\sigma(f))$ or in the *interior spectrum* $\textrm{Int}(\sigma(f))$, depending on whether or not the multifunction $f(\lambda;\cdot)^{-}$ arises as the graphical limit of a net of functions in either $\rho(f)$ or $R\sigma(f)$. This is suggested by the spectra arising from the second row of Table \[Table: spectrum\] (injective $\mathcal{L}_{\lambda}$ and discontinuous $\mathcal{L}_{\lambda}^{-1}$) that lies sandwiched in the $\lambda$-plane between the two components arising from the first and third rows, see @Naylor1971 Sec. 6.6, for example. According to this simple scheme, the spectral set is a closed set with its boundary and interior belonging to $\textrm{Bdy}(\sigma(f))$ and $\textrm{Int}(\sigma(f))$ respectively. Table \[Table: Appel\_multi\] shows this division for the examples in Fig. \[Fig: Appel\]. Because $0$ is no more significant than any other point in the domain of a nonlinear map in inducing non-injectivity, the division of the spectrum into the traditional sets would be as shown in Table \[Table: Appel\_multi\]; compare also with the conventional linear point spectrum of Table \[Table: Appel\_spectra\]. In this nonlinear classification, the point spectrum consists of any $\lambda$ for which the inverse $f(\lambda;\cdot)^{-}$ is set-valued, irrespective of whether this is produced at $0$ or not, while the continuous and residual spectra together comprise the boundary spectrum. Thus a $\lambda$ can be both in the point and the continuous or residual spectra which need not be disjoint. The continuous and residual spectra are included in the boundary spectrum which may also contain parts of the point spectrum. Function $\textrm{Int}(\sigma(f))$ $\textrm{Bdy}(\sigma(f))$ $P\sigma(f)$ $C\sigma(f)$ $R\sigma(f)$ ------------------ --------------------------- --------------------------- -------------- -------------- -------------- $f_{\textrm{a}}$ $(0,1)$ $\{0,1\}$ $[0,1]$ $\{1\}$ $\{0\}$ $f_{\textrm{b}}$ $(0,1)$ $\{0,1\}$ $[0,1]$ $\{1\}$ $\{0\}$ $f_{\textrm{c}}$ $(0,\infty)$ $\{0\}$ $[0,\infty)$ $\{0\}$ $\emptyset$ $f_{\textrm{d}}$ $(0,2)$ $\{0,2\}$ $(0,2)$ $\{0,2\}$ $\emptyset$ $f_{\textrm{e}}$ $(0,1)$ $\{0,1\}$ $(0,1)$ $\{1\}$ $\{0\}$ $f_{\textrm{f}}$ $(0,1)$ $\{0,1\}$ $(0,1)$ $\{0,1\}$ $\emptyset$ : \[Table: Appel\_multi\][Nonlinear spectra of functions of Fig. \[Fig: Appel\]. Compare the present point spectra with the usual linear spectra of Table \[Table: Appel\_spectra\].]{} **Example 6.1.** To see how these concepts apply to linear mappings, consider the equation $(D-\lambda)y(x)=r(x)$ where $D=d/dx$ is the differential operator on $L^{2}[0,\infty)$, and let $\lambda$ be real. For $\lambda\neq0$, the unique solution of this equation in $L^{2}[0,\infty)$, is $$\begin{aligned} y(x)= & \left\{ \begin{array}{ll} {\displaystyle e^{\lambda x}\left(y(0)+\int_{0}^{x}e^{-\lambda x^{\prime}}r(x^{\prime})dx^{\prime}\right)}, & \lambda<0\\ {\displaystyle e^{\lambda x}\left(y(0)-\int_{x}^{\infty}e^{-\lambda x^{\prime}}r(x^{\prime})dx^{\prime}\right),} & \lambda>0\end{array}\right.\end{aligned}$$ showing that for $\lambda>0$ the inverse is functional so that $\lambda\in(0,\infty)$ belongs to the resolvent of $D$. However, when $\lambda<0$, apart from the $y=0$ solution (since we are dealing a with linear problem, only $r=0$ is to be considered), $e^{\lambda x}$ is also in $L^{2}[0,\infty)$ so that all such $\lambda$ are in the point spectrum of $D$. For $\lambda=0$ and $r\neq0$, the two solutions are not necessarily equal unless $\int_{0}^{\infty}r(x)=0$, so that the range $\mathcal{R}(D-I)$ is a subspace of $L^{2}[0,\infty)$. To complete the problem, it is possible to show [@Naylor1971] that $0\in C\sigma(D)$, see Ex. 2.2; hence the continuous spectrum forms at the boundary of the functional solution for the resolvent-$\lambda$ and the multifunctional solution for the point spectrum. With a slight variation of problem to $y(0)=0$, all $\lambda<0$ are in the resolvent set, while $\lambda>0$ the inverse is bounded but must satisfy $y(0)=\int_{0}^{\infty}e^{-\lambda x}r(x)dx=0$ so that $\textrm{Cl}(\mathcal{R}(D-\lambda))\neq L^{2}[0,\infty)$. Hence $\lambda>0$ belong to the residual spectrum. The decomposition of the complex $\lambda$-plane for these and some other linear spectral problems taken from @Naylor1971 is shown in Fig. \[Fig: spectrum\]. In all cases, the spectrum due to the second row of Table \[Table: spectrum\] acts as a boundary between that arising from the first and third rows, which justifies our division of the spectrum for a nonlinear operator into the interior and boundary components. Compare Example 2.2.$\qquad\blacksquare$ From the basic representation of the resolvent operator $(\mathbf{1}-f)^{-1}$ $$\mathbf{1}+f+f^{2}+\cdots+f^{i}+\cdots$$ in $\textrm{Multi}(X)$, if the iterates of $f$ converge to a multifunction for some $\lambda$, then that $\lambda$ must be in the spectrum of $f$, which means that the control parameter of a chaotic dynamical system is in its spectrum. Of course, the series can sum to a multi even otherwise: take $f_{\lambda}(x)$ to be identically $x$ with $\lambda=1$, for example, to get $1\in P\sigma(f)$. A comparison of Tables \[Table: spectrum\] and \[Table: Appel\_spectra\] reveal that in case (d), for example, $0$ and $2$ belong to the Lipschtiz spectrum because although $f_{\textrm{d}}^{-1}$ is not Lipschitz continuous, $\Vert f\Vert_{\textrm{Lip}}=2$. It should also be noted that the boundary between the functional resolvent and multifunctional spectral set is formed by the graphical convergence of a net of resolvent functions while the multifunctions in the interior of the spectral set evolve graphically independent of the functions in the resolvent. The chaotic states forming the boundary of the functional and multifunctional subsets of $\textrm{Multi}(X)$ marks the transition from the less efficient functional state to the more efficient multifunctional one. These arguments also suggest the following. The countably many outputs arising from the non-injectivity of $f(\lambda;\cdot)$ corresponding a given input can be interpreted to define *complexity because* *in a nonlinear system each of these possibilities constitute a experimental result in itself that may not be combined in any definite predetermined manner.* This is in sharp contrast to linear systems where a linear combination, governed by the initial conditions, always generate a unique end result; recall also the combination offered by the singular generalized eigenfunctions of neutron transport theory. This multiplicity of possibilities that have no definite combinatorial property is the basis of the diversity of nature, and is possibly responsible for Feigenbaum’s “historical prejudice”, [@Feigenbaum1992], see Prelude, 2. Thus *order* represented by the functional resolvent passes over to *complexity* of the countably multifunctional interior spectrum via the uncountably multifunctional boundary that is a prerequisite for *chaos.* We may now strengthen our hypothesis offered at the end of the previous section in terms of the examples of Figs. \[Fig: Appel\] and \[Fig: spectrum\], that nature uses chaoticity as an intermediate step to the attainment of states that would otherwise be inaccessible to it. Well-posedness of a system is an extremely inefficient way of expressing a multitude of possibilities as this requires a different input for every possible output. Nature chooses to express its myriad manifestations through the multifunctional route leading either to averaging as in the delta function case or to a countable set of well-defined states, as in the examples of Fig. \[Fig: Appel\] corresponding to the interior spectrum. Of course it is no distraction that the multifunctional states arise respectively from $f_{\lambda}$ and $f_{\lambda}^{-}$ in these examples as $f$ is a function on $X$ that is under the influence of both $f$ and its inverse. The functional resolvent is, for all practical purposes, only a tool in this structure of nature. The equation $f(x)=y$ is typically an input-output system in which the inverse images at a functional value $y_{0}$ represents a set of input parameters leading to the same experimental output $y_{0};$ this is stability characterized by a complete insensitivity of the output to changes in input. On the other hand, a continuous multifunction at $x_{0}$ is a signal for a hypersensitivity to input because the output, which is a definite experimental quantity, is a choice from the possibly infinite set $\{ f(x_{0})\}$ made by a choice function which represents the experiment at that particular point in time. Since there will always be finite differences in the experimental parameters when an experiment is repeated, the choice function (that is the experimental output) will select a point from $\{ f(x_{0})\}$ that is representative of that experiment and which need not bear any definite relation to the previous values; this is instability and signals sensitivity to initial conditions. Such a state is of high entropy as the number of available states $f_{\textrm{C}}(\{ f(x_{0})\})$ — where $f_{\textrm{C}}$ is the choice function — is larger than a functional state represented by the singleton $\{ f(x_{0})\}.$ **Epilogue** @Gleick1987 **Appendix** This Appendix gives a brief overview of some aspects of topology that are necessary for a proper understanding of the concepts introduced in this work. **A1. Convergence in Topological Spaces: Sequence, Net and Filter.** In the theory of convergence in topological spaces, *countability* plays an important role. To understand the significance of this concept, some preliminaries are needed. The notion of a basis, or base, is a familiar one in analysis: a base is a subcollection of a set which may be used to construct, in a specified manner, any element of the set. This simplifies the statement of a problem since a smaller number of elements of the base can be used to generate the larger class of every element of the set. This philosophy finds application in topological spaces as follows. Among the three properties $(\textrm{N}1)-(\textrm{N}3)$ of the neighbourhood system $\mathcal{N}_{x}$ of Tutorial4, (N1) and (N2) are basic in the sense that the resulting subcollection of $\mathcal{N}_{x}$ can be used to generate the full system by applying $(\textrm{N}3)$; this *basic neighbourhood* *system*, or *neighbourhood (local) bas*e $\mathcal{B}_{x}$ *at* $x$, is characterized by (NB1) $x$ belongs to each member $B$ of $\mathcal{B}_{x}$*.* (NB2) The intersection of any two members of **$\mathcal{B}_{x}$ **contains another member of $\mathcal{B}_{x}$: $B_{1},B_{2}\in\mathcal{B}_{x}\Rightarrow(\exists B\in\mathcal{B}_{x}\!:B\subseteq B_{1}\bigcap B_{2})$. ** Formally, compare Eq. (\[Eqn: nbd-topology\]), **Definition A1.1.** *A neighbourhood (local) base* $\mathcal{B}_{x}$ *at $x$ in a topological space $(X,\mathcal{U})$ is a subcollection of the neighbourhood system $\mathcal{N}_{x}$ having the property that each $N\in\mathcal{N}_{x}$ contains some member of* $\mathcal{B}_{x}$*.* *Thus* $$\mathcal{B}_{x}\overset{\textrm{def}}=\{ B\in\mathcal{N}_{x}\!:x\in B\subseteq N\textrm{ for each }N\in\mathcal{N}_{x}\}\label{Eqn: TBx}$$ *determines the full neighbourhood system* $$\mathcal{N}_{x}=\{ N\subseteq X\!:x\in B\subseteq N\textrm{ for some }B\textrm{ }\in\,\mathcal{B}_{x}\}\label{Eqn: TBx_nbd}$$ *reciprocally as all supersets of the basic elements.$\qquad\square$* The entire neighbourhood system $\mathcal{N}_{x}$, which is recovered from the base by forming all supersets of the basic neighbourhoods, **is trivially a local base at $x$; non-trivial examples are given below. The second example of a base, consisting as usual of a subcollection of a given collection, is the topological base $_{\textrm{T}}\mathcal{B}$ that allows the specification of the topology on a set $X$ in terms of a smaller collection of open sets. **Definition A1.2.** *A base* $_{\textrm{T}}\mathcal{B}$ *in a topological space $(X,\mathcal{U})$ is a subcollection of the topology $\mathcal{U}$ having the property that each $U\in\mathcal{U}$ contains some member of* $_{\textrm{T}}\mathcal{B}$*.* *Thus* $$_{\textrm{T}}\mathcal{B}\overset{\textrm{def}}=\{ B\in\mathcal{U}\!:B\subseteq U\textrm{ for each }U\in\mathcal{U}\}\label{Eqn: TB}$$ *determines reciprocally the topology $\mathcal{U}$ as* $$\mathcal{U}=\left\{ U\subseteq X\!:U=\bigcup_{B\in\,\!_{\textrm{T}}\mathcal{B}\,}B\right\} \qquad\square\label{Eqn: TB_topo}$$ This means that the topology on $X$ can be reconstructed form the base by taking all possible unions of members of the base, and a collection of subsets of a set $X$ is a topological base iff Eq. (\[Eqn: TB\_topo\]) of arbitrary unions of elements of $_{\textrm{T}}\mathcal{B}$ generates a topology on $X$. This topology, which is the coarsest (that is the smallest) that contains $_{\textrm{T}}\mathcal{B}$, is obviously closed under finite intersections. Since the open set $\textrm{Int}(N)$ is a neighbourhood of $x$ whenever $N$ is, Eq. (\[Eqn: TBx\_nbd\]) and the definition Eq. (\[Eqn: Def: nbd system\]) of $\mathcal{N}_{x}$ implies that *the open neighbourhood system of any point in a topological space is an example of a neighbourhood base at that point,* an observation that has often led, together with Eq. (\[Eqn: TB\]), to the use of the term “neighbourhood” as a synonym for “non-empty open set”. The distinction between the two however is significant as neighbourhoods need not necessarily be open sets; thus while not necessary, it is clearly sufficient for the local basic sets $B$ to be open in Eqs. (\[Eqn: TBx\]) and (\[Eqn: TBx\_nbd\]). If Eq. (\[Eqn: TBx\_nbd\]) holds for every $x\in N$, then the resulting $\mathcal{N}_{x}$ reduces to the topology induced by the open basic neighbourhood system $\mathcal{B}_{x}$ as given by Eq. (\[Eqn: nbd-topology\]). In order to check if a collection of subsets $_{\textrm{T}}\mathcal{B}$ of $X$ qualifies to be a basis, it is not necessary to verify properties $(\textrm{T}1)-(\textrm{T}3)$ of Tutorial4 for the class (\[Eqn: TB\_topo\]) generated by it because of the properties (TB1) and (TB2) below whose strong affinity to (NB1) and (NB2) is formalized in Theorem A1.1. **Theorem A1.1.** *A collection* $_{\textrm{T}}\mathcal{B}$ *of subsets of $X$ is a* *topological basis on* $X$ *iff* (TB1) *$X=\bigcup_{B\in\,_{\textrm{T}}\mathcal{B}}B$. Thus each $x\in X$ must belong to some* $B\in\,_{\textrm{T}}\mathcal{B}$ *which implies the existence of a* *local base* *at each point* *$x\in X$.* (TB2) *The intersection of any two members $B_{1}$ and $B_{2}$ of* $_{\textrm{T}}\mathcal{B}$ *with $x\in B_{1}\bigcap B_{2}$* ***contains another member of* $_{\textrm{T}}\mathcal{B}$: $(B_{1},B_{2}\in\,_{\textrm{T}}\mathcal{B})\wedge(x\in B_{1}\bigcap B_{2})\Rightarrow(\exists B\in\,_{\textrm{T}}\mathcal{B}\!:x\in B\subseteq B_{1}\bigcap B_{2})$.$\qquad\square$ This theorem, together with Eq. (\[Eqn: TB\_topo\]) ensures that a given collection of subsets of a set $X$ satisfying (TB1) and (TB2) induces *some* topology on $X$; compared to this is the result that *any* collection of subsets of a set $X$ is a *subbasis* for some topology on $X$. If $X$, however, already has a topology $\mathcal{U}$ imposed on it, then Eq. **(\[Eqn: TB\]) must also be satisfied in order that the topology generated by $_{\textrm{T}}\mathcal{B}$ is indeed $\mathcal{U}$. The next theorem connects the two types of bases of Defs. A1.1 and A1.2 by asserting that although a local base of a space need not consist of open sets and a topological base need not have any reference to a point of $X$, any subcollection of the base containing a point is a local base at that point. **Theorem A1.2.** *A collection of open sets* $_{\textrm{T}}\mathcal{B}$ *is a base for a topological space $(X,\mathcal{U})$ iff for each $x\in X$, the subcollection* $$\mathcal{B}_{x}=\{ B\in\mathcal{U}\!:x\in B\in\!\,_{\textrm{T}}\mathcal{B}\}\label{Eqn: base_local base}$$ *of basic sets containing $x$ is a local base at* $x$.$\qquad\square$ **Proof.** *Necessity.* Let $_{\textrm{T}}\mathcal{B}$ be a base of *$(X,\mathcal{U})$* and $N$ be a neighbourhood of $x$, so that $x\in U\subseteq N$ for some open set $U=\bigcup_{B\in\!\,_{\textrm{T}}\mathcal{B}}B$ and basic open sets $B$. Hence $x\in B\subseteq N$ shows, from Eq. (\[Eqn: TBx\]), that $B\in\mathcal{B}_{x}$ is a local basic set at $x$. *Sufficiency.* If $U$ is an open set of $X$ containing $x$, then the definition of local base Eq. (\[Eqn: TBx\]) requires $x\in B_{x}\subseteq U$ for some subcollection of basic sets $B_{x}$ in $\mathcal{B}_{x}$; hence $U=\bigcup_{x\in U}B_{x}$. By Eq. (\[Eqn: TB\_topo\]) therefore, $_{\textrm{T}}\mathcal{B}$ is a topological base for $X$.$\qquad\blacksquare$ Because the basic sets are open, (TB2) of Theorem A1.1 leads to the following physically appealing paraphrase of Thm. A1.2. **Corollary.** *A collection* $_{\textrm{T}}\mathcal{B}$ *of open sets of* $(X,\mathcal{U})$ *is a topological base that generates* $\mathcal{U}$ *iff for each open set $U$ of $X$ and each $x\in U$ there is an open set* $B\in\!\,_{\textrm{T}}\mathcal{B}$ *such that $x\in B\subseteq U$*; *that is iff* $$x\in U\in\mathcal{U}\Longrightarrow(\exists B\in\,_{\textrm{T}}\mathcal{B}\!:x\in B\subseteq U).\qquad\square$$ ****Example A1.1.** Some examples of local bases in $\mathbb{R}$ are intervals of the type $(x-\varepsilon,x+\varepsilon)$, $[x-\varepsilon,x+\varepsilon]$ for real $\varepsilon$, $(x-q,x+q)$ for rational $q$, $(x-1/n,x+1/n)$ for $n\in\mathbb{Z}_{+}$, while for a metrizable space with the topology induced by a metric $d$, each of the following is a local base at $x\in X$: $B_{\varepsilon}(x;d):=\{ y\in X:d(x,y)<\varepsilon\}$ and $D_{\varepsilon}(x;d):=\{ y\in X:d(x,y)\leq\varepsilon\}$ for $\varepsilon>0$, $B_{q}(x;d)$ for $\mathbb{Q}\ni q>0$ and $B_{1/n}(x;d)$ for $n\in\mathbb{Z}_{+}$. In $\mathbb{R}^{2}$, two neighbourhood bases at any $x\in\mathbb{R}^{2}$ are the disks centered at $x$ and the set of all squares at $x$ with sides parallel to the axes. Although these bases have no elements in common, they are nevertheless equivalent in the sense that they both generate the same (usual) topology in $\mathbb{R}^{2}$. Of course, the entire neighbourhood system at any point of a topological space is itself a (less useful) local base at that point. By Theorem A1.2, $B_{\varepsilon}(x;d)$, $D_{\varepsilon}(x;d)$, $\varepsilon>0$, $B_{q}(x;d)$, $\mathbb{Q}\ni q>0$ and $B_{1/n}(x;d)$, $n\in\mathbb{Z}_{+}$, for all $x\in X$ are examples of bases in a metrizable space with topology induced by a metric $d$.$\qquad\square$ In terms of local bases and bases, it is now possible to formulate the notions of first and second countability as follows. **Definition A1.3.** *A topological space is* *first countable* *if each $x\in X$ has some countable neighbourhood base, and is* *second countable* *if it has a countable base.* $\qquad\square$ Every metrizable space $(X,d)$ is first countable as both $\{ B(x,q)\}_{\mathbb{Q}\ni q>0}$ and $\{ B(x,1/n)\}_{n\in\mathbb{Z}_{+}}$ are examples of countable neighbourhood bases at any $x\in(X,d)$; hence $\mathbb{R}^{n}$ is first countable. It should be clear that although every second countable space is first countable, *only a countable first countable space can be second countable*, and a common example of a uncountable first countable space that is also second countable is provided by $\mathbb{R}^{n}$. Metrizable spaces need not be second countable: any uncountable set having the discrete topology is as an example. **Example A1.2.** The following is an important example of a space that is not first countable as it is needed for our pointwise biconvergence of Section 3. Let $\textrm{Map}(X,Y)$ be the set of all functions between the uncountable spaces $(X,\mathcal{U})$ and $(Y,\mathcal{V})$. Given any integer $I\geq1$, and any *finite* collection of points $(x_{i})_{i=1}^{I}$ of $X$ and of open sets $(V_{i})_{i=1}^{I}$ in $Y$, let $$B((x_{i})_{i=1}^{I};(V_{i})_{i=1}^{I})=\{ g\in\textrm{Map}(X,Y)\!:(g(x_{i})\in V_{i})(i=1,2,\cdots,I)\}\label{Eqn: point}$$ be the functions in $\textrm{Map}(X,Y)$ whose graphs pass through each of the sets $(V_{i})_{i=1}^{I}$ at $(x_{i})_{i=1}^{I}$, and let $_{\textrm{T}}\mathcal{B}$ be the collection of all such subsets of $\textrm{Map}(X,Y)$ for every choice of $I$, $(x_{i})_{i=1}^{I}$, and $(V_{i})_{i=1}^{I}$. The existence of a unique topology $\mathcal{T}$ — the *topology of pointwise convergence* on $\textrm{Map}(X,Y)$ — that is generated by the open sets $B$ of the collection $_{\textrm{T}}\mathcal{B}$ now follows because (TB1) is satisfied: For any $f\in\textrm{Map}(X,Y)$ there must be some $x\in X$ and a corresponding $V\subseteq Y$ such that $f(x)\in V$, and (TB2) is satisfied because $$B((s_{i})_{i=1}^{I};(V_{i})_{i=1}^{I})\bigcap B((t_{j})_{j=1}^{J};(W_{j})_{j=1}^{J})=B((s_{i})_{i=1}^{I},(t_{j})_{j=1}^{J};(V_{i})_{i=1}^{I},(W_{j})_{j=1}^{J})$$ implies that a function simultaneously belonging to the two open sets on the left must pass through each of the points defining the open set on the right. We now demonstrate that $(\textrm{Map}(X,Y),\mathcal{T})$ is not first countable by verifying that it is not possible to have a countable local base at any $f\in\textrm{Map}(X,Y)$. If this is not indeed true, let $B_{f}^{I}((x_{i})_{i=1}^{I};(V_{i})_{i=1}^{I})=\{ g\in\textrm{Map}(X,Y)\!:(g(x_{i})\in V_{i})_{i=1}^{I}\}$, which denotes those members of $_{\textrm{T}}\mathcal{B}$ that contain $f$ with $V_{i}$ an open neighbourhood of $f(x_{i})$ in $Y$, be a countable local base at $f$, see Thm. A1.2. Since $X$ is uncountable, it is now possible to choose some $x^{*}\in X$ different from any of the $(x_{i})_{i=1}^{I}\textrm{ }$ (for example, let $x^{*}\in\mathbb{R}$ be an irrational for rational $(x_{i})_{i}^{I}\textrm{ }$), and let $f(x^{*})\in V^{*}$ where $V^{*}$ is an open neighbourhood of $f(x^{*})$. Then $B(x^{*};V^{*})$ is an open set in $\textrm{Map}(X,Y)$ containing $f$; hence from the definition of the local base, Eq. (\[Eqn: TBx\]), or equivalently from the Corollary to Theorem A1.2, there exists some (countable) $I\in\mathbb{N}$ such that $f\in B^{I}\subseteq B(x^{*};V^{*})$. However, $$\begin{array}{ccc} f^{*}(x) & = & \begin{cases} y_{i}\in V_{i}, & \textrm{if }x=x_{i},\textrm{ and }1\leq i\leq I\\ y^{*}\in V^{*} & \textrm{if }x=x^{*}\\ \textrm{arbitrary}, & \textrm{otherwise}\end{cases}\end{array}$$ is a simple example of a function on $X$ that is in $B^{I}$ (as it is immaterial as to what values the function takes at points other than those defining $B^{I}$), but not in $B(x^{*};V^{*})$. From this it follows that *a sufficient condition for the topology of pointwise convergence to be first countable is that $X$ be countable.*$\qquad\blacksquare$ Even though it is not first countable, $(\textrm{Map}(X,Y),\mathcal{T})$ is a Hausdorff space when $Y$ is Hausdorff. Indeed, if $f,g\in(\textrm{Map}(X,Y),\mathcal{T})$ with $f\neq g$, then $f(x)\neq g(x)$ for some $x\in X$. But then as $Y$ is Hausdorff, it is possible to choose disjoint open intervals $V_{f}$ and $V_{g}$ at $f(x)$ and $g(x)$ respectively. With this background on first and second countability, it is now possible to go back to the question of nets, filters and sequences. Technically, a sequence on a set $X$ is a map $x\!:\mathbb{N}\rightarrow X$ from the set of natural numbers to $X$; instead of denoting this is in the usual functional manner of $x(i)\textrm{ with }i\in\mathbb{N}$, it is the standard practice to use the notation $(x_{i})_{i\in\mathbb{N}}$ for the terms of a sequence. However, if the space $(X,\mathcal{U})$ is not first countable (and as seen above this is not a rare situation), it is not difficult to realize that sequences are inadequate to describe convergence in $X$ simply because it can have only countably many values whereas the space may require uncountably many neighbourhoods to completely define the neighbourhood system at a point. The resulting uncountable generalizations of a sequence in the form of *nets* and *filters* is achieved through a corresponding generalization of the index set $\mathbb{N}$ to the directed set $\mathbb{D}$. **Definition A1.4.** *A* *directed set* *$\mathbb{D}$ is a preordered set for which the order $\preceq$, known as a* *direction of* $\mathbb{D}$, *satisfies* \(a) *$\alpha\in\mathbb{D}$ $\Rightarrow$ $\alpha\preceq\alpha$* (that is $\preceq$ is reflexive)*.* \(b) **$\alpha,\beta,\gamma\in\mathbb{D}\textrm{ such that }(\alpha\preceq\beta\wedge\beta\preceq\gamma)$ $\Rightarrow$ $\alpha\preceq\gamma$ (that is $\preceq$ is transitive). \(c) $\alpha,\beta\in\mathbb{D}$ $\Rightarrow$ $\exists\gamma\in\mathbb{D}\textrm{ such that }(\alpha\preceq\gamma)\wedge(\beta\preceq\gamma)$*.$\qquad\square$* While the first two properties are obvious enough and constitutes the preordering of $\mathbb{{D}}$, the third which replaces antisymmetry, ensures that for any finite number of elements of the directed set (recall that a preordered set need not be fully ordered), there is always a successor. Examples of directed sets can be both straight forward, as any totally ordered set like $\mathbb{N}$, $\mathbb{R}$, $\mathbb{Q}$, or $\mathbb{Z}$ and all subsets of a set $X$ under the superset or subset relation (that is $(\mathcal{P}(X),\supseteq)$ or $(\mathcal{P}(X),\subseteq)$ that are directed by their usual ordering, and not quite so obvious as the following examples which are significantly useful in dealing with convergence questions in topological spaces, amply illustrate. The neighbourhood system $$_{\mathbb{D}}N=\{ N\!:N\in\mathcal{N}_{x}\}$$ at a point $x\in X$, directed by the reverse inclusion direction $\preceq$ defined as $$M\preceq N\Longleftrightarrow N\subseteq M\qquad\textrm{for }M,N\in\mathcal{N}_{x},\label{Eqn: Direction1}$$ is a fundamental example of a *natural direction of $\mathcal{N}_{x}$*. In fact while reflexivity and transitivity are clearly obvious, (c) follows because for any $M,N\in\mathcal{N}_{x}$, $M\preceq M\bigcap N$ and $N\preceq M\bigcap N$. Of course, this direction is not a total ordering on $\mathcal{N}_{x}$. A more naturally useful directed set in convergence theory is $$_{\mathbb{D}}N_{t}=\{(N,t)\!:(N\in\mathcal{N}_{x})(t\in N)\}\label{Eqn: Directed}$$ under its *natural direction* $$(M,s)\preceq(N,t)\Longleftrightarrow N\subseteq M\qquad\textrm{for }M,N\in\mathcal{N}_{x};\label{Eqn: Direction2}$$ **$_{\mathbb{D}}N_{t}$ is more useful than $_{\mathbb{D}}N$ because, unlike the later, $_{\mathbb{D}}N_{t}$ does not require a simultaneous choice of points from every $N\in\mathcal{N}_{x}$ that implicitly involves a simultaneous application of the Axiom of Choice; see Examples A1.2(2) and (3) below. The general indexed variation $$_{\mathbb{D}}N_{\beta}=\{(N,\beta)\!:(N\in\mathcal{N}_{x})(\beta\in\mathbb{D})(x_{\beta}\in N)\}\label{Eqn: DirectedIndexed}$$ of Eq. (\[Eqn: Directed\]), with natural direction $$(M,\alpha)\leq(N,\beta)\Longleftrightarrow(\alpha\preceq\beta)\wedge(N\subseteq M),\label{Eqn: DirectionIndexed}$$ often proves useful in applications as will be clear from the proofs of Theorems A1.3 and A1.4. **Definition A1.5.** ***Net.*** *Let $X$ be any set and $\mathbb{D}$ a directed set. A net $\chi\!:\mathbb{D}\rightarrow X$* *in $X$* *is a function* *on the directed set $\mathbb{D}$ with values in $X$.$\qquad\square$* A net, to be denoted as $\chi(\alpha)$, $\alpha\in\mathbb{D}$, is therefore a function indexed by a directed set. We adopt the convention of denoting nets in the manner of functions and do not use the sequential notation $\chi_{\alpha}$ that can also be found in the literature. Thus, while every sequence is a special type of net, $\chi:\!\mathbb{Z}\rightarrow X$ is an example of a net that is not a sequence. Convergence of sequences and nets are described most conveniently in terms of the notions of being *eventually in* and *frequently in* every neighbourhood of points. We describe these concepts in terms of nets which apply to sequences with obvious modifications. **Definition A1.6.** *A net* $\chi\!:\mathbb{D}\rightarrow X$ *is said to be* \(a) *Eventually in* *a subset $A$* *of* *$X$ if its tail is eventually in $A$*: *$(\exists\beta\in\mathbb{D})\!:(\forall\gamma\succeq\beta)(\chi(\gamma)\in A).$* \(b) *Frequently in* *a subset $A$* *of* *$X$ if for any index $\beta\in\mathbb{D}$, there is a successor index $\gamma\in\mathbb{D}$ such that $\chi(\gamma)$* is in $A$: *$(\forall\beta\in\mathbb{D})(\exists\gamma\succeq\beta)\!:(\chi(\gamma)\in A).\qquad\square$* It is not difficult to appreciate that \(i) A net eventually in a subset is also frequently in it but not conversely, \(ii) A net eventually (respectively, frequently) in a subset cannot be frequently (respectively, eventually) in its complement. With these notions of eventually in and frequently in, convergence characteristics of a net may be expressed as follows. **Definition A1.7.** *A net* *$\chi\!:\mathbb{D}\rightarrow X$ converges to $x\in X$ if it is eventually in every neighbourhood of $x$, that is* $$(\forall N\in\mathcal{N}_{x})(\exists\mu\in\mathbb{D})(\chi(\nu\succeq\mu)\in N).$$ *The point $x$ is known as the* *limit* *of $\chi$ and the collection of all limits of a net is the* *limit set* $$\textrm{lim}(\chi)=\{ x\in X\!:(\forall N\in\mathcal{N}_{x})(\exists\mathbb{R}_{\beta}\in\textrm{Res}(\mathbb{D}))(\chi(\mathbb{R}_{\beta})\subseteq N)\}\label{Eqn: lim net}$$ *of $\chi$, with the set of* *residuals* $\textrm{Res}(\mathbb{D})$ *in $\mathbb{D}$ given by* $$\textrm{Res}(\mathbb{D})=\{\mathbb{R}_{\alpha}\in\mathcal{P}(\mathbb{D})\!:\mathbb{R}_{\alpha}=\{\beta\in\mathbb{D}\textrm{ for all }\beta\succeq\alpha\in\mathbb{D}\}\}.\label{Eqn: residual}$$ *The net* *adheres at* *$x\in X$*[^27] *if it is frequently in every neighbourhood of $x$, that is* $$((\forall N\in\mathcal{N}_{x})(\forall\mu\in\mathbb{D}))((\exists\nu\succeq\mu)\!:\chi(\nu)\in N).$$ *The point $x$ is known as the* *adherent* *of $\chi$ and the collection of all adherents of $\chi$ is the* *adherent set of the net, which* *may be expressed in terms of the* *cofinal subset* *of $\mathbb{D}$* $$\textrm{Cof}(\mathbb{D})=\{\mathbb{C}_{\alpha}\in\mathcal{P}(\mathbb{D})\!:\mathbb{C}_{\alpha}=\{\beta\in\mathbb{D}\textrm{ for some }\beta\succeq\alpha\in\mathbb{D}\}\}\label{Eqn: cofinal}$$ (thus $\mathbb{D}_{\alpha}$ is cofinal in $\mathbb{D}$ iff it intersects every residual in $\mathbb{D}$), *as* $$\textrm{adh}(\chi)=\{ x\in X\!:(\forall N\in\mathcal{N}_{x})(\exists\mathbb{C}_{\beta}\in\textrm{Cof}(\mathbb{D}))(\chi(\mathbb{C}_{\beta})\subseteq N)\}.\label{Eqn: adh net1}$$ *This recognizes, in keeping with the limit set, each subnet of a net to be a net in its own right, and is equivalent to* $${\textstyle \textrm{adh}(\chi)=\{ x\in X\!:(\forall N\in\mathcal{N}_{x})(\forall\mathbb{R}_{\alpha}\in\textrm{Res}(\mathbb{D}))(\chi(\mathbb{R}_{\alpha})\bigcap N\neq\emptyset)\}.\qquad\square}\label{Eqn: adh net2}$$ Intuitively, a sequence is eventually in a set $A$ if it is always in it after a finite number of terms (of course, the concept of a *finite number of terms* is unavailable for nets; in this case the situation may be described by saying that a net is eventually in $A$ if its *tail is in* $A$) and it is frequently in $A$ if it always returns to $A$ to leave it again. It can be shown that a net is eventually (resp. frequently) in a set iff it is not frequently (resp.eventually) in its complement. The following examples illustrate graphically the role of a proper choice of the index set $\mathbb{D}$ in the description of convergence. **Example A1.3.** (1) Let $\gamma\in\mathbb{D}$. The eventually constant net $\chi(\delta)=x$ for $\delta\succeq\gamma$ converges to $x$. \(2) Let $\mathcal{N}_{x}$ be a neighbourhood system at a point $x$ in $X$ and suppose that the net $(\chi(N))_{N\in\mathcal{N}_{x}}$ is defined by $$\chi(M)\overset{\textrm{def}}=s\in M;\label{Eqn: Def: Net1}$$ here the directed index set $_{\mathbb{D}}N$ is ordered by the natural direction (\[Eqn: Direction1\]) of $\mathcal{N}_{x}$. Then $\chi(N)\rightarrow x$ because given any $x$-neighbourhood $M\in\!\:_{\mathbb{D}}N$, it follows from $$M\preceq N\in\,{}_{\mathbb{D}}N\Longrightarrow\chi(N)=t\in N\subseteq M\label{Eqn: DirectedNet1}$$ that a point in any subset of $M$ is also in $M$; $\chi(N)$ is therefore eventually in every neighbourhood of $x$. \(3) This slightly more general form of the previous example provides a link between the complimentary concepts of nets and filters that is considered below. For a point $x\in X$, and $M,N\in\mathcal{N}_{x}$ with the corresponding directed set $_{\mathbb{D}}M_{s}$ of Eq. (\[Eqn: Directed\]) ordered by its natural order (\[Eqn: Direction2\]), the net $$\chi(M,s)\overset{\textrm{def}}=s\label{Eqn: Def: Net2}$$ converges to $x$ because, as in the previous example, for any given $(M,s)\in\:\!_{\mathbb{D}}N_{s}$, it follows from $$(M,s)\preceq(N,t)\in\!\:_{\mathbb{D}}M_{s}\Longrightarrow\chi(N,t)=t\in N\subseteq M\label{Eqn: DirectedNet2}$$ that $\chi(N,t)$ is eventually in every neighbourhood $M$ of $x$. The significance of the directed set $_{\mathbb{D}}N_{t}$ of Eq. (\[Eqn: Directed\]), as compared to $_{\mathbb{D}}N$, is evident from the net that it induces *without using the Axiom of Choice*: For a subset $A$ of $X$, the net $\chi(N,t)=t\in A$ indexed by the directed set $${\textstyle _{\mathbb{D}}N_{t}=\{(N,t)\!:(N\in\mathcal{N}_{x})(t\in N\bigcap A)\}}\label{Eqn: Closure_Directed}$$ under the direction of Eq. (\[Eqn: Direction2\]), converges to $x\in X$ with all such $x$ defining the closure $\textrm{Cl}(A)$ of $A$. Furthermore taking the directed set to be $${\textstyle _{\mathbb{D}}N_{t}=\{(N,t)\!:(N\in\mathcal{N}_{x})(t\in N\bigcap A-\{ x\})\}}\label{Eqn: Der_Directed}$$ which, unlike Eq. (\[Eqn: Closure\_Directed\]), excludes the point $x$ that may or may not be in the subset $A$ of $X$, induces the net $\chi(N,t)=t\in A-\{ x\}$ converging to $x\in X$, with the set of all such $x$ yielding the derived set $\textrm{Der}(A)$ of $A$. In contrast, Eq. (\[Eqn: Closure\_Directed\]) also includes the isolated points $t=x$ of $A$ so as to generate its closure. Observe how neighbourhoods of a point, which define convergence of nets and filters in a topological space $X$, double up here as index sets to yield a self-consistent tool for the description of convergence. As compared with sequences where, the index set is restricted to positive integers, the considerable freedom in the choice of directed sets as is abundantly borne out by the two preceding examples, is not without its associated drawbacks. Thus as a trade-off, the wide range of choice of the directed sets may imply that induction methods, so common in the analysis of sequences, need no longer apply to arbitrary nets. \(4) The non-convergent nets (actually these are sequences) \(a) $(1,-1,1,-1,\cdots)$ adheres at $1$ and $-1$ and \(b) $\begin{array}{ccl} x_{n} & = & {\displaystyle \left\{ \begin{array}{lcl} n & & \textrm{if }n\textrm{ is odd}\\ 1-1/(1+n) & & \textrm{if }n\textrm{ is even}\end{array},\right.}\end{array}$ adheres at $1$ for its even terms, but is unbounded in the odd terms.$\qquad\blacksquare$ A converging sequence or net is also adhering but, as examples (4) show, the converse is false. Nevertheless it is true, as again is evident from examples (4), that in a first countable space where sequences suffice, a sequence $(x_{n})$ adheres at $x$ iff some subsequence $(x_{n_{m}})_{m\in\mathbb{N}}$ of $(x_{n})$ converges to $x$. If the space is not first countable this has a corresponding equivalent formulation for nets with subnets replacing subsequences as follows. Let $(\chi(\alpha))_{\alpha\in\mathbb{D}}$ be a net. A *subnet* of $\chi(\alpha)$ is the net $\zeta(\beta)=\chi(\sigma(\beta))$, $\beta\in\mathbb{E}$, where $\sigma\!:(\mathbb{E},\leq)\rightarrow(\mathbb{D},\preceq)$ is a function that captures the essence of the subsequential mapping $n\mapsto n_{m}$ in $\mathbb{N}$ by satisfying (SN1) $\sigma$ is an increasing order-preserving function: it respects the order of $\mathbb{E}$: $\sigma(\beta)\preceq\sigma(\beta^{\prime})$ for every $\beta\leq\beta^{\prime}\in\mathbb{E}$, and (SN2) For every $\alpha\in\mathbb{D}$ there exists a $\beta\in\mathbb{E}$ such that $\alpha\preceq\sigma(\beta)$. These generalize the essential properties of a subsequence in the sense that (1) Even though the index sets $\mathbb{D}$ and $\mathbb{E}$ may be different, it is necessary that the values of $\mathbb{E}$ be contained in $\mathbb{D}$, and (2) There are arbitrarily large $\alpha\in\mathbb{D}$ such that $\chi(\alpha=\sigma(\beta))$ is a value of the subnet $\zeta(\beta)$ for some $\beta\in\mathbb{E}$. Recalling the first of the order relations Eq. (\[Eqn: FunctionOrder\]) on $\textrm{Map}(X,Y)$, we will denote a subnet $\zeta$ of $\chi$ by $\zeta\preceq\chi$. We now consider the concept of filter on a set $X$ that is very useful in visualizing the behaviour of sequences and nets, and in fact filters constitute an alternate way of looking at convergence questions in topological spaces. A filter $\mathcal{F}$ on a set $X$ is a collection of *nonempty* subsets of $X$ satisfying properties $(\textrm{F}1)-(\textrm{F}3)$ below that are simply those of a neighbourhood system $\mathcal{N}_{x}$ without specification of the reference point $x$. (F1) The empty set $\emptyset$ does not belong to $\mathcal{F}$, (F2) The intersection of any two members of a filter is another member of the filter: $F_{1},F_{2}\in\mathcal{F}\Rightarrow F_{1}\bigcap F_{2}\in\mathcal{F}$, (F3) Every superset of **a member of a filter belongs to the filter: $(F\in\mathcal{F})\wedge(F\subseteq G)\Rightarrow G\in\mathcal{F}$; in particular $X\in\mathcal{F}$. **Example A1.4.** (1) The *indiscrete filter* is the smallest filter on $X$. \(2) The neighbourhood system $\mathcal{N}_{x}$ is the important *neighbourhood filter at $x$ on $X$,* and any local base at $x$ is also a filter-base for $\mathcal{N}_{x}$. In general for any subset $A$ of $X$, $\{ N\subseteq X\!:A\subseteq\textrm{Int}(N)\}$ is a filter on $X$ at $A$. \(3) All subsets of $X$ containing a point $x\in X$ is the *principal filter* $_{\textrm{F}}\mathcal{P}(x)$ *on $X$ at $x$.* More generally, if $\mathcal{F}$ consists of all supersets of a *nonempty* subset $A$ of $X$, then $\mathcal{F}$ is the *principal filter* $_{\textrm{F}}\mathcal{P}(A)=\{ N\subseteq X\!:A\subseteq\textrm{Int}(N)\}$ *at $A$. By adjoining the empty set to this filter give the $p$-inclusion and $A$-inclusion topologies on $X$ respectively.* The single element sets $\{\{ x\}\}$ and $\{ A\}$ are particularly simple examples of filter-bases that generate the principal filters at $x$ and $A$. \(4) For an uncountable (resp. infinite) set $X$, all cocountable (resp. cofinite) subsets of $X$ constitute the *cocountable* (resp. *cofinite* or *Frechet*) filter on $X$. Again, adding to these filters the empty set gives the respective topologies.$\qquad\blacksquare$ Like the topological and local bases $_{\textrm{T}}\mathcal{B}$ and $\mathcal{B}_{x}$ respectively, a subclass of $\mathcal{F}$ may be used to define a filter-base $_{\textrm{F}}\mathcal{B}$ that in turn generate $\mathcal{F}$ on $X$, just as it is possible to define the concepts of limit and adherence sets for a filter to parallel those for nets that follow straightforwardly from Def. A1.7, taken with Def. A1.11. **Definition A1.8.** *Let $(X,\mathcal{T})$ be a topological space and $\mathcal{F}$ a filter on $X$. Then*$$\textrm{lim}(\mathcal{F})=\{ x\in X\!:(\forall N\in\mathcal{N}_{x})(\exists F\in\mathcal{F})(F\subseteq N)\}\label{Eqn: lim filter}$$ and $${\textstyle \textrm{adh}(\mathcal{F})=\{ x\in X\!:(\forall N\in\mathcal{N}_{x})(\forall F\in\mathcal{F})(F\bigcap N\neq\emptyset)\}}\label{Eqn: adh filter}$$ *are respectively the sets of* *limit points* *and* *adherent* *points* *of $\mathcal{F}$*[^28]*.$\qquad\square$* A comparison of Eqs. (\[Eqn: lim net\]) and (\[Eqn: adh net2\]) with Eqs. (\[Eqn: lim filter\]) and (\[Eqn: adh filter\]) respectively demonstrate their formal similarity; this inter-relation between filters and nets will be made precise in Definitions A1.10 and A1.11 below. It should be clear from the preceding two equations that $$\textrm{lim}(\mathcal{F})\subseteq\textrm{adh}(\mathcal{F}),\label{Eqn: lim/adh(fil)}$$ with a similar result $$\textrm{lim}(\chi)\subseteq\textrm{adh}(\chi)\label{Eqn: lim/adh(net)}$$ holding for nets because of the duality between nets and filters as displayed by Defs. A1.9 and A1.10 below, with the equality in Eqs. (\[Eqn: lim/adh(fil)\]) and (\[Eqn: lim/adh(net)\]) being true (but not characterizing) for ultrafilters and ultranets respectively, see Example 4.2(3) for an account of this notion . It should be clear from the equations of Definition A1.8 that $$\textrm{adh}(\mathcal{F})=\{ x\in X\!:(\exists\textrm{ a finer filter }\mathcal{G}\supseteq\mathcal{F}\textrm{ on }X)\textrm{ }(\mathcal{G}\rightarrow x)\}\label{Eqn: filter adh}$$ consists of all the points of $X$ to which some finer filter $\mathcal{G}$ (in the sense that $\mathcal{F}\subseteq\mathcal{G}$ implies every element of $\mathcal{F}$ is also in $\mathcal{G}$) converges in $X$; thus $${\textstyle \textrm{adh}(\mathcal{F})=\bigcup\lim(\mathcal{G}\!:\mathcal{G}\supseteq\mathcal{F}),}$$ which corresponds to the net-result of Theorem A1.5 below, that a net *$\chi$* adheres at *$x$* iff there is some subnet of *$\chi$* that converges to *$x$* in *$X$*. Thus if $\zeta\preceq\chi$ is a subnet of $\chi$ and $\mathcal{F}\subseteq\mathcal{G}$ is a filter coarser than $\mathcal{G}$ then $$\begin{aligned} \lim(\chi)\subseteq\lim(\zeta) & & \lim(\mathcal{F})\subseteq\lim(\mathcal{G})\\ \textrm{adh}(\zeta)\subseteq\textrm{adh}(\chi) & & \textrm{adh}(\mathcal{G})\subseteq\textrm{adh}(\mathcal{F});\end{aligned}$$ a filter $\mathcal{G}$ finer than a given filter $\mathcal{F}$ corresponds to a subnet $\zeta$ of a given net $\chi$. The implication of this correspondence should be clear from the association between nets and filters contained in Definitions A1.10 and A1.11. A filter-base in $X$ is a *nonempty* family $(B_{\alpha})_{\alpha\in\mathbb{D}}=\!\,_{\textrm{F}}\mathcal{B}$ of subsets of $X$ characterized by (FB1) There are no empty sets in the collection $_{\textrm{F}}\mathcal{B}$: $(\forall\alpha\in\mathbb{D})(B_{\alpha}\neq\emptyset)$ (FB2) The intersection of any two members of **$_{\textrm{F}}\mathcal{B}$ **contains another member of $_{\textrm{F}}\mathcal{B}$: $B_{\alpha},B_{\beta}\in\,_{\textrm{F}}\mathcal{B}\Rightarrow(\exists B\in\,_{\textrm{F}}\mathcal{B}\!:B\subseteq B_{\alpha}\bigcap B_{\beta})$; hence any class of subsets of $X$ that does not contain the empty set and is closed under finite intersections is a base for a unique filter on $X$; compare the properties (NB1) and (NB2) of a local basis given at the beginning of this Appendix. Similar to Def. A1.1 for the local base, it is possible to define **Definition A1.9.** *A filter-base* $_{\textrm{F}}\mathcal{B}$ *in a set $X$ is a subcollection of the filter* $\mathcal{F}$ *on $X$ having the property that each $F\in\mathcal{F}$ contains some member of* $_{\textrm{F}}\mathcal{B}$*.* *Thus* $$_{\textrm{F}}\mathcal{B}\overset{\textrm{def}}=\{ B\in\mathcal{F}\!:B\subseteq F\textrm{ for each }F\in\mathcal{F}\}\label{Eqn: FB}$$ *determines the filter* $$\mathcal{F}=\{ F\subseteq X\!:B\subseteq F\textrm{ for some }B\textrm{ }\in\!\,_{\textrm{F}}\mathcal{B}\}\label{Eqn: filter_base}$$ *reciprocally as all supersets of the basic elements.$\qquad\square$* This is the smallest filter on $X$ that contains $_{\textrm{F}}\mathcal{B}$ and is said to be *the filter generated by its filter-base* $_{\textrm{F}}\mathcal{B}$; alternatively $_{\textrm{F}}\mathcal{B}$ is the filter-base of $\mathcal{F}$. The entire neighbourhood system $\mathcal{N}_{x}$, the local base $\mathcal{B}_{x}$, $\mathcal{N}_{x}\bigcap A$ for $x\in\textrm{Cl}(A)$, and the set of all residuals of a directed set $\mathbb{D}$ are among the most useful examples of filter-bases on $X$, $A$ and $\mathbb{D}$ respectively. Of course, every filter is trivially a filter-base of itself, and *the singletons $\{\{ x\}\}$, $\{ A\}$ are filter-bases that generate the principal filters $_{\textrm{F}}\mathcal{P}(x)$ and $_{\textrm{F}}\mathcal{P}(A)$ at $x$, and $A$ respectively*. Paralleling the case of topological subbase $_{\textrm{T}}\mathcal{S}$, a filter subbase $_{\textrm{F}}\mathcal{S}$ can be defined on $X$ to be any collection of subsets of $X$ *with the finite intersection property* (as compared with $_{\textrm{T}}\mathcal{S}$ where no such condition was necessary, this represents the fundamental point of departure between topology and filter) and it is not difficult to deduce that the filter generated by **$_{\textrm{F}}\mathcal{S}$ on $X$ is obtained by taking all finite intersections $_{\textrm{F}}\mathcal{S}_{\wedge}$ of members of $_{\textrm{F}}\mathcal{S}$ followed by their supersets $_{\textrm{F}}\mathcal{S}_{\Sigma\wedge}$. $\mathcal{F}(_{\textrm{F}}\mathcal{S}):=\,_{\textrm{F}}\mathcal{S}_{\Sigma\wedge}$ is the smallest filter on $X$ that contains $_{\textrm{F}}\mathcal{S}$ and is the filter *generated by* $_{\textrm{F}}\mathcal{S}$. Equation (\[Eqn: adh filter\]) can be put in the more useful and transparent form given by **Theorem A1.3.** *For a filter $\mathcal{F}$ in a space $(X,\mathcal{T})$* $$\begin{aligned} {\displaystyle \textrm{adh}(\mathcal{F})} & = & {\displaystyle \bigcap_{F\in\mathcal{F}}\textrm{Cl}(F)}\label{Eqn: filter adh*}\\ & = & {\displaystyle \bigcap_{B\in\,_{\textrm{F}}\mathcal{B}}\textrm{Cl}(B)},\nonumber \end{aligned}$$ *and dually* $\textrm{adh}(\chi)$, *are closed set*s.$\qquad\square$ **Proof.** Follows immediately from the definitions for the closure of a set Eq. (\[Eqn: Def: Closure\]) and the adherence of a filter Eq. (\[Eqn: adh filter\]). As always, it is a matter of convenience in using the basic filters **$_{\textrm{F}}\mathcal{B}$** instead of $\mathcal{F}$ to generate the adherence set.$\qquad\blacksquare$ It is infact true that the limit sets $\lim(\mathcal{F})$ and $\lim(\chi)$ are also closed set of $X$; the arguments involving ultrafilters are omitted. Similar to the notion of the adherence set of a filter is its *core —* a concept that unlike the adherence, is purely set-theoretic being the infimum of the filter and is not linked with any topological structure of the underlying (infinite) set $X$ — defined as $${\displaystyle \textrm{core }(\mathcal{F})=\bigcap_{F\in\mathcal{F}}F.}\label{Eqn: core}$$ From Theorem A1.3 and the fact that the closure of a set $A$ is the smallest closed set that contains $A$, see Eq. (\[Eqn: closure\]) at the end of Tutorial4, it is clear that in terms of filters$$\begin{aligned} A & = & \textrm{core}(\,_{\textrm{F}}\mathcal{P}(A))\nonumber \\ \textrm{Cl}(A) & = & \textrm{adh}(\,_{\textrm{F}}\mathcal{P}(A))\label{Eqn: PrinFil_Cl(A)}\\ & = & \textrm{core}(\textrm{Cl}(\,_{\textrm{F}}\mathcal{P}(A)))\nonumber \end{aligned}$$ where $_{\textrm{F}}\mathcal{P}(A)$ is the principal filter at $A$; thus *the core and adherence sets of the principal filter at $A$ are equal respectively to $A$ and* $\textrm{Cl}(A)$ *—* a classic example of equality in the general relation $\textrm{Cl}(\bigcap A_{\alpha})\subseteq\bigcap\textrm{Cl}(A_{\alpha})$ — but both are empty, for example, in the case of an infinitely decreasing family of rationals centered at any irrational (leading to a principal filter-base of rationals at the chosen irrational). This is an important example demonstrating that *the infinite intersection of a non-empty family of (closed) sets with the finite intersection property may be empty,* *a situation that cannot arise on a finite set or an infinite compact set*. Filters on $X$ with an empty core are said to be *free,* and are *fixed* otherwise: notice that by its very definition filters cannot be free on a finite set, and a free filter represents an additional feature that may arise in passing from finite to infinite sets. Clearly $(\textrm{adh}(\mathcal{F})=\emptyset)\Rightarrow(\textrm{core}(\mathcal{F})=\emptyset)$, but as the important example of the rational space in the reals illustrate, the converse need not be true. Another example of a free filter of the same type is provided by the filter-base $\{[a,\infty)\!:a\in\mathbb{R}\}$ in $\mathbb{R}$. Both these examples illustrate the important property that *a filter is free iff it contains the cofinite filter,* and the cofinite filter is the smallest possible free filter on an infinite set. The free cofinite filter, as these examples illustrate, may be typically generated as follows. Let $A$ be a subset of $X$, $x\in\textrm{Bdy}_{X-A}(A)$, and consider the directed set Eq. (\[Eqn: Closure\_Directed\]) to generate the corresponding net in $A$ given by $\chi(N\in\mathcal{N}_{x},t)=t\in A$. Quite clearly, the core of any Frechet filter based on this net must be empty as the point $x$ does not lie in $A$. In general, the intersection is empty because if it were not so then the complement of the intersection — which is an element of the filter — would be infinite in contravention of the hypothesis that the filter is Frechet. It should be clear that every filter finer than a free filter is also free, and any filter coarser than a fixed filter is fixed. Nets and filters are complimentary concepts and one may switch from one to the other as follows. **Definition A1.10.** *Let $\mathcal{F}$ be a filter on $X$ and let $_{\mathbb{D}}F_{x}=\{(F,x)\!:(F\in\mathcal{F})(x\in F)\}$ be a directed set with its natural direction $(F,x)\preceq(G,y)\Rightarrow(G\subseteq F)$. The net $\chi_{\mathcal{F}}\,\!:\,_{\mathbb{D}}F_{x}\rightarrow X$ defined by* $$\chi_{\mathcal{F}}(F,x)=x$$ *is said to be* *associated with* *the filter* *$\mathcal{F}$, see Eq. (\[Eqn: DirectedNet2\]).$\qquad\square$* **Definition A1.11.** *Let $\chi\!:\mathbb{D}\rightarrow X$ be a net and $\mathbb{R}_{\alpha}=\{\beta\in\mathbb{D}\!:\beta\succeq\alpha\in\mathbb{D}\}$ a residual in $\mathbb{D}$. Then* $$_{\textrm{F}}\mathcal{B}_{\chi}\overset{\textrm{def}}=\{\chi(\mathbb{R}_{\alpha})\!:\textrm{Res}(\mathbb{D})\rightarrow X\textrm{ for all }\alpha\in\mathbb{D}\}$$ *is the* *filter-base associated with* *$\chi$, and the corresponding filter $\mathcal{F}_{\chi}$ obtained by taking all supersets of the elements of* $_{\textrm{F}}\mathcal{B}_{\chi}$ *is the* *filter* *associated with* *$\chi$.$\qquad\square$* $_{\textrm{F}}\mathcal{B}_{\chi}$ is a filter-base in $X$ because $\chi(\bigcap\mathbb{R}_{\alpha})\subseteq\bigcap\chi(\mathbb{R}_{\alpha})$, that holds for any functional relation, proves (FB2). It is not difficult to verify that \(i) $\chi$ is eventually in $A\Longrightarrow A\in\mathcal{F}_{\chi}$, and \(ii) $\chi$ is frequently in $A\Longrightarrow(\forall\mathbb{R}_{\alpha}\in\textrm{Res}(\mathbb{D}))(A\bigcap\chi(\mathbb{R}_{\alpha})\neq\emptyset)$ $\Longrightarrow A\bigcap\mathcal{F}_{\chi}\neq\emptyset$ . Limits and adherences are obviously preserved in switching between nets (respectively, filters) and the filters (respectively, nets) that they generate: $$\begin{aligned} \lim(\chi)=\lim(\mathcal{F}_{\chi}), & & \textrm{adh}(\chi)=\textrm{adh}(\mathcal{F}_{\chi})\label{Eqn: net-fil}\\ \lim(\mathcal{F})=\lim(\chi_{\mathcal{F}}), & & \textrm{adh}(\mathcal{F})=\textrm{adh}(\chi_{\mathcal{F}}).\label{Eqn: fil-net}\end{aligned}$$ The proofs of the two parts of Eq. (\[Eqn: net-fil\]), for example, go respectively as follows. $x\in\lim(\chi)\Leftrightarrow\chi\textrm{ is eventually in }\mathcal{N}_{x}\Leftrightarrow(\forall N\in\mathcal{N}_{x})(\exists F\in\mathcal{F}_{\chi})\textrm{ such that }(F\subseteq N)\Leftrightarrow x\in\lim(\mathcal{F}_{\chi})$, and $x\in\textrm{adh}(\chi)\Leftrightarrow\chi\textrm{ is frequently in }\mathcal{N}_{x}\Leftrightarrow(\forall N\in\mathcal{N}_{x})(\forall F\in\mathcal{F}_{\chi})\textrm{ }(N\bigcap F\neq\emptyset)\Leftrightarrow x\in\textrm{adh}(\mathcal{F}_{\chi})$; here $F$ is a superset of $\chi(\mathbb{R}_{\alpha})$. Some examples of convergence of filters are \(1) Any filter on an indiscrete space $X$ converges to every point of $X$. \(2) Any filter on a space that coincides with its topology (minus the empty set, of course) converges to every point of the space. \(3) For each $x\in X$, the neighbourhood filter $\mathcal{N}_{x}$ converges to $x$; this is the smallest filter on $X$ that converges to $x$. \(4) The *indiscrete* filter $\mathcal{F}=\{ X\}$ converges to no point in the space $(X,\{\emptyset,A,X-A,X\})$, but converges to every point of $X-A$ if $X$ has the topology $\{\emptyset,A,X\}$ because the only neighbourhood of any point in $X-A$ is $X$ which is contained in the filter. One of the most significant consequences of convergence theory of sequences and nets, as shown by the two theorems and the corollary following, is that this can be used to describe the topology of a set. The proofs of the theorems also illustrate the close inter-relationship between nets and filters. **Theorem A1.4.** *For a subset $A$ of a topological space $X$,* $$\textrm{Cl}(A)=\{ x\in X\!:(\exists\textrm{ a net }\chi\textrm{ in }A)\textrm{ }(\chi\rightarrow x)\}.\qquad\square\label{Eqn: net closure}$$ **Proof.** *Necessity.* For **$x\in\textrm{Cl}(A)$, construct a **net **$\chi\rightarrow x$ in *$A$* as **follows. Let $\mathcal{B}_{x}$ be a topological local base at $x$, which by definition is the collection of all open sets of $X$ containing $x$. For each $\beta\in\mathbb{D}$, the sets $$N_{\beta}=\bigcap_{\alpha\preceq\beta}\{ B_{\alpha}\!:B_{\alpha}\in\mathcal{B}_{x}\}$$ form a nested decreasing local neighbourhood filter base at $x$. With respect to the directed set $_{\mathbb{D}}N_{\beta}=\{(N_{\beta},\beta)\!:(\beta\in\mathbb{D})(x_{\beta}\in N_{\beta})\}$ of Eq. (\[Eqn: DirectedIndexed\]), define the desired net in $A$ by $${\textstyle \chi(N_{\beta},\beta)=x_{\beta}\in N_{\beta}\bigcap A}$$ where the family of nonempty decreasing subsets $N_{\beta}\bigcap A$ of $X$ constitute the filter-base in $A$ as required by the directed set $_{\mathbb{D}}N_{\beta}$. It now follows from Eq. (\[Eqn: DirectionIndexed\]) and the arguments in Example A1.3(3) that $x_{\beta}\rightarrow x$; compare the directed set of Eq. (\[Eqn: Closure\_Directed\]) for a more compact, yet essentially identical, argument. Carefully observe the dual roles of $\mathcal{N}_{x}$ as a neighbourhood filter base at $x$. *Sufficiency.* Let $\chi$ be a net in $A$ that converges to $x\in X$. For any $N_{\alpha}\in\mathcal{N}_{x}$, there is a $\mathbb{R}_{\alpha}\in\textrm{Res}(\mathbb{D})$ of Eq. (\[Eqn: residual\]) such that $\chi(\mathbb{R}_{\alpha})\subseteq N_{\alpha}$. Hence the point $\chi(\alpha)=x_{\alpha}$ of $A$ belongs to $N_{\alpha}$ so that $A\bigcap N_{\alpha}\neq\emptyset$ which means, from Eq. (\[Eqn: Def: Closure\]), that $x\in\textrm{Cl}(A)$.$\qquad\blacksquare$ **Corollary.** Together with Eqs. (\[Eqn: Def: Closure\]) and (\[Eqn: Def: Derived\]), is follows that $$\textrm{Der}(A)=\{ x\in X\!:(\exists\textrm{ a net }\zeta\textrm{ in }A-\{ x\})(\zeta\rightarrow x)\}\qquad\square\label{Eqn: net derived}$$ The filter forms of Eqs. (\[Eqn: net closure\]) and (\[Eqn: net derived\]) $$\begin{aligned} \textrm{Cl}(A) & = & \{ x\in X\!:(\exists\textrm{ a filter }\mathcal{F}\textrm{ on }X)(A\in\mathcal{F})(\mathcal{F}\rightarrow x)\}\label{Eqn: filter cls_der}\\ \textrm{Der}(A) & = & \{ x\in X\!:(\exists\textrm{ a filter }\mathcal{F}\textrm{ on }X)(A-\{ x\}\in\mathcal{F})(\mathcal{F}\rightarrow x)\}\nonumber \end{aligned}$$ then follows from Eq. (\[Eqn: Def: LimFilter\]) and the finite intersection property (F2) of $\mathcal{F}$ so that every neighbourhood of $x$ must intersect $A$ (respectively $A-\{ x\}$) in Eq. (\[Eqn: filter cls\_der\]) to produce the converging net needed in the proof of Theorem A1.3. We end this discussion of convergence in topological spaces with a proof of the following theorem which demonstrates the relationship that “eventually in” and “frequently in” bears with each other; Eq. (\[Eqn: net adh\]) below is the net-counterpart of the filter equation (\[Eqn: filter adh\]). **Theorem A1.5.** *If $\chi$ is a net in a topological space $X$, then* $x\in\textrm{adh}(\chi)$ *iff some subnet $\zeta(\beta)=\chi(\sigma(\beta))$ of $\chi(\alpha)$, with $\alpha\in\mathbb{D}$ and $\beta\in\mathbb{E}$ , converges in $X$ to $x$; thus* $$\textrm{adh}(\chi)=\{ x\in X\!:(\exists\textrm{ a subnet }\zeta\preceq\chi\textrm{ in }X)(\zeta\rightarrow x)\}.\qquad\square\label{Eqn: net adh}$$ ****Proof.** *Necessity.* Let $x\in\textrm{adh}(\chi)$. Define a subnet function $\sigma\!:\,_{\mathbb{D}}N_{\alpha}\rightarrow\mathbb{D}$ by $\sigma(N_{\alpha},\alpha)=\alpha$ where $_{\mathbb{D}}N_{\alpha}$ is the directed set of Eq. (\[Eqn: DirectedIndexed\]): (SN1) and (SN2) are quite evidently satisfied according to Eq. (\[Eqn: DirectionIndexed\]). Proceeding as in the proof of the preceding theorem it follows that $x_{\beta}=\chi(\sigma(N_{\alpha},\alpha))=\zeta(N_{\alpha},\alpha)\rightarrow x$ is the required converging subnet that exists from Eq. (\[Eqn: adh net1\]) and the fact that $\chi(\mathbb{R}_{\alpha})\bigcap N_{\alpha}\neq\emptyset$ for every $N_{\alpha}\in\mathcal{N}_{x}$, by hypothesis. *Sufficiency.* Assume now that $\chi$ has a subnet $\zeta(N_{\alpha},\alpha)$ that converges to $x$. If $\chi$ does not adhere at $x$, there is a neighbourhood $N_{\alpha}$ of $x$ not frequented by it, in which case $\chi$ must be eventually in $X-N_{\alpha}$. Then $\zeta(N_{\alpha},\alpha)$ is also eventually in $X-N_{\alpha}$ so that $\zeta$ cannot be eventually in $N_{\alpha}$, a contradiction of the hypothesis that $\zeta(N_{\alpha},\alpha)\rightarrow x$.[^29]$\qquad\blacksquare$ Eqs. (\[Eqn: net closure\]) and (\[Eqn: net adh\]) imply that the closure of a subset $A$ of $X$ is the class of $X$-adherences of all the (sub)nets of $X$ that are eventually in $A$. This includes both the constant nets yielding the isolated points of $A$ and the non-constant nets leading to the cluster points of $A$, and implies the following physically useful relationship between convergence and topology that can be used as defining criteria for open and closed sets having a more appealing physical significance than the original definitions of these terms. Clearly, the term “net” is justifiably used here to include the subnets too. The following corollary of Theorem A1.5 summarizes the basic topological properties of sets in terms of nets (respectively, filters). **Corollary.** Let $A$ be a subset of a topological space $X$. Then \(1) $A$ is closed in $X$ iff every convergent net of $X$ that is eventually in $A$ actually converges to a point in $A$ (respectively, iff the adhering points of each filter-base on $A$ all belong to $A$). Thus no $X$-convergent net in a closed subset may converge to a point outside it. \(2) $A$ is open in $X$ iff every convergent net of $X$ that converges to a point in $A$ is eventually in $A$. Thus no $X$-convergent net outside an open subset may converge to a point in the set. \(3) $A$ is closed-and-open (clopen) in $X$ iff every convergent net of $X$ that converges in $A$ is eventually in $A$ and conversely. \(4) $x\in\textrm{Der}(A)$ iff some net (respectively, filter-base) in $A-\{ x\}$ converges to $x$; this clearly eliminates the isolated points of $A$ and $x\in\textrm{Cl}(A)$ iff some net (respectively, filter-base) in $A$ converges to $x$.$\qquad\square$ **Remark.** The differences in these characterizations should be fully appreciated: If we consider the cluster points $\textrm{Der}(A)$ of a net $\chi$ in $A$ as the *resource generated by* $\chi$, then a closed subset of $X$ can be considered to be *selfish* as its keeps all its resource to itself: $\textrm{Der}(A)\cap A=\textrm{Der}(A)$. The opposite of this is a *donor* set that donates all its generated resources to its neighbour: $\textrm{Der}(A)\cap X-A=\textrm{Der}(A)$, while for a *neutral* set, both $\textrm{Der}(A)\cap A\neq\emptyset$ and $\textrm{Der}(A)\cap X-A\neq\emptyset$ implying that the convergence resources generated in $A$ and $X-A$ can be deposited only in the respective sets. The clopen sets (see diagram 2-2 of Fig. \[Fig: DerSets\]) are of some special interest as they are boundary less so that no net-resources can be generated in this case as any such limit are required to be simultaneously in the set and its complement. **Example A1.1, Continued.** This continuation ****of Example A1.2 illustrates how sequential convergence is inadequate in spaces that are not first countable like the uncountable set with cocountable topology. In this topology, a sequence can converge to a point $x$ in the space iff it has only a finite number of distinct terms, and is therefore eventually constant. Indeed, let the complement $$G\overset{\textrm{def}}=X-F,\qquad F=\{ x_{i}\!:x_{i}\neq x,\textrm{ }i\in\mathbb{N}\}$$ of the countably closed sequential set $F$ be an open neighbourhood of $x\in X$. Because a sequence $(x_{i})_{i\in\mathbb{N}}$ in $X$ converges to a point $x\in X$ iff it is eventually in *every* neighbourhood (including $G$) of $x$, the sequence represented by the set $F$ cannot converge to $x$ unless it is of the uncountable type[^30]$$(x_{0},x_{1},\cdots,x_{I},x_{I+1},x_{I+1},\cdots)\label{Eqn: cocount}$$ with only a finite number $I$ of distinct terms actually belonging to the closed sequential set $F=X-G$, and $x_{I+1}=x$. Note that as we are concerned only with the eventual behaviour of the sequence, we may discard all distinct terms from $G$ by considering them to be in $F$, and retain only the constant sequence $(x,x,\cdots)$ in $G$. In comparison with the cofinite case that was considered in Sec. 4, the entire countably infinite sequence can now lie outside a neighbourhood of $x$ thereby enforcing the eventual constancy of the sequence. This leads to a generalization of our earlier cofinite result in the sense that a cocountable filter on a cocountable space converges to every point in the space. It is now straightforward to verify that for a point $x_{0}$ in an uncountable cocountable space $X$ \(a) Even though no sequence in the open set $G=X-\{ x_{0}\}$ can converge to $x_{0}$, yet $x_{0}\in\textrm{Cl}(G)$ since the intersection of any (uncountable) open neighbourhood $U$ of $x_{0}$ with $G$, being an uncountable set, is not empty. \(b) By corollary (1) of Theorem A1.5, the uncountable open set $G=X-\{ x_{0}\}$ is also closed in $X$ because if any sequence $(x_{1},x_{2},\cdots)$ in $G$ converges to some $x\in X$, then $x$ must be in $G$ as the sequence must be eventually constant in order for it to converge. But this is a contradiction as $G$ cannot be closed since it is not countable.[^31] By the same reckoning, although $\{ x_{0}\}$ is not an open set because its complement is not countable, nevertheless it follows from Eq. (\[Eqn: cocount\]) that should any sequence converge to the only point $x_{0}$ of this set, then it must eventually be in $\{ x_{0}\}$ so by corollary (2) of the same theorem, $\{ x_{0}\}$ becomes an open set. \(c) The identity map $\mathbf{1}\!:X\rightarrow X_{\textrm{d}}$, where $X_{\textrm{d}}$ is $X$ with discrete topology, is not continuous because the inverse image of any singleton of $X_{\textrm{d}}$ is not open in $X$. Yet if a sequence converges in $X$ to $x$, then its image $(\mathbf{1}(x))=(x)$ must actually converge to $x$ in $X_{\textrm{d}}$ because a sequence converges in a discrete space, as in the cofinite or cocountable spaces, iff it is eventually constant; this is so because each element of a discrete space being clopen is boundaryless. This pathological behaviour of sequences in a non Hausdorff, non first countable space does not arise if the discrete indexing set of sequences is replaced by a continuous, uncountable directed set like $\mathbb{R}$ for example, leading to nets in place of sequences. In this case the net can be in an open set without having to be constant-valued in order to converge to a point in it as the open set can be defined as the complement of a closed countable part of the uncountable net. The careful reader could not have failed to notice that the burden of the above arguments, as also of that in the example following Theorem 4.6, is to formalize the fact that since *a closed set is already defined as a countable (respectively finite) set,* the closure operation cannot add further points to it from its complement, and any sequence that converges in an open set in these topologies must necessarily be eventually constant at its point of convergence, a restriction that no longer applies to a net. The cocountable topology thus has the very interesting property of filtering out a countable part from an uncountable set, as for example the rationals in $\mathbb{R}$.$\qquad\blacksquare$ This example serves to illustrate the hard truth that in a space that is not first countable, the simplicity of sequences is not enough to describe its topological character, and in fact “sequential convergence will be able to describe only those topologies in which the number of (basic) neighbourhoods around each point is no greater than the number of terms in the sequences”, @Willard1970. It is important to appreciate the significance of this interplay of convergence of sequences and nets (and of continuity of functions of Appendix A1) and the topology of the underlying spaces. A comparison of the defining properties (T1), (T2), (T3) of topology $\mathcal{T}$ with (F1), (F2), (F3) of that of the filter $\mathcal{F}$, shows that a filter is very close to a topology with the main difference being with regard to the empty set which must always be in $\mathcal{T}$ but never in $\mathcal{F}$. Addition of the empty set to a filter yields a topology, but removal of the empty set from a topology need not produce the corresponding filter as the topology may contain nonintersecting sets. The distinction between the topological and filter-bases should be carefully noted. Thus \(a) While the topological base may contain the empty set, a filter-base cannot. \(b) From a given topology, form a common base by dropping all basic open sets that do not intersect. Then a (coarser) topology can be generated from this base by taking all unions, and a filter by taking all supersets according to Eq. (\[Eqn: filter\_base\]). For any given filter this expression may be used to extract a subclass $_{\textrm{F}}\mathcal{B}$ as a base for $\mathcal{F}$. **A2. Initial and Final topology** The commutative diagram of Fig. \[Fig: GenInv\] contains four sub-diagrams $X-X_{\textrm{B}}-f(X)$, $Y-X_{\textrm{B}}-f(X)$, $X-X_{\textrm{B}}-Y$ and $X-f(X)-Y$. Of these, the first two are especially significant as they can be used to conveniently define the topologies on $X_{\textrm{B}}$ and $f(X)$ from those of $X$ and $Y$, so that $f_{\textrm{B}}$, $f_{\textrm{B}}^{-1}$ and $G$ have some desirable continuity properties; we recall that a function $f\!:X\rightarrow Y$ is continuous if inverse images of open sets of $Y$ are open in $X$. This simple notion of continuity needs refinement in order that topologies on $X_{\textrm{B}}$ and $f(X)$ be unambiguously defined from those of $X$ and $Y$, a requirement that leads to the concepts of the so-called *final* and *initial topologies.* To appreciate the significance of these new constructs, note that if $f\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$ is a continuous function, there may be open sets in $X$ that are not inverse images of open — or for that matter of any — subset of $Y$, just as it is possible for non-open subsets of $Y$ to contribute to $\mathcal{U}$. When the triple $\{\mathcal{U},f,\mathcal{V}\}$ are tuned in such a manner that these are impossible, the topologies so generated on $X$ and $Y$ are the initial and final topologies respectively; they are the smallest (coarsest) and largest (finest) topologies on $X$ and $Y$ that make $f\!:X\rightarrow Y$ continuous. It should be clear that every image and preimage continuous function is continuous, but the converse is not true. Let $\textrm{sat}(U):=f^{-}f(U)\subseteq X$ be the saturation of an open set $U$ of $X$ and $\textrm{comp}(V):=ff^{-}(V)=V\bigcap f(X)\in Y$ be the component of an open set $V$ of $Y$ on the range $f(X)$ of $f$. Let $\mathcal{U}_{\textrm{sat}}$, $\mathcal{V}_{\textrm{comp}}$ denote respectively the saturations $U_{\textrm{sat}}=\{\textrm{sat}(U)\!:U\in\mathcal{U}\}$ of the open sets of $X$ and the components $V_{\textrm{comp}}=\{\textrm{comp}(V)\!:V\in\mathcal{V}\}$ of the open sets of $Y$ whenever these are also open in $X$ and $Y$ respectively. Plainly, $\mathcal{U}_{\textrm{sat}}\subseteq\mathcal{U}$ and $\mathcal{V}_{\textrm{comp}}\subseteq\mathcal{V}$. **Definition A2.1.** *For a function* $e\!:X\rightarrow(Y,\mathcal{V})$, *the* *preimage* *or* *initial topology of* $X$ *based on (generated by)* **$e$** *and $\mathcal{V}$* *is* $$\textrm{IT}\{ e;\mathcal{V}\}\overset{\textrm{def}}=\{ U\subseteq X\!:U=e^{-}(V)\textrm{ if }V\in\mathcal{V}_{\textrm{comp}}\},\label{Eqn: IT}$$ *while for $q\!:(X,\mathcal{U})\rightarrow Y$, the* *image* *or* *final topology of* $Y$ *based on (generated by) $\mathcal{U}$ and* **$q$** *is* $$\textrm{FT}\{\mathcal{U};q\}\overset{\textrm{def}}=\{ V\subseteq Y\!:q^{-}(V)=U\textrm{ if }U\in\mathcal{U}_{\textrm{sat}}\}.\qquad\square\label{Eqn: FT'}$$ Thus, the topology of $(X,\textrm{IT}\{ e;\mathcal{V}\})$ consists of, and only of, the $e$-saturations of all the open sets of $e(X)$, while the open sets of $(Y,\textrm{FT}\{\mathcal{U};q\})$ are the $q$-images *in* $Y$ (and not just in $q(X)$) of all the $q$-saturated open sets of $X$.[^32] The need for defining (\[Eqn: IT\]) in terms of $\mathcal{V}_{\textrm{comp}}$ rather than $\mathcal{V}$ will become clear in the following. The subspace topology $\textrm{IT}\{ i;\mathcal{U}\}$ of a subset $A\subseteq(X,\mathcal{U})$ is a basic example of the initial topology **by the inclusion map $i\!:X\supseteq A\rightarrow(X,\mathcal{U})$, and we take its generalization $e\!:(A,\textrm{IT}\{ e;\mathcal{V}\})\rightarrow(Y,\mathcal{V})$ that embeds a subset $A$ of $X$ into $Y$ as the prototype of a preimage continuous map. Clearly the topology of $Y$ may also contain open sets not in $e(X)$, and any subset in $Y-e(X)$ may be added to the topology of $Y$ without altering the preimage topology of $X$: *open sets of $Y$ not in $e(X)$ may be neglected in obtaining the preimage topology* as $e^{-}(Y-e(X))=\emptyset$. The final topology on a quotient set by the quotient map $Q\!:(X,\mathcal{U})\rightarrow X/\sim$, which is just the collection of $Q$-images of the $Q$-saturated open sets of $X$, known as the *quotient topology of $X/\sim$,* is the basic example of the image topology and the resulting space $(X/\sim,\textrm{FT}\{\mathcal{U};Q\})$ is called the *quotient space.* We take the generalization $q\!:(X,\mathcal{U})\rightarrow(Y,\textrm{FT}\{\mathcal{U};q\})$ of $Q$ as the prototype of a image continuous function. The following results are specifically useful in dealing with initial and final topologies; compare the corresponding results for open maps given later. **Theorem A2.1.** *Let $(X,\mathcal{U})$ and $(Y_{1},\mathcal{V}_{1})$ be topological spaces and let $X_{1}$ be a set. If $f\!:X_{1}\rightarrow(Y_{1},\mathcal{V}_{1})$, $q\!:(X,\mathcal{U})\rightarrow X_{1}$, and $h=f\circ q\!:(X,\mathcal{U})\rightarrow(Y_{1},\mathcal{V}_{1})$ are functions with the topology $\mathcal{U}_{1}$ of $X_{1}$ given by* $\textrm{FT}\{\mathcal{U};q\}$, *then* \(a) *$f$ is continuous iff $h$ is continuous.* \(b) *$f$ is image continuous iff* $\mathcal{V}_{1}=\textrm{FT}\{\mathcal{U};h\}$.$\qquad\square$ **Theorem A2.2.** *Let $(Y,\mathcal{V})$ and $(X_{1},\mathcal{U}_{1})$ be topological spaces and let $Y_{1}$ be a set. If $f\!:(X_{1},\mathcal{U}_{1})\rightarrow Y_{1}$, $e\!:Y_{1}\rightarrow(Y,\mathcal{V})$ and $g=e\circ f\!:(X_{1},\mathcal{U}_{1})\rightarrow(Y,\mathcal{V})$ are function with the topology $\mathcal{V}_{1}$ of $Y_{1}$ given by* $\textrm{IT}\{ e;\mathcal{V}\}$*, then* \(a) *$f$ is continuous iff $g$ is continuous.* \(b) *$f$ is preimage continuous iff* $\mathcal{U}_{1}=\textrm{IT}\{ g;\mathcal{V}\}$.$\qquad\square$ As we need the second part of these theorems in our applications, their proofs are indicated below. The special significance of the first parts is that they ensure the converse of the usual result that the composition of two continuous functions is continuous, namely that one of the components of a composition is continuous whenever the composition is so. **Proof of Theorem A2.1.** If $f$ be image continuous, $\mathcal{V}_{1}=\{ V_{1}\subseteq Y_{1}\!:f^{-}(V_{1})\in\mathcal{U}_{1}\}$ and $\mathcal{U}_{1}=\{ U_{1}\subseteq X_{1}\!:q^{-}(U_{1})\in\mathcal{U}\}$ are the final topologies of $Y_{1}$ and $X_{1}$ based on the topologies of $X_{1}$ and $X$ respectively. Then $\mathcal{V}_{1}=\{ V_{1}\subseteq Y_{1}\!:q^{-}f^{-}(V_{1})\in\mathcal{U}\}$ shows that $h$ is image continuous. Conversely, when $h$ is image continuous, $\mathcal{V}_{1}=\{ V_{1}\subseteq Y_{1}\!:h^{-}(V_{1})\}\in\mathcal{U}\}=\{ V_{1}\subseteq Y_{1}\!:q^{-}f^{-}(V_{1})\}\in\mathcal{U}\}$, with $\mathcal{U}_{1}=\{ U_{1}\subseteq X_{1}\!:q^{-}(U_{1})\in\mathcal{U}\}$, proves $f^{-}(V_{1})$ to be open in $X_{1}$ and thereby $f$ to be image continuous. **Proof of Theorem A2.2.** If $f$ be preimage continuous, $\mathcal{V}_{1}=\{ V_{1}\subseteq Y_{1}\!:V_{1}=e^{-}(V)\textrm{ if }V\in\mathcal{V}\}$ and $\mathcal{U}_{1}=\{ U_{1}\subseteq X_{1}\!:U_{1}=f^{-}(V_{1})\textrm{ if }V_{1}\in\mathcal{V}_{1}\}$ are the initial topologies of $Y_{1}$ and $X_{1}$ respectively. Hence from $\mathcal{U}_{1}=\{ U_{1}\subseteq X_{1}\!:U_{1}=f^{-}e^{-}(V)\textrm{ if }V\in\mathcal{V}\}$ it follows that $g$ is preimage continuous. Conversely, when $g$ is preimage continuous, $\mathcal{U}_{1}=\{ U_{1}\subseteq X_{1}\!:U_{1}=g^{-}(V)\textrm{ if }V\in\mathcal{V}\textrm{ }\}=\{ U_{1}\subseteq X_{1}\!:U_{1}=f^{-}e^{-}(V)\textrm{ if }V\in\mathcal{V}\}$ and $\mathcal{V}_{1}=\{ V_{1}\subseteq Y_{1}\!:V_{1}=e^{-}(V)\textrm{ if }V\in\mathcal{V}\}$ shows that $f$ is preimage continuous.$\qquad\blacksquare$ Since both Eqs. (\[Eqn: IT\]) and (\[Eqn: FT’\]) are in terms of inverse images (the first of which constitutes a direct, and the second an inverse, problem) the image $f(U)=\textrm{comp}(V)$ for $V\in\mathcal{V}$ is of interest as it indicates the relationship of the openness of $f$ with its continuity. This, and other related concepts are examined below, where the range space $f(X)$ is always taken to be a subspace of $Y$. Openness of a function *$f\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$* is the “inverse” of continuity, when images of open sets of $X$ are required to be open in $Y$; such a function is said to be *open.* Following are two of the important properties of open functions. \(1) *If $f\!:(X,\mathcal{U})\rightarrow(Y,f(\mathcal{U}))$ is an open function, then so is* $f_{<}\!:(X,\mathcal{U})\rightarrow(f(X),\textrm{IT}\{ i;f(\mathcal{U})\})$*. The converse is true if $f(X)$ is an open set of $Y$; thus openness of* $f_{<}\!:(X,\mathcal{U})\rightarrow(f(X),f_{<}(\mathcal{U}))$ *implies tha*t *of $f\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$ whenever $f(X)$ is open in $Y$ such that $f_{<}(U)\in\mathcal{V}$ for $U\in\mathcal{U}$.* The truth of this last assertion follows easily from the fact that if $f_{<}(U)$ is an open set of $f(X)\subset Y$, then necessarily $f_{<}(U)=V\bigcap f(X)$ for some $V\in\mathcal{V}$, and the intersection of two open sets of $Y$ is again an open set of $Y$. \(2) *If $f\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$ and $g\!:(Y,\mathcal{V})\rightarrow(Z,\mathcal{W})$ are open functions then $g\circ f\!:(X,\mathcal{U})\rightarrow(Z,\mathcal{W})$* *is also open.* It follows that the condition in (1) on $f(X)$ can be replaced by the requirement that the inclusion $i\!:(f(X),\textrm{IT}\{ i;\mathcal{V}\})\rightarrow(Y,\mathcal{V})$ be an open map. This interchange of $f(X)$ with its inclusion $i\!:f(X)\rightarrow Y$ into $Y$ is a basic result that finds application in many situations. Collected below are some useful properties of the initial and final topologies that we need in this work. ***Initial Topology.*** In Fig. \[Fig: Initial-Final\](b), consider $Y_{1}=h(X_{1})$, $e\rightarrow i$ and $f\rightarrow h_{<}\!:X_{1}\rightarrow(h(X_{1}),\textrm{IT}\{ i;\mathcal{V}\})$. From $h^{-}(B)=h^{-}(B\bigcap h(X_{1}))$ for any $B\subseteq Y$, it follows that for an open set $V$ of $Y$, $h^{-}(V_{\textrm{comp}})=h^{-}(V)$ is an open set of $X_{1}$ which, if the topology of $X_{1}$ is $\textrm{IT}\{ h;\mathcal{V}\}$, are the only open sets of $X_{1}$. Because $V_{\textrm{comp}}$ is an open set of $h(X_{1})$ in its subspace topology, this implies that *the preimage topologies* $\textrm{IT}\{ h;\mathcal{V}\}$ and $\textrm{IT}\{ h_{<};\textrm{IT}\{ i;\mathcal{V}\}\}$ *of $X_{1}$ generated by $h$ and* $h_{<}$ *are the same.* Thus the preimage topology of $X_{1}$ is not affected if $Y$ is replaced by the subspace $h(X_{1})$, the part $Y-h(X_{1})$ contributing nothing to $\textrm{IT}\{ h;\mathcal{V}\}$. *A preimage continuous function* $e\!:X\rightarrow(Y,\mathcal{V})$ *is not necessarily an open function.* Indeed, if $U=e^{-}(V)\in\textrm{IT}\{ e;\mathcal{V}\}$, it is almost trivial to verify along the lines of the restriction of open maps to its range, that $e(U)=ee^{-}(V)=e(X)\bigcap V$, $V\in\mathcal{V}$, is open in $Y$ (implying that $e$ is an open map) iff $e(X)$ is an open subset of $Y$ (because finite intersections of open sets are open). A special case of this is the important consequence that *the restriction* $e_{<}\!:(X,\textrm{IT}\{ e;\mathcal{V}\})\rightarrow(e(X),\textrm{IT}\{ i;\mathcal{V}\})$ *of* $e\!:(X,\textrm{IT}\{ h;\mathcal{V}\})\rightarrow(Y,\mathcal{V})$ *to its range is an open map.* Even though a preimage continuous map need not be open, it is true that *an injective, continuous and open map $f\!:X\rightarrow(Y,\mathcal{V})$ is preimage continuous.* Indeed, from its injectivity and continuity, inverse images of all open subsets of $Y$ are saturated-open in $X$, and openness of $f$ ensures that these are the only open sets of $X$ the condition of injectivity being required to exclude non-saturated sets from the preimage topology. It is therefore possible to rewrite Eq. (\[Eqn: IT\]) as $$U\in\textrm{IT}\{ e;\mathcal{V}\}\Longleftrightarrow e(U)=V\textrm{ if }V\in\mathcal{V}_{\textrm{comp}},\label{Eqn: IT'}$$ and to compare it with the following criterion for an *injective, open-continuous* *map* $f\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$ that necessarily satisfies $\textrm{sat}(A)=A$ for all $A\subseteq X$ $$U\in\mathcal{U}\Longleftrightarrow(\{\{ f(U)\}_{U\in\mathcal{U}}=\mathcal{V}_{\textrm{comp}})\wedge(f^{-1}(V)|_{V\in\mathcal{V}}\in\mathcal{U}).\label{Eqn: OCINJ}$$ ***Final Topology.*** Since it is necessarily produced on the range $\mathcal{R}(q)$ of $q$, the final topology is often considered in terms of a surjection. This however is not necessary as, much in the spirit of the initial topology, $Y-q(X)\neq\emptyset$ inherits the discrete topology without altering anything, thereby allowing condition (\[Eqn: FT’\]) to be restated in the following more transparent form $$V\in\textrm{FT}\{\mathcal{U};q\}\Longleftrightarrow V=q(U)\textrm{ if }U\in\mathcal{U}_{\textrm{sat}},\label{Eqn: FT}$$ and to compare it with the following criterion for a *surjective, open-continuous* *map* $f\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$ that necessarily satisfies $_{f}B=B$ for all $B\subseteq Y$ $$V\in\mathcal{V}\Longleftrightarrow(\mathcal{U}_{\textrm{sat}}=\{ f^{-}(V)\}_{V\in\mathcal{V}})\wedge(f(U)|_{U\in\mathcal{U}}\in\mathcal{V}).\label{Eqn: OCSUR}$$ As may be anticipated from Fig. \[Fig: Initial-Final\], the final topology does not behave as well for subspaces as the initial topology does. This is so because in Fig. \[Fig: Initial-Final\](a) the two image continuous functions $h$ and $q$ are connected by a preimage continuous inclusion $f$, whereas in Fig. \[Fig: Initial-Final\](b) all the three functions are preimage continuous. Thus quite like open functions, although image continuity of $h\!:(X,\mathcal{U})\rightarrow(Y_{1},\textrm{FT}\{\mathcal{U};h\})$ implies that of $h_{<}\!:(X,\mathcal{U})\rightarrow(h(X),\textrm{IT}\{ i;\textrm{FT}\{\mathcal{U};h\}))$ for a subspace $h(X)$ of $Y_{1}$, the converse need not be true unless — entirely like open functions again — either $h(X)$ is an open set of $Y_{1}$ or $i\!:(h(X),\textrm{IT}\{ i;\textrm{FT}\{\mathcal{U};h\}))\rightarrow(X,\textrm{FT}\{\mathcal{U};h\})$ is an open map. Since an open preimage continuous map is image continuous, this makes $i\!:h(X)\rightarrow Y_{1}$ an ininal function and hence all the three legs of the commutative diagram image continuous. Like preimage continuity, *an image continuous function $q\!:(X,\mathcal{U})\rightarrow Y$ need not be open.* However, although *the restriction of an image continuous function to the saturated open sets of its domain is an open function*, $q$ is unrestrictedly open iff the saturation of every open set of $X$ is also open in $X$. Infact it can be verified without much effort that a continuous, open surjection is image continuous. Combining Eqs. (\[Eqn: IT’\]) and (\[Eqn: FT\]) gives the following criterion for ininality $$U\textrm{ and }V\in\textrm{IFT}\{\mathcal{U}_{\textrm{sat}};f;\mathcal{V}\}\Longleftrightarrow(\{ f(U)\}_{U\in\mathcal{U}_{\textrm{sat}}}=\mathcal{V})(\mathcal{U}_{\textrm{sat}}=\{ f^{-}(V)\}_{V\in\mathcal{V}}),\label{Eqn: INI}$$ which reduces to the following for a homeomorphism $f$ that satisfies both $\textrm{sat}(A)=A$ for $A\subseteq X$ and $_{f}B=B$ for $B\subseteq Y$ $$U\textrm{ and }V\in\textrm{HOM}\{\mathcal{U};f;\mathcal{V}\}\Longleftrightarrow(\mathcal{U}=\{ f^{-1}(V)\}_{V\in\mathcal{V}})(\{ f(U)\}_{U\in\mathcal{U}}=\mathcal{V})\label{Eqn: HOM}$$ and compares with $$\begin{gathered} U\textrm{ and }V\in\textrm{OC}\{\mathcal{U};f;\mathcal{V}\}\Longleftrightarrow(\textrm{sat}(U)\in\mathcal{U}\!:\{ f(U)\}_{U\in\mathcal{U}}=\mathcal{V}_{\textrm{comp}})\wedge\\ \wedge(\textrm{comp}(V)\in\mathcal{V}\!:\{ f^{-}(V)\}_{V\in\mathcal{V}}=\mathcal{U}_{\textrm{sat}})\label{Eqn: OC}\end{gathered}$$ for an open-continuous $f$. The following is a slightly more general form of the restriction on the inclusion that is needed for image continuity to behave well for subspaces of $Y$. **Theorem A2.3.** *Let* $q\!:(X,\mathcal{U})\rightarrow(Y,\textrm{FT}\{\mathcal{U};q\})$ *be an image continuous* *function. For a subspace* $B$ of $(Y,\textrm{FT}\{\mathcal{U};q\})$,$$\textrm{FT}\{\textrm{IT}\{ j;\mathcal{U}\};q_{<}\}=\textrm{IT}\{ i;\textrm{FT}\{\mathcal{U};q\}\}$$ *where* $q_{<}\!:(q^{-}(B),\textrm{IT}\{ j;\mathcal{U}\})\rightarrow(B,\textrm{FT}\{\textrm{IT}\{ j;\mathcal{U}\};q_{<}\})$, *if either $q$ is an* *open map or $B$ is an open set of* $Y$.$\qquad\square$ In summary we have the useful result that an open preimage continuous function is image continuous and an open image continuous function is preimage continuous, where the second assertion follows on neglecting non-saturated open sets in $X$; this is permitted in as far as the generation of the final topology is concerned, as these sets produce the same images as their saturations. Hence *an image continuous function* $q\!:X\rightarrow Y$ *is preimage continuous iff every open set in $X$ is saturated with respect to $q$,* and *a preimage continuous function* $e\!:X\rightarrow Y$ *is image continuous iff the $e$-image of every open set of $X$ is open in $Y$.* **A3. More on Topological Spaces** This Appendix — which completes the review of those concepts of topological spaces begun in Tutorial4 that are needed for a proper understanding of this work — begins with the following summary of the different possibilities in the distribution of $\textrm{Der}(A)$ and $\textrm{Bdy}(A)$ between sets $A\subseteq X$ and its complement $X-A$, and follows it up with a few other important topological concepts that have been used, explicitly or otherwise, in this work. **Definition A3.1.** ***Separation, Connected Space*.** *A* *separation* *(disconnection)* *of $X$ is a pair of mutually disjoint nonempty open (and therefore closed) subsets $H_{1}$ and $H_{2}$ such that $X=H_{1}\cup H_{2}$* *A space $X$ is said to be* *connected* *if it has no separation, that is if it cannot be partitioned into two open or two closed nonempty subsets. $X$ is* *separated (disconnected)* *if it is not connected.$\qquad\square$* It follows from the definition, that for a disconnected space $X$ the following are equivalent statements. \(a) There exist a pair of disjoint nonempty open subsets of $X$ that cover $X$. \(b) There exist a pair of disjoint nonempty closed subsets of $X$ that cover $X.$ \(c) There exist a pair of disjoint nonempty clopen subsets of $X$ that cover $X.$ \(d) There exists a nonempty, proper, clopen subset of $X$. By a *connected subset* is meant a subset of $X$ that is connected *when provided with its relative topology making it a subspace of $X$.* Thus any connected subset of a topological space must necessarily be contained in any clopen set that might intersect it: if $C$ and $H$ are respectively connected and clopen subsets of $X$ such that $C\bigcap H\neq\emptyset$, then $C\subset H$ because $C\bigcap H$ is a nonempty clopen set in $C$ which must contain $C$ because $C$ is connected. For testing whether a subset of a topological space is connected, the following relativized form of (a)$-$(d) is often useful. **Lemma A3.1.** *A subset $A$ of $X$ is disconnected iff there are disjoint open sets $U$ and $V$ of $X$ satisfying* $${\textstyle U\bigcap A\neq\emptyset\neq V\bigcap A\textrm{ such that }A\subseteq U\bigcup V,\;\textrm{with }U\bigcap V\bigcap A=\emptyset}\label{Eqn: SubDisconnect1}$$ *or there are disjoint closed sets $E$ and $F$ of $X$ satisfying* $${\textstyle E\bigcap A\neq\emptyset\neq F\bigcap A\textrm{ such that }A\subseteq E\bigcup F,\;\textrm{with }E\bigcap F\bigcap A=\emptyset.}\label{Eqn: SybDisconnect2}$$ *Thus $A$ is disconnected iff there are disjoint clopen subsets in the relative topology of $A$ that cover $A$.$\qquad\square$* **Lemma A3.2.** *If $A$ is a subspace of $X$, a* *separation of* *$A$ is a pair of disjoint nonempty subsets $H_{1}$ and $H_{2}$ of $A$ whose union is $A$ neither of which contains a cluster point of the other. $A$ is connected iff there is no separation of $A.$* *$\qquad\square$* **Proof.** Let $H_{1}$ and $H_{2}$ be a separation of $A$ so that they are clopen subsets of $A$ whose union is $A$. As $H_{1}$ is a closed subset of $A$ it follows that $H_{1}=\textrm{Cl}_{X}(H_{1})\bigcap A$, where $\textrm{Cl}_{X}(H_{1})\bigcap A$ is the closure of $H_{1}$ in $A$; hence $\textrm{Cl}_{X}(H_{1})\bigcap H_{2}=\emptyset$. But as the closure of a subset is the union of the set and its adherents, an empty intersection signifies that $H_{2}$ cannot contain any of the cluster points of $H_{1}$. A similar argument shows that $H_{1}$ does not contain any adherent of $H_{2}$. Conversely suppose that neither $H_{1}$ nor $H_{2}$ contain an adherent of the other: $\textrm{Cl}_{X}(H_{1})\bigcap H_{2}=\emptyset$ and $\textrm{Cl}_{X}(H_{2})\bigcap H_{1}=\emptyset$. Hence $\textrm{Cl}_{X}(H_{1})\bigcap A=H_{1}$ and $\textrm{Cl}_{X}(H_{2})\bigcap A=H_{2}$ so that both $H_{1}$ and $H_{2}$ are closed in $A.$ But since $H_{1}=A-H_{2}$ and $H_{2}=A-H_{1}$, they must also be open in the relative topology of $A$. *$\qquad\blacksquare$* Following are some useful properties of connected spaces. (c1) The closure of any connected subspace of a space is connected. In general, every $B$ satisfying $$A\subseteq B\subseteq\textrm{Cl}(A)$$ is connected. Thus any subset of $X$ formed from $A$ by adjoining to it some or all of its adherents is connected so that *a topological space with a dense connected subset is connected.* (c2) The union of any class of connected subspaces of $X$ with nonempty intersection is a connected subspace of $X$. (c3) A topological space is connected iff there is a covering of the space consisting of connected sets with nonempty intersection. Connectedness is a topological property: Any space homeomorphic to a connected space is itself connected. (c4) If $H_{1}$ and $H_{2}$ is a separation of $X$ and $A$ is any connected subset $A$ of $X$, then either $A\subseteq H_{1}$ or $A\subseteq H_{2}$*.* While the real line $\mathbb{R}$ is connected, a subspace of $\mathbb{R}$ is connected iff it is an interval in $\mathbb{R}$. The important concept of total disconnectedness introduced below needs the following **Definition A3.2.** ***Component*.** *A* *component $C^{*}$* *of a space $X$ is a maximally* (with respect to inclusion) *connected subset of $X$.* *$\qquad\square$* Thus a component is a connected subspace which is not properly contained in any larger connected subspace of $X$. The maximal element need not be unique as there can be more than one component of a given space and a “maximal” criterion rather than “maximum” is used as the component need not contain every connected subsets of $X$; it simply must not be contained in any other connected subset of $X$. Components can be constructively defined as follows: Let $x\in X$ be any point. Consider the collection of all connected subsets of $X$ to which $x$ belongs Since $\{ x\}$ is one such set, the collection is nonempty. As the intersection of the collection is nonempty, its union is a nonempty connected set $C$. This the largest connected set containing $x$ and is therefore a component containing $x$ and we have (C1) Let $x\in X$. The unique component of $X$ containing $x$ is the union of all the connected subsets of $X$ that contain *$x$.* Conversely **any nonempty connected subset $A$ of $X$ **is contained in that unique component of $X$ to which each of the points of $A$ belong*.* Hence *a* *topological space is connected iff it is the unique component of itself.* (C2) Each component $C^{*}$ of $X$ is a closed set of $X$: By property (c1) above, $\textrm{Cl}(C^{*})$ is also connected and from $C^{*}\subseteq\textrm{Cl}(C^{*})$ it follows that $C^{*}=\textrm{Cl}(C^{*})$. Components need not be open sets of $X$: an example of this is the space of rationals $\mathbb{Q}$ in reals in which the components are the individual points which cannot be open in $\mathbb{R}$; see Example (2) below. (C3) Components of $X$ are equivalence classes of $(X,\sim)$ with $x\sim y$ iff they are in the same component: while reflexivity and symmetry are obvious enough, transitivity follows because if $x,y\in C_{1}$ and $y,z\in C_{2}$ with $C_{1}$, $C_{2}$ connected subsets of $X$, then $x$ and $z$ are in the set $C_{1}\bigcup C_{2}$ which is connected by property c(2) above as they have the point $y$ in common. Components are connected disjoint subsets of $X$ whose union is $X$ (that is they form a partition of $X$ with each point of $X$ contained in exactly one component of $X$) such that any connected subset of $X$ can be contained in only one of them. Because a connected subspace cannot contain in it any clopen subset of $X$, it follows that *every clopen connected subspace must be a component of $X$.* Even when a space is disconnected, it is always possible to decompose it into pairwise disjoint connected subsets. If $X$ is a discrete space this is the only way in which $X$ may be decomposed into connected pieces. If $X$ is not discrete, there may be other ways of doing this. For example, the space $$X=\{ x\in\mathbb{R}\!:(0\leq x\leq1)\vee(2<x<3)\}$$ has the following three distinct decomposition into connected subsets: $$\begin{array}{rcl} {\displaystyle X} & = & [0,1/2)\bigcup[1/2,1]\bigcup(2,7/3]\bigcup(7/3,3)\\ X & = & \{0\}\bigcup{\displaystyle \left(\bigcup_{n=1}^{\infty}\left(\frac{1}{n+1},\frac{1}{n}\right]\right)}\bigcup(2,3)\\ X & = & [0,1]\bigcup(2,3).\end{array}$$ Intuition tells us that only in the third of these decompositions have we really broken up $X$ into its connected pieces. What distinguishes the third from the other two is that neither of the pieces $[0,1]$ or $(2,3)$ can be enlarged into bigger connected subsets of $X$. As connected spaces, the empty set and the singleton are considered to be *degenerate* and any connected subspace with more than one point is *nondegenerate.* At the opposite extreme of the largest possible component of a space $X$ which is $X$ itself, are the singletons $\{ x\}$ for every $x\in X$. This leads to the extremely important notion of a **Definition A3.3.** ***Totally disconnected space.*** *A space $X$ is* *totally disconnected* *if every pair of distinct points in it can be separated by a disconnection of $X$.$\qquad\square$* $X$ is totally disconnected iff the components in $X$ are single points with the only nonempty connected subsets of $X$ being the one-point sets: If $x\neq y\in A\subseteq X$ are distinct points of a subset $A$ of $X$ then $A=(A\bigcap H_{1})\bigcup(A\bigcap H_{2})$, where $X=H_{1}\bigcup H_{2}$ with $x\in H_{1}$ and $y\in H_{2}$ is a disconnection of $X$ (it is possible to choose $H_{1}$ and $H_{2}$ in this manner because $X$ is assumed to be totally disconnected), is a separation of $A$ that demonstrates that any subspace of a totally disconnected space with more than one point is disconnected. A totally disconnected space has interesting physically appealing separation properties in terms of the (separated) Hausdorff spaces; here a topological space $X$ is *Hausdorff, or $T_{2}$,* iff each two distinct points of $X$ can be *separated* by disjoint neighbourhoods, so that for every $x\neq y\in X$, there are neighbourhoods $M\in\mathcal{N}_{x}$ and $N\in\mathcal{N}_{y}$ such that $M\bigcap N=\emptyset$. This means that for any two distinct points $x\neq y\in X$, it is impossible to find points that are arbitrarily close to both of them. Among the properties of Hausdorff spaces, the following need to be mentioned. (H1) $X$ is Hausdorff iff for each $x\in X$ and any point $y\neq x$, there is a neighbourhood $N$ of $x$ such that $y\not\in\textrm{Cl}(N)$. This leads to the significant result that for any $x\in X$ the closed singleton $$\{ x\}=\bigcap_{N\in\mathcal{N}_{x}}\textrm{Cl}(N)$$ *is the intersection of the closures of any local base at that point,* which in the language of nets and filters (Appendix A1) means that a net in a Hausdorff space cannot converge to more than one point in the space and the adherent set $\textrm{adh}(\mathcal{N}_{x})$ of the neighbourhood filter at $x$ is the singleton $\{ x\}$. (H2) Since each singleton is a closed set, each finite set in a Hausdorff space is also closed in $X$. Unlike a cofinite space, however, there can clearly be infinite closed sets in a Hausdorff space. (H3) Any point $x$ in a Hausdorff space $X$ is a cluster point of $A\subseteq X$ iff every neighbourhood of $x$ contains infinitely many points of $A$, a fact that has led to our mental conditioning of the points of a (Cauchy) sequence piling up in neighbourhoods of the limit. Thus suppose for the sake of argument that although some neighbourhood of $x$ contains only a finite number of points, $x$ is nonetheless a cluster point of $A$. Then there is an open neighbourhood $U$ of $x$ such that $U\bigcap(A-\{ x\})=\{ x_{1},\cdots,x_{n}\}$ is a finite closed set of $X$ not containing $x$, and $U\bigcap(X-\{ x_{1},\cdots,x_{n}\})$ being the intersection of two open sets, is an open neighbourhood of $x$ not intersecting $A-\{ x\}$ implying thereby that $x\not\in\textrm{Der}(A)$; infact $U\bigcap(X-\{ x_{1},\cdots,x_{n}\})$ is simply $\{ x\}$ if $x\in A$ or belongs to $\textrm{Bdy}_{X-A}(A)$ when $x\in X-A$. Conversely if every neighbourhood of a point of $X$ intersects $A$ in infinitely many points, that point must belong to $\textrm{Der}(A)$ by definition. Weaker separation axioms than Hausdorffness are those of $T_{0}$, respectively $T_{1}$, spaces in which for every pair of distinct points *at least one,* respectively *each one,* has some neighbourhood not containing the other; the following table is a listing of the separation properties of some useful spaces. Space $T_{0}$ $T_{1}$ $T_{2}$ ------------------------- -------------- -------------- -------------- Discrete $\checkmark$ $\checkmark$ $\checkmark$ Indiscrete $\times$ $\times$ $\times$ $\mathbb{R}$, standard $\checkmark$ $\checkmark$ $\checkmark$ left/right ray $\checkmark$ $\times$ $\times$ Infinite cofinite $\checkmark$ $\checkmark$ $\times$ Uncountable cocountable $\checkmark$ $\checkmark$ $\times$ $x$-inclusion/exclusion $\checkmark$ $\times$ $\times$ $A$-inclusion/exclusion $\times$ $\times$ $\times$ : \[Table: separation\][Separation properties of some useful spaces.]{} It should be noted that that as none of the properties (H1)–(H3) need neighbourhoods of both the points simultaneously, it is sufficient for $X$ to be $T_{1}$ for the conclusions to remain valid. From its definition it follows that any totally disconnected space is a Hausdorff space and is therefore both $T_{1}$ and $T_{0}$ spaces as well. However, if a Hausdorff space has a base of clopen sets then it is totally disconnected; this is so because if $x$ and $y$ are distinct points of $X$, then the assumed property of $x\in H\subseteq M$ for every $M\in\mathcal{N}_{x}$ and some clopen set $M$ yields $X=H\bigcup(X-H)$ as a disconnection of $X$ that separates $x$ and $y\in X-H$; note that the assumed Hausdorffness of $X$ allows $M$ to be chosen so as not to contain $y$. **Example A3.1.** (1) Every indiscrete space is connected; every subset of an indiscrete space is connected. Hence if $X$ is empty or a singleton, it is connected. A discrete space is connected iff it is either empty or is a singleton; the only connected subsets in a discrete space are the degenerate ones. This is an extreme case of lack of connectedness, and a discrete space is the simplest example of a total disconnected space. \(2) $\mathbb{Q}$, the set of rationals considered as a subspace of the real line, is (totally) disconnected because all rationals larger than a given irrational $r$ is a clopen set in $\mathbb{Q}$, and $${\textstyle \mathbb{Q}=((-\infty,r)\bigcap\mathbb{Q})\bigcup(\mathbb{Q}\bigcap(r,\infty))\qquad r\textrm{ is an irrational}}$$ is the union of two disjoint clopen sets in the relative topology of $\mathbb{Q}$. The sets *$(-\infty,r)\cap\mathbb{Q}$* and $\mathbb{Q}\cap(r,\infty)$ are clopen in $\mathbb{Q}$ because neither contains a cluster point of the other. Thus for example, any neighbourhood of the second must contain the irrational $r$ in order to be able to cut the first which means that any neighbourhood of a point in either of the relatively open sets cannot be wholly contained in the other. The only connected sets of $\mathbb{Q}$ are one point subsets consisting of the individual rationals. In fact, a connected piece of $\mathbb{Q}$, being a connected subset of $\mathbb{R}$, is an interval in $\mathbb{R}$, and a nonempty interval cannot be contained in $\mathbb{Q}$ unless it is a singleton. It needs to be noted that the individual points of the rational line are not (cl)open because any open subset of $\mathbb{R}$ that contains a rational must also contain others different from it. This example shows that a space need not be discrete for each of its points to be a component and thereby for the space to be totally disconnected. In a similar fashion, the set of irrationals is (totally) disconnected because all the irrationals larger than a given rational is an example of a clopen set in $\mathbb{R}-\mathbb{Q}$. \(3) The $p$-inclusion ($A$-inclusion) topology is connected; a subset in this topology is connected iff it is degenerate or contains $p$. For, a subset inherits the discrete topology if it does not contain $p$, and $p$-inclusion topology if it contains $p$. \(4) The cofinite (cocountable) topology on an infinite (uncountable) space is connected; a subset in a cofinite (cocountable) space is connected iff it is degenerate or infinite (countable). \(5) Removal of a single point may render a connected space disconnected and even totally disconnected. In the former case, the point removed is called a *cut point* and in the second, it is a *dispersion point.* Any real number is a cut point of $\mathbb{R}$ and it does not have any dispersion point only. \(6) Let $X$ be a topological space. Considering components of $X$ as equivalence classes by the equivalence relation $\sim$ with $Q\!:X\rightarrow X/\sim$ denoting the quotient map, $X/\sim$ is totally disconnected: As $Q^{-}([x])$ is connected for each $[x]\in X/\sim$ in a component class of $X$, and as any open or closed subset $A\subseteq X/\sim$ is connected iff $Q^{-}(A)$ is open or closed, it must follow that $A$ can only be a singleton.$\qquad\blacksquare$ The next notion of compactness in topological spaces provides an insight of the role of nonempty adherent sets of filters that lead in a natural fashion to the concept of attractors in the dynamical systems theory that we take up next. **Definition A3.4.** ***Compactness.*** *A topological space $X$ is* *compact* *iff every open cover of $X$ contains a finite subcover of $X$*. *$\qquad\square$* This definition of compactness has an useful equivalent contrapositive reformulation: *For any given collection of open sets of $X$ if none of its finite subcollections cover $X$, then the entire collection also cannot cover $X$.* The following theorem is a statement of the fundamental property of compact spaces in terms of adherences of filters in such spaces, the proof of which uses this contrapositive characterization of compactness. **Theorem A3.1.** *A topological space $X$ is compact iff each class of closed subsets of $X$ with finite intersection property has nonempty intersection.* *$\qquad\square$* **Proof.** *Necessity.* Let $X$ be a compact space. Let $\mathcal{F}=\{ F_{\alpha}\}_{\alpha\in\mathbb{D}}$ be a collection of closed subsets of $X$ with finite FIP, and let $\mathcal{G}=\{ X-F_{\alpha}\}_{\alpha\in\mathbb{D}}$ be the corresponding open sets of $X$. If $\{ G_{i}\}_{i=1}^{N}$ is a nonempty finite subcollection from $\mathcal{G}$, then $\{ X-G_{i}\}_{i=1}^{N}$ is the corresponding nonempty finite subcollection of $\mathcal{F}$. Hence from the assumed finite intersection property of $\mathcal{F}$, it must be true that $$\begin{array}{ccl} {\displaystyle X-\bigcup_{i=1}^{N}G_{i}} & = & {\displaystyle \bigcap_{i=1}^{N}(X-G_{i})}\qquad(\textrm{DeMorgan}'\textrm{s Law})\\ & \neq & \emptyset,\end{array}$$ so that no finite subcollection of $\mathcal{G}$ can cover $X$. Compactness of $X$ now implies that $\mathcal{G}$ too cannot cover $X$ and therefore $$\bigcap_{\alpha}F_{\alpha}=\bigcap_{\alpha}(X-G_{\alpha})=X-\bigcup_{\alpha}G_{\alpha}\neq\emptyset.$$ The proof of the converse is a simple exercise of reversing the arguments involving the two equations in the proof above.$\qquad\blacksquare$ Our interest in this theorem and its proof lies in the following corollary — *which essentially means that for every filter $\mathcal{F}$ on a compact space the adherent set* $\textrm{adh}(\mathcal{F})$ *is not empty —* from which follows that every net in a compact space must have a convergent subnet. **Corollary.** *A space $X$ is compact iff for every class $\mathcal{A}=(A_{\alpha})$ of nonempty subsets of $X$ with* FIP*,* $\textrm{adh}(\mathcal{A})=\bigcap_{A_{\alpha}\in\mathcal{A}}\textrm{Cl}(A_{\alpha})\neq\emptyset$*.*$\qquad\square$ The proof of this result for nets given by the next theorem illustrates the general approach in such cases which is all that is basically needed in dealing with attractors of dynamical systems; compare Theorem A1.3. **Theorem A3.2.** *A topological space $X$ is compact iff each net in $X$ adheres in $X$*.$\qquad\square$ **Proof.** *Necessity.* Let $X$ be a compact space, $\chi\!:\mathbb{D}\rightarrow X$ a net in $X$, and $\mathbb{R}_{\alpha}$ the residual of $\alpha$ in the directed set $\mathbb{D}$. For the filter-base $(_{\textrm{F}}\mathcal{B}_{\chi(\mathbb{R}_{\alpha})})_{\alpha\in\mathbb{D}}$ of nonempty, decreasing, nested subsets of $X$ associated with the net $\chi$, compactness of $X$ requires from $\bigcap_{\alpha\preceq\delta}\textrm{Cl}(\chi(\mathbb{R}_{\alpha})\supseteq\chi(\mathbb{R}_{\delta})\neq\emptyset$, that the uncountably intersecting subset $$\textrm{adh}(_{\textrm{F}}\mathcal{B}_{\chi}):=\bigcap_{\alpha\in\mathbb{D}}\textrm{Cl}(\chi(\mathbb{R}_{\alpha}))$$ of $X$ be non-empty. If $x\in\textrm{adh}(_{\textrm{F}}\mathcal{B}_{\chi})$ then because $x$ is in the closure of $\chi(\mathbb{R}_{\beta})$, it follows from Eq. (\[Eqn: Def: Closure\]) that $N\bigcap\chi(\mathbb{R}_{\beta})\neq\emptyset$[^33] for every $N\in\mathcal{N}_{x}$, $\beta\in\mathbb{D}$. Hence $\chi(\gamma)\in N$ for some $\gamma\succeq\beta$ so that $x\in\textrm{adh}(\chi)$; see Eq. (\[Eqn: adh net2\]). *Sufficiency.* Let *$\chi$* be a net in $X$ that adheres at $x\in X$. From any class $\mathcal{F}$ of closed subsets of $X$ with FIP, construct as in the proof of Thm. A1.4, a decreasing nested sequence of closed subsets $C_{\beta}=\bigcap_{\alpha\preceq\beta\in\mathbb{D}}\{ F_{\alpha}\!:F_{\alpha}\in\mathcal{F}\}$ and consider the directed set $_{\mathbb{D}}C_{\beta}=\{(C_{\beta},\beta)\!:(\beta\in\mathbb{D})(x_{\beta}\in C_{\beta})\}$ with its natural direction (\[Eqn: DirectionIndexed\]) to define the net $\chi(C_{\beta},\beta)=x_{\beta}$ in $X$; see Def. A1.10. From the assumed adherence of $\chi$ at some $x\in X$, it follows that $N\bigcap F\neq\emptyset$ for every $N\in\mathcal{N}_{x}$ and $F\in\mathcal{F}$. Hence $x$ belongs to the closed set $F$ so that $x\in\textrm{adh}(\mathcal{F})$; see Eq. (\[Eqn: adh filter\]). Hence $X$ is compact.$\qquad\blacksquare$ Using Theorem A1.5 that specifies a definite criterion for the adherence of a net, this theorem reduces to the useful formulation that *a space is compact iff each net in it has some convergent subnet.* An important application is the following: Since every decreasing sequence $(F_{m})$ of nonempty sets has FIP (because $\bigcap_{m=1}^{M}F_{m}=F_{M}$ for every finite $M$), *every decreasing sequence of nonempty* closed *subsets* *of a compact spac*e *has nonempty intersection.* For a complete metric space this is known as the *Nested Set Theorem,* and for $[0,1]$ and other compact subspaces of $\mathbb{R}$ as the *Cantor Intersection Theorem.*[^34] For subspaces $A$ of $X$, it is the relative topology that determines as usual compactness of $A$; however the following criterion renders this test in terms of the relative topology unnecessary and shows that the topology of $X$ itself is sufficient to determine compactness of subspaces: *A subspace $K$ of a topological space $X$ is compact iff each open cover of $K$ in $X$ contains a finite cover of $K$.* A proper understanding of the distinction between compactness and closedness of subspaces — which often causes much confusion to the non-specialist — is expressed in the next two theorems. As a motivation for the first that establishes that not every subset of a compact space need be compact, mention may be made of the subset $(a,b)$ of the compact closed interval $[a,b]$ in $\mathbb{R}$. **Theorem A3.3.** *A closed subset $F$ of a compact space $X$ is compact.* *$\qquad\square$* **Proof.** Let $\mathcal{G}$ be an open cover of $F$ so that an open cover of $X$ is $\mathcal{G}\bigcup(X-F)$, which because of compactness of $X$ contains a finite subcover $\mathcal{U}$. Then $\mathcal{U}-(X-F)$ is a finite collection of $\mathcal{G}$ that covers $F$.*$\qquad\blacksquare$* It is not true in general that a compact subset of a space is necessarily closed. For example, in an infinite set $X$ with the cofinite topology, let $F$ be an infinite subset of $X$ with $X-F$ also infinite. Then although $F$ is not closed in $X$, it is nevertheless compact because $X$ is compact. Indeed, let $\mathcal{G}$ be an open cover of $X$ and choose any nonempty $G_{0}\in\mathcal{G}$. If $G_{0}=X$ then $\{ G_{0}\}$ is the required finite cover of $X$. If this is not the case, then because $X-G_{0}=\{ x_{i}\}_{i=1}^{n}$ is a finite set, there is a $G_{i}\in\mathcal{G}$ with $x_{i}\in G_{i}$ for each $1\leq i\leq n$, and therefore $\{ G_{i}\}_{i=0}^{n}$ is the finite cover that demonstrates the compactness of the cofinite space $X$. Compactness of $F$ now follows because the subspace topology on $F$ is the induced cofinite topology from $X$. The distinguishing feature of this topology is that it, like the cocountable, is not Hausdorff: If $U$ and $V$ are any two nonempty open sets of $X$, then they cannot be disjoint as the complements of the open sets can only be finite and if $U\bigcap V$ were to be indeed empty, then $${\textstyle X=X-\emptyset={X-(U\bigcap V)=(X-U)\bigcup(X-V)}}$$ would be a finite set. An immediate fallout of this is that in an infinite cofinite space, a sequence $(x_{i})_{i\in\mathbb{N}}$ (and even a net) with $x_{i}\neq x_{j}$ for $i\neq j$ behaves in an extremely unusual way: *It converges,* as in the indiscrete space, *to* *every point of the space.* Indeed if $x\in X$, where $X$ is an infinite set provided with its cofinite topology, and $U$ is any neighbourhood of $x$, any infinite sequence $(x_{i})_{i\in\mathbb{N}}$ in $X$ must be eventually in $U$ because $X-U$ is finite, and ignoring of the initial set of its values lying in $X-U$ in no way alters the ultimate behaviour of the sequence (note that this implies that the filter induced on $X$ by the sequence agrees with its topology). Thus $x_{i}\rightarrow x$ for any $x\in X$ is a reflection of the fact that there are no small neighbourhoods of any point of $X$ with every neighbourhood being almost the whole of $X$, except for a null set consisting of only a finite number of points. This is in sharp contrast with Hausdorff spaces where, although every finite set is also closed, every point has arbitrarily small neighbourhoods that lead to unique limits of sequences. A corresponding result for cocountable spaces can be found in Example A1.2 Continued. This example of the cofinite topology motivates the following “converse” of the previous theorem. **Theorem A3.4.** *Every compact subspace of a Hausdorff space is closed.$\qquad\square$* **Proof.** Let $K$ be a nonempty compact subset of $X$, Fig. \[Fig: cmpct\_clsd\], and let $x\in X-K$. Because of the separation of $X$, for every $y\in K$ there are disjoint open subsets $U_{y}$ and $V_{y}$ of $X$ with $y\in U_{y}$, and $x\in V_{y}$. Hence $\{ U_{y}\}_{y\in K}$ is an open cover for $K$, and from its compactness there is a finite subset $A$ of $K$ such that $K\subseteq\bigcup_{y\in A}U_{y}$ with $V=\bigcap_{y\in A}V_{y}$ an open neighbourhood of $x$; $V$ is open because each $V_{y}$ is a neighbourhood of $x$ and the intersection is over finitely many points $y$ of $A$. To prove that $K$ is closed in $X$ it is enough to show that $V$ is disjoint from $K$: If there is indeed some $z\in V\bigcap K$ then $z$ must be in some $U_{y}$ for $y\in A$. But as $z\in V$ it is also in $V_{y}$ which is impossible as $U_{y}$ and $V_{y}$ are to be disjoint. **This last part of the argument infact shows that *if $K$ is a compact subspace of a Hausdorff space $X$ and $x\notin K$, then there are disjoint open sets $U$ and $V$ of $X$ containing $x$ and $K$.$\qquad\blacksquare$* The last two theorems may be combined to give the obviously important **Corollary.** *In a compact Hausdorff space, closedness and compactness of its subsets are equivalent concepts.$\qquad\square$* In the absence of Hausdorffness, it is not possible to conclude from the assumed compactness of the space that every point to which the net may converge actually belongs to the subspace. **Definition A3.5.** *A subset $D$ of a topological space* *$(X,\mathcal{U})$* *is* *dense in $X$ if* $\textrm{Cl}(D)=X$*. Thus the closure of $D$ is the largest open subset of $X$, and every neighbourhood of any point of $X$ contains a point of $D$ not necessarily distinct from it; refer to the distinction between Eqs. (\[Eqn: Def: Closure\]) and (\[Eqn: Def: Derived\]).$\qquad\square$* Loosely, $D$ is dense in $X$ iff every point of $X$ has points of $D$ arbitrarily close to it. A *self-dense* (*dense in itself*) set is a set without any isolated points; hence $A$ is self-dense iff $A\subseteq\textrm{Der}(A)$. A closed self-dense set is called a *perfect set* so that a closed set $A$ is perfect iff it has no isolated points. Accordingly **$$A\textrm{ is perfect}\Longleftrightarrow A=\textrm{Der}(A),$$ means that the closure of a set without any isolated points is a perfect set. **Theorem A3.5.** *The following are equivalent statements.* \(1) *$D$ is dense in $X$*. \(2) *If $F$ is any closed set of $X$ with $D\subseteq F$, then $F=X$*; *thus the only closed superset of $D$ is $X$.* \(3) *Every nonempty (basic) open set of $X$ cuts $D;$ thus the only open set disjoint from $D$ is the empty set $\emptyset$.* \(4) *The exterior of $D$ is empty.$\qquad\square$* **Proof.** (3) If $U$ indeed is a nonempty open set of $X$ with $U\bigcap D=\emptyset$, then $D\subseteq X-U\neq X$ leads to the contradiction $X=\textrm{Cl}(D)\subseteq\textrm{Cl}(X-U)=X-U\neq X$, which also incidentally proves (2). From (3) it follows that for any open set $U$ of $X$, $\textrm{Cl}(U)=\textrm{Cl}(U\bigcap D)$ because if $V$ is any open neighbourhood of $x\in\textrm{Cl}(U)$ then $V\bigcap U$ is a nonempty open set of $X$ that must cut $D$ so that $V\bigcap(U\bigcap D)\neq\emptyset$ implies $x\in\textrm{Cl}(U\bigcap D)$. Finally, $\textrm{Cl}(U\bigcap D)\subseteq\textrm{Cl}(U)$ completes the proof.$\qquad\blacksquare$ **Definition A3.6.** (a) *A set $A\subseteq X$ is said to be* *nowhere dense* *in* ***$X$ if* $\textrm{Int}(\textrm{Cl}(A))=\emptyset$ *and* *residual* *in* ***$X$ if* $\textrm{Int}(A)=\emptyset$*.$\qquad\square$* $A$ is nowhere dense in $X$ iff $$\textrm{Bdy}(X-\textrm{Cl}(A))=\textrm{Bdy}(\textrm{Cl}(A))=\textrm{Cl}(A)$$ so that $${\textstyle \textrm{Cl}(X-\textrm{Cl}(A))={(X-\textrm{Cl}(A))\bigcup\textrm{Cl}(A)=X}}$$ from which it follows that $$A\textrm{ is nwd in }X\Longleftrightarrow X-\textrm{Cl}(A)\textrm{ is dense in }X$$ and $$A\textrm{ is residual in }X\Longleftrightarrow X-A\textrm{ is dense in }X.$$ Thus $A$ is nowhere dense iff $\textrm{Ext}(A):=X-\textrm{Cl}(A)$ **is dense in *$X$,* and in particular a closed set is nowhere dense in $X$ iff its complement is open dense in $X$ with open-denseness being complimentarily dual to closed-nowhere denseness. The rationals in reals is an example of a set that is residual but not nowhere dense. The following are readily verifiable properties of subsets of $X$. \(1) A set $A\subseteq X$ is nowhere dense in $X$ iff it is contained in its own boundary, iff it is contained in the closure of the complement of its closure, that is $A\subseteq\textrm{Cl}(X-\textrm{Cl}(A))$. In particular a closed subset $A$ is nowhere dense in $X$ iff $A=\textrm{Bdy}(A)$, that is iff it contains no open set. \(2) From $M\subseteq N\Rightarrow\textrm{Cl}(M)\subseteq\textrm{Cl}(N)$ it follows, with $M=X-\textrm{Cl}(A)$ and $N=X-A$, that a nowhere dense set is residual, but a residual set need not be nowhere dense unless it is also closed in $X$. \(3) Since $\textrm{Cl}(\textrm{Cl}(A))=\textrm{Cl}(A)$, $\textrm{Cl}(A)$ is nowhere dense in $X$ iff $A$ is. \(4) For any $A\subseteq X$, both $\textrm{Bdy}_{A}(X-A):=\textrm{Cl}(X-A)\bigcap A$ and $\textrm{Bdy}_{X-A}(A):=\textrm{Cl}(A)\bigcap(X-A)$ are residual sets and as Fig. \[Fig: DerSets\] shows $$\textrm{Bdy}_{X}(A)=\textrm{Bdy}_{X-A}(A)\bigcup\textrm{Bdy}_{A}(X-A)$$ is the union of these two residual sets. When $A$ is closed (or open) in $X$, its boundary consisting of the only component $\textrm{Bdy}_{A}(X-A)$ (or $\textrm{Bdy}_{X-A}(A)$) as shown by the second row (or column) of the figure, being a closed set of $X$ is also nowhere dense in $X$; infact *a closed nowhere dense set is always the boundary of some open set.* Otherwise, the boundary components of the two residual parts — as in the donor-donor, donor-neutral, neutral-donor and neutral-neutral cases — need not be individually closed in $X$ (although their union is) and their union is a residual set that need not be nowhere dense in $X$: the union of two nowhere dense sets is nowhere dense but the union of a residual and a nowhere dense set is a residual set. One way in which a two-component boundary can be nowhere dense is by having $\textrm{Bdy}_{A}(X-A)\supseteq\textrm{Der}(A)$ or $\textrm{Bdy}_{X-A}(A)\supseteq\textrm{Der}(X-A)$, so that it is effectively in one piece rather than in two, as shown in Fig. \[Fig: DerSets1\](b). **Theorem A3.6.** *$A$ is nowhere dense in $X$ iff each non-empty open set of $X$ has a non-empty open subset disjoint from* Cl *$\qquad\square$* **Proof.** If $U$ is a nonempty open set of $X$, then $U_{0}=U\cap\textrm{Ext}(A)\neq\emptyset$ as $\textrm{Ext}(A)$ is dense in $X$; $U_{0}$ is the open subset that is disjoint from $\textrm{Cl}(A)$. It clearly follows from this that each non-empty open set of $X$ has a non-empty open subset disjoint from a nowhere dense set $A$.$\qquad\blacksquare$ What this result (which follows just from the definition of nowhere dense sets) actually means is that no point in $\textrm{Bdy}_{X-A}(A)$ can be isolated in it. **Corollary.** $A$ *is nowhere dense in $X$ iff* Cl$(A)$ *does not contain any nonempty open set of $X$* *any nonempty open set that contains $A$ also contains its closure.* *$\qquad\square$* **Example A3.2.** Each finite subset of $\mathbb{R}^{n}$ is nowhere dense in $\mathbb{R}^{n}$; the set $\{1/n\}_{n=1}^{\infty}$ is nowhere dense in $\mathbb{R}$. The Cantor set $\mathcal{C}$ is nowhere dense in $[0,1]$ because every neighbourhood of any point in $\mathcal{C}$ must contain, by its very construction, a point with $1$ in its ternary representation. That the interior and the interior of the closure of a set are not necessarily the same is seen in the example of the rationals in reals: The set of rational numbers $\mathbb{Q}$ has empty interior because any neighbourhood of a rational number contains irrational numbers (so also is the case for irrational numbers) and $\mathbb{R}=\textrm{Int}(\textrm{Cl}(\mathbb{Q}))\supseteq\textrm{Int}(\mathbb{Q})=\emptyset$ justifies the notion of a nowhere dense set.$\qquad\blacksquare$ The following properties of $\mathcal{C}$ can be taken to define any subset of a topological space as a Cantor set; set-theoretically it should be clear from its classical middle-third construction that the Cantor set consists of all points of the closed interval $[0,1]$ whose infinite triadic (base 3) representation, expressed so as not to terminate with an infinite string of $1$’s, does not contain the digit $1$. Accordingly, any end-point of the infinite set of closed intervals whose intersection yields the Cantor set, is represented by a repeating string of either $0$ or $2$ while a non end-point has every other arbitrary collection of these two digits. Recalling that any number in $[0,1]$ is a rational iff its representation in any base is terminating or recurring — thus any decimal that neither repeats or terminates but consists of all possible sequences of all possible digits represents an irrational number — it follows that both rationals and irrationals belong to the Cantor set. ($\mathcal{C}1$) ***$\mathcal{C}$ is*** *******totally disconnected.*** If possible, let $\mathcal{C}$ have a component containing points $a$ and $b$ with $a<b$. Then $[a,b]\subseteq\mathcal{C}\Rightarrow[a,b]\subseteq C_{i}$ for all $i$. But this is impossible because we may choose $i$ large enough to have $3^{-i}<b-a$ so that $a$ and $b$ must belong to two different members of the pairwise disjoint closed $2^{i}$ subintervals each of length $3^{-i}$ that constitutes $C_{i}$. Hence $$[a,b]\textrm{ is not a subset of any }C_{i}\Longrightarrow[a,b]\textrm{ is not a subset of }\mathcal{C}.$$ ($\mathcal{C}2$) ***$\mathcal{C}$ is perfect*** so that for any $x\in\mathcal{C}$ every neighbourhood of $x$ must contain some other point of $\mathcal{C}$. Supposing to the contrary that the singleton $\{ x\}$ is an open set of $\mathcal{C}$, there must be an $\varepsilon>0$ such that in the usual topology of $\mathbb{R}$$${\textstyle \{ x\}=\mathcal{C}\bigcap(x-\varepsilon,x+\varepsilon).}\label{Eqn: Cantor_Perfect}$$ Choose a positive integer $i$ large enough to satisfy $3^{-i}<\varepsilon$. Since $x$ is in every $C_{i}$, it must be in one of the $2^{i}$ pairwise disjoint closed intervals $[a,b]\subset(x-\varepsilon,x+\varepsilon)$ each of length $3^{-i}$ whose union is $C_{i}$. As $[a,b]$ is an interval, at least one of the endpoints of $[a,b]$ is different from $x$, and since an endpoint belongs to $\mathcal{C}$, $\mathcal{C}\cap(x-\varepsilon,x+\varepsilon)$ must also contain this point thereby violating Eq. (\[Eqn: Cantor\_Perfect\]). ($\mathcal{C}3$) ***$\mathcal{C}$ is nowhere dense*** because each neighbourhood of any point of $\mathcal{C}$ intersects $\textrm{Ext}(\mathcal{C})$; see Thm. A3.6. ($\mathcal{C}4$) ***$\mathcal{C}$ is compact*** because it is a closed subset contained in the compact subspace $[0,1]$ of $\mathbb{R}$, see Thm. A3.3. The compactness of $[0,1]$ follows from the Heine-Borel Theorem which states that any subset of the real line is compact iff it is both closed and bounded with respect to the Euclidean metric on $\mathbb{R}$. Compare ($\mathcal{C}1$) and ($\mathcal{C}2$) with the essentially similar arguments of Example A3.1(2) for the subspace of rationals in $\mathbb{R}$. **A4. Neutron Transport Theory** This section introduces the reader to the basics of the *linear* neutron transport theory where graphical convergence approximations to the singular distributions, interpreted here as multifunctions, led to the present study of this work. The one-speed (that is mono-energetic) neutron transport equation in one dimension and plane geometry, is $$\mu\frac{\partial\Phi(x,\mu)}{\partial x}+\Phi(x,\mu)=\frac{c}{2}\int_{-1}^{1}\Phi(x,\mu^{\prime})d\mu^{\prime},\,0<c<1,\,-1\leq\mu\leq1\label{Eqn: NeutronTransport}$$ where $x$ is a non-dimensional physical space variable that denotes the location of the neutron moving in a direction $\theta=\cos^{-1}(\mu)$, $\Phi(x,\mu)$ is a neutron density distribution function such that $\Phi(x,\mu)dxd\mu$ is the expected number of neutrons in a distance $dx$ about the point $x$ moving at constant speed with their direction cosines of motion in $d\mu$ about $\mu$, and $c$ is a physical constant that will be taken to satisfy the restriction shown above. Case’s method starts by assuming the solution to be of the form $\Phi_{\nu}(x,\mu)=e^{-x/\mu}\phi(\mu,\nu)$ with a normalization integral constraint of $\int_{-1}^{1}\phi(\mu,\nu)d\mu=1$ to lead to the simple equation $$(\nu-\mu)\phi(\mu,\nu)=\frac{c\nu}{2}\label{Eqn: case_eigen}$$ for the unknown function $\phi(\nu,\mu)$. Case then suggested, see @Case1967, the non-simple complete solution of this equation to be $$\phi(\mu,\nu)=\frac{c\nu}{2}\mathcal{P}\frac{1}{\nu-\mu}+\lambda(v)\delta(v-\mu),\label{Eqn: singular_eigen}$$ where $\lambda(\nu)$ is the usual combination coefficient of the solutions of the homogeneous and non-homogeneous parts of a linear equation, $\mathcal{P}(\cdot)$ is a principal value and $\delta(x)$ the Dirac delta, to lead to the full-range $-1\leq\mu\leq1$ solution valid for $-\infty<x<\infty$ $$\Phi(x,\mu)=a(\nu_{0})e^{-x/\nu_{0}}\phi(\mu,\nu_{0})+a(-\nu_{0})e^{x/\nu_{0}}\phi(-\nu_{0},\mu)+\int_{-1}^{1}a(\nu)e^{-x/\nu}\phi(\mu,\nu)d\nu\label{Eqn: CaseSolution_FR}$$ of the one-speed neutron transport equation (\[Eqn: NeutronTransport\]). Here the real $\nu_{0}$ and $\nu$ satisfy respectively the integral constraints $$\frac{c\nu_{0}}{2}\ln\frac{\nu_{0}+1}{\nu_{0}-1}=1,\qquad\mid\nu_{0}\mid>1$$ $$\lambda(\nu)=1-\frac{c\nu}{2}\ln\frac{1+\nu}{1-\nu},\qquad\nu\in[-1,1],$$ with $$\phi(\mu,\nu_{0})=\frac{c\nu_{0}}{2}\frac{1}{\nu_{0}-\mu}$$ following from Eq. (\[Eqn: singular\_eigen\]). It can be shown [@Case1967] that the eigenfunctions **$\phi(\nu,\mu)$** satisfy the full-range orthogonality condition $$\int_{-1}^{1}\mu\phi(\nu,\mu)\phi(\nu^{\prime},\mu)d\mu=N(\nu)\delta(\nu-\nu^{\prime}),$$ where the odd normalization constants $N$ are given by $$\begin{array}{ccl} {\displaystyle N(\pm\nu_{0})} & = & {\displaystyle \int_{-1}^{1}\mu\phi^{2}(\pm\nu_{0},\mu)d\mu}\qquad\textrm{for }\mid\nu_{0}\mid>1\\ & = & {\displaystyle \pm\frac{c\nu_{0}^{3}}{2}\left(\frac{c}{\nu_{0}^{2}-1}-\frac{1}{\nu_{0}^{2}}\right)},\end{array}$$ and$$N(\nu)=\nu\left(\lambda^{2}(\nu)+\left(\frac{\pi c\nu}{2}\right)^{2}\right)\qquad\textrm{for }\nu\in[-1,1].$$ With a source of particles $\psi(x_{0},\mu)$ located at $x=x_{0}$ in an infinite medium, Eq. (\[Eqn: CaseSolution\_FR\]) reduces to the boundary condition, with $\mu,\textrm{ }\nu\in[-1,1]$, $$\psi(x_{0},\mu)=a(\nu_{0})e^{-x_{0}/\nu_{0}}\phi(\mu,\nu_{0})+a(-\nu_{0})e^{x_{0}/\nu_{0}}\phi(-\nu_{0},\mu)+\int_{-1}^{1}a(\nu)e^{-x_{0}/\nu}\phi(\mu,\nu)d\nu\label{Eqn: BC_FR}$$ for the determination of the expansion coefficients $a(\pm\nu_{0}),\textrm{ }\{ a(\nu)\}_{\nu\in[-1,1]}$. Use of the above orthogonality integrals then lead to the complete solution of the problem to be $$a(\nu)=\frac{e^{x_{0}/\nu}}{N(\nu)}\int_{-1}^{1}\mu\psi(x_{0},\mu)\phi(\mu,\nu)d\mu,\qquad\nu=\pm\nu_{0}\textrm{ or }\nu\in[-1,1].$$ For example, in the infinite-medium Greens function problem with $x_{0}=0$ and $\psi(x_{0},\mu)=\delta(\mu-\mu_{0})/\mu$, the coefficients are $a(\pm\nu_{0})=\phi(\mu_{0},\pm\nu_{0})/N(\pm\nu_{0})$ when $\nu=\pm\nu_{0}$, and $a(\nu)=\phi(\mu_{0},\nu)/N(\nu)$ for $\nu\in[-1,1]$. For a half-space $0\leq x<\infty$, the obvious reduction of Eq. (\[Eqn: CaseSolution\_FR\]) to $$\Phi(x,\mu)=a(\nu_{0})e^{-x/\nu_{0}}\phi(\mu,\nu_{0})+\int_{0}^{1}a(\nu)e^{-x/\nu}\phi(\mu,\nu)d\nu\label{Eqn: CaseSolution_HR}$$ with boundary condition, $\mu,\textrm{ }\nu\in[0,1]$, $$\psi(x_{0},\mu)=a(\nu_{0})e^{-x_{0}/\nu_{0}}\phi(\mu,\nu_{0})+\int_{0}^{1}a(\nu)e^{-x_{0}/\nu}\phi(\mu,\nu)d\nu,\label{Eqn: BC_HR}$$ leads to an infinitely more difficult determination of the expansion coefficients due to the more involved nature of the orthogonality relations of the eigenfunctions in the half-interval $[0,1]$ that now reads for $\nu,\textrm{ }\nu^{\prime}\in[0,1]$ [@Case1967] $$\begin{aligned} \int_{0}^{1}W(\mu)\phi(\mu,\nu^{\prime})\phi(\mu,\nu)d\mu & = & \frac{W(\nu)N(\nu)}{\nu}\delta(\nu-\nu^{\prime})\nonumber \\ \int_{0}^{1}W(\mu)\phi(\mu,\nu_{0})\phi(\mu,\nu)d\mu & = & 0\nonumber \\ \int_{0}^{1}W(\mu)\phi(\mu,-\nu_{0})\phi(\mu,\nu)d\mu & = & c\nu\nu_{0}X(-\nu_{0})\phi(\nu,-\nu_{0})\nonumber \\ \int_{0}^{1}W(\mu)\phi(\mu,\pm\nu_{0})\phi(\mu,\nu_{0})d\mu & = & \mp\left(\frac{c\nu_{0}}{2}\right)^{2}X(\pm\nu_{0})\label{Eqn: HR Ortho}\\ \int_{0}^{1}W(\mu)\phi(\mu,\nu_{0})\phi(\mu,-\nu)d\mu & = & \frac{c^{2}\nu\nu_{0}}{4}X(-\nu)\nonumber \\ \int_{0}^{1}W(\mu)\phi(\mu,\nu^{\prime})\phi(\mu,-\nu)d\mu & = & \frac{c\nu^{\prime}}{2}(\nu_{0}+\nu)\phi(\nu^{\prime},-\nu)X(-\nu)\nonumber \end{aligned}$$ where the half-range weight function $W(\mu)$ is defined as $$W(\mu)=\frac{c\mu}{2(1-c)(\nu_{0}+\mu)X(-\mu)}\label{Eqn: W(mu)}$$ in terms of the $X$-function $$X(-\mu)=\textrm{exp}-\left\{ \frac{c}{2}\int_{0}^{1}\frac{\nu}{N(\nu)}\left[1+\frac{c\nu^{2}}{1-\nu^{2}}\right]\ln(\nu+\mu)d\nu\right\} ,\qquad0\leq\mu\leq1,$$ that is conveniently obtained from a numerical solution of the nonlinear integral equation $$\Omega(-\mu)=1-\frac{c\mu}{2(1-c)}\int_{0}^{1}\frac{\nu_{0}^{2}(1-c)-\nu^{2}}{(\nu_{0}^{2}-\nu^{2})(\mu+\nu)\Omega(-\nu)}d\nu\label{Eqn: Omega(-mu)}$$ to yield $$X(-\mu)=\frac{\Omega(-\mu)}{\mu+\nu_{0}\sqrt{1-c}},$$ and the $X(\pm\nu_{0})$ satisfy $$X(\nu_{0})X(-\nu_{0})=\frac{\nu_{0}^{2}(1-c)-1}{2(1-c)v_{0}^{2}(\nu_{0}^{2}-1)}.$$ Two other useful relations involving the $W$-function are given by $\int_{0}^{1}W(\mu)\phi(\mu,\nu_{0})d\mu=c\nu_{0}/2$ and $\int_{0}^{1}W(\mu)\phi(\mu,\nu)d\mu=c\nu/2$. The utility of these full and half range orthogonality relations lie in the fact that a suitable class of functions of the type that is involved here can always be expanded in terms of them, see @Case1967. An example of this for a full-range problem has been given above; we end this introduction to the generalized — traditionally known as singular in neutron transport theory — eigenfunction method with two examples of half-range orthogonality integrals to the half-space problems A and B of Sec. 5. **Problem A: The Milne Problem.** In this case there is no incident flux of particles from outside the medium at $x=0$, but for large $x>0$ the neutron distribution inside the medium behaves like $e^{x/\nu_{0}}\phi(-\nu_{0},\mu)$. Hence the boundary condition (\[Eqn: BC\_HR\]) at $x=0$ reduces to $$-\phi(\mu,-\nu_{0})=a_{\textrm{A}}(\nu_{0})\phi(\mu,\nu_{0})+\int_{0}^{1}a_{\textrm{A}}(\nu)\phi(\mu,\nu)d\nu\qquad\mu\geq0.$$ Use of the fourth and third equations of Eq. (\[Eqn: HR Ortho\]) and the explicit relation Eq. (\[Eqn: W(mu)\]) for $W(\mu)$ gives respectively the coefficients $$\begin{aligned} {\displaystyle a_{\textrm{A}}(\nu_{0})} & = & X(-\nu_{0})/X(v_{0})\nonumber \\ a_{\textrm{A}}(\nu) & = & -\frac{1}{N(\nu)}\textrm{ }c(1-c)\nu_{0}^{2}\nu X(-\nu_{0})X(-\nu)\label{Eqn: Milne_Coeff}\end{aligned}$$ The extrapolated end-point $z_{0}$ of Eq. (\[Eqn: extrapolated\]) is related to $a_{\textrm{A}}(\nu_{0})$ of the Milne problem by $a_{\textrm{A}}(\nu_{0})=-\exp(-2z_{0}/\nu_{0})$. **Problem B: The Constant Source Problem.** Here ****the boundary condition at $x=0$ is $$1=a_{\textrm{B}}(\nu_{0})\phi(\mu,\nu_{0})+\int_{0}^{1}a_{\textrm{B}}(\nu)\phi(\mu,\nu)d\nu\qquad\mu\geq0$$ which leads, using the integral relations satisfied by $W$, to the expansion coefficients $$\begin{aligned} {\displaystyle a_{\textrm{B}}(\nu_{0})} & = & -2/c\nu_{0}X(v_{0})\label{Eqn: Constant_Coeff}\\ a_{\textrm{B}}(\nu) & = & \frac{1}{N(\nu)}\textrm{ }(1-c)\nu(\nu_{0}+\nu)X(-\nu)\nonumber \end{aligned}$$ where the $X(\pm\nu_{0})$ are related to Problem A as $$\begin{aligned} X(\nu_{0}) & = & \frac{1}{\nu_{0}}\sqrt{\frac{\nu_{0}^{2}(1-c)-1}{2a_{\textrm{A}}(\nu_{0})(1-c)(\nu_{0}^{2}-1)}}\\ X(-\nu_{0}) & = & \frac{1}{\nu_{0}}\sqrt{\frac{a_{A}(\nu_{0})\left(\nu_{0}^{2}(1-c)-1\right)}{2(1-c)(\nu_{0}^{2}-1)}}.\end{aligned}$$ This brief introduction to the singular eigenfunction method should convince the reader of the great difficulties associated with half-space, half-range methods in particle transport theory; note that the $X$-functions in the coefficients above must be obtained from numerically computed tables. In contrast, full-range methods are more direct due to the simplicity of the weight function $\mu$, which suggests the full-range formulation of half-range problems presented in Sec. 5. Finally it should be mentioned that this singular eigenfunction method is based on the theory of singular integral equations. **Acknowledgment** It is a pleasure to thank the referees for recommending an enlarged Tutorial and Review revision of the original submission *Graphical Convergence, Chaos and Complexity*, **and the Editor Professor Leon O Chua for suggesting a pedagogically self-contained, jargonless no-page limit version accessible to a wider audience for the present form of the paper. Financial assistance during the initial stages of this work from the National Board for Higher Mathematics is also acknowledged. [^1]: \[Foot: UNConf\]A partial listing of papers is as follows: *Chaos and Politics: Application of nonlinear dynamics to socio-political issues; Chaos in Society: Reflections on the impact of chaos theory on sociology; Chaos in neural networks; The impact of chaos on mathematics; The impact of chaos on physics; The impact of chaos on economic theory; The impact of chaos on engineering; The impact of chaos on biology; Dynamical disease:* and *The impact of nonlinear dynamics and chaos on cardiology and medicine.* [^2]: \[Foot: ScienceMag\]The eight Viewpoint articles are titled: *Simple Lessons from Complexity; Complexity in Chemistry; Complexity in Biological Signaling Systems; Complexity and the Nervous System; Complexity, Pattern, and Evolutionary Trade-Offs in Animal Aggregation; Complexity in Natural Landform Patterns; Complexity and Climate* and *Complexity and the Economy*. [^3]: [\[Foot: reln&graph\]We do not distinguish between a relation and its graph although technically they are different objects. Thus although a functional relation, strictly speaking, is the triple $(X,f,Y)$ written traditionally as $f\!:X\rightarrow Y$, we use it synonymously with the graph $f$ itself. Parenthetically, the word]{} *functional* [in this work is not necessarily employed for a scalar-valued function, but is used in a wider sense to distinguish between a function and an arbitrary relation (that is a multifunction). Formally, whereas an arbitrary relation from $X$ to $Y$ is a subset of $X\times Y$, a functional relation must satisfy an additional restriction that requires $y_{1}=y_{2}$ whenever $(x,y_{1})\in f$ and $(x,y_{2})\in f$. In this subset notation, $(x,y)\in f\Leftrightarrow y=f(x)$. ]{} [^4]: [\[Foot: EquivRel\]An useful alternate way of expressing these properties for a relation $\mathscr{M}$ on $X$ are]{} [$\quad$(ER2) $\mathscr{M}$ is symmetric iff $\mathscr{M}=\mathscr M^{-}$ ]{} [$\quad$(ER3) $\mathscr{M}$ is transitive iff $\mathscr{M}\circ\mathscr{M}\subseteq\mathscr{M}$, ]{} [with $\mathscr{M}$ an equivalence relation only if $\mathscr{M}\circ\mathscr{M}=\mathscr{M}$, where for $\mathscr{M}\subseteq X\times Y$ and $\mathscr{N}\subseteq Y\times Z$, the composition $\mathscr{N}\circ\mathscr{M}:=\{(x,z)\in X\times Z\!:(\exists y\in Y)\textrm{ }((x,y)\in\mathscr{M})\wedge((y,z)\in\mathscr{N})\}$]{} [^5]: [\[Foot: family\]A function $\chi\!:\mathbb{D}\rightarrow X$ will be called a]{} *family* [in $X$ indexed by $\mathbb{D}$ when reference to the domain $\mathbb{D}$ is of interest, and a]{} *net* [when it is required to focus attention on its values in $X$.]{} [^6]: [\[Foot: extension\]Observe that it is]{} *not* [being claimed that $f$ belongs to the same class as $(f_{k})$. This is the single most important cornerstone on which this paper is based: the need to “complete” spaces that are topologically “incomplete”. The classical high-school example of the related problem of having to enlarge, or extend, spaces that are not big enough is the solution space of algebraic equations with real coefficients like $x^{2}+1=0$. ]{} [^7]: [\[Foot: support\]By definition, the support (or supporting interval) of $\varphi(x)\in\mathcal{C}_{0}^{\infty}[\alpha,\beta]$ is $[\alpha,\beta]$ if $\varphi$ and all its derivatives vanish for $x\leq\alpha$ and $x\geq\beta$. ]{} [^8]: [\[Foot: integral\]Both Riemann and Lebesgue integrals can be formulated in terms of the so-called]{} *step functions* [$s(x)$, which are piecewise constant functions with values $(\sigma_{i})_{i=1}^{I}$on a finite number of bounded subintervals $(J_{i})_{i=1}^{I}$ (which may reduce to a point or may not contain one or both of the end-points) of a bounded or unbounded interval $J$, with integral $\int_{J}s(x)dx\overset{\textrm{def}}=\sum_{i=1}^{I}\sigma_{i}|J_{i}|$. While the Riemann integral of a bounded function $f(x)$ on a bounded interval $J$ is defined with respect to sequences of step functions $(s_{j})_{j=1}^{\infty}$ and $(t_{j})_{j=1}^{\infty}$ satisfying $s_{j}(x)\leq f(x)\leq t_{j}(x)$ on $J$ with $\int_{J}(s_{j}-t_{j})\rightarrow0$ as $j\rightarrow\infty$ as $R\int_{J}f(x)dx=\lim\int_{J}s_{j}(x)dx=\lim\int_{J}t_{j}(x)dx$, the less restrictive Lebesgue integral is defined for arbitrary functions $f$ over bounded or unbounded intervals $J$ in terms of Cauchy sequences of step functions $\int_{J}|s_{i}-s_{k}|\rightarrow0$, $i,k\rightarrow\infty$, converging to $f(x)$ as $$s_{j}(x)\rightarrow f(x)\textrm{ pointwise almost everywhere on }J,$$ ]{} [to be $$\int_{J}f(x)dx\overset{\textrm{def}}=\lim_{j\rightarrow\infty}\int_{J}s_{j}(x)dx.$$ ]{} [That the Lebesgue integral is more general (and therefore is the proper candidate for completion of function spaces) is illustrated by the example of the function defined over $[0,1]$ to be $0$ on the rationals and $1$ on the irrationals for which an application of the definitions verify that whereas the Riemann integral is undefined, the Lebesgue integral exists and has value $1$. The Riemann integral of a bounded function over a bounded interval exists and is equal to its Lebesgue integral. Because it involves a larger family of functions, all integrals in integral convergences are to be understood in the Lebesgue sense. ]{} [^9]: [\[Foot: delta\]The observant reader cannot have failed to notice how mathematical ingenuity successfully transferred the “troubles” of $(\delta_{k})_{k=1}^{\infty}$ to the sufficiently differentiable benevolent receptor $\varphi$ so as to be able to work backward, via the resultant trouble free $(\delta_{k}^{(-m)})_{k=1}^{\infty}$, to the final object $\delta$. This necessarily hides the true character of $\delta$ to allow only a view of its integral manifestation on functions. This unfortunately is not general enough in the strongly nonlinear physical situations responsible for chaos, and is the main reason for constructing the multifunctional extension of function spaces that we use. ]{} [^10]: [\[Foot: cont=3Dbound\]Recall that for a linear operator continuity and boundedness are equivalent concepts. ]{} [^11]: [\[Foot: OrthoMatrix\]A real matrix $A$ is an orthogonal projector iff $A^{2}=A$ and $A=A^{\textrm{T}}$. ]{} [^12]: [\[Foot: class\]In this sense, a]{} *class* [is a set of sets. ]{} [^13]: [\[Foot: interval\]By definition, an interval $I$ in a totally ordered set $X$ is a subset of $X$ with the property $$(x_{1},x_{2}\in I)\wedge(x_{3}\in X\!:x_{1}\prec x_{3}\prec x_{2})\Longrightarrow x_{3}\in I$$ ]{} [so that any element of $X$ lying between two elements of $I$ also belongs to $I$.]{} [^14]: [\[Foot: entropy\]Although we do not pursue this point of view here, it is nonetheless tempting to speculate that the answer to the question]{} *“Why* [does the entropy of an isolated system increase?” may be found by exploiting this line of reasoning that seeks to explain the increase in terms of a visible component associated with the usual topology as against a different latent workplace topology that governs the dynamics of nature.]{} [^15]: [\[Foot: subspace\]In a subspace $A$ of $X$, a subset $U_{A}$ of $A$ is open iff $U_{A}=A\bigcap U$ for some open set $U$ of $X$. The notion of subspace topology can be formalized with the help of the inclusion map $i\!:A\rightarrow(X,\mathcal{U})$ that puts every point of $A$ back to where it came from, thus $$\begin{array}{ccl} \mathcal{U}_{A} & = & \{ U_{A}=A\bigcap U\!:U\in\mathcal{U}\}\\ & = & \{ i^{-}(U)\!:U\in\mathcal{U}\}.\end{array}$$ ]{} [^16]: [\[Foot: assoc&embed\]A surjective function is an]{} *association* [iff it is image continuous and an injective function is an]{} *embedding* [iff it is preimage continuous. ]{} [^17]: [\[Foot: 0=3Dphi\]If $y\notin\mathcal{R}(f)$ then $f^{-}(\{ y\}):=\emptyset$ which is true for any subset of $Y-\mathcal{R}(f)$. However from the set-theoretic definition of natural numbers that requires $0:=\emptyset$, $1=\{0\}$, $2=\{0,1\}$ to be defined recursively, it follows that $f^{-}(y)$ can be identified with $0$ whenever $y$ is not in the domain of $f^{-}$. Formally, the successor set $A^{+}=A\bigcup\{ A\}$ of $A$ can be used to write $0:=\emptyset$, $1=0^{+}=0\bigcup\{0\}$, $2=1^{+}=1\bigcup\{1\}=\{0\}\bigcup\{1\}$ $3=2^{+}=2\bigcup\{2\}=\{0\}\bigcup\{1\}\bigcup\{2\}$ etc. Then the set of natural numbers $\mathbb{N}$ is defined to be the intersection of all the successor sets, where a successor set $\mathcal{S}$ is any set that contains $\emptyset$ and $A^{+}$ whenever $A$ belongs to $\mathcal{S}$. Observe how in the successor notation, countable union of singleton integers recursively define the corresponding sum of integers. ]{} [^18]: [See footnote \[Foot: 0=3Dphi\] for a justification of the definition when $b$ is not in $\mathcal{R}(a)$.]{} [^19]: [\[Foot: subnet\]A subnet is the generalized uncountable equivalent of a subsequence; for the technical definition, see Appendix A1. ]{} [^20]: [\[Foot: point\_inter\]Equation (\[Eqn: func\_bi\]) is essentially the intersection of the pointwise topologies (\[Eqn: point\]) due to $f$ and $f^{-}$. ]{} [^21]: [\[Foot: strict reln\]If $\preceq$ is an order relation in $X$ then the]{} *strict relation $\prec$ in $X$* [corresponding to $\preceq$, given by $x\prec y\Leftrightarrow(x\preceq y)\wedge(x\neq y)$,]{} *is not an order relation* [because unlike $\preceq$, $\prec$ is not reflexive even though it is both transitive and asymmetric.]{} ** [^22]: [\[Foot: infinite\]This makes $T$, and hence $X$, inductively defined infinite sets. It should be realized that (ST3)]{} *does not mean* [that every member of $T$ is obtained from $g$, but only ensures that the immediate successor of any element of $T$ is also in $T.$ The infimum $_{\rightarrow}T$ of these towers satisfies the additional property of being totally ordered (and is therefore essentially a sequence or net) in $(X,\preceq)$ to which (ST2) can be applied. ]{} [^23]: [\[Foot: Hausdorff\]Recall that this means that if there is a totally ordered chain $C$ in $(X,\preceq)$ that succeeds $C_{+}$, then $C$ must be $C_{+}$ so that no chain in $X$ can be strictly larger than $C_{+}$. The notation adopted here and below is the following: If $X=\{ x,y\}$ is a non-empty set, then $\mathcal{X}:=\mathcal{P}(X)=\{ A\!:A\subseteq X\}=\{\emptyset,\{ x\},\{ y\},\{ x,y\}\}$ is the set of subsets of $X$, and $\mathfrak{X}:=\mathcal{P}^{2}(X)=\{\mathcal{A}:\mathcal{A}\subseteq\mathcal{X}\}$, the set of all subsets of $\mathcal{X}$, consists of the $16$ elements $\emptyset$, $\{\emptyset\}$, $\{\{ x\}\}$, $\{\{ y\}\}$, $\{\{ x,y\}\}$, $\{\{\emptyset\},\{ x\}\}$, $\{\{\emptyset\},\{ y\}\}$, $\{\{\emptyset\},\{ x,y\}\}$, $\{\{ x\},\{ y\}\}$, $\{\{ x\},\{ x,y\}\}$, $\{\{ y\},\{ x,y\}\}$, $\{\{\emptyset\},\{ x\},\{ y\}\}$, $\{\{\emptyset\},\{ x\},\{ x,y\}\}$, $\{\{\emptyset\},\{ y\},\{ x,y\}\}$, $\{\{ x\},\{ y\},\{ x,y\}\}$, and $\mathcal{X}$: an element of $\mathcal{P}^{2}(X)$ is a subset of $\mathcal{P}(X)$, any element of which is a subset of $X$. Thus if $C=\{0,1,2\}$ is a chain in $(X=\{0,1,2\},\leq)$, then $\mathcal{C}=\{\{0\},\{0,1\},\{0,1,2\}\}\subseteq\mathcal{P}(X)$ and $\mathfrak{C}=\{\{\{0\}\},\{\{0\},\{0,1\}\},\{\{0\},\{0,1\},\{0,1,2\}\}\}\subseteq\mathcal{P}^{2}(X)$ represent chains in $(\mathcal{P}(X),\subseteq)$ and $(\mathcal{P}^{2}(X),\subseteq)$ respectively . ]{} [^24]: [\[Foot: supremum\]A similar situation arises in the following more intuitive example. Although the subset $A=\{1/n\}_{n\in Z_{+}}$ of the interval $I=[-1,1]$ has no a smallest or minimal elements, it does have the infimum 0. Likewise, although $A$ is bounded below by any element of $[-1,0)$, it has no greatest lower bound in $[-1,0)\bigcup(0,1]$. ]{} [^25]: [\[Foot: omega-limit\]How does this happen for $A=\{ f^{i}(x_{0})\}_{i\in\mathbb{N}}$ that is not the constant sequence $(x_{0})$ at a fixed point? As $i\in\mathbb{N}$ increases, points are added to $\{ x_{0},f(x_{0}),\cdots,f^{I}(x_{0})\}$ not, as would be the case in a normal sequence, as a piled-up Cauchy tail, but as points generally lying between those already present; recall a typical graph as of Fig. \[Fig: tent4\] for example.]{} [^26]: \[Foot: gen\_eigen\][The technical definition of a generalized eigenvalue is as follows. Let $\mathcal{L}$ be a linear operator such that there exists in the domain of $\mathcal{L}$ a sequence of elements $(x_{n})$ with $\Vert x_{n}\Vert=1$ for all $n$. If $\lim_{n\rightarrow\infty}\Vert(\mathcal{L}-\lambda)x_{n}\Vert=0$ for some $\lambda\in\mathbb{C}$, then this $\lambda$ is a]{} *generalized eigenvalue* [of $\mathcal{L}$, the corresponding eigenfunction $x_{\infty}$ being a]{} *generalized eigenfunction.* [^27]: \[Foot: cluster\]This is also known as a *cluster point*; we shall, however, use this new term exclusively in the sense of the elements of a derived set, see Definition 2.3. [^28]: \[Foot: Filter\_conv\][The restatement $$\mathcal{F}\rightarrow x\Longleftrightarrow\mathcal{N}_{x}\subseteq\mathcal{F}\label{Eqn: Def: LimFilter}$$ of Eq. (\[Eqn: lim filter\]) that follows from (F3), and sometimes taken as the definition of convergence of a filter, is significant as it ties up the algebraic filter with the topological neighbourhood system to produce the filter theory of convergence in topological spaces. From the defining properties of $\mathcal{F}$ it follows that for each $x\in X$, $\mathcal{N}_{x}$ is the coarsest (that is smallest) filter on $X$ that converges to $x$.]{} [^29]: \[Foot: adh\_seq\][In a first countable space, while the corresponding proof of the first part of the theorem for sequences is essentially the same as in the present case, the more direct proof of the converse illustrates how the convenience of nets and directed sets may require more general arguments. Thus if a sequence $(x_{i})_{i\in\mathbb{N}}$ has a subsequence $(x_{i_{k}})_{k\in\mathbb{N}}$ converging to $x$, then a more direct line of reasoning proceeds as follows. Since the subsequence converges to $x$, its tail $(x_{i_{k}})_{k\geq j}$ must be in every neighbourhood $N$ of $x$. But as the number of such terms is infinite whereas $\{ i_{k}\!:k<j\}$ is only finite, it is necessary that for any given $n\in\mathbb{N}$, cofinitely many elements of the sequence $(x_{i_{k}})_{i_{k}\geq n}$ be in $N$. Hence $x\in\textrm{adh}((x_{i})_{i\in\mathbb{N}})$. ]{} [^30]: \[Foot: seq xxx\][This is uncountable because interchanging any two eventual terms of the sequence does not alter the sequence. ]{} [^31]: [Note that $\{ x\}$ is a $1$-point set but $(x)$ is an uncountable sequence.]{} [^32]: \[Foot: e&q\][We adopt the convention of denoting arbitrary preimage and image continuous functions by $e$ and $q$ respectively even though they are not be injective or surjective; recall that the embedding $e\!:X\supseteq A\rightarrow Y$ and the association $q\!:X\rightarrow f(X)$ are $1:1$ and onto respectively. ]{} [^33]: \[Foot: fil-nbd\][This is of course a triviality if we identify each $\chi(\mathbb{R}_{\beta})$ (or $F$ in the proof of the converse that follows) with a neighbourhood $N$ of $X$ that generates a topology on $X$.]{} [^34]: **Nested-set theorem.** *If $(E_{n})$ is a decreasing sequence of nonempty, closed, subsets of a complete metric space $(X,d)$ such that* [$\lim_{n\rightarrow\infty}\textrm{dia}(E_{n})=0$]{}*, then there is a unique point* [$$x\in\bigcap_{n=0}^{\infty}E_{n}.$$ The uniqueness arises because the limiting condition on the diameters of $E_{n}$ imply, from property (H1), that $(X,d)$ is a Hausdorff space. ]{}

--- author: - Nachi Gupta - Raphael Hauser bibliography: - 'ieee-tac3.bib' title: Kalman Filtering with Equality and Inequality State Constraints --- Introduction ============ Kalman Filtering [@Kalman1960] is a method to make real-time predictions for systems with some known dynamics. Traditionally, problems requiring Kalman Filtering have been complex and nonlinear. Many advances have been made in the direction of dealing with nonlinearities (e.g., Extended Kalman Filter [@BLK2001], Unscented Kalman Filter [@JU1997]). These problems also tend to have inherent state space [*equality*]{} constraints (e.g., a fixed speed for a robotic arm) and state space [*inequality*]{} constraints (e.g., maximum attainable speed of a motor). In the past, less interest has been generated towards constrained Kalman Filtering, partly because constraints can be difficult to model. As a result, constraints are often neglected in standard Kalman Filtering applications. The extension to Kalman Filtering with known equality constraints on the state space is discussed in [@SAP1988; @TS1988; @SC2002; @WCC2002; @Gupta2007]. In this paper, we discuss two distinct methods to incorporate constraints into a Kalman Filter. Initially, we discuss these in the framework of equality constraints. The first method, projecting the updated state estimate onto the constrained region, appears with some discussion in [@SC2002; @Gupta2007]. We propose another method, which is to restrict the optimal Kalman Gain so the updated state estimate will not violate the constraint. With some algebraic manipulation, the second method is shown to be a special case of the first method. We extend both of these concepts to Kalman Filtering with inequality constraints in the state space. This generalization for the first approach was discussed in [@SS2005].[^1] Constraining the optimal Kalman Gain was briefly discussed in [@Q1989]. Further, we will also make the extension to incorporating state space constraints in Kalman Filter predictions. Analogous to the way a Kalman Filter can be extended to solve problems containing non-linearities in the dynamics using an Extended Kalman Filter by linearizing locally (or by using an Unscented Kalman Filter), linear inequality constrained filtering can similarly be extended to problems with nonlinear constraints by linearizing locally (or by way of another scheme like an Unscented Kalman Filter). The accuracy achieved by methods dealing with nonlinear constraints will naturally depend on the structure and curvature of the nonlinear function itself. In the two experiments we provide, we look at incorporating inequality constraints to a tracking problem with nonlinear dynamics. Kalman Filter {#sec::kf} ============= A discrete-time Kalman Filter [@Kalman1960] attempts to find the best running estimate for a recursive system governed by the following model[^2]: $$\label{kfsm} x_{k} = F_{k,k-1} x_{k-1} + u_{k,k-1}, \qquad u_{k,k-1} \sim \mathcal{N}\left(0,Q_{k,k-1}\right)$$ $$\label{kfmm} z_{k} = H_{k} x_{k} + v_{k}, \qquad v_{k} \sim \mathcal{N}\left(0,R_{k}\right)$$ Here $x_{k}$ is an $n$-vector that represents the true state of the underlying system and $F_{k,k-1}$ is an $n \times n$ matrix that describes the transition dynamics of the system from $x_{k-1}$ to $x_{k}$. The measurement made by the observer is an $m$-vector $z_{k}$, and $H_{k}$ is an $m \times n$ matrix that transforms a vector from the state space into the appropriate vector in the measurement space. The noise terms $u_{k,k-1}$ (an $n$-vector) and $v_{k}$ (an $m$-vector) encompass known and unknown errors in $F_{k,k-1}$ and $H_{k}$ and are normally distributed with mean 0 and covariances given by $n \times n$ matrix $Q_{k,k-1}$ and $m \times m$ matrix $R_{k}$, respectively. At each iteration, the Kalman Filter makes a state prediction for $x_k$, denoted $\hat{x}_{k|k-1}$. We use the notation ${k|k-1}$ since we will only use measurements provided until time-step $k-1$ in order to make the prediction at time-step $k$. The state prediction error $\tilde{x}_{k|k-1}$ is defined as the difference between the true state and the state prediction, as below. $$\label{se1} \tilde{x}_{k|k-1} = x_{k} - \hat{x}_{k|k-1}$$ The covariance structure for the expected error on the state prediction is defined as the expectation of the outer product of the state prediction error. We call this covariance structure the error covariance prediction and denote it $P_{k|k-1}$.[^3] $$\label{P-outer1} P_{k|k-1} = \mathbb{E}\left[\left(\tilde{x}_{k|k-1}\right)\left(\tilde{x}_{k|k-1}\right)'\right]$$ The filter will also provide an updated state estimate for $x_{k}$, given all the measurements provided up to and including time step $k$. We denote these estimates by $\hat{x}_{k|k}$. We similarly define the state estimate error $\tilde{x}_{k|k}$ as below. $$\label{se2} \tilde{x}_{k|k} = x_{k} - \hat{x}_{k|k}$$ The expectation of the outer product of the state estimate error represents the covariance structure of the expected errors on the state estimate, which we call the updated error covariance and denote $P_{k|k}$. $$\label{P-outer2} P_{k|k} = \mathbb{E}\left[\left(\tilde{x}_{k|k}\right)\left(\tilde{x}_{k|k}\right)'\right]$$ At time-step $k$, we can make a prediction for the underlying state of the system by allowing the state to transition forward using our model for the dynamics and noting that $\mathbb{E}\left[u_{k,k-1}\right] = 0$. This serves as our state prediction. $$\label{kfsp} \hat{x}_{k|k-1} = F_{k,k-1} \hat{x}_{k-1|k-1}$$ If we expand the expectation in Equation , we have the following equation for the error covariance prediction. $$\label{kfcp} P_{k|k-1} = F_{k,k-1} P_{k-1|k-1} F_{k,k-1}' + Q_{k,k-1}$$ We can transform our state prediction into the measurement space, which is a prediction for the measurement we now expect to observe. $$\label{kfmp} \hat{z}_{k|k-1} = H_{k} \hat{x}_{k|k-1}$$ The difference between the observed measurement and our predicted measurement is the measurement residual, which we are hoping to minimize in this algorithm. $$\label{kfi} \nu_{k} = z_{k} - \hat{z}_{k|k-1}$$ We can also calculate the associated covariance for the measurement residual, which is the expectation of the outer product of the measurement residual with itself, $\mathbb{E}\left[\nu_k \nu_k'\right]$. We call this the measurement residual covariance. $$\label{kfic} S_{k} = H_{k} P_{k|k-1} H_{k}' + R_{k}$$ We can now define our updated state estimate as our prediction plus some perturbation, which is given by a weighting factor times the measurement residual. The weighting factor, called the Kalman Gain, will be discussed below. $$\label{kfsu} \hat{x}_{k|k} = \hat{x}_{k|k-1} + K_{k} \nu_{k}$$ Naturally, we can also calculate the updated error covariance by expanding the outer product in Equation .[^4] $$\label{kfcu} P_{k|k} = \left(\operatorname{I}- K_{k} H_{k}\right) P_{k|k-1} \left(\operatorname{I}- K_{k} H_{k}\right)' + K_k R_k K_k'$$ Now we would like to find the Kalman Gain $K_k$, which minimizes the mean square state estimate error, $\mathbb{E}\left[\left|\tilde{x}_{k|k}\right|^2\right]$. This is the same as minimizing the trace of the updated error covariance matrix above.[^5] After some calculus, we find the optimal gain that achieves this, written below.[^6] $$\label{kfkg} K_{k} = P_{k|k-1} H_{k}' S_{k}^{-1}$$ The covariance matrices in the Kalman Filter provide us with a measure for uncertainty in our predictions and updated state estimate. This is a very important feature for the various applications of filtering since we then know how much to trust our predictions and estimates. Also, since the method is recursive, we need to provide an initial covariance that is large enough to contain the initial state to ensure comprehensible performance. For a more detailed discussion of Kalman Filtering, we refer the reader to the following book [@BLK2001]. Equality Constrained Kalman Filtering ===================================== A number of approaches have been proposed for solving the equality constrained Kalman Filtering problem [@TS1988; @SAP1988; @WCC2002; @SC2002; @Gupta2007]. In this paper, we show two different methods. The first method will restrict the state at each iteration to lie in the equality constrained space. The second method will start with a constrained prediction, and restrict the Kalman Gain so that the estimate will lie in the constrained space. Our equality constraints in this paper will be defined as below, where $A$ is a $q \times n$ matrix, $b$ a $q$-vector, and $x_k$, the state, is a $n$-vector.[^7] $$\label{constraints} A x_k = b$$ So we would like our updated state estimate to satisfy the constraint at each iteration, as below. $$\label{kfsu-con} A \hat{x}_{k|k} = b$$ Similarly, we may also like the state prediction to be constrained, which would allow a better forecast for the system. $$A \hat{x}_{k|k-1} = b$$ In the following subsections, we will discuss methods for constraining the updated state estimate. In Section \[sec::aic\], we will extend these concepts and formulations to the inequality constrained case, and in Section \[sec::csp\], we will address the problem of constraining the prediction, as well. Projecting the state to lie in the constrained space {#sec::pue} ---------------------------------------------------- We can solve the following minimization problem for a given time-step $k$, where $\hat{x}_{k|k}^{P}$ is the constrained estimate, $W_k$ is any positive definite symmetric weighting matrix, and $\hat{x}_{k|k}$ is the unconstrained Kalman Filter updated estimate. $$\label{eq-proj-problem} \hat{x}_{k|k}^{P} = \operatorname*{arg\,min}_{x \in \mathbb{R}^n} \ \left\{\left(x - \hat{x}_{k|k} \right)' W_k \left(x - \hat{x}_{k|k} \right) : A x = b\right\}$$ The best constrained estimate is then given by $$\label{bce-xP} \hat{x}_{k|k}^{P} = \hat{x}_{k|k} - W_k^{-1} A' \left( A W_k^{-1} A' \right)^{-1} \left(A \hat{x}_{k|k} - b \right)$$ To find the updated error covariance matrix of the equality constrained filter, we first define the matrix $\Upsilon$ below.[^8] $$\Upsilon = W_k^{-1} A' \left(A W_k^{-1} A' \right)^{-1}$$ Equation can then be re-written as following. $$\label{xeq} \hat{x}_{k|k}^P = \hat{x}_{k|k} - \Upsilon\left(A \hat{x}_{k|k} - b \right)$$ We can find a reduced form for $x_k - \hat{x}_{k|k}^P$ as below. $$\begin{aligned} x_k - \hat{x}_{k|k}^P &= x_k - \hat{x}_{k|k} +\Upsilon \left(A \hat{x}_{k|k} - b - \left(A x_k - b \right)\right) \\ &= x_k - \hat{x}_{k|k} +\Upsilon \left(A \hat{x}_{k|k} - A x_k\right) \\ &= -\left(\operatorname{I}- \Upsilon A \right) \left(\hat{x}_{k|k} - x_k\right)\end{aligned}$$ Using the definition of the error covariance matrix, we arrive at the following expression. \[bce-PP\] $$\begin{aligned} P_{k|k}^P &= \mathbb{E}\left[\left(x_k - \hat{x}_{k|k}^P\right)\left(x_k - \hat{x}_{k|k}^P\right)'\right] \\ &= \mathbb{E}\left[\left(\operatorname{I}- \Upsilon A \right) \left(\hat{x}_{k|k} - x_k\right) \left(\hat{x}_{k|k} - x_k\right)' \left(\operatorname{I}- \Upsilon A \right)'\right] \\ &= \left(\operatorname{I}- \Upsilon A \right) P_{k|k} \left(\operatorname{I}- \Upsilon A \right)' \\ &= P_{k|k} - \Upsilon A P_{k|k} - P_{k|k} A' \Upsilon' + \Upsilon A P_{k|k} A' \Upsilon' \\ &= P_{k|k} - \Upsilon A P_{k|k} \\ &= \label{Peq} \left(\operatorname{I}- \Upsilon A \right) P_{k|k} \end{aligned}$$ It can be shown that choosing $W_k = P_{k|k}^{-1}$ results in the smallest updated error covariance. This also provides a measure of the information in the state at $k$.[^9] Restricting the optimal Kalman Gain so the updated state estimate lies in the constrained space ----------------------------------------------------------------------------------------------- Alternatively, we can expand the updated state estimate term in Equation using Equation . $$A \left( \hat{x}_{k|k-1} + K_{k} \nu_{k} \right) = b$$ Then, we can choose a Kalman Gain $K_k^R$, that forces the updated state estimate to be in the constrained space. In the unconstrained case, we chose the optimal Kalman Gain $K_k$, by solving the minimization problem below which yields Equation . $$K_k = \operatorname*{arg\,min}_{K \in \mathbb{R}^{n \times m}} {\ensuremath{\textnormal{trace}}}\left[ \left(\operatorname{I}- K H_{k}\right) P_{k|k-1} \left(\operatorname{I}- K H_{k}\right)' + K R_k K'\right]$$ Now we seek the optimal $K_k^R$ that satisfies the constrained optimization problem written below for a given time-step $k$. $$\label{min-con} \begin{split} K_k^R = \operatorname*{arg\,min}_{K \in \mathbb{R}^{n \times m}} & {\ensuremath{\textnormal{trace}}}\left[ \left(\operatorname{I}- K H_{k}\right) P_{k|k-1} \left(\operatorname{I}- K H_{k}\right)' + K R_k K'\right] \\ \textnormal{s.t. } & A \left( \hat{x}_{k|k-1} + K \nu_{k} \right) = b \end{split}$$ We will solve this problem using the method of Lagrange Multipliers. First, we take the steps below, using the vec notation (column stacking matrices so they appear as long vectors, see Appendix \[app::kv\]) to convert all appearances of $K$ in Equation into long vectors. Let us begin by expanding the following term.[^10] $$\begin{gathered} \nonumber{\ensuremath{\textnormal{trace}}}\left[\left(\operatorname{I}- K H_{k}\right) P_{k|k-1} \left(\operatorname{I}- K H_{k}\right)' + K R_k K' \right] \qquad \qquad \qquad \qquad \qquad \qquad \qquad\\ \begin{aligned} &\stackrel{\hphantom{\eqref{kfic}}}{=}{\ensuremath{\textnormal{trace}}}\left[ P_{k|k-1} - K H_{k} P_{k|k-1} - P_{k|k-1} H_{k}' K' + K H_{k} P_{k|k-1} H_{k}' K' + K R_k K' \right] \\ &\stackrel{\eqref{kfic}}{=} {\ensuremath{\textnormal{trace}}}\left[ P_{k|k-1} - K H_{k} P_{k|k-1} - P_{k|k-1} H_{k}' K' + K S_k K' \right] \\ &\stackrel{\hphantom{\eqref{kfic}}}{=}\label{trace-separated}{\ensuremath{\textnormal{trace}}}\left[ P_{k|k-1} \right] - {\ensuremath{\textnormal{trace}}}\left[ K H_{k} P_{k|k-1} \right] - {\ensuremath{\textnormal{trace}}}\left[ P_{k|k-1} H_{k}' K' \right] + {\ensuremath{\textnormal{trace}}}\left[ K S_k K' \right] \end{aligned}\end{gathered}$$ We now expand the last three terms in Equation one at a time.[^11] $$\label{KHP} \begin{aligned} {\ensuremath{\textnormal{trace}}}\left[ K H_{k} P_{k|k-1} \right] \stackrel{\eqref{tr-ab}}{=} {\ensuremath{\textnormal{vec}\left[{\left(H_k P_{k|k-1}\right)'}\right]}}' {\ensuremath{\textnormal{vec}\left[{K}\right]}} \\ = {\ensuremath{\textnormal{vec}\left[{P_{k|k-1} H_k'}\right]}}' {\ensuremath{\textnormal{vec}\left[{K}\right]}} \end{aligned}$$ $${\ensuremath{\textnormal{trace}}}\left[ P_{k|k-1} H_{k}' K' \right] \stackrel{\eqref{tr-ab}}{=} {\ensuremath{\textnormal{vec}\left[{K}\right]}}' {\ensuremath{\textnormal{vec}\left[{P_{k|k-1} H_k'}\right]}}$$ $$\label{KSK} \begin{aligned} {\ensuremath{\textnormal{trace}}}\left[ K S_k K' \right] &\stackrel{\eqref{tr-ab}}{=} {\ensuremath{\textnormal{vec}\left[{K}\right]}}' {\ensuremath{\textnormal{vec}\left[{K S_k}\right]}} \\ &\stackrel{\eqref{vec-ab}}{=} {\ensuremath{\textnormal{vec}\left[{K}\right]}}' {\ensuremath{\left({S}\otimes{\operatorname{I}}\right)}} {\ensuremath{\textnormal{vec}\left[{K}\right]}} \end{aligned}$$ Remembering that ${\ensuremath{\textnormal{trace}}}\left[ P_{k|k-1} \right]$ is constant, our objective function can be written as below. $$\begin{aligned} {\ensuremath{\textnormal{vec}\left[{K}\right]}}' \left(\operatorname{I}\otimes S_k \right) {\ensuremath{\textnormal{vec}\left[{K'}\right]}} &- {\ensuremath{\textnormal{vec}\left[{P_{k|k-1} H_k'}\right]}}' {\ensuremath{\textnormal{vec}\left[{K}\right]}}\\ &- {\ensuremath{\textnormal{vec}\left[{K}\right]}}' {\ensuremath{\textnormal{vec}\left[{P_{k|k-1} H_k'}\right]}} \end{aligned}$$ Using Equation on the equality constraints, our minimization problem is the following. $$\begin{split} K_k^R = \operatorname*{arg\,min}_{K \in \mathbb{R}^{n \times m}}& \ {\ensuremath{\textnormal{vec}\left[{K}\right]}}' {\ensuremath{\left({S_k}\otimes{\operatorname{I}}\right)}} {\ensuremath{\textnormal{vec}\left[{K}\right]}} \\ &- {\ensuremath{\textnormal{vec}\left[{P_{k|k-1} H_k'}\right]}}' {\ensuremath{\textnormal{vec}\left[{K}\right]}} \\ & - {\ensuremath{\textnormal{vec}\left[{K}\right]}}' {\ensuremath{\textnormal{vec}\left[{P_{k|k-1} H_k'}\right]}} \\ \textnormal{s.t. } & \left( \nu_{k}' \otimes A \right) {\ensuremath{\textnormal{vec}\left[{K}\right]}} = b - A \hat{x}_{k|k-1} \end{split}$$ Further, we simplify this problem so the minimization problem has only one quadratic term. We complete the square as follows. We want to find the unknown variable $\mu$ which will cancel the linear term. Let the quadratic term appear as follows. Note that the non-“${\ensuremath{\textnormal{vec}\left[{K}\right]}}$" term is dropped as is is irrelevant for the minimization problem. $$\left({\ensuremath{\textnormal{vec}\left[{K}\right]}} + \mu \right)' {\ensuremath{\left({S_k}\otimes{\operatorname{I}}\right)}} \left( {\ensuremath{\textnormal{vec}\left[{K}\right]}} + \mu \right)$$ The linear term in the expansion above is the following. $${\ensuremath{\textnormal{vec}\left[{K}\right]}}' {\ensuremath{\left({S_k}\otimes{\operatorname{I}}\right)}} \mu + \mu' {\ensuremath{\left({S_k}\otimes{\operatorname{I}}\right)}} {\ensuremath{\textnormal{vec}\left[{K}\right]}}$$ So we require that the two equations below hold. $$\begin{aligned} {\ensuremath{\left({S_k}\otimes{\operatorname{I}}\right)}} \mu &= -{\ensuremath{\textnormal{vec}\left[{P_{k|k-1} H_k'}\right]}} \\ \mu' {\ensuremath{\left({S_k}\otimes{\operatorname{I}}\right)}} &= -{\ensuremath{\textnormal{vec}\left[{P_{k|k-1} H_k'}\right]}}' \end{aligned}$$ This leads to the following value for $\mu$. $$\begin{aligned} \mu &\stackrel{\eqref{kron-inv}}{=} - {\ensuremath{\left({S_k^{-1}}\otimes{\operatorname{I}}\right)}} {\ensuremath{\textnormal{vec}\left[{P_{k|k-1} H_k'}\right]}} \\ &\stackrel{\eqref{vec-abc}}{=} -{\ensuremath{\textnormal{vec}\left[{P_{k|k-1} H_k' S_k^{-1}}\right]}} \\ &\stackrel{\eqref{kfkg}}{=} -{\ensuremath{\textnormal{vec}\left[{K_k}\right]}} \end{aligned}$$ Using Equation , our quadratic term in the minimization problem becomes the following. $$\left({\ensuremath{\textnormal{vec}\left[{K - K_k}\right]}} \right)' {\ensuremath{\left({S_k}\otimes{\operatorname{I}}\right)}} \left( {\ensuremath{\textnormal{vec}\left[{K - K_k}\right]}} \right)$$ Let $l = {\ensuremath{\textnormal{vec}\left[{K - K_k}\right]}}$. Then our minimization problem becomes the following. $$\begin{aligned} K_k^R = \operatorname*{arg\,min}_{l \in \mathbb{R}^{mn}} & \ l' {\ensuremath{\left({S_k}\otimes{\operatorname{I}}\right)}} l \\ \textnormal{s.t. }& \left( \nu_{k}' \otimes A \right) \left(l + {\ensuremath{\textnormal{vec}\left[{K_{k}}\right]}}\right) = b - A \hat{x}_{k|k-1} \end{aligned}$$ We can then re-write the constraint taking the ${\ensuremath{\textnormal{vec}\left[{K_k}\right]}}$ term to the other side as below. $$\begin{aligned} \left( \nu_{k}' \otimes A \right) l & = b - A \hat{x}_{k|k-1} - \left( \nu_{k}' \otimes A \right) {\ensuremath{\textnormal{vec}\left[{K_{k}}\right]}} \\ & \stackrel{\eqref{vec-abc}}{=} b - A \hat{x}_{k|k-1} -{\ensuremath{\textnormal{vec}\left[{A K_{k} \nu_k}\right]}} \\ & = b - A \hat{x}_{k|k-1} - A K_{k} \nu_k \\ & \stackrel{\eqref{kfsu}}= b - A \hat{x}_{k|k} \end{aligned}$$ This results in the following simplified form. $$\label{first-SDPT3} \begin{aligned} K_k^R = \operatorname*{arg\,min}_{l \in \mathbb{R}^{mn}}&\ l' {\ensuremath{\left({S_k}\otimes{\operatorname{I}}\right)}} l \\ \textnormal{s.t. }& \left( \nu_{k}' \otimes A \right) l = b - A \hat{x}_{k|k} \end{aligned}$$ We form the Lagrangian $\mathcal{L}$, where we introduce $q$ Lagrange Multipliers in vector $ \lambda = \left( \lambda_1, \lambda_2, \ldots, \lambda_q \right)'$ $$\begin{aligned} \mathcal{L} = & l' {\ensuremath{\left({S_k}\otimes{\operatorname{I}}\right)}} l - \lambda' \left[\left( \nu_{k}' \otimes A \right) l - b + A \hat{x}_{k|k}\right] \end{aligned}$$ We take the partial derivative with respect to $l$.[^12] $$\label{partial1} \frac{\partial \mathcal{L}}{\partial l} = 2 l' {\ensuremath{\left({S_k}\otimes{\operatorname{I}}\right)}} - \lambda' \left( \nu_{k}' \otimes A \right) \\$$ Similarly we can take the partial derivative with respect to the vector $\lambda$. $$\frac{\partial \mathcal{L}}{\partial \lambda} = \left( \nu_{k}' \otimes A \right) l - b + A \hat{x}_{k|k}$$ When both of these derivatives are set equal to the appropriate size zero vector, we have the solution to the system. Taking the transpose of Equation , we can write this system as $Mn = p$ with the following block definitions for $M,n$, and $p$. $$\label{M-matrix} M = \begin{bmatrix} 2 {\ensuremath{{S_k}\otimes{\operatorname{I}}}} & \nu_{k} \otimes A' \\ \nu_{k}' \otimes A & 0_{{\ensuremath{\left[{q}\times{q}\right]}}} \end{bmatrix}$$ $$\label{n-vector} n = \begin{bmatrix} l \\ \lambda \end{bmatrix}$$ $$\label{p-vector} p = \begin{bmatrix} 0_{{\ensuremath{\left[{mn}\times{1}\right]}}} \\ b - A \hat{x}_{k|k} \end{bmatrix}$$ We solve this system for vector $n$ in Appendix \[app::Mnp\]. The solution for $l$ is pasted below. $$\left(\left[S_k^{-1} \nu_k \left(\nu_{k}' S_k^{-1} \nu_k \right)^{-1}\right] \otimes \left[A' \left(A A' \right)^{-1} \right]\right) \left(b - A \hat{x}_{k|k}\right)$$ Bearing in mind that $b - A \hat{x}_{k|k} = {\ensuremath{\textnormal{vec}\left[{b - A \hat{x}_{k|k}}\right]}}$, we can use Equation to re-write $l$ as below.[^13] $${\ensuremath{\textnormal{vec}\left[{A' \left(A A' \right)^{-1}\left(b - A \hat{x}_{k|k} \right) \left(\nu_{k}' S_k^{-1} \nu_k \right)^{-1} \nu_k' S_k^{-1}}\right]}}$$ The resulting matrix inside the vec operation is then an $n$ by $m$ matrix. Remembering the definition for $l$, we notice that $K - K_k$ results in an $n$ by $m$ matrix also. Since both of the components inside the vec operation result in matrices of the same size, we can safely remove the vec operation from both sides. This results in the following optimal constrained Kalman Gain $K_k^R$. $$K_k - A' \left(A A' \right)^{-1}\left(A \hat{x}_{k|k} - b \right) \left(\nu_{k}' S_k^{-1} \nu_k \right)^{-1} \nu_k' S_k^{-1}$$ If we now substitute this Kalman Gain into Equation to find the constrained updated state estimate, we end up with the following. $$\hat{x}_{k|k}^R = \hat{x}_{k|k} - A' \left(A A' \right)^{-1}\left(A \hat{x}_{k|k} - b \right)$$ This is of course equivalent to the result of Equation with the weighting matrix $W_k$ chosen as the identity matrix. The error covariance for this estimate is given by Equation .[^14] Adding Inequality Constraints {#sec::aic} ============================= In the more general case of this problem, we may encounter equality and inequality constraints, as given below.[^15] $$\label{ineq-constraints} \begin{split} A x_{k} = b\\ C x_{k} \leq d \end{split}$$ So we would like our updated state estimate to satisfy the constraint at each iteration, as below. $$\begin{split} A \hat{x}_{k|k} = b \\ C \hat{x}_{k|k} \leq d \end{split}$$ Similarly, we may also like the state prediction to be constrained, which would allow a better forecast for the system. $$\begin{split} A \hat{x}_{k|k-1} = b \\ C \hat{x}_{k|k-1} \leq d \end{split}$$ We will present two analogous methods to those presented for the equality constrained case. In the first method, we will run the unconstrained filter, and at each iteration constrain the updated state estimate to lie in the constrained space. In the second method, we will find a Kalman Gain $\check{K}_k^R$ such that the the updated state estimate will be forced to lie in the constrained space. In both methods, we will no longer be able to find an analytic solution as before. Instead, we use numerical methods. By Projecting the Unconstrained Estimate {#sec::pue-ineq} ---------------------------------------- Given the best unconstrained estimate, we could solve the following minimization problem for a given time-step $k$, where $\check{x}_{k|k}^{P}$ is the inequality constrained estimate and $W_k$ is any positive definite symmetric weighting matrix. $$\begin{aligned} \check{x}_{k|k}^{P} = \operatorname*{arg\,min}_{x} &\ \left(x - \hat{x}_{k|k} \right)' W_k \left(x - \hat{x}_{k|k} \right) \\ \textnormal{s.t. } & A x = b \\ & C x \leq d \end{aligned}$$ For solving this inequality constrained optimization problem, we can use a variety of standard methods, or even an out-of-the-box solver, like `fmincon` in Matlab. Here we use an active set method [@Fletcher1981]. This is a common method for dealing with inequality constraints, where we treat a subset of the constraints (called the active set) as additional equality constraints. We ignore any inactive constraints when solving our optimization problem. After solving the problem, we check if our solution lies in the space given by the inequality constraints. If it doesn’t, we start from the solution in our previous iteration and move in the direction of the new solution until we hit a set of constraints. For each iteration, the active set is made up of those inequality constraints with non-zero Lagrange Multipliers. We first find the best estimate (using Equation for the equality constrained problem with the equality constraints given in Equation plus the active set of inequality constraints. Let us call the solution to this $\check{x}_{k|k,j}^{P*}$ since we have not yet checked if the solution lies in the inequality constrained space.[^16] In order to check this, we find the vector that we moved along to reach $\check{x}_{k|k,j}^{P*}$. This is given by the following. $$s = \check{x}_{k|k,j}^{P*} - \check{x}_{k|k,j-1}^P$$ We now iterate through each of our inequality constraints, to check if they are satisfied. If they are all satisfied, we choose $\tau_{\max}=1$. If they are not, we choose the largest value of $\tau_{\max}$ such that $\hat{x}_{k|k,j-1} + \tau_{\max} s$ lies in the inequality constrained space. We choose our estimate to be $$\check{x}_{k|k,j}^P = \check{x}_{k|k,j-1}^{P} + \tau_{\max} s$$ If we find the solution has converged within a pre-specified error, or we have reached a pre-specified maximum number of iterations, we choose this as the updated state estimate to our inequality constrained problem, denoted $\check{x}_{k|k}^P$. If we would like to take a further iteration on $j$, we check the Lagrange Multipliers at this new solution to determine the new active set.[^17] We then repeat by finding the best estimate for the equality constrained problem including the new active set as additional equality constraints. Since this is a Quadratic Programming problem, each step of $j$ guarantees the same estimate or a better estimate. When calculating the error covariance matrix for this estimate, we can also add on the safety term below. $$\left(\check{x}_{k|k,j}^P - \check{x}_{k|k,j-1}^{P}\right)\left(\check{x}_{k|k,j}^P - \check{x}_{k|k,j-1}^{P}\right)'$$ This is a measure of our convergence error and should typically be small relative to the unconstrained error covariance. We can then use Equation to project the covariance matrix onto the constrained subspace, but we only use the defined equality constraints. We do not incorporate any constraints in the active set when computing Equation since these still represent inequality constraints on the state. Ideally we would project the error covariance matrix into the inequality constrained subspace, but this projection is not trivial. By Restricting the Optimal Kalman Gain -------------------------------------- We could solve this problem by restricting the optimal Kalman gain also, as we did for equality constraints previously. We seek the optimal $K_k$ that satisfies the constrained optimization problem written below for a given time-step $k$. $$\label{min-con} \begin{aligned} \check{K}^R_k = \operatorname*{arg\,min}_{K \in \mathbb{R}^{n \times m}} & {\ensuremath{\textnormal{trace}}}\left[\left(\operatorname{I}- K H_{k}\right) P_{k|k-1} \left(\operatorname{I}- K H_{k}\right)' + K R_k K'\right] \\ \textnormal{s.t. } & A \left( \hat{x}_{k|k-1} + K_{k} \nu_{k} \right) = b \\ & C \left( \hat{x}_{k|k-1} + K_{k} \nu_{k} \right) \leq d \end{aligned}$$ Again, we can solve this problem using any inequality constrained optimization method (e.g., `fmincon` in Matlab or the active set method used previously). Here we solved the optimization problem using SDPT3, a Matlab package for solving semidefinite programming problems [@TTT1999]. When calculating the covariance matrix for the inequality constrained estimate, we use the restricted Kalman Gain. Again, we can add on the safety term for the convergence error, by taking the outer product of the difference between the updated state estimates calculated by the restricted Kalman Gain for the last two iterations of SDPT3. This covariance matrix is then projected onto the subspace as in Equation using the equality constraints only. Dealing with Nonlinearities {#sec::nl} =========================== Thus far, in the Kalman Filter we have dealt with linear models and constraints. A number of methods have been proposed to handle nonlinear models (e.g., Extended Kalman Filter [@BLK2001], Unscented Kalman Filter [@JU1997]). In this paper, we will focus on the most widely used of these, the Extended Kalman Filter. Let’s re-write the discrete unconstrained Kalman Filtering problem from Equations and below, incorporating nonlinear models. $$\label{kfsm-nl} x_{k} = f_{k,k-1} \left(x_{k-1}\right) + u_{k,k-1}, \qquad u_{k,k-1} \sim \mathcal{N}\left(0,Q_{k,k-1}\right)$$ $$\label{kfmm-nl} z_{k} = h_{k} \left(x_{k}\right) + v_{k}, \qquad v_{k} \sim \mathcal{N}\left(0,R_{k}\right)$$ In the above equations, we see that the transition matrix $F_{k,k-1}$ has been replaced by the nonlinear vector-valued function $f_{k,k-1}\left(\cdot\right)$, and similarly, the matrix $H_k$, which transforms a vector from the state space into the measurement space, has been replaced by the nonlinear vector-valued function $h_k\left(\cdot\right)$. The method proposed by the Extended Kalman Filter is to linearize the nonlinearities about the current state prediction (or estimate). That is, we choose $F_{k,k-1}$ as the Jacobian of $f_{k,k-1}$ evaluated at $\hat{x}_{k-1|k-1}$, and $H_k$ as the Jacobian of $h_k$ evaluated at $\hat{x}_{k|k-1}$ and proceed as in the linear Kalman Filter of Section \[sec::kf\].[^18] Numerical accuracy of these methods tends to depend heavily on the nonlinear functions. If we have linear constraints but a nonlinear $f_{k,k-1}\left(\cdot\right)$ and $h_k\left(\cdot\right)$, we can adapt the Extended Kalman Filter to fit into the framework of the methods described thus far. Nonlinear Equality and Inequality Constraints --------------------------------------------- Since equality and inequality constraints we model are often times nonlinear, it is important to make the extension to nonlinear equality and inequality constrained Kalman Filtering for the methods discussed thus far. Without loss of generality, our discussion here will pertain only to nonlinear inequality constraints. We can follow the same steps for equality constraints.[^19] We replace the linear inequality constraint on the state space by the following nonlinear inequality constraint $c\left(x_k\right) = d$, where $c\left(\cdot\right)$ is a vector-valued function. We can then linearize our constraint, $c\left(x_k\right) = d$, about the current state prediction $\hat{x}_{k|k-1}$, which gives us the following.[^20] $$c\left(\hat{x}_{k|k-1}\right) + C \left(x_k - \hat{x}_{k|k-1} \right) \lessapprox d$$ Here $C$ is defined as the Jacobian of $c$ evaluated at $\hat{x}_{k|k-1}$. This indicates then, that the nonlinear constraint we would like to model can be approximated by the following linear constraint $$\label{puenl} C x_k \lessapprox d + C \hat{x}_{k|k-1} - c\left(\hat{x}_{k|k-1}\right)$$ This constraint can be written as $\tilde{C} x_k \leq \tilde{d}$, which is an approximation to the nonlinear inequality constraint. It is now in a form that can be used by the methods described thus far. The nonlinearities in both the constraints and the models, $f_{k,k-1}\left(\cdot\right)$ and $h_k\left(\cdot\right)$, could have been linearized using a number of different methods (e.g., a derivative-free method, a higher order Taylor approximation). Also an iterative method could be used as in the Iterated Extended Kalman Filter [@BLK2001]. Constraining the State Prediction {#sec::csp} ================================= We haven’t yet discussed whether the state prediction (Equation ) also should be constrained. Forcing the constraints should provide a better prediction (which is used for forecasting in the Kalman Filter). Ideally, the transition matrix $F_{k,k-1}$ will take an updated state estimate satisfying the constraints at time $k-1$ and make a prediction that will satisfy the constraints at time $k$. Of course this may not be the case. In fact, the constraints may depend on the updated state estimate, which would be the case for nonlinear constraints. On the downside, constraining the state prediction increases computational cost per iteration. We propose three methods for dealing with the problem of constraining the state prediction. The first method is to project the matrix $F_{k,k-1}$ onto the constrained space. This is only possible for the equality constraints, as there is no trivial way to project $F_{k,k-1}$ to an inequality constrained space. We can use the same projector as in Equation so we have the following.[^21] $$F_{k,k-1}^P = \left(\operatorname{I}- \Upsilon A \right) F_{k,k-1}$$ Under the assumption that we have constrained our updated state estimate, this new transition matrix will make a prediction that will keep the estimate in the equality constrained space. Alternatively, if we weaken this assumption, i.e., we are not constraining the updated state estimate, we could solve the minimization problem below (analogous to Equation ). We can also incorporate inequality constraints now. $$\begin{aligned} \check{x}_{k|k-1}^{P} = \operatorname*{arg\,min}_{x} &\ \left(x - \hat{x}_{k|k-1} \right)' W_k \left(x - \hat{x}_{k|k-1} \right) \\ \textnormal{s.t. } & A x = b \\ & C x \leq d \end{aligned}$$ We can constrain the covariance matrix here also, in a similar fashion to the method described in Section \[sec::pue-ineq\]. The third method is to add to the constrained problem the additional constraints below, which ensure that the chosen estimate will produce a prediction at the next iteration that is also constrained. $$\begin{aligned} A_{k+1} F_{k+1,k} x_k &= b_{k+1} \\ C_{k+1} F_{k+1,k} x_k &\leq d_{k+1} \end{aligned}$$ If $A_{k+1}, b_{k+1}, C_{k+1}$ or $d_{k+1}$ depend on the estimate (e.g., if we are linearizing nonlinear functions $a\left(\cdot\right)$ or $b\left(\cdot\right)$, we can use an iterative method, which would resolve $A_{k+1}$ and $b_{k+1}$ using the current best updated state estimate (or prediction), re-calculate the best estimate using $A_{k+1}$ and $b_{k+1}$, and so forth until we are satisfied with the convergence. This method would be preferred since it looks ahead one time-step to choose a better estimate for the current iteration.[^22] However, it can be far more expensive computationally. Experiments =========== We provide two related experiments here. We have a car driving along a straight road with thickness 2 meters. The driver of the car traces a noisy sine curve (with the noise lying only in the frequency domain). The car is tagged with a device that transmits the location within some known error. We would like to track the position of the car. In the first experiment, we filter over the noisy data with the knowledge that the underlying function is a noisy sine curve. The inequality constrained methods will constrain the estimates to only take values in the interval $[-1,1]$. In the second experiment, we do not use the knowledge that the underlying curve is a sine curve. Instead we attempt to recover the true data using an autoregressive model of order 6 [@BJ1976]. We do, however, assume our unknown function only takes values in the interval $[-1,1]$, and we can again enforce these constraints when using the inequality constrained filter. The driver’s path is generated using the nonlinear stochastic process given by Equation . We start with the following initial point. $$\label{ickf1-x0} x_0 = \begin{bmatrix} 0 \text{\ m}\\ 0 \text{\ m} \end{bmatrix}$$ Our vector-valued transition function will depend on a discretization parameter $T$ and can be expressed as below. Here, we choose $T$ to be $\pi/10$, and we run the experiment from an initial time of 0 to a final time of $10 \pi$. $$f_{k,k-1} = \begin{bmatrix} \left(x_{k-1}\right)_1 + T \\ \sin \left(\left(x_{k-1}\right)_1 + T \right) \end{bmatrix}$$ And for the process noise we choose the following. $$Q_{k,k-1} = \begin{bmatrix} 0.1 \text{\ m}^2 & 0 \\ 0 & 0 \text{\ m}^2 \end{bmatrix}$$ The driver’s path is drawn out by the second element of the vector $x_k$ – the first element acts as an underlying state to generate the second element, which also allows a natural method to add noise in the frequency domain of the sine curve while keeping the process recursively generated. First Experiment ---------------- To create the measurements, we use the model from Equation , where $H_k$ is the square identity matrix of dimension 2. We choose $R_k$ as below to noise the data. This considerably masks the true underlying data as can be seen in Fig. \[fig-ickf1\].[^23] $$\label{ickf1-R} R_{k} = \begin{bmatrix} 10 \text{\ m}^2 & 0 \\ 0 & 10 \text{\ m}^2 \end{bmatrix}$$ ![We take our sine curve, which is already noisy in the frequency domain (due to process noise), and add measurement noise. The underlying sine curve is significantly masked.[]{data-label="fig-ickf1"}](ickf.ps){width="\columnwidth"} For the initial point of our filters, we choose the following point, which is different from the true initial point given in Equation . $$\hat{x}_{0|0} = \begin{bmatrix} 0 \text{\ m}\\ 1 \text{\ m} \end{bmatrix}$$ Our initial covariance is given as below.[^24]. $$P_{0|0} = \begin{bmatrix} 1 \text{\ m}^2 & 0.1\\ 0.1 & 1 \text{\ m}^2 \end{bmatrix}$$ In the filtering, we use the information that the underlying function is a sine curve, and our transition function $f_{k,k-1}$ changes to reflect a recursion in the second element of $x_k$ – now we will add on discretized pieces of a sine curve to our previous estimate. The function is given explicitly below. $$f_{k,k-1} = \begin{bmatrix} \left(x_{k-1}\right)_1 + T \\ \left(x_{k-1}\right)_1 + \sin \left(\left(x_{k-1}\right)_1 + T \right) - \sin \left(\left(x_{k-1}\right)_1\right) \end{bmatrix}$$ For the Extended Kalman Filter formulation, we will also require the Jacobian of this matrix denoted $F_{k,k-1}$, which is given below. $$F_{k,k-1} = \begin{bmatrix} 1 & 0 \\ \cos \left(\left(x_{k-1}\right)_1 + T \right) - \cos \left(\left(x_{k-1}\right)_1\right) & 1 \end{bmatrix}$$ The process noise $Q_{k,k-1}$, given below, is chosen similar to the noise used in generating the simulation, but is slightly larger to encompass both the noise in our above model and to prevent divergence due to numerical roundoff errors. The measurement noise $R_k$ is chosen the same as in Equation . $$Q_{k,k-1} = \begin{bmatrix} 0.1 \text{\ m}^2 & 0 \\ 0 & 0.1 \text{\ m}^2 \end{bmatrix}$$ The inequality constraints we enforce can be expressed using the notation throughout the chapter, with $C$ and $d$ as given below. $$C = \begin{bmatrix} 0 & 1 \\ 0 & -1 \end{bmatrix}$$ $$d = \begin{bmatrix} 1\\ 1 \end{bmatrix}$$ These constraints force the second element of the estimate $x_{k|k}$ (the sine portion) to lie in the interval $[-1,1]$. We do not have any equality constraints in this experiment. We run the unconstrained Kalman Filter and both of the constrained methods discussed previously. A plot of the true position and estimates is given in Fig. \[fig-ickf2\]. Notice that both constrained methods force the estimate to lie within the constrained space, while the unconstrained method can violate the constraints. ![We show our true underlying state, which is a sine curve noised in the frequency domain, along with the estimates from the unconstrained Kalman Filter, and both of our inequality constrained modifications. We also plotted dotted horizontal lines at the values -1 and 1. Both inequality constrained methods do not allow the estimate to leave the constrained space.[]{data-label="fig-ickf2"}](ickf2.ps){width="\columnwidth"} Second Experiment ----------------- In the previous experiment, we used the knowledge that the underlying function was a noisy sine curve. If this is not known, we face a significantly harder estimation problem. Let us assume nothing about the underlying function except that it must take values in the interval $[-1,1]$. A good model for estimating such an unknown function could be an autoregressive model. We can compare the unconstrained filter to the two constrained methods again using these assumption and an autoregressive model of order 6, or AR(6) as it is more commonly referred to. In the previous example, we used a large measurement noise $R_k$ to emphasize the gain achieved by using the constraint information. Such a large $R_k$ is probably not very realistic, and when using an autoregressive model, it will be hard to track such a noisy signal. To generate the measurements, we again use Equation , this time with $H_k$ and $R_k$ as given below. $$H_k = \begin{bmatrix} 0 & 1 \end{bmatrix}$$ $$R_k = \begin{bmatrix} 0.5 \text{\ m}^2 \end{bmatrix}$$ Our state will now be defined using the following 13-vector, in which the first element is the current estimate, the next five elements are lags, the six elements afterwards are coefficients on the current estimate and the lags, and the last element is a constant term. $$\hat{x}_{k|k} = \begin{bmatrix} y_k & y_{k-1} & \cdots & y_{k-5} & \alpha_1 & \alpha_2 & \cdots & \alpha_7 \end{bmatrix}'$$ Our matrix $H_k$ in the filter is a row vector with the first element 1, and all the rest as 0, so $y_{k|k-1}$ is actually our prediction $\hat{z}_{k|k-1}$ in the filter, describing where we believe the expected value of the next point in the time-series to lie. For the initial state, we choose a vector of all zeros, except the first and seventh element, which we choose as 1. This choice for the initial conditions leads to the first prediction on the time series being 1, which is incorrect as the true underlying state has expectation 0. For the initial covariance, we choose $\operatorname{I}_{{\ensuremath{\left[{13}\times{13}\right]}}}$ and add $0.1$ to all the off-diagonal elements.[^25] The transition function $f_{k,k-1}$ for the AR(6) model is given below. $$\begin{bmatrix} \min\left(1, \max\left(-1, \alpha_1 y_{k-1} + \cdots + \alpha_6 y_{k-6} + \alpha_7 \right) \right)\\ \min\left(1, \max\left(-1,y_{k-1} \right) \right)\\ \min\left(1, \max\left(-1,y_{k-2} \right) \right)\\ \min\left(1, \max\left(-1,y_{k-3} \right) \right)\\ \min\left(1, \max\left(-1,y_{k-4} \right) \right)\\ \min\left(1, \max\left(-1,y_{k-5} \right) \right)\\ \alpha_1 \\ \alpha_2 \\ \vdots \\ \alpha_6 \\ \alpha_7 \end{bmatrix}$$ Putting this into recursive notation, we have the following. $$\begin{bmatrix} \min\left(1, \max\left(-1, \left(x_{k-1}\right)_7 \left(x_{k-1}\right)_1 + \cdots + \left(x_{k-1}\right)_{13} \right) \right)\\ \min\left(1, \max\left(-1, \left(x_{k-1}\right)_1 \right) \right)\\ \min\left(1, \max\left(-1, \left(x_{k-1}\right)_2 \right) \right)\\ \min\left(1, \max\left(-1, \left(x_{k-1}\right)_3 \right) \right)\\ \min\left(1, \max\left(-1, \left(x_{k-1}\right)_4 \right) \right)\\ \min\left(1, \max\left(-1, \left(x_{k-1}\right)_5 \right) \right)\\ \left(x_{k-1}\right)_7 \\ \left(x_{k-1}\right)_8 \\ \vdots \\ \left(x_{k-1}\right)_{12} \\ \left(x_{k-1}\right)_{13} \end{bmatrix}$$ The Jacobian of $f_{k,k-1}$ is given below. We ignore the $\min \left( \cdot \right)$ and $\max \left( \cdot \right)$ operators since the derivative is not continuous across them, and we can reach the bounds by numerical error. Further, when enforced, the derivative would be 0, so by ignoring them, we are allowing our covariance matrix to be larger than necessary as well as more numerically stable. $$\begin{bmatrix} \begin{BMAT}{c.c}{c.c} \begin{BMAT}{c.c}{c.c} \begin{BMAT}{cc}{c} \left(x_{k-1}\right)_7 & \cdots \end{BMAT} & \left(x_{k-1}\right)_{12} \\ \operatorname{I}_{{\ensuremath{\left[{5}\times{5}\right]}}} & 0_{{\ensuremath{\left[{5}\times{1}\right]}}} \end{BMAT} & \begin{BMAT}{c}{c.c} \begin{BMAT}{cccc}{c} \left(x_{k-1}\right)_{1} & \cdots & \left(x_{k-1}\right)_{6} & 1 \\ \end{BMAT} \\ 0_{{\ensuremath{\left[{5}\times{7}\right]}}} \end{BMAT} \\ 0_{{\ensuremath{\left[{7}\times{6}\right]}}} & \operatorname{I}_{{\ensuremath{\left[{7}\times{7}\right]}}} \end{BMAT} \end{bmatrix}$$ For the process noise, we choose $Q_{k,k-1}$ to be a diagonal matrix with the first entry as 0.1 and all remaining entries as $10^{-6}$ since we know the prediction phase of the autoregressive model very well. The inequality constraints we enforce can be expressed using the notation throughout the chapter, with $C$ as given below and $d$ as a 12-vector of ones. $$C = \begin{bmatrix} \begin{BMAT}{c.c}{c} \begin{BMAT}{c}{c.c} \operatorname{I}_{{\ensuremath{\left[{6}\times{6}\right]}}} \\ -\operatorname{I}_{{\ensuremath{\left[{6}\times{6}\right]}}} \end{BMAT} & 0_{{\ensuremath{\left[{12}\times{7}\right]}}} \end{BMAT} \end{bmatrix}$$ These constraints force the current estimate and all of the lags to take values in the range $[-1,1]$. As an added feature of this filter, we are also estimating the lags at each iteration using more information although we don’t use it – this is a fixed interval smoothing. In Fig. \[fig-ickfb\], we plot the noisy measurements, true underlying state, and the filter estimates. Notice again that the constrained methods keep the estimates in the constrained space. Visually, we can see the improvement particularly near the edges of the constrained space. ![We show our true underlying state, which is a sine curve noised in the frequency domain, the noised measurements, and the estimates from the unconstrained and both inequality constrained filters. We also plotted dotted horizontal lines at the values -1 and 1. Both inequality constrained methods do not allow the estimate to leave the constrained space.[]{data-label="fig-ickfb"}](ickfb.ps){width="\columnwidth"} Conclusions =========== We’ve provided two different formulations for including constraints into a Kalman Filter. In the equality constrained framework, these formulations have analytic formulas, one of which is a special case of the other. In the inequality constrained case, we’ve shown two numerical methods for constraining the estimate. We also discussed how to constrain the state prediction and how to handle nonlinearities. Our two examples show that these methods ensure the estimate lies in the constrained space, which provides a better estimate structure. Kron and Vec {#app::kv} ============ In this appendix, we provide some definitions used earlier in the chapter. Given matrix $A \in \mathbb{R}^{ m \times n}$ and $B \in \mathbb{R}^{p \times q}$, we can define the right Kronecker product as below.[^26] $$\left( A \otimes B \right) = \begin{bmatrix} a_{1,1} B & \cdots & a_{1,n} B \\ \vdots & \ddots & \vdots \\ a_{m,1} B & \cdots & a_{m,n} B \end{bmatrix}$$ Given appropriately sized matrices $A, B, C,$ and $D$ such that all operations below are well-defined, we have the following equalities. $$\label{kron-trans} \left( A \otimes B \right)' = \left( A' \otimes B' \right)$$ $$\label{kron-inv} \left( A \otimes B \right) ^{-1} = \left( A^{-1} \otimes B^{-1} \right)$$ $$\label{kron-dist} \left( A \otimes B \right) \left( C \otimes D \right) = \left( AC \otimes BD \right)$$ We can also define the vectorization of an ${\ensuremath{\left[{m}\times{n}\right]}}$ matrix $A$, which is a linear transformation on a matrix that stacks the columns iteratively to form a long vector of size ${\ensuremath{\left[{mn}\times{1}\right]}}$, as below. $${\ensuremath{\textnormal{vec}\left[{A}\right]}} = \begin{bmatrix} a_{1,1} \\ \vdots \\ a_{m,1} \\ a_{1,2} \\ \vdots \\ a_{m,2} \\ \vdots \\ a_{1,n} \\ \vdots \\ a_{m,n} \end{bmatrix}$$ Using the vec operator, we can state the trivial definition below. $$\label{vec-sum} {\ensuremath{\textnormal{vec}\left[{A+B}\right]}} = {\ensuremath{\textnormal{vec}\left[{A}\right]}} + {\ensuremath{\textnormal{vec}\left[{B}\right]}}$$ Combining the vec operator with the Kronecker product, we have the following. $$\label{vec-ab} {\ensuremath{\textnormal{vec}\left[{AB}\right]}} = {\ensuremath{\left({B'}\otimes{\operatorname{I}}\right)}} {\ensuremath{\textnormal{vec}\left[{A}\right]}}$$ $$\label{vec-abc} {\ensuremath{\textnormal{vec}\left[{ABC}\right]}} = \left(C' \otimes A \right) {\ensuremath{\textnormal{vec}\left[{B}\right]}}$$ We can express the trace of a product of matrices as below. $$\label{tr-ab} {\ensuremath{\textnormal{trace}\left[{AB}\right]}} = {\ensuremath{\textnormal{vec}\left[{B'}\right]}}'{\ensuremath{\textnormal{vec}\left[{A}\right]}}$$ $$\begin{aligned} {\ensuremath{\textnormal{trace}\left[{ABC}\right]}} &= \label{trace-1} {\ensuremath{\textnormal{vec}\left[{B}\right]}}' \left(\operatorname{I}\otimes C\right) {\ensuremath{\textnormal{vec}\left[{A}\right]}} \\ &= \label{trace-2} {\ensuremath{\textnormal{vec}\left[{A}\right]}}' \left(\operatorname{I}\otimes B \right) {\ensuremath{\textnormal{vec}\left[{C}\right]}} \\ &= \label{trace-3} {\ensuremath{\textnormal{vec}\left[{A}\right]}}' \left(C \otimes \operatorname{I}\right) {\ensuremath{\textnormal{vec}\left[{B}\right]}}\end{aligned}$$ For more information, please see [@LT1985]. Analytic Block Representation for the inverse of a Saddle Point Matrix {#app::spm} ====================================================================== $M_S$ is a saddle point matrix if it has the block form below.[^27] $$\label{spm} M_S = \begin{bmatrix} A_S & B_S' \\ B_S & -C_S \end{bmatrix}$$ In the case that $A_S$ is nonsingular and the Schur complement $J_S = -\left(C_S + B_S A_S^{-1} B_S'\right)$ is also nonsingular in the above equation, it is known that the inverse of this saddle point matrix can be expressed analytically by the following equation (see e.g., [@BGL2005]). $$M_S^{-1} = \begin{bmatrix} A_S^{-1} + A_S^{-1} B_S' J_S^{-1} B_S A_S^{-1} & -A_S^{-1} B_S' J_S^{-1} \\ -J_S^{-1} B_S A_S^{-1} & J_S^{-1} \end{bmatrix}$$ Solution to the system $Mn=p$ {#app::Mnp} ============================= Here we solve the system $Mn=p$ from Equations , , and , re-stated below, for vector $n$. $$\label{Mnp} \begin{bmatrix} 2 {\ensuremath{{S_k}\otimes{\operatorname{I}}}} & \nu_{k} \otimes A' \\ \nu_{k}' \otimes A & 0_{{\ensuremath{\left[{q}\times{q}\right]}}} \end{bmatrix} \begin{bmatrix} l \\ \lambda \end{bmatrix} = \begin{bmatrix} 0_{{\ensuremath{\left[{mn}\times{1}\right]}}} \\ b - A \hat{x}_{k|k} \end{bmatrix}$$ $M$ is a saddle point matrix with the following equations to fit the block structure of Equation .[^28] $$\begin{aligned} A_S & = 2 {\ensuremath{{S_k}\otimes{\operatorname{I}}}} \\ B_S & = \nu_{k}' \otimes A \\ C_S & = 0_{{\ensuremath{\left[{q}\times{q}\right]}}}\end{aligned}$$ We can calculate the term $A_S^{-1} B_S'$. $$\begin{aligned} A_S^{-1} B_S' & = \left[ 2{\ensuremath{\left({S_k}\otimes{\operatorname{I}}\right)}}\right]^{-1} \left( \nu_{k}' \otimes A \right)' \\ &\stackrel{\eqref{kron-trans}\eqref{kron-inv}}{=} \frac{1}{2} {\ensuremath{\left({S_k^{-1}}\otimes{\operatorname{I}}\right)}} \left( \nu_{k} \otimes A' \right) \\ &\stackrel{\eqref{kron-dist}}{=} \frac{1}{2} \left( S_k^{-1} \nu_k \right) \otimes A'\end{aligned}$$ And as a result we have the following for $J_S$. $$\begin{aligned} J_S & = - \frac{1}{2} \left( \nu_{k}' \otimes A \right) \left[ \left( S_k^{-1} \nu_k \right) \otimes A' \right] \\ &\stackrel{\eqref{kron-dist}}{=} - \frac{1}{2} \left( \nu_{k}' S_k^{-1} \nu_k \right) \otimes \left(A A' \right) \end{aligned}$$ $J_S^{-1}$ is then, as below. $$\begin{aligned} J_S^{-1} & = -2 \left[ \left( \nu_{k}' S_k^{-1} \nu_k \right) \otimes \left( A A' \right)\right]^{-1} \\ &\stackrel{\eqref{kron-inv}}{=} -2 \left(\nu_{k}' S_k^{-1} \nu_k \right)^{-1} \otimes \left(A A' \right)^{-1}\end{aligned}$$ For the upper right block of $M^{-1}$, we then have the following expression. $$\begin{aligned} A_S^{-1} B_S' J_S^{-1} &= \left[\left( S_k^{-1} \nu_k \right) \otimes A' \right] \left[\left(\nu_{k}' S_k^{-1} \nu_k \right)^{-1} \otimes \left(A A' \right)^{-1}\right] \\ &\stackrel{\eqref{kron-dist}}{=} \left[S_k^{-1} \nu_k \left(\nu_{k}' S_k^{-1} \nu_k \right)^{-1}\right] \otimes \left[A' \left(A A' \right)^{-1} \right]\end{aligned}$$ Since the first block element of $p$ is a vector of zeros, we can solve for $n$ to arrive at the following solution for $l$. $$\left(\left[S_k^{-1} \nu_k \left(\nu_{k}' S_k^{-1} \nu_k \right)^{-1}\right] \otimes \left[A' \left(A A' \right)^{-1} \right]\right) \left(b - A \hat{x}_{k|k}\right) \\$$ The vector of Lagrange Multipliers $\lambda$ is given below. $$-2 \left[\left(\nu_{k}' S_k^{-1} \nu_k \right)^{-1} \otimes \left(A A' \right)^{-1} \right] \left(b - A \hat{x}_{k|k}\right)$$ [^1]: The similar extension for the method of [@WCC2002] was made in [@GHJ2005]. [^2]: The subscript $k$ on a variable stands for the $k$-th time step, the mathematical notation $\mathcal{N}\left(\mu,\Sigma\right)$ denotes a normally distributed random vector with mean $\mu$ and covariance $\Sigma$, and all vectors in this paper are column vectors (unless we are explicitly taking the transpose of the vector). [^3]: We use the prime notation on a vector or a matrix to denote its transpose throughout this paper. [^4]: The $\operatorname{I}$ in Equation represents the $n \times n$ identity matrix. Throughout this paper, we use $\operatorname{I}$ to denote the same matrix, except in Appendix \[app::kv\], where $\operatorname{I}$ is the appropriately sized identity matrix. [^5]: Note that $v'v = {\ensuremath{\textnormal{trace}\left[{vv'}\right]}}$ for some vector $v$. [^6]: We could also minimize the mean square state estimate error in the $N$ norm, where $N$ is a positive definite and symmetric weighting matrix. In the $N$ norm, the optimal gain would be $K^N_k = N^{\frac{1}{2}}K_k$. [^7]: $A$ and $b$ can be different for different $k$. We don’t subscript each $A$ and $b$ to avoid confusion. [^8]: Note that $\Upsilon A$ is a projection matrix, as is $\left(\operatorname{I}- \Upsilon A\right)$, by definition. If $A$ is poorly conditioned, we can use a QR factorization to avoid squaring the condition number. [^9]: If $M$ and $N$ are covariance matrices, we say $N$ is smaller than $M$ if $M-N$ is positive semidefinite. Another formulation for incorporating equality constraints into a Kalman Filter is by observing the constraints as pseudo-measurements [@TS1988; @WCC2002]. When $W_k$ is chosen to be $P_{k|k}^{-1}$, both of these methods are mathematically equivalent [@Gupta2007]. Also, a more numerically stable form of Equation with discussion is provided in [@Gupta2007]. [^10]: Throughout this paper, a number in parentheses above an equals sign means we made use of this equation number. [^11]: We use the symmetry of $P_{k|k-1}$ in Equation and the symmetry of $S_k$ in Equation . [^12]: We used the symmetry of ${\ensuremath{\left({S_k}\otimes{\operatorname{I}}\right)}}$ here. [^13]: Here we used the symmetry of $S_k^{-1}$ and $\left(\nu_{k}' S_k^{-1} \nu_k \right)^{-1}$ (the latter of which is actually just a scalar). [^14]: We can use the unconstrained or constrained Kalman Gain to find this error covariance matrix. Since the constrained Kalman Gain is suboptimal for the unconstrained problem, before projecting onto the constrained space, the constrained covariance will be different from the unconstrained covariance. However, the difference lies exactly in the space orthogonal to which the covariance is projected onto by Equation . The proof is omitted for brevity. [^15]: $C$ and $d$ can be different for different $k$. We don’t subscript each $C$ and $d$ to simplify notation. [^16]: For the inequality constrained filter, we allow multiple iterations within each step. The $j$ subscript indexes these further iterations. [^17]: The previous active set is not relevant. [^18]: We can also do a midpoint approximation to find $F_{k,k-1}$ by evaluating the Jacobian at $\left(\hat{x}_{k-1|k-1} + \hat{x}_{k|k-1}\right)/2$. This should be a much closer approximation to the nonlinear function. We use this approximation for the Extended Kalman Filter experiments later. [^19]: We replace the ‘$\leq$’ sign with an ‘$=$’ sign and the ‘$\lessapprox$’ with an ‘$\approx$’ sign. [^20]: This method is how the Extended Kalman Filter linearizes nonlinear functions for $f_{k,k-1}\left(\cdot\right)$ and $h_k\left(\cdot\right)$. Here $\hat{x}_{k|k-1}$ can be the state prediction of any of the constrained filters presented thus far and does not necessarily relate to the unconstrained state prediction. [^21]: In these three methods, the symmetric weighting matrix $W_k$ can be different. The resulting $\Upsilon$ can consequently also be different. [^22]: Further, we can add constraints for some arbitrary $n$ time-steps ahead. [^23]: The figure only shows the noisy sine curve, which is the second element of the measurement vector. The first element, which is a noisy straight line, isn’t plotted. [^24]: Nonzero off-diagonal elements in the initial covariance matrix often help the filter converge more quickly [^25]: The bracket subscript notation is used through the remainder of this paper to indicate the size of zero matrices and identity matrices. [^26]: The indices $m,n,p$, and $q$ and all matrix definitions are independent of any used earlier. Also, the subscript notation $a_{1,n}$ denotes the element in the first row and $n$-th column of $A$, and so forth. [^27]: The subscript $S$ notation is used to differentiate these matrices from any matrices defined earlier. [^28]: We use Equation with $B_S'$ to arrive at the same term for $B_s$ in Equation .

--- abstract: 'We classify all quadratic imaginary number fields that have a Euclidean ideal class. There are seven of them, they are of class number at most two, and in each case the unique class that generates the class-group is moreover norm-Euclidean.' author: - Hester Graves - Nick Ramsey title: Euclidean Ideals in Quadratic Imaginary Fields --- Introduction ============ In [@lenstra], Lenstra generalized the notion of a Euclidean domain to that of a Dedekind domain $R$ with a Euclidean ideal $C\subseteq R$. He proved that if $C$ is a Euclidean ideal then the class-group of $R$ is cyclic and generated by $C$. Moreover, if $C=R$, then his notion reduces to that of a Euclidean domain and the above result reduces to the familiar fact that a Euclidean domain is a principal ideal domain. Building on work of Weinberger ([@weinberger]), Lenstra ([@lenstra]) showed (conditional of the generalized Riemann hypothesis) that any generator of the class group of the ring of integers in a number field with infinite unit group is Euclidean. As the only number fields with finite unit group aside from ${{\mathbb{Q}}}$, it is natural to inquire about the situation for quadratic imaginary fields. It is known that, among the nine quadratic imaginary fields of class number one, exactly five have Euclidean integer rings and in each case the norm serves as a Euclidean algorithm (see [@samuel]). The purpose of this paper is to extend this result to the setting of Euclidean ideal classes by determining all quadratic imaginary fields that have a Euclidean ideal. We record them in the following theorem. \[thm1\] The quadratic imaginary fields with a Euclidean ideal are as follows. [class number]{} [fields]{} ------------------ ---------------------------------------------------------- 1 ${{\mathbb{Q}}}(\sqrt{-D})$ for $D\in\{1, 2, 3, 7, 11\}$ 2 ${{\mathbb{Q}}}(\sqrt{-D})$ for $D\in\{5,15\}$ In each case the unique class that generates the class-group is moreover norm-Euclidean. If one is only interested in *norm*-Euclidean ideals, then this result is contained in Proposition 2.4 of [@lenstra]. Of course, as the results of Weinberger and Lenstra mentioned above conditionally demonstrate and examples of Clark ([@clark]) and Harper ([@harper]) unconditionally demonstrate (in the class number one case), there are ideals in integer rings that are are Euclidean but not with respect to the norm. In the Euclidean ring setting, a construction of Motzkin ([@motzkin]) has proven to be a fruitful tool in the study of Euclidean rings that are not norm-Euclidean. In her thesis, the first author adapts this construction to the Euclidean ideal setting. Her techniques are the main tool used to prove Theorem \[thm1\]. \ [**Convention**]{} - In this paper all Euclidean algorithms are taken to be ${{\mathbb{N}}}$-valued and ${{\mathbb{N}}}$ is taken to include $0$. A Motzkin-type construction for Euclidean ideals ================================================ Let $R$ be a Dedekind domain with fraction field $K$. We denote by $E$ the set of fractional ideals of $R$ in $K$ that contain $R$ itself. Recall from [@lenstra] that $C$ is called *Euclidean* if there exists a function $\psi:E\longrightarrow {{\mathbb{N}}}$ such that for all $I\in E$ and all $x\in IC\setminus C$, there exists $y\in C$ such that $$\psi((x+y)^{-1}IC)<\psi(I)$$ In this case $\psi$ is called a *Euclidean algorithm* for $C$. If $C$ is Euclidean then it generates the class-group of $R$. Also, if $\psi$ is a Euclidean algorithm for $C$ then it is also a Euclidean algorithm for any ideal in the same class as $C$ and no ideal in a different class than $C$. These facts are all elementary and can be found in [@lenstra]. The following definition, given in [@gravesthesis], is an adaptation of Motzkin’s construction to the Euclidean ideal setting. \[hesterconstruction\] Let $C$ be a non-zero ideal in $R$. We define a nested sequence of subsets of $E$ as follows. Set $A_{C,0} = \{R\}$ and for $i>0$ we set $$A_{C,i} = A_{C,i-1}\cup \left\{ I\in E \left| \begin{array}{c} \forall x\in IC\setminus C\ \ \exists y\in C \\ \mbox{such\ that}\ (x-y)^{-1}IC \in A_{C,i-1}\end{array}\right.\right\}$$ Finally, set $A_C = \cup_i A_{C,i}$. When the ideal $C$ is fixed or otherwise clear from the context, we will often omit it from the notation and simply use $A_i$ and $A=\cup_i A_i$. The significance of this construction is the following lemma of the first author (see [@gravesthesis]). \[lem1\] The ideal $C$ is Euclidean if and only if $A=E$. In fact, one can say more. Namely, if $A=E$, then the function $\psi: E\longrightarrow {{\mathbb{N}}}$ defined by $\psi(I) = i$ if $I\in A_i\setminus A_{i-1}$ is a Euclidean algorithm for $C$ and is minimal with respect to this property. The following two lemmas furnish constraints on the sets $A_{C,i}$ that will be useful in what follows. The first is general in nature and highlights the role of cyclicity of the class-group. \[lem2\] If $I\in A_i\setminus A_{i-1}$ then $[I] = [C^{-i}]$. This is an immediate inductive consequence of Definition \[hesterconstruction\]. By definition, any $I\in A_i\setminus A_{i-1}$ has the property that for all $x\in IC\setminus C$ there exists $y\in C$ such that $(x+y)^{-1}IC\in A_{i-1}$. However, using the previous lemma and ideal class considerations, one can often cut down the set “$A_{i-1}$” in this statement. When $R^\times$ is finite, this observation is particularly useful because one can use it to efficiently bound the norm of a new element of $A_i$, as the following lemma demonstrates. \[lem3\] Suppose that $R^\times$ is finite, and suppose that $S\subseteq A_{i-1}$ is a subset with the property that, if $I\in A_i\setminus A_{i-1}$ then for all $x\in IC\setminus C$ there exists $y\in C$ such that $$(x-y)^{-1}IC\in S$$ Then all $I\in A_i\setminus A_{i-1}$ have the property that $${{\mathrm{Nm}}}(I^{-1})\leq |R^\times||S|+1.$$ For $x\in IC\setminus C$, the condition that there exists $y\in C$ such that $(x-y)^{-1}IC$ is a particular ideal depends only on the class of $x$ in $IC/C$. Fix an ideal $I\in A_i\setminus A_{i-1}$. For each non-zero class in $IC/C$ choose a representative $x\in IC\setminus C$ and a $y\in C$ such that $(x-y)^{-1}IC\in S\subseteq A_{i-1}$. This collection of choices amounts to a (decidedly non-canonical) function $$(IC\setminus C)/C\longrightarrow S$$ Suppose that two classes in $(IC\setminus C)/C$ map to the same ideal and let $x_1$ and $x_2$ be their chosen representatives. Then there exist $y_1,y_2\in C$ such that $$(x_1-y_1)^{-1}IC = (x_2-y_2)^{-1}IC.$$ It follows that there exists a unit $u\in {{\EuScript{O}}}_K^\times$ such that $x_1-y_1=u(x_2-y_2)$, and hence $x_1-ux_2 = y_1-uy_2\in C$. That is, the classes of $x_1$ and $x_2$ in $IC/C$ differ (multiplicatively) by a unit. The upshot is that the set of nonzero classes in $IC/C$ modulo the multiplicative action of $R^\times$ injects into $S$, and hence $$|IC/C| -1 \leq |R^\times||S|$$ But since $R$ is Dedekind, the left side is simply ${{\mathrm{Nm}}}(I^{-1})-1$, so this is the desired inequality. Application to quadratic imaginary fields ========================================= For the remainder of the paper, $K$ will denote a quadratic imaginary field and ${{\EuScript{O}}}_K$ its ring of integers. One approach to classifying the Euclidean ${{\EuScript{O}}}_K$ is to break into cases according to the factorizations of small rational primes in ${{\EuScript{O}}}_K$ and use Lemmas \[lem2\] and \[lem3\] of the previous section to glean consequences about the sets $A_i$. If one uses the crutch of known lists of quadratic imaginary fields of small class number, then this approach *nearly* yields Theorem \[thm1\]. Indeed, aside from the known norm-Euclidean cases detailed in this theorem, one finds in nearly all cases that the sequence of sets $A_i$ stabilizes very quickly (one needn’t ever consider ideals with prime factors of norm larger than 7). The one vexing exception is the field $K={{\mathbb{Q}}}(\sqrt{-23})$. A bit of computation with SAGE ([@sage]) reveals that the $A_i$ in this case contain at least the inverses of every ideal of norm up to $47$. Lacking the patience to continue this computation to its end (and indeed the confidence that it had one), we decided to switch perspective. It is convenient to first dispense with the cases where ${{\EuScript{O}}}_K^\times$ is unusually large, namely $K={{\mathbb{Q}}}(\sqrt{-1})$ and $K={{\mathbb{Q}}}(\sqrt{-3})$. These two fields are well-known to have norm-Euclidean rings of integers, and for any other $K$ we have ${{\EuScript{O}}}_K^\times = \{\pm 1\}$. From this point on we assume that $K$ is among the latter fields. It then follows from Lemma \[lem3\] that any $I\in A_1\setminus A_0$ has ${{\mathrm{Nm}}}(I^{-1})\leq 3$. As a result, by Lemma \[lem2\], a Euclidean ideal class in $K$ is represented by a residue degree one prime lying over $2$ or $3$. Fix an embedding of $K$ into ${{\mathbb{C}}}$. We will freely identify $K$ with its image in ${{\mathbb{C}}}$ in what follows. Under this embedding, the field norm corresponds to the square of the complex absolute value. Note that a nonzero fractional ideal $C$ of $K$ is identified with a lattice in ${{\mathbb{C}}}$. Consider the union of the open disks of radius $\sqrt{{{\mathrm{Nm}}}(C)}$ centered about these lattice points. It is a simple consequence of the definition and the above comments that $C$ is norm-Euclidean if these disks cover all of ${{\mathbb{C}}}$ (see also [@lenstra]). The moral of the following result is that, if this covering fails too badly, then $C$ cannot possibly be Euclidean for *any* choice of algorithm. \[prop1\] Let $K$ and $C$ be as above, and let $U$ denote the union of the open disks of radius $\sqrt{{{\mathrm{Nm}}}(C)}$ centered at the elements of $C$. If the complement of $U$ in ${{\mathbb{C}}}$ contains a nonempty open set, then $C$ is not a Euclidean ideal. Before proceeding with the proof, we need the following lemma, which effectively states that inverses of fractional ideals of increasingly large norm are increasingly dense in $K$. \[lemdense\] Let $K$ be a quadratic imaginary field and let $\varepsilon>0$ be any positive real number. There exists a number $M$ such that, for all $z\in K$ and all fractional ideals $I$ with ${{\mathrm{Nm}}}(I)>M$, there exists an element $x\in I^{-1}$ such that ${{\mathrm{Nm}}}(x-z)<\epsilon$. Let $I_1, I_2, \dots, I_h$ be a set of representatives of the ideal class group of $K$. Viewing each fractional ideal $I_i^{-1}$ as a lattice in ${{\mathbb{C}}}$, we see that disks of sufficiently large radius centered at the elements of $C$ will cover ${{\mathbb{C}}}$. Thus, for each $i$ there exists a positive number $M_i$ such that, for each $z'\in K$ there exists $x'\in I_i^{-1}$ such that $${{\mathrm{Nm}}}(x'-z')=|x'-z'|^2<M_i$$ Now choose $M$ so that $M>\max_i (M_i{{\mathrm{Nm}}}(I_i)/\varepsilon)$. Let $z\in K$ and let $I$ be a fractional ideal with ${{\mathrm{Nm}}}(I)>M$. Choose $i$ so that $I=gI_i$ for some $g\in K^\times$ and pick $x'\in I_i^{-1}$ such that ${{\mathrm{Nm}}}(x'-gz)< M_i$. Then $$\begin{aligned} {{\mathrm{Nm}}}(g^{-1}x'-z) &= & {{\mathrm{Nm}}}(g^{-1}){{\mathrm{Nm}}}(x'-gz) \\ &<& \frac{{{\mathrm{Nm}}}(I_i)}{{{\mathrm{Nm}}}(I)}M_i <\varepsilon\end{aligned}$$ so that $x=g^{-1}x'\in (I_iI^{-1})I_i^{-1} = I^{-1}$ is the desired element. (of Proposition \[prop1\]) Suppose that the complement of $U$ in ${{\mathbb{C}}}$ contains a nonempty open set. Arguing by contradiction, let us suppose that $C$ is Euclidean for the algorithm $\psi:E\longrightarrow {{\mathbb{N}}}$, so by Lemma \[lem1\], $A=\cup A_i = E$. Since $K$ is dense in ${{\mathbb{C}}}$ under its embedding, the complement of $U$ contains an $\sqrt{\varepsilon}$-neighborhood of an element $z\in K$ for some $\varepsilon>0$. Let $M$ be as in Lemma \[lemdense\] for this $K$ and $\varepsilon$. Suppose that $I_0\in E$ and ${{\mathrm{Nm}}}(I_0^{-1}C^{-1})>M$. By Lemma \[lemdense\], there exists $x\in I_0C$ such that $$|x-z| = ({{\mathrm{Nm}}}(x-z))^{1/2}<\sqrt{\varepsilon}$$ It follows that $x$ lies in the complement of $U$. Since $x\in I_0C\setminus C$ and $I_0\in E=A$, there exists $y\in C$ such that $\psi((x+y)^{-1}I_0C)<\psi(I_0)$. Define $I_1 = (x+y)^{-1}I_0C$ and note that $I_1\in E$ and $\psi(I_1)<\psi(I_0)$. By the above, we also have $${{\mathrm{Nm}}}(I_1) = {{\mathrm{Nm}}}(I_0){{\mathrm{Nm}}}(C)/{{\mathrm{Nm}}}(x+y) = {{\mathrm{Nm}}}(I_0){{\mathrm{Nm}}}(C)/|x+y|^2 \leq {{\mathrm{Nm}}}(I_0)$$ since $x$ lies in the complement of $U$. Because of this norm inequality, we again have ${{\mathrm{Nm}}}(I_1^{-1}/C)>M$ and can repeat the argument with $I_0$ replaced by $I_1$ to obtain a fractional ideal $I_2\in E$ with $\psi(I_2)<\psi(I_1)$ and ${{\mathrm{Nm}}}(I_2)\leq {{\mathrm{Nm}}}(I_1)$. Proceeding in this fashion, we obtain a sequence of ideals $I_0, I_1, \dots$ in $E$ with $${{\mathrm{Nm}}}(I_0)\geq {{\mathrm{Nm}}}(I_1)\geq {{\mathrm{Nm}}}(I_2)\geq \cdots$$ and $$\psi(I_0)>\psi(I_1)>\psi(I_2)>\cdots$$ But the latter is clearly impossible, as ${{\mathbb{N}}}$ is well-ordered. We conclude that $C$ could not have been Euclidean to begin with. With the running restrictions on $K$, we know that any Euclidean ideal class is represented by a prime of norm $2$ or $3$. We are led by the above to examine the union $U$ as above for $C$ a degree one prime dividing $2$ or $3$. In determining the extent to which $U$ covers ${{\mathbb{C}}}$, it is clear that one need only consider a particular fundamental domain for $C$ in ${{\mathbb{C}}}$. Let $K = {{\mathbb{Q}}}(\sqrt{-D})$ for a square-free positive integer $D$. In each of the following cases, we identify which $D$ correspond to the case, and for such $D$ we draw a fundamental domain and the covering circles comprising $U$ that meet this fundamental domain. The pictures below are were generated with SAGE ([@sage]). As we will see, as $D$ increases, the fundamental domains we choose below get too tall to be covered entirely by these disks. In each case, we illustrate the fundamental domain and the disks comprising $U$ that meet it. We do this for the following $D$ in each class: those for which $U$ covers all of ${{\mathbb{C}}}$ and the first $D$ for which it does not (keeping in mind that we are only interested in square-free $D$). For the latter $D$, as we will see, it always happens that the complement of $U$ moreover contains an open set (as opposed to having a nonempty but discrete complement), so Proposition \[prop1\] implies that $C$ is not Euclidean. We note that in each case below, the given *ideal* generators of $C$ are also generators of $C$ as an Abelian group, as is easy to check. Thus the parallelogram that they span forms the boundary of a fundamental domain, which is the one that we consider in each case. Since there is a degree one prime dividing $2$, $2$ either ramifies or splits in $K$, corresponding to the conditions $D\equiv 1,2\pmod{4}$ and $D\equiv 7\pmod{8}$, respectively. We consider the various sub-cases separately. Here ${{\EuScript{O}}}_K$ is generated as an algebra over ${{\mathbb{Z}}}$ by $\sqrt{-D}$, so a defining polynomial of ${{\EuScript{O}}}_K$ is $x^2+D$. Modulo $2$, this is congruent to $(x+1)^2$, so the unique prime above $(2)$ is $(2,\sqrt{-D}+1)$. Thus we use $2$ and $\sqrt{-D}+1$ to span a parallelogram bounding a fundamental domain, and obtain the following pictures for increasing $D$.    $D=1$ $D=5$ $D=13$ We conclude that the only $D$ under consideration (recall that $D=1$ was treated separately because of additional units) in this class for which a degree one prime over $2$ is Euclidean is $D=5$. Again the defining polynomial of ${{\EuScript{O}}}_K$ is $x^2+D$. Modulo $2$, this is simply $x^2$, so the prime above $(2)$ is $(2,\sqrt{-D})$, and working as above we obtain the pictures.   $D=2$ $D=6$ We conclude that the only $D$ in this class for which a degree one prime over $2$ is Euclidean is $D=2$ Here ${{\EuScript{O}}}_K$ is generated as an algebra over ${{\mathbb{Z}}}$ by $\frac{1+\sqrt{-D}}{2}$. The defining polynomial is then $x^2-x+\frac{1+D}{4}$, which is congruent modulo $2$ to $x(x-1)$. The primes above $(2)$ are $(2,\frac{1+\sqrt{-D}}{2})$ and $(2,\frac{-1+\sqrt{-D}}{2})$. As these are Galois-conjugate, we need only examine the first, which gives the following pictures.    $D=7$ $D=15$ $D=23$ We conclude that the only $D$ in this class for which a degree one prime over $2$ is Euclidean are $D=7$ and $D=15$. It is worth mentioning that the complement $U$ for $D=23$ is very small. This explains the atypical behavior of the sets $A_i$ for ${{\mathbb{Q}}}(\sqrt{-23})$ in that they do not stabilize quickly. \ Next, we examine the case of a degree one primes dividing $3$. Again, this means that either $3$ ramifies or splits in $K$, but in order associate a congruence condition on $D$, we must also take into account the residue of $D$ mod $4$ since this effects the nature of the ring of integers ${{\EuScript{O}}}_K$. These conditions are equivalent to $D\equiv 6, 9\pmod{12}$. Here ${{\EuScript{O}}}_K$ is generated over ${{\mathbb{Z}}}$ by $\sqrt{-D}$, so a defining polynomial is $x^2+D$. Modulo $3$ this is $x^2$, so the prime above $(3)$ is $(3,\sqrt{-D})$. Already for $D=6$, we see that the complement of $U$ contains a nonempty open set.  $D=6$ We conclude that there are no $D$ in this class for which a degree one prime over $3$ is Euclidean. These conditions amount to $D\equiv 2, 5\pmod{12}$. Again, $x^2+D$ is a defining polynomial, which is congruent modulo $3$ to $(x-1)(x+1)$. Thus the primes above $(3)$ are $(3,\sqrt{-D}+1)$ and $(3,\sqrt{-D}-1)$. As these are Galois-conjugate, we need only consider the first, which gives the following pictures.    $D=2$ $D=5$ $D=14$ We conclude that the only $D$ in this class for which a degree one prime over $3$ is Euclidean are $D=2$ and $D=5$. This amounts to $D\equiv 3\pmod{12}$, and in this case $\frac{1+\sqrt{-D}}{2}$ generates ${{\EuScript{O}}}_K$, and $x^2-x+\frac{1+D}{4}$ is a defining polynomial. Modulo $3$, this is congruent to $(x+1)^2$, so the prime above $(3)$ is $(3,\frac{3+\sqrt{-D}}{2})$.    $D=3$ $D=15$ $D=39$ We conclude that the only $D$ under consideration (recall that $D=3$ was treated separately because of extra units) in this class for which a degree one prime over $3$ is Euclidean is $D=15$. This amounts to $D\equiv 11 \pmod{12}$. Again, ${{\EuScript{O}}}_K$ is generated by $\frac{1+\sqrt{-D}}{2}$ and $x^2-x+\frac{1+D}{4}$ is a defining polynomial. Modulo $3$, this is $x(x-1)$, so the primes above $(3)$ are $(3,\frac{1+\sqrt{-D}}{2})$ and $(3,\frac{-1+\sqrt{-D}}{2})$. As these are Galois-conjugate, we need only consider the first, which gives the following pictures.   $D=11$ $D=23$ We conclude that the only $D$ in this class for which a degree one prime over $3$ is Euclidean is $D=11$. \ The upshot of this enumeration is that the only $D$ for which ${{\mathbb{Q}}}(\sqrt{-D})$ has a Euclidean ideal class are $D\in \{1, 2, 3, 5, 7, 11, 15\}$, and each the unique generator of the class-group is in fact norm-Euclidean, establishing Theorem \[thm1\]. [1]{} – <http://www.sagemath.org/>. – [[“]{}A quadratic field which is [E]{}uclidean but not norm-[E]{}uclidean[”]{}]{}, *Manuscripta Math.* **83** (1994), no. 3-4, p. 327–330. – [[“]{}On euclidean ideal classes[”]{}]{}, *University of Michigan Thesis* (2009). – [[“]{}[$\Bbb Z[\sqrt{14}]$]{} is [E]{}uclidean[”]{}]{}, *Canad. J. Math.* **56** (2004), no. 1, p. 55–70. – [[“]{}Euclidean ideal classes[”]{}]{}, in *Journées [A]{}rithmétiques de [L]{}uminy ([C]{}olloq. [I]{}nternat. [CNRS]{}, [C]{}entre [U]{}niv. [L]{}uminy, [L]{}uminy, 1978)*, Astérisque, vol. 61, Soc. Math. France, Paris, 1979, p. 121–131. – [[“]{}The [E]{}uclidean algorithm[”]{}]{}, *Bull. Amer. Math. Soc.* **55** (1949), p. 1142–1146. – [[“]{}About [E]{}uclidean rings[”]{}]{}, *J. Algebra* **19** (1971), p. 282–301. – [[“]{}On [E]{}uclidean rings of algebraic integers[”]{}]{}, in *Analytic number theory ([P]{}roc. [S]{}ympos. [P]{}ure [M]{}ath., [V]{}ol. [XXIV]{}, [S]{}t. [L]{}ouis [U]{}niv., [S]{}t. [L]{}ouis, [M]{}o., 1972)*, Amer. Math. Soc., Providence, R. I., 1973, p. 321–332.

--- abstract: 'We show that any arbitrary time-dependent density operator of an open system can always be described in terms of an operator-sum representation (Kraus representation) regardless of its initial condition and the path of its evolution in the state space, and we provide a general expression of Kraus operators for arbitrary time-dependent density operator of an $N$-dimensional system. Moreover, applications of our result are illustrated through several examples.' address: | $^1$Department of Physics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore\ $^2$Department of Physics, Shandong Normal University, Jinan 250014, People’s Republic of China\ $^3$ National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore 639798, Singapore author: - 'D. M. Tong$^{1,2}$, L. C. Kwek$^{1,3}$, C. H. Oh$^1$, Jing-Ling Chen$^1$, and L. Ma$^1$' title: 'Operator-sum representation of time-dependent density operators and its applications' --- Arbitrary states of any quantum system, be it open or closed, can always be described by density matrices $\rho(t)$. If the system is closed, its time evolution is unitary, and there exists a unitary operator $U(t)$, such that $\rho(t)=U(t)\rho(0)U(t)^\dagger$, where $\rho(0)$ is the initial state. However, if the system is open, the evolution is not necessarily unitary and the above relation between $\rho(t)$ and $\rho(0)$ is in general not valid. To describe the evolution of an open system, one usually employs the Kraus representation[@Kraus]. If there exist operators $M_\mu(t)$, called Kraus operators, that satisfy $$\begin{aligned} \rho(t)=\sum\limits_\mu M_\mu(t)\rho(0)M_\mu(t)^\dagger, \label{rhot}\end{aligned}$$and $$\begin{aligned} \sum\limits_\mu M_\mu(t)^\dagger M_\mu(t)=I, \label{mm}\end{aligned}$$ then the evolution of $\rho(t)$ is said to have operator-sum representation or the Kraus representation. It has been known that a completely positive map always possesses the operator-sum representation while a general positive map may not do[@Kraus; @Preskill; @Pechukas; @Pomero; @Bouda; @Philip; @Gen; @Peixoto; @Mika; @Sonja]. The papers[@Peter; @Salgado; @Hayashi] furthermore pursued the existence of the Kraus representation by investigating the role of initial correlations between the open system and its environment, and showed that a map based on the reduced dynamics, in general, cannot be described as the Kraus representation in the presence of the initial correlations because an additional inhomogeneous part appears. On the other hand, we know that for a qubit system, although a non-completely positive map $\$:\rho(0)\rightarrow\rho(t)$ does not possess the Kraus representation with the Kraus operators independent of $\rho(0)$, the state $\rho(t)=\$(\rho(0))$ can still be expressed in the form of Eq. (\[rhot\]) with the operators $M_\mu(t)$ selected for $\rho(0)$[@Philip; @Gen; @Tong1]. Arbitrary density matrices $\rho(t)$ of a qubit system can always be described in terms of the operator-sum representation. This start us to wonder whether the same property is valid for $N$-dimensional systems. That is, for any arbitrary time-dependent state $\rho(t)$ and initial state $\rho(0)$ of an $N$-dimensional system, is it true that the operators $M_\mu(t) $ can always be found so that (\[rhot\]) and (\[mm\]) are fulfilled? This is a interesting issue as it extends the notion of the Kraus representation associated with a map to that associated with an evolution of the state and we find that it is useful to some physical problem. In this paper, we prove that the time-dependent density operator of an arbitrary $N$-dimensional quantum system can always be described in terms of the Kraus representation and provide a general expression of Kraus operators for the $N$-dimensional system. Some applications of our result are also discussed. Firstly, we prove that any arbitrary two density matrices $\rho_A$ and $\rho_B$ of an $N$-dimensional open system can always be connected by Kraus operators. Suppose $$\begin{aligned} \rho_A=\sum\limits_{i=1}^N p^A_i |A_i\rangle\langle A_i|=U_A\rho^d_AU_A^\dagger,\label{rhoab1}\\ \rho_B=\sum\limits_{i=1}^N p^B_i |B_i\rangle\langle B_i|=U_B\rho^d_BU_B^\dagger, \label{rhoab2}\end{aligned}$$ where $p^A_i$ ($p^B_i$) and $|A_i\rangle$ ($|B_i\rangle$) are the eigenvalues and the orthonormal eigenvectors of the density matrix $\rho_A$ ($\rho_B$) respectively, $\rho^d_A$ ($\rho^d_B$) is the diagonal matrix with the entries $p^A_i$ ($p^B_i$), and $U_A$ ($U_B$) is the unitary transformation matrix, the $i$-th column of which is just the vector $|A_i\rangle $ ($|B_i\rangle$). We want to prove that there exist the operators $M_\mu$, satisfying $$\begin{aligned} &&\rho_B=\sum\limits_\mu M_\mu\rho_AM_\mu^\dagger,\label{rhoa}\\ &&\sum\limits_\mu M_\mu^\dagger M_\mu=I. \label{mm2}\end{aligned}$$ To find the required Kraus operators, one may write $M_\mu$ as $N\times N$ matrices with unknown elements and then directly solve (\[rhoa\]) and (\[mm2\]) to determine the matrices. However, this method is not feasible for high dimensional systems due to the computational complexity. To overcome the problem, we first seek diagonal matrices, which are generally easier to manipulate. To this end, we look for operators $M'_i$ such that $$\begin{aligned} &&\rho^d_B=\sum\limits_\mu M'_\mu \rho^d_A {M'_\mu}^\dagger,\label{rhod}\\ &&\sum\limits_\mu {M'_\mu}^\dagger M'_\mu=I. \label{mm3}\end{aligned}$$ To solve the above equations, we obtain the matrix operators $M'_\mu $ ($\mu=0,1,2,...N-1$) with the entries $$\begin{aligned} (M'_\mu)_{ij}= \sqrt{p^B_i}\delta_{i,j-\mu}+\sqrt{p^B_i}\delta_{i,j-\mu+N}~~, \label{m}\end{aligned}$$ where $i,j=1,2,...N$[@Tong2]. With the matrices $M'_\mu $, we can construct the Kraus operators $M_\mu $ satisfying (\[rhoa\]) and (\[mm2\]). To see this, we use the relations, $$\begin{aligned} \rho^d_A=U_A^\dagger\rho_AU_A,\label{rhoab3}\\ \rho^d_B=U_B^\dagger\rho_BU_B. \label{rhoab4}\end{aligned}$$ Substituting (\[rhoab3\]) and (\[rhoab4\]) into (\[rhod\]), we have $$\begin{aligned} U_B^\dagger\rho_BU_B=\sum\limits_\mu M'_\mu U_A^\dagger\rho_A U_A{M'_\mu}^\dagger,\end{aligned}$$ that is, $$\begin{aligned} \rho_B=\sum\limits_\mu U_B M'_\mu U_A^\dagger\rho_A (U_BM'_\mu U_A^\dagger)^\dagger.\end{aligned}$$ Let $$\begin{aligned} M_\mu=U_BM'_\mu U_A^\dagger, \label{mmu}\end{aligned}$$ it is obvious that $M_\mu $ defined by (\[mmu\]) satisfy (\[rhoa\]) and (\[mm2\]). They are the Kraus operators connecting $\rho_A$ with $\rho_B$. Since $U_A$, $U_B$ and $M'_\mu $ can be deduced directly from $\rho_A$ and $\rho_B$, the explicit expression of $M_\mu$ is obtained for arbitrary density matrices. Secondly, having proved that any arbitrary two density matrices $\rho_A$ and $\rho_B$ of an $N$-dimensional open system can always be connected by Kraus operators, we now revert to our discussion concerning the time evolution of open systems. We show that the time-dependent state of an arbitrary open system can always be described in terms of the Kraus representation. We replace $\rho_A$ and $\rho_B$ in the above demonstration by $\rho(0)$ and $\rho(t)$ respectively. Note that for any evolution of an $N$-dimensional open system, the state $\rho(t)$ and the initial state $\rho(0)$ can always be expressed as $$\begin{aligned} \rho(t)=\sum\limits_{i=1}^N p_i(t) |\psi_i(t)\rangle\langle \psi_i(t)| ,\label{rhot1}\\ \rho(0)=\sum\limits_{i=1}^N p_i(0) |\psi_i(0)\rangle\langle \psi_i(0)|. \label{rhot2}\end{aligned}$$ By comparing (\[rhot1\]) (\[rhot2\]) with (\[rhoab1\]) (\[rhoab2\]) and using (\[mmu\]), one can immediately write down the Kraus operators $M_\mu (t)$ as $$\begin{aligned} M_\mu(t)=U(t)M'_\mu(t) U(0)^\dagger, \label{mmt}\end{aligned}$$ where $U(t)$ ($U(0)$) is the unitary transformation matrix that diagonalizes $\rho(t)$ ($\rho(0)$) given explicitly by $$\begin{aligned} &U(t)&=\left(\begin{array}{ccccc}\psi_1(t)~&\psi_2(t)~&...&...&~\psi_N(t) \end{array}\right)\label{ut},\\ &U(0)&=\left(\begin{array}{ccccc}\psi_1(0)~&\psi_2(0)~&...&...&~\psi_N(0) \end{array}\right), \label{u0}\end{aligned}$$ and $M'_\mu(t) $ ($\mu=0,1,...N-1$) are given by $$\begin{aligned} &M'_0(t)&=\left(\begin{array}{ccccc} \sqrt{p_1(t)}&0&0&...&0\\ 0&\sqrt{p_2(t)}&0&...&0\\ 0&0&\sqrt{p_3(t)}&...&0\\ &...&...&...&\\ 0&0&0&0&\sqrt{p_N(t)} \end{array}\right), \nonumber\end{aligned}$$ $$\begin{aligned} &M'_1(t)&=\left(\begin{array}{ccccc} 0&\sqrt{p_1(t)}&0&...&0\\ 0&0&\sqrt{p_2(t)}&...&0\\ &...&...&...&\\ 0&0&0&...&\sqrt{p_{N-1}(t)}\\ \sqrt{p_N(t)}&0&0&0&0 \end{array}\right),\nonumber\end{aligned}$$ $$\begin{aligned} ...~~~~~~~...~~~~~~~...~~~~~~~\nonumber\end{aligned}$$ $$\begin{aligned} &M'_{N-1}(t)&=\left(\begin{array}{ccccc} 0&0&...&0&\sqrt{p_1(t)}\\ \sqrt{p_2(t)}&0&0&...&0\\ 0&\sqrt{p_3(t)}&0&...&0\\ &...&...&...&\\ 0&0&0&\sqrt{p_N(t)}&0 \end{array}\right). \label{m0m1}\end{aligned}$$ The operators $M_\mu (t)$ defined by (\[mmt\])-(\[m0m1\]) satisfy (\[rhot\]) and (\[mm\]), giving the Kraus representation of $\rho(t)$. Generally speaking, one cannot assert that any arbitrary time-dependent density operator possesses the Kraus representation simply from the result that any two states can be connected by Kraus operators. However, here we can make the assertion because $p_i(t)$ are always positive and (\[m0m1\]) is always valid independent of time $t$. So far, we have shown that the time-dependent state of an open system can always be described in terms of the Kraus representation. A general expression of Kraus operators for any arbitrary $N$-dimensional system is given. When the evolution of the open system is given by $\rho(t)$ and $\rho(0)$, the Kraus operators can be written immediately as $M_\mu(t)=U(t)M'_\mu(t) U(0)^\dagger$, where $U(t)$, $U(0)$ and $M'_\mu(t)$ are explicitly given by (\[ut\]), (\[u0\]) and (\[m0m1\]). Since the Kraus operators are not unique, the expression (\[mmt\]) is only one set of them. The other equivalent expressions of the Kraus operators can be obtained by $\tilde{M}_\mu(t)=\sum\limits_\nu M_\nu(t) V_{\mu\nu}$, where $V_{\mu\nu}$ are the elements of an arbitrary unitary matrix $V$. Thirdly, in the following paragraphs, we will illustrate some applications of our result. The result can help to clarify some ambiguous concepts. For example, the [*[Kraus representation theorem]{}*]{}, recalling the well-known [*[representation theorem]{}*]{}, which states that a map has the Kraus representation if and only if it is linear, completely positive (CP) and trace preserving[@Kraus], one may thought that the expression $\rho'=\$(\rho)$ has no Kraus representation if the map $\$$ is not CP. However, our result shows that any two states $\rho$ and $\rho'$ can always be connected by the Kraus operators irrespective of the form of the map acting on $\rho$. This shows that although non-CP map does not possess the Kraus representation, the expression $\rho'=\$(\rho)$ can still be cast into the Kraus representation. For instance, the transposition operator $T: \rho \rightarrow \rho^T$ is positive but not completely positive. According to the [*[Kraus representation theorem]{}*]{}, there does not exist the Kraus representation of the map $T$. However, the expression $ \rho^T=T(\rho)$ can still be described by Kraus operators. Let $\rho_A=\rho$ and $\rho_B=\rho^T$, the Kraus operators can be obtained directly using (\[mmu\]). The important key for clarifying the ambiguity is to note that there is a difference between the Kraus representation of a map and the Kraus representation for a state under the action of a map. The Kraus operators describing the Kraus representation of a map are independent of the state $\rho$ while the Kraus operators describing the Kraus representation of a state $\rho$ under the action of a map may be dependent on the state. A non-CP map has no Kraus representation, but $\rho'=\$(\rho)$, be $\$$ a CP map or not, always has the Kraus representation. An important corollary of our result is that there always exists a CP map between any two quantum states and the map can be represented through $N$ Kraus operators. This corollary is obvious, because any two states can be connected by $N$ Kraus operators $M_\mu $ and the CP map can then be defined by the $N$ Kraus operators. The corollary shows that even for such two states $\rho$ and $\rho'$, where $\rho'$ is obtained from $\rho$ by a non-CP map $\$:\rho\rightarrow\rho'$, one can still find an alternative map ${\tilde{\$}}$ that is completely positive, satisfying ${\tilde{\$}}:\rho\rightarrow\rho'$. The conclusion that an arbitrary time-dependent state $\rho(t)$ can always be described by the Kraus representation could have other deep applications in the study of open systems, especially in the field of quantum information. For example, it is useful for the study on geometric phase. As we know, geometric phases of both pure state and mixed state under unitary evolutions have been clarified[@Berry; @Aharonov; @Samuel; @Mukunda; @Pati; @Sjoqvistm; @Tong]. A new issue is on the geometric phases for open systems under nonunitary evolutions. Some papers[@Peixoto; @Ericsson; @Carollo] just use the Kraus operators $M_\mu$ to define and calculate the geometric phases of open systems. As described in ref. [@Ericsson], the relative phases are defined by $\alpha_\mu=\arg tr[M_\mu(\tau)\rho(0)]$, and the geometric phases can be calculated by making polar decomposition of $M_{\mu}(t)$, such that $M_{\mu}(t)=h_{\mu}(t)u_{\mu}(t)$, where $h_\mu(t)$ are Hermitian and positive, and $u_\mu(t)$ are unitary. The relative phases $\alpha_\mu$ will then lead to the geometric phase when $u_{\mu}(t)$ satisfy the $N$ parallel transport conditions $\langle \psi_i(0)|u_{\mu}(t)^+\dot {u}_{\mu}(t)|\psi_i(0)\rangle =0,~~(i=1,2,...,N)$. With the scheme showed in the present paper, any time-dependent density matrix can be easily written as the Kraus representation. So one can transplant the notion of geometric phases defined by using Kraus operators to arbitrary evolutions of quantum systems by writing $\rho(t)$ as the Kraus representation. Another example of its applications concerns the inverse problem of the evolutions of open systems. Suppose that the density matrix of a system is given as a time-dependent function $\rho(t)$, one wants inversely to deduce the evolutional operators or Hamiltonians that evolve the initial state $\rho(0)$ to the state $\rho(t)$. This is an important issue because physicists sometimes need to prepare experimentally a quantum system that is expected to evolve along a given path in the projected Hilbert space. The approach for solving the problem is to constitute a closed system by combining the open system with an ancilla. The open system will undergo nonunitary evolution while the combined system evolves unitarily as $\varrho(t)=U_{sa}(t)\varrho(0)U_{sa}(t)^+$, where $U_{sa}(t)$ are unitary operators acting on the combined system and $\varrho(0)=\rho(0)\otimes|0_a\rangle\langle 0_a|$ is the initial state. The issue becomes to finding the unitary operators $U_{sa}(t)$ so that $tr_e\varrho(t)=\rho(t)$. It is quite a difficult problem in general. However, the present paper can provide an effective approach to obtain the unitary operators. As concluded above, evolutions of an open system always have the Kraus representation and the Kraus operators can be directly deduced by the given $\rho(t)$. With the Kraus operators $M_\mu(t)$, one can easily obtain the unitary operator $U_{sa}(t)$. In fact, in order to satisfy $tr_e\varrho(t)=\rho(t)$, the elements of $U_{sa}(t)$ in the bases $|\Psi_i(0)\rangle \otimes|j_a\rangle$ are only required to be $[U_{sa}(t)]_{ij,k0}=[M_{j}(t)]_{ik}$ while $[U_{sa}(t)]_{ij,kl}~(l\neq 0)$ are arbitrary but keeping $U_{sa}(t)$ to be unitary and $U_{sa}(t)|_{t=0}=I$, where $|j_a\rangle~ (j=0,1,...K-1)$ are the bases of the ancilla. Obviously, there are infinitely many such unitary operators $U_{sa}(t)$, so do the Hamiltonians $H=i\dot U(t)U(t)^+$. One can choose the suitable ones which can be easily performed in the laboratory. It is also interesting to note that the Kraus operators provided in the present paper have some intriguing properties. Since these Kraus operators are dependent on the eigenvectors of “input state" $\rho_A$ but independent of its eigenvalues, all the input states with distinct eigenvalues but the same eigenvectors will yield the same “output" state $\rho_B$. That is, the map defined by the Kraus operators can transform all the input states whose Bloch vectors lie on the same diameter of the Bloch sphere to the same output state. Specially, one may let $\rho_A=\frac{1}{N}I$, and then has $\rho_B=\frac{1}{N}\sum M_\mu M_\mu^\dagger$. We suppose that these properties of the Kraus operators may be useful in the study of open systems. The work was supported by NUS Research Project Grant: Quantum Entanglement with WBS R-144-000-089-112. J.L.C. acknowledges financial support from Singapore Millennium. [99]{} K. Kraus, [*[States, Effects and Operations]{}*]{} (Spring-Verlag, Berlin, 1983) J. Preskill, Lecture notes:[*[Information for Physics 219/Computer Science 219, Quantum Computation,]{}*]{} www.theory.caltech.edu/people/preskill/ph229. P. Pechukas, Phys. Rev. Lett. [**73**]{}, 1060(1994). L.D. Romero and J. P. Paz, Phys. Rev. A [**55**]{}, 4070 (1997). J. Bouda and V. Buzek, Phys. Rev. A [**65**]{}, 034304 (2003). Philip Pechukas, Phys. Rev. Lett. [**73**]{}, 1060(1994). G. Kimura, Bussei-Kenkyu, 79-1(2002); Phys. Rev. A [**66**]{}, 062113 (2002). J. G. Peixoto de Faria, A. F. R. de Toledo Piza and M. C. Nemes, Europhys. lett. [**62**]{}, 782(2003). Mika Hirvensalo, [*[Quantum Computing]{}*]{}, p137 (Spring-Verlag, Berlin, 2001) Sonja Daffer, Krzysztof Wódkiewicz, James D. Cresser, and John K. Mclver, e-print:quant-ph/0309081 v1(2003). P. Stelmachovic and V. Buzek, Phys. Rev. A [**64**]{}, 062106 (2001); [**67**]{}, 029902(E)(2003). D. Salgado, and J. L. Sánchez-Gómez, e-print:quant-ph/0211164(2002). H. Hayashi, G. Kimura and Y. Ota, Phys. Rev. A [**67**]{}, 062109 (2003). D. M. Tong, Jing-Ling Chen, L. C. Kwek, and C. H. Oh, e-print:quant-ph/0311091 v2(2003). Noticing that $\sum_{\mu=0}^{N-1}(\delta_{i,k-\mu}+\delta_{i,k-\mu+N})(\delta_{j,k-\mu}+\delta_{j,k-\mu+N})= \delta_{ij}$ $(i,j,k=1,2,...,N)$, one can verify that Eq. (9) satisfies Eqs. (7) and (8) by showing $(\sum\limits_\mu M'_\mu \rho^d_A {M'_\mu}^\dagger)_{ij}=p^B_i\delta_{ij}$ and $(\sum\limits_\mu {M'_\mu}^\dagger M'_\mu)_{ij}=\delta_{ij}$. Besides, Eq. (9) can be written as $M_\mu=\sqrt{\rho^d_B}S^\mu$, where $S$ is a shift unitarity defined by $S_{ij}=\delta_{i+1,j}+\delta_{N,j+N-1}$. With this form of $M_\mu$, the verification can also be carried out by direct matrix operations. M.V. Berry, Proc. R. Soc. London Ser. A [**392**]{}, 45 (1984). Y. Aharonov and J. Anandan, Phys. Rev. Lett. [**58**]{}, 1593 (1987). J. Samuel and R. Bhandari, Phys. Rev. Lett. [**60**]{}, 2339 (1988). N. Mukunda and R. Simon, Ann. Phys. (N.Y.) [**228**]{}, 205 (1993). A.K. Pati, Phys. Rev. A [**52**]{}, 2576 (1995). E. Sjöqvist, A.K. Pati, A. Ekert, J.S. Anandan, M. Ericsson, D.K.L. Oi, and V. Vedral, Phys. Rev. Lett. [**85**]{}, 2845 (2000). D. M. Tong, E. Sjöqvist, L. C. Kwek, C. H. Oh, and M. Ericsson, Phys. Rev. A [**68**]{}, 022106 (2003). M. Ericsson, E. Sj" oqvist,J. Brannlund, D. K. L. Oi, and A. K. Pati, Phys. Rev. A [**67**]{}, 020101(R) (2003). A. Carollo, I. Fuentes-Guridi, M. F. Santos, and V. Vedral Phys. Rev. Lett. [**90**]{}, 160402 (2003).

--- abstract: | We examine the issue of the cosmological constant in the $many$ $inflating$ branes scenario, extending on two recent models by I.Oda and Randall-Sundrum.The exact solution in a closed form is found in the slow roll approximation of the radion. Defining an effective expansion rate, which depends on the location of each brane in the fifth dimension and demanding stability for this case we show that each positive tension brane has a localized, decaying cosmological constant (the opposite process applies to the negative energy branes \[4\]) .\ The reason is that the square of the effective expansion rate enters as a source term in the Einstein equations for the branes.Thus the brane has two scale factors depending on time and the fifth dimnesion respectively .The brane will roll along the fifth dimension in order to readjust its effective expansion rate in such a way that it compensates for its internal energy changes due to inflation and possible phase transitions. author: - | Laura Mersini\ Department of Physics\ University of Wisconsin-Milwaukee\ Milwaukee, WI 53201\ [email protected]\ Wisc-Milw-99Th-14 date: 'September 20, 1999' title: 'Decaying Cosmological Constant of the Inflating Branes in the Randall-Sundrum -Oda Model' --- Setup ===== In this paper we consider and find solutions to a five dimensional model with many inflating branes along the fifth compact direction, thus extending on the recent models proposed by Randall-Sundrum and Oda \[1,2,3\]. Each brane is taken as a point in the fifth dimension (negligible thickness) $S^1$. They act as localized cosmological constants or alternatively as gravitational point sources.The whole 5D setup should be static in order to preserve Poincare invariance. That means that the 5D cosmological constant $\Lambda_5$ should cancel out the collected contribution of the 4D cosmological constants $\Lambda_{4i}$ of each brane (weighted by the appropriate conformal factor as shown below in Sect.2). Note that each $\Lambda_{4i}$ depends on the position of the i-th brane in $S^1$. The existence of solution to the Einstein equation in the many branes scenario does not allow for a singular orbifold geometry $S^1/Z_2$ but instead, it requires a smooth manifold $S^1$ \[1\]. The 5D metric ansatz considered here is: $$ds^2 = g_{MN} dx^M dx^N = u(z)^2 dt^2 - a(z,t)^2 dx^2 - b(z)^2 dz^2$$ with $z$ ranging between $0$ and $2 L$. The 5D Einstein-Hilbert action is: $$S = \frac{1}{2\kappa^2_{5}} \int d^4 x \int^{2L}_0 dz \sqrt{-g} (R+2\Lambda_5) + \sum^n_{i=1} \int_{z=L_i} d^4 x \sqrt{-g_i}({\cal L}_i + V_i)$$ where the 4D metric $g_{i\mu\nu}$ is obtained from the 5D one as follows: $g_{i\mu\nu}({\bf x},t) = g_{MN}({\bf x},t,z = L_i)$ and ${\cal L}_i, V_i$ are the lagrangian of the matter fields and the vacuum energy respectively, in the i-th brane.The 5D gravitational coupling constant $\kappa_{5}$ is related to the 4D Newton constant $G_N$ through $\kappa_5 =16 G_{N}\pi \int dz\sqrt{-g_{55}} =16 G_{N}\pi L_{phys} $.\ Many Inflating Branes Scenario. Solution ======================================== We are interested in constructing a solution for the inflating branes. Thus in Eqn. (1) we take: $$\begin{array}{ccl} a(z,t) & = & f(z) v(t)\\ b(z) & = & f(z)\\ u(z) & = & f(z) \end{array}$$ The dynamics of the boundaries is based on the slow roll assumption.The energy density of each brane is denoted by ${\cal L}_i$ where we have separated a constant vacuum energy contribution, $V_{i}$. The following equations result from varying the action in Eqn.(2) with respect to the metric, \[1,2,3\]: $$\begin{aligned} \frac{1}{f^2}\left[ (\frac{\dot{v}}{v})^2 \right] - \frac{1}{f^2} \left[\frac{f''}{f} + (\frac{f'}{f})^2 \right] & = & - \frac{\kappa^2}{3 f}[\sum_i({\cal L}_i + V_i)\delta(z - L_i)] + \frac{\Lambda_5}{3}\nonumber \\ & & \nonumber\\ \frac{1}{f^2}\left[ (2\frac{\ddot{v}}{v})+ (\frac{\dot{v}}{v})^2\right] - \frac{3}{f^2}\left[ \frac{f''}{f} + (\frac{f'}{f})^2\right] & = & - \frac{\kappa^2}{f}\left[ \sum_i({\cal L}_i + V_i)\delta(z - L_i)\right] +{\Lambda_{5}}\nonumber\\ & & \nonumber\\ \frac{1}{f^2}\left[ (\frac{2\ddot{v}}{v})+ (\frac{\dot{v}}{v})^2\right] - \frac{2}{f^2}(\frac{f'}{f})^2 & = & \frac{\Lambda_5}{3} \end{aligned}$$ where prime and dot denote differentiation with respect to z and t respectively. It is straightforward from the Eqns. (4) above (as well as the component $G_{55}=0$), that $v(t)$ satisfies: $$\frac{\ddot{v}}{v} = \frac{\dot{v}^2}{v}$$ which gives for $v$: $$v(t) = v(0)exp[Ht]$$ Thus Eqns. (4) become: $$\begin{array}{lcc} \frac{H^2}{f^2} - \frac{1}{f^2} \left[ \frac{f''}{f} + (\frac{f'}{f})^2 \right] & = & - \frac{\kappa^2}{3f} [\sum_i({\cal L}_i + V_i)]\delta(z - L_i) + \frac{\Lambda_5}{3} \\ & & \\ \frac{H^2}{f^2} - \frac{2}{f^2}(\frac{f'}{f})^2 = \frac{\Lambda_5}{6} \end{array}$$ The effective expansion rate is defined as $H_{eff} = H / f(z) $ . The effective expansion rate of each brane, located at $z = L_i$ in the extra dimension, is then given by $H_{ieff} = H / f(L_i)$. Thus each brane, depending on its position in the fifth dimension, sees a different expansion rate resulting from the fact that the canonical proper time has the following scaling on the position of the brane, $d \tau_{i} = f( L_{i}) dt$. Denote the absolute value of $\Lambda_5$ by $\Lambda$. Since we consider an $ADS_5$ geometry, $\Lambda$ is positive and satisfies the following relation: $$\Lambda = - \Lambda_5 .$$ We will find the solution by construction. Assume the ansatz in Eqn. (7) for the warp factor $f(z)$. Then its derivatives are given by Eqns. (8,9). $$\begin{aligned} f(z) = {\alpha} {\displaystyle Sinh[g(z)]^{-1} } & & \\ f'( z) = -{\alpha} \displaystyle \frac {Cosh [g(z)]}{ Sinh[g(z)]^{2} } g'(z) & & \\ f''(z) = -{\alpha } (\displaystyle \frac{ g'(z)^2 }{ Sinh[g(z)] } - - \frac { 2 Cosh[g(z)]^{2} }{ Sinh[g(z)]^{3} } {g'}^{2} + \frac { Cosh[g] }{ Sinh[g]^2 } g'')\end{aligned}$$ The function $g(z)$ \[1,3\], describing even branes (domain walls) of positive and negative energy positioned along $S^1$ is of the following form $$\begin{aligned} f(z) & =\alpha \displaystyle{Sinh[g(z]^{-1}} \\ g(z) &=\left(\displaystyle{\sum^{n-1}_{i=1}}(-1)^{i+1}|z-L_i|+L\right)(-\beta)\\ g''(z)&=2\left(\displaystyle{\sum^{n-1}_{i=1}}(-1)^{i+1} \delta |z - L_i| \right) (-\beta) \end{aligned}$$ The even $(n-1)$ branes are located at $L_i (i = 2, \dots n-1)$. One always ends up with even branes as it is clear from topological consideration. Each brane is a pointlike gravitational source in the fifth dimension, i.e. there are flux lines extending from one brane to the others (mutual interaction). Those flux lines should close in order to preserve the stability of the 5th dimension (see \[4\]). Thus each positive energy brane in $S^1$ is alternating with its counterpart (negative tension brane) so that the number of positive energy branes equals the number of negative energy branes.\ Replacing the above Eqns. (7-12) for $f(z)$ in the Einstein Eqns.(6) we get the following relation $$\begin{aligned} \nonumber (H^2 - {g'}^2)+ \frac {\Lambda /3 - 2{g'}^2}{ \displaystyle Sinh[g]^2 } &= & \frac{( \sum_i - \kappa_{5} ({\cal L}_i + V_i) - g'' {\displaystyle Cosh[g(L_i)]}) \delta(z - L_i)}{Sinh[g]} \\ \nonumber (H^2 - {g'}^2) & =& - \displaystyle \frac{\Lambda \alpha^2}{6 Sinh[g]^2} + \frac { {g'}^2 }{ Sinh[g]^2 } \end{aligned}$$ There is a solution to Eqns. (9) only if $\beta = H$ and $\alpha^2 = \frac {6 H^2}{\Lambda}$ in Eqns. (10-12).The value of $ H $ is decided by $\Lambda$ and $ L_{phys}$. The conditions on the energy densities of the branes and their respective effective expansion rates $H_{ieff}$ become: $$\begin{aligned} {\cal L}_i +V_i &=& \frac {2}{\kappa_5} (-1)^{(i+1)} \sqrt{ \frac {\Lambda}{6} }(\displaystyle Cosh[g(L_i)]) \\ H_{ieff} &=& \sqrt{ \frac {\Lambda}{6} } (\displaystyle Sinh[g(L_i)])\end{aligned}$$ The warp factor $f(z)$ found in Eqns. (10-12) satisfies Einstein equations and has a minimum at $z=0, 2L$ and a maximum at $z= L$. (For subtleties related to ensuring the periodicity of $f(z)$ and the overlap of branes located at $z=0, 2L$ see \[1\]. The physical length, $L_{phys} = 2 L$ calculated by the formula in Sect.1, depends on the energies of the branes as they are the gravitational sources causing the curvature of the fifth dimension. It is straightforward to perform the calculation explicitly ,see \[3\]). It is clear from Eqns. (14, 15) that branes positioned near the minimum of the warp factor, $z=0$ (the ’TeV branes’) in the fifth dimension, have the maximum energy and effective expansion rate allocated to them while the ones near the position of the maximum warp factor $z=L_{phys}$ (’Planck branes’), are almost empty of energy and have the lowest effective expansion rate ($H_{ieff}$ at $z=L_{phys}$ is almost but not identically zero, a result also found by \[4,10\]). Thus we end up with the following relation between the expansion rate and the energy density for the branes $$|\frac {\kappa_5 ( {\cal L}_i + V_i)}{2}|^2 - H_{ieff}^2 = \frac {\Lambda}{6}$$ Notice the unusual ’Friedman relation’ between the energy density and the expansion rate for each individual brane \[4,5,10\]. An important result of the Eqn. (16) is that each brane has a different expansion rate depending on its position in $S^1$. Clearly from (16), branes located around the position of minimum warp factor have maximum expansion rate, while the ’Planck branes’ have nearly zero expansion rate and energy. There is a deep physical reason behind the relation of Eqn. (16). It is related to the induced radion potential and the low suppression of its modes to the TeV branes and high supression to th e Planck branes. We outline this reasoning below, in Sect.3, (however a more complete and detailed treatment ofthe relation of radion dynamics and inflating branes will be reported soon in \[12\]). Discussion =========== In order to gain insight in the relation of Eqn. (16) above, we would like to take the point of view of Dvali et al.\[11\] that although the inflaton is a brane mode in the ground state, it behaves as an ’inter-brane’ mode that describes a relative separation of the branes in the extra dimension. This approach clarifies the physics of brane dynamics during inflation. It is the radion (inter-brane separation) which in the effective 4D description becomes the inflaton in the branes. The ’TeV branes” have minimum suppression of the radion modes as compared to the ’Planck branes’, thus they will have maximum inflation which is confirmed by Eqn. (16). The coupling of ’Planck branes’ have maximum suppression to the radion, hence nearly zero inflation rate since the the energy dumped into them by the radion is nearly zero. Branes inflate with different rates for the above reason. Their coupling to the radion depends on their position/angle in the $S^1$. We have avoided introducing a radion potential and assumed a slow-roll behaviour for the branes. Hence this potential is implicitly there as it is induced by the branes. The slow-roll holds for the most part except around $z=L$ where our analysis would break down. For a complete treatment we need to consider the full dynamics of the branes, radion stability and bulk fields. These will provide a mechanism whereby branes roll towards decreasing values of $H_{eff}$ (also decreasing the effective 4D cosmological constant) thus exiting inflation, reheating the universe and recovering the usual Friedman relation in late cosmology (work in progress \[12\]). It is always useful to use analogies of the system under consideration with the condensed matter systems. The above scenario is very similar to a string of molecules confined to reside in a circle. Each brane would represent a molecule with four internal degrees of freedom (branes are 4D worlds), chemical potential (gibbs free energy per molecule) $ H_{ieff}^2$, energy ${{\mathcal L}}_i + V_i$, arranged in such a configuration that the total system is in equlibrium.For the moment let’s concentrate on the positive tension branes (the opposite will go for the negative energy ones). Since each brane inflates, the energy ${{\mathcal L}}_i$ of the matter fields and the temperature tend to change towards decreasing values. The vacuum energy $V_i$ of each brane can also change towards decreasing values due to internal phase transitions driving the vacuum to a lower minimum. In our analogy, $V_i$ would be the analog of the latent heat of each molecule. But each position in the circle $S^1$ belongs to a certain $H_{ieff}$ energy level (’molecules’ with different ’chemical potential’). Thus, when the total internal energy of the brane decreases (for the reasons mentioned above) the brane moves to the next position in the fifth dimension thus changing its $H_{ieff}$ in order to accommodate (compensate for) the internal energy changes and to preserve the equlibrium configuration of the system as a whole. Each brane has the freedom to do the readjustment through rolling due to the fact that the effective expansion rate enters as a constant source term in the 4D brane equations but it is a function of the fifth dimension, $H_{eff} = Hf(z)^{-1}$.\ Therefore, each positive (negative) energy brane will move towards (away from) the lowest $H_{eff}$ (i.e. maximum (minimum) $f(z)$) to compensate for internal energy changes. In short $H_{eff}$ will $decay$ nearly $exponentially$ , (see Eqn.(15) for the exponential dependence of $f(z)$), by rolling down $f(z)^{-1}$. Ultimately all positive energy branes will locate at the $f(z)$ maximum point and the negative energy ones at the $f(z)$ minimum point.\ As the internal 4D world of each brane is not aware of the fifth dimension z , $H_{eff}^2$ plays the role of a decaying 4D cosmological constant to each brane. The reason, as mentioned, is that when ${{\mathcal L}}_i$ (which is a dynamic quantity) changes internally for each brane, then $H_{ieff}(z)^2$ ( which reflects the coupling of each 4D brane to the 5D gravity) is forced to change by displacing to a new position such that $({{\mathcal L}}_i + V_i) -\alpha H_{ieff}^2$ cancels out the flux lines/ energy allocated by $\Lambda_5$ to that position in $S^1$. (The constant $\alpha=\frac{3}{\kappa^2 }$). Obviously the mass scales and proper time $\tau_i$ of each brane also change while the brane changes position in $S^1$, thus a set of hierarchy scales.\ To put it differently, there are two competing scale factors or redshifts affecting each brane, the $z$ and $t$ factors. A redshift in $t$ forces a ’blueshift’ in $z$ because of the requirement of equlibrium configuration of the system as a whole (pictorially ,the flux line should be closed in this static 5D model. Since branes interact gravitationally with each other along the $S^1$, the flux lines originating from positive tension branes should end up in the negative tension branes. This is also consistent with the solution in Sect.3 that, in each pair of branes, while the positive energy branes shifts towards maximum $f(z)$ its counterpart (negative energy) shifts towards minimum $f(z)$ such that the flux lines remain close).\ There remains an open issue,namely: What happens to the branes as,ultimately,they all pile up to only two point of $S^1$, (stability issues are discussed in detail in \[4\]) , the positive tension ones at maximum $f(z)$ and negative tension branes at minimum $f(z)$. Naive energy arguments would suggest they recombine into a single 4D brane in either point (a chemical equlibrium situation), thus the problem would reduce to two single branes as in \[2,3\]. However this isssue remains to be investigated more closely \[12\].\ In conclusion, in this many inflating branes scenario, the equlibrium condition of the 5D model, together with inflation and phase transitions taking place internally in each 4D brane, drives a decaying cosmological constant,which while it may not depend on the internal 4 dimensions, does depend on the fifth one and thus has the freedom to rearrange thereby preserving the equlibrium configuration.\ Acknowledgment: I would like to thank Prof. L. Parker for our regular discussions.This work was supported in part by NSF Grant No. Phy-9507740. - Ichiro Oda,$hep-th/9908076$; Ichiro Oda, $hep-th/9909048$ - L. Randall and R.Sundrum, ’A large mass hierarchy from a small extra dimension’, $hep-ph/9905221$ - T.Nihei, $hep-ph/9905487$ - P.Steinhardt, $hep-ph/9907080$; L.Mersini, $gr-qc/9906106$; C.Csaki et al, $hep-ph/990651$; W.D.Goldberger and M.B.Wise, $hep-ph/9907447, hep-ph/9907218$ - A.Lukas, B.A.Ovrut and D.Waldram, $hep-th/9902071$ - N.Kaloper and A.Linde, $hep-th/9811141$ - J. Lykken and L.Randall, $hep-th/9908076$ - N. Arkani-Hamed, S, Dimopoulos, G.Dvali and N.Kaloper, $hep-th/9907209$ - N. Verlinde, $hep-th/9906182$ - C. Csaki et al $hep-ph/9906513, hep-th/9908186$, J.Cline et al $hep-ph/9906523$, J. Cline $hep-ph/9904495$ - G. Dvali and Henry Tye, $hep-ph/9812483$, Daniel J.H. Chung, Edward W. Kolb and Antonio Riotto, $hep-ph/9802238$ - P. Binetruy et al $hep-th/9905012$, L. Mersini, ’Radion potential and Brane Dynamics’, $in$ $ preparation$ \]

--- abstract: 'In this paper, we describe our method for DCASE2019 task 3: Sound Event Localization and Detection (SELD). We use four CRNN SELDnet-like single output models which run in a consecutive manner to recover all possible information of occurring events. We decompose the SELD task into estimating number of active sources, estimating direction of arrival of a single source, estimating direction of arrival of the second source where the direction of the first one is known and a multi-label classification task. We use custom consecutive ensemble to predict events’ onset, offset, direction of arrival and class. The proposed approach is evaluated on the TAU Spatial Sound Events 2019 - Ambisonic and it is compared with other participants’ submissions.' address: | Samsung R&D Institute Poland\ Artificial Intelligence\ Warsaw, 00-844, Poland\ {s.kapka, m.lewandows4}@samsung.com bibliography: - 'refs.bib' title: 'Sound source detection, localization and classification using consecutive ensemble of CRNN models' --- DCASE 2019, Sound Event Localization and Detection, CRNN, Ambisonics Introduction {#sec:intro} ============ Sound Event Localization and Detection (SELD) is a complex task which naturally appears when one wants to develop a system that possesses spatial awareness of the surrounding world using multi-channel audio signals. This year, the task 3 from the IEEE AASP Challenge on Detection and Classification of Acoustic Scenes and Events (DCASE 2019) [@dcase2019web] concerned the SELD problem. SELDnet introduced in [@Adavanne2018_JSTSP] is a single system of a good quality designed for the SELD task, and the slight modification of SELDnet was set as the baseline system [@Adavanne2019_DCASE] during the DCASE 2019 Challenge. Solely based on [@Adavanne2018_JSTSP] and [@Adavanne2019_DCASE], we develop a novel system designed for the task 3 from the DCASE 2019 Challenge. In our work, we follow the philosophy that if a complex problem can be split into simpler ones, one should do so. Thus we decompose the SELD task with up to 2 active sound sources into the following subtasks: - estimating the number of active sources (*noas*), - estimating the direction of arrival of a sound event when there is one active sound source (*doa1*), - estimating the direction of arrival of a sound event when there are two active sound sources and we posses the knowledge of the direction of arrival of one of these sound events, which we will call an *associated event* (*doa2*), - multi-label classification of sound events (*class*). For each of this subtasks, we develop a SELDnet-like convolutional recurrent neural network (CRNN) with a single output. We discuss it in detail in section \[sec:architect\]. Given such models, we develop a custom consecutive ensemble of these models. This allows us to predict the events’ onset, offset, direction of arrival and class, which we discuss in detail in section \[sec:ensemble\]. Due to the sequential nature of generating predictions in our system, errors in models’ predictions may cascade, and thus an overall error may cumulate. Despite this drawback, our system acquire very good results on the TAU Spatial Sound Events 2019 - Ambisonic database. We discuss the results in detail in section \[sec:results\]. ![An example of the normalised amplitude spectrogram in the decibel scale and the normalised phase spectrogram obtained from the first *foa* channel from some randomly selected recording. The horizontal and vertical axes denote frame numbers and frequencies respectively obtained from the STFT. Note that the values from the legends on the right are dimensionless due to the normalization used in the preprocessing.[]{data-label="fig:spectrograms"}](spectrogram_final.png){width="\columnwidth"} Features {#sec:features} ======== The DCASE 2019 task 3 provides two formats of the TAU Spatial Sound Events 2019 dataset: first order ambisonic (*foa*) and 4 channels from a microphone array (*mic*) [@Adavanne2019_DCASE]. In our method we only use the ambisonic format. Each recording is approximately 1 minute long with sampling rate of 48k. We use the short time Fourier transform (STFT) with Hann window. We use the window of length 0.4s and hop of length 0.2s in STFT to transform a raw audio associated to each *foa* channel into the complex spectrogram of size 3000x1024. If audio is longer than 1 minute, we truncate spectrograms. If an audio is shorter than 1 minute, we pad them with zeros. From each complex spectrogram we extract its module and phase point-wise, that is amplitude and phase spectrograms, respectively. We transform amplitude spectrograms to the decibel scale. Finally, we standardize all spectrograms frequency-wise to zero mean and unit variance, to obtain spectrograms as in Figure \[fig:spectrograms\]. In summary, from each recording we acquire 4 standardized amplitude spectrograms in the decibel scale and 4 standardized phase spectrograms corresponding to 4 *foa* channels. Architecture {#sec:architect} ============ As mentioned in the introduction, each of the subtasks (*noas*, *doa1*, *doa2* and *class*) has its own SELDnet-like CRNN. Each of these models is a copy of a single SELDnet node with just minor adjustments so that it fits to the specific subtask and for the regularization purpose. Each of these models takes as an input a fixed length subsequence of decibel scale amplitude spectrograms (in case of *noas* and *class* subtasks) or both decibel scale amplitude and phase spectrograms (in case of *doa1* and *doa2* subtasks) from all 4 channels. In each case, the input layers are followed by 3 convolutional layer blocks. Each block is made of a convolutional layer, batch norm, relu activation, maxpool and dropout. The output from the last convolutional block is reshaped so that it forms a multivariate sequence of a fixed length. In the case of *doa2*, we additionaly concatenate directions of arrivals of associated events with this multivariate sequence. Next, there are two recurrent layers (GRU or LSTM) with 128 units each with dropout and recurrent dropout. Next layer is a time distributed dense layer with dropout and with the number of units depending on subtask. Lastly, depending on a subtask, the model has a different output. For *noas*, the model has just a single time distributed output that corresponds to the number of active sources (0, 1 or 2). For *doa1* and *doa2*, the models have 3 time distributed outputs that corresponds to cartesian xyz coordinates as in [@Adavanne2018_JSTSP]. Cartesian coordinates are advantageous over spherical coordinates in this task due to their continuity. Lastly, for *class*, the model has 11 time distributed outputs corresponding to 11 possible classes. We present the detailed architecture in Table \[tab:parameteres\]. [\*6l]{} **Layer Type** & **Parameters** & *noas* & *doa1* & *doa2* & *class*\ Input & Shape & $256 \times 1024 \times 4$ & $128 \times 1024 \times 8$ & $128 \times 1024 \times 8$ & $128 \times 1024 \times 4$\ *ConvBlock\** & Pool & 8 & 8 & 8 & 8\ *ConvBlock\** & Pool & 8 & 8 & 8 & 8\ *ConvBlock\** & Pool & 4 & 4 & 4 & 4\ Reshape & Sequence length $\times$ features & $256 \times -1$ & $128 \times -1$ & $128 \times -1$ & $128 \times -1$\ Doa2 input & Is used & False & False & True & False\ Concatenate & Is used & False & False & True & False\ *RecBlock\*\** & Unit type & GRU & LSTM & GRU & GRU\ *RecBlock\*\** & Unit type & GRU & LSTM & GRU & GRU\ TD Dense & Number of units & $16$ & $128$ & $128$ & $16$\ Dropout & Dropout rate & $0.2$ & $0.2$ & $0.2$ & $0.2$\ TD Dense & Number of units & $1$ & $3$ & $3$ & $11$\ Activation & Function & linear & linear & linear & sigmoid\ *\*ConvBlock$(P)$*&\ Conv2D &\ BatchNorm & —\ Activation &\ MaxPooling2D &\ Dropout &\ *\*\*RecBlock$(U)$*&\ Recurrent &\ Activation &\ Dropout &\ \[tab:parameteres\] Depending on a subtask, we feed the network with the whole recordings or just their parts. For *noas*, we feed all the data. For *doa1*, we extract only those parts of the recordings where there is just one sound source active. For *doa2*, we extract only those parts of the recordings where there are exactly two active sound sources. For *class*, we extract those parts of the recordings where there are at least one active source. As for the learning process, we used mean square error loss for the *noas*, *doa1*, *doa2* subtasks and binary cross-entropy loss for the *class* subtask. For all subtasks we initialised learning process using Adam optimizer with default parameters [@Adam]. The *noas* and *class* subtasks were learned for 500 epochs with exponential learning rate decay; every 5 epochs the learning rate were multiplied by 0.95. In *doa1* and *doa2* subtasks, we run learning process for 1000 epochs without changing the initial learning rate. As for complexity, the *noas*, *doa1*, *doa2* and *class* have , , and parameters respectively, making total of parameters. Consecutive ensemble {#sec:ensemble} ==================== In this section, we introduce and describe the idea of the consecutive ensemble which is the core of our approach. This custom binding of our four models allows us to predict the events’ onset, offset, direction of arrival and class. The algorithm {#ssec:algorithm} ------------- We assume that recordings have at most 2 active sound sources at once and the sound events occur on a 10 degrees resolution grid. In our setting, the audios after feature extraction have exactly 3000 vectors corresponding to the time dimension. Henceforth we will call these vectors as frames. The algorithm itself goes as follows: 1\. We feed the features to the *noas* network to predict the number of active sources (NOAS) in each frame. 2\. We transform the predicted NOAS so that each recording starts and ends with no sound sources and the difference of NOAS between each frames is no greater than 1. 3\. From the predicted NOAS we deduce the number of events, their onsets and the list of possible offsets for each event. If NOAS in two consecutive frames increases, then we predict that a new event happened at the second frame. If in two consecutive frames NOAS decreases, then we append the first frame to all events since last time NOAS was 0 as a possible offset. 4\. In order to determine which offset corresponds to which event we use the *doa1* network. We extract chunks (intervals of equal NOAS) of audio where the predicted NOAS equals 1 and we feed it to *doa1* network. For each chunk where NOAS was 1 we predict the average azimuth and elevation, and we round it to the closest multiple of 10. If two consecutive chunks have the same azimuth and elevation then we conclude that the first event covered two chunks and the second event started and ended between those chunks. If two consecutive chunks have a different azimuth or elevation, then we conclude that the first event ended when the second chunk started and the second event continued in the second chunk. 5\. To determine the remaining information about angles we need to predict the direction of arrival (DOA) of events that start and end while the associated event is happening. We feed the chunks where NOAS is 2 to the *doa2* network with the second input being DOA of the associated event in cartesian xyz coordinates. Similarly as in step 4, we average the predicted results from chunks and round it to the closest multiple of 10. 6\. Lastly, we predict the events’ classes. If an event has chunks where the event is happening in an isolation (NOAS = 1), then all such chunks are feed to the *class* network and the most probable class (using soft voting among frames) is taken as a predicted class. If an event has no such chunks, i.e. the event is only happening with an associated event, then such chunk (NOAS = 2) is fed to the network and two most probable classes are extracted. We choose the first one which does not equal to the class of the associated event. An example {#ssec:example} ---------- The algorithm itself may seem quite complex at first glance. Hence, we investigate here a concrete example. Given a recording constituting of 3000 vectors, we predict its NOAS in each frame as in Figure \[fig:noas\]. For the sake of clarity we constrain only to a part of the recording. Consider a block with predicted NOAS as in the top plot from Figure \[fig:example2\]. According to the step 3 from the algorithm, we predict that 3 events happened here: $E_1, E_2, E_3$ with 3 corresponding onsets $On_1, On_2, On_3$. Events $E_1$ and $E_2$ may end at $Off_1, Off_2$ or $Off_3$ and event $E_3$ may end at $Off_2$ or $Off_3$ (see the bottom plot from Figure \[fig:example2\]). According to the step 4 from the algorithm, we predict DOA using *doa1* in chunks from $On_1$ to $On_2$, from $Off_1$ to $On_3$ and from $Off_2$ to $Off_3$. Based on that we deduce the events’ offsets as in Figure \[fig:example2\]. Based on step 5 from the algorithm, we predict the DOA of chunk from $On_3$ to $Off_2$ using *doa2* where the associated DOA is the DOA of $E_2$. Lastly we deduce classes of the events $E_1, E_2$ and $E_3$. According to the step 6 form the algorithm, we predict class of $E_1$ based on the chunk from $On_1$ to $On_2$, predict the class of $E_2$ based on chunks from $Off_1$ to $On_3$ and from $Off_2$ to $Off_3$. Finally, we predict the class of $E_3$ based on the chunk from $On_3$ to $Off_2$. If the predicted class of $E_3$ is the same as the class of $E_2$ then we predict it to be the second most probable class from the *class* network. ![The plot visualising the predicted number of active sources for some randomly selected recording.[]{data-label="fig:noas"}](noas_final.png){width="\columnwidth"} {width="\textwidth"} Results {#sec:results} ======= We evaluate our results on TAU Spatial Sound Events 2019 - Ambisonic dataset. This dataset constitutes of two parts: the development and evaluation sets. The development part consists of 400 recordings with predefined 4-fold cross-validation and the evaluation part consists of 100 recordings. The results from this section relate to our submission `Kapka_SRPOL_task3_2`. Development phase {#ssec:develop} ----------------- As for the development part, we used 2 splits out of 4 for training for every fold using the suggested cross-validation even though validation splits do not influence the training process. We show in Table \[tab:results\] the averaged metrics from all folds for our setting and metrics for the baseline [@Adavanne2019_DCASE]. In order to demonstrate the variance among folds, we present in Table \[tab:test\_splits\] the detailed results on the test splits from each fold. The development set provides the distinction for the files where there is up to 1 active sound source at once (ov1) and where there are up to 2 (ov2). In Table \[tab:ov1\_ov2\] we compare metrics for the ov1 and ov2 subsets. [l\*5c]{} & Error rate & F-score & DOA error & Frame recall & Seld score\ Train & 0.03 & 0.98 & 2.71 & 0.98 & 0.02\ Val. & 0.15 & 0.89 & 4.81 & 0.95 & 0.08\ Test & 0.14 & 0.90 & 4.75 & 0.95 & 0.08\ Baseline & 0.34 & 0.80 & 28.5 & 0.85 & 0.22\ \[tab:results\] [l\*5c]{} & Error rate & F-score & DOA error & Frame recall & Seld score\ Split 1 & 0.13 & 0.91 & 6.01 & 0.95 & 0.07\ Split 2 & 0.16 & 0.88 & 6.01 & 0.95 & 0.09\ Split 3 & 0.11 & 0.93 & 4.93 & 0.96 & 0.06\ Split 4 & 0.17 & 0.86 & 5.89 & 0.96 & 0.10\ \[tab:test\_splits\] [l\*5c]{} & Error rate & F-score & DOA error & Frame recall & Seld score\ ov1 & 0.07 & 0.94 & 1.28 & 0.99 & 0.04\ ov2 & 0.18 & 0.87 & 7.96 & 0.93 & 0.11\ \[tab:ov1\_ov2\] Official results {#ssec:official} ---------------- For the evaluation part, we used all 4 splits for training from the development set. We compare our final results with the selected submissions in Table \[tab:comparison\]. The idea of decomposing the SELD task into simpler ones proved to be a very popular idea among contestants. The recent two-stage approach to SELD introduced in [@Cao_oryginal] was used and developed further by many. The best submission using two-step approach `Cao_Surrey_task3_4` [@Cao] obtained results very similar to ours. `He_THU_task3_2` [@He] and `Chang_HYU_task3_3` [@Chang] outperform our submission in SED metrics and DOA error respectively. However, our approach based on estimating NOAS first allows us to outperform all contestants in frame recall. [\*2l\*4c]{} **Rank** & **Submission name** & **Error rate** & **F-score** & **DOA error** & **Frame recall**\ 1 & `Kapka_SRPOL_task3_2` & 0.08 & 94.7 & 3.7 & **96.8**\ 4 & `Cao_Surrey_task3_4` & 0.08 & 95.5 & 5.5 & 92.2\ 6 & `He_THU_task3_2` & **0.06** & **96.7** & 22.4 & 94.1\ 19 & `Chang_HYU_task3_3` & 0.14 & 91.9 & **2.7** & 90.8\ 48 & `DCASE2019_FOA_baseline` & 0.28 & 85.4 & 24.6 & 85.7\ \[tab:comparison\] Submissions {#sec:submission} =========== Overall, we created 4 submissions for the competition: - ConseqFOA (`Kapka_SRPOL_task3_2`), - ConseqFOA1 (`Kapka_SRPOL_task3_3`), - ConseqFOAb (`Kapka_SRPOL_task3_4`), - MLDcT32019 (`Lewandowski_SRPOL_task3_1`). The first three submissions use the approach described in the above sections. The only difference is that ConseqFOA is trained on all four splits from development dataset. ConseqFOA1 is trained on splits 2,3,4. ConseqFOAb is trained on all splits but the classifier in this version was trained using categorical cross-entropy instead of binary cross-entropy loss. Our MLDcT32019 submission uses a different approach. It works in the same way as the original SELDnet architecture but with the following differences: - We implemented the Squeeze-and-Excitation block [@Hu_2018_CVPR] after the last convolutional block. We pass the output from the last convolutional block through two densely connected neural layers with respectively 1 and 4 neurons, we multiply it with the output of the last convolutional block and we pass it further to recurrent layers. - We set all dropout rates to $0.2$. - We used SpecAugment [@specAug] as an augmentation technique to double the training dataset. - We replaced recurrent layer GRU units with LSTM units. Conclusion {#sec:conclusion} ========== We conclude that decomposing the SELD problem into simpler tasks is instinctive and efficient. However, we are aware that our solution has some serious limitations and it fails when one wants to consider a more general setup. For example when there are more than 2 active sources at once or when the grid resolution is more refined. Thus, we claim that the pursuit for universal and efficient SELD solutions is still open. Acknowledgement {#sec:acknowledgement} =============== We are most grateful to Zuzanna Kwiatkowska for spending her time on careful reading with a deep understanding the final draft of this paper.

--- abstract: 'We describe a novel technique for creating an artificial magnetic field for ultra-cold atoms using a periodically pulsed pair of counter propagating Raman lasers that drive transitions between a pair of internal atomic spin states: a multi-frequency coupling term. In conjunction with a magnetic field gradient, this dynamically generates a rectangular lattice with a non-staggered magnetic flux. For a wide range of parameters, the resulting Bloch bands have non-trivial topology, reminiscent of Landau levels, as quantified by their Chern numbers.' author: - Tomas Andrijauskas - 'I. B. Spielman' - Gediminas Juzeliūnas title: 'Topological lattice using multi-frequency radiation' --- Introduction ============ Ultracold atoms find wide applications in realising condensed matter phenomena [@Greiner2002; @Lewenstein2007; @Bloch2008a; @Lewenstein2012]. Since ultracold atom systems are ensembles of electrically neutral atoms, various methods have been used to simulate Lotentz-type forces, with an eye for realizing physics such as the quantum Hall effect (QHE). Lorentz forces are present in spatially rotating systems [@Matthews1999a; @Madison2001; @Abo-Shaeer2001; @Cooper2008; @Fetter2009; @Gemelke2010; @Wright13PRL] and appear in light-induced geometric potentials [@Dalibard2011; @Goldman2014]. The magnetic fluxes achieved with these methods are not sufficiently large for realizing the integer or fractional QHE. In optical lattices, larger magnetic fluxes can be created by shaking the lattice potential [@struck12; @Windpassinger2013RPP; @Jotzu2014; @Eckardt16review], combining static optical lattices along with laser-assisted spin or pseudo spin coupling [@Javanainen2003; @Jaksch2003; @Osterloh2005; @Dalibard2011; @Cooper2011PRL; @Aidelsburger:2013; @Goldman2014; @goldman16review; @miyake13harper]; current realizations of these techniques are beset with micro motion and interaction induced heating effects. Here we propose a new method that simultaneously creates large artificial magnetic fields and a lattice that may overcome these limitations. Our technique relies on a pulsed atom-light coupling between internal atomic states along with a state-dependent gradient potential that together create a two-dimensional (2D) periodic potential with an intrinsic artificial magnetic field. With no pre-existing lattice potential, there are no a priori resonant conditions that would otherwise constrain the modulation frequency to avoid transitions between original Bloch bands [@Weinberg15PRA]. For a wide range of parameters, the ground and excited bands of our lattice are topological, with nonzero Chern number. Moreover, like Landau levels the lowest several bands can all have unit Chern number. The manuscript is organized as follows. Firstly, we describe a representative experimental implementation of our technique directly suitable for alkali atoms. Secondly, because the pulsed atom-light coupling is time-periodic, we use Floquet methods to solve this problem. Specifically, we employ a stroboscopic technique to obtain an effective Hamiltonian. Thirdly, using the resulting band structure we obtain a phase diagram which includes a region of Landau level-like bands each with unit Chern number. Pulsed lattice ============== Figure 1 depicts a representative experimental realization of the proposed method. A system of ultracold atoms is subjected to a magnetic field with a strength $B(X)=B_{0}+B^{\prime}X$. This induces a position-dependent splitting $g_{F}\mu_{{\rm B}}B$ between the spin up and down states; $g_{F}$ is the Landé $g$-factor and $\mu_{{\rm B}}$ is the Bohr magneton. Additionally, the atoms are illuminated by a pair of Raman lasers counter propagating along ${\bf e}_{y}$, i.e. perpendicular to the detuning gradient. The first beam (up-going in Fig. \[fig:schematic\](a)) is at frequency $\omega^{+}=\omega_{0}$, while the second (down-going in Fig. \[fig:schematic\](a)) contains frequency components $\omega_{n}^{-}=\omega_{0}+(-1)^{n}(\delta\omega+n\omega)$; the difference frequency between these beams contains frequency combs centered at $\pm\delta\omega$ with comb teeth spaced by $2\omega$, as shown in Fig. \[fig:schematic\](b). In our proposal, the Raman lasers are tuned to be in nominal two-photon resonance with the Zeeman splitting from the large offset field $B_{0}$ such that $g_{F}\mu_{{\rm B}}B_{0}=\hbar\delta\omega_{0}$, making the frequency difference $\omega_{n=0}^{-}-\omega^{+}$ resonant at $X=0$, where $B=B_{0}$. Intuitively, each additional frequency component $\omega_{n}^{-}$ adds a resonance condition at the regularly spaced points $X_{n}=n\hbar\omega/g_{F}\mu_{{\rm B}}B^{\prime}$, however, transitions using even-$n$ side bands give a recoil kick opposite from those using odd-$n$ side bands (see Fig. \[fig:schematic\](c)). Each of these coupling-locations locally realizes synthetic magnetic field experiment performed at NIST [@Lin2009b], arrayed in a manner to give a rectified artificial magnetic field with a non-zero average that we will show is a novel flux lattice. ![Floquet flux lattice. a. Experimental schematic depicting a cold cloud of atoms in a gradient magnetic field, illuminated by a pair of counter-propagating laser beams tuned near two-photon Raman resonance. The down-going beam includes sidebands both to the red and blue of the carrier ($\omega_{0}$) in resonance at different spatial positions along ${\bf e}_{x}$. b. Level diagram showing even and odd side-bands linking the $\left|\uparrow\right\rangle $ and $\left|\downarrow\right\rangle $ states with differing detuning from resonance at $X=0$. c. Spatially dependent coupling. Bottom: different frequency components are in two-photon resonance in different $X$ positions. Top: the recoil kick associated with the Raman transition is along $\pm\mathbf{e}_{y}$ and thus alternates spatially depending on whether the Raman transition is driven from the red or blue sideband of the down-going laser beam.[]{data-label="fig:schematic"}](fig1) We formally describe our system by first making the rotating wave approximation (RWA) with respect to the large offset frequency $\omega_{0}$. This situation is modeled in terms of a spin-1/2 atom of mass $M$ and wave-vector $\bm{K}$ with a Hamiltonian $$H(t)=H_{0}+V(t).\label{eq:full-Hamiltonian}$$ The first term is $$H_{0}=\frac{\hbar^{2}\bm{K}^{2}}{2M}+\frac{\Delta(X)}{2}\sigma_{3},\label{eq:Hamiltonian0}$$ where $\Delta(X)=\Delta^{\prime}X$ describes the detuning gradient along $\mathbf{e}_{x}$ axis, and $\sigma_{3}=|\!\uparrow\rangle\langle\uparrow\!|-|\!\downarrow\rangle\langle\downarrow\!|$ is a Pauli spin operator. In the RWA only near-resonant terms are retained, giving the Raman coupling described by $$V(t)=V_{0}\sum_{n}\left[{\rm e}^{{\rm i}(K_{0}Y-2n\omega t)}+{\rm e}^{{\rm i}(-K_{0}Y-(2n+1)\omega t)}\right]|\!\downarrow\rangle\langle\uparrow\!|+{\rm H.\,c.}\,.\label{eq:Raman-coupling}$$ The first term describes coupling from the sidebands with even frequencies $2n\omega$, whereas the second term describes coupling from the sidebands with odd frequencies $\left(2n+1\right)\omega$. The recoil kick is aligned along $\pm\mathbf{e}_{y}$ with opposite sign for the even and odd frequency components. In writing Eq.(\[eq:Raman-coupling\]) we assumed that the coupling amplitude $V_{0}$ and the associated recoil wave number $K_{0}$ are the same for all frequency components. The coupling Hamiltonian $V(t)$ and therefore the full Hamiltonian $H(t)$ are time-periodic with period $2\pi/\omega$, and we accordingly apply Floquet techniques. Theoretical analysis ==================== The outline of this Section is as follows. (1) We begin the analysis of the Hamiltonian given by Eq. (\[eq:full-Hamiltonian\]) by moving to dimensionless units; (2) subsequently derive an approximate effective Hamiltonian from the single-period time evolution operator; (3) provide an intuitive description in terms of adiabatic potentials; and (4) finally solve the band structure, evaluate its topology and discuss possibilities of the experimental implementation. Dimensionless units ------------------- For the remainder of the manuscript we will use dimensionless units. All energies will be expressed in units of $\hbar\omega$, derived from the Floquet frequency $\omega$; time will be expressed in units of inverse driving frequency $\omega^{-1}$, denoted by $\tau=\omega t$; spatial coordinates will be expressed in units of inverse recoil momentum $K_{0}^{-1}$, denoted by lowercase letters $(x,y)=K_{0}(X,Y)$. In these units, the Hamiltonian (\[eq:full-Hamiltonian\]) takes the form $$h(\tau)=\frac{H(\tau/\omega)}{\hbar\omega}=E_{\text{r}}\bm{k}^{2} +\frac{1}{2}\boldsymbol{\Omega}(\tau)\cdot\boldsymbol{\sigma}\,, \label{eq:dimless-Hamiltonian}$$ where $E_{\text{r}}=\hbar^{2}K_{0}^{2}/(2M\hbar\omega)$ is the dimensionless recoil energy associated with the recoil wavenumber $K_{0}$; $\bm{k}=\bm{K}/K_{0}$ is the dimensionless wavenumber. The dimensionless coupling $$\boldsymbol{\Omega}(x,y,\tau)=\left(2{\rm Re}\,u(y,\tau),\,2{\rm Im}\,u(y,\tau),\,\beta x\right) \label{eq:full-coupling}$$ includes a combination of position-dependent detuning and Raman coupling. Here $\beta=\Delta^{\prime}/(\hbar\omega k_{0})$ describes the linearly varying detuning in dimensionless units; the function $u(y,\tau)=v_{0}\sum_{n}\left\{ \exp[{\rm i}(y-2n\tau)]+\exp[{\rm i}(-y-(2n+1)\tau)]\right\} $ is a dimensionless version of the sum in Eq. (\[eq:Raman-coupling\]) with $v_{0}=V_{0}/(\hbar\omega)$. The vector $\boldsymbol{\Omega}(x,y,\tau)$ is spatially periodic along the $y$ direction with a period $2\pi$. This period can be halved to $\pi$ by virtue of a gauge transformation $U=\exp(-{\rm i}y\sigma_{3}/2)$. Subsequently, when exploring energy bands and their topological properties, this prevents problems arising from using a twice larger elementary cell. Following this transformation the dimensionless Hamiltonian becomes $$\tilde{h}(\tau)=E_{\text{r}}\left(\bm{k}+\sigma_{3}\bm{e}_{y}/2\right)^{2} +\frac{1}{2}\tilde{\boldsymbol{\Omega}}(\tau)\cdot\boldsymbol{\sigma}$$ with $\tilde{\boldsymbol{\Omega}}(\tau)=U\boldsymbol{\Omega}(\tau)U^{-1}$. In the time domain the coupling (\[eq:full-coupling\]) is $$\frac{1}{2}\tilde{\boldsymbol{\Omega}}(\tau)\cdot\boldsymbol{\sigma}=\frac{1}{2}\beta x\sigma_{3} +\sum_{l}v_{l}(y)\delta(\tau-\pi l), \label{eq:coupling-even-odd}$$ with $$v_{l}(y)=\pi v_{0}\left[{\rm e}^{{\rm i}2y}+(-1)^{l}\right]|\!\downarrow\rangle\langle\uparrow\!|+{\rm H.\,c.}\,.$$ In this way we separated the spatial and temporal dependencies in the coupling (\[eq:coupling-even-odd\]). Effective Hamiltonian --------------------- We continue our analysis by deriving an approximate Hamiltonian that describes the complete time evolution over a single period from $\tau=0-\epsilon$ to $\tau=2\pi-\epsilon$ with $\epsilon\to0$. This evolution includes a kick $v_{0}$ at the beginning of the period $\tau_{+}=0$ and a second kick $v_{1}$ in the middle of the period $\tau_{-}=\pi$; between the kicks the evolution includes the kinetic and gradient energies. In the full time period, the complete evolution operator is a product of four terms: $$U(2\pi,0)\equiv\lim_{\epsilon\to0}U(2\pi-\epsilon,0-\epsilon)=U_{0}U_{\text{kick}}^{(1)}U_{0}U_{\text{kick}}^{(0)}. \label{eq:time-evolution}$$ Here $$U_{0}=\exp\left\{ -{\rm i}\pi\left[E_{\text{r}}\left(\bm{k}+\frac{1}{2}\sigma_{3}\bm{e}_{y}\right)^{2} +\frac{1}{2}\sigma_{3}\beta x\right]\right\} \label{eq:time-evolution-0}$$ is the evolution operator over the half period, generated by kinetic energy and gradient. The operator $$U_{\text{kick}}^{(l)}=\exp\left[-{\rm i}v_{l}(y)\right].\label{eq:kicks}$$ describes a kick at $\tau=l\pi$. We obtain an effective Hamiltonian by assuming that the Floquet frequency $\omega$ greatly exceeds the recoil frequency, $1\gg E_{\text{r}}$, allowing us to ignore the commutators between the kinetic energy and functions of coordinates in eq.(\[eq:time-evolution\]). We then rearrange terms in the full time evolution operator (\[eq:time-evolution\]) and obtain $$U_{\text{eff}}=\exp\left\{ -{\rm i}2\pi\left[E_{\text{r}}\left(\bm{k} +\sigma_{3}\bm{e}_{y}/2\right)^{2}+v_{\text{eff}}\right]\right\}, \label{eq:effective-time-evolution-op}$$ where $v_{\text{eff}}$ is an effective coupling defined by $$\exp\left(-{\rm i}2\pi v_{\text{eff}}\right)={\rm e}^{-{\rm i}\pi\sigma_{3}\beta x/2}U_{\text{kick}}^{(1)}{\rm e}^{-{\rm i} \pi\sigma_{3}\beta x/2}U_{\text{kick}}^{(0)}. \label{eq:V-eff-exponent}$$ The algebra of Pauli matrices allows us to write the effective coupling in a form $$v_{\text{eff}}(\bm{r})=\frac{1}{2}\bm{\Omega}_{\text{eff}}(\bm{r})\cdot\bm{\sigma}, \label{eq:effective-coupling}$$ where $\bm{\Omega}_{\text{eff}}=\left(\Omega_{\text{eff},1},\Omega_{\text{eff},2},\Omega_{\text{eff},3}\right)$ is a position-dependent effective Zeeman field which takes the analytic form $$\exp\left(-{\rm i}2\pi v_{\text{eff}}\right)=q_{0}-{\rm i}q_{1}\sigma_{1} -{\rm i}q_{2}\sigma_{2}-{\rm i}q_{3}\sigma_{3}. \label{eq:V-eff-exponent-in-q}$$ Here $q_{0}$, $q_{1}$, $q_{2}$ and $q_{3}$ are real functions of the coordinates $(x,y)$, allowing to express the effective Zeeman field as $$\bm{\Omega}_{\text{eff}}=\pi^{-1}\frac{\bm{q}}{||\bm{q}||}\arccos q_{0}, \label{eq:effective-magnetic-field}$$ where $\bm{q}$ is a shorthand of a three dimensional vector $(q_{1},q_{2,}q_{3})$. In general the equation (\[eq:V-eff-exponent-in-q\]) gives multiple solutions that correspond for different Floquet bands. Our choice (\[eq:effective-magnetic-field\]) picks only to the two bands that lie in the energy window from $-1/2$ to $1/2$ covering a single Floquet period. Comparing (\[eq:V-eff-exponent\]) and (\[eq:V-eff-exponent-in-q\]) and multiplying four matrix exponents give explicit expressions $$\begin{aligned} q_{0} & =\cos f_{1}\cos f_{2}\cos(\pi\beta x),\label{eq:function-q0}\\ q_{1} & =\sin f_{1}\cos f_{2}\cos(y+\pi\beta x)-\cos f_{1}\sin f_{2}\sin(y),\label{eq:function-q1}\\ q_{2} & =\sin f_{1}\cos f_{2}\sin(y+\pi\beta x)+\cos f_{1}\sin f_{2}\cos(y),\label{eq:function-q2}\\ q_{3} & =\cos f_{1}\cos f_{2}\sin(\pi\beta x)-\sin f_{1}\sin f_{2}\label{eq:function-q3}\end{aligned}$$ with $$\begin{aligned} f_{1}(y) & =2\pi v_{0}\cos(y),\label{eq:function-f1}\\ f_{2}(y) & =2\pi v_{0}\sin(y).\label{eq:function-f2}\end{aligned}$$ ![Coupling components (a) $\Omega_{1}(\bm{r})$, (b) $\Omega_{2}(\bm{r})$ and (c) $\Omega_{3}(\bm{r})$ for $v_{0}=0.25$ and $\beta=0.6$. The corresponding eigenvalues of the coupling $v_{\pm}(\bm{r})=\pm\Omega_{\text{eff}}/2$ are presented by the thick red solid lines in the fig. \[fig:floquet-spectrum\].[]{data-label="fig:coupling"}](fig2a "fig:"){width="33.00000%"}![Coupling components (a) $\Omega_{1}(\bm{r})$, (b) $\Omega_{2}(\bm{r})$ and (c) $\Omega_{3}(\bm{r})$ for $v_{0}=0.25$ and $\beta=0.6$. The corresponding eigenvalues of the coupling $v_{\pm}(\bm{r})=\pm\Omega_{\text{eff}}/2$ are presented by the thick red solid lines in the fig. \[fig:floquet-spectrum\].[]{data-label="fig:coupling"}](fig2b "fig:"){width="33.00000%"}![Coupling components (a) $\Omega_{1}(\bm{r})$, (b) $\Omega_{2}(\bm{r})$ and (c) $\Omega_{3}(\bm{r})$ for $v_{0}=0.25$ and $\beta=0.6$. The corresponding eigenvalues of the coupling $v_{\pm}(\bm{r})=\pm\Omega_{\text{eff}}/2$ are presented by the thick red solid lines in the fig. \[fig:floquet-spectrum\].[]{data-label="fig:coupling"}](fig2c "fig:"){width="33.00000%"} These explicit expressions show that the resulting effective Zeeman field (\[eq:effective-magnetic-field\]) and the associated effective coupling (\[eq:effective-coupling\]) are periodic along both $\bm{e}_{x}$ and $\bm{e}_{y}$, with spatial periods $a_{x}=2/\beta$ and $a_{y}=\pi$ respectively. Therefore, although the original Hamiltonian containing the spin-dependent potential slope $\propto x\sigma_{3}$ is not periodic along the $x$ direction, the effective Floquet Hamiltonian is. The spatial dependence of the Zeeman field components $\Omega_{\text{eff},1}$, $\Omega_{\text{eff},2}$ and $\Omega_{\text{eff},3}$ is presented in the fig. \[fig:coupling\] for $\beta=0.6$ giving an approximately square unit cell. In fig. \[fig:coupling\] we select $v_{0}=0.25$ where the absolute value of the Zeeman field $\Omega_{\text{eff}}$ is almost uniform, as is apparent from the nearly flat adiabatic bands shown in fig. \[fig:floquet-spectrum\] below. Adiabatic evolution and magnetic flux\[subsec:Adiabatic-evolution-and\] ----------------------------------------------------------------------- Before moving further to an explicit numerical analysis of the band structure, we develop an intuitive understanding by performing an adiabatic analysis of motion governed by effective Hamiltonian $$h_{\text{eff}}(\bm{r})=E_{\text{r}}\left(\bm{k}+\sigma_{3}\bm{e}_{y}/2\right)^{2} +\frac{1}{2}\bm{\Omega}_{\text{eff}}\cdot\bm{\sigma}\, \label{eq:h_eff}$$ featured in the evolution operator $U_{\text{eff}}$, Eq. (\[eq:effective-time-evolution-op\]). The coupling field $\bm{\Omega}_{\text{eff}}(\bm{r})$ is parametrized by the spherical angles $\theta(\bm{r})$ and $\phi(\bm{r})$ defined by $$\begin{aligned} \cos\theta & =\frac{\Omega_{\text{eff},3}}{\Omega_{\text{eff}}},\label{eq:spherical-cos-theta}\\ \tan\phi & =\frac{\Omega_{\text{eff},2}}{\Omega_{\text{eff},1}}.\label{eq:spherical-tan-phi}\end{aligned}$$ This gives the effective coupling [@Dalibard2011] $$\frac{1}{2}\bm{\Omega}_{\text{eff}}\cdot\bm{\sigma}= \frac{1}{2}\Omega_{\text{eff}}\left[\begin{array}{cc} \cos\theta & {\rm e}^{-{\rm i}\phi}\sin\theta\\ {\rm e}^{{\rm i}\phi}\sin\theta & -\cos\theta \end{array}\right]\,,\label{eq:eff-coupling-in-spherical-coords}$$ characterized by the position-dependent eigenstates $$\left|+\right\rangle =\left(\begin{array}{c} \cos\left(\theta/2\right)\\ {\rm e}^{{\rm i}\phi}\sin\left(\theta/2\right) \end{array}\right)\,,\qquad\left|-\right\rangle =\left(\begin{array}{c} -{\rm e}^{-{\rm i}\phi}\sin\left(\theta/2\right)\\ \cos\left(\theta/2\right) \end{array}\right)\,.\label{eq:pm-states}$$ The corresponding eigenvalues $$v_{\pm}(\bm{r})=\pm\frac{1}{2}\Omega_{\text{eff}}, \label{eq:eigenvalues-of-V-eff}$$ are shown in Fig. \[fig:floquet-spectrum\] for various value of the Raman coupling $v_{0}$. As one can see in Fig. \[fig:floquet-spectrum\], for $v_{0}=0.25$ the resulting bands $v_{\pm}(\bm{r})$ (adiabatic potentials) are flat and have a considerable gap $\approx\omega/2$, a regime suitable for a description in terms of an adiabatic motion in selected bands [@Zoller2008]. ![Adiabatic Floquet potentials for $\beta=0.6$. (a) Thin black dotted lines denote the spin-dependent gradient slopes without including the Raman coupling ($v_{0}=0$); (b) thin blue solid lines denote effective adiabatic potentials for weak Raman coupling ($v_{0}=0.05$) (c) red solid lines denote nearly flat adiabatic potentials that are achieved for stronger Raman coupling ($v_{0}=0.25$). All the curves are projected into $x$ plane for various $y$ values. A weak $y$ dependence of the adiabatic potentials is seen to appear in the strong coupling case (c) making the superimposed red lines thicker.[]{data-label="fig:floquet-spectrum"}](fig3){width="75.00000%"} As in Ref. [@Juzeliunas2012], we consider the adiabatic motion of the atom in one of these flat adiabatic bands with the projection Shrodinger equation that includes a geometric vector potential $$\bm{A}_{\pm}(\bm{r})=\pm\frac{1}{2}\left(\cos\theta-1\right)\nabla\phi\,. \label{eq:geometric-vector-potential}$$ This provides a synthetic magnetic flux density $\bm{B}_{\pm}(\bm{r})=\nabla\times\bm{A}_{\pm}(\bm{r})$. The geometric vector potential $\bm{A}_{\pm}(\bm{r})$ may contain Aharonov-Bohm type singularities, that give rise to a synthetic magnetic flux over an elementary cell $$\alpha_{\pm}=-\sum\oint_{{\rm singul}}{\rm d}\bm{r}\cdot\bm{A}_{\pm}(\bm{r}). \label{eq:synthetic-magnetic-flux}$$ The singularities appear at points where $\theta=\pi$, where the angle $\phi$ and its gradient $\nabla\phi$ are undefined and $\cos\theta=-1$. The term $\cos\theta-1$ in (\[eq:geometric-vector-potential\]) is non zero and does not remove the undefined phase $\nabla\phi$. Our unit cell contains two such singularities located at $\bm{r}=(a_{x},3a_{y})/4$ and $\bm{r}=(3a_{x},a_{y})/4$, containing the same flux, so that they do not compensate each other, giving the synthetic magnetic flux $\pm2\pi$ in each unit cell. ![Geometric flux density $\bm{B}_{+}$ computed for $v_{0}=0.25$ and $\beta=0.6$. The overall spatial structure of this flux density does not depend on the gradient $\beta$; rather it scales with the corresponding lattice constant $a_{x}=2/\beta$. []{data-label="fig:flux-density"}](fig4){width="50.00000%"} For a weak coupling (such as $v=0.05$) the geometric flux density $\bm{B}(\bm{r})\equiv\bm{B}_{\pm}(\bm{r})$ is concentrated around the intersection points of the gradient slopes shown in in Fig. \[fig:floquet-spectrum\] and has a very weak $y$ dependence. With increasing the coupling $v$, the flux extends beyond the intersection areas and acquires a $y$ dependence. Fig. \[fig:flux-density\] shows the geometric flux density $\bm{B}(\bm{r})\equiv\bm{B}_{+}(\bm{r})$ for the strong coupling ($v_{0}=0.25$) corresponding to the most flat adiabatic bands. In this regime the flux develops stripes in the $x$ direction and has a strong $y$ dependence. For the whole range of coupling strengths $0\le v_{0}\le 1/2$ the total synthetic magnetic flux per unit cell is $2\pi$ and is independent of the Floquet frequency $\omega$ and the gradient $\beta$. Band structure and Chern numbers -------------------------------- We analyze the topological properties of this Floquet flux lattice by explicitly numerically computing the band structure and associated Chern number using the effective Hamiltonian (\[eq:h\_eff\]) without making the adiabatic approximation introduced in Sec. \[subsec:Adiabatic-evolution-and\]. Again the gradient of the original magnetic field is such that we approximately get a square lattice, $\beta=0.6$. Furthermore, we choose the Floquet frequency to be ten times larger than the recoil energy, $E_{\text{r}}=0.1$. First, let us consider the case where $v_{0}=0.25$ corresponding to the most flat adiabatic potential. In this situation the Chern numbers of the first five bands appear to be equal to the unity, as one can see in the left part of Fig. \[fig:bands-chern\]. Thus the Hall current should monotonically increase when filling these bands. This resembles the Quantum Hall effect involving the Landau levels. Second, we check what happens when we leave the regime $v_{0}=0.25$ where the adiabatic potential is flat, and consider lower and higher values of the coupling strength $v_{0}$. Near $v_{0}=0.175$ we find a topological phase transition where the lowest two energy bands touch and their Chern numbers change to $c_{1}=0$ and $c_{2}=2$, while the Chern numbers of the higher bands remain unchanged, illustrated in fig. \[fig:chern\]. In a vicinity of $v_{0}=0.3$ there is another phase transition, where the second and third bands touch, leading to a new distribution of Chern numbers: $c_{1}=1$, $c_{2}=-1$, $c_{3}=3$, $c_{4}=1$. Interestingly the Chern numbers of the second and the third bands jump by two units during such a transition. ![Left: band structure given by the effective Hamiltonian (\[eq:h\_eff\]) for $v_{0}=0.25$, $\beta=0.6$ and $E_{\text{r}}=0.1$. Right: The band gap $\Delta_{12}$ between the first and second bands for $E_{\text{r}}=0.1$ and various values of $v_{0}$ and $\beta$.[]{data-label="fig:bands-chern"}](fig5){width="85.00000%"} ![Dependence of Chern number on the coupling strength $v_{0}$ for $\beta=0.6$ and $E_{\text{r}}=0.1$. Here we present the Chern numbers $c_{1}$, $c_{2}$ and $c_{3}$ of the three lowest bands.[]{data-label="fig:chern"}](fig6){width="50.00000%"} Finally, we explore the robustness of the topological bands. The right part of Fig. \[fig:bands-chern\] shows the dependence of the band gap $\Delta_{12}$ between the first and second bands on the coupling strength $v_{0}$ and the potential gradient $\beta$. One can see that the band gap is maximum for $v_{0}=0.25$ when the adiabatic potential is the most flat. The gap increases by increasing the gradient $\beta$, simultaneously extending the range of the $v_{0}$ values where the band gap is nonzero. Therefore to observe the topological bands, one needs to take a proper value of the Raman coupling $v_{0}\approx0.25$ and a sufficiently large gradient $\beta$, such as $\beta=0.6$. We now make some numerical estimates to confirm that this scheme is reasonable. We consider an ensemble of $^{87}{\rm Rb}$ atoms, with $\left|{\uparrow}\right>=\left|f=2,m_{F}=2\right>$ and $\left|\downarrow\right>=\left|{f=1,m_{F}=1}\right>$; the relative magnetic moment of these hyperfine states is $\approx2.1\ {\rm MHz}/{\rm G}$, where $1\ {\rm G}=10^{-4}\ {\rm T}$. For a reasonable magnetic field gradient of $300\ \mathrm{G}/\mathrm{cm}$, this leads to the $\Delta^{\prime}/\hbar\approx2\pi\times600\ \mathrm{MHz}/\mathrm{cm}=2\pi\times60\ {\rm kHz}/\mu{\rm m}$ detuning gradient. For $^{87}{\rm Rb}$ with $\lambda=790\ {\rm nm}$ laser fields the recoil frequency is $\omega_{r}/2\pi=3.5\ \mathrm{kHz}$. Along with the driving frequency $\omega=10\omega_{r}$, this provides the dimensionless energy gradient $\beta=\Delta^{\prime}/(\hbar\omega k_{0})\approx1.3$, allowing easy access to the topological bands displayed in Fig. \[fig:bands-chern\]. Loading into dressed states --------------------------- Adiabatic loading into this lattice can be achieved by extending the techniques already applied to loading in to Raman dressed states [@Lin2009a]. The loading technique begins with a BEC in the lower energy $\downarrow$ state in a uniform magnetic field $B_{0}$. Subsequently one slowly ramps on a single off resonance RF coupling field and the adiabatically ramp the RF field to resonance (at frequency $\delta\omega$). This RF dressed state can be transformed into a resonant Raman dressed by ramping on the Raman lasers (with only the $\omega_{0}+\delta\omega$ frequency on the $k^{-}$ laser beam) while ramping off the RF field. The loading procedure then continues by slowly ramping on the remaining frequency components on the $k^{-}$ beam, and finally by ramping on the magnetic field gradient (essentially according in the lattice sites from infinity). This procedure leaves the BEC in the $q=0$ crystal momentum state in a single Floquet band. Conclusions =========== Initial proposals [@Juzeliunas2006; @Spielman2009; @Gunter2009] and experiments [@Lin2009b] with geometric gauge potentials were limited by the small spatial regions over which these existed. Here we described a proposal that overcomes these limitations using laser coupling reminiscent of a frequency comb: temporally pulsed Raman coupling. Typically, techniques relying on temporal modulation of Hamiltonian parameters to engineer lattice parameters suffer from micro-motion driven heating. Because our method is applied to atoms initially in free space, with no optical lattice present, there are no a priori resonant conditions that would otherwise constrains the modulation frequency to avoid transitions between original Bloch bands [@Weinberg15PRA]. Still, no technique is without its limitations, and this proposal does not resolve the second standing problem of Raman coupling techniques: spontaneous emission process from the Raman lasers. Our new scheme extends the spatial zone where gauge fields are present by adding side-bands to Raman lasers, ultimately leading to a $\propto\sqrt{N}$ increase in the required laser power (where $N$ is the number of frequency tones), and therefore the spontaneous emission rate. As a practical consequence it is likely that this technique would not be able reach the low entropies required for many-body topological matter in alkali systems [@Goldman2014], but straightforward implementations with single-lasers on alkaline-earth clock transitions [@Fallani16PRL; @Kolkowitz2016socSr] are expected to be practical. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Immanuel Bloch, Egidijus Anisimovas and Julius Ruseckas for helpful discussions. This research was supported by the Lithuanian Research Council (Grant No. MIP-086/2015). I. B. S. was partially supported by the ARO’s Atomtronics MURI, by AFOSR’s Quantum Matter MURI, NIST, and the NSF through the PCF at the JQI. [34]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop [****, ()]{} [****,  ()](\doibase 10.1080/00018730701223200) [****,  ()](\doibase 10.1103/RevModPhys.80.885) [**](https://www.amazon.com/Ultracold-Atoms-Optical-Lattices-Simulating/dp/0199573123) (, ) @noop [****, ()]{} @noop [****,  ()]{} @noop [****,  ()]{} [****,  ()](http://www.informaworld.com/10.1080/00018730802564122 %@ 0001-8732 %[ August 24, 2009) [****,  ()](\doibase 10.1103/RevModPhys.81.647) @noop [“,” ]{} (),  @noop [****,  ()]{} [****,  ()](\doibase 10.1103/RevModPhys.83.1523) [****,  ()](\doibase 10.1088/0034-4885/77/12/126401) [****,  ()](\doibase 10.1103/PhysRevLett.108.225304) [****,  ()](\doibase 10.1088/0034-4885/76/8/086401) [****,  ()](\doibase 10.1038/nature13915) @noop [****,  ()]{} [****, ()](\doibase 10.1103/PhysRevLett.91.150404) [****,  ()](\doibase 10.1088/1367-2630/5/1/356) [****,  ()](\doibase 10.1103/PhysRevLett.95.010403) @noop [**** ()]{} [****,  ()](\doibase 10.1103/PhysRevLett.111.185301) [****,  ()](\doibase 10.1038/nphys3803) @noop [****,  ()]{} @noop [****, ()]{} @noop [****,  ()]{} @noop [****,  ()]{} [****,  ()](http://stacks.iop.org/1367-2630/14/i=12/a=123022) @noop [****, ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{}

Introduction ============ The study of manganite materials continues attracting considerable attention due to their potential technological applications and interesting physical properties dominated by its colossal magnetoresistance effect [@jin]. Ferromagnetism in models for manganites has been studied since the 1950s, and it is widely assumed that the Double-Exchange (DE) ideas are sufficient to understand the stabilization of a magnetic state at low-temperatures upon hole-doping the manganites [@zener]. The main reason for this stabilization is that the kinetic energy of the holes is substantially improved in a fully aligned spin background. These concepts are similar to those that lead to the well-known Nagaoka phase in the context of the $t-J$ model for the cuprates. However, experimental work has revealed a phase diagram for manganites that is far more complicated than simple DE models predict [@co]. In particular, at densities corresponding to the undoped compound (referred to as “half-filling” ($n=1$) in the language of the one-orbital model) an A-type antiferromagnet (AF) is stabilized. After a small amount of holes is introduced, some manganites enter a regime where ferromagnetic droplets have been recently observed [@hennion]. Upon further doping the DE ferromagnetic metallic regime is reached, including the widely studied $x \sim 1/3$ concentration. In the other limit of intermediate- and small-electron densities, a charge-ordered state is stabilized in $\rm Ca$-doped manganites [@co]. In addition, at densities corresponding to the DE ground state at low-temperatures, a metal-insulator transition occurs as the temperature is raised. This insulating state is likely responsible for the colossal magnetoresistance effects in manganites. Recent work has initiated the computational analysis of models for these compounds [@yunoki; @dago; @yunoki2; @yunoki3], allowing for calculations that go beyond simple mean-field approximations. The techniques used for such computational studies are borrowed in part from the field of cuprates, where they were applied to the analysis of $t-J$-like models in recent years [@review]. The use of unbiased techniques in models for manganites has already led to a fairly good understanding of the zero temperature phase diagram of the so-called Ferromagnetic Kondo model (FKM) which is based on the assumption that only one $e_g$-orbital is relevant. Although Jahn-Teller (JT) effects cannot be neglected in a quantitative study of manganites [@millis], it is important to clarify the properties of this simplest one-orbital model before addressing more complicated two-orbital Hamiltonians with JT-phonons. Following this computational approach, Yunoki et al. [@yunoki; @dago; @yunoki2] recently reported the phase diagram of the FKM addressing dimensions $D=1,2,3$ and $\infty$. The computational techniques used were the Monte Carlo method in $D=1,2,3$ with classical $t_{2g}$-spins, the dynamical mean-field approximation in $D=\infty$ [@furu], as well as the Exact Diagonalization (ED) and Density-Matrix Renormalization Group (DMRG) [@white] techniques in one dimension for quantum-mechanical localized $t_{2g}$-spins. The phase diagram was found to contain three dominant regimes: (i) a ferromagnetic phase induced by the DE mechanism, (ii) a spin incommensurate regime at intermediate and small Hund coupling [@inoue] predominantly due to the RKKY interaction, and (iii) a region of “phase separation” (PS) near density $n=1$ between ferromagnetic hole-rich and antiferromagnetic hole-undoped phases. The latter regime (PS) is unexpected since for a long time it has been assumed that the transition from the AF regime, characteristic of the undoped region, to the FM regime at $x \sim 1/3$ should occur through a spin-canted state [@gennes]. However, the results of Ref. [@yunoki] showed that this is incorrect and actually the transition from AF to FM occurs through the creation of “islands” of one phase embedded into the other, growing in size as the overall density changes. Work by other groups [@also] is in agreement with the main conclusions of Ref. [@yunoki]. Previous studies in 1D models also indicated the relevance of phase separation as one moves from copper to the left in the transition-metal row of the periodic table of elements [@riera]. On the experimental side, a growing body of evidence suggests that the transition from undoped $\rm LaMnO_3$ to the doped ferromagnetic compounds indeed occurs through an inhomogeneous process. For instance, the existence of quasi-static magnetic droplets at small hole-density in $\rm La_{1-x} Ca_x Mn O_3$ has recently been emphasized using elastic neutron scattering techniques at $x=0.05$ and $0.08$ below the critical temperature for the magnetic state [@hennion]. Phase separation using NMR techniques has also been observed for the same compound at low temperatures [@allodi]. Working at $x=1/3$ and at temperatures above the ferromagnetic critical temperature $T_c^{FM}$, neutron scattering experiments were interpreted as providing evidence for two distinct phases in the system [@lynn], while small-angle neutron scattering and magnetic susceptibility data suggested the presence of magnetic clusters ($\sim 12 \AA$ in size) in the same regime  [@deteresa]. X-ray and powder neutron scattering methods applied to $\rm Pr_{0.7} Ca_{0.3} Mn O_3$ also revealed the existence of ferromagnetic clusters [@cox]. Neutron studies of $\rm La_{1-x} Sr_x Mn O_3$ at $x=0.10$ and $0.15$ were interpreted as corresponding to a polaron-ordered arrangement [@yamada]. Experimental studies of the 2D compound $\rm Sr_{2-x} La_x Mn O_4$ at small electronic density have also been considered as evidence of phase separation in manganites [@bao]. Many other papers have reported results compatible with charge segregation tendencies at temperatures and densities that surround the low-temperature ferromagnetic phase of the manganites [@otroexp; @perring; @otro]. This interesting agreement between theory and experiments reinforces the notion that simple electronic models for manganites with a strong Hund-coupling may contain the essence of the physics needed to describe these materials. However, to confirm this assumption it is important to proceed further with the computational calculations in two main directions: (i) first, the influence of Jahn-Teller phonons and the use of two $e_g$-orbitals is important for a qualitative comparison theory-experiment. An effort in this direction has been recently reported by two of the authors and A. Moreo [@yunoki3]. The existence of PS with JT-phonons was confirmed, including a tendency to a novel “orbital” phase separation regime [@yunoki3]. PS tendencies were found to be as robust in the phase diagram as in the case of a one-orbital Kondo model. This shows that studies of one-orbital Hamiltonians can capture at least part of the physics of more sophisticated multi-orbital models for manganites; (ii) second, Coulomb interactions beyond the on-site term are important to avoid the accumulation of charge suggested by the phase separation process described in Ref. [@yunoki]. It is likely that in the region of “unstable” densities the large clusters of the hole-rich ferromagnetic phase will be divided into small clusters due to this Coulomb repulsion. Actually previous work in the context of the cuprates has shown the tendency to form extended charge-ordered structures [@emery] once a long-range Coulomb interaction is added to a phase separated state. It is precisely one of the purposes of the present paper to study the influence of Coulomb interactions beyond the on-site $U$-term on a model for manganites that has the tendency to phase separate in its ground state. The selection of the model for the present investigation is important for technical reasons. The Ferromagnetic Kondo model with classical $t_{2g}$-spins analyzed in Ref. [@yunoki] is difficult to study with Coulomb interactions since in this case the Monte Carlo method will need the addition of Hubbard-Stratonovich (HS) degrees of freedom to decouple the four-fermion terms in the Hamiltonian. In addition to the intrinsic complexity of this procedure, the HS decoupling will lead to “sign-problems” [@review] that likely will prevent the study of low-temperature properties. To complicate matters even more, techniques that deal directly with the Hilbert space of the problem, such as ED and DMRG, cannot be easily applied to a model with classical spins. Then, in order to analyze the influence of intersite Coulomb terms it would be better to use $quantum$ $t_{2g}$-spins and ED or DMRG methods for their analysis. In this context there are no “sign-problems”. However, the huge Hilbert spaces that need to be studied in finite size chains impose constraints on the value of the quantum localized spins $S$. In order to carry out a study on a cluster large enough to reach bulk properties $S$ must be restricted to $1/2$, rather than the $3/2$ realistic value corresponding to manganites [@zang]. Previous calculations [@dago] have shown that qualitatively $S=1/2$, $3/2$ and $\infty$ (classical spins) give very similar phase diagrams in the absence of Coulomb interactions, and there is no reason to expect that this situation will change when these interactions are added. In addition, again due to the large size of the Hilbert spaces involved in the problem our analysis must be restricted to one-dimensional systems. Previous work has shown the existence of many similarities between the results obtained in all dimensions [@yunoki] including $1$. Thus, the restriction of working on chains should not be considered severe, and the analysis below is expected to capture the main qualitative aspects of the problem under investigation. Note that calculations using $S=1/2$ localized spins on chains have relevance not only in the context of manganites but also for recently synthesized one-dimensional materials such as ${\rm Y_{2-x} Ca_x Ba Ni O_5}$ which have two active electrons per Ni-ion. This compound has been studied experimentally [@batlogg] and theoretically [@nio], and upon doping interesting properties have been observed including a metal-insulator transition. To facilitate the understanding of the main results of this paper the phase diagram in one-dimension of the Ferromagnetic Kondo model (without Coulombic interactions and using localized $S=1/2$ spins) is reproduced in Fig.1 from Ref. [@dago]. As explained before, three main regimes were found using ED and DMRG techniques. The region labeled FM corresponds to saturated ferromagnetism (the ground state spin is maximized). The IC regime suggests the existence of incommensurate spin correlations, at least of short-range, and it manifests itself through the behavior of the spin structure factor $S(k)$ which has a peak that moves away from $k=\pi$ as holes are added to the “half-filled” system. Finally, the regime labeled PS corresponds to phase separation which was studied calculating the density $ n $ vs the chemical potential $\mu$, as well as the inverse compressibility, searching for unstable densities [@dago]. The goal of the present manuscript is to analyze how the phase diagram of Fig.1 changes when Coulomb interactions are added to the one-orbital Kondo model. The organization of the paper is as follows. In section II the details of the model, technique, and calculated observables are given. In Sec. III the phase diagram is constructed when a nonzero on-site repulsion is considered, but without including nearest-neighbor Coulomb interactions. In Sec.IV these longer-range interactions are added. The tendency to form charge-ordered states (at least at short distances) reported in this section is the main result of this paper. Implications for experiments are discussed in the conclusions (Sec.V). Model and Computational Technique ================================= As explained in the Introduction, the model used in the present study is the one-orbital Ferromagnetic Kondo model with localized (spin-1/2) degrees of freedom that mimic the effect of the $t_{2g}$-electrons. Coulomb interactions in the $e_g$-band are also incorporated in the model. The Hamiltonian defined on a chain with $L$ sites is $$H = -t \sum_{{\langle ij \rangle}\sigma} ( c^\dagger_{i \sigma} c_{j \sigma} + h.c.) - J_H \sum_i { {{\bf S}_{if}}\cdot{{\bf s}_{ic}} } +~~~$$ $$~~+ J' \sum_{\langle i j \rangle} { {{\bf S}_{if}}\cdot{{\bf S}_{jf}} } + U \sum_i n_{i \uparrow} n_{i \downarrow} + V \sum_{\langle i j \rangle} n_i n_j. \eqno(1)$$ The first term is the electron-transfer between Mn-ions. $\langle ij \rangle$ denotes nearest-neighbor sites and $t$ is the hopping amplitude that will be set to 1 in most of the manuscript. In the second term, $J_H>0$ is the ferromagnetic Hund-coupling. The spin-1/2 operator for the conduction electron is defined as $ {\bf s}_{ic} = \sum_{\alpha \beta} c^\dagger_{i \alpha} {\bf \sigma}_{\alpha \beta} c_{i \beta}, $ while ${\bf S}_{if}$ represents a localized spin-1/2 at site $i$. A strong Hund-coupling will be used throughout the paper, and its value will be fixed to $J_H = 40$, in units of $t$, unless otherwise stated. $J'$ is the strength of a direct Heisenberg coupling between the localized spins. This term is needed on phenomenological grounds since in fully doped manganites (e.g. when all $\rm La$ has been replaced by $\rm Ca$) a finite Néel temperature is experimentally observed. $U$ is the strength of the on-site electronic repulsion, with $n_{i \sigma}$ the number operator at site $i$ with spin $\sigma$. $V$ regulates the Coulomb repulsion at a distance of one lattice spacing. The rest of the notation in Eq.(1) is standard. The technique used in this paper to analyze ground state properties of the Hamiltonian Eq.(1) in one-dimension is the DMRG method. The finite-system variation of DMRG was used, working with open boundary conditions [@comm101]. All results were obtained keeping a number of states $m=100$ in the iterations, with the exception of densities $n=1$ and $0$ where a smaller number of states produced accurate enough results. With this value of $m$ a truncation error of order $10^{-6}$ or smaller was obtained throughout the results shown in the next sections. In order to characterize the ground state properties of Eq.(1), a variety of expectation values have been calculated. The spin structure-factor defined as $$S(k) = {{1}\over{L}}\sum_{j,m} \langle { {{\bf S}_{jf}}\cdot{{\bf S}_{mf}} } \rangle e^{i(j-m)k}, \eqno(2)$$ and the charge structure-factor $$N(k) = {{1}\over{L}}\sum_{j,m} \langle n_j n_m \rangle e^{i(j-m)k}, \eqno(3)$$ are among the measured quantities. In addition, the inverse compressibility defined as $$1/\kappa = {{N^2}\over{4L}} [ E(N+2,L) + E(N-2,L) - 2E(N,L) ], \eqno(4)$$ was also calculated. Here $E(N,L)$ is the ground state energy of a chain of $L$ sites with $N$ electrons, and density $n=N/L$. Finally, the charge correlation $C(i) = \langle n_{j} n_{m} \rangle - \langle n_j \rangle \langle n_m \rangle$, where $i=| j -m|$, was studied in some special cases. Here the $\langle \rangle$ notation not only denotes expectation value in the ground state but it also indicates that for a given distance $i$ all possible pairs of sites $j,m$ of the cluster compatible with $i=|j-m|$ have been used. The reason is that open boundary conditions are needed in the DMRG technique and, thus, the correlations are different at, e.g., the center and near the chain end. Such an average procedure uses information from the whole chain, and in practice it produces smoothly changing results as the distance $i$ and the couplings are varied. Results at $V=0$ ================= Let us begin the study for the case without a repulsion at a distance of one lattice-spacing (i.e. working at $V=0$). The results obtained here will be later compared with those of the following section for $V \neq 0$. Zero on-site repulsion $U$ --------------------------- In Fig.2a, the inverse compressibility $1/\kappa$ is shown as a function of $n$, with both the on-site and intersite repulsions $U$ and $V$ equal to zero, in order to study the dependence of the results with $J'$. A similar analysis was carried out by Yunoki and Moreo [@yunoki2] but in the classical localized spin limit. This study found that $J'$ is an important parameter in determining the low-temperature properties of models for manganites. Fig.2a shows that $1/\kappa$ at $J'=0.04$ is small or even slightly negative both in the limits of small and large density, in good qualitative agreement with Ref. [@yunoki2]. For larger values of $J'$, Fig.2a suggests that the phase separation regime near $n=1$ is lost (i.e. all densities are stable), while the results close to $n\sim 0$ are only slightly affected. Then, in order to study the effect of a nearest-neighbor (NN) repulsion $V$ over a phase separated regime, relatively small values of $J'/t$ must be selected. Note that this is not a problem since experimental results have actually shown that $J' \sim 0.05$ in units of $t=0.2eV$ is a realistic value for the exchange coupling between the localized spins [@yunoki2; @perring]. Then, in the rest of the paper $J'=0.05$ will be used, unless otherwise stated. Let us now analyze the behavior of the spin structure-factor $S(k)$ as a function of density. The results are shown in Fig.2b. As expected, in the limits $n=1$ and $0$ a clear signal for strong antiferromagnetic correlations is observed since $S(k)$ is peaked at $k=\pi$ [@comment99]. In the intermediate regime where double-exchange tendencies leads to a ferromagnetic ground state, $S(k)$ is maximized at $k=0$ also as expected based on previous literature. The most interesting results in this context arise close to $n=1$ and also $0$. In this regime, $S(k)$ has important weight $both$ at $k=0$ and $\pi$ signaling the coexistence of FM and AF domains as expected in a phase-separated regime. This is in agreement with the previous work reported in Ref. [@yunoki], with the discussion in the Introduction, and with the inverse compressibility data shown in Fig.2a. Note that the densities $n$ that correspond to the phase separation regime can be studied in detail since the analysis presented here is in the $canonical$-ensemble formalism. If a grand-canonical approach would be used, as in Ref. [@yunoki], then the regimes $n \sim 0.1$ and $\sim 0.9$ would not be reachable. Non-zero on-site repulsion $U$ ------------------------------ For the Hund-coupling used throughout this paper, moderate values of the on-site Coulomb interaction $U$ are not expected to play an important role since double occupancy is naturally suppressed by a large $J_H$. To illustrate this statement, in Fig.3 the inverse compressibility is shown at $n=0.9$, namely in a regime with phase separation at $U=V=0$, as discussed in the previous subsection. When $U$ is switched on and increased to a large value in units of the hopping amplitude ($U=16$), the plots of $1/\kappa$ vs $J'$ change only slightly. As a consequence, in the rest of the paper $U$ will be fixed to $16$ to avoid the proliferation of free parameters. The conclusions of our paper are not expected to change as long as $J_H$ and $U$ are the largest scales in the problem. The compressibility at $V=0$ and $U=16$ as a function of $n$ is shown in Fig.4a. It is clear that both at small and large $n$, this quantity is either $\sim 0$ or negative signaling the instability of the ground state towards the formation of two different phases. In between, where the system is expected to be ferromagnetic, the ground state is stable. In Fig.4b the spin structure factor $S(k)$ is shown to illustrate the coexistence of FM and AF features in the ground state of the “unstable” regime. This occurs at $n=0.85$ where peaks at both $k=0$ and $\pi$ are observed. In the other cases, $n=1$ and $0.75$, there is only one dominant peak at the antiferromagnetic and ferromagnetic locations, respectively. To further confirm if at small and large $n$ phase separation is indeed observed, the on-site densities $\langle n_i \rangle$ (expectation value in the ground state of the local density operator) have also been monitored. Note here that the DMRG method works using open boundary conditions and, as a consequence, the on-site density changes from site to site. In Fig.5a results at $n=0.75$, where the compressibility is positive and $S(k)$ peaks at $k=0$, show that aside from a boundary effect involving about 3 sites at each chain-end the local density only slightly Friedel oscillates around the global density. On the other hand, in the presumed to be phase-separated regime the local density changes substantially as a function of the site position $i$. Most of the holes are near the boundary and the local density changes abruptly between two values, which are the extreme stable densities $\sim 0.75$ and $\sim 1$ as expected. Also the charge structure factor $N(k)$ (Fig.5b) has the characteristics corresponding to phase separation, namely at $n=0.75$ it behaves as a non-interacting spinless fermionic system due to the ferromagnetic character of the ground state, while at $n=0.85$ it develops structure at small wavenumbers related with the inhomogeneous distributions of charge. In short, the results of this subsection have shown that the Kondo model with localized $S=1/2$ spins, and with the addition of a $J'$-coupling among them, has a qualitative phase diagram similar to the results presented before in the classical limit for the $t_{2g}$-spins in Refs. [@yunoki; @yunoki2]. This conclusion does not change even if an on-site repulsion $U$ of moderate strength is incorporated in the problem. Then, the present analysis have allowed us to fix the parameters $J_H,U,J'$ such that the physics under investigation, focussed on phase separation and ferromagnetism, is realized in the ground state of the model Eq.(1). Thus, the problem is now ready for the analysis of the influence of $V$ on the phase diagram. Results Including a Repulsion $V$ ================================= Influence of $V$ on the phase separated regime ----------------------------------------------- The $V$-term will now be switched on at a density such that phase separation occurs in the ground state. In Fig.6a the inverse compressibility is shown as a function of $V$ using a 40-sites chain. The results show that the unstable region observed at $V=0$ in the previous section now becomes stable when $V > 0.5$. Then, in agreement with the introductory discussion, phase separation is severely affected by Coulombic interactions beyond the on-site term. However, here it is interesting to observe that vestiges of phase separation survive even up to large couplings $V$. For instance, consider $S(k)$ which is shown in Fig.6b. At $V=1$, a double-peaked structure is observed, as it occurs at $V=0$, but now with maxima deviated from $0$ and $\pi$ forming incommensurate structure. Results at other values of $V >1$ (not shown) are very similar to those at $V=1$. Then, the ground state properties do not seem to change abruptly with $V$ but smoothly. Even ground states that have been stabilized by the Coulomb interaction in this regime contain a spin structure-factor with remnants of FM and AF domains. It is only the macroscopic accumulation of charge that is penalized by $V$. In Fig.7a the local density is shown for the case of 6 holes on a 40-site chain. The accumulation of charge near the boundary characteristic of phase separation at $V=0$ is replaced by a fairly clear periodic distribution of charge at $V=4$. This occurs not only at $V=4$ but in a wide range of couplings. The replacement of phase separation by a state with charge-ordering tendencies was expected based on the discussion given in the Introduction. The density where the “holes” are mostly located is close to $0.7$. There is no clustering of charge at large $V$. From Fig.7b it can be observed that the charge structure-factor $N(k)$ at $V=4$ develops a very sharp peak at $k=2\pi n$ due to the periodic arrangement of charge in the ground state. It is interesting to notice that the positions where the “holes” are in Fig.7a ($V=4$) are actually made out of four sites instead of one. Then, the holes are not fully static but they have some mobility, and it is natural that to enhance this mobility the spins must be aligned. Then, the large FM regions at small $V$ in the regime of phase separation are now replaced by periodically distributed small regions resembling magneto-polarons. This is among the most important results discussed in this paper. Influence of $V$ on the ferromagnetic regime -------------------------------------------- After analyzing in the previous subsection the influence of $V$ on the properties of a phase-separated ground state near $n=1$, let us study what occurs when a NN-Coulomb interaction is switched on in a fully ferromagnetic ground state. Such a state can be easily obtained in the present model Eq.(1) simply using a density, e.g., $n=0.6$ where the double-exchange ideas are operative. The results for $S(k)$ are shown in Fig.8a. While at $V=1$ (and smaller values) the peak at $k=0$ characteristic of ferromagnetism remains strong, at $V=8$ it has reduced substantially its intensity and moved slightly from $k=0$. At $V=12$ a very broad peak is the only remnant of the ferromagnetic structure at small $V$. To understand the reason for the complicated behavior of $S(k)$, the local density at $n=0.6$ is shown in Fig.8b. At $V=0$ it is almost uniform as expected in a ferromagnetic state. However, at $V=8$ a charge-ordered pattern is observed at $k=2 \pi n$. This is in agreement with the previously discussed results at $n=0.85$ which showed a similar tendency. However, note that now most minima in the local density involves only one site, instead of four as in Fig.7a. For this lattice and number of electrons (12 of them on a 20-site chain), the charges cannot arrange themselves in a periodic structure, causing the incommensurate-like peak in $S(k)$ at $V=8$. The replacement of a metallic FM state by a state with charge-ordering tendencies is also clear in Fig.9a where $N(k)$ is shown. At $V=8$ a large peak is observed. Also the real-space density-density correlations $C(i)$ shown in Fig.9b (for its definition see Sec.II) indicate strong effects in the charge sector at short distances: while at $V=0$ these correlations are not negligible only at distances $i=0$ and $1$, at $V=8$ the same correlations have been enhanced and they are now robust also at distances 2 and 3. Working at exactly $n=0.5$ the incommensurate structures observed at $n=0.6$ should disappear. In Fig.10a the local density is shown for a 20-site cluster and several values of $V$. At $V=0$ the density is exactly uniform for symmetry reasons, but as soon as $V$ is switched on a charge-ordered pattern clearly emerges. At this density a soliton at the center of the chain appears justifying the reduction in $\langle n_i \rangle$ towards that center. $N(k)$ in Fig.10b also presents a clear peak at $k=\pi$ which grows with $V$, in agreement with the previous discussion. Then, from the analysis of the densities $n=0.6$ and $0.5$ it is concluded that a tendency to charge ordering, at least at short-distances, is obtained once an intersite Coulomb interaction $V$ is introduced in a fully ferromagnetic state. This state (which here is referred to as charge-ordered (CO) for simplicity although the large distance behavior is difficult to analyze) occurs $within$ the ferromagnetic phase i.e. through a calculation of $S(k)$ the spins were found to be fully polarized unless $V$ reaches larger values than those shown in Fig10a-b. A similar result was reported in Ref. [@shen] using a path-integral approach. In order to monitor the development of charge correlations in the ground state the mean value of $n_{cd} = \sum_{i=odd} ( n_i - n_{i+1} )$ has been studied before in other models [@mutou]. Its expectation value should be zero in a metallic state, but nonzero in a system with charge order. Results for a chain of 20 sites are shown in Fig.11a. Although nonzero for all values of $V \neq 0$, this result suggests only a tendency towards the formation of a charge-ordered state. For a proper analysis of the critical $V$ leading to charge ordering a careful finite-size study is needed, beyond the scope (and numerical accuracy) of the present paper. However, in order to gain at least some insight from the 20-site cluster, the first derivative of $n_{cd}$ with respect to $V$ is shown in the inset of Fig.11a. Previous calculations studying variations around the $t-J$ model showed [@mutou] that the critical point is located at the inflection point of the $n_{cd}$ vs $V$ curve. The calculation shown in Fig.11a suggests that the critical coupling is roughly estimated to be at $V_c \sim 2$, which is in good agreement with the critical coupling corresponding to a model of spinless fermions with NN repulsion [@luther]. The charge gap $\Delta_c(N,L) = E(N+2,L) + E(N-2,L) -2E(N,L) = {{4L}/{N^2 \kappa}}$ was also studied (Fig.11b). The second derivative of this quantity with respect to $V$ is also peaked around $\sim 2$, as in the case of $dn_{cd}/dV$. Then, this (rough) analysis suggests that a critical coupling $V_c \sim 2$ separates two ferromagnetic regions, one charge-disordered and the other with at least short-distance charge-ordering tendencies. However, further work is needed to establish $V_c$ more accurately. Phase Separation in the Small Electronic Density Region ======================================================= In the low $e_g$-density region, a nearest-neighbor repulsion is not expected to affect the ground state properties substantially since the mean distance between electrons is large. Coulombic interactions at distances larger than one lattice spacing would be more important in this regime (but they are difficult to study with the DMRG method). To verify that indeed $V$ is not playing an important role at small $ n $, in Fig.12a the inverse compressibility is shown for three values of $V$. While phase separation at large density $n \sim 1$ is affected by $V$ (as discussed in previous sections) and $\kappa$ changes substantially at intermediate densities, the results at low density $ n \leq 0.20$ are almost $V$-independent. Let us investigate the properties of the phase-separated regime at small $ n $ and $V=0$. The results for the spin structure factor are shown in Fig.12b. At $n=0.20$, a clear signal of ferromagnetism is observed, with $1/\kappa$ being slightly positive (stable). However, at $n=0.10$ and $0.05$ a coexistence of weight both at $k=0$ and $\pi$ appears, signaling the expected regime that separates (i) electron-rich spin-ferro and (ii) electron-undoped spin-antiferromagnetic states. The presence of PS can also be inferred from the local density $\langle n_i \rangle$ shown in Fig.13a. At the stable density $n=0.25$, and leaving aside boundary effects involving about three sites at each end, the density presents Friedel oscillations around the density $n$ (actually the results at this density could have been obtained directly from those of Fig.5a at $n=0.75$ since in the fully spin-aligned state low- and high-densities are exactly related by symmetry). However, at $n=0.10$ the 4 electrons present in the $L=40$ system accumulate at the center, leaving about 10 sites (a quarter of the lattice) virtually empty on each end of the chain. This is the way in which phase separation seems to manifest itself when clusters with open boundary conditions are used. The central cluster of electrons (which has $n \sim 0.2$, i.e. the lower limit of the densities which are stable according to Fig.12a) was found to be spin ferromagnetic as expected. Finally, to further confirm that $V$ does not influence severely on the low-density regime, in Fig.13b $S(k)$ is shown now at $V=8$ for two densities. If $n=0.20$, the system is ferromagnetic, while for $n=0.10$, once again a coexistence of FM and AF features is observed, as in the absence of the intersite Coulomb interaction. Conclusions =========== In this paper the influence of a nearest-neighbor Coulomb repulsion $V$ between $e_g$-electrons was studied using the Ferromagnetic Kondo model. The main goal of the paper was to analyze the evolution with $V$ of the ground states in the regimes of (i) the recently computationally discovered phase-separation [@yunoki; @dago; @yunoki2] and (ii) with ferromagnetism induced by the double-exchange mechanism. Spin and charge correlations were studied in detail. The computational work using DMRG was made possible by restricting the spatial dimension to 1, the localized spin value to 1/2, and using only one-orbital per site. These limitations prevent us from making detailed quantitative statements about the effect of Coulomb interactions on real manganite compounds. Actually it is unrealistic to expect that accurate numerical work will be possible in the near future for large enough clusters in dimensions larger than one, considering the Coulombic interactions included here. However, several qualitative features have emerged that seem robust enough to survive an increase in dimensionality. The overall phase diagram in the $V$-$n$ plane found in this study is presented in Fig.14. In the limit $V=0$, two regimes of phase separation were observed near $n=1$ (PS1) and $0$ (PS2) (in excellent agreement with Ref. [@yunoki2]). In between, a robust ferromagnetic phase was observed with no indications of strong charge ordering tendencies. However, when the Coulomb repulsion $V$ is included in the calculation the PS1 regime rapidly becomes unstable due to the expected energy penalization caused by the charge difference between the two competing phases. At these densities, the $V$-term induces a regular arrangement of charge which resembles a polaron lattice. Each hole is spread over four lattice sites in the regime of parameters investigated in Fig.7a. This picture gives support to the intuitive notion that visualizes a phase separated state with extended Coulomb interactions included as a collection of small “islands” of one phase embedded into the other. This results is in good agreement with a large body of experimental work in manganites [@hennion; @allodi; @lynn; @deteresa; @cox; @yamada; @bao; @otroexp; @perring]. Vestiges of the phase separated regime are observed in the spin structure factor which has weak incommensurate-like peaks both near $k=\pi$ and $0$. In the other extreme of low electronic density, PS2 is not affected by $V$ since the mean distance between carriers is large at low density. Then, here phase separation persists up to large values of $V$. Presumably longer-range Coulombic terms are needed to melt this regime, and induce the charge-ordered pattern found in experiments for manganites. Another of the main results of the paper is the stabilization by NN-Coulomb interactions of a ferromagnetic state with charge periodically distributed at least at short-distances. This state is expected to be an insulator but a calculation of the Drude weight is needed to confirm this conjecture. This occurs in the vicinity of $n=0.5$ (Fig.14). The electrons gain kinetic energy by keeping the spin background fully polarized and they avoid the Coulomb repulsion by forming a charge pattern which is approximately periodic. For $n \neq 0.5$, the charge structure-factor peaks away from $k=\pi$, signaling a (natural) tendency to form incommensurate charge arrangements, but its strength is weaker than at $n=0.5$. Then, the present calculation suggests that the ferromagnetic phase of the manganites may coexist with charge ordering tendencies which are maximized at $n=0.5$. Likely these tendencies are more dynamic than static. Note that recent experimental results by C. H. Chen and S-W. Cheong (Ref. [@chen]) have reported the existence of weakly incommensurate charge ordering in $\rm La_{0.5} Ca_{0.5} Mn O_3$ using electron diffraction techniques. Our results suggest that this phenomenon may originate on the influence of NN-Coulomb interactions on double-exchange induced ferromagnetic phases of the manganites. acknowledgments =============== A. L. M. acknowledges the financial support of the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq-Brazil), as well as partial support from the NHMFL In-House Research Program, supported under grant DMR-9527035. S. Y. and E. D. are supported by the NSF grant DMR-9520776. S. Jin, et al., Science [**264**]{}, 413 (1994), and references therein. See also J. M. D. Coey, M. Viret, and S. von Molnar, [*Mixed Valence Manganites*]{}, Adv. Phys. 1998, in press. C. Zener, Phys. Rev. [**82**]{}, 403 (1951). A. P. Ramirez et al., Phys. Rev. Lett. [**76**]{}, 3188 (1996); C. H. Chen and S-W. Cheong, Phys. Rev. Lett. [**76**]{}, 4042 (1996); P. G. Radaelli, D.E. Cox, M. Marezio, and S-W. Cheong, Phys. Rev. [**B 55**]{}, 3015 (1997). See also S-W. Cheong and C. H. Chen, in [*Colossal Magnetoresistance and Related Properties*]{}, ed. by B. Raveau and C. N. R. Rao (World Scientific). M. Hennion et al., Phys. Rev. Lett. [**81**]{}, 1957 (1998). S. Yunoki, J. Hu, A. Malvezzi, A. Moreo, N. Furukawa, and E. Dagotto, Phys. Rev. Lett. [**80**]{}, 845 (1998). E. Dagotto, S. Yunoki, A. L. Malvezzi, A. Moreo, J. Hu, S. Capponi, D. Poilblanc, and N. Furukawa, Phys. Rev. B[**58**]{}, 6414 (1998). S. Yunoki and A. Moreo, Phys. Rev. B [**58**]{}, 6403 (1998). S. Yunoki, A. Moreo, and E. Dagotto, cond-mat/9807149, preprint, and references therein. E. Dagotto, Rev. Mod. Phys. [**66**]{}, 763 (1994); and references therein. A. J. Millis, P. B. Littlewood, and B. I. Shraiman, Phys. Rev. Lett. [**74**]{}, 5144 (1995); H. Röder, J. Zang and A. R. Bishop, Phys. Rev. Lett. [**76**]{}, 1356 (1996); A. J. Millis, B. Shraiman and R. Mueller, Phys. Rev. Lett. [**77**]{}, 175 (1996). N. Furukawa, J. Phys. Soc. Jpn. [**63**]{}, 3214 (1994). S. R. White, Phys. Rev. Lett. [**69**]{}, 2863 (1992). J. Inoue and S. Maekawa, Phys. Rev. Lett. [**74**]{}, 3407 (1995). P. G. de Gennes, Phys. Rev. [**118**]{}, 141 (1960). E. L. Nagaev, Phys. Status Solidi (b) [**186**]{}, 9 (1994); D. Arovas and F. Guinea, cond-mat/9711145; M. Yu. Kagan, et al., cond-mat/9804213; M. Yamanaka, W. Koshibae, and S. Maekawa, cond-mat/9807173. J. Riera, K. Hallberg and E. Dagotto, Phys. Rev. Lett. [**79**]{}, 713 (1997). G. Allodi et al., Phys. Rev. [**B 56**]{}, 6036 (1997); Phys. Rev. [**B 57**]{}, 1024 (1998). J. W. Lynn et al., Phys. Rev. Lett. [**76**]{}, 4046 (1996). J. M. De Teresa et al., Nature (London) [**386**]{}, 256 (1997). See also J. B. Goodenough and J.-S. Zhou, Nature (London) [**386**]{}, 229 (1997). D. E. Cox et al., Phys. Rev. [**B 57**]{}, 3305 (1998). Y. Yamada et al., Phys. Rev. Lett. [**77**]{}, 904 (1996). Wei Bao et al., Solid State Comm. [**98**]{}, 55 (1996). J. J. Rhyne, H. Kaiser, H. Luo, Gang Xiao, and M. L. Gardel, J. Appl. Phys. [**93**]{}, 7339 (1998); M. Roy, J. F. Mitchell, A. P. Ramirez, and P. Schiffer, Phys. Rev. [**B 58**]{}, 5185 (1998); C. H. Booth, F. Bridges, G. H. Kwei, J. M. Lawrence, A. L. Cornelius, and J. J. Neumeier, Phys. Rev. Lett. [**80**]{}, 853 (1998); Phys. Rev. [**B 57**]{}, 10440 (1998); M. Jaime, P. Lin, M. B. Salamon, P. Dorsey, and M. Rubinstein, preprint; J.-S. Zhou and J. B. Goodenough, Phys. Rev. Lett. [**80**]{}, 2665 (1998); S. J. L. Billinge, R. G. DiFrancesco, G. H. Kwei, J. D. Thompson, M. F. Hundley, and J. Sarrao, proceedings of the workshop “Physics of Manganites”, Michigan State University, July 1988, Eds. T. Kaplan and S. D. Mahanti; M. R. Ibarra, G-M. Zhao, J. M. De Teresa, B. Garcia-Landa, Z. Arnold, C. Marquina, P. A. Algarabel, H. Keller, and C. Ritter, Phys. Rev. [**B 57**]{}, 7446 (1998); R. H. Heffner et al., Phys. Rev. Lett. [**77**]{}, 1869 (1996); H. L. Liu, S. L. Cooper, and S.-W. Cheong, preprint. T. G. Perring et al., Phys. Rev. Lett. [**78**]{}, 3197 (1997). In addition, J. Q. Li et al., Phys. Rev. Lett. [**79**]{}, 297 (1997) reported phase segregation in $\rm La_{1-x} Sr_x Fe O_3$. V. J. Emery, and S. A. Kivelson, Physica [**C 209**]{}, 597 (1993); U. Löw, V. J. Emery, K. Fabricius, and S. A. Kivelson, Phys. Rev. Lett. [**72**]{}, 1918 (1994); S. Haas, E. Dagotto, A. Nazarenko, and J. Riera, Phys. Rev. [**B 51**]{}, 5989 (1995). See also K. Kubo, J. Phys. Soc. Jpn. [**51**]{}, 782 (1982); E. Müller-Hartmann and E. Dagotto, Phys. Rev. [**B 54**]{}, R6819 (1996); J. Zang, H. Röder, A. R. Bishop, and S. A. Trugman, J. Phys.: Condens. Matter [**9**]{}, L157 (1997); T. A. Kaplan and S. D. Mahanti, J. Phys.: Condens. Matter [**9**]{}, L291 (1997); H. Röder, R. R. P. Singh, and J. Zang, Phys. Rev. [**B 56**]{}, 5084 (1997). B. Batlogg, S-W. Cheong, and L. W. Rupp, Jr., Physica (Amsterdam) [**194 - 196B**]{}, 173 (1994); A. P. Ramirez et al., Phys. Rev. Lett. [**72**]{}, 3108 (1994); J. F. DiTusa, et al., Phys. Rev. Lett. [**73**]{}, 1857 (1994). E. Dagotto, J. Riera, A. Sandvik and A. Moreo, Phys. Rev. Lett. [**76**]{}, 1731 (1996). The ground state was always assumed to be even under reflections with respect to the center of the chain. Note that strictly speaking at large $J_H$ and $n=1$ the system may resemble a spin-1 chain with a finite spin-gap. However, even in this state the antiferromagnetic correlations are very robust. For the purposes of the present paper only the dominant character of the spin and charge correlations is important, and not their asymptotic behavior with distance. L. Sheng and C. S. Ting, Phys. Rev. [**B 57**]{}, 5265 (1998); and references therein. T. Mutou, N. Shibata, and K. Ueda, J. Phys.: Condens. Matter [**10**]{}, L507 (1998). A. Luther and I. Peschel, Phys. Rev. [**B 12**]{}, 3908 (1975); D. M. Haldane, Phys. Rev. Lett. [**45**]{}, 1358 (1980). C. H. Chen and S-W. Cheong, Phys. Rev. Lett. [**76**]{}, 4042 (1996).

--- abstract: 'Magnetic excitations in an array of single crystals have been measured using inelastic neutron scattering. Until now, has been thought of as a two-leg antiferromagnetic Heisenberg spin ladder with chains running in the $a$-direction. The present results show unequivocally that is best described as an alternating spin-chain directed along the crystallographic $b$-direction. In addition to the expected magnon with magnetic zone-center energy gap $\Delta = 3.1$ meV, a second excitation is observed at an energy just below $2\Delta$. The higher mode may be a triplet two-magnon bound state. Numerical results in support of bound modes are presented.' address: | $^1$Oak Ridge National Laboratory, Oak Ridge, TN 37831-6393, USA\ $^2$University of Tennessee, Knoxville, TN 37996-1501, USA\ $^3$University of Florida, Gainesville, FL 32611-0448, USA\ $^4$Insituto de Fisica Rosario, 2000 Rosario, Argentina\ author: - 'D.A. Tennant$^1$, S.E. Nagler$^1$, T. Barnes$^{1,2}, $ A.W. Garrett$^3$, J. Riera$^4$, and B.C. Sales$^1$' title: '**Excitations and Possible Bound States in the $S=1/2$ Alternating Chain Compound** ' --- \#1 [**Abstract**]{} [**Keywords:** ]{} Bound Magnons, Alternating Heisenberg Chain.\ [**Corresponding Author:** ]{}\ Dr Alan Tennant\ Bldg 7962 MS 6393, Solid State Division\ Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6393, U.S.A.\ tel.: +1-423-576-7747\ fax.: +1-423-574-6268\ email: [email protected]\ The S=1/2 alternating Heisenberg chain (AHC) is a fascinating quantum system that is currently the subject of much interest. We have established [@gntsb] that the material , previously considered to be a spin ladder, is in fact an excellent realization of the AHC. The physics of the AHC is also very relevant to spin-Peierls materials such as CuGeO3 [@nishi]. Recent theoretical work [@uhrig] on the AHC underscored the potential importance of two-magnon bound modes. In this paper, we review our neutron scattering experiments on the alternating chain material . In addition to the expected one-magnon excitations, we observe an extra mode, which may be a two-magnon bound state. We follow with a discussion of some related theoretical issues. The crystal structure of is nearly orthorhombic, with a slight monoclinic distortion so that the space group is P2$_{1}$[@nhs]. The room temperature lattice parameters are $a$=7.73Å, $b$=16.59Å, $c$=9.58Åand $\beta$=89.98$^{\circ}$. The magnetic properties of arise from $S=1/2$ V$^{4+}$ ions situated within distorted VO$_6$ octahedra. Face-sharing pairs of VO$_6$ octahedra are stacked in two-leg ladder structures oriented along the $a$-axis. The ladders are separated by large, covalently bonded PO$_4$ complexes. The structure is illustrated schematically in figure 1. The susceptibility of powder [@jjgj] can be accurately reproduced by either a spin ladder (with $J_\| \approx J_\perp$) or by an alternating chain [@jjgj; @br], but the expectation that the PO$_4$ group would provide a weak superexchange path led to a general acceptance of the spin ladder interpretation of . Pulsed inelastic neutron scattering measurements on powders [@ebbj] showed a spin gap of 3.7 meV, which was interpreted as further support for the ladder model. Beltrán-Porter [*et al.*]{}[@euro_chemists] examined the superexchange pathways in several vanadyl phosphate compounds, and were led to question the spin-ladder interpretation of . Instead they proposed that an alternating V-O-V-PO$_4$-V chain in the $b$-direction was a more likely magnetic system. The observation of a second spin excitation near 6 meV (not predicted by the ladder model) in a recent triple-axis neutron scattering experiment [@gnbs] on powder, and the discovery of strong superexchange through PO$_4$ groups in the precursor compound [@vodpo_paper], also cast doubt on the spin ladder interpretation. For these reasons we undertook studies of the spin dynamics in single crystals. Measurements of the excitations were made using a single crystal array of approx. 200 oriented crystals of typical size 1x1x0.25 mm$^3$. The resulting sample had an effective mosaic spread of $8-10^\circ$ FWHM. Inelastic neutron scattering measurements were carried out using triple-axis spectrometers at the HFIR reactor, Oak Ridge National Laboratory; full experimental details can be found in [@gntsb]. Scans [@gntsb] at $T=10$K showed two modes of roughly equal strength at the antiferromagnetic zone-centre $(0,\pi,0)$ at energies of $\Delta_{l}=3.12(3)$ meV and $\Delta_{u}=5.75(2)$ meV. Full resolution convolutions with the fitted dispersion showed these modes to be resolution limited. The disappearance of both modes at higher temperatures confirmed their magnetic origin. The modes were found to track approximately in $Q$ close to $(0,\pi,0)$ (see Figure (2)). At the zone-boundary $(0,\pi/2,0)$ only a single mode was observed at an energy of $\approx 15$meV. Because of poor instrumental resolution it was not possible to tell whether the modes had coalesced or were simply not resolved. Fig. 2 shows the measured dispersion for both modes along $a^*,b^*$ and $c^*$. The excitation energy is almost independent of $Q_{c}$ (middle panel), implying a very weak coupling along $c$. The dependence of energy on $Q_{a}$ is much weaker than on $Q_{b}$ and is ferromagnetic. The strong $Q_{b}$ dependence implies that the exchange coupling is dominantly one-dimensional along the $b$-direction, confirming the V-O-V-PO$_4$-V alternating chain proposed in [@euro_chemists] and [@vodpo_paper]. For any exchange alternation $-$ as occurs with the two inequivalent exchanges along $b$ $-$ a gap should appear in the dispersion (as observed) and the absence of magnetic ordering in is consistent with a singlet ground state. However the observation of an extra mode requires a more thorough theoretical investigation. Since the high temperature limit of the magnetic susceptibility[@jjgj] is consistent with expectations for simple $S=1/2$, $g=2$ spins, the possibility that the upper mode is an additional low lying single ion excitation can be ruled out. Two other plausible explanations of the second peak are (a) splitting due to an exchange anisotropy, and (b) a physical two-magnon bound state. Although a pseudo-Boson calculation including exchange anisotropy gave an excellent fit to the dispersion [@gntsb] $-$ the solid line in Figure (2) is a fit to this model [@gntsb] $-$ considerable exchange anisotropy ($\approx 15 \%$) was necessary to account for the mode splitting. Recent single crystal magnetic susceptibility measurements [@thompson] were quantitatively consistent with the previous powder results[@jjgj] and found little if any evidence for anisotropy. Also the coupling in the precursor compound was found to be consistent with isotropic exchange [@vodpo_paper] suggesting that one should seek another explanation for the second mode, and because the energy of the upper mode at $(0,\pi,0)$, $\Delta_{u}$, is just below $2\Delta_{l}$, a bound two-magnon mode may provide a good explanation. In support of this explanation, preliminary high-field measurements show splitting of both modes which is consistent with both modes being triplets [@vopo_field]. To gain some insight into the formation of two-magnon modes in we have studied the $S=1/2$ AHC using numerical techniques. The isotropic AHC Hamiltonian is $$H = \sum_{i=1}^{L/2} \ J \; {\vec S_{2i-1}} \cdot {\vec S_{2i}} + \alpha J \; {\vec S_{2i}} \cdot {\vec S_{2i+1}} \ ,$$ where $J>0$ and $1 \ge \alpha \ge 0$. Equation (1) has been studied analytically and numerically over many years, but it had not been appreciated until recently that $S=0$ and $S=1$ bound magnon states may form. Uhrig and Schulz [@uhrig] have used field theory and RPA methods to study these modes at $k=\pi/2$ and $k=0,\pi$. The existence of these bound states depends subtly on the kinetic and potential energies of pair formation, and occur for only certain values of $k$. Perturbation theory in $\alpha$ about the dimer limit ($\alpha=0$) provides a quantitative basis for understanding the excitations for small $\alpha$, and also provides insight into the competition between potential and kinetic energy effects in bound states [@brt]. Figure (3) shows the one- and two-magnon excitation spectra calculated within a simplified approximate first order (one- and two-magnon manifold) treatment of the AHC. At $k=\pi/2$ there is a node in the two-magnon continuum which corresponds to a degeneracy in the total kinetic energy $\omega(k_{1})+\omega(\pi/2-k_{1})$ of two magnons. The $S=0$ and $S=1$ bound states lie well below the continuum lower boundary. However at $k=0$ and $\pi$ only the $S=0$ bound state is seen. The continuum is much broader at $k=0,\pi$ indicating larger mixing effects which disrupt the $S=1$ bound state. Although no $S=1$ bound state forms, the attractive potential still leads to a strongly enhanced scattering cross-section $S(Q,\omega)$ at the continuum lower boundary [@brt], see dashed line in Figure (4). The $S=1$ bound state appears clearly at the $k=\pi/2$ point (solid line in Figure (4)). It should be noted that the neutron scattering cross-section for the $S=0$ mode is zero, however this mode may be visible by light scattering [@brt]. Harris [@abh] used a reciprocal space perturbation theory to calculate the ground state and excited state energy up to $O(\alpha^3)$. This gives a $k=0,\pi$ energy gap of $E_{gap}=J(1-\alpha/2-3 \alpha^2 /8 + \alpha^3 /32)$. However these results can be derived more easily in real space [@brt], and in the case of the ground state energy, we have extended the calculation to $O(\alpha^5)$, $$\begin{aligned} e_0(\alpha)/J=-3/2^3-(3/2^6) \cdot \alpha^2 - (3/2^8) \cdot \alpha^3 \nonumber \\ - (13/2^{12}) \cdot \alpha^4 - (95/3) \cdot (1/2^{14}) \cdot \alpha^5 - O(\alpha^6) .\end{aligned}$$ The perturbation series appears to be rapidly converging for $\alpha \le 0.5$, and may have a radius of convergence of unity. Because has $\alpha \approx 0.8$ [@gntsb], we have used a numerical Lanczos algorithm on finite $L=4n$ lattices of up to $L=28$ and with approximately 14 place accuracy to study the ground states and binding energies up to similar values of $\alpha$. Full details will be given elsewhere [@brt]. Figure (4) shows the calculated binding energies of the $S=0$ bound mode at $k=\pi/2$, and $k=0$, as well as those for the $S=1$ bound mode at $k=\pi/2$. The results show strong binding at $\alpha=0.8$ of the $S=0$ mode at $k=\pi/2$ but the situation is not clear for $k=0$. They also suggest weak binding for the S=1 mode at $\pi/2$ at the alternation for . Unfortunately finite size effects precluded an accurate determination of this binding energy. In order to make a quantitative comparison with $S(Q,\omega)$ is required for the bound modes and continuum. We are currently undertaking calculations to quantify this. The effects of interchain coupling have been neglected and these may enhance the binding. Next-nearest neighour exchange within the chains may have a similar effect. The $\alpha$ perturbation theory provides a useful quantitative guide to such effects, and further theoretical studies are in progress. We also note that similar dynamics are also important in many other low-dimensional Hamiltonians such as spin ladders and we shall present some work on those in the future. In conclusion, we have measured an extra mode in the alternating chain system . The evidence suggests that this is a two-magnon bound state. Perturbation theory and Lanczos calculations give an insight into the formation of bound modes. We thank J.Thompson for sharing his susceptibility results with us prior to publication. Oak Ridge National Laboratory is managed for the U.S. D.O.E. by Lockheed Martin Energy Research Corporation under contract DE-AC05-96OR22464. Work at U.F. is supported by the U.S. D.O.E. under contract DE-FG05-96ER45280. A.W. Garrett, S.E. Nagler, D.A. Tennant, B.C. Sales and T. Barnes, Phys. Rev. Lett. 79, 745 (1997). M. Nishi [*et al.*]{} Phys. Rev. B 32, 6508 (1994). G.S.Uhrig and H.J.Schulz, Phys. Rev. B 54, R9624 (1996). P.T. Nguyen, [*et al.*]{} Mat. Res. Bull. 30, 1055 (1995). D.C. Johnston [*et al.*]{}, Phys. Rev. B 35, 219 (1987). T.Barnes and J.Riera, Phys. Rev. B 50, 6817 (1994). R.S.Eccleston [*et al.*]{} Phys. Rev. Lett. 73, 2626 (1994). D. Beltrán-Porter P. Amorós, R. Ibáñez, E. Martínez, A. Beltrán-Porter, A. LeBail, G. Ferey, and G. Villeneuve, Solid State Ionics 32-33, 57 (1989). A.W.Garrett, S.E.Nagler, T.Barnes and B.C.Sales, Phys. Rev. B 55, 3631 (1997). D.A.Tennant, S.E.Nagler, A.W.Garrett, T.Barnes, and C.C.Torardi, Phys. Rev. Lett. 78, 4998 (1997). J.R. Thompson and K.J. Song, private communication S.E. Nagler, D.A. Tennant, A.W. Garrett, B.Sales, and T.Barnes, in preparation. T. Barnes, J. Riera, and D.A. Tennant, in preparation. A.B. Harris, Phys. Rev. B. 7, 3166 (1973). [Figure Captions]{} [Figure 1. Schematic depiction of the structure and magnetic interactions in VOPO. The spin ladder model previously thought to describe VOPO has nearest neighbor exchange constants $J_{\parallel}$ along the $a$ (“ladder”) direction and $J_{\perp}$ along the $b$ (“rung”) direction. In the alternating chain model, nearest neighbor V$^{4+}$ ions are alternately coupled by constants $J_{1}$ and $J_{2}$ along the $b$ (chain) direction. Neighboring spins in adjacent chains are coupled by $J_{a}$. Magnetic coupling in the $c$ direction is negligible. ]{} [Figure 2. Measured dispersion of magnetic excitations in VOPO at T = 10K. When not visible error bars are smaller than the size of the plotted symbols. Filled circles (open diamonds) are points from the lower (upper) energy mode. The solid lines are dispersion curves calculated using parameters obtained by fitting to a pseudo-Boson model [@gntsb]. Wavevectors are plotted in units corresponding to the VOPO reciprocal lattice. ]{} [Figure 3. Schematic depiction of the one- and two-magnon excitation spectra of the $S=1/2$ AHC with an alternation of $\alpha=0.2$. An $S=1$ bound mode appears below the continuum at $k \approx \pi/2$. The more deeply bound $S=0$ mode (dashed line) is not visible to neutrons scattering. ]{} [Figure 4. Calculated $S(Q,\omega)$ for constant-Q scans at $k=\pi/2$ (solid line) and $k=\pi$ (dashed line) using the first order perturbation approach with $\alpha=0.2$. ]{} [Figure 5. Calculated binding energies of the $S=0$ and $S=1$ bound states using a Lanczos method [@brt] at $k=0$ and $\pi/2$. The binding energies are given in units of $J$. ]{}

--- abstract: 'We apply the machinery of relative tensor triangular Chow groups [@1510.00211] to the action of $\mathcal{T}=\mathrm{D}(\Qcoh(X))$, the derived category of quasi-coherent sheaves on a noetherian scheme $X$, on the derived category of quasi-coherent $\mathcal{A}$modules $\mathcal{K}=\mathrm{D}(\Qcoh(\mathcal{A}))$, where $\mathcal{A}$ is a (not necessarily commutative) quasi-coherent $\mathcal{O}_X$algebra. When $\mathcal{A}$ is commutative and coherent, we recover the tensor triangular Chow groups of $\mathbf{Spec}(\mathcal{A})$. We also obtain concrete descriptions for integral group algebras and hereditary orders over curves, and we investigate the relation of these invariants to the classical ideal class group of an order. An important tool for these computations is a new description of relative tensor triangular Chow groups as the image of a map in the $\mathrm{K}$-theoretic localization sequence associated to a certain Verdier localization.' author: - 'Pieter Belmans[^1]' - 'Sebastian Klein[^2]' bibliography: - 'arxiv.bib' - 'bibliography.bib' - 'mr.bib' title: Relative tensor triangular Chow groups for coherent algebras --- Introduction ============ In [@1510.00211], the second author defined and began the study of *relative tensor triangular Chow groups*, a family of $\mathrm{K}$-theoretic invariants attached to a compactly generated triangulated category $\mathcal{K}$ with an action of a rigidly-compactly generated tensor triangulated category $\mathcal{T}$ in the sense of [@MR3181496]. While in [@1510.00211], they were used to improve upon and extend results of [@MR3423452], the initial observation of the present work is that they allow us to enter the realm of *noncommutative* algebraic geometry: if $X$ is a noetherian scheme and $\mathcal{A}$ a (possibly noncommutative) quasi-coherent $\mathcal{O}_X$-algebra, then the derived category $\mathcal{K}:=\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))$ admits an action by $\mathcal{T}:=\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))$ which is obtained by deriving the tensor product functor $$\begin{aligned} \mathrm{Qcoh}(\mathcal{A}) \times \mathrm{Qcoh}(\mathcal{O}_X) &\to \mathrm{Qcoh}(\mathcal{A})\\ (M,F) &\mapsto M \otimes_{\mathcal{O}_X} F~.\end{aligned}$$ In this situation, the general machinery of [@1510.00211] gives us abelian groups $\mathrm{Z}^{\Delta}_{i}(X,\mathcal{A})$ and $\mathrm{CH}^{\Delta}_i(X,\mathcal{A})$, the dimension $i$ tensor triangular cycle and Chow groups of $\mathcal{K}$ relative to the action of $\mathcal{T}$. In the test case where $\mathcal{A}$ is coherent and commutative, and hence $\mathcal{A}$ corresponds to a scheme $\mathbf{Spec}(\mathcal{A})$ and a finite morphism $\mathbf{Spec}(\mathcal{A}) \to X$, we show that $\mathrm{Z}^{\Delta}_{i}(X,\mathcal{A})$ and $\mathrm{CH}^{\Delta}_i(X,\mathcal{A})$ agree with the dimension $i$ tensor triangular cycle and Chow groups of $\mathrm{D^{perf}}(\mathbf{Spec}(\mathcal{A}))$ as defined in [@MR3423452], and hence with the usual dimension $i$ cycle and Chow groups of $\mathrm{Z}_i(\mathbf{Spec}(\mathcal{A})),\mathrm{CH_i}(\mathbf{Spec}(\mathcal{A}))$ when $\mathbf{Spec}(\mathcal{A})$ is a regular algebraic variety (see Theorem \[thmcommrecover\]). This computation serves as a motivation to study the groups $\mathrm{Z}^{\Delta}_{i}(X,\mathcal{A})$ and $\mathrm{CH}^{\Delta}_i(X,\mathcal{A})$ for noncommutative coherent $\mathcal{A}$. We obtain computations of both invariants when $\mathcal{A}$ is a sheaf of hereditary orders on a curve in Section \[sectionorders\], and in particular $\mathrm{CH}^{\Delta}_i(X,\mathcal{A})$ recovers the classical *stable class group* in this case. We also briefly touch upon the subjects of maximal orders on a surface and orders over a singular base, in the context of noncommutative resolutions of singularities. The case of a finite group algebra over $\mathrm{Spec}(\mathbb{Z})$ is discussed as a final example. Let us highlight that the main ingredient for the calculations carried out in this article is a new exact sequence which is established in Section \[sectionexseq\] for a general rigidly-compactly generated tensor triangulated category $\mathcal{T}$ acting on a compactly generated triangulated category $\mathcal{K}$, and for the case $\mathcal{K}:=\mathrm{D}(\mathrm{Qcoh}(\mathcal{A})), \mathcal{T}:=\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))$ gives $$0 \to \mathrm{CH}^{\Delta}_{p}(X, \mathcal{A}) \to \mathrm{K}_0\left(\left(\mathcal{K}_{(p+1)}/\mathcal{K}_{(p-1)}\right)^c\right) \to \mathrm{Z}^{\Delta}_{p+1}(X, \mathcal{A})~.$$ The middle term of the sequence is the Grothendieck group of the subcategory of compact objects of a subquotient of the filtration of $\mathcal{K}$ by dimension of support in $\mathrm{Spc}(\mathcal{T}^c)$. The article is structured as follows: in Section \[sectionttprelims\] we recall all relevant notions from tensor triangular geometry and the definition of relative tensor triangular cycle and Chow groups. We then establish the exact sequence mentioned above in Section \[sectionexseq\]. In Section \[sectiondercatalg\] we prove some auxilliary results concerning the categories $\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))$ and $\mathrm{D^b}(\mathrm{Coh}(\mathcal{A}))$, most of which should be known to the experts. In Section \[sectionrelgroupsalg\] we discuss the action of $\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))$ on $\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))$ and contemplate the definition of tensor triangular cycle and Chow groups in this more specific context, including a map $\mathrm{CH}^{\Delta}_i(X,\mathcal{A}) \to \mathrm{CH}_i(X)$ for regular $X$, induced by the forgetful functor $\mathrm{D}(\mathrm{Qcoh}(\mathcal{A})) \to \mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))$. We then have a look at commutative coherent $\mathcal{O}_X$-algebras in Section \[sectioncommcohalg\] and carry out our computations for orders in Section \[sectionorders\]. #### Acknowledgements We would like to thank Greg Stevenson for interesting discussions. Pieter Belmans was supported by a Ph.D. fellowship of the Research Foundation–Flanders (FWO). Sebastian Klein was supported by the ERC grant no. 257004–HHNcdMir Tensor triangular preliminaries {#sectionttprelims} =============================== In this section we recall the categorical notions we need. None of the following material is new, our main sources are [@MR2827786; @MR2806103; @MR3181496; @1510.00211]. Tensor triangular geometry {#subsection:ttgeometry} -------------------------- Let us quickly recall the basics of Balmer’s tensor triangular geometry. See e.g. [@MR2827786] for a reference that covers all the material we need (and much more). A *tensor triangulated category* is an essentially small triangulated category $\mathcal{C}$ equipped with a symmetric monoidal structure $\otimes$ with unit $\mathbb{I}$ such that the functors $a \otimes -$ are exact for all objects $a \in \mathcal{C}$. \[dfnsmallttcat\] To every tensor triangulated category $\mathcal{C}$, we associate its *Balmer spectrum* $\mathrm{Spc}(\mathcal{C})$, a topological space that is constructed in analogy with the prime ideal spectrum of a commutative ring. By construction of $\mathrm{Spc}(\mathcal{C})$, every object $a \in \mathcal{C}$ has a closed *support* $\mathrm{supp}(a) \subset \mathrm{Spc}(\mathcal{C})$, which satisfies the identities - $\mathrm{supp}(0) = \emptyset$ and $\mathrm{supp}(\mathbb{I}) = \mathrm{Spc}(\mathcal{C})$, - $\mathrm{supp}(\Sigma a) = \mathrm{supp}(a)$, - $\mathrm{supp}(a \oplus b) = \mathrm{supp}(a) \cup \mathrm{supp}(b)$, - $\mathrm{supp}(a \otimes b) = \mathrm{supp}(a) \cap \mathrm{supp}(b)$, - $\mathrm{supp}(b) \subset \mathrm{supp}(a) \cup \mathrm{supp}(c)$ whenever there is a distinguished triangle $$a \to b \to c \to \Sigma a~.$$ for all objects $a,b,c \in \mathcal{C}$. One can show that, in a precise sense, the space $\mathrm{Spc}(\mathcal{C})$ and the support function $\mathrm{supp}$ are optimal among all pairs of spaces and support functions satisfying the above criteria. If $X$ is a quasi-compact, quasi-separated scheme, then $\mathcal{C} = \mathrm{D^{perf}}(X)$, the derived category of perfect complexes on $X$, is a tensor triangulated category with tensor product $\otimes_{\mathcal{O}_X}^{\mathrm{L}}$. We have $\mathrm{Spc}(\mathcal{C}) \cong X$ and under this identification the support $\mathrm{supp}(C^{\bullet})$ of some complex $C^{\bullet}$ is identified with the complement of the set of points $x \in X$ such that $C^{\bullet}_x$ is acyclic, or equivalently with the support of the total cohomology sheaf $\mathrm{H}^*(C^{\bullet}):= \bigoplus_{i}\mathrm{H}^i(C^{\bullet})$. \[exschemereconstruct\] The spectrum $\mathrm{Spc}(\mathcal{C})$ is always a *spectral* topological space, i.e. it is homeomorphic to the prime ideal spectrum of some (usually unknown) commutative ring. Hence, it makes sense to talk about the Krull (co)-dimension of points in $\mathrm{Spc}(\mathcal{C})$. For a subset $S \subset \mathrm{Spc}(\mathcal{C})$, we define $$\mathrm{dim} (S) := \max_{P \in S} \mathrm{dim}(P) \quad \text{and} \quad \mathrm{codim} (S) := \min_{P \in S} \mathrm{codim}(P)~,$$ where we set $\mathrm{dim}(\emptyset) = - \infty, \mathrm{codim}(\emptyset) = \infty$. Supports in large categories ---------------------------- Let $\mathcal{T}$ be a triangulated category. The category $\mathcal{T}$ is called a *rigidly-compactly generated tensor triangulated category* if 1. *$\mathcal{T}$ is compactly generated.* We implicitly assume here that $\mathcal{T}$ has set-indexed coproducts. Note that this implies that $\mathcal{T}$ is not essentially small. 2. *$\mathcal{T}$ is equipped with a compatible closed symmetric monoidal structure $$\otimes: \mathcal{T} \times \mathcal{T} \to \mathcal{T}$$ with unit object $\mathbb{I}$*. Here, a symmetric monoidal structure on $\mathcal{T}$ is *closed* if for all objects $A \in \mathcal{T}$ the functor $A \otimes -$ has a right adjoint $\underline{\mathrm{hom}}(A,-)$. A *compatible* closed symmetric monoidal structure on $\mathcal{T}$ is one such that the functor $\otimes$ is exact in both variable and such that the two ways of identifying $\Sigma(x) \otimes \Sigma(y)$ with $\Sigma^2(x \otimes y)$ are the same up to a sign. Since adjoints of exact functors are exact (see [@MR1812507 lemma 5.3.6]) we automatically have that the functor $\underline{\mathrm{hom}}(A,-)$ is exact for all objects $A \in \mathcal{T}$. 3. *$\mathbb{I}$ is compact and all compact objects of $\mathcal{T}$ are rigid.* Let $\mathcal{T}^c \subset \mathcal{T}$ denote the full subcategory of compact objects of $\mathcal{T}$. Then we require that $\mathbb{I} \in \mathcal{T}^c$ and that all objects $A$ of $\mathcal{T}^c$ are rigid, i.e. for every object $B \in \mathcal{T}$ the natural map $$\underline{\circ}: \underline{\mathrm{hom}}(A,\mathbb{I}) \otimes B \cong \underline{\mathrm{hom}}(A,\mathbb{I}) \otimes \underline{\mathrm{hom}}(\mathbb{I},B) \to \underline{\mathrm{hom}}(A,B)~,$$ is an isomorphism. The subcategory $\mathcal{T}^c$ of a rigidly-compactly generated tensor triangulated category $\mathcal{T}$ is a tensor triangulated category in the sense of Definition \[dfnsmallttcat\]. Hence, it makes sense to talk about the spectrum $\Spc(\mathcal{T}^c)$. Throughout this section we assume that $\mathcal{T}$ is a compactly-rigidly generated tensor triangulated category. *We also assume that $\Spc(\mathcal{T}^c)$ is a noetherian topological space.* \[convbasiccat\] If $X$ is a quasi-compact, quasi-separated scheme, then $\mathcal{T} = \mathrm{D_{Qcoh}}(X)$, the derived category of complexes of $\mathcal{O}_X$-modules with quasi-coherent cohomology is a rigidly-compactly generated tensor triangulated category with tensor product $\otimes_{\mathcal{O}_X}^{\mathrm{L}}$. The rigid-compact objects are the perfect complexes in $\mathcal{T}$. By Example \[exschemereconstruct\], $\Spc(\mathcal{T}^c) = X$ and the condition of Convention \[convbasiccat\] hence holds whenever the space $|X|$ is noetherian, e.g. when $X$ is noetherian. If $X$ is noetherian and separated, $\mathrm{D_{Qcoh}}(X)$ is equivalent to $\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))$. Rigidly-compactly generated tensor triangulated categories come with an associated support theory that extends the notion of support in an essentially small tensor triangulated category. Let us briefly review the theory as introduced in [@MR2806103]. First recall the concepts of Bousfield and smashing subcategories: A thick triangulated subcategory $\mathcal{I} \subset \mathcal{T}$ is *Bousfield* if the Verdier quotient functor $\mathcal{T} \to \mathcal{T}/\mathcal{I}$ exists and has a right adjoint. A Bousfield subcategory $\mathcal{I} \subset \mathcal{T}$ is called *smashing* if the right-adjoint of the Verdier quotient functor $\mathcal{T} \to \mathcal{T}/\mathcal{I}$ preserves coproducts. If $\mathcal{I}$ is a Bousfield subcategory, there exists a localization functor $L_{\mathcal{I}}: \mathcal{T} \to \mathcal{T}$ (given as the composition of the Verdier quotient $\mathcal{T} \to \mathcal{T}/\mathcal{I}$ and its right-adjoint) such that $\mathcal{I} = \mathrm{ker}(L_{\mathcal{I}})$ and the composition of functors $$\mathcal{I}^{\perp} \rightarrow \mathcal{T} \rightarrow \mathcal{T}/\mathcal{I}$$ is an exact equivalence, where $\mathcal{I}^{\perp}$ is the full subcategory consisting of those $t \in \mathrm{Ob}(\mathcal{T})$ such that $\mathrm{Hom}(c,t) = 0$ for all $c \in \mathrm{Ob}(\mathcal{I})$. A quasi-inverse of the equivalence is given by the right adjoint of the Verdier quotient functor $\mathcal{T} \to \mathcal{T}/\mathcal{I}$. This says that we can actually realize the Verdier quotient $\mathcal{T}/\mathcal{I}$ inside of $\mathcal{T}$ and we will freely (and slightly abusively) confuse $\mathcal{T}/\mathcal{I}$ with $\mathcal{I}^{\perp}$. Also recall, that for every object $a \in \mathcal{T}$ there is a distinguished *localization triangle* $$\Gamma_{\mathcal{I}}(a) \to a \to L_{\mathcal{I}}(a) \to \Sigma(\Gamma_{\mathcal{I}}(a))$$ which is unique among triangles $x \to a \to y \to \Sigma(x)$ with $x \in \mathcal{I}$ and $y \in \mathcal{I}^{\perp}$, up to unique isomorphism of triangles that restrict to the identity on $a$. This defines a functor $\Gamma_{\mathcal{I}}(-)$ on $\mathcal{T}$ with essential image $\mathcal{I}$. The functor $\Gamma_{\mathcal{I}}$ is a *colocalization functor*, i.e. $\Gamma_{\mathcal{I}}^{\mathrm{op}}$ is a localization functor on $\mathcal{T}^{\mathrm{op}}$. A triangulated subcategory $\mathcal{I} \subset \mathcal{T}$ is called - *$\otimes$-ideal* if $\mathcal{T} \otimes \mathcal{I} = \mathcal{I}$. - *smashing ideal* if it is a $\otimes$-ideal, a Bousfield subcategory and $\mathcal{I}^{\perp} \subset \mathcal{T}$ is also a $\otimes$-ideal. Smashing ideals are well-behaved: as they are Bousfield subcategories there exists a unique triangle $$\Gamma_{\mathcal{I}}(\mathbb{I}) \to \mathbb{I} \to L_{\mathcal{I}}(\mathbb{I}) \to \Sigma(\Gamma_{\mathcal{I}}(\mathbb{I}))~,$$ and by tensoring this triangle with $a \in \mathcal{T}$, we see that we must have $L_{\mathcal{I}}(a) = L_{\mathcal{I}}(\mathbb{I}) \otimes a$ and $\Gamma_{\mathcal{I}}(a) = \Gamma_{\mathcal{I}}(\mathbb{I}) \otimes a$. Smashing ideals are smashing subcategories: $L_{\mathcal{I}} = \Gamma_{\mathcal{I}}(\mathbb{I}) \otimes -$ preserves coproducts since it has a right adjoint by definition of a rigidly-compactly generated tensor triangulated category. It follows that the Verdier quotient functor $\mathcal{T} \to \mathcal{T}/\mathcal{I}$ must preserve coproducts as well. An important tool for extending the notion of support from $\mathcal{T}^c$ to $\mathcal{T}$ is the following theorem: Let $\mathcal{S} \subset \mathcal{T}^c$ be a thick $\otimes$-ideal in $\mathcal{T}^c$ (i.e. $\mathcal{T}^c \otimes \mathcal{S} = \mathcal{S}$). Let $\langle \mathcal{S} \rangle$ denote the smallest triangulated subcategory of $\mathcal{T}$ that is closed under taking arbitrary coproducts (in $\mathcal{T}$). Then $\langle \mathcal{S} \rangle$ is a smashing ideal in $\mathcal{T}$ and $\langle\mathcal{S}\rangle^c = \mathcal{S}$. \[thmmiller\] Let $V \subset \Spc(\mathcal{T}^c)$ be a specialization-closed subset. We denote by $\mathcal{T}_{V}$ the smashing ideal $\langle \mathcal{T}^c_V \rangle$, where $\mathcal{T}^c_V \subset \mathcal{T}^c$ is the thick $\otimes$-ideal $\lbrace a \in \mathcal{T}^c: \supp(a) \subset V \rbrace$. We denote the two associated localization and acyclization functors by $L_V$ and $\Gamma_V$. \[deftelescopelocacyc\] Now let $x \in \mathrm{Spc}(\mathcal{T}^c)$ be a point. The sets $\overline{\lbrace x \rbrace}$ and $Y_x:= \lbrace y: x \notin \overline{\lbrace y \rbrace}\rbrace$ are both specialization-closed. Let $x \in \mathrm{Spc}(\mathcal{T}^c)$ and let $\Gamma_x$ denote the functor given as the composition $L_{Y_x}\Gamma_{\overline{\lbrace x \rbrace}}$. Then, for an object $a \in \mathcal{T}$, we define its *support* as $$\supp(a) := \lbrace x \in \mathrm{Spc}(\mathcal{T}^c) : \Gamma_x(a) \neq 0\rbrace~.$$ \[dfnbigsupport\] Suppose $X = \mathrm{Spec}(A)$ is an affine scheme with $A$ a noetherian ring. Then $\mathrm{D_{Qcoh}}(\mathrm{Spec}(A)) \cong \mathrm{D}(\mathrm{Mod}(A))$ and $$\mathrm{Spc}(\mathrm{D}(\mathrm{Mod}(A))^c) = \mathrm{Spc}(\mathrm{D}^{\mathrm{perf}}(A)) = \mathrm{Spec}(A)~.$$ Let $\mathfrak{p} \in \mathrm{Spec}(A)$ be a prime ideal. Then the functor $\Gamma_{\mathfrak{p}}$ is given as $\mathrm{K}_{\infty}(\mathfrak{p}) \otimes A_{\mathfrak{p}} \otimes -$, where $\mathrm{K}_{\infty}(\mathfrak{p})$ is the *stable Koszul complex* of the prime ideal $\mathfrak{p}$. In particular, if $\mathrm{Supp}(C^{\bullet})$ denotes the complement of the set of points where $C^{\bullet}$ is acyclic, then we see that $\mathrm{supp}(C^{\bullet}) \subset \mathrm{Supp}(C^{\bullet})$. The set $\mathrm{supp}(C^{\bullet})$ is sometimes known as the *small support* of $C^{\bullet}$ and coincides with the set of prime ideals $\mathfrak{p}$ such that $\mathrm{k}(\mathfrak{p}) \otimes^{\mathrm{L}} C^{\bullet} \neq 0$. In comparison to the notion of support of an essentially small tensor triangulated category, the support of an object of $\mathcal{T}$ is still a well-behaved construction. For example, we have $\mathrm{supp}(\bigoplus_i a_i) = \bigcup_i \mathrm{supp}(a_i)$, but $\mathrm{supp}(a)$ needs not be closed. If $a \in \mathcal{T}^c$, then $\mathrm{supp}(a)$ coincides with the notion of support from Section \[subsection:ttgeometry\] and hence it will be closed. Relative supports and tensor triangular Chow groups {#subsection:relative-tensor-triangular-chow-groups} --------------------------------------------------- We shall now adapt to a situation where we consider triangulated categories $\mathcal{K}$ that don’t necessarily have a symmetric monoidal structure themselves, but rather admit an *action* by a tensor triangulated category $\mathcal{T}$. Let us recall from [@MR3181496] what it means for $\mathcal{T}$ to have an action $\ast$ on $\mathcal{K}$. We are given a biexact bifunctor $$\ast: \mathcal{T} \times \mathcal{K} \to \mathcal{K}$$ that commutes with coproducts in both variables, whenever they exist. Furthermore we are given natural isomorphisms $$\begin{aligned} \alpha_{x,y,a}:& (X \otimes Y) \ast a \overset{\sim}{\longrightarrow} x \ast (y \ast a)\\ l_a:& \mathbb{I} \ast a \overset{\sim}{\longrightarrow} a \end{aligned} \label{eqnassocunitiso}$$ for all objects $x,y \in \mathcal{T}, a \in \mathcal{K}$. These natural isomorphisms should satisfy a list of natural coherence relations that we omit here, but rather refer the reader to [@MR3181496]. Any rigidly-compactly generated tensor triangulated category has an action on itself via its monoidal structure. Let us now assume that we are given a tensor triangulated category $\mathcal{T}$ with an action $\ast$ on a triangulated category $\mathcal{K}$, where $\mathcal{K}$ is assumed to be compactly generated as well (and so we implicitly mean that it has all coproducts). As in the previous section, we still assume that $\mathrm{Spc}(\mathcal{T}^c)$ is a noetherian topological space. Let us first describe a procedure to construct smashing subcategories of $\mathcal{K}$. Suppose $V \subset \mathrm{Spc}(\mathcal{T})$ is a specialization-closed subset. Then the full subcategory $$\Gamma_V(\mathbb{I}) \ast \mathcal{K} = \lbrace a \in \mathcal{K}: a \cong \Gamma_V(\mathbb{I}) \ast b~\text{for some}~b \in \mathcal{K}\rbrace$$ is smashing. The corresponding localization and colocalization functors are given by $L_V(\mathbb{I}) \ast -$ and $\Gamma_V(\mathbb{I}) \ast -$, respectively. \[lmarelativesmash\] It is shown in [@MR3181496 Lemma 4.4] that the subcategory $\Gamma_V(\mathbb{I}) \ast \mathcal{K}$ is Bousfield with $$(\Gamma_V(\mathbb{I}) \ast \mathcal{K})^{\perp} = L_V(\mathbb{I}) \ast \mathcal{K} := \lbrace a \in \mathcal{K}: a \cong L_V(\mathbb{I}) \ast b~\text{for some}~b \in \mathcal{K}\rbrace~.$$ Both $\Gamma_V(\mathbb{I}) \ast \mathcal{K}$ and $L_V(\mathbb{I}) \ast \mathcal{K}$ are $\mathcal{T}$-submodules, and we have a localization triangle $$\Gamma_V(\mathbb{I}) \to \mathbb{I} \to L_V(\mathbb{I}) \to \Sigma(\Gamma_V(\mathbb{I}))~.$$ Applying the functor $- \ast a$ to this triangle shows that the localization and colocalization functors associated to the Bousfield subcategory are given by $L_V(\mathbb{I}) \ast -$ and $\Gamma_V(\mathbb{I}) \ast -$, respectively. Since $L_V(\mathbb{I}) \ast -$ preserves coproducts by defintion of an action, it follows that $\Gamma_V(\mathbb{I}) \ast \mathcal{K} $ is indeed smashing. Following [@MR3181496], we can now assign to any object $a \in \mathcal{K}$ a support in $\Spc(\mathcal{T}^c)$ as follows: Let $x \in \mathrm{Spc}(\mathcal{T}^c)$. Then, for an object $a \in \mathcal{K}$, we define its *support* as $$\supp_{\mathcal{T}}(a) := \lbrace x \in \mathrm{Spc}(\mathcal{T}^c) : \Gamma_x(\mathbb{I}) \ast a \neq 0\rbrace~.$$ If there is no risk of confusion, we will usually drop the subscript $\mathcal{T}$ and write $\supp(a)$ instead. Furthermore, we will abbreviate the expression $\Gamma_x(\mathbb{I}) \ast a $ by $\Gamma_x a$. Let us state two important properties of the support. Let $V$ be a specialization-closed subset of $\Spc(\mathcal{T}^c)$ and $a$ an object of $\mathcal{K}$. Then $$\supp(\Gamma_V(a)) = \supp(a) \cap V$$ and $$\supp(L_V(a)) = \supp(a) \cap (\Spc(\mathcal{T})^c \setminus V)~.$$ \[propstevesuppprop\] For every specialization-closed subset $V\subset \Spc(\mathcal{T}^c)$, the subcategory $\mathcal{K}_V$ is defined as the essential image of the functor $\Gamma_V(\mathbb{I}) \ast -$. The subcategory $\mathcal{K}_{(p)}$ is defined as $\Gamma_{V_\leq p}(\mathbb{I}) \ast \mathcal{K}$, where $V_{\leq p} \subset \Spc(\mathcal{T}^c)$ is the subset of all points $x$ such that $\mathrm{dim}(x) \leq p$. \[dfnsubsetsubcat\] In [@1510.00211], $\mathcal{K}_{(p)}$ is defined differently, namely as the full subcategory of $\mathcal{K}$ on the collection of objects $\lbrace a \in \mathcal{K}: \mathrm{dim}(\supp(a)) \leq p \rbrace$. This coincides with Definition \[dfnsubsetsubcat\] whenever $\mathrm{supp}$ detects vanishing, i.e. whenever $\mathrm{supp}(a) = \emptyset \Leftrightarrow a= 0$ holds. Indeed, if $a \in \Gamma_{V_\leq p}(\mathbb{I}) \ast \mathcal{K}$, then $a \cong \Gamma_{V \leq p}(\mathbb{I}) \ast b$ for some $b \in \mathcal{K}$ and it follows from Proposition \[propstevesuppprop\] that $\mathrm{supp}(a) \subset V_{\leq p}$. Conversely, if $$\mathrm{dim}(\mathrm{supp}(a)) \leq p \Leftrightarrow \mathrm{supp}(a) \subset V_{\leq p}~,$$ we have a localization triangle $$\Gamma_{V_{\leq p}}(\mathbb{I}) \ast a \to a \to L_{V_{\leq p}}(\mathbb{I}) \ast a \to \Sigma(\Gamma_{V_{\leq p}}(\mathbb{I}))~,$$ and it follows from Proposition \[propstevesuppprop\] that $\mathrm{supp}(L_{V_{\leq p}}(\mathbb{I})) = \emptyset$ and hence $L_{V_{\leq p}}(\mathbb{I}) = 0$. This implies $\Gamma_{V_{\leq p}}(\mathbb{I}) \ast a \cong a$ and shows that $a \in \Gamma_{V_{\leq p}}(\mathbb{I}) \ast \mathcal{K}$. By [@MR3181496 Theorem 6.9], $\mathrm{supp}$ detects vanishing when the action of $\mathcal{T}$ on $\mathcal{K}$ satisfies the *local-to-global principle*, see Remark \[remlocglob\]. \[remdifferentsubcatdef\] Let $V\subset \Spc(\mathcal{T}^c)$ be specialization-closed. The category $\mathcal{K}_V$ is compactly generated. \[propsubsetsubcatcompgen\] We now come to the definition of the central invariant that is studied in this article. For a triangulated category $\mathcal{C}$, we shall denote by $\mathcal{C}^{\natural}$ its *idempotent completion*, a triangulated category with a fully faithful inclusion $\mathcal{C} \to \mathcal{C}^{\natural}$ which is universal for the property that all idempotents in $\mathcal{C}^{\natural}$ split (see [@MR1813503] for a detailed discussion). Let us first write down a diagram of Grothendieck groups: $$\xymatrix{ \Knought(\mathcal{K}^c_{(p)}) \ar[r]^-{q^{\natural}}\ar[d]^{i} &\Knought((\mathcal{K}^c_{(p)}/\mathcal{K}^c_{(p-1)})^{\natural}) ~(= \Knought((\mathcal{K}_{(p)}/\mathcal{K}_{(p-1)})^c))\\ \Knought(\mathcal{K}^c_{(p+1)}) }$$ Here, $q^{\natural}$ is the map induced by the composition of the Verdier quotient functor $\mathcal{K}^c_{(p)} \to \mathcal{K}^c_{(p)}/\mathcal{K}^c_{(p-1)}$ and the inclusion into the idempotent completion of the latter category. The morphism $i$ is induced by the inclusion functor. The identification $$(\mathcal{K}^c_{(p)}/\mathcal{K}^c_{(p-1)})^{\natural} \cong (\mathcal{K}_{(p)}/\mathcal{K}_{(p-1)})^c$$ holds by [@MR2681709 Theorem 5.6.1] since $\mathcal{K}_{(p-1)}$ is compactly generated by Proposition \[propsubsetsubcatcompgen\]. The *dimension $p$ tensor triangular cycle group* of $\mathcal{K}$ relative to the action $\ast$ is defined as $$\Cyc^{\Delta}_{p}(\mathcal{T},\mathcal{K}) := \Knought((\mathcal{K}^c_{(p)}/\mathcal{K}^c_{(p+1)})^{\natural})~.$$ The *dimension $p$ tensor triangular Chow group* of $\mathcal{K}$ relative to the action $\ast$ is defined as $$\CH^{\Delta}_{p}(\mathcal{K}) := \Cyc^{\Delta}_{p}(\mathcal{K})/q^{\natural}(\ker(i))~.$$ \[definition:1510.00211\] In [@1510.00211], the definition of relative tensor triangular cycle and Chow groups was given under the assumption that another technical condition, the *local-to-global principle*, is satisified. While it is not necessary for the statement of Definition \[definition:1510.00211\], the local-to-global principle makes dealing with these invariants easier (see Remark \[remdifferentsubcatdef\]), and it is satisfied very often. In particular, it will be satisfied in our main case of interest by [@MR3181496 Theorem 6.9], when we consider actions of the derived category of quasi-coherent sheaves on a noetherian separated scheme. In order to keep the exposition of the article at hand a bit lighter, we will not go into further details concerning this topic. \[remlocglob\] Let us illustrate our definitions with an example that explains the name “tensor triangular Chow group”. The following theorem is a slight variation of [@1510.00211 Corollary 3.6]. Let $X$ be a separated regular scheme of finite type over a field. Consider the action of $\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))$ on itself via $\otimes^{\mathrm{L}}$. Then for all $p \geq 0$, we have isomorphisms $$\begin{aligned} \Cyc^{\Delta}_{p}(\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X)),\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))) &\cong \Cyc_{p}(X) \\ \CH^{\Delta}_{p}(\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X)),\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))) &\cong \CH_{p}(X)~, \end{aligned}$$ where $\Cyc_{p}(X)$ and $\CH_{p}(X)$ denote the dimension $p$ cycle and Chow groups of $X$. \[thmchowrecover\] This is [@1510.00211 Corollary 3.6], with codimension replaced by dimension. The former statement is proved by showing that the groups $\Cyc_{\Delta}^{p}(\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X)),\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X)))$ and $\CH_{\Delta}^{p}(\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X)),\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X)))$ which are defined analogously via a filtration by *co*dimension of support, are isomorphic to certain terms on the $E^1$ and $E^2$ page of Quillen’s coniveau spectral sequence associated to $X$, which happen to be isomorphic to $\Cyc^{p}(X)$ and $\CH^{p}(X)$, respectively. In order to prove the “dimension” version of the statement, we see that the same argument shows that $\Cyc^{\Delta}_{p}(\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X)),\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X)))$ and $\CH^{\Delta}_{p}(\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X)),\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X)))$ are isomorphic to the terms $E^1_{p,-p}$ and $E^2_{p,-p}$ of the *niveau* spectral sequence of $X$, which happen to be isomorphic to $\Cyc_{p}(X)$ and $\CH_{p}(X)$ (see e.g. [@MR2648734] for the identification of $E^1_{p,-p}$ and $E^2_{p,-p}$ with $\Cyc_{p}(X)$ and $\CH_{p}(X)$). We can actually do better and also recover $\CH_{p}(X)$ for singular schemes. In order to do so, one lets $\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))$ act on $\mathrm{K}(\mathrm{Inj}(X))$, the homotopy category of quasi-coherent injective sheaves on $X$, instead of considering the action of $\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))$ on itself. Later on, we shall be interested in the action of $\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))$ on the derived category of a quasi-coherent $\mathcal{O}_X$-algebra. \[remcoderived\] An exact sequence {#sectionexseq} ================= In this section we derive an exact sequence that will give us a new description of $\mathrm{CH}^{\Delta}_{p}(\mathcal{T}, \mathcal{K})$ as an image of a map in a $\mathrm{K}$-theoretic localization sequence. It will be especially useful for computing $\mathrm{CH}^{\Delta}_{0}(\mathcal{T}, \mathcal{K})$ when $\mathrm{dim}(\mathrm{Spc}(\mathcal{T}^c)) =1$. Let $\mathcal{T}$ be a rigidly-compactly generated triangulated category that has an action $\ast$ on a compactly generated triangulated category $\mathcal{K}$ and assume that $\mathrm{Spc}(\mathcal{T}^c)$ is a noetherian topological space. Then we know that $\mathcal{K}_{(p)}$ is a compactly generated subcategory of $\mathcal{K}$ for all $p \geq 0$ and we have an exact sequence of triangulated categories $$\mathcal{K}_{(p)}/\mathcal{K}_{(p-1)} \to \mathcal{K}_{(p+1)}/\mathcal{K}_{(p-1)} \to \mathcal{K}_{(p+1)}/\mathcal{K}_{(p)}~. \label{eqnexseqdimdiff2}$$ Since the inclusion $\mathcal{K}_{(p)} \to \mathcal{K}_{(p+1)}$ admits a coproduct-preserving right-adjoint $\Gamma_{V_{\leq p}}(\mathbb{I}) \ast -$, the same is true for both functors in the sequence (\[eqnexseqdimdiff2\]). Hence it restricts to a sequence of compact objects $$\left(\mathcal{K}_{(p)}/\mathcal{K}_{(p-1)}\right)^c \to \left(\mathcal{K}_{(p+1)}/\mathcal{K}_{(p-1)}\right)^c \to \left(\mathcal{K}_{(p+1)}/\mathcal{K}_{(p)}\right)^c$$ which is exact up to factors. Applying $\mathrm{K}_0$ to this diagram yields a sequence of abelian groups $$\mathrm{Z}^{\Delta}_{p}(\mathcal{T}, \mathcal{K}) \xrightarrow{\iota} \mathrm{K}_0\left(\left(\mathcal{K}_{(p+1)}/\mathcal{K}_{(p-1)}\right)^c\right) \xrightarrow{\pi} \mathrm{Z}^{\Delta}_{p+1}(\mathcal{T}, \mathcal{K})$$ which is exact at the middle term. The map $\pi$ is surjective if and only if $\mathcal{K}_{(p+1)}^c/\mathcal{K}_{(p)}^c$ is idempotent complete. \[lmaidcompsurj\] We have $\left(\mathcal{K}_{(p+1)}/\mathcal{K}_{(p)}\right)^c = \left(\mathcal{K}_{(p+1)}^c/\mathcal{K}_{(p)}^c\right)^{\natural}$ and hence $\mathcal{K}_{(p+1)}^c/\mathcal{K}_{(p)}^c$ is a dense triangulated subcategory of $\left(\mathcal{K}_{(p+1)}/\mathcal{K}_{(p)}\right)^c$. Thomason’s classification of these subcategories (see [@MR1436741]) then shows that $\mathrm{im}(\pi)$ is maximal if and only if the inclusion $\mathcal{K}_{(p+1)}^c/\mathcal{K}_{(p)}^c \hookrightarrow \left(\mathcal{K}_{(p+1)}/\mathcal{K}_{(p)}\right)^c$ is essentially surjective which happens if and only if the former category is idempotent complete. We shall now be concerned with the kernel of $\iota$. Our goal is to prove the following statement: \[propchowexseq\] In the notation of Definition \[definition:1510.00211\], we have $\mathrm{ker}(\iota) = q^{\natural}(\ker(i))$. Hence, we obtain an exact sequence $$0 \to \mathrm{CH}^{\Delta}_{p}(\mathcal{T}, \mathcal{K}) \xrightarrow{\overline{\iota}} \mathrm{K}_0\left(\left(\mathcal{K}_{(p+1)}/\mathcal{K}_{(p-1)}\right)^c\right) \xrightarrow{\pi} \mathrm{Z}^{\Delta}_{p+1}(\mathcal{T}, \mathcal{K})$$ which is exact on the right if and only if $\mathcal{K}_{(p+1)}^c/\mathcal{K}_{(p)}^c$ is idempotent complete. Let $\mathcal{K}$ be a triangulated category and $\mathcal{L} \subset \mathcal{K}$ a triangulated subcategory. Consider the full triangulated subcategories $\mathcal{L}^{\natural}, \mathcal{K} \subset \mathcal{K}^{\natural}$. Then $\mathcal{L}^{\natural} \cap \mathcal{K} = \mathcal{L}$ as full subcategories of $\mathcal{K}^{\natural}$. \[lmacapsubcats\] It is clear that an object $A \in \mathcal{L}$ is both contained in $\mathcal{L}^{\natural}$ and $\mathcal{K}$. For the converse inclusion, suppose that $A$ is in $\mathcal{L}^{\natural} \cap \mathcal{K}$. Any object $A \in \mathcal{L}^{\natural}$ can be written as a pair $(A',e)$, where $A'$ is an object of $\mathcal{L}$ and $e$ is an idempotent endomorphism $A' \to A'$ in $\mathcal{L}$. Similarly, the objects $B$ of $\mathcal{K}$ in $\mathcal{K}^{\natural}$ are identified with exactly the pairs $(B', \mathrm{id}_B)$. It follows that $A$ can be written in the form $(A',\mathrm{id}_{A'})$ with $A' \in \mathcal{L}$. Hence, $A$ is in the image of the inclusion functor $\mathcal{L}^{\natural} \to \mathcal{K}^{\natural}$. In the situation of Lemma \[lmacapsubcats\], assume that $\mathcal{L}, \mathcal{K}$ are essentially small and consider the diagram of Grothendieck groups $$\begin{tikzcd} \mathrm{K}_0(\mathcal{L}) \arrow{r}{\alpha} \arrow{d}{\rho} & \mathrm{K}_0(\mathcal{K}) \arrow{d}{\sigma} \\ \mathrm{K}_0(\mathcal{L}^{\natural}) \arrow{r}{\beta} & \mathrm{K}_0(\mathcal{K}^{\natural}) \end{tikzcd}$$ induced by the inclusion functors. Then $\mathrm{ker}(\beta) = \rho(\mathrm{ker}(\alpha))$. \[lmaidcompkers\] By the commutativity of the diagram, it is clear that $\mathrm{ker}(\beta) \supseteq \rho(\mathrm{ker}(\alpha))$, so let us prove the converse inclusion. Consider an element $[a] \in \mathrm{ker}(\beta)$, i.e. $[a] = 0$ in $\mathrm{K}_0(\mathcal{K}^{\natural})$. By Thomason’s classification of dense triangulated subcategories (see [@MR1436741]) applied to $\mathcal{K} \subset \mathcal{K}^{\natural}$, we have $$\mathcal{K} = \lbrace x \in \mathcal{K}^{\natural}: [x] \in \mathrm{im}(\sigma) \rbrace~.$$ Since $0 \in \mathrm{im}(\sigma)$, we must have $a \in \mathcal{K} \subset \mathcal{K}^{\natural}$, and by Lemma \[lmacapsubcats\] it follows that $a \in \mathcal{L}$. Thus, $[a] \in \mathrm{im}(\rho)$ and since $\sigma$ is injective (see [@MR1436741 Corollary 2.3]), it follows that $[a] \in \mathrm{ker}(\alpha)$. Consider the commutative diagram $$\begin{gathered} \xymatrix{ \mathrm{K}_0(\mathcal{K}_{(p)}^c) \ar[r]^i \ar[d]^q& \mathrm{K}_0(\mathcal{K}_{(p+1)}^c) \ar[d]^h\\ \mathrm{K}_0\left(\mathcal{K}_{(p)}^c/\mathcal{K}_{(p-1)}^c\right) \ar[r]^{k} \ar[d]^j & \mathrm{K}_0\left(\mathcal{K}_{(p+1)}^c/\mathcal{K}_{(p-1)}^c\right) \ar[d]^l \\ {\underbrace{\mathrm{K}_0\left(\left(\mathcal{K}_{(p)}^c/\mathcal{K}_{(p-1)}^c\right)^{\natural}\right)}_{ = \mathrm{Z}^{\Delta}_p(\mathcal{T}, \mathcal{K})}} \ar[r]^{\iota} & {\underbrace{\mathrm{K}_0\left(\left(\mathcal{K}_{(p+1)}^c/\mathcal{K}_{(p-1)}^c\right)^{\natural}\right)}_{ = \mathrm{K}_0\left(\left(\mathcal{K}_{(p+1)}/\mathcal{K}_{(p-1)}\right)^c\right)}} } \end{gathered} \label{eqncommdiagsubcats}$$ where all maps are induced by inclusions of subcategories or Verdier quotient functors and in particular, we have $q^{\natural} = j \circ q$. Since $\mathrm{ker}(h) = i(\mathrm{ker}(q))$, we obtain that $\mathrm{ker}(k) = q(\mathrm{ker}(i))$. Therefore, it suffices to show that $\mathrm{ker}(\iota) = j(\mathrm{ker}(k))$, which follows from Lemma \[lmaidcompkers\]. The last statement of the proposition is Lemma \[lmaidcompsurj\]. When $\mathrm{dim}(\mathrm{Spc}(\mathcal{T}^c)) =1$, Proposition \[propchowexseq\] exhibits $\mathrm{CH}^{\Delta}_0(\mathcal{T},\mathcal{K})$ as a subgroup of $\mathrm{K}_0(\mathcal{K}^c)$. If $X$ is a regular algebraic curve, $\mathcal{T} = \mathcal{K} =\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))$, then we recover the well-known isomorphism $$\mathrm{K}_0(X) \cong \mathrm{CH}_0(X) \oplus \mathrm{Z}_1(X)$$ using Theorem \[thmchowrecover\]: the map $\pi$ is surjective by Lemma \[lmaidcompsurj\], since $$\begin{aligned} \mathrm{D^{perf}}(\mathrm{Coh}(X))_{(1)}/\mathrm{D^{perf}}(\mathrm{Coh}(X))_{(0)} &\cong \mathrm{D^b}(\mathrm{Coh}(X))/\mathrm{D^b}(\mathrm{Coh}(X))_{(0)} \\ &\cong \mathrm{D^b}(\mathrm{Coh}(X)/\mathrm{Coh}(X)_{\leq 0})~, \end{aligned}$$ (see [@MR3423452 §3.2], compare Corollary \[corverdiervsserre\]) and the latter category is idempotent complete since it is the bounded derived category of an abelian category (see [@MR1813503]). Furthermore, $\mathrm{Z}_1(X)$ is free abelian and hence the exact sequence splits. Again, as in Remark \[remcoderived\], we can drop the regularity assumption and consider the action of $\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))$ on $\mathrm{K}(\mathrm{Inj}(X))$ instead. Then we obtain $$\mathrm{G}_0(X) \cong \mathrm{CH}_0(X) \oplus \mathrm{Z}_1(X)~.$$ Derived categories of quasi-coherent OX-algebras {#sectiondercatalg} ================================================ In this section, we first recall some well-known facts about the categories of quasi-coherent right $\mathcal{A}$-modules $\mathrm{Qcoh}(\mathcal{A})$ and about $\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))$. We show how to realize the functor $\mathrm{D}(\mathrm{Qcoh}(\mathcal{A})) \to \mathrm{D}(\mathrm{Mod}(\mathcal{A}_x))$ that takes stalks at $x \in X$ as a localization of $\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))$ and prove a technical result about the filtration of $\mathrm{D^b}(\mathrm{Coh}(\mathcal{A}))$ by dimension of support. Basics of quasi-coherent modules over quasi-coherent OX-algebras ---------------------------------------------------------------- Let $X$ be a scheme. In this section we recall some basic facts about modules over an $\mathcal{O}_X$-algebra $\mathcal{A}$. The material we present here should be well-known (or at least hardly surprising) to most experts. An $\mathcal{O}_X$-algebra $\mathcal{A}$ is a sheaf of $\mathcal{O_X}$-modules $\mathcal{A}$ together with a multiplication map $\mathcal{A} \times \mathcal{A} \to \mathcal{A}$ that is associative and has unit, and is $\mathcal{O}_X$-bilinear[^3]. An $\mathcal{O}_X$-algebra $\mathcal{A}$ is *quasi-coherent*, if it is so as an $\mathcal{O}_X$-module. The pair $(X,\mathcal{A})$ is a ringed space, and hence it makes sense to talk about quasi-coherent right $\mathcal{A}$-modules. It is not hard to show that if $\mathcal{A}$ is a quasi-coherent $\mathcal{O}_X$-algebra, then a right $\mathcal{A}$-module is quasi-coherent if and only if it is quasi-coherent as an $\mathcal{O}_X$-module. Furthermore, quasi-coherent right $\mathcal{A}$-modules over a quasi-coherent $\mathcal{O}_X$-algebra $\mathcal{A}$ have a local description analogous to quasi-coherent $\mathcal{O}_X$-modules. Let $\mathcal{A}$ be a quasi-coherent $\mathcal{O}_X$-algebra, $U \subset X$ an affine open and $A := \Gamma(U,\mathcal{A})$. Then the functor $\Gamma(U,-)$ induces an equivalence of categories $$\lbrace \text{quasi-coherent right $\mathcal{A}|_{U}$-modules} \rbrace \xrightarrow{\sim} \lbrace \text{right $A$-modules} \rbrace~.$$ Since the notion of coherence is general as well, it applies to right $\mathcal{A}$-modules. We shall primarily be interested in the case where $X$ is noetherian and $\mathcal{A}$ is a *coherent $\mathcal{O}_X$-algebra*, i.e. one that is coherent as an $\mathcal{O}_X$-module. Suppose $X$ is noetherian and $\mathcal{A}$ is a coherent $\mathcal{O}_X$-algebra. Then a right $\mathcal{A}$-module $M$ is coherent if and only if it is coherent as an $\mathcal{O}_X$-module. Let us first notice that under the given conditions, $\mathcal{A}$ is a sheaf of right-noetherian rings. A right $\mathcal{A}$-module is hence coherent if and only if it is locally of finite type. Therefore, it suffices to show that a right $\mathcal{A}$-module is locally of finite type over $\mathcal{A}$ if and only if it is so over $\mathcal{O}_X$, which is straightforward. The category $\mathrm{Qcoh}(\mathcal{A})$ is Grothendieck abelian. \[corqcohAGrothendieck\] The category $\mathrm{Qcoh}(\mathcal{A})$ is exactly the category of modules over the right-exact monad corresponding to the adjunction $\mathcal{A} \otimes_{\mathcal{O}_X} - \dashv U$. Then [@MR3161097 lemma A.3] applies and shows that $\mathrm{Qcoh}(\mathcal{A})$ is Grothendieck abelian, since $\mathrm{Qcoh}(\mathcal{O}_X)$ is so. The following notion is central for our further considerations: Let $M \in \mathrm{Qcoh}(\mathcal{O}_X)$. The *support $\mathrm{Supp}(M)$ of $M$* is the set of points $P \in X$ such that $M_P \neq 0$. If $N \in \mathrm{Qcoh}(\mathcal{A})$, then $\mathrm{Supp}(N) := \mathrm{Supp}(U(N)) \subset X$. The derived category of a quasi-coherent OX-algebra {#subsection:derived-quasi-coherent-algebras} --------------------------------------------------- In the following, *we shall always assume that $X$ is a noetherian separated scheme* and that $\mathcal{A}$ is a quasi-coherent $\mathcal{O}_X$-algebra. While both assumptions on $X$ can certainly be weakened at certain points, we feel that this choice makes some arguments and notations easier and still provides a fairly general framework. ### Basic properties In this section we study the category $\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))$, the derived category of quasi-coherent right-$\mathcal{A}$-modules. Let us first note that $\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))$ exists, since $\mathrm{Qcoh}(\mathcal{A})$ is Grothendieck abelian by Corollary \[corqcohAGrothendieck\]. Furthermore, since the forgetful functor $U$ is exact, it directly descends to give a functor $U : \mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))\to \mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))$. Its right adjoint $\mathcal{A} \otimes_{\mathcal{O}_X} -$ induces a left-derived functor $$\mathcal{A} \otimes^{\mathrm{L}}_{\mathcal{O}_X} - : \mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X)) \to \mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))$$ which is computed by first taking $\mathrm{K}$-flat resolutions in $\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))$ and then applying $\mathcal{A} \otimes_{\mathcal{O}_X} -$. There is an adjunction $(\mathcal{A} \otimes^{\mathrm{L}}_{\mathcal{O}_X} -) \dashv U$. This is a consequence of a general criterion for the adjointness of derived functors, see Stacks Project, 09T5. The category $\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))$ is compactly generated, and a complex in $\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))$ is compact if and only if it is perfect, i.e. it is locally quasi-isomorphic to a bounded complex of projective modules of finite rank. \[thmcompgen\] This can be shown using Rouquier’s cocoverings. See [@1501.06023 theorem 3.14]. In the following, we shall denote the full subcategory of perfect complexes over $\mathcal{A}$ by $\mathrm{D^{perf}}(\mathcal{A}) \subset \mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))$. Whenever $S \subset |X|$ is a subset, we shall denote by $\mathrm{D}_S(\mathcal{A})$ ($\mathrm{D}^{\mathrm{b}}_S(\mathrm{Coh}(\mathcal{A})), \mathrm{D}^{\mathrm{perf}}_S(\mathcal{A})$) the corresponding full subcategories consisting of complexes $C^{\bullet}$ with $\mathrm{Supp}(\mathrm{H}^*(C^{\bullet})) \subset S$. If $S=V_{\leq p}$, the subset of all points of dimension $\leq p$, we shall replace the subscript “$V_{\leq p}$” by “$\leq p$”. ### Taking stalks Let us consider a point $x \in X$ and the inclusion $\mathrm{Spec}(\mathcal{O}_{X,x}) \to X$. If we equip $\mathrm{Spec}(\mathcal{O}_{X,x})$ with the sheaf of rings $\mathcal{A}_x$, we obtain a morphism of ringed spaces $$i_x: (\mathrm{Spec}(\mathcal{O}_{X,x}, \mathcal{A}_x) \to (X,\mathcal{A})$$ and the general theory of ringed spaces gives us a pair of adjoint functors $$\begin{tikzcd} \mathrm{Mod}(\mathcal{A}_x) \arrow[bend right,swap]{d}{(i_x)_*}\\ \mathrm{Qcoh}(\mathcal{A}) \arrow[bend right,swap]{u}{(i_x)^*} \end{tikzcd}$$ which fits into a commutative diagram $$\begin{tikzcd} \mathrm{Mod}(\mathcal{O}_{X,x}) \arrow[bend right,swap]{d}{(i_x)_*} & \mathrm{Mod}(\mathcal{A}_x) \arrow[bend right, swap]{d}{(i_x)_*} \arrow[swap,]{l}{U}\\ \mathrm{Qcoh}(\mathcal{O}_X) \arrow[bend right,swap]{u}{(i_x)^*} &\mathrm{Qcoh}(\mathcal{A}) \arrow[bend right, swap]{u}{(i_x)^*} \arrow{l}{U} \end{tikzcd}$$ and satisfies $(i_x)^* \circ (i_x)_* = \mathrm{id}$. The map $\mathrm{Spec}(\mathcal{O}_{X,x}) \to X$ is quasi-separated and quasi-compact (recall that we assumed that $X$ noetherian). Therefore the functor $(i_x)_*$ indeed produces quasi-coherent $\mathcal{O}_X$-modules, and hence also quasi-coherent $\mathcal{A}$-modules, since quasi-coherence can be checked after applying $U$. Since $X$ was separated, the map $i_x$ is affine and thus the functor $(i_x)_*$ is exact on the level of $\mathcal{O}_{X,x}$-modules. Since $U$ preserves and reflects exactness, it follows that $(i_x)_*$ is exact on the level of $\mathcal{A}_x$-modules as well. Furthermore, the map $\mathrm{Spec}(\mathcal{O}_{X,x}) \to X$ is flat and hence $(i_x)^*$ is exact on both levels as well. By a general criterion for adjoints of derived functors (see Stacks Project, 09T5), we obtain an adjoint pair $$\begin{tikzcd} \mathrm{D}(\mathrm{Mod}(\mathcal{A}_x)) \arrow[bend right, swap]{d}{(i_x)_*}\\ \mathrm{D}(\mathrm{Qcoh}(\mathcal{A})) \arrow[bend right, swap]{u}{(i_x)^*} \end{tikzcd}$$ which still satisfies $(i_x)^* \circ (i_x)_* = \mathrm{id}$ since there was no need to derive any of the two functors. Let $X$ be a noetherian separated scheme and $\mathcal{A}$ a quasi-coherent $\mathcal{O}_X$-algebra. Let $x \in X$ and $\mathrm{D}_{Y_x}(\mathcal{A}) \subset \mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))$ be the full subcategory of complexes $C^{\bullet}$ such that $\mathrm{Supp}(\mathrm{H}^*(C^{\bullet})) \subset Y_x = \lbrace y \in X | x \notin \overline{\lbrace y \rbrace} \rbrace$. Then $\mathrm{D}_{Y_x}(\mathcal{A}) = \mathrm{ker}(i_x)^*$ and the functor $(i_x)^*$ induces an exact equivalence $$\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))/\mathrm{D}_{Y_x}(\mathcal{A}) \xrightarrow{\sim} \mathrm{D}(\mathrm{Mod}(\mathcal{A}_x))~.$$ \[propstalkquot\] The first part follows from the identity $\mathrm{H}^*((i_x)^*C^{\bullet}) = (i_x)^*(\mathrm{H}^*(C^{\bullet}))$. Since $(i_x)^* \circ (i_x)_* = \mathrm{id}$, we must have that $(i_x)_*$ is fully faithful. It is well-known (see e.g. [@MR2681712 lemma 3.4]) that we therefore get an exact sequence of triangulated categories $$\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))/\mathrm{ker}(i_x)^* \xrightarrow{\sim} \mathrm{D}(\mathrm{Mod}(\mathcal{A}_x))~,$$ which finishes the proof by the first part of the proposition. ### Filtrations of the bounded derived category of coherent sheaves Let us now assume that $X$ is a noetherian scheme and that $\mathcal{A}$ is a coherent $\mathcal{O}_X$-algebra. We record the following, essentially trivial Lemma for later use. Let $\mathcal{J} \subset \mathcal{O}_X$ be an ideal sheaf and $M$ an $\mathcal{A}$-module. Then $$\mathcal{J} M = 0 \Leftrightarrow (\mathcal{A} \cdot \mathcal{J}) M = 0~.$$\[lmaidealannihilate\] Easy local computation. A sheaf of ideals $\mathcal{I} \subset \mathcal{A}$ is called *central*, if for any open $U \subset X$, the ideal $\mathcal{I}(U) \subset \mathcal{A}(U)$ can be generated by central elements. Let $$0 \to A \to B \to C \to 0$$ be an exact sequence of coherent $\mathcal{A}$-modules with $\mathrm{Supp}(A) = V \subset X$. Then there exists a commutative diagram of $\mathcal{A}$-modules $$\begin{tikzcd} 0 \arrow{r} & A \arrow{r} \arrow{d}{\mathrm{id}} & B \arrow{r} \arrow{d} & C \arrow{d} \arrow{r} & 0 \\ 0 \arrow{r} & A \arrow{r} & B' \arrow{r} & C' \arrow{r} & 0 \end{tikzcd}$$ with exact rows and such that $\mathrm{Supp}(B'), \mathrm{Supp}(C') \subset V$. \[propexseqext\] Let $\mathcal{J} \subset \mathcal{O}_X$ denote the radical ideal corresponding to the closed subset $V$. Then there exists $n_0 \in \mathbb{N}$ such that $\mathcal{J}^n A= 0$ for all $n \geq n_0$, and by Lemma \[lmaidealannihilate\] it follows that $(\mathcal{A} \cdot \mathcal{J}^n) A = (\mathcal{A} \cdot \mathcal{J})^n A =0$ all $n \geq n_0$. For each $n$, we obtain a commutative diagram wit exact rows $$\begin{tikzcd} 0 \arrow{r}& A \arrow{r}{\iota} \arrow{d}{\mathrm{id}}& B \arrow{r}{\pi} \arrow{d}& C \arrow{d} \arrow{r} & 0 \\ & A \arrow{r}{\overline{\iota}} & B/(\mathcal{A} \cdot \mathcal{J})^n B \arrow{r}{\overline{\pi}}& C/(\mathcal{A} \cdot \mathcal{J})^n C \arrow{r} & 0 \end{tikzcd} \label{eqexseq}$$ where $\overline{\iota},\overline{\pi}$ are induced by $\iota,\pi$ respectively and the non-labeled vertical maps are the canoncial projections. We claim that for $n$ large enough, $\overline{\iota}$ is a monomorphism. As we can check injectivity locally, let $X= \bigcup_{i=1}^r U_i$ with $U_{i} = \mathrm{Spec}(R_i)$ open affine. Then, on each $U_i$, the problem looks as follows: we are given an $R_i$-algebra $S_i$, an ideal $J_i \subset R_i$, an exact of $S_i$-modules $$0 \to A_i \to B_i \to C_i \to 0$$ and we know that for all $n \geq n_i$, $J^n A =0$. Diagram (\[eqexseq\]) translates as $$\xymatrix{ 0 \ar[r]& A_i \ar[r]^{\iota_i} \ar[d]^{\mathrm{id}}& B_i \ar[r]^{\pi_i} \ar[d]& C_i \ar[d] \ar[r] & 0 \\ & A_i \ar[r]^-{\overline{\iota_i}} & B_i/(S_i \cdot J_i)^n B_i \ar[r]^-{\overline{\pi_i}}& C/(S_i \cdot J_i)^n C_i \ar[r] & 0 }$$ We will now use the Artin-Rees lemma, which is in general not valid for non-commutative rings, but does hold for central ideals like $S_i \cdot J_i$ (see [@MR0231816 Chapter 7.2, Theorem 1]): there exists $q_i \in \mathbb{N}$ such that for all $m_i \geq q_i$ we have $$A_i \cap (S_i \cdot J_i)^n B_i = (S_i \cdot J_i)^{n-q_i} (A_i \cap (S_i \cdot J_i)^{q_i} B_i)~.$$ Now note that $\ker(\overline{\iota_i}) = A_i \cap (S_i \cdot J_i)^n B_i$, and thus the Artin-Rees lemma tells us that if we choose $m_i$ such that $n-q \geq n_i$, then $\ker(\overline{\iota_i}) = 0$, i.e. $\overline{\iota_i}$ is injective. Now, if we choose $n = \max_i m_i$, then $\overline{\iota_i}$ will be injective for all $i$, proving that $\overline{\iota}$ is a monomorphism. To conclude the proof, note that for any coherent $\mathcal{A}$-module $M$, we have that $\mathrm{Supp}(M) = \mathrm{V}(\mathrm{Ann}_{\mathcal{O}_X}(M))$ since $M$ is also $\mathcal{O}_X$-coherent. But by Lemma \[lmaidealannihilate\], we know that $\mathcal{J}^n$ annihilates $M/(\mathcal{A} \cdot \mathcal{J})^n M = M/(\mathcal{A} \cdot \mathcal{J}^n) M$ as $\mathcal{A} \cdot \mathcal{J}^n$ does so. It follows that $$\mathrm{Supp}(B/(\mathcal{A} \cdot \mathcal{J})^n B), \mathrm{Supp}(C/(\mathcal{A} \cdot \mathcal{J})^n C) \subset \mathrm{V}(\mathcal{J}^n) = \mathrm{V}(\mathcal{J}) = V~.$$ For $p \in \mathbb{Z}$, denote by $\mathrm{Coh}(\mathcal{A})_{\leq p}$ the full subcategory of $\mathrm{Coh}(\mathcal{A})$ consisting of those $\mathcal{A}$-modules $M$ with $\dim(\mathrm{Supp}(M)) \leq p$. The properties of $\mathrm{Supp}(-)$ easily imply that $\mathrm{Coh}(\mathcal{A})_{\leq p}$ is a Serre subcategory of $\mathrm{Coh}(\mathcal{A})_{\leq q}$ if $p \leq q$. \[corollary:abelian-derived-filtration\] The natural functors $$\begin{aligned} \derived^{\mathrm{b}}(\mathrm{Coh}(\mathcal{A})_{\leq p}) &\to \derived^{\mathrm{b}}_{\leq p}(\mathrm{Coh}(\mathcal{A})) \\ \derived^{\mathrm{b}}(\mathrm{Coh}(\mathcal{A})_{\leq p})/\derived^{\mathrm{b}}(\mathrm{Coh}(\mathcal{A})_{\leq p -1}) &\to \derived^{\mathrm{b}}(\mathrm{Coh}(\mathcal{A})_{\leq p}/\mathrm{Coh}(\mathcal{A})_{\leq p-1}) \end{aligned}$$ are equivalences of categories. \[corverdiervsserre\] The statement of Proposition \[propexseqext\] is exactly the condition of [@MR1667558 Section 1.15, Lemma (c1)] which makes the above functors equivalences. Relative tensor triangular Chow groups of a quasi-coherent OX-algebra {#sectionrelgroupsalg} ===================================================================== In this section, we obtain a definition of the relative tensor triangular cycle and Chow groups of a (quasi-)coherent $\mathcal{O}_X$algebra $\mathcal{A}$ by means of an action of the derived category of quasi-coherent $\mathcal{O}_X$-modules $\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))$ on the derived category of quasi-coherent right $\mathcal{A}$-modules $\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))$. We then derive some basic properties of these groups, including a group homomorphism induced by the forgetful functor that relates $\mathrm{CH}^{\Delta}_i(X,\mathcal{A})$ to $\mathrm{CH}_i(X)$ when $X$ is regular. The general approach we use for the relative tensor triangular Chow groups works for all quasi-coherent $\mathcal{O}_X$-algebras $\mathcal{A}$ but as we will see below, the coherent case will turn out to be more manageable, since then two notions of support will agree for bounded complexes of coherent $\mathcal{A}$-modules. Later on, this will allows us to actually do some concrete computations. The action of D(Qcoh OX) on D(Qcoh A) ------------------------------------- The bifunctor $$- \otimes_{\mathcal{O}_X} - : \mathrm{Qcoh}(\mathcal{O}_X) \times \mathrm{Qcoh}(\mathcal{A}) \to \mathrm{Qcoh}(\mathcal{A})$$ gives rise to a bifunctor $$- \otimes^{\mathrm{L}}_{\mathcal{O}_X} - : \mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X)) \times \mathrm{D}(\mathrm{Qcoh}(\mathcal{A})) \to \mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))$$ by taking $\mathrm{K}$-flat resolution in the first variable and applying $- \otimes_{\mathcal{O}_X} -$. This defines an action of $\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))$ on $\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))$, where the unitor and associator isomorphisms (\[eqnassocunitiso\]) are induced by those on the level of complexes, i.e. the natural isomorphisms $$\begin{aligned} (A^{\bullet} \otimes_{\mathcal{O}_X} B^{\bullet}) \otimes_{\mathcal{O}_X} X^{\bullet} &\xrightarrow{\sim} A^{\bullet} \otimes_{\mathcal{O}_X} (B^{\bullet} \otimes_{\mathcal{O}_X} X^{\bullet})\\ \mathcal{O}_X \otimes_{\mathcal{O}_X} X^{\bullet} &\xrightarrow{\sim} X^{\bullet}\end{aligned}$$ for $A^{\bullet},B^{\bullet}$ complexes of quasi-coherent $\mathcal{O}_X$-modules and $X^{\bullet}$ a complex of quasi-coherent right $\mathcal{A}$-modules. The action of $\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))$ on $\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))$ satisfies the local-to-global principle (see Remark \[remlocglob\]) since the action $\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))$ on itself does so. We will now continue to derive some properties of the notion of support that the action of $\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))$ on $\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))$ induces on objects of the latter category. Let $V \subset X$ be a specialization-closed subset. Then $\mathrm{D}_V(\mathrm{Qcoh}(\mathcal{A}))$ coincides with the subcategory $\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))_V$ of all complexes $C^{\bullet} \in \mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))$ such that $\mathrm{supp}(C^{\bullet}) \subset V$. In particular, the subcategories $\mathrm{D}_V(\mathrm{Qcoh}(\mathcal{A}))$ are smashing. \[propsubsetsubcatcoincide\] If $C^{\bullet}$ is a complex of quasi-coherent right $\mathcal{A}$-modules, then we need to show that $\mathrm{supp}(C^{\bullet}) \subset V \Leftrightarrow \mathrm{Supp}(C^{\bullet}) \subset V$. If $X = \bigcup_i U_i$ is an open cover, then it suffices to show that $\mathrm{supp}(C^{\bullet}) \cap U_i \subset V \cap U_i \Leftrightarrow \mathrm{Supp}(C^{\bullet}) \cap U_i \subset V \cap U_i$ for all $i$. Let $U_i = \mathrm{Spec}(R_i), i= 1, \ldots, n$ be a cover of $X$ by affine opens with closed complements $Z_i$ and set $V_i := U_i \cap V$. Notice that the sets $V_i$ are still specialization-closed in $U_i$. We have $\mathrm{supp}(C^{\bullet}|_{U_i}) = \mathrm{supp}(L_{Z_i}\mathcal{O}_X \ast C^{\bullet}) = \mathrm{supp}(C^{\bullet}) \cap U_i$ by Proposition \[propstevesuppprop\] and $\mathrm{Supp}(C^{\bullet}|_{U_i}) = \mathrm{Supp}(L_{Z_i}\mathcal{O}_X \ast C^{\bullet}) = \mathrm{Supp}(C^{\bullet}) \cap U_i$ since localization is exact. Hence we have reduced to showing that $$\mathrm{supp}(C^{\bullet}|_{U_i}) \subset V_i \Leftrightarrow \mathrm{Supp}(C^{\bullet}|_{U_i}) \subset V_i$ for $i =1, \ldots n~.$$ But now, we can assume that $\mathcal{A}$ is given as an $R_i$-algebra $A$ and $C^{\bullet}|_{U_i}$ a complex of right $A$-modules. Since both $\mathrm{supp}$ and $\mathrm{Supp}$ can be computed by first applying the forgetful functor $\mathrm{D}(\mathrm{Qcoh}(\mathcal{A})) \to \mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))$, the result follows from [@1401.6925 Proposition 3.14], where it is shown that for the complex of $R_i$-modules $C^{\bullet}|_{U_i}$, the sets $\mathrm{supp}(C^{\bullet}|_{U_i})$ and $\mathrm{Supp}(C^{\bullet}|_{U_i}) $ have the same minimal elements. The last statement follows from the first and Lemma \[lmarelativesmash\]. Let us show that $\mathrm{supp}$ and $\mathrm{Supp}$ coincide for small complexes. Let $C^{\bullet} \in \mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))$ such that $\mathrm{H}^{\ast}(C^{\bullet})$ is bounded and coherent. Then $\mathrm{supp}(C^{\bullet}) = \mathrm{Supp}(C^{\bullet})$. \[propsupp=Supp\] As in the proof of Proposition \[propsubsetsubcatcoincide\], we notice that if $X = \bigcup_i U_i$ is a cover by affine opens with complements $Z_i$, then it suffices to show that $$\underbrace{\mathrm{supp}(C^{\bullet}) \cap U_i}_{= \mathrm{supp}(C^{\bullet}|_{U_i})} = \underbrace{\mathrm{Supp}(C^{\bullet}) \cap U_i}_{=\mathrm{Supp}(C^{\bullet}|_{U_i})}$$ for all $i$. Hence, we have reduced to the affine case, where the result is implied from the corresponding one for complexes in $\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))$. But the latter is well known, see e.g. [@MR2489634]. Given a coherent $\mathcal{O}_X$algebra $\mathcal{A}$ (on which $\mathcal{O}_X$ acts by centrally by assumption) we can consider $\mathrm{Z}(\mathcal{A})$ as a commutative coherent $\mathcal{O}_X$algebra. Let $$\pi\colon Z\coloneqq\mathbf{Spec}_X\mathrm{Z}(\mathcal{A})\to X$$ be the relative affine scheme defined by $\mathrm{Z}(\mathcal{A})$. We can consider $\pi^*(\mathcal{A})$ as a coherent $\mathcal{O}_Z$algebra, and by [@1501.06023 proposition 3.5] we have that $\Qcoh\mathcal{A}\cong\Qcoh\pi^*(\mathcal{A})$. The action of $\derived(\Qcoh X)$ and $\derived(\Qcoh Z)$ will be different in general. Unwinding the definitions {#subsection:main-result} ------------------------- With all the technical material we have assembled so far, let us look once more at Definition \[definition:1510.00211\]. Let $\mathcal{T} = \mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))$ and $\mathcal{K} = \mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))$. We will write $$\mathrm{Z}^{\Delta}_i(X,\mathcal{A}) \quad \text{and} \quad \mathrm{CH}^{\Delta}_i(X,\mathcal{A})$$ for the groups $\mathrm{Z}^{\Delta}_i(\mathcal{T},\mathcal{K})$ and $\mathrm{CH}^{\Delta}_i(\mathcal{T},\mathcal{K})$, respectively. We have $$\mathrm{Z}^{\Delta}_{i} (X,\mathcal{A}) = \mathrm{K}_0\left((\mathcal{K}_{(i)}/\mathcal{K}_{(i-1)})^c\right)$$ by definition, and both categories $\mathcal{K}_{(i)},\mathcal{K}_{(i+1)}$ are compactly generated. Hence, we have that $$(\mathcal{K}_{(i)}/\mathcal{K}_{(i-1)})^c = \left((\mathcal{K}_{(i)})^c/(\mathcal{K}_{(i-1)})^c\right)^{\natural}$$ by [@MR2681709 Theorem 5.6.1]. Furthermore, $(\mathcal{K}_{(i)})^c$ coincides with the full subcategory of $\mathcal{K}^c$ consisting of objects with support in codimension $\geq i$ by [@1510.00211 Proposition 2.23]. From Theorem \[thmcompgen\], we have $$\mathcal{K}^c = \mathrm{D^{perf}}(\mathcal{A}) \subset \mathrm{D^b}(\mathrm{Coh}(\mathcal{A}))~.$$ If we assume $\mathcal{A}$ coherent, then $\mathrm{Supp}$ and $\mathrm{supp}$ coincide for objects of $\mathrm{D^b}(\mathrm{Coh}(\mathcal{A}))$ by Proposition \[propsupp=Supp\]. It follows that $$\mathrm{Z}^{\Delta}_{i} (X,\mathcal{A}) = \mathrm{K}_0\left(\left(\mathrm{D}^{\mathrm{perf}}_{\leq i}(\mathcal{A})/ \mathrm{D}^{\mathrm{perf}}_{\leq i-1}(\mathcal{A}) \right)^{\natural} \right)~.$$ If $\mathcal{A}$ is additionally of finite global dimension, $\mathrm{D}^{\mathrm{perf}}(\mathcal{A}) = \mathrm{D^b}(\mathrm{Coh}(\mathcal{A}))$ and we get from Corollary \[corverdiervsserre\] that $$\mathrm{Z}^{\Delta}_{i} (X,\mathcal{A})= \mathrm{K}_0\left(\mathrm{D^b}\left(\mathrm{Coh}_{\leq i}(\mathcal{A})/\mathrm{Coh}_{\leq i-1}(\mathcal{A})\right)\right) = \mathrm{K}_0\left(\mathrm{Coh}_{\leq i}(\mathcal{A})/\mathrm{Coh}_{\leq i-1}(\mathcal{A})\right)~.$$ Similarly, we deduce in this case an isomorphism of sequences of abelian groups $$\begin{gathered} \begin{tikzcd} \mathrm{Z}^{\Delta}_{i} (X,\mathcal{A}) \arrow{r} \arrow[equals]{d} & \mathrm{K}_0\left((\mathcal{K}_{(i+1)}/\mathcal{K}_{(i-1)})^c\right) \arrow{r} \arrow[equals]{d} & \mathrm{Z}^{\Delta}_{i+1} (X,\mathcal{A}) \arrow[equals]{d} \\ \mathrm{K}_0\left(\mathrm{Coh}_{\leq i}(\mathcal{A})/\mathrm{Coh}_{\leq i-1}(\mathcal{A})\right) \arrow{r}{\iota} & \mathrm{K}_0\left(\mathrm{Coh}_{\leq i+1}(\mathcal{A})/\mathrm{Coh}_{\leq i-1}(\mathcal{A})\right) \arrow{r}{\pi} & \mathrm{K}_0\left(\mathrm{Coh}_{\leq i+1}(\mathcal{A})/\mathrm{Coh}_{\leq i}(\mathcal{A})\right) \end{tikzcd} \end{gathered}$$ which are exact in the middle. Hence, we deduce from Proposition \[propchowexseq\] an isomorphism $\mathrm{CH}^{\Delta}_{i}(X,\mathcal{A}) \cong \mathrm{im}(\iota) = \mathrm{ker}(\pi)$ for this situation. The lower sequence is the end of the $\mathrm{K}$-theory long exact localization sequence for the Serre localization $$\mathrm{Coh}_{\leq i}(\mathcal{A})/\mathrm{Coh}_{\leq i-1}(\mathcal{A}) \to \mathrm{Coh}_{\leq i+1}(\mathcal{A})/\mathrm{Coh}_{\leq i-1}(\mathcal{A}) \to \mathrm{Coh}_{\leq i+1}(\mathcal{A})/\mathrm{Coh}_{\leq i}(\mathcal{A})$$ and hence $$\label{equation:chow-as-cokernel} \mathrm{CH}^{\Delta}_{i} (X,\mathcal{A}) \cong \mathrm{coker}\left(\mathrm{K}_1\left( \mathrm{Coh}_{\leq i+1}(\mathcal{A})/\mathrm{Coh}_{\leq i}(\mathcal{A}) \right) \to \mathrm{K}_0\left( \mathrm{Coh}_{\leq i}(\mathcal{A})/\mathrm{Coh}_{\leq i-1}(\mathcal{A}) \right)\right)~.$$ There is also a local description of $\mathrm{Z}^{\Delta}_{i} (X,\mathcal{A})$. Abstractly, it follows from [@1206.2721] and [@1510.00211 Proposition 2.18, Lemma 2.19], that $$\mathrm{Z}^{\Delta}_{i} (X,\mathcal{A}) = \coprod_{x \in X_{(i)}} \mathrm{K}_0\left((\Gamma_x \mathcal{K})^c\right)~, \label{eqnabstractsplitting}$$ where $X_{i}$ is the set of points $x \in X$ such that $\mathrm{dim}(x) = i$. Suppose $\mathcal{A}$ is coherent. Then $$(\Gamma_x \mathcal{K})^c \cong \mathrm{D^{perf}_{\lbrace x \rbrace}}(\mathcal{A}_x)~.$$ \[lmasplitcomp\] Since for any object $A \in \mathcal{K}$ we have, by definition, $\Gamma_x A = \Gamma_{\overline{\lbrace x \rbrace}} L_{Y_x} \mathcal{O}_X \otimes_{\mathcal{O}_X}^{\mathrm{L}} A$, it follows that $$\Gamma_x \mathcal{K} = \Gamma_{\overline{\lbrace x \rbrace}} \mathcal{O}_X \ast \left(L_{Y_x} \mathcal{O}_X \ast \mathcal{K}\right)~.$$ The subcategory $\mathrm{D}_{Y_x}(\mathcal{A})$ is smashing by Proposition \[propsubsetsubcatcoincide\] and it follows from Lemma \[lmarelativesmash\] and Proposition \[propstalkquot\] that $L_{Y_x} \mathcal{O}_X \ast \mathcal{K} = \mathrm{D}(\mathrm{Mod}(\mathcal{A}_x))$. The compact objects of $\Gamma_x \mathcal{K}$ are given by the compact objects $a$ of $L_{Y_x} \mathcal{O}_X \ast \mathcal{K}$ with $\mathrm{supp}(a) \subset \overline{ \lbrace x \rbrace}$: the inclusion functor $I: \Gamma_x \mathcal{K} \to L_{Y_x} \mathcal{O}_X \ast \mathcal{K}$ has a coproduct preserving right-adjoint $\Gamma_{\overline{\lbrace x\rbrace}}(\mathbb{I}) \ast -$ and hence preserves compactness. Thus, the compact objects of $\Gamma_x \mathcal{K}$ embed into the compact objects of $L_{Y_x} \mathcal{O}_X \ast \mathcal{K}$ with support in $\overline{\lbrace x\rbrace}$. On the other hand, if $a$ is a compact object of $L_{Y_x} \mathcal{O}_X \ast \mathcal{K}$ with support in $\overline{\lbrace x\rbrace}$, then the localization triangle $$\Gamma_{\overline{\lbrace x\rbrace}}(\mathbb{I}) \ast a \to a \to L{\overline{\lbrace x\rbrace}}(\mathbb{I}) \ast a \to \Sigma\left(\Gamma_{\overline{\lbrace x\rbrace}}(\mathbb{I}) \ast a\right)$$ and Proposition \[propstevesuppprop\] show that $\Gamma_{\overline{\lbrace x\rbrace}}(\mathbb{I}) \ast a \cong a$, and hence $a$ belongs to the essential image of the embedding $I$. Since $\mathrm{D}(\mathrm{Mod}(\mathcal{A}_x))^c = \mathrm{D^{perf}}(\mathcal{A}_x)$ and $\mathrm{supp} = \mathrm{Supp}$ for its objects by Proposition \[propsupp=Supp\], the desired description follows. Let $(R,\mathfrak{m})$ be a commutative noetherian local ring and $A$ a (module-)finite $R$-algebra. Then a right $A$-module $M$ has finite length over $A$ if and only if it has finite length over $R$. \[lmaflsame\] Recall that a right module has finite length if and only if it is both artinian and noetherian. Hence, if $M$ has finite length over $R$, it must also have finite length over $A$, since every chain of $A$-submodules of $M$ is also a chain of $R$-submodules. In order to prove that right $A$-modules of finite $A$-length also have finite $R$-length, it suffices to show that all simple right $A$-modules have finite $R$-length: one can then refine finite composition series over $A$ to finite composition series over $R$. In order to study simple right $A$-modules it suffices to consider simple modules over $A/\mathrm{J}(A)$, since the Jacobson radical annihilates all simple modules, by definition. We have $\mathrm{J}(R) = \mathfrak{m}$ and by [@MR1125071 Corollary 5.9], it follows that $\mathfrak{m}A \subset \mathrm{J}(A)$, and hence we have a surjection $A/\mathfrak{m}A \twoheadrightarrow A/\mathrm{J}(A)$. By assumption, $A/\mathfrak{m}A$ is a finite $R$-module with support contained in $\lbrace \mathfrak{m} \rbrace$ and hence has finite length over $R$. It follows that $A/\mathrm{J}(A)$ has finite $R$-length as well. Hence, the finite length right modules over $A/\mathrm{J}(A)$ have finite length over $R$, which holds in particular for the simple ones. Suppose $\mathcal{A}$ is coherent. Then $$\mathrm{Z}^{\Delta}_{i} (X,\mathcal{A}) = \coprod_{x \in X_{(i)}} \mathrm{K}_0\left(\mathrm{D^{perf}_{fl.}}(\mathcal{A}_x)\right)~.$$ where $\mathrm{D^{perf}_{fl.}}(\mathcal{A}_x) \subset \mathrm{D^{perf}}(\mathcal{A}_x)$ denotes the full subcategory of complexes with finite length cohomology. If furthermore $\mathcal{A}$ has finite global dimension, then $$\mathrm{Z}^{\Delta}_{i} (X,\mathcal{A}) = \coprod_{x \in X_{(i)}} \mathrm{K}_0\left(\mathrm{D^b}(\mathcal{A}_x\mathrm{-fl.})\right)~,$$ where $\mathcal{A}_x\mathrm{-fl.}$ denotes the abelian category of right $\mathcal{A}_x$-modules of finite length. \[corsplitcycles\] For the first statement, it suffices to prove that $\mathrm{D^{perf}_{\lbrace x \rbrace}}(\mathcal{A}_x) = \mathrm{D^{perf}_{fl.}}(\mathcal{A}_x)$ by Lemma \[lmasplitcomp\]. This follows from Lemma \[lmaflsame\] since a complex $C^{\bullet} \in \mathrm{D^{perf}}(\mathcal{A}_x)$ has support in $\lbrace x \rbrace$ iff $\mathrm{Supp}(\mathrm{H}^{\ast}(C^{\bullet})) \subset \lbrace x \rbrace$ iff $\mathrm{H}^{\ast}(C^{\bullet})$ has finite $\mathcal{O}_{X,x}$-length iff $\mathrm{H}^{\ast}(C^{\bullet})$ has finite $\mathcal{A}_x$-length. For the second assertion, Corollary \[corverdiervsserre\] gives $$\mathrm{D^{perf}_{\lbrace x \rbrace}}(\mathcal{A}_x) = \mathrm{D}^{\mathrm{b}}_{\lbrace x \rbrace}(\mathrm{mod}(\mathcal{A}_x)) = \mathrm{D^{b}}(\mathrm{mod}(\mathcal{A}_x)_{\lbrace x \rbrace})~$$ Now a finitely generated right $A_{x}$-module has support in $\lbrace x \rbrace$ iff it has finite length as an $R$-modules iff it has finite length as a right $\mathcal{A}_x$-module by Lemma \[lmaflsame\]. This shows that $\mathrm{mod}(\mathcal{A}_x)_{\lbrace x \rbrace} = \mathcal{A}_x\mathrm{-fl.}$ and finishes the proof. Corollary \[corsplitcycles\] makes it possible to give a computation of $\mathrm{Z}_i^{\Delta}(X,\mathcal{A})$ in large generality. \[theorem:cycle-groups\] Let $X$ be a noetherian scheme and $\mathcal{A}$ a coherent $\mathcal{O}_X$-algebra of finite global dimension. Then $$\mathrm{Z}^{\Delta}_{i} (X,\mathcal{A}) = \bigoplus_{x \in X_{(i)}} \mathbb{Z}^{r_x}$$ where $r_x < \infty$ is the number of isomorphism classes of simple right modules of $\mathcal{A}_x$. By Corollary \[corsplitcycles\], it suffices to show that $\mathrm{K}_0(\mathrm{D^b}(\mathcal{A}_x\mathrm{-fl.})) = \mathrm{K}_0(\mathcal{A}_x\mathrm{-fl.})=\mathbb{Z}^{r_x}$ with $r_x < \infty$. From the proof of Lemma \[lmaflsame\] we see that the simple $\mathcal{A}_x$-modules correspond to the simple $\mathcal{A}_x/\mathrm{J}(\mathcal{A}_x)$-modules, and that the latter algebra is of finite length over $\mathcal{O}_{X,x}$. This implies that $\mathcal{A}_x/\mathrm{J}(\mathcal{A}_x)$ is right Artinian and hence has $r_x < \infty$ simple right modules (all of them occur in a composition series of $\mathcal{A}_x$ over itself by the Jordan-Hölder theorem). A standard induction on the composition multiplicities of these simple modules shows that $\mathrm{K}_0(\mathcal{A}_x\mathrm{-fl.})=\mathbb{Z}^{r_x}$ as desired. Let us finish the section with an easy observation concerning the vanishing of $\mathrm{Z}^{\Delta}_i(X,\mathcal{A})$ and $\mathrm{CH}^{\Delta}_i(X,\mathcal{A})$. \[proposition:vanishing-outside-range\] Suppose $\mathrm{dim}(\mathrm{supp}(\mathcal{A})) = n$. Then $$\mathrm{Z}^{\Delta}_i (X,\mathcal{A})= \mathrm{CH}^{\Delta}_i(X,\mathcal{A}) = 0$$ for all $i > n$. If $i>n$, then $\mathcal{K}_{i} = \mathcal{K}_{i-1} = \mathcal{K}$ and hence $$\mathrm{Z}^{\Delta}_{i} (X,\mathcal{A})= \mathrm{K}_0\left((\mathcal{K}_{(i)}/\mathcal{K}_{(i-1)})^c\right) = 0~,$$ which also implies $\mathrm{CH}^{\Delta}_i(X,\mathcal{A}) = 0$. Comparison to Chow groups of X for coherent OX-algebras on regular schemes -------------------------------------------------------------------------- Suppose that $\mathcal{A}$ is a coherent $\mathcal{O}_X$-algebra and that $X$ is regular. By definition of $\mathrm{supp}$, the forgetful functor $U: \mathrm{D}(\mathrm{Qcoh}(\mathcal{A})) \to \mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))$ induces functors $$\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))_{(p)} \to \mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))_{(p)}$$ for all $p \geq 0$. If $C^{\bullet}$ is a perfect complex in $\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))$, then $U(C^{\bullet})$ will be an object of $\mathrm{D^b}(\mathrm{Coh}(X)) = \mathrm{D^{perf}}(X)$ and hence $U$ preserves compactness. Hence, we obtain a commutative diagram of functors $$\begin{tikzcd} \mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))_{(p)}^c \arrow{r} \arrow{d} \arrow{dr}& \underbrace{(\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))_{(p)}^c/\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))_{(p-1)}^c)^{\natural}}_{= (\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))_{(p)}/\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))_{(p-1)})^c} \arrow{dr} &\\ \mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))_{(p+1)}^c \arrow{dr} & \mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))_{(p)}^c \arrow{r} \arrow{d}& \underbrace{(\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))_{(p)}^c/\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))_{(p-1)}^c)^{\natural}}_{= (\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))_{(p)}/\mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))_{(p-1)})^c}\\ & \mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))_{(p+1)}^c & \end{tikzcd} \label{eqncomptobase}$$ in which the horizontal arrows are given by the Verdier quotient followed by the inclusion into the idempotent completion, the vertical arrows are inclusions and the diagonal ones are induced by $U$. It is possible to construct the above diagram without assuming $X$ to be regular: the main obstruction is for $U$ to preserve compactness. This happens for example, when $U$ admits a coproduct-preserving right-adjoint. But the functor $\mathrm{R}\mathcal{H}\mathrm{om}_{\mathcal{O}_X}(U(\mathcal{A}), -)$ is always right-adjoint to $U$. It will preserve coproducts if $U(\mathcal{A})$ is a perfect complex over $X$ by [@MR1308405 proof right after Example 1.13]. Hence, we see that, instead of assuming that $X$ is regular, it suffices that $U(\mathcal{A})$ is perfect. If $X$ is regular this is, of course, always the case. \[remcomparisonsingular\] Suppose that $\mathcal{A}$ is a coherent $\mathcal{O}_X$-algebra on a noetherian regular scheme $X$. Let $\mathcal{T} = \mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))$ and $\mathcal{K} = \mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))$. Then the forgetful functor $\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))_{(p)} \to \mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))_{(p)}$ induces group homomorphisms $$\mathrm{Z}^{\Delta}_{p} (X,\mathcal{A}) \to \mathrm{Z}^{\Delta}_{p} (X,\mathcal{O}_X) = \mathrm{Z}_p(X) \quad \text{and} \quad \mathrm{CH}^{\Delta}_{p} (X,\mathcal{A}) \to \mathrm{CH}^{\Delta}_{p} (X,\mathcal{O}_X) = \mathrm{CH}_p(X)$$ for all $p \geq 0$. \[propcomparisonmap\] This follows immediately from Theorem \[thmchowrecover\] and the definitions of $\mathrm{Z}^{\Delta}_{p}(X,\mathcal{A})$ and $\mathrm{CH}^{\Delta}_{p} (X,\mathcal{A})$ by applying $\mathrm{K}_0(-)$ to (\[eqncomptobase\]). If in Proposition \[propcomparisonmap\] we only assume that $U(\mathcal{A})$ is perfect instead of $X$ being regular (see Remark \[remcomparisonsingular\]), then $U$ still gives group homomorphisms $$\mathrm{Z}^{\Delta}_{p} (X,\mathcal{A}) \to \mathrm{Z}^{\Delta}_{p} (X,\mathcal{O}_X) \quad \text{and} \quad \mathrm{CH}^{\Delta}_{p} (X,\mathcal{A}) \to \mathrm{CH}^{\Delta}_{p} (X,\mathcal{O}_X)$$ for all $p \geq 0$. Extension of scalars $\mathcal{A} \otimes^{\mathrm{L}}_{\mathcal{O}_X} -$ has a a coproduct preserving right-adjoint $U$ and hence preserves compact object. For $C^{\bullet} \in \mathrm{D^{perf}}(X)$, we have $$\mathrm{supp}(\mathcal{A} \otimes^{\mathrm{L}}_{\mathcal{O}_X} C^{\bullet}) = \mathrm{supp}(\mathcal{A}) \cap \mathrm{supp}(C^{\bullet})$$ from which we deduce that $\mathcal{A} \otimes^{\mathrm{L}}_{\mathcal{O}_X} -$ restricts to $$\mathrm{D^{perf}}(\mathcal{O}_X)_{(p)} \to \mathrm{D^{perf}}(\mathcal{A})_{(p)}$$ for all $p \geq 0$. Hence, by a similar argument as for $U$, we obtain that extension of scalars induces morphisms $\mathrm{CH}^{\Delta}_{p} (X,\mathcal{O}_X) \to \mathrm{CH}^{\Delta}_{p} (X,\mathcal{A})$. Note however, that if $\mathrm{dim}(\mathrm{supp}(\mathcal{A})) = q$, then these morphisms are necessarily trivial for $p >q$ since $\mathrm{Z}^{\Delta}_{p} (X,\mathcal{A}) = \mathrm{CH}^{\Delta}_{p} (X,\mathcal{A}) = 0$ in this case by Proposition \[proposition:vanishing-outside-range\]. The case of coherent commutative OX-algebras {#sectioncommcohalg} ============================================ In the following, we will show, how the framework we have set up lets us deal with finite morphisms between noetherian schemes. Let $X$ be a noetherian separated scheme and $\mathcal{A}$ a *commutative* $\mathcal{O}_X$-algebra which is *coherent* as an $\mathcal{O}_X$-module. Then $\mathcal{A}$ corresponds to an affine morphism $\varphi: Y := \mathbf{Spec}(\mathcal{A}) \to X$ and there is an equivalence of categories $\Theta:\mathrm{Qcoh}(\mathcal{A}) \cong \mathrm{Qcoh}(\mathcal{O}_Y)$ that makes the following diagram commute up to natural isomorphism: $$\begin{tikzcd} \mathrm{Qcoh}(\mathcal{A}) \arrow{rr}{\Theta} \arrow{rd}{U} & & \mathrm{Qcoh}(\mathcal{O}_Y) \arrow{ld}{\varphi_*} \\ & \mathrm{Qcoh}(\mathcal{O}_X)& \end{tikzcd} \label{eqncoherentcommutative}$$ Let us note that $\Theta$ also restricts to an equivalence between the subcategories of coherent modules and the restriction makes a diagram similar to (\[eqncoherentcommutative\]) commute, with $\mathrm{Qcoh}(-)$ replaced by $\mathrm{Coh}(-)$. The following three results should be well-known. The morphism $\varphi$ is finite. In particular, $Y$ is noetherian and separated. \[lmacohalgfinmor\] This is an immediate consequence of the construction of $\mathbf{Spec}(\mathcal{A})$: over each open affine $U = \mathrm{Spec}(R)$ of $X$ lies an open affine $\mathrm{Spec}(\mathcal{A}(U))$, and $\mathcal{A}(U)$ is a finite $R$-module since $\mathcal{A}$ was assumed to be a coherent sheaf on $X$. Let $f: Y \to X$ be a morphism of schemes and assume $X$ locally noetherian. 1. For any coherent $\mathcal{O}_X$-module $M$, we have $\mathrm{Supp}(f^*(M)) = f^{-1}(\mathrm{Supp}(M))$. 2. Suppose $f$ is finite. For any closed subset $Z \subset \mathrm{im}(f)$, we have $$\mathrm{dim}(f^{-1}(Z)) = \mathrm{dim}(Z)$$ and for any closed set $W \subset Y$, we have $$\mathrm{dim}(f(W)) = \mathrm{dim}(W)$$ \[lmadimfinmor\] For the first assertion we can assume that $X,Y$ are affine, in this case the statement is proved in Atiyah-MacDonald, Chap. 3, exercise 19(viii). For the second statement, we consider the fibre square $$\begin{tikzcd} f^{-1}Z \arrow{r} \arrow{d} & Y \arrow{d}{f} \\ Z \arrow{r} & \mathrm{im}(f) \end{tikzcd}$$ and use that for finite and surjective morphisms, domain and codomain have the same Krull dimension. The last assertion follows from the second one by considering the composition $f|_W: W \to Y \xrightarrow{f} X$. Suppose $X$ is a locally noetherian scheme and $f: X \to Y$ is an affine closed morphism and $M$ a quasi-coherent $\mathcal{O}_X$-module. Then $\mathrm{Supp}(f_*M) = f(\mathrm{Supp}(M))$. \[propsuppdirectcommute\] We shall compute the stalks of the sheaf $f_*M$ at $y \in Y$. Since $f$ is closed, this can be done using all opens on $X$, i.e. $(f_*M)_y = \varinjlim_{V \supset f^{-1}(y)} M(V)$. The set $f^{-1}(y)$ will be contained in an affine open $\mathrm{Spec}(R) \subset X$ because $f$ is affine and hence, we can assume that $M$ is an $R$-module and $f^{-1}(y) =: P$ is a set of prime ideals of $R$. We rewrite $$(f_*M)_y = \varinjlim_{V \supset f^{-1}(y)} M(V) = \varinjlim_{D(r) \supset P} M_r,$$ where $D(r)$ runs over the basic opens of $\mathrm{Spec}(R)$ that contain $P$. From this, we see that $(f_*M)_y = S^{-1}M$, where $S := R \setminus \bigcup_{\mathfrak{p} \in P} \mathfrak{p}$. It follows that $(f_*M)_y = 0$ if and only if $M_{\mathfrak{p}} = 0$ for all $\mathfrak{p} \in P = f^{-1}(y)$, which proves the claim. The equivalence $\Theta: \mathrm{Qcoh}(\mathcal{A}) \to \mathrm{Qcoh}(\mathcal{O}_Y)$ respects dimension of support: if $M \in \mathrm{Qcoh}(\mathcal{A})$, then $\mathrm{dim}(\mathrm{Supp}_X(M)) = \mathrm{dim}(\mathrm{Supp}_Y(\Theta(M))) $. Hence, $\Theta$ induces exact equivalences $$\mathrm{Qcoh}(\mathcal{A})_{\leq p} \xrightarrow{\sim} \mathrm{Qcoh}(\mathcal{O}_Y)_{\leq p}$$ for all $p \geq 0$. \[corfinitedimpres\] By definition and (\[eqncoherentcommutative\]), we have $$\mathrm{dim}(\mathrm{Supp}(M)) = \mathrm{dim}(\mathrm{Supp}_X(U(M))) = \mathrm{dim}(\mathrm{Supp}_X(\varphi_*(\Theta(M))))~.$$ Since $\mathcal{A}$ was assumed to be coherent, $\varphi$ is finite by Lemma \[lmacohalgfinmor\] and it follows from Proposition \[propsuppdirectcommute\] that $$\mathrm{dim}(\mathrm{Supp}_X(\varphi_*(\Theta(M)))) = \mathrm{dim}(\varphi(\mathrm{Supp}_X(\Theta(M))))$$ as finite morphisms are in particular affine and (universally) closed. By Lemma \[lmadimfinmor\], the latter quantity is equal to $\mathrm{dim}(\mathrm{Supp}_Y(\Theta(M)))$, which proves the claim. The functor $\Theta$ induces an equivalence $$\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))_{(p)} \cong \mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_Y))_{(p)}$$ for all $p \geq 0$. \[corthetasubcatequiv\] The equivalence $\Theta$ is exact (as any equivalence of abelian categories) and hence induces and equivalence $\mathrm{D}(\mathrm{Qcoh}(\mathcal{A})) \cong \mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_Y))$. Now, it suffices to remark that for $C^{\bullet} \in \mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))$ we have $$\begin{aligned} C^{\bullet} \in \mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))_{(p)} &\Leftrightarrow \mathrm{H}^{\ast}(C^{\bullet}) \in \mathrm{Qcoh}(\mathcal{A})_{\leq p} \\ &\Leftrightarrow \mathrm{H}^{\ast}(\Theta(C^{\bullet})) \in \mathrm{Qcoh}(\mathcal{O}_Y)_{\leq p} \\ &\Leftrightarrow \Theta(C^{\bullet}) \in \mathrm{D}(\mathrm{Qcoh}(\mathcal{O}_X))_{(p)} \end{aligned}$$ where we used Proposition \[propsubsetsubcatcoincide\] and Lemma \[corfinitedimpres\]. Let $X$ be a separated scheme of finite type over a field and $\mathcal{A}$ a coherent sheaf of *commutative* $\mathcal{O}_X$-algebras. Then $$\mathrm{CH}^{\Delta}_{p}(X,\mathcal{A}) \cong \mathrm{CH}^{\Delta}_{p}(Y,\mathcal{O}_Y)$$ for all $p \geq 0$. In particular if $\mathbf{Spec}(\mathcal{A})$ is regular ($\Leftrightarrow \mathcal{A}$ has finite global dimension), then $$\mathrm{CH}^{\Delta}_{p}(X,\mathcal{A}) \cong \mathrm{CH}_p(\mathbf{Spec}(\mathcal{A}))~.$$ \[thmcommrecover\] There is a diagram $$\xymatrix{ \mathrm{K}_0\left((\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))_{(p)})^c\right) \ar[r] \ar[dr] \ar[d]& \mathrm{K}_0\left((\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))_{(p)}/\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))_{(p+1)})^c\right) \ar[dr] & \\ \mathrm{K}_0\left((\mathrm{D}(\mathrm{Qcoh}(\mathcal{A}))_{(p-1)})^c\right) \ar[dr] & \mathrm{K}_0\left((\mathrm{D}(\mathcal{O}_Y)_{(p)})^c\right) \ar[r] \ar[d]& \mathrm{K}_0\left((\mathrm{D}(\mathcal{O}_Y)_{(p)}/\mathrm{D}(\mathcal{O}_Y)_{(p+1)})^c\right) \\ & \mathrm{K}_0\left((\mathrm{D}(\mathcal{O}_Y)_{(p-1)})^c\right) }$$ where all diagonal arrows are isomorphisms induced by $\Theta$, as follows from Corollary \[corthetasubcatequiv\]. This immediately gives the desired isomorphisms of Chow groups. Relative tensor triangular Chow groups for orders {#sectionorders} ================================================= In this section we study relative tensor triangular Chow groups for a special class of coherent $\mathcal{O}_X$algebras: orders. These are particularly well-behaved *noncommutative* algebras, whose definition we recall in \[subsection:orders-preliminaries\]. In their modern incarnation they were defined in [@MR0117252] and the main reference is [@MR0393100]. The main goal is to show that they coincide with other invariants in the literature, as is the case in the commutative setting where tensor triangular Chow groups agree with the classical Chow groups, see [@MR3423452; @1510.00211]. We give some general results on cycle groups in \[subsection:cycle-groups\], based on \[theorem:cycle-groups\]. We get a description of the top degree cycle groups for any order in \[proposition:highest-cycle-group\]. Finally we will use the structure theory for hereditary orders over discrete valuation rings to describe all cycle groups of hereditary orders and the codimension one cycle groups of tame orders, making the result in \[theorem:cycle-groups\] concrete in a well-known example. In \[subsection:chow-groups\] we discuss the Chow groups for orders. An easy corollary of the theory is a description of the top degree Chow group in \[corollary:highest-chow-group\]. More importantly, we recall the definition of various class groups in the theory of orders, and we show that these classical invariants agree with the appropriate tensor triangular Chow groups. In \[subsection:group-rings\] we study the Chow groups of group rings over Dedekind domains, for which it is again possible to relate the tensor triangular Chow groups to classical invariants. We give some explicit examples on how one can compute them for integral group rings, using tools from algebraic number theory and representation theory. Preliminaries on orders {#subsection:orders-preliminaries} ----------------------- In this section we will introduce some basic results about orders on schemes. There are no new results here, but the literature at this level of generality is somewhat scattered. Observe that for most of this section we will assume that we are working in a central simple algebra. This corresponds to the more geometric approach to the theory of orders. In \[subsection:group-rings\] we will relax this condition, and consider algebras which are only separable over the generic point, as is common in representation theory and algebraic number theory. We will explain how the results of \[subsection:cycle-groups,subsection:chow-groups\] change in this more general situation. \[definition:order\] Let $X$ be an integral normal noetherian scheme with function field $K$. Let $A_K$ be a central simple $K$algebra. An *$\mathcal{O}_X$order* $\mathcal{A}$ in $A_K$ is a torsion-free coherent $\mathcal{O}_X$algebra whose generic fibre is $A_K$. We say that $\mathcal{A}$ is a *maximal order* if it is not properly contained in another order. In [@MR0393100] (maximal) orders are studied in both the geometric and arithmetic setting, mostly in the case of dimension 1. The behaviour of orders in higher dimension quickly becomes more and more complicated. We will need two more classes of orders, besides just the maximal ones. Recall that Auslander–Goldman characterized maximal orders as those orders which are reflexive as $\mathcal{O}_X$modules, and for which $\mathcal{A}_{\eta_Y}$ is a maximal order over the discrete valuation ring $\mathcal{O}_{X,\eta_Y}$, for all $\eta_Y$ a point of codimension 1. In dimension one there is a larger class of orders whose behaviour is as nice as that of the maximal orders. Assume that $X$ is regular and of dimension 1. Then we say that $\mathcal{A}$ is an *hereditary order* if $\mathcal{A}(U)$ is of global dimension 1 for every affine open $U\subseteq X$. For hereditary (and maximal) orders in dimension 1 there exists an extensive structure theory. Inspired by the Auslander–Goldman maximality criterion we can introduce a final class of orders, for which one can bootstrap the structure theory of hereditary orders. We say that $\mathcal{A}$ is a *tame order* if it is reflexive as an $\mathcal{O}_X$module, and $\mathcal{A}_{\eta_Y}$ is an hereditary order over the discrete valuation ring $\mathcal{O}_{X,\eta_Y}$, for all $\eta_Y$ a point of codimension 1. We now give some examples of orders for which we can describe the tensor triangular cycle and Chow groups. The easiest examples of maximal orders are matrix algebras and their étale twisted forms: Azumaya algebras. An example of an hereditary but non-maximal order on $\mathbb{P}_k^1$ is $$\mathcal{A}\coloneqq \begin{pmatrix} \mathcal{O}_{\mathbb{P}_k^1} & \mathcal{O}_{\mathbb{P}_k^1} \\ \mathcal{O}_{\mathbb{P}_k^1}(-p) & \mathcal{O}_{\mathbb{P}_k^1} \end{pmatrix}$$ where $p\in\mathbb{P}_k^1$ is a closed point. The algebra structure is induced from the embedding in $\Mat_2(\mathcal{O}_{\mathbb{P}_k^1})$. For each closed point $q\neq p$ we see that $\mathcal{A}_q$ is isomorphic to the matrix ring over $\mathcal{O}_{\mathbb{P}_k^1,q}$, whereas for the point $p$ we get the non-maximal order $$\mathcal{A}_p\cong \begin{pmatrix} \mathcal{O}_{\mathbb{P}_k^1,p} & \mathcal{O}_{\mathbb{P}_k^1,p} \\ \mathfrak{m} & \mathcal{O}_{\mathbb{P}_k^1,p} \end{pmatrix}.$$ It is precisely this non-maximality that will contribute to the structure of the relative Chow group, see \[corollary:quasiprojective-curve-matrix\]. Cycle groups {#subsection:cycle-groups} ------------ Using \[theorem:cycle-groups\] we have a complete description of cycle groups of coherent $\mathcal{O}_X$algebras. In this section we discuss what happens in the special case of orders. First we observe that the top-dimensional Chow group always is of the same form. \[proposition:highest-cycle-group\] Let $X$ be an integral normal noetherian scheme of dimension $n$. Let $\mathcal{A}$ be an order on $X$. Then $$\ZZ_n^\Delta(X,\mathcal{A})\cong\mathbb{Z}.$$ Let $\eta$ be the unique generic point of $X$. Then $\mathcal{A}_\eta$ is a central simple algebra over the function field $\mathcal{O}_{X,\eta}$ and by Morita theory we can conclude from \[theorem:cycle-groups\], as there is a unique simple for a division algebra. There are several issues in computing the cycle and Chow groups for orders in other degrees: 1. there is no general structure theory for (maximal) orders on local rings in arbitrary dimension; 2. even if there is such a description (as will be the case in dimension 1) the non-splitness of the central simple algebra over the generic point will play an important role, because the higher K-theory of central simple algebras (let alone orders) is different in general from the K-theory of the center. Nevertheless, in the one-dimensional case we can obtain an explicit description. First we consider the complete local case, for which there exists an explicit description of hereditary orders [@MR0393100 §39]. In this affine situation we will use ring-theoretical notation from op. cit. In particular, we consider a (complete) discrete valuation ring $(R,\mathfrak{m})$ whose field of fractions is denoted $K$, and an hereditary $R$order $\Lambda$ in a central simple $K$algebra $A\cong\Mat_n(D)$, where $D$ is a division algebra over $K$. Then there exists a unique maximal $R$order $\Delta$ in $D$, and we have a block decomposition $$\Lambda= \begin{pmatrix} \Delta & \rad\Delta & \rad\Delta & \ldots & \rad\Delta \\ \Delta & \Delta & \rad\Delta & \ldots & \rad\Delta \\ \Delta & \Delta & \Delta & \ldots & \rad\Delta \\ \ldots & & & & \ldots \\ \Delta & \Delta & \Delta & \ldots & \Delta \\ \end{pmatrix}^{n_1,\ldots,n_r}$$ where the block decomposition is given by putting $\Mat_{n_i\times n_j}(\Delta)$ (resp. $\Mat_{n_i\times n_j}(\rad\Delta)$) if $i\geq j$ (resp. $i<j$). In particular, $\sum_{i=1}^rn_i=n$. The number of blocks $r$ in the block decomposition is the *type* of $\Lambda$. The following result can be proved along the same lines as \[theorem:quasiprojective-curve\], but we give an alternative proof here using dévissage in algebraic K-theory [@MR0338129 §5]. \[proposition:cDVR-type\] Let $R$ be a complete discrete valuation ring, with fraction field $K$ and residue field $k$. Let $\Lambda$ be an hereditary $R$order in the central simple $K$algebra $A$. Then $$\ZZ_0^\Delta(R,\Lambda)\cong\mathbb{Z}^r$$ where $r$ is the type of $\Lambda$. By dévissage for algebraic K-theory and the invariance of K-theory under nilpotent thickenings applied to [@MR0393100 corollary 39.18(iii)] we have that $$\Knought(\fl\Lambda)\cong\Knought(\Lambda/\rad\Lambda).$$ By [@MR0393100 (39.17)] we have $$\Knought(\Lambda/\rad\Lambda)\cong\bigoplus_{i=1}^r\Knought(\Mat_{n_i}(\Delta/\rad\Delta))\cong\mathbb{Z}^{\oplus r}$$ where $\Delta/\rad\Delta$ is a skew field over $k$. Similarly one can by dévissage appeal to [@MR0393100 corollary 39.18(v)] for the conclusion. Chow groups in the regular case {#subsection:chow-groups} ------------------------------- In this section we prove the main results for orders: \[corollary:reduced-projective-class-group-affine\] shows that for an hereditary order over a Dedekind domain the 0th relative Chow group agrees with the reduced projective class group, and if the order is moreover maximal it agrees with the ideal class group. These are classical invariants that will be introduced below. In the setting of a quasiprojective curve over a field we get the analogous result in \[corollary:reduced-projective-class-group-projective\], from which we obtain \[theorem:quasiprojective-curve\]. As an immediate corollary to \[proposition:highest-cycle-group\] and the description of the rational equivalence we have the following general result. \[corollary:highest-chow-group\] With notation and assumptions as in \[proposition:highest-cycle-group\] we have that $$\CH_n^\Delta(X,\mathcal{A})\cong\mathbb{Z}.$$ We have that $q^\natural(\ker(i))$ from is zero because $i$ is an isomorphism if $p\geq n$. A similar proof of course works for every coherent $\mathcal{O}_X$algebra, where the cycle group is given by the Grothendieck group of a certain finite-dimensional algebra over the function field, in particular it is easy to construct examples for which $$\CH_n^\Delta(X,\mathcal{A})\neq\mathbb{Z},$$ e.g. by taking $\mathcal{A}=\mathcal{O}_X\oplus\mathcal{O}_X$. #### Classical invariants In the 1dimensional case the only other tensor triangular Chow group we need to describe is $\CH_0^\Delta$, see \[proposition:vanishing-outside-range\]. We will do this using \[propchowexseq\], which allows us to interpret the tensor triangular Chow groups in terms of classical invariants such as the ideal class group and the reduced projective class group, whose definitions we now recall in the affine setting. Let $R$ be a Dedekind domain, and denote its quotient field by $K$. Let $\Lambda$ be an $R$order in a central simple $K$algebra $A$. Let $M,N$ be left $\Lambda$modules. We say that they are *stably isomorphic* if there exists an integer $r$ and an isomorphism $M\oplus\Lambda^{\oplus r}\cong N\oplus\Lambda^{\oplus r}$. The *ideal class group* (or *stable class group*) $\Cl\Lambda$ of $\Lambda$ consists of the stable isomorphism classes of left $\Lambda$ideals (i.e. those submodules $I$ such that $KI=A$), where the group structure is defined in [@MR0393100 theorem 35.5]. It is a one-sided generalisation of the usual class group (or Picard group). There also exists a two-sided version, which is different in general, see \[remark:one-vs-two-sided\]. Because we are only considering the module structure on one side, it is the former and not the latter that is important to us. In this case the localisation sequence that is used to define rational equivalence in the zeroth Chow group as in is also known as the *Bass–Tate sequence* [@MR712062; @MR925271]. We will now recall the description from [@MR0404410 §2]. In the relevant degrees the localization sequence takes on the form $$\label{equation:bass-tate} \Kone(\Lambda)\to\Kone(A)\to\Knought(\fl\Lambda)\to\Knought(\Lambda)\to\Knought(A)\to 0.$$ We can also apply dévissage to the term $\Knought(\fl\Lambda)$, and obtain $$\Knought(\fl\Lambda)\cong\bigoplus_{\mathfrak{p}\in\Spec R\setminus\{0\}}\Knought(\fl\Lambda_{\mathfrak{p}}).$$ The *reduced projective class group* $\widetilde{\Knought}(\Lambda)$ of $\Lambda$ is the kernel of the morphism $\Knought(\Lambda)\twoheadrightarrow\Knought(A)$ in . In some texts the reduced projective class group is also denoted $\mathrm{SK}_0$. Observe that the reduced projective class group is the kernel of a *split* epimorphism, because $\Knought(A)\cong\mathbb{Z}$ is projective. So to compute the reduced projective class group it suffices to compute $\Knought(\Lambda)$. The connection between these two types of class groups is given by [@MR0393100 theorem 36.3] and [@MR0404410 (2.9)]. The first result says that for a maximal order we have that $$\Cl\Lambda\cong\widetilde{\Knought}(\Lambda),$$ whilst the latter describes the ideal class group in general as a *subgroup* of the reduced projective class group via the short exact sequence $$\label{equation:SES-Cl-K0red} 0\to\Cl\Lambda\to\widetilde{\Knought}(\Lambda)\overset{\lambda_0}{\to}\bigoplus_{\mathfrak{p}\in\Spec R\setminus\{(0)\}}\widetilde{\Knought}(\Lambda_{\mathfrak{p}})\to 0$$ In particular, if $\Lambda$ is maximal, then $\lambda_0$ is the zero map: by [@MR0393100 theorem 18.7] we have indeed that $\Cl\Lambda_{\mathfrak{p}}=\widetilde{\Knought}(\Lambda_{\mathfrak{p}})$ is zero. Moreover, we know by Jacobinski that $\Cl\Lambda\cong\Cl\Lambda'$, for $\Lambda\subseteq\Lambda'$ an inclusion of *hereditary* orders [@MR0393100 theorem 40.16]. In particular it suffices to compute the ideal class group of a maximal order containing $\Lambda$, provided one starts with an hereditary order. It is possible to reprove Jacobinski’s result using and the results used in the proof of \[proposition:correct-reiten-vandenbergh\]: if $\Lambda$ is an hereditary order, then $\Knought(\Lambda_{\mathfrak{p}})\cong\mathbb{Z}^{\oplus r-1}$ for $\mathfrak{p}$ a maximal ideal of $R$, where $r$ is the type of $\Lambda_{\mathfrak{p}}$, because the last terms of reduce to the *split* short exact sequence $$0\to\mathbb{Z}^{\oplus r-1}\to\mathbb{Z}^{\oplus r}\to\mathbb{Z}\to 0.$$ As an immediate corollary to \[propchowexseq\] we have the following main result. In particular, by the above discussion we obtain an explicit description of the relative tensor triangular Chow groups in the case of an order $\Lambda$ over a Dedekind domain $R$. \[corollary:reduced-projective-class-group-affine\] We have that $$\CH_0^\Delta(R,\Lambda)\cong\widetilde{\Knought}(\Lambda).$$ If $\Lambda$ is moreover hereditary, then $$\CH_0^\Delta(R,\Lambda)\cong\widetilde{\Knought}(\Lambda)\cong\Cl\Lambda'\oplus\mathbb{Z}^{r-1}$$ where $\Lambda'$ is a maximal order containing $\Lambda$ and $r$ is the maximal length of a chain of inclusions of orders. In \[subsection:group-rings\] we will encounter another situation in which we can express the relative tensor triangular Chow groups in terms of class groups of orders, but there the behaviour with respect to inclusions in maximal orders is different. \[remark:one-vs-two-sided\] In [@MR0393100 theorem 40.9] a description of the (two-sided) Picard group is given. It combines information about the local type (see \[proposition:cDVR-type\]) and the ramification. This differs from the tensor triangular Chow groups, for which the local type shows up as copies of $\mathbb{Z}$, not in the form of torsion quotients. #### Hereditary orders on curves Up to now we only looked at hereditary orders on Dedekind domains. In [@MR1825805; @1412.0290] the case of hereditary orders on smooth (quasi)projective curves over a field $k$ is studied, mostly from a representation theory point of view. Let $C$ be a quasiprojective curve over $\Spec k$. Let $\mathcal{A}$ be an hereditary order in the central simple $k(C)$algebra $A$. \[corollary:reduced-projective-class-group-projective\] We have that $$\CH_0^\Delta(C,\mathcal{A})\cong\ker\left( \Knought(\mathcal{A})\twoheadrightarrow\Knought(A)\cong\mathbb{Z} \right).$$ One can use the results of [@MR1825805] to compute Grothendieck groups of hereditary orders in this setting. The results in this paper are stated only for $k$ algebraically closed. In this case we have by Tsen’s theorem that $\Br(k)=\Br(k(C))=0$, which means that the central simple $k(C)$algebra $A$ is always of the form $\Mat_n(k(C))$, i.e. it is unramified. If $k$ is not algebraically closed, then one should change the definition of $r$ in [@MR1825805 proposition 2.1]: it should only incorporate the local types of the hereditary order, not the ramification of a maximal order containing it. The reason why the definition using ramification works in the algebraically closed case is because every central simple $k(C)$algebra is automatically unramified, and so is every maximal order. But if $\Br(k(C))\neq 0$ there are ramified maximal orders. The correct definition should only account for the length of a chain of orders containing $\mathcal{A}$ and terminating in a maximal order $\overline{\mathcal{A}}$. If $\mathcal{A}$ is itself already maximal we will say that this length is 0. \[proposition:length-of-chain\] Let $\mathcal{A}$ be a sheaf of hereditary $\mathcal{O}_C$orders. Let $r_p$ be the type of the hereditary $\mathcal{O}_{C,p}$order $\mathcal{A}_p$. Then the maximal length of a chain of orders containing $\mathcal{A}$ is independent of the maximal order in which it terminates and is equal to $$\sum_{p\in C}(r_p-1).$$ This follows from the proof of [@MR0393100 theorem 40.8]. We can now formulate [@MR1825805 proposition 2.1] in such a way that it is also valid over non-algebraically closed fields. By the discussion above the formulation of loc. cit. can be misinterpreted if one does not assume throughout that $k$ is algebraically closed. \[proposition:correct-reiten-vandenbergh\] Let $\mathcal{A}$ be a sheaf of hereditary $\mathcal{O}_C$orders in a central simple $k(C)$algebra $A$. Let $\overline{\mathcal{A}}$ be a maximal order containing $\mathcal{A}$. Then $$\Knought(\mathcal{A})\cong\Knought(\overline{\mathcal{A}})\oplus\mathbb{Z}^{\oplus\rho}$$ where $\rho\coloneqq\sum_{p\in C_{(0)}}(r_p-1)$. This follows from \[proposition:length-of-chain\] and [@MR978602 theorem 1.14]. We are now ready to prove the main result for hereditary orders on quasiprojective curves. \[theorem:quasiprojective-curve\] Let $\mathcal{A}$ be a sheaf of hereditary $\mathcal{O}_C$orders in a central simple $k(C)$algebra $A$. Let $\overline{\mathcal{A}}$ be a maximal order containing $\mathcal{A}$. Then $$\begin{aligned} \CH_0^\Delta(C,\mathcal{A})&\cong\Cl(\overline{\mathcal{A}})\oplus\mathbb{Z}^{\oplus\rho} \\ \CH_1^\Delta(C,\mathcal{A})&\cong\mathbb{Z} \end{aligned}$$ where $\rho\coloneqq\sum_{p\in C_{(0)}}(r_p-1)$. By [@MR1825805 proposition 2.1] we obtain that $$\Knought(\mathcal{A})\cong\Knought(\overline{\mathcal{A}})\oplus\mathbb{Z}^{\oplus\rho}.$$ Now we apply \[corollary:reduced-projective-class-group-projective\] to conclude. We now discuss some situation in which these Chow groups can be described more explicitly, which reduces to having an explicit description of the ideal class group of a maximal order in this geometric setting. \[corollary:quasiprojective-curve-matrix\] Let $k$ be algebraically closed. Then for every $\mathcal{A}$ as in \[theorem:quasiprojective-curve\] we have that $$\CH_0^\Delta(C,\mathcal{A})\cong\Pic C\oplus\mathbb{Z}^{\oplus\rho}.$$ If $k$ is not algebraically closed the same description holds as long as $A\cong\Mat_n(k(C))$. By Tsen’s theorem we know that $\Br(k(C))=0$, so $A\cong\Mat_n(k(C))$. The maximal orders in $A$ are all of the form $\End_X(\mathcal{E})$ for $\mathcal{E}$ a vector bundle of rank $n$, and by Morita theory we can conclude because $\Knought(\overline{\mathcal{A}})\cong\Knought(\mathcal{O}_C)\cong\Pic(C)\oplus\mathbb{Z}$. It would be interesting to develop the notion of functoriality for relative tensor triangular Chow groups, as was done for the non-relative case in [@MR3423452]. One example would be the observation that the functor $$-\otimes_R\Mat_n(R)\colon R\mhyphen\mathrm{mod}\to\Mat_n(R)\mhyphen\mathrm{mod}$$ induces multiplication by $n$ on the level of Grothendieck groups. In more general settings (e.g. inclusions of orders) one expects similar interesting behaviour. If $k$ is not algebraically closed we have an inclusion $$\label{equation:brauer-inclusion} \Br C\hookrightarrow\Br k(C)$$ sending an Azumaya algebra to the central simple algebra at the generic point of $C$. In the special case of $C=\mathbb{P}_k^1$ we moreover have that $\Br(\mathbb{P}_k^1)\cong\Br(k)$. If the class of the central simple $k(C)$algebra $\mathcal{A}_\eta$ in the Brauer group $\Br(k(C))$ actually comes from $\Br(C)$ in the inclusion  we say that it is *unramified*. Because $C$ is nonsingular of dimension 1 we have that every maximal order in the unramified central simple algebra $\mathcal{A}_\eta$ is actually an Azumaya algebra [@MR3461057; @MR0121392], and we can describe the Chow groups up to *controlled* torsion. The situation of \[corollary:quasiprojective-curve-matrix\] is a special case of this where the Azumaya algebra is split, where $n=1$. \[corollary:quasiprojective-curve-azumaya\] Let $\mathcal{A}$ be an hereditary order as in \[theorem:quasiprojective-curve\] such that $\mathcal{A}_\eta$ is an unramified central simple $k(C)$algebra, and denote $\rho=\sum(e_i-1)$. Let $n$ be the degree of $\mathcal{A}_\eta$ over $k(C)$. Then $$\CH_0^\Delta(C,\mathcal{A})\otimes_{\mathbb{Z}}\mathbb{Z}[1/n]\cong\left( \Pic C\oplus\mathbb{Z}^{\oplus\rho} \right)\otimes_{\mathbb{Z}}\mathbb{Z}[1/n].$$ Denote by $\overline{\mathcal{A}}$ any maximal order containing $\mathcal{A}$. By the assumptions it is necessarily an Azumaya algebra. Using [@MR3056551 corollary 1.2] we have that there exists an isomorphism $$\label{equation:CH-maximal-up-to-torsion} \Knought(C)\otimes_{\mathbb{Z}}\mathbb{Z}[1/n]\cong\Knought(\overline{\mathcal{A}})\otimes_{\mathbb{Z}}\mathbb{Z}[1/n],$$ and by \[theorem:quasiprojective-curve\] we can conclude. In this case op. cit. gives that the map induced on $\Knought$ by $-\otimes_{\mathcal{O}_C}\mathcal{A}$ has torsion (co)kernel of exponent $n^4$. #### Maximal orders on surfaces There is another invariant in the literature which is a special case of relative Chow groups for orders [@artin-dejong §3.7]. In op. cit. these are defined for a (terminal) maximal order $\mathcal{A}$ on a (smooth) projective surface $X$ over an algebraically closed field $k$. Here we don’t need a precise definition of a terminal maximal order, only that it has finite global dimension [@artin-dejong corollary 3.3.5]. Using this, we have that the filtration obtained by the tensor action is the same as the filtration by dimension of support \[corverdiervsserre\] on the abelian level, which is precisely the filtration used in op. cit. They define a divisor group for $\mathcal{A}$, and as the filtrations are the same we see that $$\mathrm{Div}(\mathcal{A})\cong\Cyc_1^\Delta(X,\mathcal{A}).$$ Moreover, they define a group $\mathrm{G}_1(\mathcal{A})$ (not to be confused with higher K-theory of coherent sheaves), using the localization sequence , as the two-dimensional analogue of the reduced projective class group. In particular, combining and [@artin-dejong proposition 3.7.8] we have that $$\mathrm{G}_1(\mathcal{A})\cong\CH_1^\Delta(X,\mathcal{A}).$$ Moreover, in [@artin-dejong proposition 3.7.12] an explicit description of $\mathrm{G}_1(\mathcal{A})$ (and hence the codimension-one Chow group) is given in their situation as $$0\to k(X)^\times/\det D^\times\to\CH_1^\Delta(X,\mathcal{A})\to\Pic X\to 0$$ where $D$ is the division algebra over $k(X)$ Morita equivalent to $\mathcal{A}_\eta$. A point not addressed here is the relationship between relative tensor triangular Chow groups for hereditary orders on smooth quasiprojective curves and various Chow groups for “orbifold curves”. By [@MR2018958] there exists a correspondence between these when working over an algebraically closed base field of characteristic zero. Observe that by [@MR1005008] the Chow groups of the orbifold curve are (up to torsion) the same as the Chow groups of the coarse moduli space. Hence the relative tensor triangular Chow groups of an hereditary order on a smooth quasiprojective curve are different from the Chow groups of its associated orbifold curve, because the stackiness shows up as copies of $\mathbb{Z}$ and not as torsion. This raises at least two questions: 1. is there a purely commutative (relative) setup that recovers the relative Chow groups of the order from the orbifold curve? 2. is there an analogue of [@MR3423452] identifying the Chow group defined by Vistoli with the tensor triangular Chow group of its derived category? Chow groups of (integral) group rings {#subsection:group-rings} ------------------------------------- In this section we consider the situation where the scheme $X$ is $\Spec R$ for a Dedekind domain $R$, and the coherent $\mathcal{O}_X$algebra is given by (the sheafification of) the integral group ring $RG$, for a finite group $G$ of order $n$. Observe that in this situation the global dimension of $RG$ is often infinite. Especially the case where $R$ is the ring of integers in an algebraic number field is interesting, where it combines the representation theory of finite groups and algebraic number theory. As in \[subsection:chow-groups\] we obtain that we can express in the relative tensor triangular Chow groups in terms of classical invariants, see \[theorem:degree-zero-integral-group-ring\]. If we denote $K$ the field of fractions of $R$, then we will relax \[definition:order\] by allowing $KG$ to be a separable $K$algebra. By Maschke’s theorem this will be the case if the characteristic of $K$ does not divide $n$ and $K$ is a perfect field. We will assume this throughout, and it is of course satisfied in the case where $K$ is an algebraic number field. By the Artin–Wedderburn decomposition theorem we have that $KG$ has a direct product decomposition $$KG\cong\prod_{i=1}^t\Mat_{n_i}(D_i)$$ whose factors are matrix rings over division rings over $K$. In particular we allow the conditions in \[definition:order\] to be relaxed in two directions: we can have multiple factors, and the division algebras can have centers which are larger than $K$. This allows us to describe the top degree cycle and Chow groups. \[theorem:top-chow-group-group-ring\] Let $R$ be a Dedekind domain such that $RG$ defines an order in $KG$. Then $$\Cyc_1^\Delta(R,RG)\cong\CH_1^\Delta(R,RG)\cong\mathbb{Z}^{\oplus t}$$ where $t$ is the number of simple factors in the Artin–Wedderburn decomposition of $KG$. This is a straightforward generalisation of \[proposition:highest-cycle-group,corollary:highest-chow-group\], taking the more general notion of order into account. An easy example of the dependence on the field of fractions is given by considering the group rings $\mathbb{Z}\mathrm{Cyc}_p$ and $\mathbb{Z}[\zeta_p]\mathrm{Cyc}_p$, for a cyclic group of prime order $p\geq 3$, where $\zeta_p$ is a primitive $p$th root of unity. \[example:cyclic-group-rings\] We have that $\mathbb{Q}\mathrm{Cyc}_p\cong\mathbb{Q}\times\mathbb{Q}(\zeta_p)$, so $$\label{equation:chow-group-cyclic-group} \CH_1^\Delta(\mathbb{Z},\mathbb{Z}\mathrm{Cyc}_p)\cong\mathbb{Z}^{\oplus2}.$$ On the other hand $\mathbb{Q}(\zeta_p)\mathrm{Cyc}_p\cong\prod_{i=0}^{p-1}\mathbb{Q}(\zeta_p)$, hence $$\CH_1^\Delta(\mathbb{Z}[\zeta_p],\mathbb{Z}[\zeta_p]\mathrm{Cyc}_p)\cong\mathbb{Z}^{\oplus p}.$$ More generally we have that the integral group ring $\mathbb{Z}G$ considered as a sheaf of algebras over $\Spec\mathbb{Z}$ has highest Chow group $$\CH_1^\Delta(\mathbb{Z},\mathbb{Z}G)\cong\mathbb{Z}^t$$ where $t$ is the number of conjugacy classes of cyclic subgroups of $G$ [@MR0450380 corollary 13.1.2]. For the zero-dimensional Chow groups we obtain a result similar to \[corollary:reduced-projective-class-group-affine\]. We will not cover the zero-dimensional cycle groups explicitly: there is no uniform description possible but the techniques of \[theorem:top-chow-group-group-ring\] go through. \[theorem:degree-zero-integral-group-ring\] Let $R$ be a Dedekind domain such that $RG$ defines an order in $KG$. Then $$\CH_0^\Delta(R,RG)\cong\widetilde{\Knought}(RG)\cong\Cl RG.$$ The first isomorphism follows from \[propchowexseq\]. The second isomorphism is [@MR892316 remarks 49.11(iv)]. The second isomorphism is indeed somewhat special to the situation of group rings: for an hereditary order $\Lambda$ we had that $\Cl\Lambda\cong\Cl\Lambda'$ if $\Lambda\subseteq\Lambda'$ is an inclusion of orders, reducing the computation of the class group to that of a maximal order. To compute the class group of a group ring, observe that $RG$ is maximal if and only if it is hereditary, which happens if and only if $n\in R^\times$ [@MR0393100 theorem 41.1]. Moreover, the inclusion of $RG$ into a maximal order $\Lambda'$ usually only induces an epimorphism of class groups. In particular one obtains a short exact sequence $$0\to\mathrm{D}(RG)\to\Cl(RG)\cong\widetilde{K}_0(RG)\to\Cl(\Lambda')\to 0$$ as in [@MR892316 (49.33)], independent of the choice of $\Lambda'$. In the case where $R$ is the ring of integers in an algebraic number field, we get by the Jordan–Zassenhaus theorem that $\Cl RG$ (and therefore $\CH_0^\Delta(R,RG)$) is a finite abelian group, generalising the theory of class groups and class numbers of $R$ to the situation of group rings. This is significantly different from the situation for hereditary orders, where the inclusion in a maximal order was responsible for copies $\mathbb{Z}$ in the Chow groups. More information and some explicit expressions can be found in [@MR0175935; @MR0404410]. To end this discussion we give some examples of explicit computations of $\Cl\mathbb{Z}G$. If one considers the situation of \[example:cyclic-group-rings\], then the (necessarily unique) maximal order in $\mathbb{Q}\times\mathbb{Q}(\zeta_p)$ is $\mathbb{Z}\times\mathbb{Z}[\zeta_p]$, and we [@MR892316 theorem 50.2] we obtain the following $$\CH_0^\Delta(\mathbb{Z},\mathbb{Z}\mathrm{Cyc}_p)\cong\Cl(\mathbb{Z}[\zeta_p]).$$ The order of this group is the class number of the cyclotomic field $\mathbb{Q}(\zeta_p)$. For example if $p=23$ then $\CH_0^\Delta(\mathbb{Z},\mathbb{Z}\mathrm{Cyc}_{23})\cong\mathbb{Z}/3\mathbb{Z}$. Using the class numbers of cyclotomic fields it is possible to give a complete classification of the finite abelian groups for which $\Cl(\mathbb{Z}G)$ (and therefore $\CH_0^\Delta(\mathbb{Z},\mathbb{Z}G)$) is zero: by [@MR892316 corollary 50.17] this is only the case if $G$ is cyclic of order $\leq 11$, cyclic of order $13,14,17,19$ or the Klein group of order 4. Chow groups in the singular case -------------------------------- Finally we discuss a single example where the base is singular, but the order is a noncommutative resolution and in particular has finite global dimension. Observe that this case is covered by the general results in \[subsection:main-result\]. By no means is this a complete discussion, it is given to suggest possible future research. We will work in the setting of [@MR2854109 remark 2.7]. Consider $$\begin{aligned} R_1\coloneqq k[[x,y]]/(xy), R_2\coloneqq k[[x,y]]/(y^2-x^3) \end{aligned}$$ which are the complete local rings for the nodal (resp. cuspidal) curve singularity, with maximal ideals $\mathfrak{m}_i$. Denote their normalizations by $\widetilde{R}_i$. Then the *Auslander order* is introduced in op. cit., and it is given by $$A_i\coloneqq \begin{pmatrix} \widetilde{R}_i & \mathfrak{m}_i \\ \widetilde{R}_i & R_i \end{pmatrix}$$ It can be seen that these orders have 3 (resp. 2) simple modules, in particular we get the following description of the cycle groups in dimension 0 $$\begin{aligned} \Knought(A_1\mhyphen\mathrm{fl})&\cong\mathbb{Z}^{\oplus3}, \\ \Knought(A_2\mhyphen\mathrm{fl})&\cong\mathbb{Z}^{\oplus2}. \end{aligned}$$ [^1]: `[email protected]`; Universiteit Antwerpen, Middelheimlaan 1, Antwerpen [^2]: `[email protected]`; Universiteit Antwerpen, Middelheimlaan 1, Antwerpen [^3]: This last condition implies that $\mathcal{O}_X$ acts centrally on $\mathcal{A}$.

--- abstract: 'This paper deals with the impact of fault prediction techniques on checkpointing strategies. We extend the classical analysis of Young and Daly in the presence of a fault prediction system, which is characterized by its recall and its precision, and which provides either exact or window-based time predictions. We succeed in deriving the optimal value of the checkpointing period (thereby minimizing the waste of resource usage due to checkpoint overhead) in all scenarios. These results allow to analytically assess the key parameters that impact the performance of fault predictors at very large scale. In addition, the results of this analytical evaluation are nicely corroborated by a comprehensive set of simulations, thereby demonstrating the validity of the model and the accuracy of the results.' author: - | Guillaume Aupy$^{1}$,Yves Robert$^{1,2}$, Frédéric Vivien$^{1}$ and Dounia Zaidouni$^{1}$\ $1.$ Ecole Normale Supérieure de Lyon & INRIA, France\ [{Guillaume.Aupy | Yves.Robert | Frederic.Vivien | Dounia.Zaidouni}@ens-lyon.fr]({Guillaume.Aupy | Yves.Robert | Frederic.Vivien | Dounia.Zaidouni}@ens-lyon.fr)\ $2.$ University of Tennessee Knoxville, USA bibliography: - 'biblio.bib' title: Impact of fault prediction on checkpointing strategies --- Introduction {#sec.intro} ============ In this paper, we assess the impact of fault prediction techniques on checkpointing strategies. We assume to have jobs executing on a platform subject to faults, and we let $\mu$ be the mean time between faults (MTBF) of the platform. In the absence of fault prediction, the standard approach is to take periodic checkpoints, each of length [C]{}, every period of duration [$T$]{}. In steady-state utilization of the platform, the value [$T_{\text{opt}}$]{}of [$T$]{}that minimizes the (expectation of the) waste of resource usage due to checkpointing is easily approximated as ${\ensuremath{T_{\text{opt}}}\xspace}= \sqrt{2 \mu{C\xspace}}$, or ${\ensuremath{T_{\text{opt}}}\xspace}= \sqrt{2 (\mu +{R\xspace}){C\xspace}}$ (where [R]{}is the duration of the recovery). The former expression is the well-known Young’s formula [@young74], while the latter is due to Daly [@daly04]. Now, when some fault prediction mechanism is available, can we compute a better checkpointing period to decrease the expected waste? and to what extent? Critical parameters that characterize a fault prediction system are its recall [$r$]{}, which is the fraction of faults that are indeed predicted, and its precision [$p$]{}, which is the fraction of predictions that are correct (i.e., correspond to actual faults). The major objective of this paper is to refine the expression of the expected waste as a function of these new parameters, and to design efficient checkpointing policies that take predictions into account. We deal with two problem instances, one where the predictor system provides exact dates for predicted events, and another where it only provides time windows during which events take place. The key contributions of this paper are the following: (i) The design of several checkpointing policies, their analysis, and a new formula for the checkpointing period that extends Young’s and Daly’s to take predictions into account; (ii) The analytical characterization of the best policy for each set of parameters; (iii) The validation of the theoretical results via extensive simulations, for both Exponential and Weibull failure distributions; (iv) The demonstration that even a poor predictor can lead to a significant reduction of application execution time; and (v) The demonstration that recall is far more important than precision, hence giving insight into the design of future predictors. The rest of the paper is organized as follows. We first detail the framework in Section \[sec.framework\]. We deal with exact date predictions in Section \[sec.no.intervals\], and with time-window based predictions in Section \[sec.intervals\]. Section \[sec.simulations\] is devoted to simulations. Finally, we provide concluding remarks in Section \[sec.conclusion\]. Framework {#sec.framework} ========= Checkpointing strategy ---------------------- We consider a *platform* subject to faults. Our work is agnostic of the granularity of the platform, which may consist either of a single processor, or of several processors that work concurrently and use coordinated checkpointing. The key parameter is $\mu$, the mean time between faults (MTBF) of the platform. If the platform is made of $N$ components whose individual MTBF is $\mu_{ind}$, then $\mu = \frac{\mu_{ind}}{N}$. Checkpoints are taken at regular intervals, or periods, of length [$T$]{}. We use [C]{}, [D]{}, and [R]{}for the duration of the checkpoint, downtime and recovery (respectively). We must enforce that ${C\xspace}\leq {\ensuremath{T}\xspace}$, and useful work is done only during ${\ensuremath{T}\xspace}-{C\xspace}$ units of time for every period of length [$T$]{}, if no fault occurs. The *waste* due to checkpointing in a fault-free execution is ${\ensuremath{\textsc{Waste}}\xspace}= \frac{{C\xspace}}{{\ensuremath{T}\xspace}}$. In the following, the *waste* always denote the fraction of time that the platform is not doing useful work. Fault predictor --------------- A fault predictor is a mechanism that is able to predict that some faults will take place, either at a certain point in time, or within some time-interval window. The accuracy of the fault predictor is characterized by two quantities, the *recall* and the *precision*. The recall [$r$]{}is the fraction of faults that are predicted while the precision [$p$]{}is the fraction of fault predictions that are correct. Traditionally, one defines three types of *events*: (i) *True positive* events are faults that the predictor has been able to predict (let $\textit{True}_P$ be their number); (ii) *False positive* events are fault predictions that did not materialize as actual faults (let $\textit{False}_P$ be their number); and (iii) *False negative* events are faults that were not predicted (let $\textit{False}_N$ be their number). With these definitions, we have ${\ensuremath{r}\xspace}= \frac{\textit{True}_P}{\textit{True}_P+\textit{False}_N}$ and ${p\xspace}= \frac{\textit{True}_P}{\textit{True}_P+ \textit{False}_P}$. Fault rates ----------- In addition to $\mu$, the platform MTBF, let ${\ensuremath{\mu_{P}}\xspace}$ be the mean time between predicted events (both true positive and false positive), and let ${\ensuremath{\mu_{NP}}\xspace}$ be the mean time between unpredicted faults (false negative). Finally, we define the mean time between events as ${\ensuremath{\mu_e}\xspace}$ (including all three event types). The relationships between $\mu$, ${\ensuremath{\mu_{P}}\xspace}$, ${\ensuremath{\mu_{NP}}\xspace}$, and ${\ensuremath{\mu_e}\xspace}$ are the following: - $\frac{{1-{\ensuremath{r}\xspace}}}{\mu} = \frac{1}{{\ensuremath{\mu_{NP}}\xspace}}$ (here, $1-{\ensuremath{r}\xspace}$ is the fraction of faults that are unpredicted); - $ \frac{{\ensuremath{r}\xspace}}{\mu} = \frac{{\ensuremath{p}\xspace}}{{\ensuremath{\mu_{P}}\xspace}}$ (here, ${\ensuremath{r}\xspace}$ is the fraction of faults that are predicted, and ${\ensuremath{p}\xspace}$ is the fraction of fault predictions that are correct); - $\frac{1}{{\ensuremath{\mu_e}\xspace}}=\frac{1}{{\ensuremath{\mu_{P}}\xspace}}+\frac{1}{{\ensuremath{\mu_{NP}}\xspace}}$ (here, events are either predicted (true or false), or not). Predictor with exact event dates {#sec.no.intervals} ================================ In this section, we present an analytical model to assess the impact of prediction on periodic checkpointing strategies. We consider the case where the predictor is able to provide exact prediction dates, and to generate such predictions at least ${C\xspace}$ seconds in advance, so that a checkpoint can indeed be taken before the event (otherwise the prediction cannot be used, because there is not enough time to take proactive actions). We consider the following algorithm:\ (1) While no fault prediction is available, checkpoints are taken periodically with period ${\ensuremath{T}\xspace}$;\ (2) When a fault is predicted, we decide whether to take the prediction into account or not. This decision is randomly taken: with probability [$q$]{}, we trust the predictor and take the prediction into account, and, with probability $1-{\ensuremath{q}\xspace}$, we ignore the prediction. If we take the prediction into account, there are two cases. If we have enough time before the prediction date, we take a checkpoint as late as possible, i.e., so that it completes right at the time where the fault is predicted to happen. After the checkpoint, we then complete the execution of the period (see Figure \[fig.enoughtime\](a)). Otherwise, if we do not have enough time to take an extra checkpoint (because we are already checkpointing), then we do some extra work during $\varepsilon$ seconds (see Figure \[fig.no\_enoughtime\](b)). We account for this work as idle time in the expression of the waste, to ease the analysis. Our expression of the waste is thus an upper bound. The rationale for not always trusting the predictor is to avoid taking useless checkpoints too frequently. Intuitively, the precision ${\ensuremath{p}\xspace}$ of the predictor must be above a given threshold for its usage to be worthwhile. In other words, if we decide to checkpoint just before a predicted event, either we will save time by avoiding a costly re-execution if the event does correspond to an actual fault, or we will lose time by unduly performing an extra checkpoint. We need a larger proportion of the former cases, i.e., a good precision, for the predictor to be really useful. The following analysis will determine the optimal value of ${\ensuremath{q}\xspace}$ as a function of the parameters ${C\xspace}$, $\mu$, ${\ensuremath{r}\xspace}$, and ${\ensuremath{p}\xspace}$. \[fig.no\_enoughtime\] Computing the waste {#sec.nointalg} ------------------- Our goal in this section is to compute a formula for the expected waste. Recall that the waste is the fraction of time that the processors do not perform useful computations, either because they are checkpointing, or because a failure has struck. There are four different sources of waste (see Figure \[fig.waste-exact\]):\ (1) **Checkpoints:** During a fault-free execution, the fraction of resources used in checkpointing is $ \frac{{C\xspace}}{{T}}$.\ (2) **Unpredicted faults:** This overhead occurs each time a unpredicted fault strikes, that is, on average, once every ${\ensuremath{\mu_{NP}}\xspace}$ seconds. The time wasted because of the unpredicted fault is then the time elapsed between the last checkpoint and the fault, plus the downtime and the time needed for the recovery. The expectation of the time elapsed between the last checkpoint and the fault is equal to half the period of checkpoints, because the time where the fault hits the system is independent of the checkpointing algorithm. Finally, the waste due to unpredicted faults is: $ \frac{1}{{\ensuremath{\mu_{NP}}\xspace}} \left[ \frac{{T}}{2} + {D\xspace}+ {R\xspace}\right]$.\ (3) **Predictions taken into account:** Now we have to compute the execution overhead due to a prediction which we trust (hence we checkpoint just before its date). This overhead occurs each time a prediction is made by the predictor, that is, on average, once every ${\ensuremath{\mu_{P}}\xspace}$ seconds, and that we decide to trust it, with probability ${\ensuremath{q}\xspace}$. If the predicted event is an actual fault, we waste ${C\xspace}+{D\xspace}+{R\xspace}$ seconds: we waste ${D\xspace}+ {R\xspace}$ seconds because the predicted event corresponds to an actual fault, and if we have enough time before the prediction date, we waste ${C\xspace}$ seconds because we take an extra checkpoint as late as possible before the prediction date (see Figure \[fig.enoughtime\](a)). Note that if we do not have enough time to take an extra checkpoint (see Figure \[fig.no\_enoughtime\](b)), we overestimate the waste as ${C\xspace}$ seconds. If the predicted event is not an actual fault, we waste ${C\xspace}$ seconds. An actual fault occurs with probability ${\ensuremath{p}\xspace}$, and a false prediction is made with probability $(1-{\ensuremath{p}\xspace})$. Averaging with these probabilities, we waste an expected amount of $\left [ {p\xspace}({C\xspace}+ {D\xspace}+ {R\xspace}) + (1-{p\xspace}) {C\xspace}\right] $ seconds. Finally, the corresponding overhead is $\frac{1}{{\ensuremath{\mu_{P}}\xspace}} {\ensuremath{q}\xspace}\left [ {p\xspace}({C\xspace}+ {D\xspace}+ {R\xspace}) + (1-p) {C\xspace}\right]$.\ (4) **Ignored predictions:** The final source of waste is for predictions that we do not trust. This overhead occurs each time a prediction is made by the predictor, that is, on average, once every ${\ensuremath{\mu_{P}}\xspace}$ seconds, and that we decide to trust it, with probability $1-{\ensuremath{q}\xspace}$. If the predicted event corresponds to an actual fault, we waste $(\frac{{T}}{2} +{D\xspace}+ {R\xspace})$ seconds (as for an unpredicted fault). Otherwise there is no fault and we took no extra checkpoint, and thus we lose nothing. An actual fault occurs with a probability [$p$]{}. The corresponding overhead is $\frac{1}{{\ensuremath{\mu_{P}}\xspace}} (1-{\ensuremath{q}\xspace}) \left [ {p\xspace}(\frac{{T}}{2} + {D\xspace}+ {R\xspace}) + (1-{p\xspace}) 0 \right] $.\ Summing up the overhead over the four different sources, and after simplification, we obtain the following equation for the waste: $${\ensuremath{\textsc{Waste}}\xspace}= \frac{{C\xspace}}{{T}} + \frac{1}{\mu} \left[ (1- {\ensuremath{r}\xspace}{\ensuremath{q}\xspace}) \frac{{T}}{2} + {D\xspace}+ {R\xspace}+ \frac{{\ensuremath{q}\xspace}{\ensuremath{r}\xspace}}{{p\xspace}} {C\xspace}\right] \label{eq.waste}$$ Validity of the analysis {#sec.validity} ------------------------ Equation (\[eq.waste\]) is accurate only when two events (an event being a prediction (true or false) or an unpredicted fault) do not take place within the same period. To ensure that this condition is met with a high probability, we bound the length of the period: without predictions, or when predictions are not taken into account, we enforce the condition ${T}< \alpha \mu$; otherwise, with predictions, we enforce the condition ${T}< \alpha {\ensuremath{\mu_e}\xspace}$. Here, $\alpha$ is some tuning parameter chosen as follows. The number of events during a period of length ${T}$ can be modeled as a Poisson process of parameter $\beta = \frac{{T}}{\mu}$ (without prediction) or $\beta = \frac{{T}}{{\ensuremath{\mu_e}\xspace}}$ (with prediction). The probability of having $k \geq 0$ faults is $P(X=k) = \frac{\beta^{k}}{k!} e^{-\beta}$, where $X$ is the number of faults. Hence the probability of having two or more faults is $\pi = P(X\geq2) = 1 -( P(X=0) + P(X=1)) = 1 - (1+\beta) e^{-\beta}$. If we assume $\alpha=0.27$ then $\pi \leq 0.03$, hence a valid approximation when bounding the period range accordingly. Indeed, with such a conservative value for $\alpha$, we have overlapping faults for only $3\%$ of the checkpointing segments in average, so that the model is quite reliable. In addition to the previous constraint, we must always enforce the condition ${C\xspace}\leq {T}$, by construction of the periodic checkpointing policy. Finally, the optimal waste may never exceed $1$; when the waste is equal to $1$, the application no longer makes any progress. Waste minimization {#sec.minwaste} ------------------ We differentiate twice Equation  with respect to [T]{}: $${\ensuremath{\textsc{Waste}}\xspace}'({T}) = \frac{-{C\xspace}}{{T}^{2}} + \frac{1}{\mu} \left[ (1- {\ensuremath{r}\xspace}{\ensuremath{q}\xspace}) \frac{1}{2}\right]$$ $${\ensuremath{\textsc{Waste}}\xspace}''({T}) = \frac{2 {C\xspace}}{{T}^{3} } > 0$$ We obtain that ${\ensuremath{\textsc{Waste}}\xspace}''({T}) $ is strictly positive, hence ${\ensuremath{\textsc{Waste}}\xspace}({T}) $ is a convex function of ${T}$ and admits a unique minimum on its domain. We also compute ${\ensuremath{T_{{\ensuremath{\text{extr}}}}^{\{q\}}}\xspace}$, the extremum value of ${T}$ that is the unique zero of the function ${\ensuremath{\textsc{Waste}}\xspace}'({T})$, as ${\ensuremath{T_{{\ensuremath{\text{extr}}}}^{\{q\}}}\xspace}=\sqrt{ \frac{2 \mu {C\xspace}}{1-{\ensuremath{r}\xspace}{\ensuremath{q}\xspace}}}$. Note that this Equation makes sense even when $1-{\ensuremath{r}\xspace}{\ensuremath{q}\xspace}=0$. Indeed this would mean that both ${\ensuremath{r}\xspace}=1$ and ${\ensuremath{q}\xspace}=1$: the predictor predicts every fault, and we take proactive action for each one of them, there should never be any periodic checkpointing! Finally, note that ${\ensuremath{T_{{\ensuremath{\text{extr}}}}^{\{q\}}}\xspace}$ may well not belong to the admissible domain $[{C\xspace}, \alpha {\ensuremath{\mu_e}\xspace}]$. The optimal waste ${\ensuremath{\textsc{Waste}_{\text{opt}}}\xspace}$ is determined via the following case analysis. We rewrite the waste as an affine function of ${\ensuremath{q}\xspace}$: $${\ensuremath{\textsc{Waste}}\xspace}({\ensuremath{q}\xspace}) = \frac{{\ensuremath{r}\xspace}{\ensuremath{q}\xspace}}{\mu}\left (\frac{{C\xspace}}{{p\xspace}}-\frac{{T}}{2}\right )+\left ( \frac{{C\xspace}}{{T}}+\frac{{T}}{2 \mu}+\frac{{D\xspace}+ {R\xspace}}{\mu} \right )$$ For any value of [T]{}, we deduce that ${\ensuremath{\textsc{Waste}}\xspace}({\ensuremath{q}\xspace})$ is minimized either for ${\ensuremath{q}\xspace}=0$ or for ${\ensuremath{q}\xspace}=1$. This (somewhat unexpected) conclusion is that the predictor should sometimes be always trusted, and sometimes never, but no in-between value for ${\ensuremath{q}\xspace}$ will do a better job. Thus we need to minimize the two functions ${\ensuremath{\textsc{Waste}}\xspace}^{\{0\}}$ and ${\ensuremath{\textsc{Waste}}\xspace}^{\{1\}}$ over the domain of admissible values for [T]{}, and to retain the best result. We have ${\ensuremath{\textsc{Waste}}\xspace}^{\{0\}}(T)= \frac{{C\xspace}}{{T}} + \frac{1}{\mu} \left[ \frac{{T}}{2} + {D\xspace}+ {R\xspace}\right]$. We recognize here the waste function of Young [@young74] and write ${\ensuremath{\textsc{Waste}_Y}\xspace}= \frac{{C\xspace}}{{T}} + \frac{1}{\mu} \left[ \frac{{T}}{2} + {D\xspace}+ {R\xspace}\right]$. The function ${\ensuremath{\textsc{Waste}_Y}\xspace}(T)$ is a convex function and reaches its minimum for ${\ensuremath{T_{\text{Y}}}\xspace}$ in the interval $[{C\xspace},\alpha \mu]$: - If (${C\xspace}<{\ensuremath{T_{{\ensuremath{\text{extr}}}}^{\{0\}}}\xspace}<\alpha \mu$): ${\ensuremath{T_{\text{Y}}}\xspace}={\ensuremath{T_{{\ensuremath{\text{extr}}}}^{\{0\}}}\xspace}=\sqrt{2 \mu {C\xspace}}$ - If (${\ensuremath{T_{{\ensuremath{\text{extr}}}}^{\{0\}}}\xspace}<{C\xspace}$): ${\ensuremath{T_{\text{Y}}}\xspace}={C\xspace}$ - If (${\ensuremath{T_{{\ensuremath{\text{extr}}}}^{\{0\}}}\xspace} \geq \alpha \mu$): ${\ensuremath{T_{\text{Y}}}\xspace}=\alpha \mu$ Thus, [$\textsc{Waste}_Y$]{}($ = {\ensuremath{\textsc{Waste}}\xspace}^{\{0\}}$) is minimized for: $${\ensuremath{T_{\text{Y}}}\xspace}=\min \left ( \alpha \mu,\max(\sqrt{2 \mu {C\xspace}}, {C\xspace}) \right )$$ Similarly, we have: ${\ensuremath{\textsc{Waste}}\xspace}^{\{1\}}({T})=\frac{{C\xspace}}{{T}} + \frac{1}{\mu} \left[ (1- {\ensuremath{r}\xspace}) \frac{{T}}{2} + {D\xspace}+ {R\xspace}+ \frac{{\ensuremath{r}\xspace}}{{p\xspace}} {C\xspace}\right] $. The function ${\ensuremath{\textsc{Waste}}\xspace}^{\{1\}}(T)$ is a convex function and reaches its minimum for [$T_{1} $]{} in the interval $[{C\xspace},\alpha {\ensuremath{\mu_e}\xspace}]$. - If (${C\xspace}<{\ensuremath{T_{{\ensuremath{\text{extr}}}}^{\{1\}}}\xspace}<\alpha {\ensuremath{\mu_e}\xspace}$): ${\ensuremath{T_{1} }\xspace}={\ensuremath{T_{{\ensuremath{\text{extr}}}}^{\{1\}}}\xspace}=\sqrt{ \frac{2 \mu {C\xspace}}{1-{\ensuremath{r}\xspace}}}$ - If (${\ensuremath{T_{{\ensuremath{\text{extr}}}}^{\{1\}}}\xspace}<{C\xspace}$): ${\ensuremath{T_{1} }\xspace}={C\xspace}$ - If (${\ensuremath{T_{{\ensuremath{\text{extr}}}}^{\{1\}}}\xspace} \geq \alpha {\ensuremath{\mu_e}\xspace}$): ${\ensuremath{T_{1} }\xspace}=\alpha {\ensuremath{\mu_e}\xspace}$ Thus, ${\ensuremath{\textsc{Waste}}\xspace}^{\{1\}}$ is minimized for: $${\ensuremath{T_{1} }\xspace}=\min \left ( \alpha {\ensuremath{\mu_e}\xspace},\max(\sqrt{ \frac{2 \mu {C\xspace}}{1-{\ensuremath{r}\xspace}} }, {C\xspace}) \right )$$ Finally, the optimal waste is: $${\ensuremath{\textsc{Waste}_{\text{opt}}}\xspace}= \min \left ({\ensuremath{\textsc{Waste}_Y}\xspace}({\ensuremath{T_{\text{Y}}}\xspace}),{\ensuremath{\textsc{Waste}}\xspace}^{\{1\}}({\ensuremath{T_{1} }\xspace}) \right )$$ Prediction and preventive migration {#sec.migration} ----------------------------------- In this section, we make a short digression and briefly present an analytical model to assess the impact of prediction and preventive migration on periodic checkpointing strategies. As before, we consider a predictor that is able to predict exactly when faults happen, and to generate these predictions at least ${C\xspace}$ seconds before the event dates. The idea of migration consists in moving a task for execution on another node, when a fault is predicted to happen on the current node in the near future. Note that the faulty node can later be replaced, in case of a hardware fault, or software rejuvenation can be used in case of a software fault. We consider the following algorithm, which is very similar to that used in Section \[sec.nointalg\]: 1. When no fault prediction is available, checkpoints are taken periodically with period ${\ensuremath{T}\xspace}$. 2. When a fault is predicted, we decide whether to execute the migration or not. The decision is a random one: with probability [$q$]{}we trust the predictor and do the migration and, with probability 1-[$q$]{}, we ignore the prediction. If we take the prediction into account, we execute the migration as late as possible, so that it completes right at the time when the fault is predicted to happen. As before, we have four different sources of waste. Summing the overhead of the execution of these different sources, we obtain the following equation for the waste (where ${M}$ is the duration of a migration): $$\begin{aligned} {\ensuremath{\textsc{Waste}}\xspace}&= \frac{{C\xspace}}{{T}} \\&+ \frac{1}{{\ensuremath{\mu_{NP}}\xspace}} \left[ \frac{{T}}{2} + {D\xspace}+ {R\xspace}\right] \\&+ \frac{1}{{\ensuremath{\mu_{P}}\xspace}} {\ensuremath{q}\xspace}\left [ {p\xspace}({M}) + (1-p) {M}\right] \\&+ \frac{1}{{\ensuremath{\mu_{P}}\xspace}} (1-{\ensuremath{q}\xspace}) \left [ {p\xspace}(\frac{{T}}{2} + {D\xspace}+ {R\xspace}) + (1-{p\xspace}) 0 \right] $$ After simplification, we get:$${\ensuremath{\textsc{Waste}}\xspace}= \frac{{C\xspace}}{{T}} + \frac{1}{\mu} \left[ (1- {\ensuremath{r}\xspace}{\ensuremath{q}\xspace}) \left (\frac{{T}}{2}+{D\xspace}+ {R\xspace}\right ) + \frac{{\ensuremath{q}\xspace}{\ensuremath{r}\xspace}}{{p\xspace}} {M}\right] \label{eq.wasteM}$$ Equation  is very similar to Equation , and the minimization of the waste proceeds exactly as in Section \[sec.minwaste\]. In a nutshell, ${\ensuremath{\textsc{Waste}}\xspace}(T) $ is again a convex function and admits a unique minimum over its domain $[{C\xspace}, \alpha {\ensuremath{\mu_e}\xspace}]$, the unique zero of the derivative has the same value ${\ensuremath{T_{{\ensuremath{\text{extr}}}}^{\{q\}}}\xspace}=\sqrt{ \frac{2 \mu {C\xspace}}{1-{\ensuremath{r}\xspace}{\ensuremath{q}\xspace}}}$, and for any value of $T$, the waste is minimized for either ${\ensuremath{q}\xspace}=0$ or ${\ensuremath{q}\xspace}=1$. We conduct the very same case analysis as in Section \[sec.minwaste\]. Predictor with a prediction window {#sec.intervals} ================================== In the previous section, we supposed that the predictor was able to predict exactly when faults will strike. Here, we suppose (maybe more realistically) that the predictor gives a *prediction window*, that is an interval of time of length [$I$]{}during which the predicted fault is likely to happen. As before in Section \[sec.no.intervals\]: (i) We suppose that we have enough time to checkpoint before the beginning of the prediction window; and (ii) When a prediction is made, we enforce that the scheduling algorithm has the choice either to take or not to take this prediction into account, with probability [$q$]{}. We start with a description of the strategies that can be used, depending upon the (relative) length [$I$]{}of the prediction window. Let us define two *modes* for the scheduling algorithm:\ **Regular**: This is the mode used when no fault prediction is available, or when a prediction is available but we decide to ignore it (with probability $1-{\ensuremath{q}\xspace}$). In regular mode, we use periodic checkpointing with period [$T_{\text{R}}$]{}. Intuitively, [$T_{\text{R}}$]{}corresponds to the checkpointing period $T$ of Section \[sec.no.intervals\].\ **Proactive**: This is the mode used when a fault prediction is available and we decide to trust it, a decision taken with probability [$q$]{}. Consider such a trusted prediction made with the prediction window $[t_0,t_0+{\ensuremath{I}\xspace}]$. Several strategies can be envisioned:\ (1) [<span style="font-variant:small-caps;">Instant</span>]{}, for *Instantaneous–* The first strategy is to ignore the time-window and to execute the same algorithm as if the predictor had given an exact date prediction at time $t_{0}$. Just as described in Section \[sec.no.intervals\], the algorithm interrupts the current period (of scheduled length [$T_{\text{R}}$]{}), checkpoints during the interval $[t_{0}-C,t_{0}]$, and then returns to regular mode: at time $t_{0}$, it resumes the work needed to complete the interrupted period of the regular mode.\ (2) [<span style="font-variant:small-caps;">NoCkptI</span>]{}, for *No checkpoint during prediction window–* The second strategy is intended for a short prediction window: instead of ignoring it, we acknowledge it, but make the decision not to checkpoint during it. As in the first strategy, the algorithm interrupts the current period (of scheduled length [$T_{\text{R}}$]{}), and checkpoints during the interval $[t_{0}-C,t_{0}]$. But here, we return to regular mode only at time $t_0+{\ensuremath{I}\xspace}$, where we resume the work needed to complete the interrupted period of the regular mode. During the whole length of the time-window, we execute work without checkpointing, at the risk of losing work if a fault indeed strikes. But for a small value of [$I$]{}, it may not be worthwhile to checkpoint during the prediction window (if at all possible, since there is no choice if ${\ensuremath{I}\xspace}< C$).\ (3) [<span style="font-variant:small-caps;">WithCkptI</span>]{}, for *With checkpoints during prediction window–* The third strategy is intended for a longer prediction window and assumes that ${C\xspace}\leq {\ensuremath{I}\xspace}$: the algorithm interrupts the current period (of scheduled length [$T_{\text{R}}$]{}), and checkpoints during the interval $[t_{0}-C,t_{0}]$, but now decides to take several checkpoints during the prediction window. The period [$T_{\text{P}}$]{}of these checkpoints in proactive mode will presumably be shorter than [$T_{\text{R}}$]{}, to take into account the higher fault probability. To simplify the presentation, we use an integer number of periods of length [$T_{\text{P}}$]{} within the prediction window. In the following, we analytically compute the optimal number of such periods. But we take at least one period here, hence one checkpoint, which implies $C \leq I$. We return to regular mode either right after the fault strikes within the time window $[t_0,t_0+{\ensuremath{I}\xspace}]$, or at time $t_0+{\ensuremath{I}\xspace}$ if no actual fault happens within this window. Then, we resume the work needed to complete the interrupted period of the regular mode. The third strategy is the most complex to describe, and the complete behavior of the scheduling algorithm is shown in Algorithm \[algo.proactive\]. Note that for all strategies, exactly as in Section \[sec.no.intervals\], we insert some additional work for the particular case where there is not enough time to take a checkpoint before entering proactive mode (because a checkpoint for the regular mode is currently on-going, see Figure \[fig.no\_enoughtime\](b)). We account for this work as idle time in the expression of the waste, to ease the analysis. Our expression of the waste is thus an upper bound. Waste for strategy [<span style="font-variant:small-caps;">WithCkptI</span>]{} {#sec-waste-int} ------------------------------------------------------------------------------ In this section we focus on computing the waste of [<span style="font-variant:small-caps;">WithCkptI</span>]{}, the most complex strategy. We first compute the fraction of time spent in the *regular* mode (checkpointing with period [$T_{\text{R}}$]{}) and the fraction of time spent in the *proactive* mode (checkpointing with period [$T_{\text{P}}$]{}). Let [$I'$]{}be the average time spent in the *proactive* mode. When a prediction is made, we may choose to ignore it, which happens with probability $1-{\ensuremath{q}\xspace}$. In this case, the algorithm stays in regular mode and does not spend any time in the proactive mode. With probability [$q$]{}, we may decide to take the prediction into account. In this case, if the prediction is a false positive event (no actual fault strikes), which happens with probability $1-{\ensuremath{p}\xspace}$, then the algorithm spends [$I$]{}units of time in the proactive mode. Otherwise, if the prediction is a true positive event (an actual fault hits the system), which happens with probability ${\ensuremath{p}\xspace}$, then the algorithm spends an average of ${\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}$ in the proactive mode. Here ${\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}$ is the expectation of the time elapsed between the beginning of the prediction window and the time when a fault happens, knowing that a fault happens in the prediction window. Note that if faults are uniformly distributed across the prediction window, then ${\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}= \frac{{\ensuremath{I}\xspace}}{2}$. Altogether, we obtain $ {\ensuremath{I'}\xspace}= {\ensuremath{q}\xspace}\left((1-{\ensuremath{p}\xspace}){\ensuremath{I}\xspace}+ {\ensuremath{p}\xspace}{\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}\right)$. Each time there is a prediction, that is, on the average, every ${\ensuremath{\mu_{P}}\xspace}$ seconds, the algorithm spends a time ${\ensuremath{I'}\xspace}$ in the proactive mode. Therefore, Algorithm \[algo.proactive\] spends a fraction of time $\frac{{\ensuremath{I'}\xspace}}{{\ensuremath{\mu_{P}}\xspace}}$ in the proactive mode, and a fraction of time $1-\frac{{\ensuremath{I'}\xspace}}{{\ensuremath{\mu_{P}}\xspace}}$ in the regular mode. As in Section \[sec.no.intervals\], we assume that there is a single event of any type (either a prediction (true or false), or an unpredicted failure) within each interval under study. The condition $T \leq \alpha {\ensuremath{\mu_e}\xspace}$ then becomes ${\ensuremath{T_{\text{R}}}\xspace}+ {\ensuremath{I}\xspace}\leq \alpha {\ensuremath{\mu_e}\xspace}$, since ${\ensuremath{T_{\text{R}}}\xspace}+{\ensuremath{I}\xspace}$ is the longest time interval considered in the analysis of Algorithm \[algo.proactive\]. We now identify the four different sources of waste, and we analyze their respective costs:\ (1) **Waste due to periodic checkpointing.** There are two cases, depending upon the mode of Algorithm \[algo.proactive\]:\ (a) **Regular mode.** In this mode, we take periodic checkpoints. We take a checkpoint of size [C]{}each time the algorithm has processed work for a time ${\ensuremath{T_{\text{R}}}\xspace}-{C\xspace}$ in the regular mode. This remains true if, after spending some time in the regular mode, the algorithm switches to the proactive mode, and later switches back to the regular mode. This behavior is enforced by recording the amount of work performed under the regular mode (variable [$W_{\mathit{reg}}$]{}, at line \[algo.proactive.wreg\] of Algorithm \[algo.proactive\]), and by taking this value into account at line \[algo.proactive.completion\]. Given the fraction of time that Algorithm \[algo.proactive\] spends in the regular mode, this source of waste has a total cost of $\left(1 -\frac{{\ensuremath{I'}\xspace}}{{\ensuremath{\mu_{P}}\xspace}}\right)\frac{{C\xspace}}{{\ensuremath{T_{\text{R}}}\xspace}}$.\ (b) **Proactive mode.** In this mode, we take a checkpoint of size [C]{}each time the algorithm has processed work for a time ${\ensuremath{T_{\text{P}}}\xspace}-{C\xspace}$. If no fault happens while the algorithm is in the proactive mode, then the algorithm stays exactly a time [$I$]{}in this mode (thanks to the condition at line \[algo.proactive.Ilimit\]). The waste due to the periodic checkpointing is then exactly $\frac{{C\xspace}}{{\ensuremath{T_{\text{P}}}\xspace}}$ (because [$T_{\text{P}}$]{}divides [$I$]{}). If a fault happens while the algorithm is in proactive mode, then, the expectation of the waste due to the periodic checkpointing is upper-bounded by the same quantity $ \frac{{C\xspace}}{{\ensuremath{T_{\text{P}}}\xspace}}$ (this is an over-approximation of the waste in that case). Overall, taking into account the fraction of time Algorithm \[algo.proactive\] is in the proactive mode, the cost of this source of waste is $\frac{{\ensuremath{I'}\xspace}}{{\ensuremath{\mu_{P}}\xspace}}\frac{{C\xspace}}{{\ensuremath{T_{\text{P}}}\xspace}}$.\ (2) **Waste incurred when switching to the proactive mode.** Each time we take into account a prediction (which happens with probability [$q$]{}on average every [$\mu_{P}$]{}units of time), we start by doing one preliminary checkpoint if we have the time to do so (line \[algo.proactive.addC\]). If we do not have the time to take an additional checkpoint, the algorithm do not do any processing for a duration of at most [C]{} (line \[algo.proactive.wait\]). In both cases, the wasted time is at most [C]{}and this happens once every $\frac{{\ensuremath{\mu_{P}}\xspace}}{{\ensuremath{q}\xspace}}$. Hence, switching from the regular mode to the proactive one induces a waste of at most $\frac{{\ensuremath{q}\xspace}}{{\ensuremath{\mu_{P}}\xspace}}C$.\ (3) **Waste due to predicted faults.** Predicted faults happen with frequency $\frac{{\ensuremath{p}\xspace}}{{\ensuremath{\mu_{P}}\xspace}}$. As we may choose to ignore a prediction, there are still two cases depending on the mode of the algorithm at the time of the fault:\ (a) **Regular mode.** If the algorithm is in regular mode when a predicted fault hits, this means that we have chosen to ignore the prediction, a decision taken with probability $(1-{\ensuremath{q}\xspace})$. The time wasted because of the predicted fault is then the time elapsed between the last checkpoint and the fault, plus the downtime and the time needed for the recovery. The expectation of the time elapsed between the last checkpoint and the fault is equal to half the period of checkpoints, because the time where the fault hits the system is independent of the checkpointing algorithm. Therefore, the waste due to predicted faults hitting the system in regular mode is $\frac{{\ensuremath{p}\xspace}(1-{\ensuremath{q}\xspace})}{{\ensuremath{\mu_{P}}\xspace}}\left(\frac{{\ensuremath{T_{\text{R}}}\xspace}}{2}+{D\xspace}+{R\xspace}\right)$.\ (b) **Proactive mode.** If the algorithm is in proactive mode when a fault hits, then we have chosen to take the prediction into account, a decision that is taken with probability ${\ensuremath{q}\xspace}$. The time wasted because of the predicted fault is then, in addition to the downtime and the time needed for the recovery, the time elapsed between the last checkpoint and the fault or, if no checkpoint had already been taken in the proactive mode, the time elapsed between the start of the proactive mode and the fault. Here, we can no longer assume that the time the fault hits the system is independent of the checkpointing date. This is because the proactive mode starts exactly at the beginning of the prediction window. Let [$T_{\text{lost}}$]{}denote the computation time elapsed between the latest of the beginning of the proactive mode and the last checkpoint, and the fault date. Then the expectation of [$T_{\text{lost}}$]{}depends on the distribution of the fault date in the prediction window. However, we know that whatever the distribution, ${\ensuremath{T_{\text{lost}}}\xspace}\leq {\ensuremath{T_{\text{P}}}\xspace}$. Therefore we over approximate the waste in that case by $\frac{{\ensuremath{q}\xspace}{\ensuremath{p}\xspace}}{{\ensuremath{\mu_{P}}\xspace}}\left({\ensuremath{T_{\text{P}}}\xspace}+{D\xspace}+{R\xspace}\right)$.\ (4) **Waste due to unpredicted faults.** There are again two cases, depending upon the mode of the algorithm at the time the fault hits the system:\ (a) **Regular mode.** In this mode the work done is periodically checkpointed with period [$T_{\text{R}}$]{}. The time wasted because of an unpredicted fault is then the time elapsed between the last checkpoint and the fault, plus the downtime and the time needed for the recovery. As before, the expectation of this value is ${\ensuremath{T_{\text{lost}}}\xspace}= \frac{{\ensuremath{T_{\text{R}}}\xspace}}{2}$. An unexpected fault hits the system once every ${\ensuremath{\mu_{NP}}\xspace}$ seconds on the average. Taking into account the fraction of the time the algorithm is in regular mode, the waste due to unpredicted faults hitting the system in regular mode is $\left(1-\frac{{\ensuremath{I'}\xspace}}{{\ensuremath{\mu_{P}}\xspace}}\right)\frac{1}{{\ensuremath{\mu_{NP}}\xspace}}\left(\frac{{\ensuremath{T_{\text{R}}}\xspace}}{2}+{D\xspace}+{R\xspace}\right)$.\ (b) **Proactive mode.** Because of the assumption that a single event takes place within a time-interval, we do not consider the very unlikely case where a unpredicted fault strikes during a prediction window. This amounts to assume that $\frac{{\ensuremath{I'}\xspace}}{{\ensuremath{\mu_{P}}\xspace}}\frac{1}{{\ensuremath{\mu_{NP}}\xspace}}({\ensuremath{T_{\text{P}}}\xspace}+{D\xspace}+{R\xspace})$ is negligible. We gather the expressions of the six different types of waste and simplify to obtain the formula of the overall waste: $$\begin{aligned} {\ensuremath{\textsc{Waste}}\xspace}_{{\textsc{WithCkptI}\xspace}}&= \quad \left(\left (1 -\frac{{\ensuremath{I'}\xspace}}{{\ensuremath{\mu_{P}}\xspace}} \right )\frac{1}{{\ensuremath{T_{\text{R}}}\xspace}} + \frac{{\ensuremath{I'}\xspace}}{{\ensuremath{\mu_{P}}\xspace}}\frac{1}{{\ensuremath{T_{\text{P}}}\xspace}} + \frac{{\ensuremath{q}\xspace}}{{\ensuremath{\mu_{P}}\xspace}}\right){C\xspace}+ \frac{{\ensuremath{p}\xspace}(1-{\ensuremath{q}\xspace})}{{\ensuremath{\mu_{P}}\xspace}}\frac{{\ensuremath{T_{\text{R}}}\xspace}}{2} \nonumber \\ & + \frac{{\ensuremath{p}\xspace}{\ensuremath{q}\xspace}}{{\ensuremath{\mu_{P}}\xspace}} {\ensuremath{T_{\text{P}}}\xspace}+\left (1 -\frac{{\ensuremath{I'}\xspace}}{{\ensuremath{\mu_{P}}\xspace}} \right ) \frac{1}{{\ensuremath{\mu_{NP}}\xspace}} \frac{{\ensuremath{T_{\text{R}}}\xspace}}{2} \nonumber \\ & + \left(\frac{{\ensuremath{p}\xspace}}{{\ensuremath{\mu_{P}}\xspace}}+\left(1-\frac{{\ensuremath{I'}\xspace}}{{\ensuremath{\mu_{P}}\xspace}}\right)\frac{1}{{\ensuremath{\mu_{NP}}\xspace}}\right)\left({D\xspace}+{R\xspace}\right) \label{eq.proa.waste}\end{aligned}$$ Waste of the other strategies {#sec-waste-other} ----------------------------- The waste of the first strategy (*Instantaneous*) is very close to the one given in Equation . The difference lies in [$T_{\text{lost}}$]{}, the expectation of the work lost when a fault is predicted and the prediction is taken into account. When a prediction is taken into account and the predicted event is an actual fault, the waste in Equation  was $\frac{{\ensuremath{q}\xspace}{p\xspace}}{{\ensuremath{\mu_{P}}\xspace}}({C\xspace}+ {D\xspace}+ {R\xspace})$. Because the prediction was exact, [$T_{\text{lost}}$]{}was equal to 0. However in our new Equation, the waste for this part is now $\frac{{\ensuremath{q}\xspace}{p\xspace}}{{\ensuremath{\mu_{P}}\xspace}}({C\xspace}+ {\ensuremath{T_{\text{lost}}}\xspace}+ {D\xspace}+ {R\xspace})$. On average, the fault occurs after a time [$\mathbb{E}_{I}^{(f)}$]{}. However, because we do not know the relation between [$\mathbb{E}_{I}^{(f)}$]{}and [$T_{\text{R}}$]{}, then [$T_{\text{lost}}$]{}has expectation $\frac{{\ensuremath{T_{\text{R}}}\xspace}}{2}$ if $\frac{{\ensuremath{T_{\text{R}}}\xspace}}{2} \leq {\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}$. The new waste is then: $$\begin{aligned} {\ensuremath{\textsc{Waste}}\xspace}_{{\textsc{Instant}\xspace}} = \frac{{C\xspace}}{{\ensuremath{T_{\text{R}}}\xspace}} + \frac{1}{\mu} \left[ (1- {\ensuremath{r}\xspace}{\ensuremath{q}\xspace}) \frac{{\ensuremath{T_{\text{R}}}\xspace}}{2} + {D\xspace}+ {R\xspace}\right. \left. + \frac{{\ensuremath{q}\xspace}{\ensuremath{r}\xspace}}{{p\xspace}} {C\xspace}+{\ensuremath{q}\xspace}{\ensuremath{r}\xspace}\min \left ( {\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}, \frac{{\ensuremath{T_{\text{R}}}\xspace}}{2} \right) \right] \label{eq.waste-instant}\end{aligned}$$ As for the second strategy (*No checkpoint during prediction window*), we do no longer incur the waste due to checkpointing in proactive mode as we no longer checkpoint in proactive mode. Furthermore, the value of [$T_{\text{lost}}$]{}in proactive mode becomes [$\mathbb{E}_{I}^{(f)}$]{}instead of [$T_{\text{P}}$]{}. Consequently, the total waste when there is no checkpoint during the proactive mode is: $$\begin{aligned} {\ensuremath{\textsc{Waste}}\xspace}_{\text{noCkpt}} &=\left (1 -\frac{{\ensuremath{I'}\xspace}}{{\ensuremath{\mu_{P}}\xspace}} \right )\frac{{C\xspace}}{{\ensuremath{T_{\text{R}}}\xspace}} + \frac{{\ensuremath{q}\xspace}}{{\ensuremath{\mu_{P}}\xspace}}{C\xspace}+ \frac{{\ensuremath{p}\xspace}(1-{\ensuremath{q}\xspace})}{{\ensuremath{\mu_{P}}\xspace}}\left (\frac{{\ensuremath{T_{\text{R}}}\xspace}}{2} + {D\xspace}+{R\xspace}\right) \\ & + \frac{{\ensuremath{p}\xspace}{\ensuremath{q}\xspace}}{{\ensuremath{\mu_{P}}\xspace}} \left ({\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}+ {D\xspace}+{R\xspace}\right) +\left (1 -\frac{{\ensuremath{I'}\xspace}}{{\ensuremath{\mu_{P}}\xspace}} \right ) \frac{1}{{\ensuremath{\mu_{NP}}\xspace}} \left( \frac{{\ensuremath{T_{\text{R}}}\xspace}}{2} + {D\xspace}+{R\xspace}\right) \nonumber \\\end{aligned}$$ which we rewrite as $$\begin{aligned} {\ensuremath{\textsc{Waste}}\xspace}_{{\textsc{NoCkptI}\xspace}} &=\left(\left (1 -\frac{{\ensuremath{I'}\xspace}}{{\ensuremath{\mu_{P}}\xspace}} \right )\frac{1}{{\ensuremath{T_{\text{R}}}\xspace}} + \frac{{\ensuremath{q}\xspace}}{{\ensuremath{\mu_{P}}\xspace}}\right){C\xspace}+ \frac{{\ensuremath{p}\xspace}(1-{\ensuremath{q}\xspace})}{{\ensuremath{\mu_{P}}\xspace}}\frac{{\ensuremath{T_{\text{R}}}\xspace}}{2} \nonumber \\ & + \frac{{\ensuremath{p}\xspace}{\ensuremath{q}\xspace}}{{\ensuremath{\mu_{P}}\xspace}} {\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}+\left (1 -\frac{{\ensuremath{I'}\xspace}}{{\ensuremath{\mu_{P}}\xspace}} \right ) \frac{1}{{\ensuremath{\mu_{NP}}\xspace}} \frac{{\ensuremath{T_{\text{R}}}\xspace}}{2} \nonumber \\ & + \left(\frac{{\ensuremath{p}\xspace}}{{\ensuremath{\mu_{P}}\xspace}}+\left(1-\frac{{\ensuremath{I'}\xspace}}{{\ensuremath{\mu_{P}}\xspace}}\right)\frac{1}{{\ensuremath{\mu_{NP}}\xspace}}\right)\left({D\xspace}+{R\xspace}\right) \label{eq.proa.noCkpt.waste}\end{aligned}$$ Note that when ${\ensuremath{I}\xspace}=0$, [<span style="font-variant:small-caps;">Instant</span>]{}and [<span style="font-variant:small-caps;">NoCkptI</span>]{}are identical. Indeed, we have ${\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}=0$ if ${\ensuremath{I}\xspace}=0$, and we check that Equations  and  are identical in that case. Waste minimization {#sec-opt-int} ------------------ In this section we aim at minimizing the waste of the three strategies, and then we find conditions to characterize which one is better. Recall that : $${\ensuremath{I'}\xspace}= {\ensuremath{q}\xspace}\left ( (1 - {\ensuremath{p}\xspace}) {\ensuremath{I}\xspace}+ {\ensuremath{p}\xspace}{\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}\right )$$ **[<span style="font-variant:small-caps;">WithCkptI</span>]{}.** In order to compute the optimal value for [$T_{\text{P}}$]{}, let us find the portion of the waste that depends on [$T_{\text{P}}$]{}: $${\ensuremath{\textsc{Waste}}\xspace}_{{\ensuremath{T_{\text{P}}}\xspace}} = \frac{{\ensuremath{r}\xspace}{\ensuremath{q}\xspace}}{ \mu}\left ( \frac{ (1 - {\ensuremath{p}\xspace}) {\ensuremath{I}\xspace}+ {\ensuremath{p}\xspace}{\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}}{{\ensuremath{p}\xspace}} \frac{ {C\xspace}}{{\ensuremath{T_{\text{P}}}\xspace}} + {\ensuremath{T_{\text{P}}}\xspace}\right )$$ As we can see, the optimal value for [$T_{\text{P}}$]{}is independent from [$q$]{}, but also from $\mu$. The optimal value for [$T_{\text{P}}$]{}is thus: $$\label{tp.opt.int} {\ensuremath{T_{\text{P}}}\xspace}^{{\ensuremath{\text{extr}}}}=\sqrt{ \dfrac{(1 - {\ensuremath{p}\xspace}) {\ensuremath{I}\xspace}+ {\ensuremath{p}\xspace}{\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}}{p} {C\xspace}}$$ However, for our algorithm to be correct, we want $\frac{{\ensuremath{I}\xspace}}{{\ensuremath{T_{\text{P}}}\xspace}} \in \mathbb{N}$ (the interval [$I$]{} is partitioned in $k$ intervals of length [$T_{\text{P}}$]{}, for some integer $k$). We choose ${\ensuremath{T_{\text{P}}}\xspace}^{{\ensuremath{\text{opt}}}}$ equal to either $\frac{{\ensuremath{I}\xspace}}{\left \lfloor \frac{{\ensuremath{I}\xspace}}{{\ensuremath{T_{\text{P}}}\xspace}^{{\ensuremath{\text{extr}}}}}\right \rfloor}$ or $\frac{{\ensuremath{I}\xspace}}{\left \lfloor \frac{{\ensuremath{I}\xspace}}{{\ensuremath{T_{\text{P}}}\xspace}^{{\ensuremath{\text{extr}}}}}\right \rfloor +1}$, depending on the value that minimizes ${\ensuremath{\textsc{Waste}}\xspace}_{{\ensuremath{T_{\text{P}}}\xspace}}$. Note that we also have the constraint ${\ensuremath{T_{\text{P}}}\xspace}^{{\ensuremath{\text{opt}}}} \geq {C\xspace}$, hence if both values are lower than [C]{}, then ${\ensuremath{T_{\text{P}}}\xspace}^{{\ensuremath{\text{opt}}}}={C\xspace}$. Now that we know that ${\ensuremath{T_{\text{P}}}\xspace}^{{\ensuremath{\text{opt}}}}$ is independent from both [$q$]{}and [$T_{\text{R}}$]{}, we can see the waste in Equation  as a function of two variables. One can see from Equation  that the waste is an affine function of [$q$]{}. This means that the minimum is always reached for either ${\ensuremath{q}\xspace}=0$ or ${\ensuremath{q}\xspace}=1$. We now consider the two functions ${\ensuremath{\textsc{Waste}}\xspace}_{\text{withCkpt}\{{\ensuremath{q}\xspace}=0\}}$ and ${\ensuremath{\textsc{Waste}}\xspace}_{\text{withCkpt}\{{\ensuremath{q}\xspace}=1\}}$ in order to minimize them with respect to [$T_{\text{R}}$]{}. First we have: $$\label{waste.int.q0} {\ensuremath{\textsc{Waste}}\xspace}_{\text{withCkpt}\{{\ensuremath{q}\xspace}=0\}} =\frac{{C\xspace}}{{\ensuremath{T_{\text{R}}}\xspace}} + \frac{1}{\mu}\left ( \frac{{\ensuremath{T_{\text{R}}}\xspace}}{2} + {D\xspace}+{R\xspace}\right )$$ As expected, this is exactly the equation without prediction, the study of the optimal solution has been done in Section \[sec.no.intervals\], it is minimized when ${\ensuremath{T_{\text{R}}}\xspace}^{{\ensuremath{\text{opt}}}_0} =\min \left( \alpha {\ensuremath{\mu_e}\xspace}- {\ensuremath{I}\xspace}, \max \left ( \sqrt{2 {C\xspace}\mu}, {C\xspace}\right )\right )$. Next we have: $$\begin{aligned} \label{waste.int.q1} {\ensuremath{\textsc{Waste}}\xspace}_{\text{withCkpt}\{{\ensuremath{q}\xspace}=1\}} &= \left (1 -\frac{{\ensuremath{r}\xspace}\left ( (1 - {\ensuremath{p}\xspace}) {\ensuremath{I}\xspace}+ {\ensuremath{p}\xspace}{\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}\right )}{{\ensuremath{p}\xspace}\mu} \right ) \left ( \frac{{C\xspace}}{{\ensuremath{T_{\text{R}}}\xspace}} + \frac{1-{\ensuremath{r}\xspace}}{\mu}\frac{{\ensuremath{T_{\text{R}}}\xspace}}{2} \right )\nonumber \\ & +\frac{{\ensuremath{r}\xspace}}{ \mu}\left ( \frac{\left ( (1 - {\ensuremath{p}\xspace}) {\ensuremath{I}\xspace}+ {\ensuremath{p}\xspace}{\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}\right )}{{\ensuremath{p}\xspace}} \frac{{C\xspace}}{{\ensuremath{T_{\text{P}}}\xspace}^{{\ensuremath{\text{opt}}}}} + {\ensuremath{T_{\text{P}}}\xspace}^{{\ensuremath{\text{opt}}}} \right ) + \frac{{\ensuremath{r}\xspace}}{{\ensuremath{p}\xspace}\mu}{C\xspace}\nonumber \\ & + \left(\frac{{\ensuremath{r}\xspace}}{\mu}+\left (1 -\frac{{\ensuremath{r}\xspace}\left ( (1 - {\ensuremath{p}\xspace}) {\ensuremath{I}\xspace}+ {\ensuremath{p}\xspace}{\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}\right )}{{\ensuremath{p}\xspace}\mu} \right )\frac{1-{\ensuremath{r}\xspace}}{\mu}\right)\left({D\xspace}+{R\xspace}\right)\end{aligned}$$ This equation is minimized when $${\ensuremath{T_{\text{R}}}\xspace}^{{\ensuremath{\text{opt}}}_1} = \sqrt{ \dfrac{2 \mu{C\xspace}}{(1-{\ensuremath{r}\xspace})}}$$ One can remark that this value is equal to the result without intervals (Section \[sec.no.intervals\]). Actually, the only impact of the prediction interval [$I$]{}is the moment when we should take a pre-emptive action. Note that when ${\ensuremath{r}\xspace}=0$ (this means that there is no prediction), we have ${\ensuremath{T_{\text{R}}}\xspace}^{{\ensuremath{\text{opt}}}_1} = {\ensuremath{T_{\text{R}}}\xspace}^{{\ensuremath{\text{opt}}}_0} $, and we retrieve Young’s formula [@young74]. Finally, we know that the waste is defined for ${C\xspace}\leq {\ensuremath{T_{\text{R}}}\xspace}\leq \alpha {\ensuremath{\mu_e}\xspace}- {\ensuremath{I}\xspace}$. Hence, if ${\ensuremath{T_{\text{R}}}\xspace}^{{\ensuremath{\text{opt}}}_1} \notin [{C\xspace},\alpha {\ensuremath{\mu_e}\xspace}- {\ensuremath{I}\xspace}]$, this solution is not satisfiable. However Equation  is convex, so the optimal solution is [C]{}if ${\ensuremath{T_{\text{R}}}\xspace}^{{\ensuremath{\text{opt}}}_1} < {C\xspace}$, and $\alpha {\ensuremath{\mu_e}\xspace}- {\ensuremath{I}\xspace}$ if ${\ensuremath{T_{\text{R}}}\xspace}^{{\ensuremath{\text{opt}}}_1} > \alpha {\ensuremath{\mu_e}\xspace}$. Hence, when ${\ensuremath{q}\xspace}=1$, the optimal solution should be $$\label{tnp.opt.int} \min \left (\alpha {\ensuremath{\mu_e}\xspace}- {\ensuremath{I}\xspace},\max \left (\sqrt{ \dfrac{2 \mu{C\xspace}}{(1-{\ensuremath{r}\xspace})}},{C\xspace}\right )\right).$$ **[<span style="font-variant:small-caps;">Instant</span>]{}**. The derivation is similar . The optimal value for [$q$]{}is either $0$ or $1$, thus we consider ${\ensuremath{\textsc{Waste}}\xspace}_{{\textsc{Instant}\xspace}}^{\{0\}} = {\ensuremath{\textsc{Waste}_Y}\xspace}$ and ${\ensuremath{\textsc{Waste}}\xspace}_{{\textsc{Instant}\xspace}}^{\{1\}}$. If ${\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}>\frac{{\ensuremath{T_{\text{R}}}\xspace}}{2}$, then ${\ensuremath{\textsc{Waste}}\xspace}_{{\textsc{Instant}\xspace}}^{\{0\}} < {\ensuremath{\textsc{Waste}}\xspace}_{{\textsc{Instant}\xspace}}^{\{1\}}$, so we can assume $\min({\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}, \frac{{\ensuremath{T_{\text{R}}}\xspace}}{2}) = {\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}$. Then we derive that ${\ensuremath{\textsc{Waste}}\xspace}_{{\textsc{Instant}\xspace}}^{\{1\}}$ is minimized for ${\ensuremath{T_{\text{R}}}\xspace}^{{\ensuremath{\text{opt}}}_1}$ as before.\ **[<span style="font-variant:small-caps;">NoCkptI</span>]{}**. One can see that Equation  and Equation  only differ by the quantity : $$\frac{{\ensuremath{q}\xspace}{\ensuremath{r}\xspace}}{\mu}\left ( \frac{(1 - {\ensuremath{p}\xspace}) {\ensuremath{I}\xspace}+ {\ensuremath{p}\xspace}{\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}}{{\ensuremath{p}\xspace}} \frac{{C\xspace}}{{\ensuremath{T_{\text{P}}}\xspace}^{{\ensuremath{\text{opt}}}}} + {\ensuremath{T_{\text{P}}}\xspace}^{{\ensuremath{\text{opt}}}} - {\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}\right )$$ This value is linear in [$q$]{}and a constant with regards to [$T_{\text{R}}$]{}. Hence the minimization is almost the same. Once again we can see that the optimal value for [$q$]{}is either 0 or 1. We can consider the two functions ${\ensuremath{\textsc{Waste}}\xspace}_{\text{noCkpt}\{{\ensuremath{q}\xspace}=0\}}$ and ${\ensuremath{\textsc{Waste}}\xspace}_{\text{noCkpt}\{{\ensuremath{q}\xspace}=1\}}$. We remark that ${\ensuremath{\textsc{Waste}}\xspace}_{\text{noCkpt}\{{\ensuremath{q}\xspace}=0\}} = {\ensuremath{\textsc{Waste}}\xspace}_{\text{withCkpt}\{{\ensuremath{q}\xspace}=0\}}$, and hence that the study has already been done. As for ${\ensuremath{\textsc{Waste}}\xspace}_{\text{noCkpt}\{{\ensuremath{q}\xspace}=1\}}$, it is also minimized when ${\ensuremath{T_{\text{R}}}\xspace}^{{\ensuremath{\text{opt}}}} = \sqrt{ \dfrac{2 \mu{C\xspace}}{(1-{\ensuremath{r}\xspace})}}$. Finally, the last step of this study is identical to the previous minimization, and the optimal solution when ${\ensuremath{q}\xspace}=1$ is defined by : $${\ensuremath{T_{\text{R}}}\xspace}^{{\ensuremath{\text{opt}}}_1}=\min \left (\alpha {\ensuremath{\mu_e}\xspace}- {\ensuremath{I}\xspace},\max \left (\sqrt{ \dfrac{2 \mu{C\xspace}}{(1-{\ensuremath{r}\xspace})}},{C\xspace}\right )\right)$$ **Summary**. Finally in this section, we consider the waste for the two algorithms that take the prediction window into account (the one that does not checkpoint during the prediction window, and the one that checkpoints during the prediction window), and try to find conditions of dominance of one strategy over the other. Since the equation of the waste is identical when ${\ensuremath{q}\xspace}=0$, let us consider the case when ${\ensuremath{q}\xspace}=1$. We have seen that: $$\begin{aligned} \label{diff.waste.algo} ({\ensuremath{\textsc{Waste}}\xspace}_{\text{withCkpt}\{{\ensuremath{q}\xspace}=1\}} - {\ensuremath{\textsc{Waste}}\xspace}_{\text{noCkpt}\{{\ensuremath{q}\xspace}=1\}}) & = \frac{{\ensuremath{r}\xspace}\left ( (1 - {\ensuremath{p}\xspace}) {\ensuremath{I}\xspace}+ {\ensuremath{p}\xspace}{\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}\right )}{{\ensuremath{p}\xspace}\mu}\frac{{C\xspace}}{{\ensuremath{T_{\text{P}}}\xspace}^{{\ensuremath{\text{opt}}}}} \nonumber \\ & + \frac{{\ensuremath{r}\xspace}}{\mu} \left ( {\ensuremath{T_{\text{P}}}\xspace}^{{\ensuremath{\text{opt}}}} - {\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}\right ) \end{aligned}$$ We want to know when Equation  is nonnegative (meaning that it is beneficial not to take any checkpoints during proactive mode). We know that this value is minimized when ${\ensuremath{T_{\text{P}}}\xspace}^{{\ensuremath{\text{extr}}}}=\sqrt{ \dfrac{ (1 - {\ensuremath{p}\xspace}) {\ensuremath{I}\xspace}+ {\ensuremath{p}\xspace}{\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}}{p}{C\xspace}}$ (Equation ), then a sufficient condition would be to study the equation : $${\ensuremath{\textsc{Waste}}\xspace}_{\text{withCkpt}\{{\ensuremath{q}\xspace}=1\}} - {\ensuremath{\textsc{Waste}}\xspace}_{\text{noCkpt}\{{\ensuremath{q}\xspace}=1\}} \geq 0$$ with ${\ensuremath{T_{\text{P}}}\xspace}^{{\ensuremath{\text{extr}}}}$ instead of ${\ensuremath{T_{\text{P}}}\xspace}^{{\ensuremath{\text{opt}}}}$. That is: $$\begin{aligned} &\frac{{\ensuremath{r}\xspace}(1 - {\ensuremath{p}\xspace}) {\ensuremath{I}\xspace}+ {\ensuremath{p}\xspace}{\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}}{{\ensuremath{p}\xspace}\mu}\frac{{C\xspace}}{\sqrt{ \dfrac{ (1 - {\ensuremath{p}\xspace}) {\ensuremath{I}\xspace}+ {\ensuremath{p}\xspace}{\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}}{p} {C\xspace}}} + \frac{{\ensuremath{r}\xspace}}{\mu} \left ( \sqrt{ \dfrac{ (1 - {\ensuremath{p}\xspace}) {\ensuremath{I}\xspace}+ {\ensuremath{p}\xspace}{\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}}{p} {C\xspace}} - {\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}\right ) &\geq 0 \nonumber\\ & \Leftrightarrow 2\sqrt{ \dfrac{ (1 - {\ensuremath{p}\xspace}) {\ensuremath{I}\xspace}+ {\ensuremath{p}\xspace}{\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}}{p} {C\xspace}} \geq {\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}\!^2 \label{cond.noCkpt}\end{aligned}$$ Consequently, we can say that if Equation  is matched, then ${\ensuremath{\textsc{Waste}}\xspace}_{\text{noCkpt}}$ $\leq {\ensuremath{\textsc{Waste}}\xspace}$, the algorithm where we do not checkpoint during the proactive mode has a better solution than Algorithm \[algo.proactive\]. For example, if we assume that faults strike uniformly during the prediction window $[t_{0}, t_{0}+{\ensuremath{I}\xspace}]$, in other words, if $0 \leq x \leq {\ensuremath{I}\xspace}$, the probability that the fault occurs in the interval $[t_{0}, t_{0}+x]$ is $\frac{x}{{\ensuremath{I}\xspace}}$, then ${\ensuremath{\mathbb{E}_{I}^{(f)}}\xspace}=\frac{I}{2}$, and our condition becomes $${\ensuremath{I}\xspace}\leq 16 \frac{1 - \sfrac{{\ensuremath{p}\xspace}}{2}}{{\ensuremath{p}\xspace}}{C\xspace}.$$ We can now finish our study by saying that in order to find the optimal solution, one should compute both optimal solutions for ${\ensuremath{q}\xspace}=0$ and ${\ensuremath{q}\xspace}= 1$, for both algorithms, and choose the one that minimizes the waste, as was done in Section \[sec.no.intervals\], except when Equation  is valid, then we can focus on the computation of the waste of the algorithms that does not checkpoint during proactive mode. Simulation results {#sec.simulations} ================== In order to validate our model, we have instantiated it with several scenarios. The experiments use parameters that are representative of current and forthcoming large-scale platforms [@j116; @Ferreira2011]. We have $C=R=10mn$, and $D=1mn$. The individual (processor) MTBF $\mu_{ind} = 125$ years, and the total number of processors $N$ varies from $N=16,384$ to $N=524,288$, so that the platform MTBF $\mu$ varies from $\mu=4,000mn$ (about $1.5$ day) down to $\mu=125mn$ (about $2$ hours). For instance the Jaguar platform, with $N=45,208$ processors, is reported to experience about one failure per day [@6264677], which leads to $\mu_{ind} = \frac{45,208}{365}\approx 125$ years. We have analytically computed the optimal value of the waste for each strategy (using the formulas of Section \[sec-opt-int\]) using a computer algebra software. In order to check the accuracy of our model, we have compared the results with those from simulations using a fault generator. Our simulation engine generates a random trace of failures, parameterized either by an Exponential failure distribution or by a Weibull distribution law with shape parameter $0.5$ and $0.7$; Exponential failures are widely used for theoretical studies, while Weibull failures are representative of the behavior of real-world platforms [@Weibull1; @Weibull2; @Heien:2011:MTH:2063384.2063444]. With probability [$r$]{}, we decide if a failure is predicted or not. In both cases, the distribution is scaled so that its expectation corresponds to the platform MTBF $\mu$. Then the simulation engine generates another random trace of false predictions (whose distribution is identical to the first trace or a uniform distributions). This second distribution is scaled so that its expectation is $\frac{{\ensuremath{p}\xspace}\mu}{{\ensuremath{r}\xspace}(1-{\ensuremath{p}\xspace})}$, the inter-arrival time of false predictions. Finally, both traces are merged to derive the final trace with all events. Each value reported for the simulations is the average of $100$ randomly generated experiments. In the simulations, we compare up to ten checkpointing strategies. Here is the list:\ $\bullet$ [<span style="font-variant:small-caps;">Young</span>]{}is the periodic checkpointing strategy of period ${\ensuremath{T_{{\ensuremath{\text{extr}}}}^{\{0\}}}\xspace} = \sqrt{2 \mu {C\xspace}}$ given in [@young74]. Note that Daly’s formula [@daly04] leads to the same results.\ $\bullet$ [<span style="font-variant:small-caps;">ExactPrediction</span>]{}is derived from the strategy Section \[sec.no.intervals\] (with exact prediction dates). However, in the simulations, we always take prediction into account and use an uncapped period ${\ensuremath{T_{{\ensuremath{\text{extr}}}}^{\{1\}}}\xspace} = \sqrt{ \dfrac{2 \mu{C\xspace}}{1-{\ensuremath{r}\xspace}}}$ instead of ${\ensuremath{T_{1} }\xspace} = \min(\alpha {\ensuremath{\mu_e}\xspace}- {\ensuremath{I}\xspace}, \max({C\xspace}, {\ensuremath{T_{{\ensuremath{\text{extr}}}}^{\{1\}}}\xspace}))$.\ $\bullet$ Similarly, [<span style="font-variant:small-caps;">Instant</span>]{}, [<span style="font-variant:small-caps;">NoCkptI</span>]{}and [<span style="font-variant:small-caps;">WithCkptI</span>]{}are the three strategies described in Section \[sec.intervals\], with the same modification: we always take prediction into account and use an uncapped period ${\ensuremath{T_{{\ensuremath{\text{extr}}}}^{\{1\}}}\xspace} $ instead of ${\ensuremath{T_{1} }\xspace}$ in regular mode.\ $\bullet$ To assess the quality of each strategy, we compare it with its [<span style="font-variant:small-caps;">BestPeriod</span>]{}counterpart, defined as the same strategy but using the best possible period ${\ensuremath{T_{\text{R}}}\xspace}$. This latter period is computed via a brute-force numerical search for the optimal period. The rationale for modifying the strategies described in the previous sections is of course to better assess the impact of prediction. For the computer algebra plots, in addition to the waste with the *capped periods* given in Section \[sec-opt-int\], i.e., with ${\ensuremath{T_{0} }\xspace}= {\ensuremath{T_{\text{Y}}}\xspace}= \min(\alpha \mu, \max({C\xspace}, {\ensuremath{T_{{\ensuremath{\text{extr}}}}^{\{0\}}}\xspace}))$, and ${\ensuremath{T_{1} }\xspace} = \min(\alpha {\ensuremath{\mu_e}\xspace}- {\ensuremath{I}\xspace}, \max({C\xspace}, {\ensuremath{T_{{\ensuremath{\text{extr}}}}^{\{1\}}}\xspace}))$, we also report the waste obtained for the *uncapped periods*, i.e., using ${\ensuremath{T_{0} }\xspace} = {\ensuremath{T_{{\ensuremath{\text{extr}}}}^{\{0\}}}\xspace}$ without prediction and ${\ensuremath{T_{1} }\xspace} = {\ensuremath{T_{{\ensuremath{\text{extr}}}}^{\{1\}}}\xspace} = \sqrt{ \dfrac{2 \mu{C\xspace}}{1-{\ensuremath{r}\xspace}}}$ with prediction. The objective is twofold: (i) Assess whether the validity of the model can be extended; and (ii) Provide an exact match with the simulations, which mimic a real-life execution and do allow for an arbitrary number of faults per period. Predictors from the literature ------------------------------ We first experiment with two predictors from the literature: one accurate predictor with high recall and precision [@5958823], namely with ${\ensuremath{p}\xspace}=0.82$ and ${\ensuremath{r}\xspace}=0.85$, and another predictor with more limited recall and precision [@5542627], namely with ${\ensuremath{p}\xspace}=0.4$ and ${\ensuremath{r}\xspace}=0.7$. In both cases, we use two different time-windows, ${\ensuremath{I}\xspace}=300s$ and ${\ensuremath{I}\xspace}=3,000s$. The former value does not allow for checkpointing within the prediction window, while the latter values allow for several checkpoints. Note that we always compare the results with [<span style="font-variant:small-caps;">ExactPrediction</span>]{}, the strategy that assumes exact prediction dates. Figures \[fig.082.085\] and \[fig.04.07\] show the average waste degradation of the ten heuristics for both predictors, as a function of the number of processors $N$. We draw the plots as a function of the number of processors $N$ rather than of the platform MTBF $\mu = \mu_{ind}/N$ , because it is more natural to see the waste increase with larger platforms; however, this work is agnostic of the granularity of the processors and intrinsically focuses on the impact of the MTBF on the waste. The first observation is that the prediction is always useful for the whole set of parameters under study! The second observation is the good correspondence between analytical results and simulations in Figures \[fig.082.085\] and \[fig.04.07\] (compare subfigures (a) and (b) with (c), (d) and (e), and subfigures (f) and (g) with (h), (i) and (j)). This shows the validity of the model for the whole range of distributions (Exponential and both Weibull shapes). More precisely: (i) The capped model overestimates the waste for large platforms (or small MTBFs), in particular for large values of ${\ensuremath{I}\xspace}$ (see Figures \[fig.082.085\](f) and \[fig.04.07\](f)), but this was the price to pay for mathematical rigor; (ii) The uncapped model is accurate for the whole range of the study. Another striking result is that all strategies taking prediction into account have the same waste as their [<span style="font-variant:small-caps;">BestPeriod</span>]{}counterpart, which demonstrates that our formula ${\ensuremath{T_{{\ensuremath{\text{extr}}}}^{\{1\}}}\xspace} = \sqrt{\frac{2 \mu {C\xspace}}{1-{\ensuremath{r}\xspace}}}$ is indeed the best possible checkpointing period in regular mode. Unsurprisingly, [<span style="font-variant:small-caps;">ExactPrediction</span>]{}is better than the heuristics that use a time window instead of exact prediction dates, especially with a high number of processors. However, interval based heuristics achieve close results when ${\ensuremath{I}\xspace}=300s$, or when ${\ensuremath{I}\xspace}=3,000s$ and a small number of processors ($N<2^{16}$). \ \ \ \ \ In order to compare the heuristics without prediction to those with prediction, we report job execution times in Table \[makespan.300.tab\]. For the strategies with prediction, we compute the gain (expressed in percentage) over [<span style="font-variant:small-caps;">Young</span>]{}, the reference strategy without prediction. For ${\ensuremath{I}\xspace}=300s$, the three strategies are identical. But for ${\ensuremath{I}\xspace}=3,000s$, [<span style="font-variant:small-caps;">WithCkptI</span>]{}has often better results. First, with ${\ensuremath{p}\xspace}=0.85$ and ${\ensuremath{r}\xspace}=0.82$ and ${\ensuremath{I}\xspace}=3,000s$, we save $25\% $ of the total time with $N=2^{19}$, and $14\%$ with $N=2^{16}$ using strategy [<span style="font-variant:small-caps;">WithCkptI</span>]{}. With ${\ensuremath{I}\xspace}=300s$, we save up to $44\%$ with $N=2^{19}$, and $18\%$ with $N=2^{16}$ using any strategy (though [<span style="font-variant:small-caps;">NoCkptI</span>]{}is slightly better than [<span style="font-variant:small-caps;">Instant</span>]{}). Then, with ${\ensuremath{p}\xspace}=0.4$ and ${\ensuremath{r}\xspace}= 0.7$, we still save $32\%$ of the execution time when ${\ensuremath{I}\xspace}=300s$ and $N=2^{19}$, and $13\%$ with $N=2^{16}$. The gain gets smaller with ${\ensuremath{I}\xspace}=3,000s$, but remains non negligible since we can save up to $9.7\%$ with $N=2^{19}$, and $7.6\%$ with $N=2^{16}$. Unexpectedly in this last case, the strategy that is the most efficient is [<span style="font-variant:small-caps;">Instant</span>]{}and not [<span style="font-variant:small-caps;">WithCkptI</span>]{}. We observe that the size of the prediction-window [$I$]{}plays an important role too: we have better results for ${\ensuremath{I}\xspace}=300$ and $({\ensuremath{p}\xspace},{\ensuremath{r}\xspace})=(0.4,0.7)$, than for ${\ensuremath{I}\xspace}=3000$ and $({\ensuremath{p}\xspace},{\ensuremath{r}\xspace})=(0.82,0.85)$. In Table \[makespan.300.tab\], we report the job execution times for Weibull distributions with $k=0.5$. For ${\ensuremath{I}\xspace}=300s$, the three strategies are identical. But for ${\ensuremath{I}\xspace}=3,000s$, [<span style="font-variant:small-caps;">WithCkptI</span>]{}has often better results. First, with ${\ensuremath{p}\xspace}=0.85$ and ${\ensuremath{r}\xspace}=0.82$ and ${\ensuremath{I}\xspace}=3,000s$, we save $61\% $ of the total time with $N=2^{19}$, and $30\%$ with $N=2^{16}$ using strategy [<span style="font-variant:small-caps;">WithCkptI</span>]{}. With ${\ensuremath{I}\xspace}=300s$, we save up to $74\%$ with $N=2^{19}$, and $38\%$ with $N=2^{16}$ using any strategy (though [<span style="font-variant:small-caps;">NoCkptI</span>]{}is slightly better than [<span style="font-variant:small-caps;">Instant</span>]{}). Then, with ${\ensuremath{p}\xspace}=0.4$ and ${\ensuremath{r}\xspace}= 0.7$, we still save $66\%$ of the execution time when ${\ensuremath{I}\xspace}=300s$ and $N=2^{19}$, and $33\%$ with $N=2^{16}$. The gain gets smaller with ${\ensuremath{I}\xspace}=3,000s$, but we can save up to $52\%$ with $N=2^{19}$, and $22\%$ with $N=2^{16}$. Using a Weibull failure distribution with shape parameter 0.5, we observe that the gain due to prediction is twice larger than the gain computed with a Weibull failure distribution with shape parameter 0.7. We can conclude the same remark from Figures \[fig.082.085\](e), \[fig.082.085\](j), \[fig.04.07\](e) and \[fig.04.07\](j). We also performed simulations with a trace of false predictions parametrized by a uniform distribution and we observe that the result (Figures  \[fig.082.085.UNIF\] and  \[fig.04.07.UNIF\]) are similar to the result (Figures  \[fig.082.085\] and  \[fig.04.07\]) with simulations with a trace of false predictions parametrized by a distribution identical to the distribution of the trace of failures. ------------------------------------------------------------------- ---------------- ---------------- ---------------- ---------------- ${\ensuremath{I}\xspace}=300$ $2^{16}$ procs $2^{19}$ procs $2^{16}$ procs $2^{19}$ procs [<span style="font-variant:small-caps;">Young</span>]{} 81.3 30.1 81.2 30.1 [<span style="font-variant:small-caps;">ExactPrediction</span>]{} 65.9 (19%) 15.9 (47%) 69.7 (14%) 19.3 (36%) [<span style="font-variant:small-caps;">NoCkptI</span>]{} 66.5 (18%) 16.9 (44%) 70.3 (13%) 20.5 (32%) [<span style="font-variant:small-caps;">Instant</span>]{} 66.5 (18%) 17.0 (44%) 70.3 (13%) 20.7 (31%) ------------------------------------------------------------------- ---------------- ---------------- ---------------- ---------------- : Comparing job execution times for a Weibull distribution ($k=0.7$), and reporting the gain when comparing to [<span style="font-variant:small-caps;">Young</span>]{}.[]{data-label="makespan.300.tab"} \ ------------------------------------------------------------------- ---------------- ---------------- ---------------- ---------------- ${\ensuremath{I}\xspace}=3,000$ $2^{16}$ procs $2^{19}$ procs $2^{16}$ procs $2^{19}$ procs [<span style="font-variant:small-caps;">Young</span>]{} 81.2 30.1 81.2 30.1 [<span style="font-variant:small-caps;">ExactPrediction</span>]{} 66.0 (19%) 15.9 (47%) 69.8 (14%) 19.3 (36%) [<span style="font-variant:small-caps;">NoCkptI</span>]{} 71.1 (12%) 24.6 (18%) 75.2 (7.3%) 28.9 (4.0%) [<span style="font-variant:small-caps;">WithCkptI</span>]{} 70.0 (14%) 22.6 (25%) 75.4 (7.1%) 27.2 (9.7%) [<span style="font-variant:small-caps;">Instant</span>]{} 71.2 (12%) 24.2 (20%) 75.0 (7.6%) 28.3 (6.0%) ------------------------------------------------------------------- ---------------- ---------------- ---------------- ---------------- : Comparing job execution times for a Weibull distribution ($k=0.7$), and reporting the gain when comparing to [<span style="font-variant:small-caps;">Young</span>]{}.[]{data-label="makespan.300.tab"} ------------------------------------------------------------------- ---------------- ---------------- ---------------- ---------------- ${\ensuremath{I}\xspace}=300$ $2^{16}$ procs $2^{19}$ procs $2^{16}$ procs $2^{19}$ procs [<span style="font-variant:small-caps;">Young</span>]{} 125.4 171.8 125.5 171.7 [<span style="font-variant:small-caps;">ExactPrediction</span>]{} 75.8 (40%) 39.4 (77%) 82.9 (34%) 51.8(70%) [<span style="font-variant:small-caps;">NoCkptI</span>]{} 77.3 (38%) 44.8 (74%) 84.6 (33%) 58.2 (66%) [<span style="font-variant:small-caps;">Instant</span>]{} 77.4 (38%) 45.1 (74%) 84.7 (33%) 59.1 (66%) ------------------------------------------------------------------- ---------------- ---------------- ---------------- ---------------- : Comparing job execution times for a Weibull distribution ($k=0.5$), and reporting the gain when comparing to [<span style="font-variant:small-caps;">Young</span>]{}.[]{data-label="makespan.300.tab"} \ ------------------------------------------------------------------- ---------------- ---------------- ---------------- ---------------- ${\ensuremath{I}\xspace}=3,000$ $2^{16}$ procs $2^{19}$ procs $2^{16}$ procs $2^{19}$ procs [<span style="font-variant:small-caps;">Young</span>]{} 125.4 171.9 125.4 172.0 [<span style="font-variant:small-caps;">ExactPrediction</span>]{} 76.1 (39%) 39.4 (77%) 83.0 (34%) 51.7 (70%) [<span style="font-variant:small-caps;">NoCkptI</span>]{} 90.0 (28%) 71.8 (58%) 98.3 (22%) 84.5 (51%) [<span style="font-variant:small-caps;">WithCkptI</span>]{} 87.8 (30%) 66.6 (61%) 98.0 (22%) 82.2 (52%) [<span style="font-variant:small-caps;">Instant</span>]{} 89.8 (28%) 70.9 (59%) 98.2 (22%) 83.2 (52%) ------------------------------------------------------------------- ---------------- ---------------- ---------------- ---------------- : Comparing job execution times for a Weibull distribution ($k=0.5$), and reporting the gain when comparing to [<span style="font-variant:small-caps;">Young</span>]{}.[]{data-label="makespan.300.tab"} Recall vs. precision {#sec.impact} -------------------- In this section, we assess the impact of the two key parameters of the predictor, its recall [$r$]{}and its precision ${\ensuremath{p}\xspace}$. To this purpose, we conduct simulations where one parameter is fixed, and we let the other vary. We choose two platforms, a smaller one with $N=2^{16}$ processors (or a MTBF $\mu=1,000mn$) and the other with $N=2^{19}$ processors (or a MTBF $\mu=125mn$). In both cases, we use a prediction-window of size ${\ensuremath{I}\xspace}=300s$, and a Weibull failure distribution with shape parameter $k=0.7$ (we have similar results (Figures  \[fig.recall.19.05\] and  \[fig.precision.19.05\]) for $k=0.5$). In Figure \[fig.recall.19\], we fix the value of [$r$]{}(either ${\ensuremath{r}\xspace}=0.4$ or ${\ensuremath{r}\xspace}=0.8$) and we let ${\ensuremath{p}\xspace}$ vary from $0.3$ to $0.99$. In the four plots, we observe that the precision has a minor impact on the waste. In Figure \[fig.precision.19\], we conduct the opposite experiment and fix the value of [$p$]{}(either $ {\ensuremath{p}\xspace}=0.4$ or ${\ensuremath{p}\xspace}=0.8$), letting ${\ensuremath{r}\xspace}$ vary from $0.3$ to $0.99$. Here we observe that increasing the recall can significantly improve the performance. Altogether we conclude that it is more important (for the design of future predictors) to focus on improving the recall [$r$]{}rather than the precision [$p$]{}, and our results can help quantify this statement. We provide an intuitive explanation as follows: unpredicted failures prove very harmful and heavily increase the waste, while unduly checkpointing due to false predictions turns out to induce a smaller overhead. Related work {#sec.related} ============ Considerable research has been conducted on fault prediction using different models (system log analysis [@5958823], event-driven approach [@GainaruIPDPS12; @5958823; @5542627], support vector machines [@LiangZXS07; @Fulp:2008:PCS:1855886.1855891]), nearest neighbors [@LiangZXS07], …). In this section we give a brief overview of the results obtained by predictors. We focus on their results rather than on their methods of prediction. The authors of [@5542627] introduce the *lead time*, that is the time between the prediction and the actual fault. This time should be sufficient to take proactive actions. They are also able to give the location of the fault. While this has a negative impact on the precision (see the low value of [$p$]{}in Table \[rel.work.tab\]), they state that it has a positive impact on the checkpointing time (from 1500 seconds to 120 seconds). The authors of [@5958823] also consider a lead time, and introduce a *prediction window* when the predicted fault should happen. The authors of [@LiangZXS07] study the impact of different prediction techniques with different prediction window sizes. They also consider a lead time, but do not state its value. These two latter studies motivate the work of Section \[sec.intervals\], even though [@5958823] does not provide the size of their prediction window. Unfortunately, much of the work done on prediction does not provide information that could be really useful for the design of efficient algorithms. These informations are those stated above, namely the lead time and the size of the prediction window, but other information that could be useful would be: (i) the distribution of the faults in the prediction window; (ii) the precision as a function of the recall (see our analysis); and (iii) the precision and recall as functions of the prediction window (what happens with a larger prediction window). While many studies on fault prediction focus on the conception of the predictor, most of them consider that the proactive action should simply be a checkpoint or a migration right in time before the fault. However, in their paper [@Fu:2007:EEC:1362622.1362678], Li et al. consider the mathematical problem to determine when and how to migrate. In order to be able to use migration, they stated that at every time, 2% of the resources are available. This allowed them to conceive a Knapsack-based heuristic. Thanks to their algorithm, they were able to save 30% of the execution time compared to an heuristic that does not take the reliability into account, with a precision and recall of 70%, and with a maximum load of 0.7. Finally, to the best of our knowledge, this work is the first to focus on the mathematical aspect of fault prediction, and to provide a model and a detailed analysis of the waste due to all three types of events (true and false predictions and unpredicted failures). Conclusion {#sec.conclusion} ========== In this work, we have studied the impact of prediction, either with exact dates or window-based, on checkpointing strategies. We have designed several algorithms that decide when to trust these predictions, and when it is worth taking preventive checkpoints. We have introduced an analytical model to capture the waste incurred by each strategy, and provided the optimal solution to the corresponding optimization problems. We have been able to derive some striking conclusions:\ $\bullet$ The model is quite accurate and its validity goes beyond the conservative assumption that requires capping checkpointing periods to diminish the probability of having several faults within the same period;\ $\bullet$ A unified formula for the optimal checkpointing period is $\sqrt{ \dfrac{2 \mu{C\xspace}}{1-{\ensuremath{r}\xspace}{\ensuremath{q}\xspace}}}$, which unifies both cases with and without prediction, and nicely extends the work of Young and Daly to account for prediction;\ $\bullet$ The simulations fully validate the model, and show that: (i) A significant gain is induced by using predictions, even for mid-range values of recall and precision; and (ii) The best period (found by brute-force search) is always very close to the one predicted by the model and given by the previous unified formula; this holds true both for Exponential and Weibull failure distributions;\ $\bullet$ The recall has more impact on the waste than the precision: *better safe than sorry*, or better prepare for a false event than miss an actual failure! Altogether, the analytical model and the comprehensive results provided in this work enable to fully assess the impact of fault prediction on optimal checkpointing strategies. Future work will be devoted to refine the assessment of the usefulness of prediction with trace-based failure and prediction logs from current large-scale supercomputers. [*Acknowledgments.*]{} The authors are with Université de Lyon, France. Y. Robert is with the Institut Universitaire de France. This work was supported in part by the ANR [*RESCUE*]{} project.

--- abstract: 'A particular initial state for the construction of a perturbative QCD expansion is investigated. It is formed as a coherent superposition of zero momentum gluon pairs and shows Lorentz as well as global $SU(3)$ symmetries. The general form of the Wick theorem is discussed, and it follows that the gluon and ghost propagators determined by the proposed vacuum state, coincides with the ones used in an alternative of the usual perturbation theory proposed in a previous work, and reviewed here. Therefore, the ability of such a procedure of producing a finite gluon condensation parameter already in the first orders of perturbation theory is naturally explained. It also follows that this state satisfies the physicality condition of the BRST procedure in its Kugo and Ojima formulation. A brief review of the canonical quantization for gauge fields, developed by Kugo and Ojima, is done and the value of the gauge parameter $\alpha$ is fixed to $\alpha=1$ where the procedure is greatly simplified. Therefore, after assuming that the adiabatic connection of the interaction does not take out the state from the interacting physical space, the predictions of the perturbation expansion for the physical quantities, at the value $\alpha=1$, should have meaning. The validity of this conclusion solves the gauge dependence indeterminacy remained in the proposed perturbation expansion.' author: - '[**Author: Marcos Rigol Madrazo**]{}\' - '[**Advisor: Dr. Alejandro Cabo Montes de Oca**]{}' date: Havana 1999 title: | Instituto Superior de Ciencias y Tecnología Nucleares\ Dissertation Diploma Thesis\ --- Introduction ============ Quantum Chromodynamics (QCD) was discovered in the seventies and it has been considered as the fundamental theory for the strong interactions. A theory of such sort, showing a non abelian invariance group, was first suggested by Yang and Mills [@Yang]. The main idea in it is the principle of local gauge invariance, which for example, in Quantum Electrodynamics (QED) means that the phase of the wave function can be defined in an arbitrary way at any point of the space-time. In a non abelian theory, the arbitrary phase is generalized to an arbitrary transformation in an internal symmetry group, for QCD the internal symmetry group is $SU(3)$. The discovery of this theory generated radical changes in the character of the Modern Theoretical Physics and as a consequence it has been deeply investigated in the last years. It is believed at the present time that all the interactions in Nature are gauge invariant [@Green]. In one limit, the smallness of the coupling constant at high momentum (asymptotic freedom) made possible the theoretical investigation of the so-called hard processes by using the familiar perturbative language. This so called perturbative QCD (PQCD) was satisfactorily developed. However, relevant phenomena associated with the strong interactions can’t be described by the standard perturbative methods and the development of the so-called non-perturbative QCD is at the moment one of the great challenges of Theoretical Physics. One of the most peculiar characteristics of the strong interactions is the color confinement. According to this philosophy colored objects, like quarks and gluons, can’t be observed as free particles in contrast with hadrons that are colorless composite states and effectively detected. The physical nature of such phenomenon remains unclear. Qualitatively, it is compared with the Meissner effect in superconductors, in which the magnetic field is expelled from the bulk in which the Cooper pairs condensate exists. It is considered that the QCD vacuum expels the color fields from it. Numerous attempts to explain this property have been made, for example explicit calculations in which the theory is regularized in a spatial lattice [@Creutz], and also through the construction of phenomenological models. One of this models, the so called MIT Bag Model [@Chodos], assumes that a bag or bubble is formed around the objects having color in such a manner that they could not escape from it, because their effective mass is smaller inside the bag volume and very high outside. The dimensional quantity introduced in this model is $B$, the bag constant, which is the pressure that the vacuum makes on the color field. Another approach is the so called String Model [@Gervais] which is based in the assumption that the interaction forces between quarks and antiquarks grow with the distance, in such a way that the energy increases linearly with the string length $E(L)=kL$. The main parameter introduced in this theory is the string tension $k$ which determines the strength of the confining interaction potential. A fundamental problem in QCD is the nature of its ground state [@Shuryak]. This state is imagined as a very dense state of matter, composed of gluons and quarks interacting in a complicated way. Its properties are not easy accessible in experiments, because quarks and gluons fields can’t be directly observed, only the color neutral hadrons are detected. Furthermore, the interactions between quarks can’t be directly determine, because their scattering amplitudes can’t be measured. It is known, from the experience in solid-state physics, that a good understanding of the ground state structure implies a natural explanation of many of the phenomenological facts concerning to its excitations. The theory of superconductivity is a good example, up to the moment in which a good theory of the ground state was at hand the description of its excitations remained basically phenomenological. It is already accepted that in QCD the zero-point oscillations of the coupled modes produce a finite energy density, such effects are called non-perturbative ones. Obviously such an energy density can be subtracted by definition, however this procedure does not solve the problem, because soft modes are rearranged in the excited states and the variation of their energy should be unavoidable considered. This energy density is determined phenomenologically and its numerical estimate is [@Shuryak] $$E_{vac}\simeq -f\langle 0\mid g^2G^2\mid 0\rangle \simeq 0.5GeV/fm^3,$$ where the so-called non-perturbative gluonic condensate $\langle 0\mid g^2G^2\mid 0\rangle$ was introduced and phenomenologically evaluated by Shiftman, Vainshtein and Zakharov [@Zakharov]. The negative sign of $ E_{vac}$ means that the non-perturbative vacuum energy is lower that the one associated to the perturbative vacuum. Some of the QCD vacuum models, developed to explain the above-mentioned properties, are mentioned below. These models can be classified considering the dimension of the manifold in which the non-perturbative field fluctuations are concentrated. 1- The “instanton” model, in which it is assumed that the field is gathered in some localized regions of the space and time as instantaneous fluctuations. These are considered as fluctuations concentrated in zero dimensional manifolds. 2- The “soliton” model, in which it is assumed that the non linear gauge fields create some kind of stable particles or solitons (i.e. glueballs [@Hansson] or monopoles [@Mandelstam]) in the space. The space-time manifold to be considered for these models is one-dimensional. 3- The “string” model, in which closed strings (field created between color charges shows a form resembling a flux tube or string) are present in the vacuum. In space-time the history of these strings is a 2-dimensional surface, so in this picture the fluctuations are concentrated in closed surfaces. 4- The last model to be mentioned is the simplest one. It will be discussed here in more detail because it furnished the starting roots of the present discussion. This model is the “homogeneous” vacuum model, in which it is assumed that a magnetic field exist in the vacuum [@Savv1]. In the homogeneous vacuum field model, the existence of a constant magnetic abelian field $H$ is assumed. A simple calculation in the one loop approximation gives as result the following energy density [@Shuryak] $$E\left(H\right) =\frac{H^2}2\left(1+\frac{bg^2}{16\pi ^2}\ln \left(\frac H{\Lambda ^2}\right) \right).$$ This formula predicts negative energy values for small values of the field $ H $, so the usual perturbative ground state with $H=0$ is unstable with respect to the formation of a state with a non vanishing field intensity [@Shuryak] $$H_{vac}=\Lambda ^2\exp \left(-\frac{16\pi ^2}{bg^2}-\frac 12\right),$$ at which the energy $E\left(H\right)$ has a minimum. With the use of this model an extensive number of physical problems, related with the hadron structure, confinement, etc. have been investigated. Nevertheless, after some time its intense study was abandoned. The main reason were: 1\. The perturbative relation giving $E_{vac}$ would be only valid if the second order of the perturbative expansion is relatively small. 2\. The specific spatial and color directions of the magnetic field break the now seemingly indispensable Lorentz and $SU(3)$ invariance of the ground state. 3\. The magnetic moment of the vector particle (gluon) is such that its energy in the presence of the field has a negative eigenvalue, which also makes unstable the homogeneous magnetic field $H$. Before presenting the objectives of the present work it should be stressed that QCD quantization [@Faddeev-Popov] is realized in the same way as that in QED, and it can be shown that QCD is renormalizable. The quadratic field terms in the QCD Lagrangian ($L_{QCD}$), which depend on the quark and gluon fields, have the same form that the ones corresponding to the electrons and photons in QED. However, in connection with the interaction, there appears a substantial difference due to the coupling of the gluon to itself. In order to assure the unitarity of the quantum theory of gauge fields, it was necessary to introduce fictitious particles called the Faddeev-Popov ghosts, which carry color charge, behaves as fermions (their fields anticommute) in spite of their boson like propagation. These particles cancel out the contributions of the non-physical gauge field degrees of freedom, and in physical calculations only appear as internal lines of the Feynman diagrams. As it is well known, a perturbative expansion depends on the initial conditions at $t\rightarrow \pm \infty $ or what is the same on the states in which the expansion is based. The perturbation theory at finite orders differs in attention to the ground state selected, or from a functional point of view what boundary conditions are chosen. The perturbation theory in QED (PQED) is in excellent correspondence with the experimental facts. In this theory the expansion is based on a perturbative vacuum state that is the empty of the Fock space, excluding the presence of fermion and boson particles. This is a radical simplification of the exact perturbative ground state that should be a complex combination of states on the Fock basis. Formally the expansion around the Fock vacuum contains all the effects associated to the exact vacuum, but it would require from infinite orders of the expansion in the coupling constant for describing them. The rapid convergence of the perturbative series in PQED indicates that the higher excited states of the Fock basis expansion, in the real vacuum, have a short life and a small influence on physical observable. In QCD the color confinement indicates that the ground state has a non-trivial structure, which in terms of a Fock expansion could be represented with the formation of a gluon “condensate”. Therefore, it should not be surprising that the PQCD fails to describe the low energy physics where the propagator of gluons could be affected by the presence of the condensate, even under the validity of a modified perturbative expansion. Such a perturbative condensate could generate all the effects over the physical observable, which in the standard expansion could require an infinite number of terms of the series. In a previous work [@Cabo], following the above ideas, the construction of a modified perturbation theory for QCD was implemented. This construction retained the main invariance of the theory (the Lorentz and $SU(3)$ ones), and it was also able to reproduce some of the main physical predictions of the chromomagnetic field models. The central idea in that work was to modify the perturbative expansion in such a way that the effects of a gluon condensate could be incorporated. Such a modification is needed to be searched through the connection of the interaction on an alternative state in the Fock space designed to incorporate the presence of the gluon condensate. It is not excluded that this procedure could be also a crude approximation of the reality as in the case in which the connection is done on the Fock vacuum (QED). However, this procedure could produce a reasonable if not good description of the low energy physics. If such is the case the low and high-energy descriptions of QCD could be unified in a common unified perturbation theory. In particular, in that previous work [@Cabo] the results had the interesting outcome of producing a non vanishing mean value for the relevant quantity $G^2$. In addition the effective potential, in terms of the condensation parameter at a first order approximation, showed a minimum at non-vanishing values of that parameter. Therefore, the procedure was able to reproduce at least some central predictions of the chromomagnetic models and general QCD analysis. The main objective of the present work is to search the foundations of the mentioned perturbation theory. The concrete aim is to find a physical state in the Fock space of the non-interacting theory being able to generate that expansion. The canonical quantization formalism for gauge fields, developed by Kugo and Ojima is employed. The exposition will be organized as follows: The Chapter 2 is divided in three sections. In the first one a review of the former work [@Cabo] is done, by also establishing the needs for the present one and the objectives which are planned to be analyzed and solved. In the second section the operational quantization method for gauge fields developed by Kugo and Ojima is discussed. Starting from it, in the third section it is exposed the ansatz for the Fock space state that generates the desired form of the perturbative expansion. The proof that the state satisfies the physical state condition is also given in this section. The Chapter 3 is divided in three sections. In the first one an analysis for the general form of the generating functional in an arbitrary ground state is made. In the second section it is shown that the proposed state can generate the desired modification for the gluon propagator by a proper selection of the parameters at hand. In the third section the modification of the propagator for the ghost particles is investigated, such propagator was not modified in the work [@Cabo] and here this procedure is justified as compatible within the present description. Finally, two appendices are introduced for a detailed analysis of the most elaborated parts in the calculation of transverse, longitudinal and scalar modes contribution to the gluon propagator modification. Ground State Ansatz =================== The previous work [@Cabo] is reviewed, as motivation for the present discussion, and the objectives for the present work stated. It is also reviewed the canonical quantization method for gauge fields developed by Kugo and Ojima (K.O.). Finally the QCD modified vacuum state is proposed and it is shown that this state satisfies the BRST physicality conditions imposed by the K.O. formalism. Motivation ---------- In this section a review of a previous work [@Cabo] is made. The main properties of this approach, as was mentioned in the introduction, were: a\) The ability to produce a gluon-condensation parameter value $\left\langle G^2\right\rangle $ directly in the first approximation. b\) The prediction of a minimum of the effective action for non-vanishing values of the condensation parameter. The discussion in [@Cabo] opened the possibility of reproducing some interesting physical implications of the early chromomagnetic field models for the QCD vacuum [@Savv1; @Savv2] by also solving some of their main shortcoming: The breaking of Lorentz and $SU\left(3\right)$ invariance. However, the discussion in [@Cabo] had also a limitation; that is it was unknown if the state that generated the proposed modification to the gluon propagator was a physical state of the theory. This shortcoming, could be expressed in the gauge parameter dependence of the calculated gluon mass. Below it is reminded the main analysis in [@Cabo]. The exposition was referred to the Euclidean space and the followed conventions were used, $$\begin{aligned} \nabla _\mu ^{ab} &=&\delta ^{ab}\partial _\mu +gf^{abc}A_\mu ^c, \\ F_{\mu \nu }^a &=&\partial _\mu A_\nu ^a-\partial _\nu A_\mu ^a+f^{abc}A_\mu ^cA_\nu ^b,\end{aligned}$$ where $g$ is the coupling constant and $f^{abc}$ are the structure constant of $SU(3)$. The action for the problem, including the auxiliary sources for all the fields was taken as, $$\begin{aligned} S_T\left[ A,\overline{C},C\right] &=&\int d^4x\left\{ -\frac 14F_{\mu \nu }^aF_{\mu \nu }^a+\frac 1{2\alpha }\partial _\mu A_\mu ^a\partial _\nu A_\nu ^a+\overline{C}^a\nabla _\mu ^{ab}\partial _\mu C^b\right. \\ &&\text{ \qquad \qquad \qquad \qquad \qquad \qquad }\left. +J_\mu ^aA_\mu ^a+ \overline{\xi }^aC^a+\overline{C}^a\xi ^a\right\},\end{aligned}$$ where $A_\mu,$ $\overline{C},C$ are the gauge and ghost fields and $\alpha $ is the gauge fixing parameter [@Faddeev] for the Lorentz gauge. The generating functional for the Green functions was expressed in the form $$Z_T\left[ J,\xi,\overline{\xi }\right] =\frac 1N\int D\left( A,\overline{C},C\right) \exp \left\{ S_T\left[ A,\overline{C},C\right] \right\},$$ which through the usual Legendre transformation led to the effective action, $$\Gamma \left[ \Phi \right] =\ln Z\left[ J\right] -J_i\Phi _i,\text{ \quad with ~}\Phi _i=\frac{\delta \ln Z\left[ J\right] }{\delta J_i}, \label{efec}$$ $\Phi _i$ denoted the mean values of the fields, and the compact notation of DeWitt [@Daemi], $$\Phi _i\equiv \left(A,\overline{C},C\right) ;\text{ \qquad }J_i\equiv \left(J,\xi,\overline{\xi }\right),$$ was used. In which $\Phi _i$ and $J_i$ indicate all the fields and sources at a space-time point, respectively. Repeated indices imply space-time integration as well as summation over all the field types and over their Lorentz and color components. The one-loop effective action and the corresponding “quantum” Lagrange equations, in the compact notation, were considered as, $$\begin{aligned} \Gamma \left[ \Phi \right] &=&S\left[ \Phi \right] +\frac 12\ln DetD\left[ \Phi \right], \label{acc} \\ \Gamma _{,i}\left[ \Phi \right] &=&S_{,i}\left[ \Phi \right] +\frac 12S_{,ikj}D_{kj}=-J_i, \label{ecmo}\end{aligned}$$ the functional derivatives were denoted by $$L_{,i}\left[ \Phi \right] =\frac{\delta L\left[ \Phi \right] }{\delta \Phi _i }$$ and the action defined by $S_T=S+J_i\Phi _i$. The $\Phi$ dependent propagator $D$ was defined, as usual, through $$D_{ij}=-S_{,ij}^{-1}\left[ \Phi \right], \label{D}$$ After considering a null mean value for the vector field $\Phi$, as requires the $SO(4)$ invariance, the propagator relation (\[D\]) took the form $$D_{ij}=-S_{,ij}^{-1}\left[ 0\right].$$ In this case the only non vanishing second derivatives of the action were, $$\begin{aligned} \frac{\delta ^2S}{\delta A_\mu ^a\left(x\right) \delta A_\nu ^b\left(x^{\prime }\right) }\left[ 0\right] &=&\delta ^{ab}\left( \partial _{x}^2\delta _{\mu \nu }-\left(1+\frac 1\alpha \right) \partial _\mu ^{x}\partial _\nu ^{x}\right) \delta \left(x-x^{\prime }\right), \label{Sglu} \\ \frac{\delta ^2S}{\delta C^a\left(x\right) \delta \overline{C}^b\left( x^{\prime }\right) }\left[ 0\right] &=&\delta ^{ab}\partial _x^2\delta \left(x-x^{\prime }\right). \label{Sgho}\end{aligned}$$ The gluon and ghost propagators are the inverse kernels of (\[Sglu\]) and (\[Sgho\]). Here, the alternative for a perturbative description of gluon condensation appeared. As (\[Sglu\]) consist of derivatives only, the inverse kernel of the gluon propagator could include coordinate independent terms reflecting a sort of gluon condensation. It should be noticed that the propagator is a $SO\left(4\right)$ tensor (not a vector) then a constant term in it does not led necessary to a breaking of the $SO\left(4\right)$ invariance [@Cabo]. Accordingly with the above remark gluon and ghost propagators were selected as $$\begin{aligned} D_{\mu \nu }^{ab}\left(x\right) &=&\int \frac{dp}{\left(2\pi \right) ^4} \left[ C\delta ^{ab}\delta _{\mu \nu }\delta \left( p\right) +\frac{\delta ^{ab}}{p^2}\left(\delta _{\mu \nu }-\left( 1+\alpha \right) \frac{p_\mu p_\nu }{p^2}\right) \right] \exp \left(ipx\right), \label{Dglu} \\ D_G^{ab}\left(x\right) &=&\int \frac{dp}{\left(2\pi \right) ^4}\frac{ \delta ^{ab}}{p^2}\exp \left(ipx\right), \label{Dgho}\end{aligned}$$ and it was checked that the equations of motion (\[ecmo\]), considering (\[Dglu\]) and (\[Dgho\]) and taking vanishing gluon and ghost fields, were satisfied. After that some implications of the modified gluon propagator, in the standard perturbative calculations, were analyzed [@Cabo]. The first interesting result obtained was the standard one loop polarization tensor. It was modified by a massive term, depending on the condensate parameter, with the form $$m^2=\frac{3g^2}{\left(2\pi \right) ^4}C\left(1-\alpha \right). \label{mass}$$ This result had a dependence on the gauge parameter $\alpha$; which as was mentioned above is one of the shortcomings of the discussion [@Cabo] because it was unknown if this mass term was generated by a non-physical vacuum state. In the present work the idea is to solve this difficulty by explicitly constructing a perturbative state leading to the considered form of the propagator, but also satisfying the BRST physical state condition in the non-interacting limit. The mean value of the squared field intensity operator was also calculated [@Cabo], within the simplest approximation (the tree approximation), with the use of the proposed propagator. That is, it was evaluated the expression $$\langle 0\mid S_g\left[ \Phi \right] \mid 0\rangle \equiv \frac 1N\left[ \int D\left(\Phi \right) S_g\left[ \Phi \right] \exp S_T\left[ \Phi \right] \right] _{J_i=0},$$ with $$S_g\left[ \Phi \right] \equiv \int d^4x\left\{ -\frac 14F_{\mu \nu }^a\left(x\right) F_{\mu \nu }^a\left(x\right) \right\},$$ and the following result was obtained, $$\langle 0\mid S_g\left[ \Phi \right] \mid 0\rangle =-\frac{72g^2C^2}{\left(2\pi \right) ^8}\int d^4x.$$ Then the mean value of $G^2$ took the form $$G^2\equiv \langle 0\mid F_{\mu \nu }^a\left(x\right) F_{\mu \nu }^a\left(x\right) \mid 0\rangle =\frac{288g^2C^2}{\left(2\pi \right) ^8}. \label{G2}$$ The substitution of Eq. (\[G2\]) in Eq. (\[mass\]) gave a rough estimate of the gluon mass. It was selected a particular value of $\alpha =0$ and assumed the more or less accepted value of $g^2G^2$ in the physical vacuum $$g^2G^2\cong 0.5\left(\frac{GeV}{c^2}\right) ^4,$$ then the estimated value of the gluon mass became $$m=0.35\frac{GeV}{c^2}.$$ Finally, an evaluation for the contribution to the effective potential of all the one-loop graphs, having only mass term insertions in the polarization tensor, was done. The result, in terms of $G^2$ (\[G2\]), turned to be of the form [@Cabo], $$V\left(G^2\right) =\frac{G^2}4+\frac 3{16\pi ^2}g^2\frac{G^2}{32}\ln \frac{ g^2G^2}{\mu ^4}, \label{VG}$$ where $\mu $ is the dimensional parameter included by the renormalization procedure. As it can be noticed in (\[VG\]), the effective potential indicates the spontaneous generation of a $G^2$ condensate from the usual perturbative vacuum $\left(G^2=0\right) $. This occurred in close analogy with the chromomagnetic fields. Then from the reviewed functional treatment, there are some interesting features that allow believing that the above procedure could describe relevant phenomena of the low energy region through a perturbative expansion. However some questions needed to be answered and taken as objectives of the present work are: 1- To determine under what conditions the new gluon propagator (\[Dglu\]) corresponds to a modified vacuum satisfying the physical state condition. This could also help in the understanding of the $\alpha$ dependence in the gluonic mass term. 2- To investigate the form of the ghost propagator in the modified vacuum state, because in the previous work [@Cabo] it was taken the as same of the usual perturbative theory. Operational Quantization Formalism ---------------------------------- As it is well known the non-abelian character of Yang-Mills fields determines the asymptotic freedom property, and the quark-confinement problem of QCD. This character simultaneously makes difficult the quantization of such theories. The first approach to this quantization was made by Faddeev and Popov in the path integral formalism [@Faddeev-Popov], with the resulting correct Feynman rules including the Faddeev-Popov ghost fields and the renormalizability of the theory. But this approach has the problem of the absence of notions about the state vector space and the Heisemberg operators. In this case due the non-abelian character of the theory it is not possible to use the operators formalism developed by Gupta-Bleuler [@Gupta] or the more general Nakanishi-Lautrup version [@Nakanishi], which can be used only for the abelian case. This situation occurs because de S-Matrix calculated with those procedures is not unitary in the non abelian case, as it was first mentioned by Feynman [@Feynman]. In the present work the operator formalism developed by T. Kugo and I. Ojima [@Kugo], for the first consistent quantization of the Yang-Mills fields, is considered. This formulation uses the Lagrangian invariance under a global symmetry operation called the BRST transformation [@BRST]. In the following a brief review of the K.O. work is done and the following conventions are used. Let $G$ be a compact Lie group, and $\Lambda$ any matrix in the adjoin representation of its associated Lie Algebra. The matrix $\Lambda$ can be represented as a linear combination of the form $$\Lambda =\Lambda ^aT^a,$$ were $T^a$ are the generators $(a=1,...,$Dim$G=n)$, which can be chosen as Hermitian ones and satisfying $$\left[ T^a,T^b\right] =if^{abc}T^c.$$ The field variations under infinitesimal gauge transformations are given by $$\begin{aligned} \delta _\Lambda A_\mu ^a\left(x\right) &=&\partial _\mu \Lambda ^a\left(x\right) +gf^{acb}A_\mu ^c\left(x\right) \Lambda ^b\left( x\right) =D_\mu ^{ab}\left(x\right) \Lambda ^b, \\ D_\mu ^{ab}\left(x\right) &=&\partial _\mu \delta ^{ab}+gf^{acb}A_\mu ^c\left(x\right).\end{aligned}$$ The metric $g_{\mu \nu }$ is taken in the convention $$g_{00}=-g_{ii}=1\qquad \text{for}\quad i=1,2,3.$$ The complete G.D. Lagrangian to be considered is the one employed in the operator quantization approach [@OjimaTex]. Its explicit form is given by $$\begin{aligned} \mathcal{L} &=&\mathcal{L}_{YM}+\mathcal{L}_{GF}+\mathcal{L}_{FP} \label{Lag} \\ \mathcal{L}_{YM} &=&-\frac 14F_{\mu \nu }^a\left( x\right) F^{\mu \nu,a}\left(x\right), \label{YM} \\ \mathcal{L}_{GF} &=&-\partial ^\mu B^a\left(x\right) A_\mu ^a\left(x\right) +\frac \alpha 2B^a\left(x\right) B^a\left( x\right), \label{GF} \\ \mathcal{L}_{FP} &=&-i\partial ^\mu \overline{c}^a\left(x\right) D_\mu ^{ab}\left(x\right) c^b\left( x\right), \label{FP}\end{aligned}$$ where field intensity is $$F_{\mu \nu }^a\left(x\right) =\partial _\mu A_\nu ^a\left( x\right) -\partial _\nu A_\mu ^a\left(x\right) +gf^{abc}A_\mu ^b\left(x\right) A_\nu ^c\left(x\right).$$ Relation (\[YM\]) defines the standard Yang-Mills Lagrangian, Eq. (\[GF\]) defines the gauge fixing term which can be also rewritten in the form $$\mathcal{L}_{GF}=-\frac 1{2\alpha }\left(\partial ^\mu A_\mu ^a\left(x\right) \right) ^2+\frac \alpha 2\left(B^a\left(x\right) +\frac 1\alpha \partial ^\mu A_\mu ^a\left(x\right) \right) ^2-\partial ^\mu \left( B^a\left(x\right) A_\mu ^a\left(x\right) \right),$$ equivalent to the more familiar $-\frac 1{2\alpha }\left(\partial ^\mu A_\mu ^a\left(x\right) \right) ^2$, at the equations of motion level [@Faddeev] and Feynman diagram expansion. Finally, Eq. (\[FP\]) describes the non-physical Faddeev-Popov ghost sector. The definition for such fields in the Kugo and Ojima (K.O.) approach is satisfying $$\overline{c}^{\dagger }=\overline{c},\text{ \qquad }c^{\dagger }=c.$$ That is, the ghost fields are Hermitian. In the Faddeev-Popov formalism [@Faddeev] they satisfy $$C^{\dagger }=\overline{C},\text{ \qquad }\overline{C}^{\dagger }=C.$$ However, a simple change of variables is able to transform between the ghost fields satisfying both kind of conjugation conditions. The selected conjugation properties, for this sector, allowed Kugo and Ojima to solve various formal problems existing for the application of the BRST operator quantization method to QCD, for example the hermiticity of the Lagrangian, which guarantees the unitarity of the S-Matrix. The physical state conditions in the BRST procedure [@OjimaTex] are given by $$\begin{aligned} &&Q_B\mid phys\rangle =0, \nonumber \\ &&Q_C\mid phys\rangle =0,\end{aligned}$$ where $$Q_B=\int d^3x\left[ B^a\left(x\right) \overleftrightarrow{\partial _0} c^a\left(x\right) +gB^a\left(x\right) f^{abc}A_0^b\left( x\right) c^c\left(x\right) +\frac i2g\partial _0\left( \overline{c}^a\right) f^{abc}c^b\left(x\right) c^c\left(x\right) \right],$$ with $$f\left(x\right) \overleftrightarrow{\partial _0}g\left(x\right) \equiv f\left(x\right) \partial _0g\left(x\right) -\partial _0\left(f\left(x\right) \right) g\left(x\right).$$ The BRST charge is conserved as a consequence of the BRST symmetry of the Lagrangian (\[Lag\]). The also conserved charge $Q_C$ is given by $$Q_C=i\int d^3x\left[ \overline{c}^a\left(x\right) \overleftrightarrow{ \partial _0}c^a\left(x\right) +g\overline{c}^a\left(x\right) f^{abc}A_0^b\left(x\right) c^c\left(x\right) \right],$$ its conservation comes from the Noether theorem, due to the Lagrangian invariance (\[Lag\]) under the phase transformation $c\rightarrow e^\theta c,\ \overline{c}\rightarrow e^{-\theta }\overline{c}$. This charge defines the so called “ghost number” as the difference between the number of ghost $c$ and $\overline{c}$. The analysis here is restricted to the Yang-Mills Theory without spontaneous breaking of the gauge symmetry. The quantization for the theory defined by the Lagrangian (\[Lag\]), considering the interacting free limit $g\rightarrow 0$, leads to the following commutation relations between the free fields, $$\begin{aligned} \left[ A_\mu ^a\left(x\right),A_\nu ^b\left(y\right) \right] &=&\delta ^{ab}\left(-ig_{\mu \nu }D\left(x-y\right) +i\left( 1-\alpha \right) \partial _\mu \partial _\nu E\left(x-y\right) \right), \nonumber \\ \left[ A_\mu ^a\left(x\right),B^b\left( y\right) \right] &=&\delta ^{ab}\left(-i\partial _\mu D\left( x-y\right) \right), \nonumber \\ \left[ B^a\left( x\right),B^b\left(y\right) \right] &=&\left\{ \overline{c} ^a\left(x\right),\overline{c}^b\left(y\right) \right\} =\left\{ c^a\left(x\right),c^b\left(y\right) \right\} =0, \nonumber \\ \left\{ c^a\left(x\right),\overline{c}^b\left(y\right) \right\} &=&-D\left(x-y\right), \label{com}\end{aligned}$$ The $E$ functions are defined by [@OjimaTex] $$E_{\left(.\right) }\left(x\right) =\frac 12\left(\nabla ^2\right) ^{-1}\left(x_0\partial ^0-1\right) D_{\left(.\right) }\left( x\right).$$ The equations of motion for the non-interacting fields take the simple form $$\begin{aligned} \Box A_\mu ^a\left(x\right) -\left(1-\alpha \right) \partial _\mu B^a\left(x\right) &=&0, \\ \partial ^\mu A_\mu ^a\left(x\right) +\alpha B^a\left(x\right) &=&0, \label{liga1} \\ \Box B^a\left( x\right) =\Box c^a\left(x\right) =\Box \overline{c} ^a\left( x\right) &=&0.\end{aligned}$$ This equations can be solved for an arbitrary values of the $\alpha$ parameter. However, the discussion will be restricted to the case $\alpha =1$ which corresponds to the situation in which all the gluon components satisfy the D’Alambert equation. This selection, as considered in the framework of the usual perturbative expansion, implies that you are not able to check the $\alpha$ independence of the physical quantities. This simplification is a necessary requirement. In the present discussion, the aim is to construct a perturbative state that satisfies the BRST physical state condition, in order to connect adiabatically the interaction. Then, the physical character of all the prediction will follow whenever the former assumption that adiabatic connection do not take the state out of the physical subspace at any intermediate state. The consideration of different values of $\alpha $, would be also a convenient recourse for checking the $\alpha$ independent perturbative expansion. However, at this stage it is preferred to delay this more technical issue for future work. In that way the field equations for the $\alpha =1$ are $$\begin{aligned} \Box A_\mu ^a\left(x\right)=\Box B^a\left(x\right) =\Box c^a\left( x\right) =\Box \overline{c} ^a\left(x\right) &=&0, \label{movi1} \\ \partial ^\mu A_\mu ^a\left(x\right) +B^a\left(x\right) &=&0. \label{movi2}\end{aligned}$$ The solutions of the set (\[movi1\]), (\[movi2\]) can be written as $$\begin{aligned} A_\mu ^a\left(x\right) &=&\sum\limits_{\vec{k},\sigma }\left( A_{\vec{k},\sigma }^af_{k,\mu }^\sigma \left(x\right) +A_{\vec{k},\sigma }^{a+}f_{k,\mu }^{\sigma *}\left(x\right) \right), \nonumber \\ B^a\left(x\right) &=&\sum\limits_{\vec{k}}\left(B_{\vec{k}}^ag_k\left(x\right) +B_{\vec{k}}^{a+}g_k^{*}\left(x\right) \right), \nonumber \\ c^a\left(x\right) &=&\sum\limits_{\vec{k}}\left( c_{\vec{k}}^ag_k\left(x\right) +c_{\vec{k}}^{a+}g_k^{*}\left( x\right) \right), \nonumber \\ \overline{c}^a\left(x\right) &=&\sum\limits_{\vec{k}}\left(\overline{c}_{ \vec{k}}^ag_k\left( x\right) +\overline{c}_{\vec{k}}^{a+}g_k^{*}\left(x\right) \right).\end{aligned}$$ The wave packets system, for non-massive scalar and vector fields, are taken in the form $$\begin{aligned} g_k\left(x\right) &=&\frac 1{\sqrt{2Vk_0}}\exp \left(-ikx\right), \nonumber \\ f_{k,\mu }^\sigma \left(x\right) &=&\frac 1{\sqrt{2Vk_0}}\epsilon _\mu ^\sigma \left(k\right) \exp \left( -ikx\right). \label{pol}\end{aligned}$$ The polarization vectors, in Eq. (\[pol\]) are defined by $$\vec{k}\cdot \vec{\epsilon}_\sigma \left(k\right) =0,\ \epsilon _\sigma ^0\left(k\right) =0,$$ and satisfy $$\vec{\epsilon}_\sigma \left(k\right) \cdot \vec{\epsilon}_\tau \left(k\right) =\delta _{\sigma \tau },$$ where $\sigma,\tau =1,2$ are the transverse modes. For the longitudinal $L$ and scalar $S$ modes the definitions are $$\begin{aligned} \epsilon _L^\mu \left(k\right) &=&-ik^\mu =-i\left(\left| \vec{k}\right|, \vec{k}\right),\ \epsilon _L^{\mu *}\left( k\right) =-\epsilon _L^\mu \left(k\right), \\ \epsilon _S^\mu \left(k\right) &=&-i\frac{\overline{k}^\mu }{2\left| \vec{k} \right| ^2}=\frac{-i\left(\left| \vec{k}\right|,-\vec{k}\right) }{2\left| \vec{k}\right| ^2},\ \epsilon _S^{\mu *}\left(k\right) =-\epsilon _S^\mu \left(k\right),\end{aligned}$$ and satisfy $$\begin{aligned} \epsilon _L^{\mu *}\left(k\right) \cdot \epsilon _{L,\mu }\left( k\right) &=&\epsilon _S^{\mu *}\left(k\right) \cdot \epsilon _{S,\mu }\left(k\right) =0, \\ \epsilon _L^{\mu *}\left(k\right) \cdot \epsilon _{S,\mu }\left(k\right) &=&1.\end{aligned}$$ The scalar product of the defined polarizations define the metric matrix $$\widetilde{\eta }_{\sigma \tau }=\epsilon _\sigma ^{\mu *}\left(k\right) \cdot \epsilon _{\tau,\mu }\left(k\right)\equiv \left( \begin{array}{cccc} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array} \right).$$ Now it is possible to introduce the contravariant (in the polarization index) polarizations $$\epsilon ^{\sigma,\mu }\left(k\right) =\sum\limits_{1,2,L,S}\widetilde{ \eta }^{\sigma \tau }\cdot \epsilon _\tau ^\mu \left(k\right),$$ satisfying $$\sum\limits_\sigma \epsilon ^{\sigma,\mu }\left(k\right) \cdot \epsilon _\sigma ^{\nu *}\left(k\right) =\sum\limits_{\sigma,\tau }\widetilde{\eta } ^{\sigma \tau }\cdot \epsilon _\tau ^\mu \left( k\right) \cdot \epsilon _\sigma ^{\nu *}\left(k\right) =g^{\mu \nu }$$ and $$\begin{aligned} \epsilon ^{\sigma,\mu }\left(k\right) \cdot \epsilon _\mu ^{\tau *}\left(k\right) &=&\widetilde{\eta }^{\sigma \tau }, \\ \widetilde{\eta }^{\sigma \tau ^{\prime }}\cdot \widetilde{\eta }_{\tau ^{\prime }\tau } &=&\delta _\tau ^\sigma.\end{aligned}$$ After that, it follows for the vector functions $$\sum\limits_{\vec{k},\sigma }f_{k,\sigma }^\mu \left(x\right) \cdot f_k^{\sigma,\nu *}\left(y\right) =g^{\mu \nu }D_{+}\left( x-y\right).$$ As it can be seen from (\[movi2\]) the $A_{\vec{k},\sigma }^a$ and $B_{\vec{k}}^a$ modes are not all independent. Indeed, it follows from (\[movi2\]) that $$B_{\vec{k}}^a=A_{\vec{k}}^{S,a}=A_{\vec{k},L}^a.$$ Then excluding the scalar mode, the free Heisemberg fields expansion takes the form $$A_\mu ^a\left(x\right) =\sum\limits_{\vec{k}}\left( \sum\limits_{\sigma =1,2}A_{\vec{k},\sigma }^af_{k,\mu }^\sigma \left(x\right) +A_{\vec{k} }^{L,a}f_{k,L,\mu }\left(x\right) +B_{\vec{k}}^af_{k,S,\mu }\left(x\right) \right)+h.c.,$$ where $h.c.$ represents the Hermitian conjugate of the first term. In order to satisfy the commutations relations (\[com\]) the creation and annihilation operator, associated to the Fourier components of the field, should obey $$\begin{aligned} \left[ A_{\vec{k},\sigma }^a,A_{\vec{k}^{\prime },\sigma ^{\prime }}^{a^{\prime }+}\right] &=&-\delta ^{aa^{\prime }}\delta _{\vec{k}\vec{k} ^{\prime }}\eta _{\sigma \sigma ^{\prime }}, \nonumber \\ \left\{ c_{\vec{k}}^a,\overline{c}_{\vec{k}^{\prime }}^{a^{\prime }+}\right\} &=&i\delta ^{aa^{\prime }}\delta _{\vec{k}\vec{k}^{\prime }}, \nonumber \\ \left\{ \overline{c}_{\vec{k}}^a,c_{\vec{k}^{\prime }}^{a^{\prime }+}\right\} &=&-i\delta ^{aa^{\prime }}\delta _{\vec{k}\vec{k}^{\prime }}\end{aligned}$$ and all the other vanish. In a symbolic matrix form these relations can be arranged as follows $$\begin{array}{cccccc} & A_{\vec{k}^{\prime },\sigma ^{\prime }}^{a^{\prime }+} & A_{\vec{k} ^{\prime }}^{L,a^{\prime }+} & B_{\vec{k}^{\prime }}^{a^{\prime }+} & c_{ \vec{k}^{\prime }}^{a+} & \overline{c}_{\vec{k}^{\prime }}^{a+} \\ A_{\vec{k},\sigma }^a & \delta ^{aa^{\prime }}\delta _{\vec{k}\vec{k} ^{\prime }}\delta _{\sigma \sigma ^{\prime }} & 0 & 0 & 0 & 0 \\ A_{\vec{k}}^{L,a} & 0 & 0 & -\delta ^{aa^{\prime }}\delta _{\vec{k}\vec{k} ^{\prime }} & 0 & 0 \\ B_{\vec{k}}^a & 0 & -\delta ^{aa^{\prime }}\delta _{\vec{k}\vec{k}^{\prime }} & 0 & 0 & 0 \\ c_{\vec{k}}^a & 0 & 0 & 0 & 0 & i\delta ^{aa^{\prime }}\delta _{\vec{k}\vec{k }^{\prime }} \\ \overline{c}_{\vec{k}}^a & 0 & 0 & 0 & -i\delta ^{aa^{\prime }}\delta _{\vec{ k}\vec{k}^{\prime }} & 0 \end{array} \label{commu}$$ The above commutation rules and equation of motions define the quantized non-interacting limit of G.D. Then, it is possible now to define the alternative interacting free ground state to be considered for the adiabatic connection of the interaction. As discussed before, the expectation is that the physics of the perturbation theory to be developed will be able to furnish good description of some low energy physical effects. It is interesting to comment now that one of the first tasks proposed for the present work was to construct a state, in quantum electrodynamics, able to generate a modification for the photon propagator similar to the one proposed in [@Cabo] for gluons. It was used the quantification operator method developed by Gupta and Bleuler (GB), however was impossible to find any state generating this covariant propagator modification and satisfying the physical state condition imposed by this formalism. In the GB formalism the physical state condition is given by $$\partial ^\mu A_\mu ^{+}\left(x\right) \mid \Phi \rangle =0,$$ or in terms of the annihilation operators [@Sokolov], by $$k_0\left(A_{\vec{k},3}-A_{\vec{k},0}\right) \mid \Phi \rangle =0.$$ The more general state satisfying this condition is [@GuptaTex] $$\mid \Phi \rangle =\sum\limits_{m,n_{1,}n_2}B_{n_{1,}n_2,m}\mid \Phi \left(n_{1,}n_2,m\right) \rangle,$$ with $$\mid \Phi \left(n_{1,}n_2,m\right) \rangle =\left(m!\right) ^{-\frac 12}\left(A_{\vec{k},3}^{+}-A_{\vec{k},0}^{+}\right) ^m\left(n_1!n_2!\right) ^{-\frac 12}\left( A_{\vec{k},1}^{+}\right) ^{n_1}\left(A_{ \vec{k},2}^{+}\right) ^{n_2}\mid 0\rangle,$$ where $B_{n_{1,}n_2,m}$ are arbitrary constants. This general form of the state is the one that disabled to find a covariant modification to the propagator. The alternative vacuum state ---------------------------- In the present section the construction of a relativistic invariant ground state in the non-interacting limit of QCD is considered. It will be required that the proposed state satisfies the BRST physical state conditions. Then this state will have an opportunity to furnish the gluodynamics ground state under the adiabatic connection of the interaction. After beginning to work in the K.O. formalism some indications were found, that the appropriate state obeying the physical state conditions in this procedure, and with possibilities for generating the modification to the gluon propagator proposed in the previous work, could have the general structure $$\mid \phi \rangle =\exp \sum\limits_a\left(\sum\limits_{\sigma =1,2}\frac 12C_\sigma \left(\left| \vec{p}\right| \right) A_{\vec{p},\sigma }^{a+}A_{ \vec{p},\sigma }^{a+}+C_3\left(\left| \vec{p}\right| \right) \left(B_{\vec{ p}}^{a+}A_{\vec{p}}^{L,a+}+i\overline{c}_{\vec{p}}^{a+}c_{\vec{p} }^{a+}\right) \right) \mid 0\rangle, \label{Vacuum}$$ where $\vec{p}$ is an auxiliary momentum chosen as one of the few smallest momenta of the quantized theory in a finite volume $V$. This value will be taken later in the limit $V\rightarrow \infty $ for recovering Lorentz invariance. From here the sum on the color index $a$ will be explicit. The parameters $C_i\left(\left| \vec{p}\right| \right)$, $i=1,2,3,$ will be fixed below from the condition that the free propagator associated to a state satisfying the BRST physical state condition, coincides with the one proposed in the previous work [@Cabo]. The solution of this problem, then would give a more solid foundation to the physical implications of the discussion in that work. It should also be noticed that the state defined by Eq. (\[Vacuum\]) has some similarity with the coherent states [@Itzykson]. However, in the present case, the creation operators appear in squares. Thus, the argument of the exponential creates pairs of physical and non-physical particles. An important property of this function is that its construction in terms of creation operator pairs determines that the mean value of an odd number of field operators vanishes. This at variance with the standard coherent state in which the mean values of the fields are nonzero. The vanishing of the mean fields is a property in common with the standard perturbative vacuum, in which Lorentz invariance could be broken by a non-vanishing expectation value of a 4-vector the gauge field. It should be also stressed that this state as formed by the superposition of gluons state pairs suggests a connection with some recent works in the literature that consider the formation gluons pairs due to the color interactions. Now it is checked that the state (\[Vacuum\]) satisfies the BRST physical state conditions $$\begin{aligned} &&Q_B\mid \Phi \rangle =0, \nonumber \\ &&Q_C\mid \Phi \rangle =0.\end{aligned}$$ The expressions for these charges in the interaction free limit [@OjimaTex] are $$\begin{aligned} &&Q_B=i\sum\limits_{\vec{k},a}\left( c_{\vec{k}}^{a+}B_{\vec{k}}^a-B_{\vec{k} }^{a+\ }c_{\vec{k}}^a\right), \nonumber \\ &&Q_C=i\sum\limits_{\vec{k},a}\left( \overline{c}_{\vec{k}}^{a+}c_{\vec{k} }^a+c_{\vec{k}}^{a+}\overline{c}_{\vec{k}}^a\right).\end{aligned}$$ Considering first the action of $Q_B$ on the proposed state, [ $$\begin{aligned} &&Q_B\mid \Phi \rangle =i\exp \left\{ \sum\limits_{\sigma,a}\frac 12C_\sigma \left(\left| \vec{p}\right| \right) A_{\vec{p},\sigma }^{a+}A_{\vec{p},\sigma }^{a+}\right\} \times \nonumber \\ &&\times \left(\exp \left\{ \sum\limits_aC_3\left(\left| \vec{p}\right| \right) i\overline{c}_{\vec{p}}^{a+}c_{\vec{p}}^{a+}\right\} \sum\limits_{ \vec{k},b}c_{\vec{k}}^{b+}B_{\vec{k}}^b\exp \left\{ \sum\limits_aC_3\left(\left| \vec{p}\right| \right) B_{\vec{p}}^{a+}A_{\vec{p}}^{L,a+}\right\} \right. \\ &&-\left. \exp \left\{ \sum\limits_aC_3\left(\left| \vec{p}\right| \right) B_{\vec{p}}^{a+}A_{\vec{p}}^{L,a+}\right\} \sum\limits_{\vec{k},b}B_{\vec{k} }^{b+}c_{\vec{k}}^b\exp \left\{ \sum\limits_aC_3\left(\left| \vec{p}\right| \right) i\overline{c}_{\vec{p}}^{a+}c_{\vec{p}}^{a+}\right\} \right) \mid 0\rangle =0, \nonumber\end{aligned}$$ ]{}where the identity $$\left[ B_{\vec{k}}^b,\exp \sum\limits_aC_3\left(\left| \vec{p}\right| \right) B_{\vec{p}}^{a+}A_{\vec{p}}^{L,a+}\right] =-C_3\left(\left| \vec{p} \right| \right) B_{\vec{p}}^{b+}\delta _{\vec{k},\vec{p}}\exp \sum\limits_aC_3\left(\left| \vec{p}\right| \right) B_{\vec{p}}^{a+}A_{\vec{ p}}^{L,a+}, \label{ident1}$$ was used. For the action of $Q_C$ on the considered state it follows [ $$\begin{aligned} &&Q_C\mid \Phi \rangle =i\exp \left\{ \sum\limits_{\sigma,a}\frac 12C_\sigma \left(\left| \vec{p}\right| \right)A_{\vec{p},\sigma }^{a+}A_{\vec{p},\sigma }^{a+}+\sum\limits_aC_3\left(\left| \vec{p}\right| \right) B_{\vec{p} }^{a+}A_{\vec{p}}^{L,a+}\right\} \\ &&\times \left[ \sum\limits_{\vec{k},b} \overline{c}_{\vec{k}}^{b+}c_{\vec{k} }^b\left( 1+\sum\limits_aiC_3\left(\left| \vec{p}\right| \right) \overline{ c }_{\vec{p}}^{a+}c_{\vec{p}}^{a+}\right) +\sum\limits_{\vec{k},b}c_{\vec{k} }^{b+}\overline{c}_{\vec{k}}^b\left(1+\sum\limits_aiC_3\left( \left| \vec{p} \right| \right) \overline{c}_{\vec{p}}^{a+}c_{\vec{p}}^{a+}\right) \right] \mid 0\rangle =0 \nonumber\end{aligned}$$ ]{}which vanishes due to the commutation rules of the ghost operators (\[commu\]). Next, the evaluation of norm of the proposed state is considered, which due to the commutation properties of the operator can be written as $$\begin{aligned} \langle \Phi \mid \Phi \rangle =\prod\limits_{a=1,..,8} &\prod\limits_{\sigma =1,2}&\langle 0\mid \exp \left\{ \frac 12C_\sigma ^{*}\left(\left| \vec{p}\right| \right) A_{\vec{p},\sigma }^aA_{\vec{p},\sigma }^a\right\} \exp \left\{ \frac 12C_\sigma \left(\left| \vec{p} \right| \right) A_{\vec{p},\sigma }^{a+}A_{\vec{p},\sigma }^{a+}\right\} \mid 0\rangle \nonumber \\ &\times &\langle 0\mid \exp \left\{ C_3^{*}\left(\left| \vec{p}\right| \right) A_{\vec{p}}^{L,a}B_{\vec{p}}^a\right\} \exp \left\{ C_3\left( \left| \vec{p}\right| \right) B_{\vec{p}}^{a+}A_{\vec{p}}^{L,a+}\right\} \mid 0\rangle \nonumber \\ &\times &\langle 0\mid \left(1-iC_3^{*}\left(\left| \vec{p}\right| \right) c_{\vec{p}}^a\overline{c}_{\vec{p}}^a\right) \left(1+iC_3\left( \left| \vec{ p}\right| \right) \overline{c}_{\vec{p}}^{a+}c_{\vec{p}}^{a+}\right) \mid 0\rangle.\end{aligned}$$ For the product of the factors associated with transverse modes and the eight values of the color index, after expanding the exponential in series, it follows that $$\begin{aligned} &&\left[ \langle 0\mid \exp \left\{ \frac 12C_\sigma ^{*}\left( \left| \vec{p }\right| \right) A_{\vec{p},\sigma }^aA_{\vec{p},\sigma }^a\right\} \exp \left\{ \frac 12C_\sigma \left(\left| \vec{p}\right| \right) A_{\vec{p},\sigma }^{a+}A_{\vec{p},\sigma }^{a+}\right\} \mid 0\rangle \right] ^8 \nonumber \\ &&=\left[ \langle 0\mid \sum\limits_{m=0}^\infty \left| \frac 12C_\sigma \left(\left| \vec{p}\right| \right) \right| ^{2m}\frac{\left(A_{\vec{p},\sigma }^a\right) ^{2m}\left( A_{\vec{p},\sigma }^{a+}\right) ^{2m}}{\left(m!\right) ^2}\mid 0\rangle \right] ^8 \nonumber \\ &&=\left[ \sum\limits_{m=0}^\infty \left| \frac 12C_\sigma \left(\left| \vec{p}\right| \right) \right| ^{2m}\frac{\left(2m\right) !}{\left(m!\right) ^2}\right] ^8, \label{normT}\end{aligned}$$ where the identity $$\langle 0\mid \left(A_{\vec{p},\sigma }^a\right) ^{2m}\left( A_{\vec{p},\sigma }^{a+}\right) ^{2m}\mid 0\rangle =\left( 2m\right) !,$$ was used. The factors linked with the scalar and longitudinal modes can be transformed as follows $$\begin{aligned} &&\left[ \langle 0\mid \exp \left\{ C_3^{*}\left(\left| \vec{p}\right| \right) A_{\vec{p}}^{L,a}B_{\vec{p}}^a\right\} \exp \left\{ C_3\left(\left| \vec{p}\right| \right) B_{\vec{p}}^{a+}A_{\vec{p}}^{L,a+}\right\} \mid 0\rangle \right] ^8 \nonumber \\ &&=\left[ \langle 0\mid \sum\limits_{m=0}^\infty \left| C_3\left(\left| \vec{p}\right| \right) \right| ^{2m}\frac{\left(A_{\vec{p}}^{L,a}B_{\vec{p} }^a\right) ^m\left( B_{\vec{p}}^{a+}A_{\vec{p}}^{L,a+}\right) ^m}{\left(m!\right) ^2}\mid 0\rangle \right] ^8 \nonumber \\ &&=\left[ \sum\limits_{m=0}^\infty \left| C_3\left(\left| \vec{p}\right| \right) \right| ^{2m}\right] ^8=\left[ \frac 1{\left(1-\left| C_3\left(\left| \vec{p}\right| \right) \right| ^2\right) }\right] ^8\text{ for}\quad \left| C_3\left(\left| \vec{p}\right| \right) \right| <1, \label{normLS}\end{aligned}$$ in which the identity $$\langle 0\mid \left(A_{\vec{p}}^{L,a}B_{\vec{p}}^a\right) ^m\left( B_{\vec{p }}^{a+}A_{\vec{p}}^{L,a+}\right) ^m\mid 0\rangle =\left( m!\right) ^2,$$ was employed. Finally the factor connected with the ghost fields can be calculated as follows $$\begin{aligned} &&\left[ \langle 0\mid \left(1-iC_3^{*}\left(\left| \vec{p}\right| \right) c_{\vec{p}}^a\overline{c}_{\vec{p}}^a\right) \left( 1+iC_3\left(\left| \vec{ p}\right| \right) \overline{c}_{\vec{p}}^{a+}c_{\vec{p}}^{a+}\right) \mid 0\rangle \right] ^8 \nonumber \\ &&=\left[ 1+\left| C_3\left(\left| \vec{p}\right| \right) \right| ^2\langle 0\mid c_{\vec{p}}^a\overline{c}_{\vec{p}}^a\overline{c}_{\vec{p}}^{a+}c_{ \vec{p}}^{a+}\mid 0\rangle \right] =\left[ 1-\left| C_3\left( \left| \vec{p} \right| \right) \right| ^2\right] ^8. \label{normG}\end{aligned}$$ After substituting all the calculated factors, the norm of the state can be written as $$N=\langle \Phi \mid \Phi \rangle =\prod\limits_{\sigma =1,2}\left[ \sum\limits_{m=0}^\infty \left| C_\sigma \left(\left| \vec{p}\right| \right) \right| ^{2m}\frac{\left(2m\right) !}{\left(m!\right) ^2}\right] ^8.$$ Therefore, it is possible to define the normalized state $$\mid \widetilde{\Phi }\rangle =\frac 1{\sqrt{N}}\mid \Phi \rangle.$$ Note that, as it should be expected, the norm is not dependent on the $ C_3\left(\left| \vec{p}\right| \right) $ parameter which defines the non-physical particle operators entering in the definition of the proposed vacuum state. Propagator Modifications ======================== The general form for generating functionals and propagators, for boson and fermion particles in an arbitrary vacuum state, are analyzed. The modification for the gluon and ghost propagators, introduced by the vacuum state defined in the previous chapter, are calculated. General Form of the Propagator ------------------------------ As it is well known in the Quantum Field Theory to calculate any element of the S-Matrix, after applying the reduction formulas, it is necessary to obtain the vacuum expectation value of the temporal ordering of Heisemberg operators [@Gasiorowicz]. That is it is needed to calculate $$\langle \Psi \mid T\left(\hat{A}_H\left(x_1\right) \hat{A}_H\left( x_2\right) \hat{A}_H\left(x_3\right)...\right) \mid \Psi \rangle, \label{orden1}$$ where $\mid \Psi \rangle $ is the real vacuum of the interacting theory. For simplifying the exposition it is considered a scalar field, the generalization for vector fields is straightforward. Using the relations between the operators in the Interaction and Heisemberg representations $$\begin{aligned} &&\hat{A}_H\left(x\right) =\hat{U}\left(0,t\right) \hat{A}_I\left( x\right) \hat{U}\left(t,0\right), \\ &&\hat{U}\left( t_1,t_2\right) \hat{U}\left(t_2,t_3\right) =\hat{U}\left( t_1,t_3\right)\end{aligned}$$ and assuming that the real vacuum interacting state can be obtained from the non-interacting one under the adiabatic connection of the interaction. The expression (\[orden1\]) takes the form [@Gasiorowicz] $$\frac{\langle \Phi \mid T\left\{ \hat{A}_I\left(x_1\right) \hat{A}_I\left(x_2\right) \hat{A}_I\left(x_3\right)...\exp \left( -\int\limits_{-\infty }^\infty H_i\left(t\right) dt\right) \right\} \mid \Phi \rangle }{\langle \Phi \mid T\left\{ \exp \left(-\int\limits_{-\infty }^\infty H_i\left(t\right) dt\right) \right\} \mid \Phi \rangle }, \label{orden2}$$ where $\Phi $ is the non interacting vacuum of the theory. To evaluate these quantities it is needed to develop the exponential in series of perturbation theory and calculate the vacuum expectation values of the temporal ordering of fields in the interaction representation ($ \hat{A}_I\left(x\right) $), but in this representation the field operators are like free fields ($\hat{A}^0\left(x\right))$ about which much is known. $$\hat{A}_I\left(x\right) =\hat{A}^0\left(x\right).$$ And it is necessary to evaluate terms of the form $$\langle \Phi \mid T\left(\hat{A}^0\left(x_1\right) \hat{A}^0\left( x_2\right) \hat{A}^0\left(x_3\right)...\right) \mid \Phi \rangle.$$ Introducing the auxiliary generating functional $$Z\left[ J\right] \equiv \langle \Phi \mid T\left(\exp \left\{ i\int d^4xJ\left(x\right) A^0\left(x\right) \right\} \right) \mid \Phi \rangle, \label{genfun}$$ it is possible to write for the relevant expectation values the expression $$\langle \Phi \mid T\left(\hat{A}^0\left(x_1\right) \hat{A}^0\left( x_2\right) \hat{A}^0\left(x_3\right)...\right) \mid \Phi \rangle =\left(\frac 1i\frac \delta {\delta J\left(x_1\right) }\frac 1i\frac \delta {\delta J\left(x_1\right) }\frac 1i\frac \delta {\delta J\left(x_1\right) }...Z\left[ J\right] \right) _{J=0}.$$ Considering now the auxiliary functional $$Z\left[ J;t\right]\equiv \langle \Phi \mid T\left(\exp \left\{ i\int\limits_{-\infty }^tdt\int d^3xJ\left(x\right) A^0\left( x\right) \right\} \right) \mid \Phi \rangle$$ and defining $W\left(t\right)$ through the relation $$\begin{aligned} &&T\left(\exp \left\{ i\int\limits_{-\infty }^tdt\int d^3xJ\left( x\right) A^0\left(x\right) \right\} \right)\nonumber \\ && =T\left(\exp \left\{ i\int\limits_{-\infty }^tdt\int d^3xJ\left( x\right) A^{0-}\left(x\right) \right\} \right) W\left(t\right), \label{T11}\end{aligned}$$ where $A^{0-}\left(x\right)$ and $A^{0+}\left( x\right)$ are the negative (creation) and positive (annihilation) frequency parts, respectively. The $t$ differentiation on the expression (\[T11\]), takes the form $$\begin{aligned} &i&\int\limits_{x_0=t}d^3xJ\left(x\right) A^0\left(x\right) T\left(\exp \left\{ i\int\limits_{-\infty }^tdt\int d^3xJ\left( x\right) A^{0-}\left(x\right) \right\} \right) W\left(t\right) \nonumber \\ &=&T\left(\exp \left\{ i\int\limits_{-\infty }^tdt\int d^3xJ\left(x\right) A^{0-}\left(x\right) \right\} \right) \frac{dW\left(t\right) }{dt}+ \nonumber \\ &&+i\int\limits_{x_0=t}d^3xJ\left(x\right) A^{0-}\left(x\right) T\left(\exp \left\{ i\int\limits_{-\infty }^tdt\int d^3xJ\left( x\right) A^{0-}\left(x\right) \right\} \right) W\left(t\right).\end{aligned}$$ Keeping in mind that the free field creation operators commute, for all times, the following relation holds $$\left[ A^{0-}\left(x\right),A^{0-}\left(y\right) \right] =0,$$ then the $T$ instruction can be eliminated and after some algebra is obtained [ $$\begin{aligned} \frac{dW\left(t\right) }{dt} &=&i\exp \left\{ -i\int\limits_{-\infty }^tdt\int d^3xJ\left(x\right) A^{0-}\left( x\right) \right\} \int\limits_{x_0=t}d^3xJ\left(x\right) A^{0+}\left(x\right) \times \nonumber \\ &&\times \exp \left\{ i\int\limits_{-\infty }^tdt\int d^3xJ\left(x\right) A^{0-}\left( x\right) \right\} W\left(t\right) \nonumber \\ &=&i\int\limits_{y_0=t}d^3yJ\left(y\right) \left\{ A^{0+}\left( y\right) -i\int\limits_{-\infty }^td^4xJ\left(x\right) \left[ A^{0-}\left(x\right),A^{0+}\left(y\right) \right] \right\} W\left( t\right). \label{dif1}\end{aligned}$$]{} The initial condition on $W\left(t\right)$ is $$W\left(-\infty \right) =1.$$ Then the solution of (\[dif1\]) is $$\begin{aligned} W\left(t\right) &=&\exp \left\{ i\int\limits_{-\infty }^td^4yJ\left(y\right) A^{0+}\left(y\right) \right\}\nonumber \\&& \times \exp \left\{ \int\limits_{-\infty }^td^4y \int\limits_{-\infty }^{y_0}d^4xJ\left(y\right) J\left(x\right) \left[ A^{0-}\left(x\right),A^{0+}\left(y\right) \right] \right\},\end{aligned}$$ when $t\rightarrow \infty $ this expression takes the form $$\begin{aligned} W\left(\infty \right) &=&\exp \left\{ i\int d^4yJ\left(y\right) A^{0+}\left(y\right) \right\} \times \\ &&\times \exp \left\{ \int d^4xd^4y\theta \left(y_0-x_0\right) J\left(y\right) J\left( x\right) \left[ A^{0-}\left(x\right),A^{0+}\left(y\right) \right] \right\}.\end{aligned}$$ Therefore, the generating functional (\[genfun\]) can be written in the following way [@Gasiorowicz] $$\begin{aligned} Z\left[ J\right] &\equiv &\langle \Phi \mid \exp \left\{ i\int d^4xJ\left(x\right) A^{0-}\left(x\right) \right\} \exp \left\{ i\int d^4yJ\left(y\right) A^{0+}\left(y\right) \right\} \mid \Phi \rangle \nonumber \\ &&\times \exp \left\{ \frac i2\int d^4xd^4yJ\left(x\right) D(x-y)J\left(y\right) \right\}, \label{bosones}\end{aligned}$$ where $D(x-y)$ is the usual propagator for an scalar particle. In case that is needed to calculate a similar matrix element for fermions the following functional is defined $$Z\left[ \eta,\bar{\eta}\right] \equiv \langle \Phi \mid T\left( \exp \left\{ i\int d^4x\left[ \bar{\eta}\left(x\right) \psi ^0\left(x\right) + \bar{\psi}^0\left(x\right) \eta \left(x\right) \right] \right\} \right) \mid \Phi \rangle.$$ Because of the anticommuting properties of $\bar{\psi},\ \psi$ fields the introduced sources $\bar{\eta},\ \eta$ satisfy anticommuting relations between then and with the field operators. Here is assumed the left differentiation convention, then the S-Matrix element can be calculate by the following expression $$\begin{aligned} &&\langle \Phi \mid T\left(\psi ^0\left(y_1\right) \bar{\psi}^0\left(z_1\right) \psi ^0\left(y_2\right) ...\bar{\psi}^0\left(z_k\right) \right) \mid \Phi \rangle \nonumber \\ &&=\left(\frac 1i\frac \delta {\delta \eta \left( z_k\right) }...\frac 1i\frac \delta {\delta \bar{\eta}\left( y_2\right) }\frac 1i\frac \delta {\delta \eta \left(z_1\right) }\frac 1i\frac \delta {\delta \bar{\eta} \left(y_1\right) }Z\left[ \eta,\bar{\eta}\right] \right) _{\eta,\bar{\eta} =0}.\end{aligned}$$ Now, in the same way that for the bosons, the following auxiliary functional is defined by $$Z\left[ \eta,\bar{\eta};t\right] \equiv \langle \Phi \mid T\left( \exp \left\{ i\int\limits_{-\infty }^tdt\int d^3x\left[ \bar{\eta}\left(x\right) \psi ^0\left(x\right) +\bar{\psi}^0\left( x\right) \eta \left(x\right) \right] \right\} \right) \mid \Phi \rangle$$ and the corresponding $G\left(t\right)$ functional by $$\begin{aligned} &&T\left(\exp \left\{ i\int\limits_{-\infty }^tdt\int d^3x\left[ \bar{\eta} \left(x\right) \psi ^0\left(x\right) +\bar{\psi}^0\left(x\right) \eta \left(x\right) \right] \right\} \right) \nonumber \\ &&=T\left(\exp \left\{ i\int\limits_{-\infty }^tdt\int d^3x\left[ \bar{\eta} \left(x\right) \psi ^{0-}\left( x\right) +\bar{\psi}^{0-}\left(x\right) \eta \left(x\right) \right] \right\} \right) G\left(t\right). \label{ferm1}\end{aligned}$$ Manipulations completely parallel to those leading to (\[dif1\]) give [ $$\begin{aligned} \frac{dG\left(t\right) }{dt}=i &&\left[ \int\limits_{y_0=t}d^3y\left(\bar{ \eta}\left(y\right) \psi ^{0+}\left(y\right) +\bar{\psi}^{0+}\left(y\right) \eta \left( y\right) \right) \right. \nonumber \\ &&+i\int\limits_{-\infty }^td^4x\int\limits_{y_0=t}d^3y\bar{\eta}\left(y\right) \left\{ \psi ^{0+}\left(y\right),\bar{\psi}^{0-}\left(x\right) \right\} \eta \left(x\right) \nonumber \\ &&\left. -i\int\limits_{-\infty }^td^4x\int\limits_{y_0=t}d^3y\bar{\eta} \left(x\right) \left\{ \psi ^{0-}\left(x\right),\bar{\psi}^{0+}\left(y\right) \right\} \eta \left(y\right) \right] G\left(t\right),\end{aligned}$$]{} This equation is easily integrated to obtain the solution $$\begin{aligned} G\left(t\right) &=&\exp \left\{ i\int\limits_{-\infty }^td^4y\left(\bar{ \eta}\left(y\right) \psi ^{0+}\left(y\right) +\bar{\psi}^{0+}\left(y\right) \eta \left(y\right) \right) \right\} \nonumber \\ &&\times \exp \left\{ -\int\limits_{-\infty }^td^4y\int\limits_{-\infty }^{y_0}d^4x\bar{\eta}\left(y\right) \left\{ \psi ^{0+}\left(y\right),\bar{ \psi}^{0-}\left(x\right) \right\} \eta \left(x\right) \right\} \nonumber \\ &&\times \exp \left\{ \int\limits_{-\infty }^td^4x\int\limits_{-\infty }^{x_0}d^4y\bar{\eta}\left(y\right) \left\{ \psi ^{0-}\left(y\right),\bar{ \psi}^{0+}\left(x\right) \right\} \eta \left(x\right) \right\},\end{aligned}$$ where in the last term the dummy variables $x$ and $y$ were interchanged. Consequently the following expression for the generating functional arise [@Gasiorowicz] [ $$\begin{aligned} Z\left[ \eta,\bar{\eta}\right] &\equiv &\langle \Phi \mid \exp \left\{ i\int d^4x\left[ \bar{\eta}\left(x\right) \psi ^{0-}\left( x\right) +\bar{ \psi}^{0-}\left(x\right) \eta \left(x\right) \right] \right\} \nonumber \\ &&\quad\times\exp \left\{ i\int d^4x\left[ \bar{\eta}\left(x\right) \psi ^{0+}\left(x\right) +\bar{\psi}^{0+}\left(x\right) \eta \left(x\right) \right] \right\} \mid \Phi \rangle \nonumber \\ &&\times \exp \left\{ i\int d^4xd^4y\bar{\eta}\left(x\right) S\left(x-y\right) \eta \left(y\right) \right\} \label{fermiones}\end{aligned}$$]{} where $S\left(x-y\right)$ is the standard fermion propagator. As much for the case of bosons as for fermions the term related with the vacuum expectation value for the usual vacuum is one. This is so because the annihilation operators are located to the right and to the left those of creation. However in the present work the vacuum expectation values generate the propagator modifications, because the vacuum state considered is not the trivial one. The other term in the generating functional expression, that is expressed by a simple exponential of c numbers, gives the usual propagator and it has the same form when is calculated by this operational method or alternatively by the functional method. Then, starting from the analysis in the present section it can be concluded that from an operation formalism point of view any modification to the usual propagators is only generated by a change in the vacuum state of the theory. And these modifications can be determined through the vacuum expectation values in (\[bosones\]) and (\[fermiones\]). From a functional formalism point of view, the propagator modifications are generated by changes in the boundary conditions. Modified Gluon Propagator ------------------------- As it follows from the general form of the Wick Theorem, analyzed in the previous section, the modification of the gluon propagator introduced by the modified vacuum state (\[Vacuum\]) is defined by the expression $$\langle \widetilde{\Phi }\mid \exp \left\{ i\int d^4xJ^{\mu ,a}\left(x\right) A_\mu ^{a-}\left(x\right) \right\} \exp \left\{ i\int d^4xJ^{\mu,a}\left(x\right) A_\mu ^{a+}\left(x\right) \right\} \mid \widetilde{\Phi } \rangle, \label{mod}$$ for each value of the color index $a$. All the different colors can be worked out independently because of the commutation relations between the annihilation and creation operators for the free theory. At the necessary point of the analysis all the color contributions will be included. The annihilation and creation fields in (\[mod\]) are given by $$\begin{aligned} A_\mu ^{a+}\left(x\right) &=&\sum\limits_{\vec{k}}\left( \sum\limits_{\sigma =1,2}A_{\vec{k},\sigma }^af_{k,\mu }^\sigma \left(x\right) +A_{\vec{k}}^{L,a}f_{k,L,\mu }\left(x\right) +B_{\vec{k} }^af_{k,S,\mu }\left(x\right) \right), \\ A_\mu ^{a-}\left(x\right) &=&\sum\limits_{\vec{k}}\left( \sum\limits_{\sigma =1,2}A_{\vec{k},\sigma }^{a+}f_{k,\mu }^{\sigma *}\left(x\right) +A_{\vec{k}}^{L,a+}f_{k,L,\mu }^{*}\left(x\right) +B_{\vec{k} }^{a+}f_{k,S,\mu }^{*}\left( x\right) \right).\end{aligned}$$ In what follows it is calculated explicitly, for each color, the action of the exponential operators $$\begin{aligned} &&\exp \left\{ i\int d^4xJ^{\mu,a}\left(x\right) A_\mu ^{a+}\left( x\right) \right\} \mid \Phi \rangle \nonumber \\ &&=\exp \left\{ i\int d^4xJ^{\mu,a}\left(x\right) \sum\limits_{\vec{k} }\left( \sum\limits_{\sigma =1,2}A_{\vec{k},\sigma }^af_{k,\mu }^\sigma \left(x\right) +A_{\vec{k}}^{L,a}f_{k,L,\mu }\left(x\right) +B_{\vec{k} }^af_{k,S,\mu }\left(x\right) \right) \right\} \nonumber \\ &&\quad\times \exp \left\{ \sum\limits_{\sigma =1,2}\frac 12C_\sigma \left(\left| \vec{p}\right| \right) A_{\vec{p},\sigma }^{a+}A_{\vec{p},\sigma }^{a+}+C_3\left(\left| \vec{p}\right| \right) \left(B_{\vec{p}}^{a+}A_{ \vec{p}}^{L,a+}+i\overline{c}_{\vec{p}}^{a+}c_{\vec{p}}^{a+}\right) \right\} \mid 0\rangle. \label{expd}\end{aligned}$$ After a systematic use of the commutation relations among the annihilation and creation operators, the exponential operators can be decomposed in products of exponential for each space-time mode. This fact allows to perform the calculation for each mode independently. Then the expression (\[expd\]) takes the form [ $$\begin{aligned} &\prod\limits_{\sigma =1,2}&\exp \left\{ i\int d^4xJ^{\mu ,a}\left(x\right) \sum\limits_{\vec{k}}A_{\vec{k},\sigma }^af_{k,\mu }^\sigma \left(x\right) \right\} \exp \left\{ \frac 12C_\sigma \left(\left| \vec{p}\right| \right) A_{\vec{p},\sigma }^{a+}A_{\vec{p},\sigma }^{a+}\right\} \mid 0\rangle \nonumber \\ &\times &\exp \left\{ i\int d^4xJ^{\mu,a}\left(x\right) \sum\limits_{\vec{k }}\left(B_{\vec{k}}^af_{k,S,\mu }\left( x\right)+A_{\vec{k} }^{L,a}f_{k,L,\mu }\left(x\right)\right) \right\} \nonumber \\ &\times &\exp \left\{ C_3\left(\left| \vec{p}\right| \right) B_{\vec{p} }^{a+}A_{\vec{p}}^{L,a+}\right\} \mid 0\rangle \exp \left\{ C_3\left(\left| \vec{p}\right| \right) i\overline{c}_{\vec{p}}^{a+}c_{\vec{p}}^{a+}\right\} \mid 0\rangle.\end{aligned}$$]{} For a transverse component, it is necessary to calculate $$\exp \left\{ i\int d^4xJ^{\mu,a}\left(x\right) \sum\limits_{\vec{k}}A_{ \vec{k},\sigma }^af_{k,\mu }^\sigma \left(x\right) \right\} \exp \left\{ \frac 12C_\sigma \left( \left| \vec{p}\right| \right) A_{\vec{p},\sigma }^{a+}A_{\vec{p},\sigma }^{a+}\right\} \mid 0\rangle \ \text{\qquad for } \sigma =1,2 \label{modT}$$ The following recourse is used to calculate this expression; calling $U$ the first exponential in (\[modT\]) this expression can be written as $$\exp \left\{ \frac 12C_\sigma\left(p\right) \left( UA_{\vec{p},\sigma }^{a+}U^{-1}\right) \left(UA_{\vec{p},\sigma }^{a+}U^{-1}\right) \right\} \mid 0\rangle, \label{modTt}$$ since $$U^{-1}\mid 0\rangle =\mid 0\rangle.$$ The inverse $U^{-1}$ is the same $U$ when in the exponential argument the sign is changed. Using the Baker-Hausdorf formula $$\exp [\hat{F}]\hat{G}\exp [-\hat{F}]=\exp \left\{ [\hat{F},\ ]\right\} \hat{G }=\sum \frac 1{n!}\left[ \hat{F},\left[ \hat{F},....,\left[ \hat{F},\hat{G} \right].....\right] \right]$$ and noticing that only the first and the second term in the expansion are non-vanishing when $\hat{F}$ and $\hat{G}$ are linear functions of annihilation and creation operators, it follows $$\exp [\hat{F}]\hat{G}\exp [-\hat{F}]=\hat{G}+\left[ \hat{F},\hat{G}\right].$$ Therefore, for the relevant commutators appearing in (\[modTt\]) it follows $$\left[ i\int d^4xJ^{\mu,a}\left(x\right) \sum\limits_{\vec{k}}A_{\vec{k},\sigma }^af_{k,\mu }^\sigma \left( x\right),A_{\vec{p},\sigma }^{a+}\right] =i\int d^4xJ^{\mu ,a}\left(x\right) f_{p,\mu }^\sigma \left(x\right).$$ Then for the expression (\[modT\]) the following result is obtained $$\exp \left\{ \frac 12C_\sigma \left(\left| \vec{p}\right| \right) \left(A_{ \vec{p},\sigma }^{a+}+i\int d^4xJ^{\mu,a}\left(x\right) f_{p,\mu }^\sigma \left(x\right) \right) ^2\right\} \mid 0\rangle \ \label{T}$$ For the longitudinal and scalar modes, following the above procedure, the result obtained is $$\begin{aligned} &&\exp \left\{i\int d^4xJ^{\mu,a}\left(x\right) \sum\limits_{\vec{k} }\left(B_{\vec{k}}^af_{k,S,\mu }\left( x\right) +A_{\vec{k} }^{L,a}f_{k,L,\mu }\left(x\right) \right) \right\}\nonumber \\ && \times \exp \left\{ C_3\left(\left| \vec{p}\right| \right) B_{\vec{p}}^{a+}A_{\vec{p}}^{L,a+}\right\} \mid 0\rangle \nonumber \\ &&=\exp \left\{ C_3\left(\left| \vec{p}\right| \right) \left( B_{\vec{p} }^{a+}-i\int d^4xJ^{\mu,a}\left(x\right) f_{p,L,\mu }\left(x\right) \right) \right. \nonumber \\&& \qquad\qquad\qquad\qquad \times \left. \left(A_{\vec{p}}^{L,a+} -i\int d^4xJ^{\mu,a}\left(x\right) f_{p,S,\mu }\left(x\right) \right) \right\} \mid 0\rangle.\label{L}\end{aligned}$$ where the expressions below were used $$\begin{aligned} &&\left[ \left(i\int d^4xJ^{\mu,a}\left(x\right) \sum\limits_{\vec{k} } A_{\vec{k} }^{L,a}f_{k,L,\mu }\left( x\right) \right),B_{\vec{p}}^{a+}\right] =-i\int d^4xJ^{\mu,a}\left(x\right) f_{p,L,\mu }\left(x\right) \\ &&\left[ \left(i\int d^4xJ^{\mu,a}\left(x\right) \sum\limits_{\vec{k} } B_{\vec{k}}^af_{k,S,\mu }\left(x\right) \right),A_{\vec{p} }^{L,a+}\right] =-i\int d^4xJ^{\mu,a}\left(x\right) f_{p,S,\mu }\left(x\right).\end{aligned}$$ For the full modification calculation (\[mod\]), it is necessary to evaluate $$\langle \Phi \mid \exp \left\{ i\int d^4xJ^{\mu,a}\left(x\right) A_\mu ^{a-}\left(x\right) \right\} =\left(\exp \left\{ -i\int d^4xJ^{\mu,a}\left(x\right) A_\mu ^{a+}\left(x\right) \right\} \mid \Phi \rangle \right) ^{\dagger }, \label{left}$$ which can be easily obtained by conjugating the result for the right hand side, through (\[T\]) and (\[L\]). Then, substituting (\[T\]), (\[L\]) and (\[left\]) in (\[mod\]), the following expression should be calculated $$\begin{aligned} &&\frac 1N\langle 0\mid \exp \left\{ \sum\limits_{\sigma =1,2}\frac 12C_\sigma ^{*}\left(\left| \vec{p}\right| \right) \left(A_{\vec{p},\sigma }^a+i\int d^4xJ^{\mu,a}\left(x\right) f_{p,\mu }^{\sigma *}\left(x\right) \right) ^2\right\} \nonumber \\ &&\qquad \times \exp \left\{ \sum\limits_{\sigma =1,2}\frac 12C_\sigma \left(\left| \vec{p}\right| \right) \left( A_{\vec{p},\sigma }^{a+}+i\int d^4xJ^{\mu,a}\left(x\right) f_{p,\mu }^\sigma \left(x\right) \right) ^2\right\} \mid 0\rangle \nonumber \\ &&\times\langle 0\mid \exp \left\{ C_3^{*}\left( \left| \vec{p}\right| \right) \left(B_{\vec{p}}^a-i\int d^4xJ^{\mu ,a}\left(x\right) f_{p,L,\mu }^{*}\left(x\right) \right) \right. \nonumber \\ &&\qquad\qquad\qquad\qquad\qquad \times \left. \left( A_{\vec{p}}^{L,a}-i\int d^4xJ^{\mu,a}\left(x\right) f_{p,S,\mu }^{*}\left(x\right) \right) \right\} \nonumber \\ &&\qquad \times \exp \left\{ C_3\left(\left| \vec{p}\right| \right) \left(B_{ \vec{p}}^{a+}-i\int d^4xJ^{\mu,a}\left(x\right) f_{p,L,\mu }\left(x\right) \right) \right. \nonumber \\ &&\qquad\qquad\qquad\qquad\qquad \times \left. \left( A_{\vec{p}}^{L,a+}-i\int d^4xJ^{\mu,a}\left(x\right) f_{p,S,\mu }\left(x\right) \right) \right\} \mid 0\rangle \nonumber \\ &&\times\langle 0\mid \exp \left(-iC_3^{*}\left(\left| \vec{p}\right| \right) c_{\vec{p}}^a\overline{c}_{\vec{p}}^a\right) \exp \left( iC_3\left(\left| \vec{p}\right| \right) \overline{c}_{\vec{p}}^{a+}c_{\vec{p}}^{a+}\right) \mid 0\rangle \label{mod1}\end{aligned}$$ In the expression (\[mod1\]) the calculated contribution, for each transverse mode, is $$\exp \left\{ -\int \frac{d^4xd^4y}{2Vp_0}J^{\mu,a}\left(x\right) J^{\nu,a}\left(y\right) \epsilon _\mu ^\sigma \left(p\right) \epsilon _\nu ^\sigma \left(p\right) \frac{\left(C_\sigma \left( \left| \vec{p}\right| \right) +C_\sigma ^{*}\left(\left| \vec{p}\right| \right) +2\left| C_\sigma \left(\left| \vec{p}\right| \right) \right| ^2\right) }{2\left(1-\left| C_\sigma \left(\left| \vec{p}\right| \right) \right| ^2\right) }\right\}, \label{T1}$$ and the longitudinal and scalar mode contribution is $$\exp \left\{ -\int \frac{d^4xd^4y}{2Vp_0}J^{\mu,a}\left(x\right) J^{\nu,a}\left(y\right) \epsilon _{S,\mu }\left(p\right) \epsilon _{L,\nu }\left(p\right) \frac{\left(C_3\left(\left| \vec{p}\right| \right) +C_3^{*}\left(\left| \vec{p}\right| \right) +2\left| C_3\left(\left| \vec{p} \right| \right) \right| ^2\right) }{\left( 1-\left| C_3\left(\left| \vec{p} \right| \right) \right| ^2\right) }\right\}, \label{L1}$$ The detailed analysis of these calculations can be found in the Appendixes 1 and 2. Therefore, after collecting the contributions of all the modes, assuming $ C_1\left(\left| \vec{p}\right| \right) =C_2\left( \left| \vec{p}\right| \right) =C_3\left(\left| \vec{p}\right| \right) $ (which follows necessarily in order to obtain Lorentz invariance) and using the properties of the defined vectors basis, the modification to the propagator becomes $$\exp \left\{ \frac 12\int \frac{d^4xd^4y}{2p_0V}J^{\mu,a}\left( x\right) J^{\nu,a}\left(y\right) g_{\mu \nu }\left[ \frac{\left( C_1\left(\left| \vec{p}\right| \right) +C_1^{*}\left(\left| \vec{p}\right| \right) +2\left| C_1\left(\left| \vec{p}\right| \right) \right| ^2\right) }{\left(1-\left| C_1\left(\left| \vec{p}\right| \right) \right| ^2\right) }\right] \right\}. \label{totmodF}$$ In the expression (\[totmodF\]), the combination of the $C_1\left(\left| \vec{p}\right| \right) $ constant is always real and nonnegative, for all $ \left| C_1\left(\left| \vec{p}\right| \right) \right| <1$. Now it is possible to perform the limit process $\vec{p}\rightarrow 0$. In doing this limit, it is considered that each component of the linear momentum $\vec{p}$ is related with the quantization volume by $$p_x\sim \frac 1a,\ p_y\sim \frac 1b,\ p_z\sim \frac 1c,\ V=abc\sim \frac 1{\left| \vec{p}\right| ^3},$$ $\ \ $ And it is necessary to calculate $$\lim_{\vec{p}\rightarrow 0}\frac{\left(C_1\left(\left| \vec{p}\right| \right) +C_1^{*}\left(\left| \vec{p}\right| \right) +2\left| C_1\left(\left| \vec{p}\right| \right) \right| ^2\right) }{4p_0V\left(1-\left| C_1\left(\left| \vec{p}\right| \right) \right| ^2\right) },\label{Lim1}$$ Then, after fixing a dependence of the arbitrary constant $C_1$ of the form $ \left| C_1\left(\left| \vec{p}\right| \right) \right| \sim \left(1-\kappa \left| \vec{p}\right| ^2\right),\kappa >0$, and $C_1\left(0\right) \neq -1$ the limit (\[Lim1\]) becomes $$\lim_{\vec{p}\rightarrow 0}\frac{\left(C_1\left(\left| \vec{p}\right| \right) +C_1^{*}\left(\left| \vec{p}\right| \right) +2\left| C_1\left(\left| \vec{p}\right| \right) \right| ^2\right) \left| \vec{p}\right| ^3\frac 1{\left(1-\left(1-\kappa \left| \vec{p}\right| ^2\right) ^2\right) }}{4p_0}=\frac C{2\left(2\pi \right) ^4}$$ where $C$ is an arbitrary real and nonnegative constant, determined by the also real and nonnegative constant $\kappa$. Therefore, the total modification to the propagator including all color values turns to be [ $$\begin{aligned} &\prod\limits_{a=1,..,8}&\langle \widetilde{\Phi }\mid \exp \left\{ i\int d^4xJ^{\mu,a}\left(x\right) A_\mu ^{a-}\left( x\right) \right\} \exp \left\{ i\int d^4xJ^{\mu,a}\left(x\right) A_\mu ^{a+}\left(x\right) \right\} \mid \widetilde{\Phi }\rangle \nonumber \\ &=&\exp \left\{ \sum\limits_{a=1,..8}\int d^4xd^4yJ^{\mu,a}\left(x\right) J^{\nu,a}\left(y\right) g_{\mu \nu }\frac C{2\left(2\pi \right) ^4}\right\}.\end{aligned}$$]{} The generating functional associated to the proposed initial state, including the usual perturbative piece for $\alpha =1$, can be written in the form $$Z[J]=\exp \left\{ \frac i2\sum\limits_{a,b=1,..8}\int d^4xd^4yJ^{\mu,a}\left(x\right) \widetilde{D}_{\mu \nu }^{ab}(x-y)J^{\nu,b}\left(y\right) \right\},$$ where $$\widetilde{D}_{\mu \nu }^{ab}(x-y)=\int \frac{d^4k}{\left(2\pi \right) ^4} \delta ^{ab}g_{\mu \nu }\left[ \frac 1{k^2}-iC\delta \left(k\right) \right] \exp \left\{ -ik\left(x-y\right) \right\} \label{propag}$$ which shows that the gluon propagator has the same form proposed in [@Cabo], for the selected gauge parameter value $\alpha =1$ (which corresponds to $\alpha =-1$ in that reference). Modified Ghost Propagator ------------------------- In the present section the possible modification to the ghost propagator will be analyzed. As was shown in Sec. 3.1 for fermionic particles the expression for the modification, introduced by a nontrivial vacuum state, is $$\begin{aligned} &&\langle \widetilde{\Phi } \mid \exp \left\{ i\int d^4x\left( \overline{\xi }^a\left(x\right) c^{a-}\left(x\right) +\overline{c}^{a-}\left(x\right) \xi ^a\left(x\right) \right) \right\} \nonumber \\&&\qquad \times \exp \left\{ i\int d^4x\left( \overline{\xi }^a\left(x\right) c^{a+}\left(x\right) +\overline{c}^{a+}\left(x\right) \xi ^a\left(x\right) \right) \right\} \mid \widetilde{\Phi }\rangle, \label{ini}\end{aligned}$$ where $$\begin{aligned} c^{a+}\left(x\right) &=&\sum\limits_{\vec{k}}c_{\vec{k}}^ag_k\left(x\right),\qquad c^{a-}\left(x\right) =\sum\limits_{\vec{k}}c_{\vec{k} }^{a+}g_k^{*}\left(x\right), \nonumber \\ \overline{c}^{a+}\left( x\right) &=&\sum\limits_{\vec{k}}\overline{c}_{\vec{k }}^ag_k\left(x\right),\qquad \overline{c}^{a-}\left(x\right) =\sum\limits_{\vec{k}}\overline{c}_{\vec{k}}^{a+}g_k^{*}\left( x\right).\end{aligned}$$ Now it is calculated explicitly the action of the exponential operator $$\begin{aligned} &&\exp \left\{ i\int d^4x\left(\overline{\xi }^a\left(x\right) c^{a+}\left(x\right) +\overline{c}^{a+}\left(x\right) \xi ^a\left( x\right) \right) \right\} \exp \left\{ C_3\left(\left| \vec{p}\right| \right) i\overline{c}_{\vec{p}}^{a+} c_{\vec{p}}^{a+}\right\} \mid 0\rangle \nonumber \\ &&=\left( 1+i\int d^4y\overline{\xi }^a\left(y\right) \sum\limits_{\vec{k} ^{\prime }}c_{\vec{k}^{\prime }}^ag_{k^{\prime }}\left(y\right) \right) \left(1+i\int d^4x\sum\limits_{\vec{k}} \overline{c}_{\vec{k}}^ag_k\left(x\right) \xi ^a\left(x\right) \right) \times \nonumber \\ &&\qquad \times \left(1+C_3\left( \left| \vec{p}\right| \right) i\overline{c}_{ \vec{p}}^{a+}c_{\vec{p}}^{a+}\right) \mid 0\rangle. \label{Ghost}\end{aligned}$$ The grassman character of the field and sources allowed expanding the exponential retaining only the first two terms in the expansion. With the use of the following relations [$$\begin{aligned} \overline{\xi }^a\left(y\right) c_{\vec{k}^{\prime }}^a\overline{c}_{\vec{p} }^{a+}c_{\vec{p}}^{a+}\mid 0\rangle &=&i\delta _{\vec{k}^{\prime },\vec{p}} \overline{\xi }^a\left( y\right) c_{\vec{p}}^{a+}\mid 0\rangle, \nonumber \\ \overline{c}_{\vec{k}}^a\xi ^a\left(x\right) \overline{c}_{\vec{p}}^{a+}c_{ \vec{p}}^{a+}\mid 0\rangle &=&i\delta _{\vec{k},\vec{p}}\overline{c}_{\vec{p} }^{a+}\xi ^a\left(x\right) \mid 0\rangle, \nonumber \\ \overline{\xi }^a\left(y\right) c_{\vec{k}^{\prime }}^ai\delta _{\vec{k}, \vec{p}}\overline{c}_{\vec{p}}^{a+}\xi ^a\left(x\right) \mid 0\rangle &=&-\delta _{\vec{k},\vec{p}}\delta _{\vec{k}^{\prime },\vec{p}}\overline{ \xi }^a\left(y\right) \xi ^a\left(x\right) \mid 0\rangle,\end{aligned}$$ ]{}the expression (\[Ghost\]) can be written as $$\begin{aligned} &&\left[ 1+C_3\left(\left| \vec{p}\right| \right) \left( i\overline{c}_{ \vec{p}}^{a+}c_{\vec{p}}^{a+}-i\int d^4xg_p\left( x\right) \left(\overline{ \xi }^a\left(x\right) c_{\vec{p}}^{a+}+\overline{c}_{\vec{p}}^{a+}\xi ^a\left(x\right) \right) \right. \right. + \nonumber \\ &&\left. \left. +i\int d^4y\int d^4xg_p\left(y\right) g_p\left(x\right) \overline{\xi }^a\left(y\right) \xi ^a\left(x\right) \right) \right] \mid 0\rangle. \label{rhs}\end{aligned}$$ In addition the formula $$\begin{aligned} &&\langle \widetilde{\Phi }\mid \exp \left\{ i\int d^4x\left( \overline{\xi } ^a\left(x\right) c^{a-}\left(x\right) +\overline{c}^{a-}\left(x\right) \xi ^a\left(x\right) \right) \right\} \nonumber \\ &&=\left[ \exp \left\{ i\int d^4x\left( \overline{\xi }^{a\dagger }\left(x\right) c^{a+}\left(x\right) +\overline{c}^{a+}\left(x\right) \xi ^{a\dagger }\left(x\right) \right) \right\} \mid \widetilde{\Phi }\rangle \right] ^{\dagger }, \label{rsh1}\end{aligned}$$ allows to calculate the left hand side of (\[ini\]) using (\[rhs\]). Then the expression (\[ini\]), substituting (\[rhs\]) and (\[rsh1\]), takes the form [ $$\begin{aligned} &\langle 0\mid &\left[ 1-C_3^{*}\left(\left| \vec{p}\right| \right) \left(ic_{\vec{p}}^a\overline{c}_{\vec{p}}^a-i\int d^4xg_p^{*}\left(x\right) \left(c_{\vec{p}}^a\overline{\xi }^a\left(x\right) +\xi ^a\left(x\right) \overline{c}_{\vec{p}}^a\right) \right. \right. \nonumber \\ &&\left. \left. +i\int d^4y\int d^4xg_p^{*}\left(y\right) g_p^{*}\left(x\right) \xi ^a\left(y\right) \bar{\xi}^a\left( x\right) \right) \right] \nonumber \\ &\times &\left[ 1+C_3\left( \left| \vec{p}\right| \right) \left(i\overline{c }_{\vec{p}}^{a+}c_{\vec{p}}^{a+}-i\int d^4xg_p\left(x\right) \left(\overline{\xi }^a\left(x\right) c_{\vec{p}}^{a+}+\overline{c}_{\vec{p} }^{a+}\xi ^a\left(x\right) \right) \right. \right. \nonumber \\ &&\left. \left. +i\int d^4y\int d^4xg_p\left(y\right) g_p\left(x\right) \overline{\xi }^a\left(y\right) \xi ^a\left(x\right) \right) \right] \mid 0\rangle. \label{ghomod}\end{aligned}$$]{} In this case, the expression (\[ghomod\]) calculus is easier than the one realized for gluons. And the result of its contribution, canceling out the normalization factor, is $$\exp \left[ \frac{i\int d^4xd^4y\overline{\xi }^a\left(x\right) \xi ^a\left(y\right) \left(C_3\left(\left| \vec{p}\right| \right) +C_3^{*}\left(\left| \vec{p}\right| \right) -2\left| C_3\left( \left| \vec{p }\right| \right) \right| ^2\right) }{2Vp_0\left( 1-\left| C_3\left(\left| \vec{p}\right| \right) \right| ^2\right) }\right],$$ which in the limit $\vec{p}\rightarrow 0$, under the same condition considered for the gluon modification limit, takes the form $$\exp \left\{ -\sum\limits_{a=1,..8}i\int d^4xd^4y\overline{\xi }^a\left(x\right) \xi ^a\left(y\right) \frac{C_G}{\left(2\pi \right) ^4}\right\}.$$ In this expression $C_G$ is a real and nonnegative constant. It is interesting to note that choosing $C_3\left(0\right) =1$, then $C_G=0$ and there is no modification to the ghost propagator as was chosen in the previous work [@Cabo]. The ghost generating functional associated to the proposed initial state, including the usual perturbative piece for $\alpha =1$, can be written in the form $$Z_G[\overline{\xi },\xi ]=\exp \left\{ i\sum\limits_{a,b=1,..8}\int d^4xd^4y \overline{\xi }^a\left( x\right) \widetilde{D}_G^{ab}(x-y)\xi ^b\left(y\right) \right\},$$ where $$\widetilde{D}_G^{ab}(x-y)=\int \frac{d^4k}{\left(2\pi \right) ^4}\delta ^{ab}\left[ \frac{\left(-i\right) }{k^2}-C_G\delta \left(k\right) \right] \exp \left\{ -ik\left(x-y\right) \right\}.$$ Summary ======= By using the operational formulation for Quantum Gauge Fields Theory developed by Kugo and Ojima, a particular state vector for QCD in the non-interacting limit, that obeys the BRST physical state condition, was constructed. The general motivation for looking this wave function is to search for a reasonably good description of low energy QCD properties, through giving foundation to the perturbative expansion proposed in [@Cabo]. The high energy QCD description should not be affected by the modified perturbative initial state. In addition it can be expected that the adiabatic connection of the color interaction starting with it as an initial condition, generate at the end the true QCD interacting ground state. In case of having the above properties, the analysis would allow to understand the real vacuum as a superposition of infinite number of soft gluon pairs. It has been checked that properly fixing the free parameters in the constructed state, the perturbation expansion proposed in the former work [@Cabo] is reproduced for the special value $\alpha =1$ of the gauge constant. Therefore, the appropriate gauge is determined for which the expansion introduced in that work is produced by an initial state, satisfying the physical state condition for the BRST quantization procedure. The fact that the non-interacting initial state is a physical one, lead to expect that the final wave-function after the adiabatic connection of the color interaction will also satisfy the physical state condition for the interacting theory. If this assumption is correct, the results for calculations of transition amplitudes and the values of physical quantities should be also physically meaningful. In future, a quantization procedure for arbitrary values of $\alpha$ will be also considered. It is expected that with its help the gauge parameter independence of the physical quantities could be implemented. It seems possible that the results of this generalization will lead to $\alpha $ dependent polarizations for gluons and ghosts and their respective propagators, which however could produce $\alpha $ independent results for the physical quantities. However, this discussion will be delayed for future consideration. It is important to mention now a result obtained during the calculation of the gluon propagator modification, in the chosen construction. It is that the arbitrary constant $C$ is determined here to be real and nonnegative. This outcome restricts an existing arbitrariness in the discussion given in the previous work. As this quantity $C$ is also determining the square of the generated gluon mass as positive or negative, real or imaginary, therefore it seems very congruent to arrive to a definite prediction of $C$ as real and positive. The modification to the standard free ghost propagator introduced by the proposed initial state, was also calculated. Moreover, after considering the free parameter in the proposed trial state as real, which it seems the most natural assumption, the ghost propagator is not be modified, as it was assumed in [@Cabo]. Some tasks which can be addressed in future works are: The study of the applicability of the Gell-Mann and Low theorem with respect to the adiabatic connection of the interaction, starting from the here proposed initial state. The investigation of zero modes quantization, that is gluon states with exact vanishing four momentum. The ability to consider them with success would allow a formally cleaner definition of the proposed state, by excluding the auxiliary momentum $\vec{p}$ recursively used in the construction carry out. Finally, the application of the proposed perturbation theory in the study of some problems related with confinement and the hadron structure. Transverse Mode Contribution ============================ The transverse mode contribution is determined by the expression $$\begin{aligned} &&\langle 0\mid \exp \left\{ \frac 12C_\sigma ^{*}\left(\left| \vec{p} \right| \right) \left(A_{\vec{p},\sigma }^a+i\int d^4xJ^{\mu,a}\left(x\right) f_{p,\mu }^{\sigma *}\left(x\right) \right) ^2\right\} \nonumber \\ &&\quad\times \exp \left\{ \frac 12C_\sigma \left(\left| \vec{p}\right| \right) \left(A_{\vec{p},\sigma }^{a+}+i\int d^4xJ^{\mu,a}\left(x\right) f_{p,\mu }^\sigma \left(x\right) \right) ^2\right\} \mid 0\rangle \label{A1}\end{aligned}$$ For simplifying the exposition, the following notation is introduced $$\begin{aligned} C^{*} &\equiv &C_\sigma ^{*}\left(\left| \vec{p}\right| \right),\ C\equiv C_\sigma \left(\left| \vec{p}\right| \right),\text{ \qquad }\hat{A} ^{+}\equiv A_{\vec{p},\sigma }^{a+},\ \hat{A}\equiv A_{\vec{p},\sigma }^a \text{\ }, \nonumber \\ a_1 &\equiv &i\int d^4xJ^{\mu,a}\left(x\right) f_{p,\mu }^{\sigma *}\left(x\right),\text{ \qquad }a_2\equiv i\int d^4xJ^{\mu ,a}\left(x\right) f_{p,\mu }^\sigma \left(x\right). \label{nota1}\end{aligned}$$ Then the expression (\[A1\]) takes the form $$\begin{aligned} &&\langle 0 \mid \exp \left\{ \frac 12C^{*}\left( \hat{A}+a_1\right) ^2\right\} \exp \left\{ \frac 12C\left( \hat{A}^{+}+a_2\right) ^2\right\} \mid 0\rangle \label{A2} \\ &&=\exp \left\{ \frac{C^{*}}2a_1^2+\frac C2a_2^2\right\} \langle 0\mid \exp \left\{ \frac{C^{*}}2\hat{A}^2 +C^{*}a_1\hat{A}\right\} \exp \left\{ \frac C2 \hat{A}^{+2}+Ca_2\hat{A}^{+}\right\} \mid 0\rangle. \nonumber\end{aligned}$$ For the action of the exponential, linear in the annihilation operator, at the left on the right the result obtained is $$\begin{aligned} &&\exp \left\{ C^{*}a_1\hat{A}\right\} \exp \left\{ \frac C2\hat{A} ^{+2}+Ca_2\hat{A}^{+}\right\} \mid 0\rangle \nonumber \\ &&=\exp \left\{ \frac C2\left(\hat{A}^{+}+C^{*}a_1\right) ^2+Ca_2\left(\hat{A}^{+}+C^{*}a_1\right) \right\} \mid 0\rangle, \label{A3}\end{aligned}$$ where the same procedure used for calculating (\[modT\]) is considered. The expression (\[A2\]), considering (\[A3\]), can be written in the form $$\exp \left\{ \frac{C^{*}}2a_1^2+\frac C2\left( C^{*}a_{1}+a_2\right) ^2\right\} \langle 0\mid \exp \left\{ \frac{C^{*}}2\hat{A}^2\right\} \exp \left\{ \frac C2\hat{A}^{+2}+C\hat{A}^{+}\left(C^{*}a_{1}+a_2\right) \right\} \mid 0\rangle \label{A4}$$ It is possible in (\[A4\]) to act with the exponential linear in the creation operator at the right on the left and the result is $$\begin{aligned} &&\exp \left\{ \frac{C^{*}}2a_1^2+\frac C2\left( C^{*}a_{1}+a_2\right) ^2\left(1+\left| C\right| ^2\right) \right\} \nonumber \\ &&\times \langle 0 \mid \exp \left\{ \frac{C^{*}}2\hat{A}^2+C^{*}C\left(C^{*}a_{1}+a_2\right) \hat{A}\right\} \exp \left\{ \frac C2\hat{A} ^{+2}\right\} \mid 0\rangle \label{A5}\end{aligned}$$ In such a way after n-steps it is possible to arrive to a recurrence relation, which can be proven by mathematical induction. This recurrence relation has the form $$\begin{aligned} &&\exp \left\{ \frac{C^{*}}2a_1^2+\frac C2\left( C^{*}a_{1}+a_2\right) ^2\sum\limits_{m=0}^n\left[ \left| C\right| ^{2\left(2m\right) }+\left| C\right| ^{2\left(2m+1\right) }\right] \right\} \nonumber \\ &&\times \langle 0 \mid \exp \left\{ \frac{C^{*}}2\hat{A}^2+C^{*n+1}C^{n+1} \left( C^{*}a_{1}+a_2\right) \hat{A}\right\} \exp \left\{ \frac C2\hat{A} ^{+2}\right\} \mid 0\rangle \label{rec1}\end{aligned}$$ Lets probe it, acting with the exponential linear in the annihilation operator at the left on the right the result is $$\begin{aligned} &&\exp \left\{ \frac{C^{*}}2a_1^2+\frac C2\left( C^{*}a_{1}+a_2\right) ^2\left(\sum\limits_{m=0}^n\left[ \left| C\right| ^{2\left(2m\right) }+\left| C\right| ^{2\left( 2m+1\right) }\right] +\left| C\right| ^{4\left(n+1\right) }\right) \right\} \nonumber \\ &&\times \langle 0 \mid \exp \left\{ \frac{C^{*}}2\hat{A}^2\right\} \exp \left\{ \frac C2\hat{A}^{+2}+C^{*n+1}C^{n+2}\left(C^{*}a_{1}+a_2\right) \hat{A}^{+}\right\} \mid 0\rangle,\end{aligned}$$ now acting on the left with the exponential linear in the creation operator is obtained the relation $$\begin{aligned} &&\exp \left\{ \frac{C^{*}}2a_1^2+\frac C2\left( C^{*}a_{1}+a_2\right) ^2\sum\limits_{m=0}^{n+1}\left[ \left| C\right| ^{2\left(2m\right) }+\left| C\right| ^{2\left( 2m+1\right) }\right] \right\} \nonumber \\ &&\times \langle 0 \mid \exp \left\{ \frac{C^{*}}2\hat{A}^2+C^{*n+2}C^{n+2} \left( C^{*}a_{1}+a_2\right) \hat{A}\right\} \exp \left\{ \frac C2\hat{A} ^{+2}\right\} \mid 0\rangle \label{A6}\end{aligned}$$ which probe the recurrence relation (\[rec1\]). At this point the limit $n\rightarrow \infty$ is taken, considering $\left| C\right| <1$ which implies that $$\begin{aligned} &&\lim_{n\rightarrow \infty }\left| C\right| ^{2n}=0, \nonumber \\ &&\lim_{n\rightarrow \infty }\sum\limits_{m=0}^n\left[ \left| C\right| ^{2\left(2m\right) }+\left| C\right| ^{2\left( 2m+1\right) }\right] =\frac 1{\left(1-\left| C\right| ^2\right) }, \label{lim}\end{aligned}$$ and the expression (\[A6\]) in this limit has the form, $$\exp \left\{ \frac{\left(C^{*}a_{1}^2+Ca_2^2+2C^{*}Ca_1a_2\right) }{2\left(1-\left| C\right| ^2\right) }\right\} \langle 0\mid \exp \left\{ \frac{C^{*}} 2\hat{A}^2\right\} \exp \left\{ \frac C2\hat{A}^{+2}\right\} \mid 0\rangle \label{A7}$$ Finally, the notation (\[nota1\]) is substituted in (\[A7\]). After that, the functions of $\vec{p}$ are expanded in the vicinity of $\vec{p}=0$, keeping in mind that the sources are located in a space finite region it is necessary to consider only the first terms in the expansion. Then for the expression (\[A7\]) it is obtained the result (\[T1\]), the renormalization factors cancel out. Longitudinal and Scalar Modes Contribution ========================================== The longitudinal and scalar modes contribution is determined by the expression $$\begin{aligned} &&\langle 0\mid \exp \left\{ C_3^{*}\left(\left| \vec{p}\right| \right) \left(B_{\vec{p}}^a-i\int d^4xJ^{\mu,a}\left(x\right) f_{p,L,\mu }^{*}\left(x\right) \right) \right. \nonumber \\ &&\qquad\qquad\qquad\qquad\qquad \times \left. \left( A_{\vec{p}}^{L,a}-i\int d^4xJ^{\mu,a}\left(x\right) f_{p,S,\mu }^{*}\left(x\right) \right) \right\} \nonumber \\ &&\qquad \times \exp \left\{ C_3\left(\left| \vec{p}\right| \right) \left(B_{ \vec{p}}^{a+}-i\int d^4xJ^{\mu,a}\left(x\right) f_{p,L,\mu }\left(x\right) \right) \right. \nonumber \\ &&\qquad\qquad\qquad\qquad\qquad \times \left. \left( A_{\vec{p}}^{L,a+}-i\int d^4xJ^{\mu,a}\left(x\right) f_{p,S,\mu }\left(x\right) \right) \right\} \mid 0\rangle \label{B1}\end{aligned}$$ introducing the following notation, $$\begin{aligned} C^{*} &\equiv &C_3^{*}\left(\left| \vec{p}\right| \right),\ C\equiv C_3\left(\left| \vec{p}\right| \right),\text{ \ \ }\hat{A}^{+}\equiv A_{ \vec{p}}^{L,a+},\ \hat{A}\equiv A_{\vec{p}}^{L,a},\text{ \ \ }\hat{B} ^{+}\equiv B_{\vec{p}}^{a+},\ \hat{B}\equiv B_{\vec{p}}^a, \nonumber \\ a_1 &\equiv &-i\int d^4xJ^{\mu,a}\left(x\right) f_{p,S,\mu }^{*}\left( x\right),\text{ \qquad }a_2\equiv -i\int d^4xJ^{\mu,a}\left( x\right) f_{p,S,\mu }\left(x\right), \nonumber \\ b_1 &\equiv &-i\int d^4xJ^{\mu,a}\left(x\right) f_{p,L,\mu }^{*}\left( x\right),\text{ \qquad }b_2\equiv -i\int d^4xJ^{\mu,a}\left( x\right) f_{p,L,\mu }\left(x\right), \label{nota2}\end{aligned}$$ the expression (\[B1\]) takes the form $$\begin{aligned} &&\langle 0\mid \exp \left\{ C^{*}\left(\hat{A}+a_1\right) \left( \hat{B} +b_1\right) \right\} \exp \left\{ C\left( \hat{B}^{+}+b_2\right) \left(\hat{A}^{+}+a_2\right) \right\} \mid 0\rangle \nonumber \\ &&=\exp \left\{ C^{*}a_1b_1+Ca_2b_2\right\} \langle 0\mid \exp \left\{ C^{*}\left(\hat{A}\hat{B} +b_1\hat{A}+a_1\hat{B}\right) \right\} \nonumber \\ &&\text{ \qquad \qquad \qquad \qquad \qquad \qquad }\times \exp \left\{ C\left(\hat{B}^{+}\hat{A}^{+}+ b_2\hat{A}^{+}+a_2\hat{B}^{+}\right) \right\} \mid 0\rangle \label{B2}\end{aligned}$$ For the action of the exponential linear in the annihilation operator at the left on the right, is obtained $$\begin{aligned} &&\exp \left\{ C^{*}\left(b_1\hat{A}+a_1\hat{B}\right) \right\} \exp \left\{ C\left( \hat{B}^{+}\hat{A}^{+}+b_2\hat{A}^{+}+a_2\hat{B}^{+}\right) \right\} \mid 0\rangle \label{B3} \\ &&=\exp \left\{ C\left[ \left(\hat{B}^{+}-C^{*}b_1\right) \left(\hat{A} ^{+}-C^{*}a_1\right) +b_2\left(\hat{A}^{+}-C^{*}a_1\right) +a_2\left(\hat{B }^{+}-C^{*}b_1\right) \right] \right\} \mid 0\rangle, \nonumber\end{aligned}$$ the same procedure used for calculating (\[L\]) is considered. Following the same steps described in the previous appended for transverse modes, in this case the recurrence relation obtained for longitudinal and scalar modes is [ $$\begin{aligned} &&\exp \left\{ C^{*}a_1b_1+C\left(C^{*}a_1-a_2\right) \left( C^{*}b_1-b_2\right) \sum\limits_{m=0}^n\left[ \left| C\right| ^{2\left(2m\right) }+\left| C\right| ^{2\left(2m+1\right) }\right] \right\}\label{B4} \\ &&\langle 0\mid \exp \left\{ C^{*}\hat{A}\hat{B}+C^{*n+1}C^{n+1}\left(\left( C^{*}b_1-b_2\right) \hat{A}+\left(C^{*}a_1-a_2\right) \hat{B}\right) \right\} \exp \left\{ C\hat{B}^{+}\hat{A}^{+}\right\} \mid 0\rangle.\nonumber\end{aligned}$$]{} For the expression (\[B4\]), in the limit $n\rightarrow \infty $ considering $\left| C\right| <1$, the following relation is obtained $$\begin{aligned} &&\exp \left\{ C^{*}a_1b_1+C\left(C^{*}a_1-a_2\right) \left( C^{*}b_1-b_2\right) \frac 1{\left(1-\left| C\right| ^2\right) }\right\} \nonumber \\ &&\quad \langle 0\mid \exp \left\{ C^{*}\hat{A}\hat{B}\right\} \exp \left\{ C \hat{B}^{+}\hat{A}^{+}\right\} \mid 0\rangle. \label{B5}\end{aligned}$$ Finally, the notation (\[nota2\]) is substituted in (\[B5\]), the functions of $\vec{p}$ are expanded in the vicinity of $\vec{p}=0$, and the result (\[L1\]) is obtained. [99]{} C. N. Yang and R. Mills, Phys. Rev. 96, 191 (1954). O. V. Greenberg, Phys. Rev. Lett. 13, 598 (1964); Y. Miyamoto, Suppl. Prog. Theor. Phys. 187 (1965); M. Y. Han and Y. Nambu, Phys. Rev. 139, 1038 (1965). M. Creutz, Phys. Rev. D21, 2308 (1980). A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn and V. Weisskopf, Phys. Rev. D9, 3471 (1974). J. L. Gervais and A. Neveu, Phys. Lett. B80 (1979). E. U. Shuryak, Phys. Rep. 115, 151 (1984); E. U. Shuryak, The QCD Vacuum, Hadrons and the Superdense Matter, World Scientific, Singapore, 1988; T. Schäfer and E. V. Shuryak, Rev. Mod. Phys. 70, 323 (1998). M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B147, 385 (1979); B147, 448 (1979); B147, 519 (1979). T. H. Hansson, K. Johnson and C. Peterson, Phys. Rev. D26, 2069 (1982). S. Mandelstam, Phys. Rep. C23, 245 (1976); N. P. Nair and C. Rosenzweig, Phys. Lett. B131, 434 (1983); B135, 450 (1984); Phys. Rev. D31, 401 (1985). G. K. Savvidi, Phys. Lett. B71, 133 (1977). L. D. Faddeev and V. N. Popov, Phys. Lett. B25, 29 (1967). A. Cabo, S. Peñaranda and R. Martinez, Mod. Phys. Lett. A10, 2413 (1995). S. G. Matinyan and G. K. Savvidi, XVIII Conference, Tbilisi (1976); S. G. Matinyan and G. K. Savvidi, Nucl. Phys. B134, 539 (1978); N. K. Nielsen and P. Olesen, Nucl. Phys. B144, 376 (1978); N. K. Nielsen and M. Ninomiya, Nucl. Phys. B169, 309 (1980); J. I. Kapusta, Thermodynamics of Chromomagnetism in Statistical Mechanics of Quarks and Hadrons, North-Holland Publishing Company, 1981. L. D. Faddeev and A. A. Slanov, Gauge Fields. Introduction to Quantum Theory, Benjamin Cummings Publishing, 1980. S. Randjbar-Daemi, Course in Quantum Field Theory, Lectures Notes ICTP Diploma Programma, ICTP, Trieste 1994. S. N. Gupta, Prc. Phys. Soc. London A63, 681 (1950). N. Nakanishi, Prog. Theor. Phys. 51, 952 (1974); N. Nakanishi, Prog. Theor. Phys. 52, 1929 (1974). R. P. Feynman, Acta Phys. Polon. 24, 697 (1963). T. Kugo and I. Ojima, Prog. Theor. Phys. 60, 1869 (1978); T. Kugo and I. Ojima, Prog. Theor. Phys. 61, 294 (1979); T. Kugo and I. Ojima, Prog. Theor. Phys. 61, 644 (1979); T. Kugo and I. Ojima, Prog. Theor. Phys. Suppl. 66, 1 (1979). C. Becchi, A. Rouet and R. Stora, Ann. Phys. 98, 287 (1976). N. Nakanishi and I. Ojima, Covariant Operator Formalism of Gauge Theories and Quantum Gravity, Singapore, Word Scientific, 1990. A. A. Sokolov, I. M. Trenov, V. CH. Zhukovski, A. V. Borísov, Quantum Electrodynamics, Moscow, Mir, 1989. Suraj N. Gupta, Quantum Electrodynamics, New York, Gordon and Breach Science Publishers, 1981. C. Itzykson and J. -B. Zuber, Quantum Field Theory, New York, McGraw-Hill, 1980. S. Gasiorowicz, Elementary Particle Physics, New York, Jonh Wiley & Sons, 1966.

--- abstract: 'We present an exactly solvable spin-orbital model based on the Gamma-matrix generalization of a Kitaev-type Hamiltonian. In the presence of small magnetic fields, the model exhibits a critical phase with a spectrum characterized by topologically protected Fermi points. Upon increasing the magnetic field, Fermi points carrying opposite topological charges move toward each other and annihilate at a critical field, signaling a phase transition into a gapped phase with trivial topology in three dimensions. On the other hand, by subjecting the system to a staggered magnetic field, an effective time-reversal symmetry essential to the existence of three-dimensional topological insulators is restored in the auxiliary free fermion problem. The nontrivial topology of the gapped ground state is characterized by an integer winding number and manifests itself through the appearance of gapless Majorana fermions confined to the two-dimensional surface of a finite system.' author: - 'Gia-Wei Chern' title: 'Three-dimensional topological phases in a layered honeycomb spin-orbital model' --- Introduction ============ Topological phases of matter are one of the most remarkable discoveries in modern condensed-matter physics. [@tknn; @avron83; @kohmoto85; @wen90; @volovik] Instead of a local order parameter describing broken symmetries of the system, this novel state of matter is characterized by a topological invariant which is insensitive to small adiabatic deformations of the Hamiltonian. A classic example is the integer quantum Hall (IQH) effect, where the quantized Hall conductance corresponds to a topological invariant called the first Chern number or the TKNN integer. [@tknn] Another distinctive property of IQH states is the appearance of zero-energy modes at the sample edge, despite the fact that all bulk excitations are fully gapped. Due to their topological nature, these conducting edge modes persist even in the presence of disorders. Recently, theoretical investigations have shown that similar topological insulators can be generalized to time-reversal invariant systems (quantum spin-Hall effect) [@kane05; @bernevig06; @moore07; @roy09a] and to three dimensions. [@fu07a; @fu07b; @roy09b] Signatures of topological insulators such as quantized conductance and protected surface Dirac cone have been reported experimentally in semiconducting alloys and quantum wells. [@konig07; @hsieh08; @xia09; @roushan09; @chen09] Following these developments, systematic classifications of topological insulators have also been proposed. [@schnyder08; @kitaev09; @qi08] Although most discussions of topological insulators are in the context of tight-binding fermionic models or mean-field superconductors, it has been shown that topological insulators can also emerge from strongly correlated electronic systems. [@jackeli09; @shitade09; @raghu08; @sun09; @zhang09] An exactly solvable example is Kitaev’s anisotropic spin-1/2 model on the two-dimensional honeycomb lattice. [@kitaev06] As shown in his seminal paper, the spin model can be reduced to a problem of free Majorana fermions coupled to a static $Z_2$ gauge field. The ground state of the Kitaev model has two distinct phases. The gapped Abelian phase is equivalent to the toric code model [@kitaev03] whose excitations are Abelian anyons. Relevant to our discussion is the non-Abelian B phase in the presence of a magnetic field. This phase is characterized by an integer winding number $\nu = \pm 1$. Similar to IQH insulators, the non-Abelian phase also supports gapless chiral edge modes (whose chirality depends on the sign of magnetic fields) except that the edge modes here are real Majorana fermions, as contrasted to complex fermions in the case of IQH states. There has been much effort to generalize Kitaev model to other trivalent lattices [@yang07; @yao07] and to three-dimensions. [@si07; @si08; @mandal09] Probably the most notable example is the discovery of a chiral spin liquid as the ground state of Kitaev model on a decorated-honeycomb lattice. [@yao07] On the other hand, despite being exactly solvable, we find that most gapped phases of 3D Kitaev model is topologically trivial. [@chern09] The fact that the model can only be defined on lattices with coordination number 3 significantly constrains the possible Majorana hopping Hamiltonian. Recently, noting that the exact solvability of Kitaev model relies on the fact that the three spin-1/2 Pauli matrices realize the simplest (dimension-2) Clifford algebra, the so-called $\Gamma$-matrix generalization of the Kitaev model offers richer possibilities of engineering exactly solvable models with unusual phases in both 2D and 3D. [@levin03; @hamma05; @yao09; @wu09; @ryu09; @nussinov09] Physically, models based on, e.g. dimension-4 $\Gamma$ matrices can be interpreted as spin-$\frac{3}{2}$ models, spin-$\frac{1}{2}\frac{1}{2}$ models, or spin-orbital models. Based on the above $\Gamma$-matrix generalization of Kitaev model, emergent topological insulators have been demonstrated on a 3D diamond lattice with coordination number 4. [@wu09; @ryu09] To construct a Kitaev-type model on such a lattice, one needs four mutually anticommuting operators for the four nearest-neighbor links. This can be easily realized using the dimension-4 $\Gamma$-matrix representation of the Clifford algebra. By exploiting the redundancy of representing two such sets of bosonic operators in terms of 6 Majorana fermions, the diamond-lattice model enjoys an effective time-reversal symmetry (TRS) which is essential to the existence of topological insulators in 3D. [@schnyder08] With only nearest-neighbor interactions, the diamond-lattice model reduces to a problem of two identical copies of free Majorana fermions sharing the same $Z_2$ gauge field. The permutation symmetry between the two fermion species thus manifests as a TRS. In the weak-pairing regime of the model, hybridization of the two Majorana species results in a gapped ground state with nontrivial topology. A natural question to ask is whether it is possible to realize the required TRS in a Kitaev-type model via generic physical mechanisms. In this paper, we provide such an example through a natural generalization of the honeycomb Kitaev model. Using a similar $\Gamma$-matrix formalism, we study a Kugel-Khomskii-type [@kk] spin-orbital Hamiltonian on a layered honeycomb lattice whose coordination number is 5. We further consider perturbations due to a weak magnetic field and single-ion spin-orbit interaction. In the weak-pairing regime of our model, the ground state remains gapless up to a critical field strength. The Fermi points of this critical phase are characterized by a nonzero topological invariant, hence are stable against weak perturbations. As the field strength is increased, Fermi points with opposite winding numbers eventually annihilate with each other and the spectrum acquires a gap above a critical field. Since the TRS is explicitly broken in the presence of a uniform magnetic field, the gapped phase represents a multilayer generalization of the 2D Kitaev model, and is topologically trivial in three dimensions. On the other hand, when the sign of magnetic field alternates between successive honeycomb planes, an effective TRS is restored in the auxiliary fermionic model, and the corresponding ground state in the weak-pairing phase is equivalent to a topological superconductor belonging to symmetry class DIII in Altland-Zirnbauer’s classification. [@az] We also show that the quantum ground state is characterized by an integer winding number, consistent with the general classification of 3D topological insulators.[@schnyder08] A physical consequence of the nontrivial ground-state topology is the appearance of surface Majorana fermions which remain gapless against perturbations respecting the discrete symmetries of the Hamiltonian. The specific multilayer geometry of our model also allows us to analytically demonstrate the existence of surface Majorana fermions obeying a Dirac-like Hamiltonian by relating the surface modes to the chiral edge modes of 2D Kitaev model. The spin-orbital model ====================== \[t\] ![\[fig:honeycomb\] A layered honeycomb lattice. The five distinct nearest-neighbor links are indicated by the corresponding exchange constant. The open and filled circles denote sites belonging to the two inequivalent sites of a unit cell. The primitive vectors of the underlying Bravais lattice are $\mathbf a_1 = (1,0,0)$, $\mathbf a_2 = (\frac{1}{2},\frac{\sqrt{3}}{2},0)$, and $\mathbf a_3 = (0,0,1)$. An arrow from site $j$ to site $k$ means the corresponding link variable $u_{jk} = +1$.](honeycomb "fig:"){width="0.8\columnwidth"} We study a Kugel-Khomskii-type[@kk] spin-orbital model defined on a layered honeycomb lattice shown in Fig. \[fig:honeycomb\]. In addition to a spin-1/2 degree of freedom, each lattice site has an extra doublet orbital degeneracy. The spin and orbital (pseudospin) operators are denoted by Pauli matrices $\sigma^\alpha$ and $\tau^\alpha$ ($\alpha = x, y, z$), respectively. As in the original honeycomb Kitaev model, nearest-neighbor links lying on a honeycomb plane are divided into three types: $x$, $y$, and $z$, depending on their orientations. The model Hamiltonian is defined as follows: $$\begin{aligned} \label{eq:h} \mathcal{H}_0 &=& -\sum_{j}\!^{\,'} \, J_{\parallel} \bigl(\tau^x_{j}\tau^x_{j + z} + \tau^y_{j}\tau^y_{j - z}\bigr) \nonumber \\ & & -\frac{1}{2} \sum_{j}\sum_{\alpha=x,y,z} J_\alpha\, \tau^z_{j}\tau^z_{j+ \bm\delta_\alpha} \,\sigma^\alpha_{j}\sigma^\alpha_{j + \bm\delta_\alpha}.\end{aligned}$$ Here $j$ runs over the lattice sites, $j\pm z$ denote nearest neighbors along the two vertical links, and $j + \bm\delta_\alpha$ denotes in-plane nearest neighbor along the link of type $\alpha$. The prime in the first term indicates that the summation runs over sites on every second honeycomb layer. The exchange constant is $J_{\parallel}$ on vertical links, and is $J_\alpha$ on $\alpha$-links lying on a honeycomb plane (see Fig. \[fig:honeycomb\]). The spin-orbital interaction within each honeycomb layer resembles the original 2D Kitaev model with spin-1/2 operator $\sigma^\alpha$ replaced by the spin-orbital operator $\tau^z\sigma^\alpha$. The inter-layer interaction in our model, on the other hand, involves only orbital operators; the orbital interaction alternates between the $\tau^x$ and $\tau^y$ types along successive vertical links. In addition to discrete lattice symmetries, the Hamiltonian is invariant under a $\pi$ rotation about $\tau^z$ axis followed by lattice translations along the $z$ direction by one layer. Since the five spin-orbital operators $\tau^x$, $\tau^y$, and $\tau^z\sigma^\alpha$ anticommute with each other, they generate a $4\times 4$ matrix representation of the Clifford algebra. With an enlarged Hilbert space, one may introduce 6 Majorana fermions $c$ and $b^\mu$ ($\mu = 1,\cdots,5$) such that: $$\begin{aligned} & &\tau^z\sigma^x = i b^1 c, \quad \tau^z\sigma^y = i b^2 c, \quad \tau^z\sigma^z = i b^3 c, \nonumber \\ & & \quad\quad\quad\quad \tau^x = i b^4 c, \quad \tau^y = i b^5 c.\end{aligned}$$ By denoting these operators as $\Gamma^\mu = i b^\mu c$, Hamiltonian (\[eq:h\]) can be recast into a Kitaev-type interaction $$\begin{aligned} \mathcal{H}_0 = -\sum_{\mu=1}^5 J_{\mu} \sum_{\mu-\mbox{\scriptsize links}} \Gamma^\mu_j \Gamma^\mu_k,\end{aligned}$$ with $J_1 = J_x$, $J_2 = J_y$, and $J_3 = J_z$ for links lying on a honeycomb plane, and $J_4 = J_5 = J_{\parallel}$ along the vertical links. The six Majorana fermions form an 8-dimensional Hilbert space which is twice as large as the local physical Hilbert space. This redundancy can be remedied by demanding the allowed physical states be eigenstate of gauge operator $D \equiv i c \,\prod_{\mu=1}^5 b^\mu$ with eigenvalue +1. This is also consistent with the identity $\tau^x\tau^y\tau^z\sigma^x\sigma^y\sigma^z = -1$. We note that the same $\Gamma$-matrix extension of the Kitaev model on a 2D decorated square lattice (Shastry-Sutherland lattice) has been studied in Ref. . Following Kitaev, [@kitaev06] we introduce the link operator $u_{jk} \equiv i b^\mu_j\,b^\mu_k$, where $\mu=\mu_{jk}$ implicitly depends on the type of link connecting sites $j$ and $k$. The Hamiltonian then becomes $$\begin{aligned} \label{eq:H-cc} \mathcal{H}_0 = \frac{i}{2} \sum_{jk} J_{\mu}u_{jk}\, c_j c_k.\end{aligned}$$ A remarkable feature of the fermionic Hamiltonian first noted by Kiatev is that the link operators $u_{jk}$ commute with each other and with the Hamiltonian. We may thus replace them by their eigenvalues $u_{jk} = \pm 1$, which act as an emergent $Z_2$ gauge field. Consequently, for a given choice of $\{u_{jk}\}$, Hamiltonian (\[eq:H-cc\]) reduces to a problem of free Majorana fermions with nearest-neighbor hopping $t_{jk} = J_{\mu} u_{jk}$. However, since $u_{jk}$ does not commute with operator $D_j$, the spectrum of the free fermion Hamiltonian depends only on gauge invariant quantities which are given by the product of link operators around the boundary of elementary plaquette: $w_p = \prod_{(jk)\in \partial p} u_{jk}$. As $w_p^2 = 1$, the flux associated with a given plaquette is also given by a $Z_2$ variable $w_p = \pm 1$. ![\[fig:flux\] Configuration of $Z_2$ gauge fields $\{u_{jk}\}$ and $\{u'_{jk}\}$ on (a) square and (b) hexagonal plaquettes; $u_{jk} = 1$ if there is an arrow pointing from site $j$ to site $k$. The numbers 1 and 2 label the two inequivalent sites within a unit cell.](flux){width="0.95\columnwidth"} The question remains of what choices of $u_{jk}$, or equivalently the $Z_2$ fluxes $w_p$, give the lowest ground-state energy of the free fermion problem. Numerically, we find that such a state is attained when all hexagons are vortex free while all squares contain a $\pi$ flux: $$\begin{aligned} w_{\varhexagon} = 1, \quad\quad w_{\Square} = -1,\end{aligned}$$ consistent with Lieb’s theorem. [@lieb] A gauge choice which gives the desired plaquette fluxes without enlarging the unit cell is shown in Fig. \[fig:flux\]. With this specific choice of $Z_2$ gauge field, the fermion spectrum can be obtained analytically using Fourier transformation. In the weak-pairing regime to be discussed below, model (\[eq:h\]) exhibits a gapless phase similar to the B phase of 2D Kitaev model. In order to explore possible topological insulators emerging from this critical phase, we consider the following single-ion perturbations: $$\begin{aligned} \label{eq:h1} \mathcal{H}_1 = -\sum_{j} \mathbf h_j\cdot\bm \sigma_j + \lambda\sum_{j} \tau^z_{j} \hat\mathbf n\cdot\bm\sigma_j.\end{aligned}$$ The first term is the Zeeman coupling to an external magnetic field $\mathbf h_j = \eta_j \mathbf h$, where the phase factor $\eta_j = \pm 1$ depends only on the $z$ coordinate of spin $j$, i.e. the field is constant within a given honeycomb plane. As mentioned previously, we shall consider two special cases in this paper: uniform field with $\eta_j = 1$ and staggered field with $\eta_j = (-1)^{z_j}$. Since spins transform as $T \sigma^\alpha T^{-1} = -\sigma^\alpha$ under time-reversal $T$, the first term above explicitly breaks TRS. The second term in Eq. (\[eq:h1\]) represents a spin-orbit-like interaction where $\lambda$ is an effective coupling constant, and $\hat\mathbf n$ is a unit vector specifying the local anisotropy axis. Such a coupling arises when the orbital basis $|\tau^z = \pm 1\rangle$ carries a nonzero orbital angular momentum $\mathbf L \sim \tau^z\hat\mathbf n$ which is parallel or antiparallel to the local anisotropy axis depending on $\tau^z = +1$ or $-1$, respectively. An explicit example is given by two degenerate orbitals with $t_{2g}$ symmetry, e.g. $|yz\rangle$ and $|zx\rangle$ in a tetragonal crystal field. By identifying $|\tau^z = \pm 1 \rangle$ as $|yz\rangle \pm i |zx \rangle$, respectively, the orbital basis has a nonzero angular momentum pointing along the symmetry axis of the tetragonal field. The effects of $\mathcal{H}_1$ can be studied following the perturbation treatments discussed in Ref. . Essentially, one constructs an effective Hamiltonian acting on the subspace which is free of vortex-type excitations. The first-order correction vanishes identically as both single-ion perturbations create $\pi$ fluxes on hexagons, [@baskaran07] whereas the second-order terms simply modify the nearest-neighbor exchange constants $J_{\alpha}$. Nontrivial corrections to the fermion spectrum arise at the third-order perturbation which involves multiple spin-orbital interactions, e.g. $$(\tau^z_j \sigma^x_j)(\tau^z_k \sigma^y_k) \,\sigma^z_l = i (\tau^z_j \sigma^x_j)(\tau^z_k \sigma^y_k) (\tau^z_l \sigma^x_l) (\tau^z_l \sigma^y_l).$$ Such terms introduce an effective second nearest-neighbor hopping for Majorana fermions $$\begin{aligned} \label{eq:H-cc2} \mathcal{H}_1 = \frac{i \kappa}{2} \sum_{jk} \eta \, u'_{jk} c_j c_k,\end{aligned}$$ where $\kappa \sim \lambda^2 h/J^2$ and $\eta = \pm 1$ is a constant within the plane containing sites $j$ and $k$. The additional second-neighbor $Z_2$ field $u'_{jk} = -u'_{kj}$ is shown by the dashed line in Fig. \[fig:flux\]. Depending on the direction of $\hat\mathbf n$ and $\mathbf h$, the hopping amplitude $\kappa$ could take different values along inequivalent second-neighbor links. As the main purpose of this term is to introduce a spectral gap, to avoid unnecessary complications, we shall assume the symmetric case in the following discussion. Uniform magnetic field {#sec:uniform-h} ====================== We first discuss the ground state in the presence of a uniform magnetic field, i.e. $\eta_j = +1$ for all layers. As will be discussed below, the gapped phase at large fields is characterized by a $\pi_2$ Chern number corresponding to a multi-layer generalization of the 2D Kitaev model, and is topologically trivial in 3D (which generally is characterized by $\pi_3$ homotopy groups). This is mainly because the TRS essential to the existence of 3D topological insulators is explicitly broken by the uniform field. However, the critical phase in the case of small fields is interesting in itself and is similar to the gapless A-phase of $^3$He discussed in Ref. ; the gaplessness of both phases are topologically protected. Since the original unit cell of the lattice is preserved by the $Z_2$ fields $u_{jk}$ and $u'_{jk}$ shown in Fig. \[fig:flux\], we express the site index as $j = (\mathbf r, s)$, where $\mathbf r$ denotes the position of the unit cell, and $s=1,2$ indicates the two inequivalent sites in a unit cell. With Fourier transformation $a_{\mathbf k,s} = \sum_{\bf r} c_{\mathbf r, s}\, e^{-i\mathbf k\cdot(\mathbf r + \mathbf d_s)}/\sqrt{2N}$, where $\mathbf d_s$ is a basis vector and $N$ is the number of unit cells, the Hamiltonian becomes $\mathcal{H} = \mathcal{H}_0 + \mathcal{H}_1 =\frac{1}{2} \sum_{\mathbf k}\,\Psi^\dagger_{\mathbf k}\,H(\mathbf k)\,\Psi_{\mathbf k} $, with $\Psi_{\mathbf k} = (a_{\mathbf k,1}, \, a_{\mathbf k, 2})^T$ and $$\begin{aligned} \label{eq:Hu} H(\mathbf k) &=& \left(\begin{array}{cc} g(k_z)+\Delta(\mathbf k_\perp) & -i f(\mathbf k_\perp) \\ i f(\mathbf k_\perp)^* & -g(k_z)-\Delta(\mathbf k_\perp) \end{array}\right), \\ & = & {\rm Im} f(\mathbf k_\perp)\,\tau^x + {\rm Re} f(\mathbf k_\perp)\,\tau^y + \bigl[g(k_z)\!+\!\Delta(\mathbf k_\perp)\bigr]\,\tau^z. \nonumber\end{aligned}$$ For convenience, we have defined the following functions: $$\begin{aligned} g(k_z) = 4 J_\parallel \sin k_z, \quad f(\mathbf k_\perp) = 2 \sum_{\alpha=1}^3 J_\alpha\,e^{i\mathbf k_{\perp}\cdot \bm \delta_\alpha}, \\ \nonumber \Delta(\mathbf k_\perp) = 8\kappa\,\sin\frac{k_x}{2}\,\Bigl(\cos\frac{\sqrt{3}k_y}{2}-\cos\frac{k_x}{2}\Bigr). \quad\end{aligned}$$ The three vectors $\bm \delta_{1,2} = (\frac{\pm 1}{2},\frac{-1}{2\sqrt{3}})$, and $\bm \delta_{3} = (0,\frac{1}{\sqrt{3}})$ connect nearest-neighbor sites in a honeycomb layer. In the following we shall focus on the emergent free fermion problem. The Pauli matrices $\tau^\mu$ appearing in the single-particle Hamiltonian now act on the sublattice index $s$ (not to be confused with orbital pseudospins). The hermitian matrix $H(\mathbf k)$ in Eq. (\[eq:Hu\]) also satisfies $$\begin{aligned} \label{eq:symD} H^T(-\mathbf k) = - H(\mathbf k),\end{aligned}$$ which is the defining property of symmetry class D in Altland-Zirnbauer’s classification. [@az] As 3D insulators in this class is topologically trivial, [@schnyder08] the gapped phase of Eq. (\[eq:Hu\]) represents a trivial multilayer generalization of the 2D Kitaev model. Nonetheless, for small magnetic fields such that the second-neighbor hopping $\kappa \sim \lambda^2 h$ is below a critical value $\kappa_c$, the fermion spectrum remains gapless in the weak-paring regime of the model. The corresponding critical phase is characterized by topologically protected Fermi points as we shall discuss below. Diagonalizing Hamiltonian (\[eq:Hu\]) yields a spectrum: $$\begin{aligned} \label{eq:e1} \epsilon_{\pm}(\mathbf k) = \pm \sqrt{\bigl[g(k_z) + \Delta(\mathbf k_\perp)\bigr]^2 + |f(\mathbf k_\perp)|^2}.\end{aligned}$$ When one of the in-plane coupling is much larger than the other two, e.g. $J_z \gg J_x, J_y$, there is no solution to equation $f(\mathbf k_\perp) = 0$, and the spectrum is always gapped irrespective of the applied magnetic field. This phase corresponds to the A phase of the 2D Kitaev model, [@kitaev06] and is referred to in the following as the strong-pairing phase based on analogy with the $p$-wave topological superconductors. [@read00] On the other hand, in the weak-pairing regime of the model where the three in-plane couplings satisfy the triangle inequalities, [@kitaev06] the model displays a possible gapless phase as two solutions $\mathbf k^*_\perp$ exist for the equation $f(\mathbf k_\perp) = 0$. To be specific, we now concentrate on the symmetric case $J_x = J_y = J_z \equiv J$, where zeros of $f(\mathbf k_\perp)$ are at the corners of the 2D hexagonal Brillouin zone $\mathbf k^*_{\perp} = (\pm \frac{4\pi}{3}, 0)$. In contrast to the 2D Kitaev model where applying a magnetic field immediately opens an energy gap, [@kitaev06] spectrum (\[eq:e1\]) of the 3D model remains gapless when $\kappa$ is less than a critical strength $|\kappa| \leq \kappa_c.$ For symmetric in-plane couplings, we find $\kappa_c \equiv 2J_\parallel/3\sqrt{3}$ and the nodes of the spectrum $\epsilon_{\pm}(\mathbf k)$ are located at $$\begin{aligned} \label{eq:fermipts} \begin{array}{llcl} \quad & \mathbf k^*_{1,+} = \bigl(\frac{4\pi}{3}, 0, -\pi+\xi\bigr), & & \mathbf k^*_{1,-} = \bigl(\frac{4\pi}{3}, 0, -\xi\bigr), \\ \quad & \mathbf k^*_{2,+} = \bigl(-\frac{4\pi}{3}, 0, \xi\bigr), & & \mathbf k^*_{2,-} = \bigl(-\frac{4\pi}{3}, 0, \pi-\xi\bigr), \end{array} \quad\end{aligned}$$ where $\xi = \arcsin(\kappa/\kappa_c)$. These Fermi points are topologically protected and are robust against weak perturbations. To see this, we first rewrite Hamiltonian (\[eq:Hu\]) as $H(\mathbf k) = \varepsilon_+(\mathbf k)\,\hat{\mathbf m}(\mathbf k)\!\cdot\!\bm\tau$, where $\hat\mathbf m(\mathbf k)$ is a unit vector. A topological invariant characterizing the singularities of the spectrum is given by the winding number of mappings from a sphere $S^2$ enclosing the Fermi point $\mathbf k^*$ to the 2-sphere of the unit vector $\hat{\mathbf m}$: [@volovik] $$\begin{aligned} \label{eq:nu1} \nu = \int_{S^2} \frac{d^2 A_\rho}{8\pi} \epsilon^{\mu\nu\rho}\, \hat{\mathbf m}\cdot\bigl(\partial_\mu\hat{\mathbf m}\times\partial_\nu\hat{\mathbf m}\bigr).\end{aligned}$$ The above definition corresponds to the second homotopy group $\pi_2(S^2) = Z$, which characterizes point defects in an $O(3)$ spin field.[@mermin79] Examples of topologically nontrivial Fermi points are given by the spectra of Weyl Hamiltonian describing a massless spin-1/2 particle: $H_{\rm Weyl} = \pm ic\tau^\mu\partial_\mu$, where $c$ is the speed of light. [@volovik] The plus and minus signs refer to left and right-handed particles, respectively. Using Eq. (\[eq:nu1\]) the winding number of Weyl spinors can be readily computed, resulting $\nu = \pm 1$ for right and left-handed particles, respectively. Take left-handed Weyl spinor for example, the expectation value of its spin is parallel to its momentum: $\langle \bm \tau \rangle \parallel \mathbf p$. In the ground state with filled negative-energy states, Fermi point with $\nu = 1$ thus looks like a magnetic monopole (a hedgehog) in momentum space. These singular points are robust in the sense that it is impossible to continuously deform a hedgehog into a uniform spin configuration (corresponding to the trivial case of $\nu = 0$). ![\[fig:k-top\] (Color online) (a) Projection of ground-state pseudospin $\langle \bm\tau\rangle = -\hat{\mathbf m}(\mathbf k)$ on a face of the Brillouin zone. The two vertical edges of the face correspond to $K = (\frac{4\pi}{3},0, k_z)$ and $K'=(\frac{2\pi}{3},\frac{2\pi}{\sqrt{3}},k_z) \equiv (-\frac{4\pi}{3},0,k_z)$. The location of the four topologically non-trivial Fermi points are indicated by black arrows. (b) Dispersion along the $K'$ edge of Brillouin zone for various $\kappa \sim \lambda^2 h$. Upon increasing magnetic field $h$, the two Fermi points carrying opposite winding numbers move in opposite directions along the $K'$ edge. At a critical field ($\kappa = \kappa_c$), the two singularities merge to form a new Fermi point with trivial winding number $\nu = 0$ at $(-\frac{4\pi}{3}, 0, \frac{\pi}{2})$. The spectrum becomes gapped above $\kappa_c$.](mk){width="0.8\columnwidth"} To compute the winding number of Fermi points in our 3D model, we expand Hamiltonian (\[eq:Hu\]) around the singular points, e.g. $$\begin{aligned} \label{eq:hw} H(\mathbf k^*_{1,\pm}+\mathbf p) = - v_\perp p_y\tau^x - v_\perp p_x\tau^y \mp v_\parallel p_z \tau^z,\end{aligned}$$ where the Fermi velocities are $v_\perp = \sqrt{3}J$ and $v_\parallel\!=\! 2 (4 J_\parallel^2 - 27 \kappa^2 )^{1/2}$. The dispersion around these points has a conic singularity: $\varepsilon_\pm\!\approx\! \pm \sqrt{(v_\perp \mathbf p_\perp)^2 + (v_\parallel p_z)^2}$. After rotation and mirror-inversion, Eq. (\[eq:hw\]) is essentially equivalent to the momentum-space Weyl Hamiltonian discussed above. The four Fermi points (\[eq:fermipts\]) form two pairs (labeled by $r = 1,2$) of singularities with opposite winding numbers $\nu = \pm 1$. The ground-state configuration of pseudospin $\langle \bm\tau\rangle = -\hat{\mathbf m}(\mathbf k)$ projected onto a face of the Brillouin zone is shown in Fig. \[fig:k-top\](a). The two pairs of singularities are located at the two inequivalent edges $K$ and $K'$ of the 3D Brillouin zone. Instead of a monopole-like configuration, vector field $\hat\mathbf m(\mathbf k)$ in the vicinity of these Fermi points has a saddle-point-like singularity. Although these Fermi points are topologically protected due to their nonzero winding numbers, a spectral gap can still be induced through mutual annihilation of Fermi points carrying opposite topological charges. This process is illustrated in Fig. \[fig:k-top\]. Take for example the two Fermi points on the $K'$-edge of the Brillouin zone. Upon increasing magnetic field $h$, the two singularities characterized by winding numbers $\nu = \pm 1$ move toward each other along the $K'$ edge, and eventually merge to form a new Fermi point when $\kappa$ reaches the critical $\kappa_c$. Because of the conservation of topological charge, the new Fermi point has a winding number $\nu = 0$, hence is topologically trivial. Above the critical field, the spectrum is fully gapped. It is also interesting to understand the critical B phase of Kitaev’s original 2D model from the perspective of topological winding number. As discussed in Ref. , the gaplessness of B phase is protected by TRS on the bipartite honeycomb lattice. This discrete symmetry essentially forces the unit vector $\hat\mathbf m(\mathbf k_\perp)$ to lie in the $xy$ plane as $\tau^z$ terms in Eq. (\[eq:Hu\]) is not allowed by TRS. For a planar unit vector $\hat\mathbf m$ whose tip lies on a circle $S^1$, the singularities are characterized by an integer topological invariant, also known as the vortex winding number, corresponding to $\pi_1(S^1) = Z$. [@mermin79] The two Fermi poins in the B phase of Kitaev’s model can be viewed as vortices carrying opposite winding numbers $\nu = \pm 1$, respectively. In the presence of perturbations breaking the TRS, the vector $\hat\mathbf m$ now lives on a 2-sphere. Since $\pi_1(S^2) = 0$, singularities of the spectrum are then topologically trivial ($\pi_2$ characterization of 2D singularities is not well defined). The gapped phase above the critical $\kappa_c$ represents a trivial generalization of the 2D Kitaev model in much the same way as the multilayer generalization of the IQH state. The topological properties of such systems are characterized by three spectral Chern numbers. [@kohmoto92] In our case, the nonzero topological invariant is given by the winding number of vector field $\hat{\mathbf m}(\mathbf k_\perp; k_z)$ which maps the in-plane hexagonal Brillouin zone (a 2-torus) to a unit 2-sphere: $$\begin{aligned} \label{eq:nu1b} \nu = \frac{1}{4\pi}\int dk_x dk_y\,\hat{\mathbf m}\cdot\bigl(\partial_x \hat{\mathbf m}\times\partial_y\hat{\mathbf m}\bigr) = \frac{\kappa}{|\kappa|}.\end{aligned}$$ By treating $k_z$ as a parameter, Hamiltonian (\[eq:Hu\]) has exactly the same form as that of the 2D Kitaev model. The first Chern number (\[eq:nu1b\]) of the corresponding ‘2D’ Hamiltonian $H_{k_z}(\mathbf k_\perp)$ is given by $\nu = {\rm sgn}\,\Delta_{k_z}$, [@kitaev06] where $\Delta_{k_z} = 6\sqrt{3}(\kappa + \kappa_c\,\sin k_z)$ is the effective gap parameter at corner $K$ of the hexagonal Brillouin zone. Therefore, as long as $|\kappa| > \kappa_c$, hence the system remains gapped, the winding number $\nu = {\rm sgn}\kappa = \pm 1$ is independent of $k_z$, and the whole Brillouin zone is characterized by the same chirality. Staggered magnetic field {#sec:stagger-h} ======================== The gapped phase discussed in the previous section is topologically trivial due to the absence of TRS. As discussed in Ref. , TRS is a prerequisite for the existence of 3D topological insulators. In this section, we show that one can introduce a gap to the fermion spectrum, while at the same time preserving an effective TRS, by subjecting the system to a staggered magnetic field, i.e. $\eta_j = (-1)^{z_j}$. Because the sign of the field alternates between successive honeycomb layers, a discrete symmetry emerges in our model system as Hamiltonian (\[eq:h1\]) is invariant under time-reversal $T$ followed by a lattice translation along $z$ axis. This additional symmetry manifests itself as a TRS in the auxiliary Majorana hopping problem. The mechanism proposed here is similar to Haldane’s model of realizing 2D quantum Hall effect without a net magnetic flux through the unit cells. [@haldane88] Due to the staggered field, the sign of second nearest-neighbor hopping $u'_{jk}$ also alternates between successive honeycomb planes. This results in a staggered gap function $ \pm \Delta(\mathbf k_{\perp})$, and a doubled unit cell along $z$ direction. We denote the fermion annihilation operators on the even and odd layers as $a_{\mathbf r, s}$ and $b_{\mathbf r, s}$, respectively, where subscript $s = 1, 2$ refers to the two inequivalent sites within a honeycomb plane. The Fourier-transformed Hamiltonian then reads $\mathcal{H} = \frac{1}{2}\sum_{\mathbf k} \Psi^\dagger_{\mathbf k} H(\mathbf k) \Psi_{\mathbf k}$, with $$\begin{aligned} \label{eq:Hs} & &H(\mathbf k) = \left(\begin{array}{cccc} \Delta(\mathbf k_{\perp}) & g(k_z) & -i f(\mathbf k_{\perp}) & 0 \\ g(k_z) & -\Delta(\mathbf k_{\perp}) & 0 & -i f(\mathbf k_{\perp}) \\ i f(\mathbf k_{\perp})^* & 0 & -\Delta(\mathbf k_{\perp}) & -g(k_z) \\ 0 & i f(\mathbf k_{\perp})^* & -g(k_z) & \Delta(\mathbf k_\perp) \end{array}\right) \quad\,\,\,\, \\ \nonumber \\ & & \,\, = \Delta(\mathbf k_{\perp}) \tau^z \sigma^z\!+\! g(k_z) \tau^z\sigma^x\! +\! {\rm Im}f(\mathbf k_{\perp})\tau^x \! +\! {\rm Re} f(\mathbf k_{\perp})\tau^y, \nonumber\end{aligned}$$ and $\Psi_{\mathbf k} = \bigl(a_{\mathbf k,1}, b_{\mathbf k_,1}, a_{\mathbf k,2}, b_{\mathbf k, 2}\bigr)^T$. The two sets of Pauli matrices $\tau^\mu$ and $\sigma^\mu$ now act on the sublattice $(1,2)$ and even-odd $(a,b)$ indices, respectively. In addition to symmetry relation (\[eq:symD\]) shared by Hamiltonians describing free Majorana fermions, the hermitian matrix (\[eq:Hs\]) also satisfies $$\begin{aligned} \label{eq:trs} \tau^z (i\sigma^y)\, H^T(\mathbf k)\, (-i\sigma^y)\, \tau^z = H(-\mathbf k),\end{aligned}$$ stemming from the generalized TRS. The extra $\tau^z$ factor can be gauged away by a $\pi/2$ rotation about the $\tau^z$ axis. Eq. (\[eq:trs\]) defines the symmetry property of DIII Hamiltonians in Altland-Zirnbauer’s classification. [@az] As discussed in Ref. , a topological invariant can be defined for Hamiltonians in this symmetry class based on the block off-diagonal representation of the Hamiltonian, or more precisely, of the spectral projection operator. In fact, the same definition can be applied to classes of Hamiltonians which possess some form of chiral symmetry arising from either the sublattice symmetry or a combination of particle-hole and time-reversal symmetries. [@schnyder08] To compute the topological winding number of our model, we first bring Hamiltonian (\[eq:Hs\]) into a block off-diagonal form through a series of unitary transformations. First, noting that the layered honeycomb lattice is bipartite in which nearest neighbors of one sublattice belong to the other one, we regroup fermions of the same sublattice into a block, e.g. $a_{\mathbf k,1}$ and $b_{\mathbf k, 2}$. Mathematically this is achieved by interchanging the 2nd and 4th entries of $\Psi_{\mathbf k}$, the transformed Hamiltonian becomes $$\begin{aligned} \label{eq:Hd1} H(\mathbf k) \to \alpha^x {\rm Im} f(\mathbf k_{\perp}) + \alpha^y {\rm Re} f(\mathbf k_{\perp}) + \alpha^z g(k_z) + \beta \Delta(\mathbf k_{\perp}),\quad\,\end{aligned}$$ where $\alpha^\mu$ and $\beta$ given by $$\begin{aligned} \alpha^\mu = \gamma^\mu = \tau^x \sigma^\mu, \quad \beta = \gamma^0 = \tau^z,\end{aligned}$$ are the standard Dirac matrices. The second part of the unitary transformation is a $\pi/2$ rotation about the new $\tau^x$ axis, which transforms $\tau^z \to \tau^y$, hence $$\begin{aligned} \label{eq:HDIII} H(\mathbf k) \to \left(\begin{array}{cc} 0 & D(\mathbf k) \\ D^\dagger(\mathbf k) & 0 \end{array}\right),\end{aligned}$$ where the upper right block is $$\begin{aligned} \label{eq:D1} D(\mathbf k) = \left(\begin{array}{cc} g(k_z) - i\Delta(\mathbf k_{\perp}) & -if(\mathbf k_{\perp}) \\ if(\mathbf k_{\perp})^* & -g(k_z) - i \Delta(\mathbf k_{\perp}) \end{array}\right).\end{aligned}$$ Noting that $f(-\mathbf k_{\perp}) = f(\mathbf k_{\perp})^*$, and $\Delta(\mathbf k_{\perp})$, $g(k_z)$ are odd functions of $\mathbf k$, matrix (\[eq:D1\]) satisfies a symmetry $D^T(\mathbf k) = -D(-\mathbf k)$. The block off-diagonal form of the Hamiltonian implies that $\epsilon(\mathbf k)^2 = D^\dagger(\mathbf k) D(\mathbf k)$, which gives rise to a fermion spectrum $$\begin{aligned} \label{eq:ek2} \epsilon_{\pm}(\mathbf k) = \pm \sqrt{\Delta(\mathbf k_{\perp})^2 + g(k_z)^2 + \bigl|f(\mathbf k_{\perp})\bigr|^2}.\end{aligned}$$ Due to the effective TRS (\[eq:trs\]), the spectrum is double degenerate at each wavevector $\mathbf k$, except at possible Fermi points. It is worth noting that, contrary to the uniform field case, the staggered field immediately opens an energy gap to the spectrum in the weak-pairing phase of the model. The topological properties of the quantum ground state (occupied Bloch states) in the gapped phase is captured by the following $Q$ matrix: [@schnyder08] $$\begin{aligned} Q(\mathbf k) = \left(\begin{array}{cc} 0 & q(\mathbf k) \\ q^\dagger(\mathbf k) & 0 \end{array}\right), \quad q(\mathbf k) = \frac{D(\mathbf k)}{\epsilon_+(\mathbf k)}.\end{aligned}$$ In fact $Q(\mathbf k)$ represents a ‘simplified’ Hamiltonian obtained by assigning an energy $\epsilon = - 1$ to all occupied states and $\epsilon = +1$ to all empty bands of Hamiltonian $H(\mathbf k)$. [@schnyder08; @qi09] As long as the system is in the same gapped phase, one can continuously deform the model such that $H(\mathbf k)$ gradually transforms to the simplified form $Q(\mathbf k)$. Not surprisingly, the $Q$ matrix is related to the spectral operator via $Q(\mathbf k) = 1- 2 P(\mathbf k)$. [@schnyder08] It is easy to see that the block matrix $q$ satisfies $$\begin{aligned} q^\dagger(\mathbf k)\, q(\mathbf k) = 1, \quad q^T(\mathbf k) = - q(-\mathbf k),\end{aligned}$$ as expected for a DIII class Hamiltonian. The topological invariant characterizing the ground state is defined as the integer winding number of mapping $q: T^3 \to U(2)$, [@schnyder08; @volovik] $$\begin{aligned} \label{eq:nu2} \nu = \int \frac{d^3 k}{24\pi^2} \epsilon^{\mu\nu\rho} {\rm tr}\bigl[(q^{-1} \partial_{\mu} q)(q^{-1} \partial_{\nu} q) (q^{-1} \partial_{\rho} q)\bigr],\end{aligned}$$ where the integral is over the three-dimensional Brillouin zone, which is essentially a 3-torus $T^3$. For DIII class Hamiltonians, the winding number $\nu$ can take on an arbitrary integer, each labels a unique topological class of the quantum ground state. \[t\] ![\[fig:phase\] (Color online) (a) Phase diagram in the plane of $J_x + J_y + J_z = $ const. The shaded triangle corresponds to the region where the in-plane couplings satisfy the triangle inequalities. In the absence of staggered field $\kappa = 0$, the shaded area represents a critical phase with zero energy gap. We define a parameter $\delta = 2 J_x - J_z$ along the $J_x = J_y$ line specifying the distance from the phase boundary. (b) Phase diagram as a function of distance $\delta$ and field strength $\kappa$. Panels (c) and (d) Numerical evaluation of the winding number $\nu$ as a function of $\delta$ and $\kappa$, respectively. The field strength $\kappa = \pm 0.02 J_z$ in the calculation of panel (c), whereas we set $J_x = J_y = J_z \equiv J$ and $\kappa$ is measured in units of $J$ in (d).](phase "fig:"){width="0.9\columnwidth"} The value of the topological winding number $\nu$ can only change in the presence of a quantum phase transition, which is usually accompanied by a vanishing bulk gap. In our case, noting that $g(0) = 0$, a prerequisite for spectrum (\[eq:ek2\]) to be gapless is that the in-plane couplings satisfy the triangle inequalities $|J_z| \le |J_x|+|J_y|$, and so on \[see Fig. \[fig:phase\](a)\], such that solutions exist for $f(\mathbf k_{\perp}) = 0$ (the weak-paring regime of the model). In the absence of magnetic field $\kappa = h = 0$, the triangle inequalities thus define a critical phase with gapless fermionic excitations. For convenience, we use $\delta$ to denote the ‘distance’ from the boundary of the critical phase. For example $\delta \equiv 2 J_x - J_z$ along the $J_x = J_y$ line shown in Fig. \[fig:phase\](a). We numerically compute the winding number $\nu$ using definition (\[eq:nu2\]) for various gapped phases of the model Hamiltonian; the resulting phase diagram as a function of $\delta$ and $\kappa$ is summarized in Fig. \[fig:phase\](b). At the phase boundary defined by $\kappa = 0$ and $\delta = 0$, the system undergoes a quantum phase transition of a topological nature. Fig. \[fig:phase\](c) shows numerical evaluation of the winding number $\nu$ as a function of $\delta = 2 J_x - J_z$ in the presence of a staggered field such that $|\kappa| = 0.02 J_z$. Depending on the sign of $\kappa$, the winding number jumps from $\nu = 0$ to $\nu = \pm 1$ when $\delta$ crosses the phase boundary from the topologically trivial phase (corresponding to the A phase in 2D Kitaev model). On the other hand, for systems inside the critical phase ($\delta < 0$), the winding number jumps from $\nu = -1$ to $\nu = +1$ as $\kappa$ changes sign. The nontrivial winding number of the quantum ground state can also be understood from the topological properties of the singular (Dirac) points in the fermion spectrum. To simplify the discussion, we focus on the symmetric case $J_x = J_y = J_z \equiv J$, where zeros of spectrum (\[eq:ek2\]) in the $\kappa \to 0$ limit are at $$\begin{aligned} \label{eq:dirac-pts} \begin{array}{lcl} \mathbf k^*_1 = (\frac{4\pi}{3},0,0), & & \mathbf k^*_2 = (-\frac{4\pi}{3}, 0, 0). \end{array}\end{aligned}$$ In the vicinity of these two points, the fermions obey a Dirac-like Hamiltonian \[c.f. Eq. (\[eq:Hd1\])\] $$\begin{aligned} \label{eq:Hc1} H(\mathbf k^* + \mathbf p) = \alpha^\mu \tilde p_\mu + m \beta,\end{aligned}$$ where the scaled momentum and mass term are $$\begin{aligned} \begin{array}{c} \tilde \mathbf p_{\perp} = \sqrt{3}J (p_y, \mp p_x), \quad \tilde p_z = 4J_{\perp} p_z, \\ \\ m=\Delta(\mathbf k^*_s) = \pm 6\sqrt{3}\kappa. \end{array}\end{aligned}$$ The plus and minus signs in the expression of $m$ correspond to $\mathbf k^*_1$ and $\mathbf k^*_2$, respectively. The spectrum of (\[eq:Hc1\]) given by $\epsilon_{\pm} = \pm \sqrt{\tilde p^2 + m^2}$ is two-fold degenerate at each $\mathbf p$. As discussed above, a $\pi/2$ rotation about $\tau^x$ brings the Dirac mass into a chiral mass term, i.e. $\beta \to i\gamma^5\beta$. The transformed Hamiltonian $H = \alpha_\mu \tilde p_\mu - i \beta \gamma^5 m$ is in a block off-diagonal form with $D = \tilde\mathbf p\cdot\bm\sigma - i m$. The $q$ matrix of the spectral projector is then given by $$\begin{aligned} \label{eq:qc1} q(\mathbf k^* + \mathbf p) = \frac{\tilde\mathbf p\cdot \bm\sigma - i m}{\sqrt{\tilde p^2 + m^2}}.\end{aligned}$$ Eqs. (\[eq:Hc1\]) and (\[eq:qc1\]) can be viewed as a continuum approximation to the low-energy physics of the model system. A direct evaluation using Eq. (\[eq:nu2\]) yields a winding number $$\begin{aligned} \nu = \pm\frac{1}{2}\frac{m}{|m|},\end{aligned}$$ with $\pm$ sign refering to $\mathbf k^*_1$ and $\mathbf k^*_2$ points, respectively. Note that when evaluating $\nu$ using the continuum description, we have extended the domain of integration to a 3-sphere in Eq. (\[eq:nu2\]). The appearance of a half-integer $\nu$ is an artifact of the continuum description; the winding number is modified once contributions from high-energy Bloch states (away from the Dirac point in the Brillouin zone) are properly included. Since the mass term has opposite sign at the two Dirac points, $m(\mathbf k^*_1) = - m(\mathbf k^*_2)$, we obtain $\nu_1 = \nu_2 = \frac{1}{2} {\rm sgn}\,\kappa$. Interestingly, the sum of these two winding numbers $\nu = \nu_1 + \nu_2$ reproduces the topological invariant of the lattice model. As recently pointed out in Ref. , stable Fermi lines generally appears in 3D topological superconductors described by Hamiltonians belonging to classes CI or DIII. An extended phase diagram of a lattice CI model [@schnyder09b] indeed shows regions of gapless phase with topologically stable Fermi lines. [@beri09] Here we show that such Fermi lines are also possible in our DIII Hamiltonian (\[eq:Hs\]). To this end, we consider perturbations which break the sublattice symmetry by introducing different second-neighbor hoppings $\kappa$ on the two inequivalent sites of the lattice: $\kappa_{1,2} = \kappa \pm \delta\kappa$. In the vicinity of the Dirac points (\[eq:dirac-pts\]), the asymmetric coupling results in an additional term of the off-diagonal block matrix $$\begin{aligned} D =\tilde\mathbf p\cdot\bm\sigma - i m - i \mu \sigma^z, \quad\quad \mu = 6\sqrt{3}\,\delta\kappa.\end{aligned}$$ The corresponding spectrum $\epsilon^2 = \tilde p_z^2 + \bigl(\mu - \sqrt{m^2 + \tilde p_{\perp}^2}\bigr)^2$ has a Fermi line specified by $$\begin{aligned} \tilde p_{\perp} = \sqrt{\mu^2 - m^2}, \quad\quad \tilde p_z = 0,\end{aligned}$$ when $\mu > m$. Finally, we point out that introducing a similar asymmetric hopping $\delta\kappa$ in the case of uniform field results in a Fermi surface in the weak-pairing regime. Gapless Surface Majorana fermions ================================= Analogous to the gapless chiral edge modes in the non-Abelian phase of 2D Kitaev model, [@kitaev06] the nontrivial topology of the $\nu = \pm 1$ quantum ground state in our 3D model manifests itself through the appearance of gapless Majorana fermions at the sample surface. To study the properties of these surface modes, we solve the Majorana hopping problem (\[eq:H-cc\]) and (\[eq:H-cc2\]) in a finite geometry with periodic boundary conditions along $x$ and $z$ directions and open boundary condition along $y$ direction. We assume that the $x$-axis is parallel to the zigzag direction of the honeycomb lattice. As Fig. \[fig:surface\] shows, in addition to gapped states in the bulk, the spectrum of a finite system contains additional surface modes crossing the bulk gap. The properties of these surface states can be understood analytically using the edge modes of 2D Kitaev model, whose existence has been demonstrated in Ref. . These boundary modes are confined to the edge of the honeycomb lattice and possess a definite chirality depending on the sign of magnetic field $h$. Without loss of generality, we assume $\kappa\sim \lambda^2 h > 0$, hence $\epsilon_{\perp}(k_x) > 0$ for positive $k_x$ in the 1D Brillouin zone. The Hamiltonian describing the edge states is given by $$\begin{aligned} \label{eq:H-edge} \mathcal{H}_{\rm edge} = \frac{1}{2}\sum_{k_x} \epsilon_{\perp}(k_x) \,\chi(-k_x)\, \chi(k_x),\end{aligned}$$ where $\chi(k_x)$ and $\chi(-k_x) = \chi^\dagger(k_x)$ for $k_x>0$ are annihilation and creation operators, respectively, of the Majorana edge modes. The spectrum of the edge states has the form $\epsilon_{\perp}(k_x) \approx 12\kappa \sin k_x$ in the vicinity of the 1D Fermi point $k^*_x = \pi$, and gradually merges with the bulk spectrum as $k_x$ moves away from $k^*_x$. [@kitaev06] ![\[fig:surface\] (Color online) Spectrum of a finite system with $L = 10$ layers along $y$ direction. We consider the symmetric case with $J_x = J_y = J_z \equiv J$. Coupling constant along vertical links is set to $J_{\parallel} = 0.35 J$, and the strength of the uniform field is $\kappa = 0.05 J$. Panels (a) and (b) show dispersion along $k_x$ and $k_z$ directions, respectively, of the 2D surface Brillouin zone.](edge "fig:"){width="0.492\columnwidth"} ![\[fig:surface\] (Color online) Spectrum of a finite system with $L = 10$ layers along $y$ direction. We consider the symmetric case with $J_x = J_y = J_z \equiv J$. Coupling constant along vertical links is set to $J_{\parallel} = 0.35 J$, and the strength of the uniform field is $\kappa = 0.05 J$. Panels (a) and (b) show dispersion along $k_x$ and $k_z$ directions, respectively, of the 2D surface Brillouin zone.](edge2 "fig:"){width="0.492\columnwidth"} It is interesting to note that the $J_{\parallel} = 0$ limit of the 3D model (\[eq:h\]) can be viewed as a collection of decoupled honeycomb layers. Due to the staggered magnetic field, the edge states on even and odd numbered layers have opposite chirality $\nu =\pm {\rm sgn}\,\kappa$; the corresponding quasiparticle operators are denoted as $\chi_+(k_x,2n)$ and $\chi_-(k_x,2n+1)$, respectively. The ensemble of the edge modes is described by Hamiltonian: $$\begin{aligned} \mathcal{H}_{\perp} = \frac{1}{2}\sum_n \sum_{k_x} \epsilon_{\perp}(k_x) \bigl[\chi_+(-k_x,2n)\chi_+(k_x,2n) \nonumber \\ -\chi_-(-k_x,2n+1)\chi_-(k_x,2n+1)\bigr].\end{aligned}$$ For odd-numbered layers, the quasiparticle annihilation operator is given by $\chi_-(k_x, 2n+1)$ with negative $k_x$. A nonzero $J_{\parallel}$ introduces coupling between adjacent layers: $$\begin{aligned} \mathcal{H}_{\parallel} = iJ_{\parallel}\sum_{n}\sum_{k_x} \bigl[\chi_-(-k_x,2n+1)\chi_+(k_x,2n) \nonumber \\ + \chi_+(-k_x,2n)\chi_-(k_x,2n-1)\bigr].\end{aligned}$$ After Fourier transformation with respect to $z$ coordinate, we obtain the Hamiltonian for surface modes $$\begin{aligned} \mathcal{H}_{\rm surf} = \frac{1}{2}\sum_{k_x,k_z} \psi^\dagger_{k_x, k_z} \bigl[\epsilon_{\perp}(k_x)\sigma^z + \epsilon_{\parallel}(k_z) \sigma^x\bigr] \psi_{k_x,k_z},\end{aligned}$$ where $\psi_{k_x,k_z} = \bigl(\chi_+(k_x,k_z), \chi_-(k_x,k_z)\bigr)^T$, and the off-diagonal coupling $\epsilon_{\parallel}(k_z) = 2J_{\parallel} \sin k_z$. The spectrum can be easily obtained $\epsilon_{\rm surf} = \bigl(\epsilon_{\parallel}^2 + \epsilon_{\perp}^2\bigr)^{1/2}$. Consistent with the numerical calculation shown in Fig. \[fig:surface\], the surface spectrum has a conic singularity at the 2D Fermi point $(k^*_x, k^*_z) = (\pi,0)$. In the continuum approximation, the low-energy surface modes close to the 2D Fermi point obeys a gapless Dirac-like Hamiltonian $H_{\rm Dirac} = -i( u_{\perp} \sigma^z \partial_x + u_{\parallel} \sigma^x \partial_z)$. The existence of gapless surface Majorana modes is a direct consequence of the nontrivial ground-state topology when the 3D bulk sample is terminated by a 2D boundary. [@callan85; @fradkin86] Due to its topological nature, the surface Dirac cone is stable against perturbations respecting the effective TRS. Discussion ========== To summarize, we have constructed an interacting bosonic model on a three-dimensional layered honeycomb lattice which displays topologically nontrivial ground states. Our approach is based on the $\Gamma$-matrix generalization of Kitaev’s original spin-1/2 model. [@levin03; @hamma05; @yao09; @wu09; @ryu09; @nussinov09] We show that the ground state in the weak-pairing phase remains gapless in the presence of a small magnetic field. The fermion spectrum of this critical phase is characterized by four topologically protected Fermi points. Upon increasing the field strength, Fermi points with opposite winding numbers move toward each other and eventually annihilate at a critical field, signaling the transition into a topologically trivial phase with a gapped spectrum. On the other hand, with the sign of magnetic field staggered along successive layers, the bulk spectrum of the weak-pairing phase immediately acquires a gap. An effective TRS is restored thanks to the staggering of the magnetic field. The corresponding quantum ground state is characterized by an integer winding invariant and possesses nontrivial topological properties, a manifestation of which is the appearance of protected gapless surface Majorana modes when the sample is terminated by a two-dimensional surface. It is worth noting that our model provides an alternative example in which a 3D topological insulator emerges from an interacting bosonic Hamiltonian. More importantly, the TRS essential to the existence of 3D topological insulators is realized in our model through a generic physical mechanism, instead of relying on special symmetries of the $\Gamma$-matrix representation in the recently proposed diamond-lattice model. [@ryu09; @wu09] We also remark that despite the experimental difficulty of realizing the Kitaev model and its variants, the study of these exactly solvable models has enriched our understanding of the physics of topological phases. Finally, we would like to point out that it remains unclear whether an emergent topological insulator can be realized in a spin-1/2 Kitaev model on a 3D trivalent lattice.\ The author acknowledges B. Béri, R. Moessner, N. Perkins, Tieyan Si, Hong Yao, Sungkit Yip for helpful discussions and comments, and also the visitors program at Max-Planck-Institut für Physik komplexer Systeme, Dresden, Germany. In particular, I would like to thank Benjamin Béri for pointing out the possibility of topologically stable Fermi lines in the DIII Hamiltonian studied in this paper. [99]{} D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. [**49**]{}, 405 (1982). J. E. Avron, R. Seiler, and B. Simon, Phys. Rev. Lett. [**51**]{}, 51 (1983). M. Kohmoto, Annals of Physics [**160**]{}, 428 (1985). X.-G. Wen, [*Quantum Field Theory of Many-Body Systems*]{}, Oxford University Press (2004). G. E. Volovik, [*The Universe in a Helium Droplet*]{}, The International Series of Monographs on Physics Vol. 117 Oxford University Press, New York (2000). C. L. Kane and E. J. Mele, Phys. Rev. Lett. [**95**]{}, 146802 (2005); Phys. Rev. Lett. [**95**]{}, 226801 (2005). B. A. Bernevig, S. C. Zhang, Phys. Rev. Lett. [**96**]{}, 106802 (2006); B. A. Bernevig, T. L. Hughes, S. C. Zhang, Science [**314**]{}, 1757 (2006). J. E. Moore and L. Balents, Phys. Rev. B [**75**]{}, 121306 (2007). R. Roy, Phys. Rev. B [**79**]{}, 195321 (2009). L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. [**98**]{}, 106803 (2007). L. Fu and C. L. Kane, Phys. Rev. B [**76**]{}, 045302 (2007). R. Roy, Phys. Rev. B [**79**]{}, 195322 (2009). M. König, S. Wiedmann, C. Bruene, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Science [**318**]{}, 766 (2007). D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. Hor, R. J. Cava, and M. Z. Hasan, Nature [**452**]{}, 970 (2008). Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Nature Phys. [**5**]{}, 398 (2009). P. Roushan, J. Seo, C. V. Parker, Y. S. Hor, D. Hsieh, D. Qian, A. Richardella, M. Z. Hasan, R. J. Cava, and A. Yazdani, Nature [**460**]{}, 1106 (2009). Y. L. Chen, J. G. Analytis, J.-H. Chu, Z. K. Liu, S.-K. Mo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. Fisher, Z. Hussain, Z.-X. Shen, Science [**325**]{}, 178 (2009). A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Phys. Rev. B [**78**]{}, 195125 (2008); AIP Conf. Proc. [**1134**]{}, 10 (2009). A. Yu Kitaev, AIP Conf. Proc. [**1134**]{}, 22 (2009). X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B [**78**]{}, 195424 (2008). G. Jackeli and G. Khaliullin, Phys. Rev. Lett. [**102**]{}, 017205 (2009). A. Shitade, H. Katsura, J. Kunes, X.-L. Qi, S.-C. Zhang, N. Nagaosa, Phys. Rev. Lett. [**102**]{}, 256403 (2009). S. Raghu, X.-L. Qi, C. Honerkamp, S.-C. Zhang, Phys.Rev.Lett. [**100**]{}, 156401 (2008). K. Sun, H. Yao, E. Fradkin, and S. A. Kivelson, Phys. Rev. Lett. [**103**]{}, 046811 (2009) Yi Zhang, Ying Ran, and A. Vishwanath, Phys. Rev. B [**79**]{}, 245331 (2009). A. Yu Kitaev, Ann. Phys. [**321**]{}, 2 (2006). A. Yu Kitaev, Ann. Phys. [**303**]{}, 2 (2003). S. Yang, D. L. Zhou, and C. P. Sun, Phys. Rev. B [**76**]{}, 180404 (2007). H. Yao and S. A. Kivelson, Phys. Rev. Lett. [**99**]{}, 247203 (2007). T. Si and Y. Yu, arXiv.0709.1302 (2007). T. Si and Y. Yu, Nucl. Phys. B [**803**]{}, 428 (2008). S. Mandal and N. Surendran, Phys. Rev. B [**79**]{}, 024426 (2009). G.-W. Chern, unpublished. M. Levin and X.-G. Wen, Phys. Rev. B [**67**]{}, 245316 (2003). A. Hamma, P. Zanardi1, and X.-G. Wen, Phys. Rev. B [**72**]{}, 035307 (2005). H. Yao, S.-C. Zhang, and S. A. Kivelson, Phys. Rev. Lett. [**102**]{}, 217202 (2009). S. Ryu, Phys. Rev. B [**79**]{}, 075124 (2009). C. Wu, D. Arovas, and H.-H. Hung, Phys. Rev. B [**79**]{}, 134427 (2009). Z. Nussinov and G. Ortiz, Phys. Rev. B [**79**]{}, 214440 (2009). A. Altland and M. R. Zirnbauer, Phys. Rev. B [**55**]{}, 1142 (1997). K. I. Kugel and D. I. Khomskii, Sov. Phys. Usp. [**231**]{}, 25 (1982). E. H. Lieb, Phys. Rev. Lett. [**73**]{}, 2158 (1994). G. Baskaran, S. Mandal, and R. Shankar, Phys. Rev. Lett. [**98**]{}, 247201 (2007). N. Read and D. Green, Phys. Rev. B [**61**]{}, 10267 (2000). N. D. Mermin, Rev. Mod. Phys. [**51**]{}, 591 (1979). M. Kohmoto, B. I. Halperin, and Y.-S. Wu, Phys. Rev. B [**45**]{}, 13488 (1992). F. D. M. Haldane, Phys. Rev. Lett. [**61**]{}, 2015 (1988). X. L. Qi, T. L. Hughes, and S. C. Zhang, arXiv:0908.3550 (unpublished). B. Béri, arXiv:0909.5680 (unpublished). A. P. Schnyder, S. Ryu, and A. W. W. Ludwig, Phys. Rev. Lett. [**102**]{}, 196804 (2009). C. G. Callan, Jr. and J. A. Harvey, Nucl. Phys. B [**250**]{}, 427 (1985). E. Fradkin, E. Dagotto, and D. Boyanovsky, Phys. Rev. Lett. [**57**]{}, 2967 (1986).

--- abstract: 'The dependence of the thermal pressure of hot galactic halos on a model parameter describing the frequency of major reheating episodes during galactic history is investigated. Pressure on the interface between interstellar medium and the halo gas is especially interesting, since empirical evidence here offers one of the simplest constraints on halo models. It is shown that two-phase model of Mo & Miralda-Escudé is sufficiently robust with respect to uncertainties in the average interval between reheating.' author: - 'MILAN M. ĆIRKOVIĆ' title: 'PRESSURE AT THE ISM-HALO INTERFACE: A REHEATING FREQUENCY DEPENDENCE' --- and INTRODUCTION ============ The existence of very extended galactic halos[^1] has been suspected for a long time (Spitzer 1956) on several grounds. In last two and a half decades, from the observations of [*Copernicus*]{} and [*IUE*]{} satellites we have learnt that highly ionized gas exists in the halo of Milky Way to much larger scale-heights than it was previously assumed (for a review, see Savage 1988). So-called high-velocity clouds were discovered high above the plane of the Galaxy, and in other nearby spiral galaxies. X-ray observations showed the presence of vast quantities of hot gas in rich clusters of galaxies (Sarazin 1986), as well as in smaller compact groups (Saracco & Ciliegi 1995; Mulchaey et al. 1996). Cooling flows phenomena were observed in cluster members (Fabian 1994), as well as in individual elliptical galaxies (Nulsen, Stewart, & Fabian 1984). The extraplanar optical recombination emission was discovered at large galactocentric distances in several nearby galaxies (Donahue, Aldering, & Stocke 1995; Pildis, Bregman, & Schombert 1994). The depletion of globular cluster gas was interpreted as the consequence of ram-pressure stripping, and consequences were drawn thereof (Frank & Gisler 1976; Ninković 1985). Especially important motivation comes from the QSO absorption line studies. It is now clear that, at least at low redshift, a bulk of both metal and Ly$\alpha$ absorption lines arise in gas which is associated with luminous galaxies (Sargent, Steidel, & Boksenberg 1988; Bechtold & Ellingson 1992; Lanzetta et al. 1995; Bowen, Blades, & Pettini 1996; Chen et al. 1998). The idea that the QSO absorbers might arise in halos of normal galaxies originated with Bahcall & Spitzer (1969), but the quantitative models were rare (e.g. Bregman 1981) until recently, when vast accumulated data on absorber statistics enabled establishing strong empirical constraints on model parameters (see discussion in Mo & Morris 1994). Very important simple two-phase model of the galactic gaseous halos was put forward recently by Mo (1994), and Mo & Miralda-Escudé (1996; hereafter MM96). Their work shows quite persuasively that formation of the halo structure as a natural consequence of the galaxy formation and subsequent evolutionary processess. Formation of galaxies implies the collapse of the baryonic content of intergalactic space and its motion through the non-dissipative dark matter halos creating deep gravitational potential well. A halo of hot gas at virial temperature ($T_v \equiv \mu V_{\rm cir}^2/2k$, where $\mu$ is the average mass per particle, $k$ is the Boltzmann constant and $V_{\rm cir}$ is the circular velocity, which is assumed to parametrize the total mass of the galaxy) will form as the kinetic energy of the infalling material is thermalized in shocks of this primordial accretion. As the gas collapses, it is shocked and it subsequently cools (for an early treatment, see White & Rees 1978). Afterwards, similar situation (although presumably on smaller scale) arises in course of major galaxy mergers: here we also have a large mass of gas accreted into the system on a short timescale and substantial reheating. In the meantime, accretion of ambiental IGM is small or entirely negligible. When hot gas cools, it forms the [ *cold phase*]{}: photoionized clouds at $T_{\rm cold} \sim 10^4$ K, in pressure equilibrium with the hot halo, slowly sinking in the galactic gravitational potential. While cold phase may dominate the total gas mass, its filling factor is always small, and we shall not deal with it here separately. The galactocentric radius $r_c$ where the cooling time of the hot gas is the [*cooling radius*]{}. Quantitatively, it is necessary to satisfy $$\label{minone} \frac{5\mu k T_v}{2 \Lambda(T_v) \rho(r_c)} = t_M,$$ where $\Lambda(T_v)$ is the cooling rate at the virial temperature, and $\rho(r_c)$ is the physical density of hot gas at the cooling radius. Inside it, the conditions of adiabaticity and hydrostatic equilibrium, lead to the equation of state $P \propto \rho^{5/3}$, where $\rho$ is the density of the gas. The ratio of the total gas mass and the virial mass is set by the dimensionless parameter $f_g$. Isothermal potential of the dark matter creates the density profile $$\label{zero} \rho \propto \left( 1- K \ln \frac{r}{r_c} \right)^{\frac{3}{2}},$$ where $K$ is the constant, in the simplest model equal to 0.8. Simple calculation shows that pressure at some distance $r$ will behave as $P/k \propto (1-K \ln \frac{r}{r_c} )^{5/2}$. We see that one of the model parameters is $t_M$, the average interval between reheating of the gas, the inverse of what we may call the reheating frequency. It is one of the most uncertain parameters. MM96 use $$\label{one} t_M =\frac{t}{1+\Omega_0}$$ approximation, where $t$ is the age of the universe, and $\Omega_0$ is the total cosmological density parameter. It is used quite successfully by MM96, although admitting that a detailed treatment is difficult (due not only to the uncertainty in $\Omega_0$, but the power spectrum and physics of structure formation as well). While not questioning the goodness of this approximation, we would like to show that actual physical quantities resulting from their model are not much affected by the choice of $t_M$. One of, from the physical point of view, most important predictions is the value of the thermal pressure of the hot halo at small galactocentric distances, where interface with the ISM occurs. Therefore, in this paper, we are investigating the changes in the pressure at this inner boundary of the hot halo, when the average time-scale for reheating of the gas is varied. This is a part of the more comprehensive study of the boundary conditions of the two-phase model, which is currently in progress (Ćirković 1998). THERMAL PRESSURE AT THE HOT HALO BOUNDARIES =========================================== Formally, thermal pressure in the adiabatic model diverges as $r\rightarrow 0$. Of course, we know, from both theoretical modelling, and observations of Galactic ISM, that the real picture is more complicated. It seems clear that assumption of (quasi) hydrostatic equilibrium must be abandoned at some fiducial galactocentric radius, comparable to the size of disk in a spiral galaxy. This is not necessarily exact ISM-halo interface, since a lot of non-equilibrium and still poorly-understood phenomena (“galactic fountain”, winds, superbubbles, etc.) will conceivably have the desired effect of “softening” the global pressure profile at $R {\:^{<}_{\sim}\:}50$ kpc. We shall neglect these dynamical effects, and following Mo (1994) regard rough size of the disk, $R_d =20$ kpc as the inner boundary of the hot halo gas. This is justified on a qualitative basis, among other arguments, by the fact that no significant large-scale pressure gradients in the ISM of Milky Way were not observed, including observations toward high-latitude stars at galactocentric distance representing significant fraction of our chosen $R_d$. It is natural to expect that approximately uniform—when averaged over individual clouds and intercloud medium—ISM pressure is in equilibrium with the thermal pressure of the envelopping halo gas for any reasonable choice of parameters (Spitzer 1978). Therefore, we have calculated pressure $P/k$ at $R_d$ (for $z=0$, i.e. at the present epoch), as a function of $T_M$, a mean interval between reheating in major mergers. As is clear from the results obtained (see Figs. 1 and 2), the model is quite robust to the variations in $t_M$. For the case of a realistic Population II metallicity ($Z=0.1 \; Z_\odot$) shown in Fig. 1, changing $t_M$ by more than an order of magnitude in the case of $L_\ast$ galaxy results in change of $P/k$ by a factor of $\sim 2.5$. Even if the metallicities are higher (MM96), the same tendency persists, as is shown in the plot in Fig. 2 (for the global gas metallicity equal to the standard solar, which is unrealistically high). For comparison, the pressure in the local ISM was estimated by Spitzer (1978) as $P/k_{\rm B} = 1700$ K cm$^{-3}$ ($\log P/k = 3.2$), or at large sample of measured clouds (Jenkins, Jura, & Loewenstein 1983) $$\label{ism} 3.4 \leq \log P/k_{\rm B} \leq 3.8.$$ We note that, especially for higher values of $t_M$ (favored also by the equation (\[one\]) and other data; see, for example, Keel & Wu 1995) theoretical values of the two-phase model are quite consistent with this empirical range. Other theoretical uncertainties, like that in a fraction of the virial mass contained in gas (parameter $f_g$ in MM96), seem to be much greater than the one considered here. DISCUSSION ========== Major weaknesses of the theoretical framework this simple, manifest themselves clearly in the fact that mergers are considered only in terms of reheating through arising shocks, neglecting the mass input which undoubtedly occurs. On the other hand, without a detailed understanding of the underlying physics of merging events, it is difficult to proceed in that direction. It is clear, though, that if the bulk of the mass deposition occurs at small enough distances (compared to the virial radius before the merger), hot gas will be just a transient phase before most of the mass goes into cold, photoionized clouds. This can be estimated as follows: if $t_M$ is large compared to the timescale given by $\tau=5 \mu k T_0/2 \Lambda(T_0) \rho(r_0)$ (where $T_0=T_v(1-K \ln \frac{r_0}{r_c})$ and $\rho(r_0)$ is given by the equation (\[zero\]), with $r_0$ being the characteristic radius for mass deposition), a quasi-stationary state is reached, and the discussion we have followed applies. We have so far discussed only rehetaing in major mergers, which undoubtedly occurs from empirical evidence (Keel & Wu 1995, and references therein). Other conceivable reheating mechanisms (starbursts or switching on of a nuclear source) may be important in the inner part of the halo and should be considered in the course of a future work. We have shown that simple model based on assumptions of adiabaticity and quasi hydrostatic equilibrium gives physically acceptable results for the pressure at the ISM-halo interface, results which are quite insensitive to the assumptions about reheating frequency $1/t_M$. Until our theoretical knowledge on merger frequencies (or other reheating mechanisms) improves, we are justified in using simple approximations like (\[one\]). The author is happy to acknowledge help of Dr. Hou Jun Mo, who kindly provided a cooling code, as well as useful discussion, and Mr. Branislav Nikolić whose hospitality and support were essential for completion of this work. Dr Luka Č. Popović was, as usual, extremely helpful with a friendly advice and technical assistance. Bahcall, J. N., & Spitzer, L. Jr. 1969, ApJ, 156, L63 Bechtold, J. & Ellingson, E. 1992, ApJ, 396, 20 Bowen, D. V., Blades, J. C., & Pettini, M. 1996, ApJ, 464, 141 Bregman, J. N. 1981, ApJ, 250, 7 Chen, H.-W., Lanzetta, K. M., Webb, J. K., & Barcons, X. 1998, ApJ, in press Ćirković, M. M. 1998, in preparation Donahue, M., Aldering, G., & Stocke, J. T. 1995, ApJ, 450, L45 Fabian, A. C. 1994, ARAA, 32, 277 Frank, J., & Gisler, G. 1976, MNRAS, 176, 533 Jenkins, E. B., Jura, M., & Loewenstein, M. 1983, ApJ, 270, 88 Keel, W. C., & Wu, W. 1995, AJ, 110, 129 Mo, H. J. 1994, MNRAS, 269, L49 Mo, H. J., & Morris, S. L. 1994, MNRAS, 269, 52 Mo, H. J., & Miralda-Escudé, J. 1996, ApJ, 469, 589 (MM96) Mulchaey J. S., Davis, D. S., Mushotzky, R. F., & Burstein, D. 1996, ApJ, 456, 80 Ninković, S. 1985, Ap & SS, 110, 379 Nulsen, P. E. J., Stewart, G. C., & Fabian, A. C. 1984, MNRAS, 208, 185 Pildis, R. A., Bregman, J. N., & Schombert, J. M. 1994, ApJ, 423, 190 Saracco, P., & Ciliegi, P. 1995, A & A, 301, 348 Sargent, W. L. W., Steidel, C. C., & Boksenberg, A. 1988, ApJ, 334, 22 Savage, B. D. 1988, in QSO Absorption Lines: Probing the Universe, eds. Blades, J. C. et al. (Cambridge University Press, Cambridge) Spitzer, L. Jr. 1956, ApJ, 124, 20 Spitzer, L. Jr. 1978, “Physical Processes in the Interstellar Medium” (John Wiley & Sons, New York) White, S. D. M., & Rees, M. J. 1978, MNRAS, 183, 341 Fig. 1. [Thermal pressure of the hot halo at galactocentric distance $R_d=20$ kpc at present epoch for $f_g=0.05$ and several characteristic values of circular velocities of a galaxy, as a function of average interval between rehetaing $t_M$. Global halo metallicity is set to $Z=0.1 \; Z_\odot$. Long-dashed line corresponds to a very massive galaxy ($V_{\rm cir}=300$ km s$^{-1}$), and shord-dashed line to low-mass galaxy ($V_{\rm cir}=100$ km s$^{-1}$). The case of a typical $L_\ast$ galactic halo is shown as the solid line ($V_{\rm cir} \approx 220$ km s$^{-1}$). We note that $P/k$ stays approximately the same with reheating frequency taking all physically realistic values.]{} Fig. 2. [The same as in Fig. 1, except for the (unrealistic) case of halo gas having global metallicity equal to standard $Z_\odot$. Average values for the pressure are slightly lower due to more efficient cooling and consequent depletion of the hot phase.]{} [^1]: The term “halo” will be preferred over such frequently used terms as “corona” or “envelope”, although it is implied that they all describe the same physical objects.

--- author: - 'C. Lataniotis, S. Marelli, B. Sudret' bibliography: - 'Lataniotis-etal-Arxiv-2018.bib' title: 'Extending classical surrogate modelling to ultrahigh dimensional problems through supervised dimensionality reduction: a data-driven approach' --- Introduction {#sec:Introduction} ============ It is nowadays a common practice to study the behaviour of physical and engineering systems through computer simulation. In a real-world setting, such systems are driven by input parameters, the values of which can be uncertain or even unknown. Uncertainty quantification (UQ) aims at identifying and quantifying the sources of uncertainty in the input parameters to assess the uncertainty they cause in the model predictions. In the context of Monte Carlo simulation, such workflow typically entails the repeated evaluation of the computational model. However, it may become intractable when a single simulation is computationally demanding, as is often the case with modern computer codes. A remedy to this problem is to substitute the model with a surrogate that accurately mimics the model response within the chosen parameter bounds, but is computationally inexpensive. An additional benefit of surrogate models is that they are often non-intrusive, i.e. their construction only depends on a training set of model evaluations, without access to the model itself. This includes the case when the model is not available, but only a pre-existent data set is, as is typical in machine learning applications. The latter setting is the focus of this paper. Popular surrogate modelling techniques (SM) include Gaussian process modelling and regression [@Sacks1989; @Rasmussen2006], polynomial chaos expansions [@Ghanembook1991; @Xiu2002; @XiuBook2010], low-rank tensor approximations [@Chevreuil2015; @KonakliJCP2016], and support vector regression [@Vapnik1995]. Parametrising and training a surrogate model, however, can become harder or even intractable as the number of input parameters increases, a well known problem often referred to as *curse of dimensionality* (see @Verleysen05). For the sake of clarity, in the following we will classify high-dimensional inputs in two broad categories, depending on their characteristics: *unstructured* or *structured*. Unstructured inputs are characterised by the lack of an intrinsic ordering, and they are commonly identified with the so-called “model parameters”, point loads on mechanical models, or resistance values in electrical circuit models. Structured inputs, on the other hand, are characterised by the existence of a natural ordering and/or a distance function (*i.e.* they show strong correlation across some physically meaningful set of coordinates), as it is typical for time-series or space-variant quantities represented by maps. Boundary conditions in complex simulations that rely on discretisation grids, *e.g.* time-dependent excitations at grid nodes, often belong to this second class. In most practical applications, unstructured inputs range in dimension in the order $\co(10^{0-2})$, while structured inputs tend to be in the order $\co(10^{2-6})$. Several strategies have been explored in the literature to deal with high dimensional problems for surrogate modelling. A common approach in dealing with unstructured inputs is input variable selection, which consists in identifying the “most important” inputs according to some importance measure, see *e.g.* @Saltelli2008 [@Iooss2015], and simply ignoring the others (*e.g.* by setting them to their nominal value). In the context of kernel-based emulators (*e.g.* Gaussian process modelling or support vector machines), some attention has been devoted to the use of simple isotropic kernels [@djolonga2013], or to the design of specific kernels for high-dimensional input vectors, sometimes including deep-learning techniques (*e.g.*, [@Lawrence2005; @durrande2012; @wilson2016]). In more complex scenarios, the more general concept of *dimensionality reduction* (DR) is applied, which essentially consists in mapping the input space to a suitable lower dimensional space using an appropriate transformation prior to the surrogate modelling stage. The latter approach is considered in this work due to its applicability to cases for which variable selection seems inadequate or insufficient ( in the presence of structured inputs). In the current literature, a two-step approach is often followed for dealing with such problems: first, the input dimension is reduced; then, the surrogate model is constructed directly in the reduced (feature-) space. The dimensionality reduction step is based on an *unsupervised* objective, an objective that only takes into account the input observations. Examples of unsupervised objectives include the minimisation of the input reconstruction error [@vincent2008], maximisation of the sample variance [@Pearson01], maximisation of statistical independence [@hyvarinen1997one], and preservation of the distances between the observations [@tenenbaum2000isomap; @roweis2000nonlinear; @hinton2003sne]. While in principle attractive due to their straightforward implementation, unsupervised approaches for dimensionality reduction may be suboptimal in this context, because the input-output map of the reduced representation may exhibit a complex topology unsuitable for surrogate modelling [@wahlstrom2015; @calandra2016]. To deal with this issue, various *supervised* techniques have been proposed, in the sense that the objective of the input compression takes into account the model outputs. One such approach that has received attention recently is based on the so-called *active subspaces* concept [@Constantine2014]. Various methods that belong to this category, provide a linear transformation of the high dimensional input space into a reduced space that is characterised by maximal variability w.r.t. the model output. However, active subspace methods often require the availability of the model gradient w.r.t. the input parameters, a limiting factor in data-driven scenarios where such information is not available and needs to be approximated [@Fornasier2012]. Moreover, the numerical computation of the gradient may be infeasible in problems that involve structured inputs such as time series or 2D maps with $\co(10^{2-6})$ components. Other data-driven supervised DR techniques have been proposed in the literature, that are dependent on the properties of a specific combination of either DR or SM techniques. @HintonSalakhutdinov2006b employ multi-layer neural networks for both the DR and the SM steps. Specifically, an unsupervised objective based on the reconstruction error is followed by a generalisation performance objective that aims at fine tuning the network weights with respect to a measure of the surrogate modelling error. Similar approaches have been proposed with other combinations of methods. In @damianou2013deep, the same idea is extended by using stacked Gaussian processes instead of multilayer neural networks. In @huang2015 [@calandra2016] this approach is extended by combining neural networks with Gaussian processes within a Bayesian framework. All of these methods demonstrate that supervised methods yield a significant accuracy advantage over the unsupervised ones, as the final goal of the supervised learner (*i.e.* surrogate model accuracy) matches the final goal of high-dimensional surrogate modelling in the first place. However, this increased accuracy comes at the cost of restricting the applicability of such methods to specific combinations of DR and SM techniques. In this paper, we propose a novel method of performing dimensionality reduction for surrogate modelling in a data-driven setting, which we name (perhaps with a lack of creative flair) DRSM. The aim of this method is to capitalise on the performance gains of supervised DR, while maintaining maximum flexibility in terms of both DR and SM methodologies. Recognising that different communities, applications and researchers have in general access to one or two preferred techniques for either DR or SM, the proposed approach is fully non-intrusive, *i.e.* both the DR and the SM stages are considered as *black boxes* under very general conditions. The novelty lies in the way the two stages are coupled into a single problem, for which dedicated solvers are proposed. This paper is structured as follows: Section 2 introduces the main ingredients required by DRSM, namely dimensionality reduction and surrogate modelling. For the sake of clarity, some of the techniques that will be specifically used in the applications section are also introduced, *i.e.* kernel principal component analysis (KPCA) for DR, Gaussian process modelling, a.k.a. Kriging, and polynomial chaos expansions (PCE) for SM. The core framework underlying DRSM is then introduced. Finally, the effectiveness of DRSM is analysed on several benchmark applications including both unstructured and structured inputs, ranging from low-dimensional analytical functions to a complex engineering 2-dimensional heat-transfer problem. Ingredients for surrogate modelling in high dimension {#sec:Methodology} ===================================================== As the name implies, DRSM consists in the combination of two families of computational tools: dimensionality reduction and surrogate modelling. This section aims at highlighting the main features of each, and how they can be exploited without resorting to intrusive, dedicated algorithms. Dimensionality reduction {#sec:Meth:DR} ------------------------ Consider a set of high-dimensional samples $\cx = \acc{\bfx^{(i)}\in \Rr^M\, , \, i=1\enu N}$. In an abstract sense, dimensionality reduction (DR) refers to the parametric mapping $g: \cx \in \Rr^M \mapsto \cz \in \Rr^m$ of the form: $$\label{eq:DR_general_form} \bfz = g(\bfx ; \bfw)$$ where $\bfz \in \cz$, $\bfx \in \cx$, and $\bfw$ is the set of parameters associated with the mapping. Dimensionality reduction occurs if $m\ll M$, if $m=\co\prt{10^{0-1}}$ whereas $M=\co\prt{10^{2-4}}$. The nature and number of the parameters $\bfw$ depends on the specific DR method under consideration. Such transformations are motivated by the assumption that the samples in $\cx$ lie on some manifold with dimensionality $m$ that is embedded within the $M$-dimensional space. This specific value of $m$ is in some applications referred to as the “intrinsic dimension” of $\cx$ [@Fukunaga2013]. From an information theory perspective, the intrinsic dimension refers to the minimum number of scalars that is required to represent $\cx$ without any loss w.r.t. an appropriate information measure. In practice it is a-priory unknown. In such cases DR is an ill-posed problem that can only be solved by assuming certain properties of $\cx$, such as its intrinsic dimension. Alternatively the later may be approximated and/or inferred from the available data by various approaches (see @Camastra2003 for a comparative overview). An important aspect of all parametric DR methods, regardless of their specificity, is that for each choice of dimension $m$ the remaining parameters $\bfw$ are estimated by minimising a suitable error measure (sometimes referred to as loss function): $$\widehat{\bfw} = \arg \underset{\cd_\bfw}{\min} \, J(\bfw;\cx),$$ where $ \widehat{\bfw}$ denotes the estimated parameters, $\cd_\bfw$ the feasible domain of $\bfw$, $J(\cdot)$ the error measure and $\cx$ the available data. The choice of the error measure depends on the specific application DR is used for. When the goal is direct compression of a high dimensional input without information loss (a common situation in telecommunication-related applications), a typical choice of $J(\cdot)$ is the so-called mean-squared reconstruction error, that reads: $$J(\bfw;\cx) = \frac{1}{N} \sum_{i=1}^{N} {\left\lVert\bfx^{(i)} - \tilde{\bfx}^{(i)}\right\rVert}^2,$$ where $\tilde{\bfx}=g^{-1}(\bfz,\bfw)$ denotes the reconstruction of the sample $\bfx$, calculated through the inverse transform $g^{-1}: \cz \in \Rr^m \mapsto \cx \in \Rr^M$. In the general case, additional parameters may be introduced in $g^{-1}$, or the inverse transform may not exist at all (see @Kwok2003). For a detailed description of the specific DR methods used in this paper to showcase the proposed methodology, namely principal component analysis (PCA) and kernel PCA, the reader is referred to [Section \[sec:Meth:Selected DR and SM\]]{}. Surrogate Modelling {#sec:Meth:SM:Intro} ------------------- In the context of UQ, the physical or computational model of a system can be seen as a black-box that performs the mapping: $$\label{eq:true_model} \ve{Y} = \cm(\ve{X}),$$ where $\ve{X}$ is a random vector that parametrises the variability of the input parameters (*e.g.* through a joint probability density function) and $\ve{Y}$ is the corresponding random vector of model responses. One of the main applications of UQ is to propagate the uncertainties from $\ve{X}$ to $\ve{Y}$ through the model $\cm$. Direct methods based on Monte-Carlo simulation may require that the computational model is run several thousands of times for different realisations $\bfx$ of the input random vector $\ve{X}$. However, most models that are used in applied sciences and engineering (*e.g.* high-resolution finite element models) can have high computational costs per model run. As a consequence, they cannot be used directly. To alleviate the associated computational burden, surrogate models have become a staple tool in all types of uncertainty quantification applications. A surrogate model $\widehat{\cm}$ is a computationally inexpensive approximation of the true model of the form: $$\label{eq:surrogate} \cm(\ve{X}) = \widehat{\cm}(\ve{X}; \bm{\theta}) + \epsilon,$$ where $\bm{\theta}$ is a set of parameters that characterise the surrogate model and $\epsilon$ refers to an error term. The parameters $\bm{\theta}$ are inferred (typically through some form of optimisation process) from a limited set of runs of the original model ${\cx = \acc{\bfx^{(1)} \enu \bfx^{(N)}}}$, called the *experimental design*. As an example, $\bfth$ denotes the set of coefficients in the case of a truncated polynomial chaos expansion, or the set of parameters of both the trend and the covariance kernel in case of Gaussian process modelling. Throughout the rest of the paper, the output of the model $\cm$ is considered scalar, $y = \cm(\bfx) \in \Rr$. Arguably the most well-known accuracy measure for most surrogates is the relative generalisation error $\varepsilon_{gen}$ that reads: $$\label{eq:epsilong_gen_ideal} \varepsilon_{gen} = \Esp{\prt{Y - \widehat{\cm}(\ve{X};\bm{\theta})}^2}/\Var{Y}.$$ This error measure (or, more precisely, one of its estimators) is also the ideal objective function for the optimisation process involved in the calibration of the surrogate parameters $\bm{\theta}$. In practical situations, however, it is not possible to calculate $\varepsilon_{gen}$ analytically. An estimator $\widehat{\varepsilon}_{gen}$ of this error can be computed by comparing the true and surrogate model responses evaluated at a sufficiently large *validation set* $\cx_v = \acc{\bfx^{(1)} \enu \bfx^{(N_v)} }$ of size $N_v$: $$\label{eq:epsilong_gen_estim} \widehat{\varepsilon}_{gen} = \frac{\sum_{i=1}^{N_v} \prt{ \cm(\bfx^{(i)}) - \widehat{\cm}(\bfx^{(i)}) }^2} {\sum_{i=1}^{N_v} \prt{\cm(\bfx^{(i)}) - \widehat{\mu}_y}^2},$$ where $\widehat{\mu}_y=\frac{1}{N} \sum_{i=1}^{N_v} \cm(\bfx^{(i)})$ is the sample mean of the validation set responses and $\widehat{\cm}(\bfx^{(i)})$ is used in place of $\widehat{\cm}(\bfx^{(i)};\bm{\theta})$ to simplify the notation. In data-driven applications, or when the computational model is expensive to evaluate, only a single set $\cs \overset{\text{def}}{=} \acc{\cx, \bfy}$ is available. The entire set is therefore used for calculating the surrogate parameters. Estimating the generalisation error by means of [Eq. (\[eq:epsilong\_gen\_estim\])]{} on the same set, however, corresponds to computing the so-called *empirical error*, which is prone to underestimate drastically the true generalisation error, due to the overfitting phenomenon. In such cases, a fair approximation of $\widehat{\varepsilon}_{gen}$ can be obtained by means of cross-validation (CV) techniques (see @Hastie2001). In $k$-fold CV, $\cs$ is randomly partitioned into $k$ mutually exclusive and collectively exhaustive sets $\cs_i$ of approximately equal size: $$\label{eq:CV_sets} \cs_i \cap \cs_j = \emptyset ~, ~ \forall (i,j) \in \lbrace 1,\ldots,k \rbrace ^2 ~ \text{ and } \bigcup_{i=1}^{k} \cs_i = \cs .$$ The $k$-fold cross-validation error $\varepsilon_{CV}$ reads: $$\label{eq:epsilon_cv} \varepsilon_{CV} = \frac{\sum_{i=1}^{k} \sum_{\bfx \in \cs_i}\prt{ \cm(\bfx) - \widehat{\cm}^{\cs\setminus\cs_i}( \bfx) }^2} {\sum_{\bfx \in \cs} \prt{\cm(\bfx) - \widehat{\mu}_y}^2},$$ where $\widehat{\cm}^{S}_{\cs \setminus \cs_i}$ denotes the surrogate model that is calculated using $\cs$ excluding $\cs_i$. The bias of the generalisation error estimator is expected to be minimal in the extreme case of *leave-one-out (LOO) cross-validation* [@Arlot2010], which corresponds to $N-$fold cross validation. The LOO error $\varepsilon_{LOO}$ is calculated as in [Eq. (\[eq:epsilon\_cv\])]{} after substituting the set $S_i$ by the singleton $\acc{\bfx^{(i)}}$ ( $k=N$): $$\label{eq:epsilon_LOO} \varepsilon_{LOO} = \frac{\sum_{i=1}^{N} \prt{ \cm(\bfx^{(i)}) - \widehat{\cm}^{\backslash i}( \bfx^{(i)}) }^2} {\sum_{i=1}^{N} \prt{\cm(\bfx^{(i)}) - \widehat{\mu}_y}^2},$$ where the term $\cm^{\backslash i}( \bfx^{(i)})$, denotes the surrogate built from the set $S \backslash \acc{\bfx^{(i)}}$, evaluated at $\bfx^{(i)}$. The calculation of $\varepsilon_{LOO}$ can be computationally expensive, because it requires the evaluation of $N$ surrogates, but it does not require any additional run of the full computational model. For Gaussian process modelling and polynomial chaos expansions, computational shortcuts are available to alleviate such costs (*e.g.* @Dubrule1983 [@BlatmanJCP2011]), in the sense that $\varepsilon_{LOO}$ in [Eq. (\[eq:epsilon\_LOO\])]{} is evaluated from a single surrogate model $\widehat{\cm}$ calculated from the full data set $\cs$. As a final step in the surrogate modelling procedure, the set of parameters $\bm{\theta}$ of the surrogate model are optimised *w.r.t.* to one of the generalisation error measures in [Eq. (\[eq:epsilon\_cv\])]{} or [Eq. (\[eq:epsilon\_LOO\])]{} directly, based on the available samples in $\cs$, : $$\label{eq:theta_optim_general} \widehat{\bfth{}} = \arg \underset{\cd_{\bfth}}{\min}\, \widehat{\epsilon}_{gen}(\bfth; \cs),$$ where $\widehat{\bfth{}} $ denotes the optimal set of parameters, $\cd_{\bfth}$ the feasible domain of parameters and $\widehat{\epsilon}_{gen}$ refers to the chosen estimator of $\epsilon_{gen}$. An important aspect of this optimisation step for many types of recent surrogates is that the number of parameters $\bm{\theta}$ scales with the number of input variables. Therefore, surrogates tend to suffer from the curse of dimensionality in two distinct ways: higher dimensional optimisation and underdetermination. Higher dimensional optimisation is linked to a complex objective-function topology, and is therefore prone to convergence to low-performing local minima. In general it requires global optimisation algorithms, such as genetic algorithms, covariance matrix adaptation, or differential evolution [@Goldberg1989; @Hansen2003; @Yang2007]. Underdetermination leads the solutions to the minimisation problem to be non-unique due to the lack of constraining data. In other words, surrogate models with more parameters require in general a larger experimental design or sparse minimisation techniques to avoid overfitting. The proposed DRSM approach {#sec:Meth:DRSM} ========================== Introduction {#sec:Meth:DRSM:intro} ------------ Consider now the experimental design $\cs = \acc{\cx, \bfy}$ introduced above, and assume that it is the only available information about the problem under investigation. Moreover, the dimensionality of the input space is high, $\bfx^{(i)} \in \Rr^M\, ,\, i=1\enu N$ where $M$ is large, say $\co\prt{10^{2-4}}$. The goal is to calculate a surrogate model that serves as an approximation of the real model solely based on the available samples. This is a key ingredient for subsequent analyses in the context of uncertainty quantification. To distinguish between various computational schemes, we denote from now on by $\widehat{\cm}|\cx,\bfy$ a surrogate model whose parameters $\bfth$ are calculated from the experimental design $\cx$ and associated model response $\bfy$. Due to the high input dimensionality, a surrogate $\widehat{\cm}|\cx,\bfy$ may lead to poor generalisation performance or it may not even be computationally tractable. To reduce the dimensionality, the class of DR methods was introduced in [Section \[sec:Meth:DR\]]{}. A DR transformation, expressed by $\cz = g(\cx; \bfw)$, can provide a compressed experimental design, $\bfz^{(i)} \in \Rr^m\, ,\, i=1\enu N$ with $m\ll M$. The surrogate $\widehat{\cm}|\cz,\bfy$ becomes tractable if $m$ is sufficiently small. The potential of $\widehat{\cm}|\cz,\bfy$ to achieve satisfactory generalisation performance depends on (i) the learning capacity of the surrogate itself and (ii) the assumption that the input-output map $\bfx \mapsto y$ can be sufficiently well approximated by a smaller set of features via the transformation $g(\cdot)$. This discussion focuses on the latter and assumes that the learning capacity of the surrogate is adequate. In case of unstructured inputs, the importance of each input variable may vary depending on the output of interest. In case of structured inputs, there is typically high correlation between the input components. Hence, in both families of problems a low-dimensional representation may often approximate well the input-output map. Traditional DR approaches are focused on the discovery of the input manifold and not the input-output manifold. Performing an input compression without taking into account the associated output values may lead to a highly complex input-output map that is difficult to surrogate. In the DRSM (dimensionality reduction for surrogate modelling) approach proposed in this paper, we capitalise on this claim to try and find an optimal input compression scheme w.r.t. the generalisation performance of $\widehat{\cm}|\cz,\bfy$. A nested optimisation problem {#sec:Meth:DRSM:1_nested_optim} ----------------------------- The goal of DRSM is to optimise the parameters $\bfw$ of the compression scheme so that the auxiliary variables $\bfz = g(\bfx;\bfw)$ are suitable to achieve an overall accurate surrogate. The general formulation of this problem reads: $$\label{eq:DRSM_general} \acc{\bfwh,\bfthh} = \underset{\bfw \in \cd_{\bfw},\,\bfth \in \cd_{\bfth} }{\arg \min} \ell \prt{\cm(\cdot),\widehat{\cm} \prt{g(\cdot;\bfw), \bfth} },$$ where $\ell$ denotes the objective function (a.k.a. loss function) that quantifies the generalisation performance of the surrogate. In practice, if a validation set is available, $\ell$ corresponds to a generalisation error estimator like the one in [Eq. (\[eq:epsilong\_gen\_estim\])]{}. In the absence of a validation set, then either the LOO estimator in [Eq. (\[eq:epsilon\_LOO\])]{} or its $k$-fold CV counterpart in [Eq. (\[eq:epsilon\_cv\])]{} are used instead. In the following, it is assumed that a validation set is not available and the generalisation error is estimated by the LOO error, hence $\ell$ is substituted by the $\varepsilon_{LOO}$ expression in [Eq. (\[eq:epsilon\_LOO\])]{}. The proposed approach for solving [Eq. (\[eq:DRSM\_general\])]{}, is related to the concept of *block-coordinate descent* [@Bertsekas1999]. During optimisation, the parameters $\bfw$ and $\bfth$ are updated in an alternating fashion. One of the main reasons for this choice is that the optimisation steps of both DR and SM techniques are often tuned ad-hoc to optimise their performance. Examples include sparse linear regression for polynomial chaos expansions [@BlatmanJCP2011], or quadratic programming for support vector machines for regression [@Vapnik1995]. A single joint optimisation, albeit potentially yielding accurate results, would require the definition of complex constraints on the different sets of parameters $\bfw$ and $\bm{\theta}$. Therefore, the problem in [Eq. (\[eq:DRSM\_general\])]{} is expressed as a nested-optimisation problem. The outer loop optimisation reads: $$\label{eq:DRSM_outer_loop} \bfwh = \underset{\bfw \in \cd_{\bfw}}{\arg \min} ~ \varepsilon_{LOO}(\bfw; \bfthh(\bfw),\cx,\bfy),$$ where $\varepsilon_{LOO}$ denotes the LOO error ([Eq. (\[eq:epsilon\_LOO\])]{}) of the surrogate $\widehat{\cm}(\bfz;\bfw,\cx,\bfy)$ evaluated at $\acc{\cx,\bfy}$ and $\bfthh(\bfw)$ denotes the optimal parameters of $\widehat{\cm}$ for that particular $\bfw$ value. The term $\bfthh(\bfw)$ is calculated by solving the inner loop optimisation problem: $$\label{eq:DRSM_inner_loop} \bfthh = \underset{\bfth \in \cd_{\bfth} }{\arg \min} ~ \varepsilon_{LOO}(\bfth;\bfw, \cx,\bfy).$$ The nested optimisation approach to DRSM comes with costs and benefits. On the one hand, each objective function evaluation of the outer-loop optimisation becomes increasingly costly w.r.t. the number of samples in the experimental design and the complexity of the surrogate model. On the other hand, the search space in each optimisation step can be significantly smaller, compared to the joint approach, due to the reduced number of optimisation variables. Moreover, this nested optimisation approach enables DRSM to be entirely non-intrusive. Off-the-shelf well-known surrogate modelling methods can be used to solve [Eq. (\[eq:DRSM\_inner\_loop\])]{}. Proxy surrogate models for the inner optimisation {#sec:Meth:DRSM:proxy} ------------------------------------------------- Albeit non-intrusive and having a relatively low dimension, the inner optimisation in [Eq. (\[eq:DRSM\_inner\_loop\])]{} is in general the driving cost of DRSM. Indeed, calculating the parameters of a single high-resolution modern surrogate may require anywhere between a few seconds and several minutes. To reduce the related computational cost, it is often possible to solve *proxy surrogate* problems, *i.e.* using simplified surrogates that, while not being as accurate as their full counterparts, are easier to parametrise. A simple example would be to prematurely stop the optimisation in the inner loop in [Eq. (\[eq:DRSM\_inner\_loop\])]{}, or to use isotropic kernels for kernel-based surrogates such as Kriging or support vector machines instead of their more accurate, but costly to train, anisotropic counterparts. Once the outer loop optimisation completes on the proxy surrogate, thus identifying the quasi-optimal DR parameters $\bfwh$, a single high-accuracy surrogate is then computed on the compressed experimental design $\acc{\cz= g(\cx;\bfwh), \bfy}$. Further discussion on this topic can be found in Sections \[sec:Meth:SM:Kriging\] and \[sec:Meth:SM:PCE\]. Selected compression and surrogate modelling techniques used in this paper {#sec:Meth:Selected DR and SM} ========================================================================== Due to the non-intrusiveness in the design of the DRSM method proposed in [Section \[sec:Meth:DRSM\]]{}, no specific dimensionality reduction or surrogate modelling technique has been introduced yet. In the following section, two well-known dimensionality reduction (namely principal component analysis and kernel-principal component analysis) and two surrogate modelling techniques (Kriging and polynomial chaos expansions) are introduced to showcase the DRSM methodology on several example applications in [Section \[sec:Applications\]]{}. Only the main concept and notation is reminded so that the paper is self-consistent. Principal component analysis {#sec:Meth:DR:PCA} ---------------------------- Principal Component Analysis (PCA) is a dimensionality reduction technique that aims at calculating a linear basis of $\ve{X}$ with reduced dimensionality that preserves the sample variance [@Pearson01]. Given a sample of the input random vector $\cx = \acc{\bfx^{(1)},\dots,\bfx^{(N)}}$, the PCA algorithm is based on the eigen-decomposition of the sample covariance matrix $\ve{C}$: $$\label{eq:PCA_Covariance} \ve{C} = \frac{1}{N} \bar{\cx}^\top \bar{\cx},$$ of the form: $$\label{eq:PCA_eigendecomp} \ve{C} \ve{v}^{(i)} = \lambda^{(i)} \ve{v}^{(i)} \, , \, i=1 \enu M$$ where $\bar{\cx}$ denotes the centred (zero mean) experimental design, $\lambda^{(i)}$ denotes each eigenvalue of $\ve{C}$ and $\ve{v}^{(i)}$ the corresponding eigenvector. The dimensionality reduction transformation reads: $$\label{eq:PCA_fwtrans} \cz = \bar{\cx} \, \mat{V}$$ where $\mat{V}$ is the $M\times m$ collection of the $m$ eigenvectors of $\mat{C}$ with maximal eigenvalues. Those eigenvectors are called the *principal components* because they correspond to the reduced basis of $\cx$ with maximal variance. Based on the general DR perspective that was presented in [Section \[sec:Meth:DR\]]{}, PCA is a linear transformation of the form $\cz = g(\cx;w)$, where the only parameter to be selected is the dimension $m$ of the reduced space, $w=m$. Kernel principal component analysis {#sec:Meth:DR:KPCA} ----------------------------------- Kernel PCA (KPCA) is the reformulation of PCA in a high-dimensional space that is constructed using a kernel function [@Scholkopf1998_KPCA]. A kernel function applied on two elements $\bfx^{(i)}, \bfx^{(j)} \in \cd_{\bfx}$ has the following form: $$\label{eq:kappa_definition} \kappa\prt{\bfx^{(i)}, \bfx^{(j)}} = \Phi\prt{\bfx^{(i)}} \cdot \Phi\prt{\bfx^{(j)}}$$ where $\Phi(\cdot)$ is a function that performs the mapping $ \Phi: \cd_{\bfx} \rightarrow \ch $ and $\ch$ is known as the feature space. Based on [Eq. (\[eq:kappa\_definition\])]{}, the so-called *kernel trick* is applied, which refers to the observation that, if the access to $\ch$ only takes place through inner products, then there is no need to explicitly define $\Phi(\cdot)$. The result of the inner product can be directly calculated using $\kappa(\cdot,\cdot)$. Kernel PCA is a non-linear extension of PCA where the kernel trick is used to perform PCA in $\ch$. The principal components in $\ch$ are obtained from the eigen-decomposition of the sample covariance matrix $\bfC_\ch$, analogously to the PCA case in [Eq. (\[eq:PCA\_Covariance\])]{}. However, in KPCA the eigen-decomposition problem: $$\label{eq:KPCA_eigenprob} \bfC_\ch \ve{v}^{(i)} = \lambda_i \ve{v}^{(i)} \, , \, i = 1 \enu N$$ is intractable, since $\bfC_\ch$ cannot in general be computed ($\ch$ might even be infinitely dimensional). This problem is by-passed by observing that each eigenvector belongs to the span of the samples $\Phi\prt{\bfx^{(1)}}, \ldots, \Phi\prt{\bfx^{(N)}}$, therefore scalar coefficients $\alpha_k^{(i)}$ exist, such that each eigenvector $\ve{v}^{(i)}$ can be expressed as the following linear combination [@Scholkopf1998_KPCA]: $$\label{eq:KPCA_v} \ve{v}^{(i)} = \sum_{k=1}^N \alpha_k^{(i)} \Phi\prt{\bfx^{(k)}} \, , \, i = 1 \enu N .$$ Based on [Eq. (\[eq:KPCA\_v\])]{} it can be shown that the eigen-decomposition problem in [Eq. (\[eq:KPCA\_eigenprob\])]{} can be cast as: $$\label{eq:KPCA_alphas} \bfK \bfal^{(i)} = \lambda^{(i)} \bfal^{(i)} \, , \, i = 1 \enu N$$ where $\bfK$ is the kernel matrix with elements: $$K_{ij} = \kappa\prt{\bfx^{(i)}, \bfx^{(j)}}.$$ As for the case of PCA, $\cz$ is calculated by projecting $\cx$ on the $m$ principal axes $\acc{\ve{v}^{(i)}\, , \, \allowbreak i = 1 \enu m}$ corresponding to the $m$ largest eigenvalues. @Scholkopf1998_KPCA showed that $\cz$ can be directly computed based only on the values of the eigenvector expansion coefficients $\alpha_k^{(i)}$ and the kernel matrix $\mat{K}$. The $k$-th component of the $i$-th sample of $\cz$, denoted by $z_k^{(i)} $ is given by; $$\label{eq:KPCA_FWtrans} z_k^{(i)} = \Phi\prt{\bfx^{(i)}}^\mathsf{T} \ve{v}^{(k)} = \sum_{j=1}^N \alpha_k^{(j)} \kappa\prt{\bfx^{(i)}, \bfx^{(j)}}$$ The key ingredient of KPCA is arguably the kernel function $\kappa$. In this paper two kernels are considered, namely the *polynomial* kernel: $$\label{eq:KPCA_kernel_poly} \kappa(\bfx, \bfx'; \bfw) = \prt{ w_1 \bfx^\mathsf{T}\bfx' + w_2}^{w_3}\, , \, w_1>0, w_2 \geq 0, w_3 \in \Nn,$$ and the *Gaussian* kernel: $$\label{eq:KPCA_kernel_gauss_aniso} \kappa(\bfx, \bfx'; \bfw) = \exp \prt{ - \frac{1}{2}\sum_{k=1}^{M}\frac{1}{w_k^2} \prt{x_k- x_k' }^2} \, , \, w_k > 0 \, , \, k = 1 \enu M.$$ A special case of the Gaussian kernel is the *isotropic* Gaussian kernel (also known as *radial basis function*) that simply assumes the same parameter value $w_k$ for all components of $\bfx$. Note that KPCA using a polynomial kernel with parameters $w_1=1$, $w_2=0$ and $w_3=1$ is identical to PCA, since $\Phi(\bfx) = \bfx$. A discussion on the equivalence between PCA and KPCA with linear kernel ($w_3=1$) for arbitrary values of $w_1,w_2$ can be found in Appendix \[sec:Appendix:PCA\_vs\_linKPCA\]. From [Eq. (\[eq:KPCA\_FWtrans\])]{} it follows that $\cz$ can be expressed as $\cz = g(\cx; \bfw)$ where $\bfw$ encompasses both the kernel parameters and the reduced space dimension $m$. In the context of unsupervised learning, two methods to infer the values of $\bfw$ from $\cx$ are considered. The *distance preservation* method aims at optimising $\bfw$ in such a way that the Euclidean distances between the samples are preserved between the original and the feature space [@Weinberger2004]. This is expressed by the following objective function: $$\label{eq:KPCA_Jdist} J_{dist} (\bfw ; \cx) = \sum_{i,j=1}^{N} \prt{d_{ij} - \delta_{ij}}^2$$ where $$\label{eq:KPCA_Jdist_d} d_{ij} = {\left\lVert\bfx^{(i)} - \bfx^{(j)}\right\rVert}$$ and $$\label{eq:KPCA_Jdist_delta_1} \delta_{ij} = {\left\lVert\Phi(\bfx^{(i)},\bfw) - \Phi(\bfx^{(j)},\bfw) \right\rVert}.$$ By expanding the norm expression in [Eq. (\[eq:KPCA\_Jdist\_delta\_1\])]{} it is straightforward to show that: $$\label{eq:KPCA_Jdist_delta_2} \delta_{ij} = \sqrt{K_{ii} + K_{jj} - 2 K_{ij}},$$ hence the value of $\delta_{ij}$ is readily available from the kernel matrix $\mat{K}$. The *reconstruction error*-based method aims at optimising $\bfw$ in such a way that the so-called pre-image, $\tilde{\bfx} = g^{-1}(\bfz,\bfw')$, of $\bfz = g(\bfx, \bfw)$ approximates $\bfx$ as close as possible [@Alam2014]. This is expressed by the following objective function: $$\label{eq:KPCA_Jrecon} J_{recon} (\bfw ; \cx) = \frac{1}{N} \sum_{i=1}^{N} {\left\lVert\bfx^{(i)} - \tilde{\bfx}^{(i)}\right\rVert}^2$$ In contrast to PCA, calculating $\tilde{\bfx}$ is non-trivial, an issue that is known as the *pre-image problem* (see @Kwok2003). The approach for dealing with this problem is the one adopted by the popular <span style="font-variant:small-caps;">python</span> package <span style="font-variant:small-caps;">scikit-learn</span> [@Pedregosa2011], which is based on @Bakir2004. After performing the KPCA transform $\cx \mapsto \cz$, the (non-unique) pre-image of a new point $\bfz$ is computed by kernel-ridge regression using a new kernel function $\kappa_{pre}$: $$\label{eq:KPCA_preimage_main} \tilde{\bfx} = \bm{\beta}^\mathsf{T} \bm{l}(\bfz) ,$$ where: $$\label{eq:KPCA_preimage_l} \bs{\ell}(\bfz) =\acc{\kappa_{pre}(\bfz, \, \bfz^{(j)}), \; j=1 \enu N},$$ and $\bm{\beta}$ are the kernel-ridge regression coefficients. They are calculated as follows: $$\label{eq:KPCA_preimage_beta} \bm{\beta} = \prt{\bm{L} + r \bm{I}_N }^{-1} \cx \qquad L_{ij} = \acc{\kappa_{pre}\prt{\bfz^{(i)},\bfz^{(j)}},\; i,j=1\enu N}$$ where $r$ is a regularisation parameter and $\bm{I}_N$ is the $N$-dimensional identity matrix. In @Pedregosa2011 and in this paper, we use for simplicity the same kernel for the pre-image problem as for KPCA, $\kappa_{pre}\prt{\cdot, \cdot}$ is chosen equal to $\kappa \prt{\cdot, \cdot}$. Note that, in the unsupervised learning literature, the reduced space dimension, $m$, is typically not part of $\bfw$, only the kernel parameters are considered when minimising the objective function in [Eq. (\[eq:KPCA\_Jdist\])]{} or [Eq. (\[eq:KPCA\_Jrecon\])]{}. Kriging {#sec:Meth:SM:Kriging} ------- Kriging, a.k.a. Gaussian process modelling, is a surrogate modelling technique which assumes that the true model response is a realisation of a Gaussian process described by the following equation [@santner_design_2003]: $$\label{eq:KrigingGeneral} \widehat{\cm}(\bfx) = \ve{\beta}^\top \ve{f}(\ve{x}) + \sigma^2 Z(\bfx)$$ where $\ve{\beta}^\top \ve{f}(\ve{x})$ is the mean value of the Gaussian process, also called *trend*, $\sigma^2$ is the Gaussian process variance and $Z(\bfx)$ is a zero-mean, unit-variance Gaussian process. This process is fully characterised by the auto-correlation function between two sample points $R(\bfx,\bfx';\bfth)$. The hyperparameters $\bfth$ associated with the correlation function $R(\cdot;\bfth)$ are typically unknown and need to be estimated from the available observations. Various correlation functions can be found in the literature [@Rasmussen2006; @santner_design_2003], including the *linear*, *exponential*, *Gaussian* (a.k.a. *squared exponential*) and *Matérn* functions. In this paper the separable Matérn correlation family is chosen: $$\label{eq:Kriging_Matern_general} R\prt{\abs{\bfx - \bfx'}; \bm{l}, \nu} = \prod_{i=1}^{M} \frac{1}{2^{\nu -1} \Gamma(\nu)} \prt{\sqrt{2\nu} \frac{\abs{x_i - x'_i}}{l_i} }^\nu \kappa_\nu \prt{\sqrt{2\nu} \frac{\abs{x_i - x'_i}}{l_i}},$$ where $\bfx$, $\bfx'$ are two samples in the input space $\cd_x$, $\bm{l} = \acc{l_i>0,\, i=1 \enu M}$ are the scale parameters (also called *correlation lengths*), $\nu \geq 1/2$ is the shape parameter, $\Gamma(\cdot)$ is the Euler Gamma function and $\kappa_\nu(\cdot)$ is the modified Bessel function of the second kind (a.k.a. Bessel function of the third kind). The values $\nu=3/2$ and $\nu=5/2$ of the shape parameter are commonly used in the literature. The *isotropic* variant of the Matérn correlation family assumes a fixed correlation length value $l$ in [Eq. (\[eq:Kriging\_Matern\_general\])]{} over all $M$ input variables. Regarding the trend part $\ve{\beta}^\top \ve{f}(\ve{x})$ in [Eq. (\[eq:KrigingGeneral\])]{}, the general formulation of *universal Kriging* is adopted, which assumes that the trend is composed of a linear combination of $P$ pre-selected functions $\acc{f_i(\bfx),\, i=1 \enu P}$, : $$\label{eq:Kriging_trend} \ve{\beta}^\top \ve{f}(\ve{x}) = \sum_{i=1}^P \beta_i f_i(\bfx),$$ where $\beta_i$ is the trend coefficient of each function. The Gaussian assumption states that the vector formed by the true model responses, ${\ensuremath{\ve{y}}}$ and the prediction, $\widehat{Y}(\bfx)$, at a new point $\bfx$, has a joint Gaussian distribution defined by: $$\bra{ \begin{matrix} \widehat{Y}(\bfx) \\ {\ensuremath{\ve{y}}}\end{matrix} } \sim \mathcal{N}_{N+1} \prt{ \bra{ \begin{matrix} \bm{f}^\top(\bfx) \bm{\beta} \\ \mat{F} \bm{\beta} \end{matrix} } , \sigma^2 \bra{ \begin{matrix} 1 & \bm{r}^\top(\bfx) \\ \bm{r}(\bfx) & \mat{R} \end{matrix} } }$$ where $\bm{F}$ is the information matrix of generic terms: $$F_{ij} = f_j(\ve{x}^{(i)})~,~i=1 \enu N,~j=1 \enu P,$$ $\bm{r}(\bfx)$ is the vector of cross-correlations between the prediction point $\bfx$ and each one of the observations whose terms read: $$r_{i}(\bfx) = R(\bfx,\bfx^{(i)};\bm{\theta}), ~i=1 \enu N. \label{eq:r0}$$ $\bm{R}$ is the correlation matrix given by: $$R_{ij} = R(\bfx^{(i)},\bfx^{(j)};\bm{\theta}), ~i,j=1 \enu N.$$ The mean and variance of the Gaussian random variate $\widehat{Y}(\bfx)$ (a.k.a. mean and variance of the Kriging predictor) can be calculated based on the best linear unbiased predictor (BLUP) from @santner_design_2003: $$\label{eq:TheoryPredicorMean} \mu_{\widehat{Y}}(\ve{x}) =\ve{f}(\ve{x})^\top \ve{\beta} + \ve{r}(\ve{x})^\top \mat{R}^{-1}\left ({\ensuremath{\ve{y}}}-\mat{F}\ve{\beta} \right )\, ,$$ $$\label{eq:TheoryPredicorVariance} \sigma_{\widehat{Y}}^2(\ve{x}) = \sigma^2 \left( 1-\ve{r}^\top(\ve{x})\mat{R}^{-1}\ve{r}(\ve{x}) + \ve{u}^\top(\ve{x}) (\mat{F}^\top\mat{R}^{-1}\mat{F})^{-1}\ve{u}(\ve{x}) \right)$$ where: $$\label{eq:AKG:TheoryCalcBeta} \ve{\beta} = \left( \mat{F}^\top \mat{R}^{-1} \mat{F} \right)^{-1}\mat{F}^\top\mat{R}^{-1} {\ensuremath{\ve{y}}}$$ is the generalised least-squares estimate of the underlying regression problem and $$\label{eq:AKG:TheoryPredicorU} \ve{u}(\ve{x}) = \mat{F}^\top \mat{R}^{-1}\ve{r}(\ve{x}) - \ve{f}(\ve{x}).$$ The mean response in [Eq. (\[eq:TheoryPredicorMean\])]{} is considered as the output of a Kriging surrogate, $\widehat{\cm}(\bfx) = \mu_{\widehat{Y}}(\bfx)$. It is important to note that the Kriging model interpolates the data, : $$\label{eq:Kriging_interpolates} \mu_{\widehat{Y}}(\bfx) = \cm(\bfx), \quad \sigma_{\widehat{Y}}^2(\bfx) = 0, \quad \forall\, \bfx \in \cx$$ The equations that were derived for the best linear unbiased Kriging predictor assumed that the covariance function $\sigma^2 R(\cdot;\bfth)$ is known. In practice however, the family and other properties of the correlation function need to be selected *a priori*. The hyperparameters $\bm{\theta}$, the regression coefficients $\bm{\beta}$ and the variance $\sigma^2$ need to be estimated based on the available experimental design. The optimal estimates of the correlation parameters $\widehat{\bfth}$ are determined by minimising the generalisation error of the Kriging surrogate, based on the leave-one-out cross-validation error [@santner_design_2003; @Bachoc2013b]: $$\label{eq:Kriging_thetaCV} \ve{\theta}_{CV} = \underset{\cd_{\ve{\theta}}}{\arg \min} \sum_{i=1}^{K} \left( \cm(\ve{x}^{(i)}) - \mu_{\widehat{Y}, (-i)}(\ve{x}^{(i)}) \right) ^2 ,$$ where $\mu_{\widehat{Y}, (-i)}(\ve{x}^{(i)})$ corresponds to the mean value of a Kriging predictor that was built from the samples $\cx \,\backslash \acc{\bfx^{(i)}, y^{(i)}}$, evaluated at $\ve{x}^{(i)}$. The computational cost for calculating the terms $\mu_{\widehat{Y}, (-i)}(\ve{x}^{(i)})$ can be significantly reduced as shown in @Dubrule1983. First, the following matrix inversion is performed: $$\label{eq:Dubrule_B} \mat{B} = \bra{ \begin{matrix} \sigma^2 \mat{R} & \mat{F} \\ \mat{F}^\mathsf{T} & \mat{0} \end{matrix}}^{-1}.$$ Then $\mu_{\widehat{Y},(-i)}$ is calculated as follows: $$\label{eq:Dubrule_mean} \mu_{\widehat{Y},(-i)} = - \sum_{j=1,j\neq i}^N \frac{\mat{B}_{ij}}{\mat{B}_{ii}} \, y^{(j)}.$$ In this work we use cross-validation for estimating the correlation parameters instead of the maximum likelihood method [@santner_design_2003]). This is motivated by the comparative study in @Bachoc2013b between maximum likelihood (ML) and CV estimation methods. The CV method is expected to perform better in cases that the correlation family of the Kriging surrogate is not identical to the one of the true model. This is typically the case in practice and in the application examples in [Section \[sec:Applications\]]{}. Determining the optimal parameters $\ve{\theta}_{CV} $ in [Eq. (\[eq:Kriging\_thetaCV\])]{} leads to a complex multi-dimensional optimisation problem. Common optimisation algorithms employed to solve [Eq. (\[eq:Kriging\_thetaCV\])]{} can be cast into two categories: local and global. Local methods are usually gradient-based, such as the BFGS algorithm [@bazaraa2013bfgs], and search locally in the vicinity of the starting point. This makes them prone to get stuck at local minima, although they can be computationally efficient due to the use of gradients. Global methods such as genetic algorithms [@Goldberg1989] do not rely on local information such as the gradient. They seek the global minimum by various adaptive resampling strategies within a bounded domain. This often leads to considerably more objective function evaluations compared to local methods. As mentioned in [Section \[sec:Meth:DRSM:proxy\]]{}, to alleviate the computational costs in the inner loop optimisation in [Eq. (\[eq:DRSM\_inner\_loop\])]{}, an inexpensive-to-calibrate Kriging surrogate is built. To this end, the isotropic version of the Matérn correlation family is used, combined with low computational budget optimisation of the correlation parameters. For calculating the final, high-accuracy, Kriging surrogate an optimisation with high-computational budget is performed instead, combined with the use of an anisotropic correlation family. The introduction of anisotropy is expected to improve the generalisation performance the metamodel, as shown for instance in the study by @MoustaphaJRUES2018. ### Polynomial chaos expansions {#sec:Meth:SM:PCE} Polynomial chaos expansions represent a different class of surrogate models that has seen widespread use in the context of uncertainty quantification due to their flexibility and efficiency. Consider that $\ve{X} \in \Rr^M$ is a random vector with independent components described by the joint PDF $f_{\ve{X}}$ and that the model output ${Y}$ in [Eq. (\[eq:true\_model\])]{} has finite variance. Then the polynomial chaos expansion of $\cm(\ve{X})$ is given by: $$\label{eqn:PCE:PCE} Y = \cm(\Ve{X}) = \sum\limits_{\ua\in\mathbb{N}^M} \theta_{\ua} {\Psi}_{\ua}(\Ve{X})$$ where the $\Psi_{\ua} (\Ve{X})$ are multivariate polynomials orthonormal with respect to $f_{\Ve{X}}$, $\ua \in \mathbb{N}^M$ is a multi-index that identifies the components of the multivariate polynomials ${\Psi}_{\ua}$ and the $\theta_{\ua} \in \mathbb{R}$ are the corresponding coefficients. In practice, the series in [Eq. (\[eqn:PCE:PCE\])]{} is truncated to a finite sum, by introducing the truncated polynomial chaos expansion: $$\label{eq:PCE_truncatedPCE} \cm(\Ve{X}) \approx \widehat{\cm}(\Ve{X}) = \sum\limits_{\ua\in\ca} \theta_{\ua} {\Psi}_{\ua}(\Ve{X}) \equiv \bfth{}^\top \ve{\Psi}(\bfx)$$ where $\ca \subset \mathbb{N}^M$ is the set of selected multi-indices of multivariate polynomials. A typical truncation scheme consists in selecting multivariate polynomials up to a total degree $p$, $\ca = \acc{\ua \in \Nn^M \, : \, {\left\lVert\ua\right\rVert}_1 \leq p }$, with ${\left\lVert\ua\right\rVert}_1 = \sum_{i=1}^{M}\alpha_i$. The corresponding number of terms in the truncated series rapidly increases with $M$, giving rise to the “curse of dimensionality”. Other truncation strategies effective in higher dimension are discussed, , in @BlatmanPEM2010 [@Jakeman2015]. The polynomial basis $\Psi_{\ua}(\Ve{X})$ in [Eq. (\[eq:PCE\_truncatedPCE\])]{} is traditionally built starting from a set of *univariate orthonormal polynomials* $\phi^{(i)}_k(x_i)$ which satisfy: $$\label{eqn:PCE:Theory:UnivariateOrthonormalPoly} \left< \phi^{(i)}_j(x_i),\phi^{(i)}_k(x_i) \right> \eqdef \int_{\cd_{X_i}} \phi^{(i)}_j(x_i)\phi^{(i)}_k(x_i) f_{X_i}(x_i)\di x_i = \delta_{jk}$$ where $i$ identifies the input variable w.r.t. which they are orthogonal, as well as the corresponding polynomial family, $j$ and $k$ the corresponding polynomial degree, $f_{X_i}(x_i)$ is the $i^{th}$-input marginal distribution and $\delta_{jk}$ is the Kronecker symbol. Note that this definition of inner product can be interpreted as the expectation value of the product of the multiplicands. The multivariate polynomials $\Psi_{\ua}(\Ve{X})$ are then assembled as the tensor product of their univariate counterparts: $$\label{eqn:PCE: multivariate polynomials} \Psi_{\ua}(\ve{x}) \eqdef \prod_{i=1}^M \phi^{(i)}_{\alpha_i} (x_i)$$ For standard distributions, such as uniform, Gaussian, gamma, beta, the associated families of orthogonal polynomials are well-known [@Xiu2002]. Orthogonal polynomials can be constructed numerically w.r.t. any distribution (including non-parametric ones like those obtained by kernel density smoothing) by means of Gram-Schmidt orthonormalisation (a.k.a. Stieltjes procedure for polynomials [@Gautschi2004]). The expansion coefficients $\bfth = \acc{\theta_{\bm{\alpha}},\, \bm{\alpha} \in \ca \subset \Nn^M } $ in [Eq. (\[eq:PCE\_truncatedPCE\])]{} are calculated by minimising the expectation of least-squares residual [@Berveiller2006]: $$\label{eq:PCE_leastSquares_min} \widehat{\bfth} = \arg \min \Esp{ \left(\bfth\tr {\Psi}(\Ve{X}) - \cm(\Ve{X}) \right)^2}.$$ In the context of DRSM, the set of input parameters $\bfw$ for a PCE surrogate consists in $\bfw = \acc{p, \bm{\theta}}$, the maximal degree of the truncated expansion and the associated coefficients. Due to the quadratic programming nature of the minimisation in [Eq. (\[eq:PCE\_leastSquares\_min\])]{} and the linearity of PCE (see [Eq. (\[eq:PCE\_truncatedPCE\])]{}), we adopt the adaptive sparse-linear regression based on least angle regression first introduced by @BlatmanJCP2011. As for the case of Kriging, the LOO error (see [Eq. (\[eq:epsilon\_LOO\])]{}) is analytically available from the expansion coefficients [@BlatmanJCP2011]: $$\label{eq:PCE_LOO} \varepsilon_{LOO} = {\sum\limits_{i = 1}^N \left( \frac{\cm(\Ve{x}^{(i)}) - \widehat{\cm}^{PC}(\Ve{x}^{(i)})}{1-h_i}\right)^2}\bigg/{\sum\limits_{i = 1}^N \left(\cm(\ve{x}^{(i)}) - \widehat{\mu}_Y\right)^2},$$ where $h_i$ is the $i^{th}$ component of the vector given by: $$\label{eq:PCE_LOO_h} \Ve{h} = \text{diag}\left(\mat{A}(\mat{A}\tr\mat{A})^{-1} \mat{A}\tr\right),$$ and $\mat{A}$ is the experimental matrix with entries $A_{ij} = \Psi_j\prt{\bfx^{(i)})}$. To calculate the *proxy PCE* surrogates used during the DRSM optimisation phase (see [Section \[sec:Meth:DRSM:proxy\]]{}), the input variables in $\bfz$, are assumed uniformly distributed and independent. The PCE coefficients are computed by solving [Eq. (\[eq:PCE\_leastSquares\_min\])]{} using the ordinary least squares method [@Berveiller2006]. To calculate the PCE coefficients of the final, high-accuracy, surrogate $\widehat{\cm}(g(\bfx;\widehat{\bfw}))$, the distributions of the input variables are fitted using kernel-smoothing, while retaining the independence assumption, motivated by the results in @TorreJCP2018. In addition, a sparse solution is obtained by solving the optimisation problem in [Eq. (\[eq:PCE\_leastSquares\_min\])]{} using least angle regression [@BlatmanJCP2011] instead of ordinary least squares. Applications {#sec:Applications} ============ The performance of DRSM is evaluated on the following applications: (i) an artificial analytic function with $20$ unstructured inputs and approximately known intrinsic dimension, (ii) a realistic electrical engineering model with $80$ unstructured inputs and unknown intrinsic dimension and, (iii) a heat diffusion model with $16,000$ structured inputs and unknown intrinsic dimension. For each example, DRSM is applied using KPCA for compression together with Kriging or polynomial chaos expansions for surrogate modelling. The surrogate performance is then compared, in terms of generalisation error, to the sequential application of unsupervised dimensionality reduction followed by surrogate modelling. To improve readability, various details regarding the implementation of the optimisation algorithms and the surrogate models calibration are omitted from the main text and given in Appendix \[sec:Appendix:details\] instead. All the surrogate modelling techniques were deployed with the <span style="font-variant:small-caps;">Matlab</span>-based uncertainty quantification software <span style="font-variant:small-caps;">UQLab</span> [@Marelli2014; @UQLabPCE; @UQLabKriging]. Sobol’ function {#sec:App:sobol} --------------- The Sobol’ function (also known as $g$-function) is a commonly used benchmark function in the context of uncertainty quantification. It reads: $$\label{eq:app_Sobol} Y = \prod_{i=1}^{M} \frac{\abs{4X_i - 2} + c_i}{1 + c_i} \,,$$ where $\bfX = \lbrace X_1 \enu X_M \rbrace$ are independent random variables uniformly distributed in the interval $[0,1]$ and $ \bfc = \lbrace c_1 \enu c_M \rbrace^\mathsf{T}$ are non-negative constants. In this application, we chose $M = 20$ and the constants $\bfc$ given by @KonakliRESS2016 [@Kersaudy2015]: $$\label{eq:app_Sobol_constants} \bfc = \lbrace 1, 2, 5, 10, 20, 50, 100, 500, 500 \enu 500\rbrace^\mathsf{T}.$$ It is straightforward to see that the effect of each input variable $X_i$ to the output $Y$ is inversely proportional to the value of $c_i$. In other words, a small (resp. large) value of $c_i$ results in a high (resp. low) contribution of $X_i$ to the value of $Y_i$. For the given values of the constants $\bfc$, one would expect that, roughly, the first $4$ to $6$ variables can provide a compressed representation of $\bfX$ with minimal information loss regarding the input-output relationship. To showcase the performance of DRSM, an experimental design $\cx$, consisting of $800$ samples, is generated by Latin Hypercube sampling of the input distribution [@McKay1979]. Based on the samples in $\cx$ and the corresponding model responses $\bfy$, several combinations of KPCA, Kriging and PCE are tested within the DRSM framework. An additional set of $10^5$ validation samples $\acc{\cx_v, \bfy_v}$ is generated for evaluating the performance of the final surrogates. The first analysis consists in comparing the generalisation performance as a function of the compressed input dimension $m$ for Kriging and PCE models combined with KPCA with different kernels. Because of the availability of a validation set, the performance of the LOO error estimator in [Eq. (\[eq:epsilon\_LOO\])]{} is also assessed by comparing it with the true validation error in [Eq. (\[eq:epsilong\_gen\_ideal\])]{}. Figures \[fig:res\_sobol\_drsm\_m\_vs\_error-kg-loo\] and \[fig:res\_sobol\_drsm\_m\_vs\_error-pce-loo\] show the LOO error estimator of the final surrogate model when using Kriging and PCE, respectively. In each panel the different curves correspond to different KPCA kernels, namely polynomial kernel ([Eq. (\[eq:KPCA\_kernel\_poly\])]{}) and isotropic (resp. anisotropic) Gaussian ([Eq. (\[eq:KPCA\_kernel\_gauss\_aniso\])]{}). Figures \[fig:res\_sobol\_drsm\_m\_vs\_error-kg-rmse\] and \[fig:res\_sobol\_drsm\_m\_vs\_error-pce-rmse\] show the corresponding validation error on the validation set for the same scenarios. At a first glance, it is clear that the top and bottom figures are remarkably similar, both in their trends and in absolute value. Therefore, it is concluded that on this example $\epsilon_{LOO}$ is a good measure of the generalisation error $\epsilon_{gen}$. This is an important observation, because in the general case a validation set is not available, while $\epsilon_{LOO}$ can always be calculated. Moreover, the intrinsic dimension identified by all the best DR-SM combinations is equal to $\widehat{m} = 6$, which is a reasonable estimate based on the values of the constants $c_i$ in [Eq. (\[eq:app\_Sobol\_constants\])]{}. The DRSM algorithm identifies the anisotropic Gaussian kernel as the best KPCA kernel to be used in conjunction with both Kriging and PCE. However, the performance of PCE is significantly better in terms of generalisation error. The optimal parameters for each case (Kriging and PCE) are highlighted by a black dot in , and their numerical values are reported in . SM method KPCA kernel $\widehat{m}$ $\varepsilon_{LOO}$ $\widehat{\varepsilon}_{gen}$ ----------- ---------------------- --------------- --------------------- ------------------------------- Kriging Anisotropic Gaussian $6$ 0.0704 0.0830 PCE Anisotropic Gaussian $6$ 0.0096 0.0083 : Sobol’ function: optimal DRSM configurations for Kriging- and PCE-based surrogate models[]{data-label="tab:res_sobol_drsm_mstar"} Subsequently, the performance of DRSM is compared against an unsupervised approach, in which dimensionality reduction is carried out first, before applying surrogate modelling. To facilitate a meaningful comparison between the various methods, the reduced dimension and the optimal KPCA kernel as determined by the first analysis (see ) is used. The results are summarised in , while the corresponding list of tested configurations for both DRSM and the sequential DR-SM is given in . [.9]{}[l X r]{} **Dim. reduction** & **Parameter tuning objective** & **Abbreviation**\ Kernel PCA & $\varepsilon_{LOO}$ of Kriging (KG) or PCE surrogate ([Eq. (\[eq:DRSM\_outer\_loop\])]{}) & DRSM\ Kernel PCA & Reconstruction error ([Eq. (\[eq:KPCA\_Jrecon\])]{}) & KPCA-RECON\ Kernel PCA & Pairwise distance preservation ([Eq. (\[eq:KPCA\_Jdist\])]{})& KPCA-DIST\ PCA & - & PCA\ The experimental design consists of $800$ samples. The performance of each method is evaluated in terms of the generalisation error of the final surrogate $\widehat{\cm}(\bfz)$ evaluated on a validation set $\acc{\cx_v, \bfy_v=\cm(\cx_v)}$ with $10^5$ samples. To evaluate the robustness of the results, this process is repeated $10$ times, each corresponding to a different set $\cx$, drawn at random using the Latin Hypercube sampling method. On the left (resp. right) panel, a Kriging (resp. PCE) surrogate is calculated using one of the methods in . Each box plot in provides summary statistics of the generalisation error that was achieved by each configuration over the $10$ repetitions. The central mark indicates the median, and the bottom and top edges of the box indicate the $25^{\text{th}}$ and $75^{\text{th}}$ percentiles, respectively. The whiskers extend to the most extreme data points up to $1.5$ times the inter-quartile range above or below the box edges. Any sample beyond that range is considered an outlier and plotted as a single point. The DRSM approach consistently shows superior performance compared to the unsupervised approaches. This performance improvement becomes more apparent in the case of PCE surrogate modelling, where the average validation error over the $10$ repetitions is reduced by almost two orders of magnitude compared to the other methods. Due to the analytical nature of the model under consideration, we further evaluate the DRSM-based input compression by means of how the most important input variables are mapped to the reduced space. We adopt the total Sobol’ sensitivity indices as a rigorous measure of the importance of each input variable. Sobol’ sensitivity analysis is a form of global sensitivity analysis based on decomposing the variance of the model output into contributions that can be directly attributed to inputs or sets of inputs [@Sobol1993]. The total Sobol’ sensitivity index of an input variable $X_i$, denoted by $S_i^{Tot} \in [0,1]$, quantifies the total effect of $X_i$ on the variance of $Y$. In this particular example, the total Sobol’ indices can be analytically derived [@Saltelli2000]. Their values are shown for reference in . It is clear from [Eq. (\[eq:app\_Sobol\])]{} and [Eq. (\[eq:app\_Sobol\_constants\])]{} that all $20$ input variables contribute to the output variability, the intrinsic dimension of the problem is $20$. However, the contribution of each input component quickly diminishes with larger values of $c_i$ (see in which the values of the $20$ total Sobol’ indices are plotted, in logarithmic scale, as horizontal bars). Compressing the inputs in this problem is expected to lead to a mapping where those first few input components have the largest contribution. In the features in the reduced space $\bfZ$ are compared against the original inputs $\bfX$. The rationale behind this heuristic analysis is simple: if the features obtained by DRSM are correctly identified, they should depend mostly on the same variables identified as important in the Sobol’ analysis in . A simple measure of dependence between the reduced space components $\acc{z_i \, , \, i = 1 \enu m}$ and the initial input space components $\acc{x_i \, , \, i = 1 \enu M}$ is provided by the metric $\abs{\rho\prt{z_i,x_i}}$, where $\rho$ denotes the Spearman correlation coefficient. Figures \[fig:res\_sobol\_drsm\_features\_corr-1\] - \[fig:res\_sobol\_drsm\_features\_corr-4\] represent graphically the quantity $\abs{\rho\prt{z_i,x_i}}$ for the best surrogate identified in , namely a PCE coupled with KPCA using an anisotropic Gaussian kernel, evaluated on the validation set $\acc{\cx_v,\bfy_v}$. Each figure corresponds to a different selection of reduced space dimension $m$. clearly shows that (i) each $z_i$ correlates strongly with a specific $x_i$, (ii) the $z_i$’s correlate with the $m$ “most important” $x_i$’s, and, (iii) the larger $m$ value leads to the discovery of a new input $z_i$ that correlates with the next “most important” component of $\bfx$. Electrical resistor network {#sec:App:resistor_networks} --------------------------- ![The resistor networks application example[]{data-label="fig:app_resnets"}](images/app_resnets.pdf){width=".75\textwidth"} The electrical resistor network in [@Jakeman2015] is considered next. It contains $80$ resistances of uncertain ohmage (model inputs) and it is driven by a voltage source providing a known potential $V_0$. The output of interest is the voltage $V$ at the node shown in . A single set of $1,000$ experimental design samples and model responses is available, courtesy of J. Jakeman. As in the previous section, the goal of the first analysis is to determine the generalisation performance of the DRSM surrogate as a function of the reduced space dimension $m$ when KPCA is combined with either Kriging or PCE. In addition, the accuracy of the LOO error in [Eq. (\[eq:epsilon\_LOO\])]{} is compared to the validation error in [Eq. (\[eq:epsilong\_gen\_estim\])]{}. The samples are randomly split into $500$ pairs $\acc{\cx, \bfy}$ used during the DRSM calibration and $500$ pairs $\acc{\cx_v, \bfy_v}$ used for validation. Figures \[fig:res\_resnets\_drsm\_m\_vs\_rmse-kg-loo\] and \[fig:res\_resnets\_drsm\_m\_vs\_rmse-pce-loo\] show the LOO error estimator of the final surrogate model (Kriging or PCE), evaluated on $\acc{\cx,\bfy}$, whereas Figures \[fig:res\_resnets\_drsm\_m\_vs\_rmse-kg-rmse\] and \[fig:res\_resnets\_drsm\_m\_vs\_rmse-pce-rmse\] show the validation error of the surrogate, evaluated on $\acc{\cx_v,\bfy_v}$. In each panel, each curve corresponds to a different KPCA kernel, namely anisotropic or isotropic Gaussian, and polynomial. Finally, the optimal configuration for each SM method is illustrated by a black dot. Similarly to the Sobol’ function, the use of an anisotropic kernel in KPCA results in significantly reduced generalisation error. Indeed this is expected from a physical standpoint. The effect of the resistors on the voltage $V$ will decay with distance (in terms of the number of preceding resistors) from $V$, which implies anisotropy in terms of the effect of each input variable to the output. As in the previous application example, the LOO error in Figures \[fig:res\_resnets\_drsm\_m\_vs\_rmse-kg-loo\] and \[fig:res\_resnets\_drsm\_m\_vs\_rmse-pce-loo\] provides a reliable proxy of the generalisation error in Figures \[fig:res\_resnets\_drsm\_m\_vs\_rmse-kg-rmse\] and \[fig:res\_resnets\_drsm\_m\_vs\_rmse-pce-rmse\] and the same optimal parameters are identified w.r.t. the two error measures. The optimal DRSM configuration for each surrogate model is given in . SM method KPCA kernel $\widehat{m}$ $\varepsilon_{LOO}$ $\widehat{\varepsilon}_{gen}$ ----------- ---------------------- --------------- --------------------- ------------------------------- Kriging Anisotropic Gaussian $24$ 2.000e-04 2.402e-04 PCE Anisotropic Gaussian $32$ 3.621e-05 3.249e-05 : Resistor networks: optimal DRSM configurations for Kriging and PCE surrogate models[]{data-label="tab:res_resnets_drsm_mstar"} Next, the performance of DRSM is compared to unsupervised approaches considering the setups in . The results of this comparative study are given in using box plots. They are obtained by the repeated random selection of $500$ samples from the available $1,000$, leading to $10$ separate surrogate models for each case. The performance of each method is determined by means of the $\widehat{\varepsilon}_{gen}$ of the final surrogate $\widehat{\cm}(\bfz)$ evaluated on the validation set $\acc{\cx_v, \bfy_v=\cm(\cx_v)}$, that corresponds to the remaining $500$ samples of each split. Hence, each box-plot provides summary statistics of the validation error over the different splits. Each of the setups is tested both for Kriging () and PCE surrogates (). In this application example the DRSM-based surrogates outperform the others by several orders of magnitude in both cases (Kriging, PCE). This highlights the difference between the unsupervised and supervised compression: compressing the input using only the information in $\cx$ appears inefficient when followed by surrogate modelling. 2D heat diffusion {#sec:App:2d_diffusion} ----------------- This last application consists in a 2-dimensional stationary heat diffusion problem. The problem is defined in a square domain, $D=[-0.5,0.5]\times[-0.5,0.5]$, where the temperature field $T(\bfv),\, \bfv \in D$ is the solution of the elliptic partial differential equation: $$\label{eq:heat_pde} - \nabla \cdot \prt{d(\bfv) \nabla{} T(\bfv)} = 500 \, I_A(\bfv),$$ with boundary conditions $T=0$ on the top boundary and $\nabla T\cdot \bm{n}=0$ on the left, right and bottom boundaries, where $\bm{n}$ denotes the vector normal to the boundary. In [Eq. (\[eq:heat\_pde\])]{}, $A$ corresponds to a square domain (see ) and $I_A$ is the indicator function equal to 1 if $\bfv \in A$ and $0$ otherwise. The diffusion coefficient $d(\bfv)$ is a lognormal random field defined by: $$\label{eq:heat_diffcoeff} d(\bfv) = \exp \prt{a_d + b_d \,g(\bfv) },$$ where $g(\bfv)$ is a Gaussian random field and the parameters $a_d$, $b_d$ are such that the mean and standard deviation of $d$ are $\mu_d = 1$ and $\sigma_d = 0.3$ respectively. The random field is characterised by a Gaussian correlation function $R(\bfv,\bfv') = \exp \prt{- {\left\lVert\bfv - \bfv'\right\rVert}^2/\ell^2}$, with $\ell=0.2$. The output of interest is the average temperature in the square domain $B$ within $D$ (see ). To solve [Eq. (\[eq:heat\_pde\])]{}, the Gaussian random field $g(\bfv)$ is first discretised using the expansion optimal linear estimation (EOLE) method [@DerKiureghian1993]. Consider a grid in $D$ with nodes $\acc{\bm{v}_1 \enu \bm{v}_n}$. By retaining the first $p$ terms in the EOLE series, $g(\bfv)$ is approximated by: $$\label{eq:heat_EOLE} \widehat{g}(\bfv) = \sum_{i=1}^{p} \frac{\xi_i}{\sqrt{l^{(i)}}}\prt{\bm{\phi}^{(i)}}^\top\mat{C}_{\bfv\bm{v}}(\bfv),$$ where $\acc{\xi_1 \enu \xi_p}$ are independent standard normal random variables, $\mat{C}_{\bfv\bm{v}}$ is a vector with elements $C_{\bfv\bm{v}}^{(k)}=R(\bfv,\bm{v}_k)$ for $k=1 \enu n$ and $\{ \prt{l^{(i)}, \allowbreak \bm{\phi}^{(i)}}, \allowbreak \, i=1 \enu n\}$ are the eigenvalues and eigenvectors of the correlation matrix $\mat{C}_{\bm{v}\bm{v}}$ with elements $C_{\bm{v}\bm{v}}^{(i,j)} = R(\bm{v}_i,\bm{v}_j)$ for $i,j= 1 \enu n$. In the following analysis the Gaussian random field realisations are computed using $p=30$ terms in the EOLE series in [Eq. (\[eq:heat\_EOLE\])]{}, which allows to represent $93.69\%$ of the variance of the original field. The underlying deterministic problem is solved with an in-house finite-element analysis code developed in [<span style="font-variant:small-caps;">Matlab</span>]{}. The mesh shown in consists of $16,000$ triangular T3 elements. shows a realisation of the diffusion coefficient random field which corresponds to the input of the model. The corresponding model output, shown in , is the mean temperature in the highlighted square region $B$. Each realisation of the diffusion coefficient random field is discretised over the mesh in . In the following analysis, the system is treated as a black-box, with the discretised heat diffusion coefficient as a high-dimensional input ($M=16,000$) and the average temperature in square B as the scalar model output. A single set of $500$ experimental design samples and model responses is available. This example mimics a realistic scenario in which various maps of spatially varying parameters measured on a regular grid, are input to a computational model that analyses some performance of the system. ----------- ------------- --------------- ----------------- --------------------- ------------------------------- ---------- ---------- SM method KPCA kernel $\widehat{m}$ $\varepsilon_{LOO}$ $\widehat{\varepsilon}_{gen}$ $\widehat{w}_1$ $\widehat{w}_2$ $\widehat{w}_3$ Kriging Polynomial $20$ $131.3681$ $112.0040$ $1$ $0.0205$ $0.0216$ PCE Polynomial $20$ $17.5225$ $15.1853$ $1$ $0.0340$ $0.0356$ ----------- ------------- --------------- ----------------- --------------------- ------------------------------- ---------- ---------- : 2D diffusion: optimal DRSM configurations for Kriging- and PCE-based surrogate models[]{data-label="tbl:res_heat_drsm_mstar"} As in the previous application examples, the goal of the first analysis is to determine the optimal DRSM configuration in terms of the KPCA kernel and the reduced space dimension, as well as test the effectiveness of the LOO error as a proxy of the validation error. In this analysis, the available samples are randomly split into $300$ pairs to be used during the DRSM optimisation and $200$ pairs to be used for validation. The results are shown in . Figures \[fig:res\_diffusion2d\_m\_vs\_rmse-KG-LOO\] and \[fig:res\_diffusion2d\_m\_vs\_rmse-PCE-LOO\] show the LOO error estimator of the final Kriging (resp. PCE) surrogate, evaluated on $\acc{\cx, \bfy}$, whereas Figures \[fig:res\_diffusion2d\_m\_vs\_rmse-KG-RMSE\] and \[fig:res\_diffusion2d\_m\_vs\_rmse-PCE-RMSE\] show the validation error of the surrogate evaluated on $\acc{\cx_v, \bfy_v}$. Each curve corresponds to a specific type of KPCA kernel, namely isotropic Gaussian and polynomial, and a specific surrogate, namely Kriging and PCE. We omitted the anisotropic Gaussian kernel for KPCA which is intractable due to the large input dimensionality. A similar convergence behaviour is observed between Kriging- and PCE- based DRSM. The corresponding optimal parameter values are highlighted in and their numerical values are reported in . The linear polynomial kernel performs best in both cases and leads to the same reduced space dimension $\widehat{m}=20$. This significantly low dimension can be interpreted by [Eq. (\[eq:heat\_EOLE\])]{}. The heat diffusion coefficient, although $16,000$- dimensional, is a non-linear combination of $p$ independent standard normal random variables. Moreover, the LOO and validation error curves show similar behaviour both in terms of their trend and their absolute value. Hence, the LOO error served as a reliable proxy of the validation error, as was observed in the previous application examples too. In the subsequent analysis we compare the performance of the DRSM approach against other sequential approaches listed in . To test each setup, we repeat the calculation process $10$ times. In each case the $500$ available samples are split randomly into $300$ samples for calculating the surrogate and $200$ samples for validation. The optimal KPCA kernel that was determined by DRSM is used in all methods that involve KPCA. Also, for the sake of comparison, the same reduced space dimension $\widehat{m}=20$ is assumed for all methods. The results of this comparative study are given in using box plots to provide summary statistics of the validation error over the different splits of the samples. In case of Kriging surrogate modelling, DRSM consistently provides superior results compared to the other methods. Notice that KPCA with linear kernel is equivalent to PCA on a scaled version of the experimental design with scaling factor $\sqrt{w_1}$ (see Appendix \[sec:Appendix:PCA\_vs\_linKPCA\] for more details). The Kriging surrogates, in contrast to the PCE ones, are affected by this scaling. This also explains the performance improvement compared to the case of PCA-based DR. In case of PCE surrogate modelling, the performance improvement gained by DRSM is marginal compared to PCA and KPCA with distance preservation- based tuning of $\bfw$. Overall, DRSM consistently provides more accurate or at least comparable results compared to the other approaches. The main difference with a standard UQ setting in which the thermal conductivity is supposed to be sampled from a random field with known properties, is that the proposed DRSM methodology is purely data-driven, it would be applied identically in a case when the input maps are given without knowing the underlying random process. Summary and Conclusions ======================= Surrogate modelling is a key ingredient of modern uncertainty quantification. Due to the detrimental effects of high input dimensionality on most recent surrogate modelling techniques, the input space needs to be compressed to make such problems tractable. We proposed a novel approach for effectively combining dimensionality reduction with surrogate modelling, called DRSM. DRSM consists of three steps: (i) the DR and SM parameters are calculated by solving a nested optimisation problem, where only low-accuracy surrogates are considered to reduce the associated computational cost, (ii) the optimal configuration parameters, including the dimension of the reduced space, are empirically estimated based on the surrogate model performance, and, (iii) a final high-accuracy surrogate is calculated using the optimal values of all the aforementioned parameters. The performance of DRSM was compared on three different benchmark problems of varying complexity against the classical approach of tuning the dimensionality reduction and surrogate modelling parameters sequentially. DRSM consistently showed superior performance compared to the others in all the benchmark applications. The novelty of the proposed methodology lies in its non-intrusive way of combining dimensionality reduction and surrogate modelling. This allows for the combination of various techniques without the need of tweaking the dedicated optimisation algorithms on which each of them capitalises. A practical implication of the non-intrusiveness of DRSM is that off-the-shelf surrogate modelling methods (or even software) with sophisticated calibration algorithms can be directly used within this framework. The focus was given to data-driven scenarios where only a limited set of observations and model responses is available. We demonstrated that the leave-one-out cross-validation error of the surrogate models can serve as a reliable proxy for estimating the generalisation error in order to tune the DR parameters, but also to assess the overall accuracy of the resulting surrogate. It is noteworthy to mention that in application-driven scenarios where the goal is to obtain a surrogate with optimal performance (regardless of its type) for that specific problem, the proposed approach could be extended in a way that the surrogate type itself is included as one of the parameters that DRSM needs to optimise. However, special care would need to be given to the error metric used during the DRSM optimisation in this case, because the LOO error estimations by different surrogates may have widely varied levels of bias (see @Tibshirani2009). In future extensions of this work, focus will be given to capitalising on available HPC resources to optimise for different combinations of surrogate models and dimensionality reduction methods. In addition, the cost of training surrogate models increases with the number of available experimental design samples. Therefore, research efforts will also be directed towards dealing with large experimental designs, possibly within a *big data* framework. Acknowledgements {#acknowledgements .unnumbered} ================ Dr John Jakeman (Sandia National Laboratories) is gratefully acknowledged for having provided the data sets used in the electrical resistor networks application example ([Section \[sec:App:resistor\_networks\]]{}). Relationship between PCA and KPCA with linear kernel {#sec:Appendix:PCA_vs_linKPCA} ==================================================== Consider the PCA-based dimensionality reduction $\bfx \in \Rr^M \mapsto \bfz \in \Rr^m$. As discussed in [Section \[sec:Meth:DR:PCA\]]{}, $\bfz$ is calculated as follows: $$\label{eq:App_pca_z} \bfz = \bfx^\top \bm{V},$$ where $\bm{V} \in \Rr^{M\times m}$ is the collection of the $m$ eigenvectors of $\bm{C} = \text{cov}\bra{\cx}$ and $\cx \in \Rr^{N\times M}$ is the experimental design. Next, consider the kernel PCA mapping $\bfx \in \Rr^M \mapsto \bm{q} \in \Rr^m$ using the linear kernel function: $$\label{eq:app_linkernel} \kappa\prt{\bm{x},\bm{x}^\prime} = a\, \bm{x}^\top \bm{x}^\prime + b.$$ It is straightforward to show that the following transformation is equivalent to the linear kernel in [Eq. (\[eq:app\_linkernel\])]{}: $$\label{eq:app_linkernel_Phi} \Phi(\bfx) = \acc{\sqrt{b},\sqrt{a}\,x_1 \enu \sqrt{a}\,x_M}^\top,$$ because $\kappa\prt{\bm{x},\bm{x}^\prime} = \Phi(\bfx)^\top \Phi(\bfx^\prime)$. A sample $\bm{q}$ in the reduced space is calculated as follows (see [Section \[sec:Meth:DR:KPCA\]]{}): $$\label{eq:app_kpca_z} \bm{q} = \Phi(\bfx)^\mathsf{T} \bm{V}_\ch,$$ where $\bm{V}_\ch$ is the collection of the $m$ eigenvectors of $\bm{C}_{\ch} = \text{cov}\bra{\Phi(\cx)}$ with maximal eigenvalues. Notice that in case of $a=1$ and $b=0$, from Eqs. (\[eq:App\_pca\_z\]), (\[eq:app\_kpca\_z\]) follows that $\bfz = \bm{q}$. The covariance matrix $\bm{C}_{\ch}$ can be expressed as: $$\label{eq:app_Ch} \bm{C}_{\ch} = \begin{bmatrix} 0 & \ldots & 0 \\ \vdots & \multicolumn{2}{c}{\multirow{2}{*}{$a\, \bm{C}$}} \\ 0 \end{bmatrix}.$$ Hence, excluding the eigenvector that corresponds to the zero eigenvalue, it is straightforward to show that $$\label{eq:app_Vh} \bm{V}_{\ch} = \begin{bmatrix} 0 & \ldots & 0 \\ \multicolumn{3}{c}{\bm{V}} \\ \end{bmatrix}.$$ Based on Eqs. (\[eq:app\_linkernel\_Phi\]) and (\[eq:app\_Vh\]), [Eq. (\[eq:app\_kpca\_z\])]{} can be written as follows: $$\begin{aligned} {3} \bm{q} & = \begin{bmatrix}\sqrt{b} & \sqrt{a}\,\bfx^\top \end{bmatrix} \begin{bmatrix} 0 & \ldots & 0 \\ \multicolumn{3}{c}{\bm{V}} \\ \end{bmatrix}\\ & = \sqrt{a} \, \bfx^\top \bm{V} \\ & = \sqrt{a} \, \bfz \quad \text{(from {Eq.~(\ref{eq:App_pca_z})})} \end{aligned}$$ Therefore, the dimensionality reduction using kernel PCA with a linear kernel provides a scaled version of standard PCA and the constant $b$ has no effect. Implementation details {#sec:Appendix:details} ====================== This section provides an extensive list of the configuration parameter values that were used to produce the results in [Section \[sec:Applications\]]{}. (resp. )lists the configuration parameters of Kriging (resp. polynomial chaos expansions) surrogate models. For each surrogate method a distinction is made, in terms of the parameters used, between the proxy ( low computational cost) surrogate and the high-accuracy one. The proxy surrogates were used for solving the nested optimisation problem of DRSM in Eqs. (\[eq:DRSM\_outer\_loop\]), (\[eq:DRSM\_inner\_loop\]). The same configuration was used to calculate the high-accuracy surrogates regardless of the input compression method (DRSM or disjoint PCA/KPCA). [p[0.25]{}| p[0.21]{} p[0.21]{} p[0.21]{}]{} Application & Sobol’ function & Resistor networks & 2D diffusion\ \ Trend & constant ($P=0$)& linear ($P=1$)& linear ($P=1$)\ Correlation family&\ Estimation method&\ Optim. method &\ Optim. constraints &\ Population size (GA) &\ Max. iterations: &\ \ Correlation family&\ Population size (GA) &\ Max. iterations: &\ [p[0.25]{}| p[0.21]{} p[0.21]{} p[0.21]{}]{} Application & Sobol’ function & Resistor networks & 2D diffusion\ \ Coeff. calculation method &\ Univariate polynomials family&\ Hyperbolic truncation $q$ [@BlatmanPEM2010] & $0.75$& $0.50$& $0.65$\ Polynomial degree (adaptive search range) & $[1,10]$ & $[1,10]$ & $[1,5]$\ \ Coeff. calculation method &\ Univariate polynomials family&\ Hyperbolic truncation $q$ [@BlatmanPEM2010]&\ Polynomial degree (adaptive search range) &\ The parameters of the DRSM-based optimisation are listed in . Note that the exact same optimisation algorithm and parameters were used for optimising $\bfw$ w.r.t. the KPCA reconstruction and point-wise distance error in the box-plots used to compare the various approaches. The optimisation constraints differ from the ones reported in when a polynomial kernel is used in KPCA, as in [Eq. (\[eq:KPCA\_kernel\_poly\])]{}, for improved numerical stability of the solver. On top of the bound constraints reported in the table, that still apply for $w_1$ and $w_2$, the variable $w_3$ (degree) is constrained to integer values $1\leq w_3 \leq 4$ instead. In addition, the following non-linear constraint is included: $$w_1 \bfx^\mathsf{T}\bfx' + w_2 > 1.$$ Application Sobol’ function Resistor networks 2D diffusion ---------------------- ------------------------------------------------------- -------------------------------------------------------- ---------------------------------------------------- Optim. method Optim. constraints Population size(GA): $20$ for isotropic KPCA kernels, $80$ for anisotropic $20$ for isotropic KPCA kernels, $100$ for anisotropic $20$ (only isotropic KPCA kernels were considered) Max. iterations: $80$ for both GA and BFGS $150$ for both GA and BFGS $80$ for both GA and BFGS

--- abstract: 'The formation of charged pion condensate under parallel electromagnetic fields is studied within the two-flavor Nambu–Jona-Lasinio model. The technique of Schwinger proper time method is extended to explore the quantity locating in the off-diagonal flavor space, i.e., charged pion. We obtain the associated effective potential as a function of the strength of the electromagnetic fields and find out that it contains a sextic term which possibly induce weakly first order phase transition. Dependence of pion condensation on model parameters is investigated.' address: - 'Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, 730000, China' - 'Institute of High Energy Physics, Chinese Academy of Sciences, Beijing, 100049, China' - 'School of Physics Sciences, University of Chinese Academy of Sciences, Beijing 100039, China' - 'Matrosov Institute for System Dynamics and Control Theory , 664033, Irkutsk, Russia' - author: - Jingyi Chao - Mei Huang - Andrey Radzhabov title: Charged pion condensation under parallel electromagnetic fields --- Introduction ============ The phase structure of Quantum Chromodynamics (QCD) at high temperature/density and other extreme conditions has attracted lots of attentions and been a main topic of heavy ion collisions. The perturbative QCD predicts a free gas of quarks and gluons at high temperature limit and a color-flavor-locking phase at very high baryon density but low temperature. However, the QCD vacuum has a rather complicated nonperturbative structure, and the QCD phase diagram is not a simple transition between the hadron phase with non-zero chiral condensate to the weakly coupled quark-gluon plasma as expected long time ago [@Cleymans:1985wb], but instead a rich structure of different phases with corresponding condensates. These phases could include different color superconducting states or inhomogeneous chiral condensates [@Buballa:2003qv; @Buballa:2014tba; @Andersen:2018osr]. Recently, QCD phase structure under strong magnetic fields has drawn great interests  [@Kharzeev:2007jp; @Skokov:2009qp; @Hattori:2016emy; @Andersen:2014xxa; @Miransky:2015ava; @Huang:2015oca]. The strong magnetic fields can be generated with the strength up to $B\sim10^{18\sim 20}$ G in the non-central heavy ion collisions [@Skokov:2009qp; @Deng:2012pc], and is expected to be on the order of $10^{18}$-$10^{20}$ G [@Duncan:1992hi; @Blaschke:2018mqw] in the inner core of magnetars. Lots of interesting phenomena under strong magnetic fields have been discussed, for example, the magnetic catalysis [@Klevansky:1989vi; @Klimenko:1990rh; @Gusynin:1995nb; @Gusynin:1999pq], inverse magnetic catalysis [@Bali:20111213; @Bali:2012zg; @Bali:2013esa] effect, the chiral magnetic effect (CME) [@Kharzeev:2007jp; @Kharzeev:2007tn; @Fukushima:2008xe] and the vacuum superconductivity [@Chernodub:2010qx; @Chernodub:2011mc]. Moreover, it was pointed out that under the parallel electromagnetic fields, the neutral pion condensation can be formed [@Cao:2015cka; @Wang:2018gmj] due to the connection of field with axial anomaly. If only QCD interaction is included, the axial isospin currents is anomaly free. It turns out that anomaly emerges associated with the coupling of quarks to electromagnetism, where the axial isospin currents is given by $${\partial}_{\mu}j_{5}^{\,\mu 3}=-\frac{e^{2}}{16\pi^{2}}{\varepsilon}^{{\alpha}{\beta}\mu\nu}F_{{\alpha}{\beta}}F_{\mu\nu}\cdot{\hbox{tr}}{\left}[\tau^{3}Q^{2}{\right}].$$ Here $Q$ is the matrix of quark electric charges and $F$ is the field strength. The corresponding process is $\pi_{0}\to {\gamma}{\gamma}$. The decay of a neutral pion into two photons, which had been a puzzle for some time in the 1960s, is the most successful proof of chiral anomaly. Above solution led to the discovery of the Adler–Bell–Jackiw anomaly [@Adler:2004qt]. In the asymmetric flavor space, one can introduce a chiral isospin chemical potential $\mu_{I}^5$ corresponding to the current $\bar{\psi}{\gamma}_{0}{\gamma}_{5}\tau_{3}\psi$, which is similar to the isospin chemical potential $\mu_{I}$ with respect to $\bar{\psi}{\gamma}_{0}\tau_{3}\psi$. It has been a long history of investigating the pion condensation under the isospin asymmetric nuclear matter. In the beginning this effect is discussed for case nuclear matter in neutron-star interiors [@Sawyer:1972cq; @Sawyer:1973fv; @Voskresensky:1980nk] or superdense and supercharged nuclei [@Migdal:1978az]. The pion condensation of charged or neutral pion modes in QCD vacuum are also considered in the frameworks of effective models with quark degrees of freedom [@Son:2000xc; @He:2006tn; @Mao:2014hga; @Khunjua:2017khh] or in lattice calculations [@Brandt:2016zdy; @Brandt:2018bwq]. The degeneracy between $\pi_{0}$ and $\pi_{\pm}$ is destroyed because of the axial isospin chemical potential. It is worth to pursuing the detailed behaviors of charged pions in a strict manner. Hence, in this work, we focus on the possibility of charged pion condensation under the parallel electromagnetic fields in the framework of the ${\mathrm}{SU}(2) \times {\mathrm}{SU}(2)$ NJL model [@Nambu:1961tp; @Nambu:1961fr]. For this purpose, we develop a full routine to derive the mean-field thermodynamical potential of the NJL model with nonzero charged pion condensate ${\langle}\bar{\psi} i \gamma_5 \tau_{\pm} \psi {\rangle}$ in the off-diagonal flavor space under the parallel electromagnetic fields. Calculations are performed with Schwinger proper time method [@Schwinger:1951nm] and the proper time regularization in the NJL model is used. Through the paper we only consider the model at zero temperature and chemical potential and restrict ourselves to the case of the electric field anti-parallel to the magnetic field. \[intr\] Lagrangian {#NJL} ========== The Lagrangian of the ${\mathrm}{SU}(2)\times {\mathrm}{SU}(2)$ NJL model is in the form of [@Nambu:1961tp; @Nambu:1961fr; @Volkov:1986zb; @Vogl:1991qt; @Klevansky:1992qe; @Hatsuda:1994pi; @Volkov:2005kw] $$\begin{aligned} \mathcal{L}_{{\mathrm}{NJL}} = \bar{\psi}\left(i \slashed{D} - m_0 \right) \psi + \mathrm{G} \left[ \left( \bar{\psi} \psi \right)^2+ \left( \bar{\psi} i\gamma_5 \tau_i \psi \right)^2 \right],\end{aligned}$$ where $\bar{\psi}(x)=(\bar{u}(x),\bar{d}(x))$ are $u$ and $d$ anti-quark fields. The limit of equal current masses for $u,d$, $m_u=m_d\equiv m_{0}$ is considered. $\gamma_i$, $\tau_{i}$ are conventional Dirac and Pauli matrices and $\tau_{0}$ is the unit matrix. $\slashed{D}$ is the covariant derivative and, in the two flavor space, expressed as $$\begin{aligned} D_{\mu}={\left}({\partial}_{\mu}-{\mathop{}\!i}QA_{\mu}{\right})\tau_{0}-{\mathop{}\!i}qA_{\mu}\tau_{3},\end{aligned}$$ where $Q=\frac{1}{2}{\left}(q_{u}+q_{d}{\right})$ and $q=\frac{1}{2}{\left}(q_{u}-q_{d}{\right})$. Introducing auxiliary bosonic fields $\pi$, $\sigma$, with the help of Hubbard-Stratonovich transformation one can integrate over the quark fields, then obtains the following effective Lagrangian: $$\begin{aligned} \mathcal{L} = \frac{\sigma^2+\vec{\pi}^2}{4\mathrm{G}}-{\mathop{}\!i}{\hbox{Tr}}\ln S^{-1}, \label{eLagrangian}\end{aligned}$$ where $S^{-1}$ is the inverse quark propagator and $$\begin{aligned} S^{-1}= {\mathop{}\!i}\slashed{D} - M, \quad M=m_{0}\tau_{0}-\sigma\tau_{0}-{\mathop{}\!i}\gamma_5\pi_{i} \tau_i. \label{Propagator}\end{aligned}$$ The auxiliary bosonic fields could have a nonzero vacuum expectation values and therefore it is necessary to shift them as $\sigma = \sigma^\prime-{\langle}\sigma{\rangle}$, $\pi_{i}=\pi_{i}^\prime- {\langle}\pi_{i}{\rangle}$. Equations of motion for mean-fields ${\langle}\sigma{\rangle}$, ${\langle}\pi_{i}{\rangle}$ are obtained from the Lagrangian (\[eLagrangian\]) after elimination from its linear terms, i.e. $$\begin{aligned} \frac{\delta \mathcal{L} }{\delta {\langle}\sigma{\rangle}}\biggl|_{\substack{ \sigma^\prime =0 \\ \pi_{i}^\prime=0 }}, \quad \frac{\delta \mathcal{L} }{\delta {\langle}\pi_{i}{\rangle}}\biggl|_{\substack{ \sigma^\prime =0 \\ \pi_{i}^\prime=0 }} =0.\end{aligned}$$ As a result, under different conditions the ${\langle}{\sigma}{\rangle}$, ${\langle}\pi_{i}{\rangle}$ condensates have non-zero values and the non-zero value of scalar condensate leads to a formation of constituent quarks with dynamical quark mass $m=m_{0}-{\langle}{\sigma}{\rangle}$. Let us denote the second term of effective Lagrangian (\[eLagrangian\]) as $\mathcal{S}_{eff}=-{\mathop{}\!i}{\hbox{Tr}}\ln S^{-1}$. Then the gap equations for ${\langle}{\sigma}{\rangle}$ and ${\langle}\pi_{i}{\rangle}$ takes the form $$\begin{aligned} m=m_{0}-2\mathrm{G}\frac{\partial\mathcal{S}_{eff}}{\partial {\langle}\sigma{\rangle}},\quad {\langle}\pi_{i}{\rangle}=-2\mathrm{G}\frac{\partial \mathcal{S}_{eff}}{\partial {\langle}\pi_{i}{\rangle}}.\end{aligned}$$ The calculation of $\mathcal{S}_{eff}$ is presented in the following section. The effective potential {#main} ======================= Without loss of generality, one can choose ${\langle}\pi_{i}{\rangle}={\left}(\pi_{1},0,0{\right})$ and therefore “mass” in quark propagator Eq.(\[Propagator\]) is $M=m\tau_{0}+{\mathop{}\!i}\pi_{1}{\gamma}_{5}\tau_{1}$. Since ${\mathrm}{Det}{\left}({\mathop{}\!i}\slashed{D}-M{\right})={\mathrm}{Det}\,{\Gamma}{\left}({\mathop{}\!i}\slashed{D}-M{\right}){\Gamma}$, where ${\Gamma}={\gamma}_{5}\tau_{3}$, the second term of the Lagrangian Eq.(\[eLagrangian\]) is replaced to $$\mathcal{S}_{eff}=-\frac{{\mathop{}\!i}}{2}\ln{\mathrm}{Det}{\left}(\slashed{\mathcal{D}}^{2}+m^{2}+\pi_{1}^{2}{\right}),$$ where $\slashed{\mathcal{D}}^{2}=\slashed{D}^{2}-{\gamma}_{5}{\gamma}^{\mu}\pi_{1}{\left}[\tau_{1},D_{\mu}{\right}]$. By using the method of proper time, we represent $\mathcal{S}_{eff}$ as following: $$\label{eqn_seff} \mathcal{S}_{eff}={\hbox{Tr}}\int\limits_{1/{\Lambda}^{2}}^{\infty}{\mathop{}\!i}\,\frac{{\mathop{}\!d}s}{2s}\int{\hbox{tr}}{\left}{\langle}x\big|{\mathop{}\!e}^{-{\mathop{}\!i}{\left}(\slashed{\mathcal{D}}^{2}+m^{2}+\pi_{1}^{2}{\right})s}\big|x'{\right}{\rangle}{\mathop{}\!d}^{4}x,$$ where the ultraviolet cutoff $1/{\Lambda}^{2}$ has been explicitly introduced, ${\hbox{tr}}$ and ${\hbox{Tr}}$ means the trace taking in the spinor and flavor space, respectively. From now on, we will work in the Euclidean space. Following notations are introduced: $$\begin{aligned} \label{AlBeLa} & {\alpha}=m^{2}+\pi_{1}^{2}-\frac{1}{2}{\sigma}^{\mu\nu}{\lambda}_{\mu\nu},\quad {\beta}_{\nu}=q\pi_{1}{\gamma}_{5}{\gamma}^{\mu}F_{\mu\nu}\tau_{2},\quad\nonumber \\ & {\lambda}_{\mu\nu}=q_{f}F_{\mu\nu},\end{aligned}$$ where $q_{f}={\mathrm}{Diag}(q_{u},q_{d})$ and ${\sigma}^{\mu\nu}=\frac{{\mathop{}\!i}}{2}{\left}[{\gamma}^{\mu},{\gamma}^{\nu}{\right}]$. In order to obtain $\mathcal{S}_{eff}$, it is then straightforward to look for the solution of $G(x,y;s)$ obeying a second order differential equation ${\left}(\slashed{\mathcal{D}}^{2}+m^{2}+\pi_{1}^{2}{\right})G{\left}(x,y;s{\right})={\delta}{\left}(x,y;s{\right})$. The explicit form is $$\begin{aligned} \slashed{\mathcal{D}}^{2}+m^{2}+\pi_{1}^{2}&={\partial}^{2}_{x}+\alpha(y)+\beta_{\mu}(y){\left}(x-y{\right})^{\mu}+\nonumber\\ & +\frac{1}{4}{\lambda}^{2}_{\mu\nu}{\left}(x-y{\right})^{\mu}{\left}(x-y{\right})^{\nu}.\end{aligned}$$ Performing the Fourier transform, one finds, $$\begin{aligned} \label{eqn_diff_p} {\left}(-p^{2}+\alpha-{\mathop{}\!i}\beta_{\mu}\frac{{\partial}}{{\partial}p_{\mu}}-\frac{1}{4}{\lambda}^{2}_{\mu\nu}\frac{{\partial}^{2}}{{\partial}p_{\mu}{\partial}p_{\nu}}{\right})G(p;s)=1.\end{aligned}$$ As suggested in the the reference [@Brown:1975bc] one can solve the equation in the form $$\begin{aligned} \label{eqn_G_pV2} G(p;s)={\mathop{}\!e}^{-\alpha s}{\mathop{}\!e}^{ p\cdot A(s)\cdot p+B(s) \cdot p+C(s)},\end{aligned}$$ whose associated descriptions of matrix $A$, vector $B$ and scalar $C$ are $$\begin{aligned} \label{eqn_sol_ABC} &A={\lambda}^{-1}\tan{\lambda}s,\quad B=-2{\mathop{}\!i}{\beta}\cdot{\lambda}^{-2}{\left}(1-\sec{\lambda}s{\right}),\\ &C=-\frac{1}{2}\,{\hbox{tr}}\ln\cos{\lambda}s-{\beta}\cdot{\lambda}^{-3}{\left}(\tan{\lambda}s-{\lambda}s{\right})\cdot{\beta}\nonumber.\end{aligned}$$ For simplicity here and below indexes are not shown. Plugging the form of ${\beta}$ in Eq.(\[AlBeLa\]) into vector $B$ and restoring indexes one has $$\begin{aligned} B_{\mu}=-2{\mathop{}\!i}q\pi_{1}\tau_{2}{\gamma}_{5}{\gamma}^{\nu}F_{\nu{\alpha}}{\left}[{\lambda}^{-2}{\left}(1-\sec{\lambda}s{\right}){\right}]^{{\alpha}}_{\mu}.\end{aligned}$$ Vector $B$ contains Dirac matrix, not commuting with ${\sigma}^{\mu\nu}$. Therefore, we emphasize that one should be careful with tracing in spinor space and integrating in momentum space. Introducing notations $P_{1}=\frac{1}{2}{\sigma}{\lambda}s$ and $P_{2}=p\cdot A(s)\cdot p+B(s) \cdot p$, one has ${\left}[{\sigma}{\lambda}s, p\cdot A(s)\cdot p{\right}]=0$ and the part with matrices in exponent Eq.(\[eqn\_G\_pV2\]) can be expanded as $$\begin{aligned} &{\mathop{}\!e}^{P_{1}+P_{2}}\simeq {\mathop{}\!e}^{P_{1}}{\mathop{}\!e}^{P_{2}}{\mathop{}\!e}^{-\frac{1}{2}[P_{1},P_{2}]}=\nonumber \\ &\quad\quad={\mathop{}\!e}^{\frac{1}{2}{\sigma}{\lambda}s}{\mathop{}\!e}^{p\cdot A(s)\cdot p+B(s) \cdot p}{\mathop{}\!e}^{-\frac{1}{4}{\left}[{\sigma}{\lambda}s,B(s) \cdot p{\right}]}.\end{aligned}$$ We denote $-\frac{1}{4}{\left}[{\sigma}{\lambda}s,B(s) \cdot p{\right}]=\frac{1}{2}q\pi_{1} Os$, where $O$ has a structure of the form $O=Q\tau_{2}O_{1}\mathbb{B}_{1}p+q\tau_{1}O_{2}\mathbb{B}_{2}p$ and $\mathbb{B}$ will render in Eq. (\[Bdefinition\]). Shorthand matrix notation is applied, i.e. $\mathbb{F}=F_{\mu}^{\nu}$. To find the eigenvalue of $O$, we square it and get $$\begin{aligned} O^{2}&=Q^{2}{\left}(\tau_{2}O_{1}\mathbb{B}_{1}p{\right})^{2}+q^{2}{\left}(\tau_{1}O_{2}\mathbb{B}_{2}p{\right})^{2} -\nonumber \\ &-{\mathop{}\!i}qQ\tau_{3}{\left}[O_{1}\mathbb{B}_{1}p,O_{2}\tilde{\mathbb{B}}_{2}p{\right}],\\ &O_{1}={\mathop{}\!i}{\left}[{\sigma}_{\mu\nu},{\gamma}_{5}{\gamma}^{{\alpha}} {\right}]=2{\gamma}_{5}g_{\nu}^{{\alpha}}{\gamma}_{\mu}-2{\gamma}_{5}g_{\mu}^{{\alpha}}{\gamma}_{\nu},\nonumber\\ &O_{2}={\left}\{{\sigma}_{\mu\nu},{\gamma}_{5}{\gamma}^{{\alpha}}{\right}\}=-2{\varepsilon}^{{\alpha}{\beta}}_{\;\mu\nu}{\gamma}_{{\beta}}.\nonumber\end{aligned}$$ With help of relation $\tau_{2}q_{f}\tau_{2}={\mathrm}{Diag}{\left}(q_{d}, q_{u}{\right})=\tilde{q}_{f}$, the $\tilde{\mathbb{B}}{\left}(\mathbb{B}{\right})$ are shown as $$\begin{aligned} \label{Bdefinition} &\tilde{\mathbb{B}}_{1}{\left}(\mathbb{B}_{1}{\right})=\frac{1}{\mathsf{q}^{2}}{\left}[1-\sec \mathsf{q}\mathbb{F}s{\right}],\quad \nonumber \\ &\tilde{\mathbb{B}}_{2}{\left}(\mathbb{B}_{2}{\right})= \frac{\bar{\mathbb{F}}\mathbb{F}}{\mathbb{F}^{2}} \frac{1}{\mathsf{q}^{2}} {\left}[1-\sec\mathsf{q}\mathbb{F}s{\right}],\end{aligned}$$ where $\mathsf{q}=\tilde{q}_{f}$ or $q_{f}$ for $\tilde{\mathbb{B}},\mathbb{B}$ respectively; $\mathbb{F}$ and $\bar{\mathbb{F}}$ are field strength tensor $F^{\mu\nu}$ and dual field strength tensor $\bar{F}^{\mu\nu}=\frac{1}{2}{\varepsilon}^{\mu\nu{\alpha}{\beta}}F_{{\alpha}{\beta}}$, in shorthand notations. Moreover, ${\left}(\tau_{2}O_{1}\mathbb{B}_{1}p{\right})^{2}=-16\mathbb{B}_{1}\tilde{\mathbb{B}}_{1}p^{2}$, ${\left}(\tau_{1}O_{2}\mathbb{B}_{2}p{\right})^{2}=16\mathbb{B}_{2}\tilde{\mathbb{B}}_{2}p^{2}$ and $[O_{1}\mathbb{B}_{1}p,O_{2}\tilde{\mathbb{B}}_{2}p]=-32{\gamma}_{5}\mathbb{B}_{1}\tilde{\mathbb{B}}_{2}p^{2}$. Applying the system that in a Lorentz frame where the electromagnetic field vectors are anti-parallel, e.g., ${\mathbf}{B}=-{\mathbf}{E}=f\hat{z}$, one gets $\mathbb{F}^{2}=f^2\,{\mathrm}{Diag}{\left}(-,+,+,-{\right})$ and $\bar{\mathbb{F}}\mathbb{F}=-f^2{\delta}_{\mu\nu}$ in Euclidean metric $(-,-,-,-)$, hence that $\bar{\mathbb{F}}\mathbb{F}/\,\mathbb{F}^{2}=f^{2}\mathbb{F}^{-2}$. Besides, ${\left}[1-\sec\mathsf{q}\mathbb{F}s{\right}]$ contains even powers of $\mathbb{F}$. It causes $O^{2}=-16Q^{2}\mathsf{p}_{1}^{2}+16q^{2}\mathsf{p}_{2}^{2}+32{\mathop{}\!i}{\gamma}_{5}Qq\mathsf{p}_{1}\cdot\mathsf{p}_{2}$ in a simply manner, where $\mathsf{p}_{1}=p_{\shortparallel}+p_{\perp}$, $\mathsf{p}_{2}=p_{\shortparallel}-p_{\perp}$, $p_{\shortparallel}=b_{\shortparallel}(p_{0},0,0,p_{3})$ and $p_{\perp}=b_{\perp}(0,p_{1},p_{2},0)$. The forms of $b_{\shortparallel}$ and $b_{\perp}$ are taken as $$\label{eqn_b03} b_{\shortparallel}=\frac{(1-\sec q_{f}s)^{\frac{1}{2}}(1-\sec\tilde{q}_{f}s)^{\frac{1}{2}}}{q_{f}\tilde{q}_{f}},$$ $$\label{eqn_b12} b_{\perp}=\frac{(1-\operatorname{sech}q_{f}s)^{\frac{1}{2}}(1-\operatorname{sech}\tilde{q}_{f}s)^{\frac{1}{2}}}{q_{f}\tilde{q}_{f}}.$$ Here and below in we rescale the integration variable as $s=s^\prime/f$ and omit prime. Because ${\gamma}_{5}^{2}=1$ associated with eigenvalue $\pm 1$, it follows that $O$ has four eigenvalues [@Schwinger:1951nm], written as $$\begin{aligned} \label{eqn_O_squar} \mathcal{O}=\pm 4{\left}({\mathop{}\!i}Q\mathsf{p}_{1}\pm q\tau_{3}\mathsf{p}_{2}{\right}).\end{aligned}$$ Let ${\theta}=q\pi_{1} s/f$, one has $$\begin{aligned} {\hbox{tr}}\,{\mathop{}\!e}^{\frac{1}{2}{\theta}O}=\mathsf{T}=\cos{\left}(2Q{\theta}\mathsf{p}_{1}{\right})\cosh{\left}(2\tau_{3}q{\theta}\mathsf{p}_{2}{\right}),\end{aligned}$$ which follow the method applied in [@dittrich2000probing]. The full statement is that $$\begin{aligned} \label{eqn_O_exp} \exp{\left}[\frac{1}{2}{\theta}O{\right}]=\mathsf{T}+{\mathop{}\!i}{\gamma}_{5}\mathsf{U}+\frac{O\mathsf{V}}{2K^{2}}+\frac{{\mathop{}\!i}{\gamma}_{5}O\mathsf{W}}{2K^{2}},\end{aligned}$$ where $K^{2}=\mathsf{p}_{1}^{2}=\mathsf{p}_{2}^{2}$. $\mathsf{T}, \mathsf{U}, \mathsf{V}$ and $\mathsf{W}$ are scalars. Similarly, $$\begin{aligned} \label{eqn_F_exp} \exp{\left}[q_{f}\frac{{\sigma}Fs}{2f}{\right}]=\mathsf{P}-{\mathop{}\!i}{\gamma}_{5}\mathsf{Q}+\frac{{\sigma}F}{2f}\,\mathsf{R}-\frac{{\mathop{}\!i}{\gamma}_{5}{\sigma}F}{2f}\,\mathsf{S}.\end{aligned}$$ Since $$\begin{aligned} {\hbox{tr}}{\left}(O^{2}{\mathop{}\!e}^{\frac{1}{2}{\theta}O}{\right})=\frac{{\partial}^{2}}{{\partial}^{2}{\theta}}{\hbox{tr}}{\left}(4{\mathop{}\!e}^{\frac{1}{2}{\theta}O}{\right})=4\frac{{\partial}^{2}\mathsf{T}}{{\partial}^{2}{\theta}},\end{aligned}$$ apply the identity of [Eq. (\[eqn\_O\_squar\])]{}, it derives that $$\begin{aligned} \mathsf{U}=\sin{\left}(2Q{\theta}\mathsf{p}_{1}{\right})\sinh{\left}(2\tau_{3}q{\theta}\mathsf{p}_{2}{\right}).\end{aligned}$$ Proceeding with the direct differentiation of the exponential function via our basic trick, we get $$\begin{aligned} &\mathsf{V}=\frac{1}{Q^{2}+q^{2}} \biggl(Q\mathsf{p}_{1}\sin{\left}(2Q{\theta}\mathsf{p}_{1}{\right})\cosh{\left}(2\tau_{3}q{\theta}\mathsf{p}_{2}{\right})+\nonumber\\ &\quad\quad\quad+\tau_{3}q\mathsf{p}_{2}\cos{\left}(2Q{\theta}\mathsf{p}_{1}{\right})\sinh{\left}(2\tau_{3}q{\theta}\mathsf{p}_{2}{\right})\biggr), \nonumber\\ &\mathsf{W}=\frac{1}{Q^{2}+q^{2}} \biggl(\tau_{3}q\mathsf{p}_{2}\sin{\left}(2Q{\theta}\mathsf{p}_{1}{\right})\cosh{\left}(2\tau_{3}q{\theta}\mathsf{p}_{2}{\right})-\nonumber\\ &\quad\quad\quad-Q\mathsf{p}_{1}\cos{\left}(2Q{\theta}\mathsf{p}_{1}{\right})\sinh{\left}(2\tau_{3}q{\theta}\mathsf{p}_{2}{\right})\biggr).\end{aligned}$$ Known in [@dittrich2000probing], one has $$\begin{aligned} &\mathsf{P}=\cos q_{f}s\cosh q_{f}s,\, \mathsf{Q}=\sin q_{f}s\sinh q_{f}s \nonumber\\ &\mathsf{R}=(\sinh q_{f}s\cos q_{f}s+\cosh q_{f}s\sin q_{f}s)/2,\\ &\mathsf{S}=(\sinh q_{f}s\cos q_{f}s-\cosh q_{f}s\sin q_{f}s)/2.\nonumber\end{aligned}$$ Then, we perform an approximate expansion $$\begin{aligned} \label{eqn_ap2bp} &{\mathop{}\!e}^{ p\cdot A(s)\cdot p+B(s)\cdot p}\simeq {\mathop{}\!e}^{p\cdot A(s)\cdot p}{\mathop{}\!e}^{B(s)\cdot p} \\ &={\mathop{}\!e}^{ p\cdot A(s)\cdot p}{\left}(\cos\varrho +B(s) \cdot p\frac{\sin\varrho}{\varrho}{\right})\nonumber \end{aligned}$$ where $\varrho=2q\pi_{1}k/f$ and $k={\left}(\mathsf{p}_{1}\cdot\mathsf{p}_{2}{\right})^{\frac{1}{2}}$. Now, it is allowed us to integrate with respect to $p$ and take the trace in the spinor space. With help of the [Eq. (\[eqn\_O\_exp\])]{}, [Eq. (\[eqn\_F\_exp\])]{} and [Eq. (\[eqn\_ap2bp\])]{}, one has $$\begin{aligned} L(s)&={\hbox{tr}}\int{\mathop{}\!e}^{\frac{1}{2f}{\sigma}{\lambda}s}{\mathop{}\!e}^{p\cdot A(s)\cdot p+B(s)\cdot p}{\mathop{}\!e}^{-\frac{1}{4f}{\left}[{\sigma}{\lambda}s,B(s) \cdot p{\right}]}{\mathop{}\!d}^{4}p \nonumber\\ &=L_{0}(s)+L_{1}(s)+L_{2}(s).\end{aligned}$$ Here ${\langle}X{\rangle}$ denotes integrating in momentum and tracing in spinor space ${\hbox{tr}}\int X{\mathop{}\!e}^{p\cdot A\cdot p}{\mathop{}\!d}^{4}p$. It gives $$\begin{aligned} &L_{0}(s)={\left}{\langle}\cos\varrho\mathsf{T}\mathsf{P}{\right}{\rangle}, \, \, L_{1}(s)={\left}{\langle}\cos\varrho\mathsf{U}\mathsf{Q}{\right}{\rangle}, \, \, L_{2}(s)=\\ &={\left}< \frac{2\tilde{q}_{f}\sin\varrho}{K^{2}k} {\left}[q_{f}K^{2}{\left}(\mathsf{W}\mathsf{S}-\mathsf{V}\mathsf{R}{\right})+\tilde{q}_{f}k^{2}{\left}(\mathsf{V}\mathsf{S}+\mathsf{W}\mathsf{R}{\right}){\right}]{\right}>. \nonumber\end{aligned}$$ The integration with respect to momentum $p$ is in the Gaussian form, which can be taken easily with result $$\begin{aligned} &{\langle}1{\rangle}=\mathcal{N}=\pi^{2}{\mathrm}{Det}A^{-\frac{1}{2}},\,A={\mathrm}{Diag}{\left}(a_{\shortparallel}, a_{\perp}, a_{\perp}, a_{\shortparallel}{\right})\nonumber\\ &{\langle}K^{2}{\rangle}=\frac{\mathcal{N}}{2}\,{\hbox{tr}}{\left}(\frac{D_{+}}{A}{\right}),\, {\langle}k^{2}{\rangle}=\frac{\mathcal{N}}{2}\,{\hbox{tr}}{\left}(\frac{D_{-}}{A}{\right})\end{aligned}$$ The matrices $D_{\pm}={\mathrm}{Diag}{\left}(b^{2}_{\shortparallel}, \pm b^{2}_{\perp}, \pm b^{2}_{\perp}, b^{2}_{\shortparallel}{\right})$, which read from [Eq. (\[eqn\_b03\])]{} and [Eq. (\[eqn\_b12\])]{}. From [Eq. (\[eqn\_sol\_ABC\])]{}, one has $a_{\shortparallel}=\tan q_{f}s/{\left}(q_{f} f{\right})$ and $a_{\perp}=\tanh q_{f}s/{\left}(q_{f} f{\right})$. The higher orders corrections ${\langle}K^{4}{\rangle}$, ${\langle}k^{4}{\rangle}$ and $ {\langle}k^{2}K^{2}{\rangle}$ can be drawn in a similar manner, which are abbreviated here. Since ${\theta}p\sim\pi_{1}ps/f\sim \pi_{1}p/{\Lambda}^{2}\ll 1$ and the integration is exponential suppressed for large $s$, it enables us to approximate $\sin(a{\theta}p)$, $\sinh(a{\theta}p)\sim a{\theta}p$ and $\cos(a{\theta}p)$, $\cosh(a{\theta}p)\sim 1$. Hence, it acquires $\mathsf{T}\sim 1$, $\mathsf{U}\sim k^{2}s$, $\mathsf{V}\sim K^{2}s$ and $\mathsf{W}\sim K^{2}k^2 s$. Finally, take the integration with respect to $s$ to get $$\begin{aligned} \label{eqn_eff_potentioal} &\mathcal{S}_{eff}= {\mathcal}{S}_{eff}^{0}+{\mathcal}{S}_{eff}^{1}+{\mathcal}{S}_{eff}^{2}, \\ &{\mathcal}{S}_{eff}^{i}=\frac{N_c }{4\pi^{2}}{\hbox{Tr}}\int_{f/{\Lambda}^{2}}^{\infty} \frac{{\mathop{}\!d}s}{2s} {\mathop{}\!e}^{-h(s)} S_{eff}^{i}(s),\quad \nonumber \end{aligned}$$ where $-h(s)=-(m^{2}+\pi_{1}^{2})s/f+{C}(s)-\frac{1}{2}\ln{\hbox{tr}}{A}$, and $$\begin{aligned} &{C}(s)-\frac{\ln{\hbox{tr}}{A}}{2}= -\ln\frac{\sin q_{f}s\sinh q_{f}s}{q_{f}^{2}f^{2}}\nonumber\\ &\quad\quad\quad-\frac{2q^{2}{\pi_{1}}^{2}}{\tilde{q}_{f}^{3}f}{\left}(2\tilde{q}_{f}s-\tan \tilde{q}_{f}s-\tanh \tilde{q}_{f}s{\right}).\end{aligned}$$ The detailed integrands $S_{eff}^{i}(s)$ are $$\begin{aligned} &S_{eff}^{0}(s) = \mathsf{P},\quad \nonumber \\ &S_{eff}^{1}(s) =4 \tau_{3} \frac{Q q^3\pi_{1}^2 s^2}{f^2{\mathcal}{N}}\frac{}{}{\left}{\langle}k^2{\right}{\rangle}\mathsf{Q} ,\quad \label{exprS2} \\ &S_{eff}^{2}(s) =\frac{8\tilde{q}_{f}q^2\pi_{1}^2 s}{f^2{\mathcal}{N}} {\left}(-q_{f}{\left}{\langle}K^2{\right}{\rangle}\mathsf{R}+\tilde{q}_{f}{\left}{\langle}k^2{\right}{\rangle}\mathsf{S}{\right}) +\nonumber\\ &\quad\quad+\tau_{3} \frac{32\tilde{q}_{f} Qq^5 \pi_{1}^4 s^3}{3f^4{\mathcal}{N}} {\left}(q_{f}{\left}{\langle}K^2k^2{\right}{\rangle}\mathsf{S}+\tilde{q}_{f}{\left}{\langle}k^4{\right}{\rangle}\mathsf{R}{\right}).\nonumber \end{aligned}$$ Eventually, we have the effective potential which takes the following form: $$\begin{aligned} \Omega = \frac{(m-m_{0})^2+\pi_1^2}{4\mathrm{G}} + \mathcal{S}_{eff}.\end{aligned}$$ Numerical results ================= The NJL model is nonrenormalizable and therefore the UV cut-off should be employed in order to get reasonable results, where a proper time regularization is applied in the work, i.e., the integration with respect to $s$ start from $f/{\Lambda}^{2}$. We perform calculations of integral expression for $\mathcal{S}_{eff}$ in Eq. (\[eqn\_eff\_potentioal\]) numerically. In the limit of zero field $f$ the expression leads to the usual proper-time regularization scheme of NJL model. Therefore, for numerical estimation we use the model parameterization from Ref. [@Inagaki:2015lma]. Namely, in [@Inagaki:2015lma] there are five sets of model parameters for proper-time regularization scheme which are fitted in favor of observable values of pion mass and weak pion decay constant. For convenience we present them in Table 1. In the set 1 the constituent quark mass is $178$ MeV and for set 5 is $372$ MeV. The constituent quark masses for other parameterizations are in between these two cases. Therefore, one can consider set 1 and set 5 as limited cases for the predictions of the NJL model. Set $m_0$\[MeV\] $\Lambda$\[MeV\] G\[GeV$^{-2}$\] $m$\[MeV\] ----- -------------- ------------------ ----------------- ------------ -- -- -- 1 3.0 1464 1.61 178 2 5.0 1097 3.07 204 3 8.0 849 5.85 245 4 10.0 755 8.13 265 5 15.0 645 17.2 372 : Parameters of the NJL model in the proper-time regularization taken from [@Inagaki:2015lma]. \[TableParameters\] The important point of calculation is that integrand of $\mathcal{S}_{eff}$ contain singularities and one should specify how to deal with them. The singularities which are generated by trigonometric functions tangent and cotangent of $q_i s$ for quark flavor i are located at real axis and by hyperbolic functions at imaginary axis. We shift $s$ to the complex plane $s+i\epsilon$, see Fig. \[ContTikz\], since we prefer to running a numerical calculation of integral instead of residues summation like what used in [@Inagaki:2003ac; @Ruggieri:2016xww]. In principle, the effective potential at finite $f$ acquires an imaginary part which correspond to pair-production because of Schwinger mechanism [@Schwinger:1951nm; @Tavares:2018poq; @Cao:2015dya]. We figure out that the imaginary part is smaller than the real part in current work. Plus, the subtle effect of Schwinger mechanism is out of the scope of the present paper and will not discuss here. ![Contour on complex $s$-plane. Singularities for a quark of flavor i which are related to tangent are shown by open circles while filled circles correspond to those of cotangent $\cot(q_i s)$.[]{data-label="ContTikz"}](ContTikz2){width="47.00000%"} In Figs. \[Scan5ff001\], \[Scan5ff010\], \[Scan5ff035\], the behavior of effective potential for Set 5 of model parameters is plotted for field values $f=0.01,0.2,0.450$ GeV$^2$, respectively. We found the following typical behavior for three regions: 1) For small field $f=0.01$ GeV$^2$ as shown in Fig. \[Scan5ff001\], the system is in usual (almost vacuum) chiral symmetry breaking phase with nonzero sigma condensate and zero pion condensate; 2) For moderate field $f=0.2$ GeV$^2$, seen in Fig. \[Scan5ff010\], the additional minima appears in the effective potential and the system takes a chiral rotation in $\sigma -\pi_1$ plane to have a nonzero pion condensate, $\pi_1$; 3) For large field $f=0.450$ GeV$^2$, read from Fig. \[Scan5ff035\], the minimum with $\pi_1=0$ is energetically favorable. There are two sources to break the chiral symmetry: spontaneous chiral symmetry breaking due to presence of quark condensate ${\langle}\bar{\psi} \psi {\rangle}$ and explicit chiral symmetry breaking due to nonzero current quark mass in the Lagrangian. Therefore, we investigate not only the reality situation but also for $m_0\to 0$. To systematically perform this task, we vary $m_{0}$ and recalculate $m$ while $\mathrm{G}$ and $\Lambda$ have the same values, i.e. we consider $m$ as a function of $m_{0}$ [@Bernard:1992mp]. In the following we denote the physical value of current quark mass as $m_{0}^\star$. The behaviors of $m$ and $\pi_1$ as a function of field $f$ are presented in Fig. \[Set1Set5MassDelta\] for different values of ratio $m_{0}/m_{0}^\star=0.01,0.1,0.5,1.0$. The left and right sides are obtained by model parameter sets 1 and 5, respectively. It is straightforward to figure out that for small current quark mass the system is more preferable to chirally rotate from zero to nonzero value $\pi_1$, leaving the total order parameter of chiral symmetry breaking $|M|=\sqrt{m^2+\pi_1^2}$ unchanged. With increasing of $m_{0}$ the situation becomes more complicated. The phase of pion condensation even never show up for $m_{0}=m_{0}^\star$ in the model parameter Set 1. Conclusions {#con} =========== In this paper the charged pion condensation under the parallel electromagnetic fields is calculated in the framework of the NJL model by using Schwinger proper-time method. The configuration of field is chosen, the electric field being anti-parallel to the magnetic one, to have a zero first Lorentz invariant, $I_{1}=\mathbf{E}^{2}-\mathbf{B}^{2}$, and a nonzero second Lorentz invariant, $I_2=\mathbf{E} \cdot \mathbf{B}$. We find that in the chiral limit the system is favorable to form a both nonzero condensation of scalar and charged pion, i.e. rotating in the chiral group. Chiral condensates aligning to pseudo mesons space has been found in [@Cao:2015cka] by the methods of $\chi{\mathrm}{PT}$ and NJL model, where the system is immediately straighten up $\pi_0$ direction in the chiral limit once the second Lorentz invariant $I_2$ turned on. The main difference of charged condensation is that the system will across a weakly first order phase transition to zero pion condensate and then a second order phase transition to chirally symmetric phase as the field strength increasing, while it, characterizing by $\pi_{0}$, is a whole second order phase transition as shown in [@Cao:2015cka]. The underlying mechanism are two folds. One is the obviously coupling between charged pions and electromagnetism. Another reason is that a more complicated influence of anomalous diagrams are implicitly included, not only $\pi_0\rightarrow\gamma\gamma$ but also $\gamma\rightarrow \pi_+\pi_-\pi_0$. Indeed, if assuming condensation in the neutral channel ${\langle}\sigma{\rangle}$ nears a second order phase transition, its effective potential has the form ${\mathcal}{S}_{eff}^{0}\sim -c_{0}M^{2}+c_{1}M^{4}/f$ according to Ginzburg-Landau theory [@Ginzburg:1950sr]. However, if we include $\pi_{\pm}$ as an additional degree of freedom and non-degenerate with $\pi_{0}$, read from [Eq. (\[eqn\_eff\_potentioal\])]{}, the potential arranges as: ${\mathcal}{S}_{eff}^{2}\sim -\tilde{c}_{1}M^{4}/f+c_{2}M^{6}/f^{2}$. As a result, we have a weakly first order phase transition and effective potential in the form of $$\begin{aligned} {\Omega}=\frac{M^{2}}{4 \mathrm{G}}-c_{0}M^{2}+\frac{{\left}(c_{1}-\tilde{c}_{1}{\right})M^{4}}{f}+\frac{c_{2}M^{6}}{f^{2}}.\end{aligned}$$ Our numerical simulations support these arguments, read from Figs. \[Scan5ff001\], \[Scan5ff010\], \[Scan5ff035\]. The mass of current quarks plays an important role and it denies our claim at some regions of the model parameters. It requires a further study via the first principle calculation, such as Dyson-Schwinger equation or functional renormalization group methods. Application of the charged pion condensation to the case heavy-ion collisions or neutron stars interior need an extension to finite temperature and/or chemical potential. We will explore this extension in future. Acknowledgments {#Ackn} =============== We are grateful to Maxim Chernodub, Nikolai Kochelev, Marco Ruggieri and Pengming Zhang for the useful discussions. J.Y.C. is supported by the NSFC under Grant number: 11605254 and Major State Basic Research Development Program in China (No. 2015CB856903). M.H. is supported by the NSFC under Grant No. 11725523, 11735007 and 11261130311(CRC 110 by DFG and NSFC). A.R. is supported by the CAS President’s international fellowship initiative (Grant No. 2017VMA0045), Council for Grants of the President of the Russian Federation (project NSh-8081.2016.9) and numerical calculations are performed on computing cluster “Akademik V.M. Matrosov” (http://hpc.icc.ru). {width="49.00000%"} {width="43.00000%"} {width="49.00000%"} {width="43.00000%"} {width="49.00000%"} {width="43.00000%"} {width="47.00000%"} {width="47.00000%"} [10]{} J. Cleymans, R. V. Gavai, E. Suhonen, Phys. Rept. [**130**]{} (1986) 217. M. Buballa, Phys. Rept. [**407**]{} (2005) 205. M. Buballa, S. Carignano, Prog. Part. Nucl. Phys. [**81**]{} (2015) 39. J. O. Andersen, P. Kneschke, [](http://arxiv.org/abs/1802.01832). D. E. Kharzeev, L. D. McLerran, H. J. Warringa, Nucl. Phys. A [**803**]{} (2008) 227. V. Skokov, A. [Yu]{}. Illarionov, V. Toneev, Int. J. Mod. Phys. A [**24**]{} (2009) 5925. K. Hattori, X.-G. Huang, Nucl. Sci. Tech. [**28**]{} (2) (2017) 26. J. O. Andersen, W. R. Naylor and A. Tranberg, Rev. Mod. Phys.  [**88**]{} (2016) 025001. V. A. Miransky and I. A. Shovkovy, Phys. Rept.  [**576**]{} (2015) 1. X. G. Huang, Rept. Prog. Phys.  [**79**]{}, no. 7 (2016) 076302. W. -T. Deng and X. -G. Huang, Phys. Rev. C [**85**]{} (2012) 044907. R. C. Duncan, C. Thompson, Astrophys. J. [**392**]{} (1992) L9. D. Blaschke, N. Chamel, [](http://arxiv.org/abs/1803.01836). S. P. Klevansky and R. H. Lemmer, Phys. Rev. D [**39**]{} (1989) 3478. K. G. Klimenko, Theor. Math. Phys.  [**89**]{} (1992) 1161 \[Teor. Mat. Fiz.  [**89**]{} (1991) 211\]. V. P. Gusynin, V. A. Miransky and I. A. Shovkovy, Nucl. Phys. B [**462**]{} (1996) 249. V. P. Gusynin, V. A. Miransky and I. A. Shovkovy, Nucl. Phys. B [**563**]{} (1999) 361. G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz, S. Krieg, A. Schafer and K. K. Szabo, JHEP [**1202**]{} (2012) 044. G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz and A. Schafer, Phys. Rev. D [**86**]{} (2012) 071502. G. S. Bali, F. Bruckmann, G. Endrodi, F. Gruber and A. Schaefer, JHEP [**1304**]{} (2013) 130. D. Kharzeev and A. Zhitnitsky, Nucl. Phys. A [**797**]{} (2007) 67. K. Fukushima, D. E. Kharzeev and H. J. Warringa, Phys. Rev.  D [**78**]{} (2008) 074033 . M. N. Chernodub, Phys. Rev. D [**82**]{} (2010) 085011. M. N. Chernodub, Phys. Rev. Lett.  [**106**]{} (2011) 142003 . G. Cao, X.-G. Huang, Phys. Lett. B [**757**]{} (2016) 1. L. Wang, G. Cao, X. G. Huang and P. Zhuang, Phys. Lett. B [**780**]{} (2018) 273. S. L. Adler, [Anomalies to all orders]{}, in: G. ’t Hooft (Ed.), 50 years of Yang-Mills theory, 2005, pp. 187–228. R. F. Sawyer, Phys. Rev. Lett. [**29**]{} (1972) 382. R. F. Sawyer, D. J. Scalapino, Phys. Rev. D [**7**]{} (1973) 953. D. N. Voskresensky, N. [Yu]{}. Anisimov, Sov. Phys. JETP [**51**]{} (1980) 13 \[Zh. Eksp. Teor. Fiz. [**78**]{} (1980) 28\]. A. B. Migdal, Rev. Mod. Phys. [**50**]{} (1978) 107. D. T. Son and M. A. Stephanov, Phys. Rev. Lett.  [**86**]{} (2001) 592. L. He, M. Jin and P. Zhuang, Phys. Rev. D [**74**]{} (2006) 036005. S. Mao, Phys. Rev. D [**89**]{} (2014) 116006. T. G. Khunjua, K. G. Klimenko, R. N. Zhokhov, V. C. Zhukovsky, Phys. Rev. D [**95**]{} (2017) 105010. B. B. Brandt, G. Endrodi, PoS [**LATTICE2016**]{} (2016) 039. B. B. Brandt, G. Endrodi, E. S. Fraga, M. Hippert, J. Schaffner-Bielich, S. Schmalzbauer, [](http://arxiv.org/abs/1802.06685). Y. Nambu, G. Jona-Lasinio, Phys. Rev. [**122**]{} (1961) 345. Y. Nambu, G. Jona-Lasinio, Phys. Rev. [**124**]{} (1961) 246. J. S. Schwinger, Phys. Rev. [**82**]{} (1951) 664. M. K. Volkov, Sov. J. Part. Nucl. [**17**]{} (1986) 186, \[Fiz. Elem. Chast. Atom. Yadra [**17**]{} (1986) 433\]. U. Vogl, W. Weise, Prog. Part. Nucl. Phys. [**27**]{} (1991) 195. S. P. Klevansky, Rev. Mod. Phys. [**64**]{} (1992) 649. T. Hatsuda, T. Kunihiro, Phys. Rept. [**247**]{} (1994) 221. M. K. Volkov, A. E. Radzhabov, Phys. Usp. [**49**]{} (2006) 551. M. R. Brown, M. J. Duff, Phys. Rev. D [**11**]{} (1975) 2124. W. Dittrich and H. Gies, [*Probing the quantum vacuum: perturbative effective action approach in quantum electrodynamics and its application*]{} (2000), Springer Science & Business Media. H. Kohyama, D. Kimura, T. Inagaki, Nucl. Phys. B [**896**]{} (2015) 682. T. Inagaki, D. Kimura, T. Murata, Int. J. Mod. Phys. A [**20**]{} (2005) 4995. M. Ruggieri, Z. Y. Lu, G. X. Peng, Phys. Rev. D [**94**]{} (2016) 116003. W. R. Tavares, S. S. Avancini, [](http://arxiv.org/abs/1801.10566). G. Cao, X.-G. Huang, Phys. Rev. D [**93**]{} (2016) 016007. V. Bernard, A. A. Osipov, U. G. Meissner, Phys. Lett. B [**285**]{} (1992) 119. V. L. Ginzburg, L. D. Landau, Zh. Eksp. Teor. Fiz. [**20**]{} (1950) 1064.

--- author: - 'Kenneth M. Lanzetta$^{1}$, H.-W. Chen$^{1}$, S. Pascarelle$^{1}$, N. Yahata$^{1}$' --- [$^{1}$ State University of New York at Stony Brook, Stony Brook, U.S.A.]{} Here we describe our attempts to establish statistically complete samples of very high redshift galaxies by obtaining photometric redshifts of galaxies in Medium Deep Survey (MDS) fields and photometric and spectroscopic redshifts of galaxies in very deep STIS slitless spectroscopy fields. On the basis of this analysis, we have identified galaxies of redshift $z = 4.92$ in an MDS field and of redshift $z = 6.68$ in a very deep STIS field. Introduction ============ Our photometric redshift technique applied to observations of the Hubble Deep Field (HDF) has identified galaxies at redshifts up to and beyond $z = 6$ ([@lyf96], [@fly98]). The recent confirmation of two of these measurements at redshifts $z > 5$ ([@weymann98], [@spinrad98]) has established that broad-band redshift measurement techniques can accurately and reliably identify high-redshift galaxies—not only in a general sense, distinguishing high- from low-redshift galaxies, but also in a specific sense, establishing redshifts to within relative errors of typically $\Delta z / (1+z) < 15\%$. Building on the success of the broad-band redshift measurement techniques, we have sought to establish statistically complete samples of very high redshift galaxies by means of four distinct methods applied to four distinct collections of observations: (1) photometric redshifts of galaxies in the HDF, (2) photometric redshifts of galaxies in Medium Deep Survey (MDS) fields, (3) photometric and spectroscopic redshifts of galaxies in very deep STIS slitless spectroscopy fields, and (4) photometric redshifts of infrared detected (and optical non-detected) galaxies in the HDF. In principle, the optical-wavelength observations are sensitive to galaxies at redshifts as large as $z \approx 7$, while the infrared-wavelength observations are sensitive to galaxies at redshifts as large as $z \approx 17$. Here we describe initial results of the second and third of these programs, namely of photometric redshifts of galaxies in MDS fields and photometric and spectroscopic redshifts of galaxies in very deep STIS slitless spectroscopy fields. Photometric redshifts of galaxies in MDS fields =============================================== First, we have sought to measure photometric redshifts of galaxies in MDS fields. The MDS ([@ratnatunga98]) currently spans more than 450 fields, which have been observed using HST with WFPC2 and the F814W, F606W, and (in some cases) the F450W filters. These observations are, of course, not as deep as the observations of the HDF. But what these observations lack in depth, they make up for in breadth, covering an angular area of $\approx 0.6$ deg$^2$ to a $5 \sigma$ point source limiting magnitude threshold of typically $AB(8140) \approx 27.0$. Our previous application of the photometric redshift technique to the HDF ([@lyf96], [@fly98]) incorporated observations spanning seven photometric bands. In contrast, our application of the photometric redshift technique to the MDS fields incorporates observations spanning only two or three photometric bands, which of course makes photometric redshift measurement more difficult. But given sufficiently small photometric errors, continuum break features indicative of high-redshift galaxies (i.e.the Lyman limit and the [Ly$\alpha$]{} decrement) can be [*unambiguously*]{} distinguished from continuum break features indicative of low-redshift galaxies (i.e. the 4000 Å break) because the amplitude of the largest continuum breaks observed in low-redshift galaxies or main-sequence stars is $\approx 3$ (see [@spinrad98] and references therein), whereas the amplitude of the Lyman continuum break is in principle infinite or close to it. The key, then, is to apply the two- or three-band photometric redshift technique only in cases where the photometric errors are small enough to distinguish the Lyman break from other continuum features to a high level of significance—i.e. to bright (through the F814W filter) galaxies or to deep images. We have so far applied this analysis to a portion of the MDS observations in order to identify candidate galaxies for confirming spectroscopy. Figure 1 shows a spectrum obtained with the KPNO 4 m telescope of a high-redshift galaxy identified in an MDS field. The spectrum is characterized by an emission line at $\lambda = 7200$ Å, which we interpret as [Ly$\alpha$]{}, and by a continuum break at $\lambda = 7200$ Å, which we interpret as the [Ly$\alpha$]{}decrement, in which case the redshift of the galaxy is $z = 4.92$. Based on an initial analysis, we expect that $\approx 65$ galaxies of redshift $z > 5$ be identified in the MDS fields. Photometric and spectroscopic redshifts of galaxies in very deep STIS slitless spectroscopy observations ======================================================================================================== Second, we have sought to measure photometric and spectroscopic redshifts of galaxies in very deep STIS sliless spectroscopy fields. At near-infrared wavelengths, where background sky light is the dominant source of noise, the Hubble Space Telescope (HST) using the Space Telescope Imaging Spectrograph (STIS) is more sensitive than the Keck telescope because from space (1) the sky is fainter and (2) the spatial resolution is higher. To exploit the unique sensitivity of STIS at near-infrared wavelengths, the Space Telescope Science Institute and the STIS instrument team at the Goddard Space Flight Center initiated the STIS Parallel Survey, in which deep STIS observations are obtained in parallel with other observations [@gardner98]. From among the observations so far obtained by the Survey, we selected for analysis very deep observations acquired in slitless spectroscopy mode, because these observations are best suited for identifying distant galaxies. The difficulty of slitless spectroscopy is that the objects can overlap along the dispersion direction. We have overcome this difficulty by developing and applying a new method of analyzing slitless spectroscopy observations that makes optimal use of the direct and dispersed images that are recorded as part of a normal sequence of observation. Specifically, we use the direct image to determine not only the exact locations but also the exact two-dimensional spatial profiles of the spectra on the dispersed image. These spatial profiles are crucial because they provide the “weights” needed to optimally extract the spectra and the models needed to deblend the overlapping spectra and determine the background sky level. Our analysis proceeds in three steps: First, we identify objects in the direct image, using standard source extraction programs. Next, we model each pixel of the dispersed image as a linear sum of contributions from relevant portions of all overlapping neighboring objects and background sky. Finally, we minimize $\chi^2$ between the model and the data with respect to the model parameters to form optimal one-dimensional spectra. We measure redshifts from the optimal one-dimensional spectra by means of [*both*]{} broad-band continuum features and narrow-band emission and absorption features. Specifically, we measure photometric redshifts using a variation of the photometric redshift technique described previously by [@lyf96], and we seek to verify the photometric redshifts by identifying confirming narrow emission and absorption features. At low redshifts, the most prominent broad-band feature is the 4000 Å break and the most prominent narrow-band features are the \[O II\] emission line and the Ca II H and K absorption lines, while at high redshifts, the most prominent broad-band feature is the [Ly$\alpha$]{}decrement and the most prominent narrow-band feature is the [Ly$\alpha$]{} emission line. Figure 2 shows examples of photometric and spectroscopic redshifts of galaxies in very deep STIS slitless spectroscopy fields. The spectrum of the highest-redshift galaxy is characterized by an emission line at $\lambda = 7200$ Å, which we interpret as [Ly$\alpha$]{}, and by a continuum break at $\lambda = 7200$ Å, which we interpret as the [Ly$\alpha$]{} decrement, in which case the redshift of the galaxy is $z = 6.68$. Based on an initial analysis, we expect that redshifts of a large fraction of galaxies in the STIS field will be identified. Lanzetta, K. M., Yahil, A., & Fernández-Soto, A 1996, Fernández-Soto, A., Lanzetta, K. M., & Yahil, A. 1998, Weymann, R. J., Stern, D., Bunker, A., Spinrad, H., Chaffee, F. H., Thompson, R. I., & Storrie-Lombardi, L. J. 1998, Spinrad, H. Stern, D., Bunker, A., Dey, A., Lanzetta, K., Yahil, A., Pascarelle, S., & Fernández-Soto, A. 1998, Ratnatunga K. U., Griffiths, R. E., & Ostrander, E. J. 1998, in preparation Gardner, J. et al. 1998,

--- abstract: 'In a conformal class of metrics with positive Yamabe invariant, we derive a necessary and sufficient condition for the existence of metrics with positive $Q$ curvature. The condition is conformally invariant. We also prove some inequalities between the Green’s functions of the conformal Laplacian operator and the Paneitz operator.' address: - 'Courant Institute, New York University, 251 Mercer Street, New York NY 10012' - 'Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton NJ 08544' author: - Fengbo Hang - 'Paul C. Yang' title: 'Sign of Green’s function of Paneitz operators and the $Q$ curvature' --- Introduction\[sec1\] ==================== Since the fundamental work [@CGY] in dimension $4$, the Paneitz operator and associated $Q$ curvature in dimension other than $4$ (see [@B; @P]) attracts much attention (see [@DHL; @GM; @HY1; @HeR1; @HeR2; @HuR; @QR] etc and the references therein). Let $\left( M,g\right) $ be a smooth compact $n$ dimensional Riemannian manifold. For $n\geq 3$, the $Q$ curvature is given by$$Q=-\frac{1}{2\left( n-1\right) }\Delta R-\frac{2}{\left( n-2\right) ^{2}}\left\vert Rc\right\vert ^{2}+\frac{n^{3}-4n^{2}+16n-16}{8\left( n-1\right) ^{2}\left( n-2\right) ^{2}}R^{2}. \label{eq1.1}$$Here $R$ is the scalar curvature, $Rc$ is the Ricci tensor. The Paneitz operator is given by$$\begin{aligned} &&P\varphi \label{eq1.2} \\ &=&\Delta ^{2}\varphi +\frac{4}{n-2}\func{div}\left( Rc\left( \nabla \varphi ,e_{i}\right) e_{i}\right) -\frac{n^{2}-4n+8}{2\left( n-1\right) \left( n-2\right) }\func{div}\left( R\nabla \varphi \right) +\frac{n-4}{2}Q\varphi . \notag\end{aligned}$$Here $e_{1},\cdots ,e_{n}$ is a local orthonormal frame with respect to $g$. For $n\neq 4$, under conformal transformation of the metric, the operator satisfies$$P_{\rho ^{\frac{4}{n-4}}g}\varphi =\rho ^{-\frac{n+4}{n-4}}P_{g}\left( \rho \varphi \right) . \label{eq1.3}$$Note this is similar to the conformal Laplacian operator, which appears naturally when considering transformation law of the scalar curvature under conformal change of metric ([@LP]). As a consequence we know$$P_{\rho ^{\frac{4}{n-4}}g}\varphi \cdot \psi d\mu _{\rho ^{\frac{4}{n-4}}g}=P_{g}\left( \rho \varphi \right) \cdot \rho \psi d\mu _{g}. \label{eq1.4}$$Here $\mu _{g}$ is the measure associated with metric $g$. Moreover$$\ker P_{g}=0\Leftrightarrow \ker P_{\rho ^{\frac{4}{n-4}}g}=0, \label{eq1.5}$$and under this assumption, the Green’s functions $G_{P}$ satisfy the transformation law$$G_{P,\rho ^{\frac{4}{n-4}}g}\left( p,q\right) =\rho \left( p\right) ^{-1}\rho \left( q\right) ^{-1}G_{P,g}\left( p,q\right) . \label{eq1.6}$$For $u,v\in C^{\infty }\left( M\right) $, we denote the quadratic form associated with $P$ as$$\begin{aligned} &&E\left( u,v\right) \label{eq1.7} \\ &=&\int_{M}Pu\cdot vd\mu \notag \\ &=&\int_{M}\left( \Delta u\Delta v-\frac{4}{n-2}Rc\left( \nabla u,\nabla v\right) +\frac{n^{2}-4n+8}{2\left( n-1\right) \left( n-2\right) }R\nabla u\cdot \nabla v\right. \notag \\ &&\left. +\frac{n-4}{2}Quv\right) d\mu \notag\end{aligned}$$and$$E\left( u\right) =E\left( u,u\right) . \label{eq1.8}$$By the integration by parts formula in (\[eq1.7\]) we know $E\left( u,v\right) $ also makes sense for $u,v\in H^{2}\left( M\right) $. To continue we recall (see [@LP]) for $n\geq 3$, on a smooth compact Riemannian manifold $\left( M^{n},g\right) $, the conformal Laplacian operator is given by$$L_{g}\varphi =-\frac{4\left( n-1\right) }{n-2}\Delta \varphi +R\varphi . \label{eq1.9}$$The Yamabe invariant is defined as$$\begin{aligned} &&Y\left( g\right) \label{eq1.10} \\ &=&\inf \left\{ \frac{\int_{M}\widetilde{R}d\widetilde{\mu }}{\left( \widetilde{\mu }\left( M\right) \right) ^{\frac{n-2}{n}}}:\widetilde{g}=\rho ^{2}g\text{ for some positive smooth function }\rho \right\} \notag \\ &=&\inf \left\{ \frac{\int_{M}L_{g}\varphi \cdot \varphi d\mu }{\left\Vert \varphi \right\Vert _{L^{\frac{2n}{n-2}}}^{2}}:\varphi \text{ is a nonzero smooth function on }M\right\} . \notag\end{aligned}$$A basic but useful fact about the scalar curvature is$$\begin{gathered} Y\left( g\right) >0\Leftrightarrow \lambda _{1}\left( L_{g}\right) >0 \label{eq1.11} \\ \Leftrightarrow g\text{ is conformal to a metric with scalar curvature}>0\text{.} \notag\end{gathered}$$Indeed more is true, namely the equivalence still holds if we replace all “$> $” by “$="$ or ”$<$". Here $\lambda _{1}\left( L_{g}\right) $ is the first eigenvalue of $L_{g}$. It is clear $Y\left( g\right) $ is a conformal invariant, on the other hand the sign of $\lambda _{1}\left( L_{g}\right) $ is also conformally invariant. The main reason that (\[eq1.11\]) holds is based on the fact the first eigenfunction of a second order symmetric differential operator does not change sign. Unfortunately such kind of property is known to be not valid for higher order operators. The following question keeps puzzling people from the beginning of research on $Q$ curvature in dimension other than $4$, namely: can we find a conformal invariant condition which is equivalent to the existence of positive $Q$ curvature in the conformal class (in the same spirit as (\[eq1.11\]))? Here we give an answer to this question for conformal class with positive Yamabe invariant. \[thm1.1\]Let $n>4$ and $\left( M^{n},g\right) $ be a smooth compact Riemannian manifold with Yamabe invariant $Y\left( g\right) >0$, then the following statements are equivalent 1. there exists a positive smooth function $\rho $ with $Q_{\rho ^{2}g}>0$. 2. $\ker P_{g}=0$ and the Green’s function $G_{P}\left( p,q\right) >0$ for any $p,q\in M,p\neq q$. 3. $\ker P_{g}=0$ and there exists a $p\in M$ such that $G_{P}\left( p,q\right) >0$ for $q\in M\backslash \left\{ p\right\} $. Along the way we also find the following comparison inequality between Green’s function of $L$ and $P$. \[prop1.1\]Assume $n>4$, $\left( M^{n},g\right) $ is a smooth compact Riemannian manifold with $Y\left( g\right) >0$, $Q\geq 0$ and not identically zero, then $\ker P=0$ and$$G_{L}^{\frac{n-4}{n-2}}\leq c_{n}G_{P}. \label{eq1.12}$$Here$$c_{n}=2^{-\frac{n-6}{n-2}}n^{\frac{2}{n-2}}\left( n-1\right) ^{-\frac{n-4}{n-2}}\left( n-2\right) \left( n-4\right) \omega _{n}^{\frac{2}{n-2}}, \label{eq1.13}$$$\omega _{n}$ is the volume of unit ball in $\mathbb{R}^{n}$. Moreover if $G_{L}^{\frac{n-4}{n-2}}\left( p,q\right) =c_{n}G_{P}\left( p,q\right) $ for some $p\neq q$, then $\left( M,g\right) $ is conformal diffeomorphic to the standard sphere. In dimension $3$ we have \[thm1.2\]Let $\left( M,g\right) $ be a smooth compact $3$ dimensional Riemannian manifold with Yamabe invariant $Y\left( g\right) >0$, then the following statements are equivalent 1. there exists a positive smooth function $\rho $ with $Q_{\rho ^{2}g}>0$. 2. $\ker P_{g}=0$ and $G_{P}\left( p,q\right) <0$ for any $p,q\in M,p\neq q$. 3. $\ker P_{g}=0$ and there exists a $p\in M$ such that $G_{P}\left( p,q\right) <0$ for $q\in M\backslash \left\{ p\right\} $. Similar to Proposition \[prop1.1\], we have \[prop1.2\]Let $\left( M,g\right) $ be a smooth compact $3$ dimensional Riemannian manifold with $Y\left( g\right) >0$, $Q\geq 0$ and not identically zero, then $\ker P=0$ and$$G_{L}^{-1}\leq -256\pi ^{2}G_{P}. \label{eq1.14}$$If for some $p,q\in M$, $G_{L}^{-1}\left( p,q\right) =-256\pi ^{2}G_{P}\left( p,q\right) $, then $\left( M,g\right) $ is conformal diffeomorphic to the standard $S^{3}$ (note here $p$ can be equal to $q$). In dimension $4$ we have the following (see Corollary \[cor5.1\]) \[prop1.3\]Assume $\left( M,g\right) $ is a smooth compact $4$ dimensional Riemannian manifold, $Y\left( g\right) >0$, then for any $p\in M$,$$\int_{M}Qd\mu +\frac{1}{2}\int_{M}\left\vert Rc_{G_{L,p}^{2}g}\right\vert _{G_{L,p}^{2}g}^{2}d\mu _{G_{L,p}^{2}g}=16\pi ^{2}. \label{eq1.15}$$In particular, $\int_{M}Qd\mu \leq 16\pi ^{2}$ and equality holds if and only if $\left( M,g\right) $ is conformal diffeomorphic to the standard $S^{4}$. It is worthwhile to point out that the proof of Theorem B in [@G], which gives the inequality in Proposition \[prop1.3\], is elementary and does not use the positive mass theorem. Our argument is also elementary and identifies the difference between $\int_{M}Qd\mu $ and $16\pi ^{2}$. Theorem \[thm1.1\] and \[thm1.2\] are motivated by works on the $Q$ curvature in dimension $5$ or higher ([@GM; @HeR1; @HeR2; @HuR]) and in dimension $3$ ([@HY1; @HY2; @HY3]). In [@HeR1; @HeR2], it was shown in some cases compactness property for solutions of the $Q$ curvature equation can be derived under the assumption that the Green’s function is positive. Recently [@GM] showed that the Green’s function is indeed positive when both scalar curvature and $Q$ curvature are positive. Theorem \[thm1.1\] says we could relax the assumption to $Y\left( g\right) >0,Q_{g}>0$. Whether these two kinds of assumptions are equivalent or not is still unknown. The main approach in [@GM] is roughly speaking by applying the maximum principle twice. In [@HY3], by replacing maximum principle with the weak Harnack inequality it was shown that for metrics with $R>0$ and $Q>0$, $P$ is invertible and $G_{P}\left( p,q\right) <0$ for $p\neq q$. Theorem [thm1.2]{} relax the assumption to $Y\left( g\right) >0$ and $Q>0$. The main new ingredient in our proof of Theorem \[thm1.1\] and \[thm1.2\] is the formula (\[eq2.1\]), which is closely related to the arguments in [HuR]{}. In [@HY4], we will apply the results on Green’s function to solution of $Q$ curvature equations. In section \[sec2\] we will prove the main formula (\[eq2.1\]). In sections \[sec3\] and \[sec4\] we will prove Theorem \[thm1.1\] and \[thm1.2\] respectively. In section [sec5]{} we will derive the corresponding formula of (\[eq2.1\]) in dimension $4$. In particular Proposition \[prop1.3\] follows from the formula. In section \[sec6\], we will show that the positive mass theorem for Paneitz operator in [@GM; @HuR] can be deduced from (\[eq2.1\]) too. An identity connecting the Green’s function of conformal Laplacian operator and Paneitz operator\[sec2\] ======================================================================================================== Here we will derive an interesting formula which illustrates the close relation between Green’s function of conformal Laplacian operator and the Paneitz operator. This identity will play a crucial role later. To motivate the discussion, we note that positive Yamabe invariant implies we have a positive Green’s function for the conformal Laplacian operator. Even though we do not know whether $P$ is invertible or not, we may still try to search for its Green’s function. Note that the possible Green’s function should have same highest order singular term as $G_{L,p}^{\frac{n-4}{n-2}}$ (modulus dimension constant), we can use $G_{L,p}^{\frac{n-4}{n-2}}$ as a first step approximation. Along this line we compute $P\left( G_{L,p}^{\frac{n-4}{n-2}}\right) $ and arrive at the interesting formula (\[eq2.1\]). \[prop2.1\]Assume $n\geq 3$, $n\neq 4$, $\left( M,g\right) $ is a $n$ dimensional smooth compact Riemannian manifold with $Y\left( g\right) >0$, $p\in M$, then we have $G_{L,p}^{\frac{n-4}{n-2}}\left\vert Rc_{G_{L,p}^{\frac{4}{n-2}}g}\right\vert _{g}^{2}\in L^{1}\left( M\right) $ and$$P\left( G_{L,p}^{\frac{n-4}{n-2}}\right) =c_{n}\delta _{p}-\frac{n-4}{\left( n-2\right) ^{2}}G_{L,p}^{\frac{n-4}{n-2}}\left\vert Rc_{G_{L,p}^{\frac{4}{n-2}}g}\right\vert _{g}^{2} \label{eq2.1}$$in distribution sense. Here$$c_{n}=2^{-\frac{n-6}{n-2}}n^{\frac{2}{n-2}}\left( n-1\right) ^{-\frac{n-4}{n-2}}\left( n-2\right) \left( n-4\right) \omega _{n}^{\frac{2}{n-2}}, \label{eq2.2}$$$\omega _{n}$ is the volume of unit ball in $\mathbb{R}^{n}$, $G_{L,p}$ is the Green’s function of conformal Laplacian operator $L=-\frac{4\left( n-1\right) }{n-2}\Delta +R$ with pole at $p$. It is worth pointing out that the metric $G_{L,p}^{\frac{4}{n-2}}g$ on $M\backslash \left\{ p\right\} $ is exactly the stereographic projection of $\left( M,g\right) $ at $p$ ([@LP]). To prove the proposition, let us first check what happens under a conformal change of the metric. If $\rho \in C^{\infty }\left( M\right) $ is a positive function, let $\widetilde{g}=\rho ^{\frac{4}{n-2}}g$, then using$$G_{\widetilde{L},p}\left( q\right) =\rho \left( p\right) ^{-1}\rho \left( q\right) ^{-1}G_{L,p}\left( q\right)$$we see$$G_{\widetilde{L},p}^{\frac{n-4}{n-2}}\left\vert Rc_{G_{\widetilde{L},p}^{\frac{4}{n-2}}\widetilde{g}}\right\vert _{\widetilde{g}}^{2}d\widetilde{\mu }=\rho \left( p\right) ^{-\frac{n-4}{n-2}}\rho ^{\frac{n-4}{n-2}}G_{L,p}^{\frac{n-4}{n-2}}\left\vert Rc_{G_{L,p}^{\frac{4}{n-2}}g}\right\vert _{g}^{2}d\mu . \label{eq2.3}$$Hence we only need to check $G_{L,p}^{\frac{n-4}{n-2}}\left\vert Rc_{G_{L,p}^{\frac{4}{n-2}}g}\right\vert _{g}^{2}\in L^{1}\left( M\right) $ for a conformal metric. By the existence of conformal normal coordinate ([@LP]) we can assume $\exp _{p}$ preserve the volume near $p$. Let $x_{1},\cdots ,x_{n}$ be a normal coordinate at $p$, denote $r=\left\vert x\right\vert $, then (see [@LP])$$G_{L,p}=\frac{1}{4n\left( n-1\right) \omega _{n}}r^{2-n}\left( 1+O^{\left( 4\right) }\left( r\right) \right) . \label{eq2.4}$$As usual, we say $f=O^{\left( m\right) }\left( r^{\theta }\right) $ to mean $f$ is $C^{m}$ in the punctured neighborhood with $\partial _{i_{1}\cdots i_{k}}f=O\left( r^{\theta -k}\right) $ for $0\leq k\leq m$. By (\[eq2.4\]) and the transformation law$$\begin{aligned} Rc_{G_{L,p}^{\frac{4}{n-2}}g} &=&Rc-2D^{2}\log G_{L,p}+\frac{4}{n-2}d\log G_{L,p}\otimes d\log G_{L,p} \label{eq2.5} \\ &&-\left( \frac{2}{n-2}\Delta \log G_{L,p}+\frac{4}{n-2}\left\vert \nabla \log G_{L,p}\right\vert ^{2}\right) g, \notag\end{aligned}$$careful calculation shows$$\left\vert Rc_{G_{L,p}^{\frac{4}{n-2}}g}\right\vert _{g}=O\left( \frac{1}{r}\right) . \label{eq2.6}$$It follows that$$G_{L,p}^{\frac{n-4}{n-2}}\left\vert Rc_{G_{L,p}^{\frac{4}{n-2}}g}\right\vert _{g}^{2}=O\left( r^{2-n}\right)$$hence it belongs to $L^{1}\left( M\right) $. To continue, we observe that equation (\[eq2.1\]) is the same as$$\int_{M}G_{L,p}^{\frac{n-4}{n-2}}P\varphi d\mu =c_{n}\varphi \left( p\right) -\frac{n-4}{\left( n-2\right) ^{2}}\int_{M}G_{L,p}^{\frac{n-4}{n-2}}\left\vert Rc_{G_{L,p}^{\frac{4}{n-2}}g}\right\vert _{g}^{2}\varphi d\mu \label{eq2.7}$$for any $\varphi \in C^{\infty }\left( M\right) $. A similar check as before shows (\[eq2.7\]) is conformally invariant. Again we assume $\exp _{p}$ preserves the volume near $p$, then for $\delta >0$ small, it follows from (\[eq2.4\]) that$$PG_{L,p}^{\frac{n-4}{n-2}}=c_{n}\delta +\text{a }L^{1}\text{ function} \label{eq2.8}$$on $B_{\delta }\left( p\right) $ in distribution sense. On the other hand, on $M\backslash \left\{ p\right\} $ using (\[eq1.2\]) and (\[eq1.3\]) we have$$\begin{aligned} P_{g}\left( G_{L,p}^{\frac{n-4}{n-2}}\right) &=&G_{L,p}^{\frac{n+4}{n-2}}P_{G_{L,p}^{\frac{4}{n-2}}g}1 \label{eq2.9} \\ &=&\frac{n-4}{2}G_{L,p}^{\frac{n+4}{n-2}}Q_{G_{L,p}^{\frac{4}{n-2}}g} \notag \\ &=&-\frac{n-4}{\left( n-2\right) ^{2}}G_{L,p}^{\frac{n-4}{n-2}}\left\vert Rc_{G_{L,p}^{\frac{4}{n-2}}g}\right\vert _{g}^{2}. \notag\end{aligned}$$Here we have used the fact $R_{G_{L,p}^{\frac{4}{n-2}}g}=0$. Combine ([eq2.8]{}) and (\[eq2.9\]) we get (\[eq2.1\]). The case dimension $n>4$\[sec3\] ================================ Throughout this section we will assume $\left( M,g\right) $ is a smooth compact Riemannian manifold with dimension $n>4$. \[lem3.1\]Assume $n>4$, $Y\left( g\right) >0$, $u\in C^{\infty }\left( M\right) $ such that $u\geq 0$ and $Pu\geq 0$. If for some $p\in M$, $u\left( p\right) =0$, then $u\equiv 0$. By (\[eq2.1\]) we have$$\int_{M}G_{L,p}^{\frac{n-4}{n-2}}Pud\mu =-\frac{n-4}{\left( n-2\right) ^{2}}\int_{M}G_{L,p}^{\frac{n-4}{n-2}}\left\vert Rc_{G_{L,p}^{\frac{4}{n-2}}g}\right\vert _{g}^{2}ud\mu .$$Hence $Pu=0$ and $\left\vert Rc_{G_{L,p}^{\frac{4}{n-2}}g}\right\vert _{g}^{2}u=0$. If $u$ is not identically zero, then by unique continuation property we know $\left\{ u\neq 0\right\} $ is dense, hence $Rc_{G_{L,p}^{\frac{4}{n-2}}g}=0$. Since $\left( M\backslash \left\{ p\right\} ,G_{L,p}^{\frac{4}{n-2}}g\right) $ is asymptotically flat, it follows from relative volume comparison theorem that $\left( M\backslash \left\{ p\right\} ,G_{L,p}^{\frac{4}{n-2}}g\right) $ is isometric to the standard $\mathbb{R}^{n}$. In particular $\left( M,g\right) $ must be locally conformally flat and simply connected compact manifold, hence it is conformal to the standard $S^{n}$ by [@K]. But in this case we have $\ker P=0$, hence $u=0$, a contradiction. \[rmk3.1\]Indeed the same argument gives us the  following statement: If $n>4$, $Y\left( g\right) >0$, $u\in L^{1}\left( M\right) $ such that $u\geq 0$ and $Pu\geq 0$ in distribution sense, for some $p\in M$, $u$ is smooth near $p$ and $u\left( p\right) =0$, then $u\equiv 0$. A straightforward consequence of Lemma \[lem3.1\] is the following useful fact. \[prop3.1\]Assume $n>4$, $Y\left( g\right) >0$, $Q\geq 0$. If $u\in C^{\infty }\left( M\right) $ such that $Pu\geq 0$ and $u$ is not identically constant, then $u>0$. If the conclusion of the proposition is false, then $u\left( p\right) =\min_{M}u\leq 0$ for some $p$. Let $\lambda =-u\left( p\right) \geq 0$, then $u+\lambda \geq 0$, $u\left( p\right) +\lambda =0$ and$$P\left( u+\lambda \right) =Pu+\frac{n-4}{2}\lambda Q\geq 0.$$It follows from the Lemma \[lem3.1\] that $u+\lambda \equiv 0$. This contradicts with the fact $u$ is not a constant function. Proposition \[prop3.1\] helps us determine the null space of $P$ without information on the first eigenvalue. \[cor3.1\]Assume $n>4$, $Y\left( g\right) >0$, $Q\geq 0$, then$$\ker P\subset \left\{ \text{constant functions}\right\} .$$If in addition, $Q$ is not identically zero, then $\ker P=0$ i.e. $0$ is not an eigenvalue of $P$. Assume $Pu=0$. If $u$ is not a constant function, then it follows from Proposition \[prop3.1\] that $u>0$ and $-u>0$, a contradiction. Now we ready to prove half of Theorem \[thm1.1\]. \[lem3.2\]Assume $n>4$, $Y\left( g\right) >0$, $Q\geq 0$ and not identically zero, then $\ker P=0$, moreover the Green’s function $G_{P,p}\left( q\right) =G_{P}\left( p,q\right) >0$ for $p\neq q$. By Corollary \[cor3.1\], we know $\ker P=0$. Hence for any $f\in C^{\infty }\left( M\right) $, there exists a unique $u\in C^{\infty }\left( M\right) $ with $Pu=f$, moreover$$u\left( p\right) =\int_{M}G_{P,p}\left( q\right) f\left( q\right) d\mu \left( q\right) .$$If $f\geq 0$, it follows from the Proposition \[prop3.1\] that $u\geq 0$. Hence $G_{P,p}\geq 0$. If $G_{P,p}\left( q\right) =0$ for some $q$, since $PG_{P,p}=\delta _{p}\geq 0$ in distribution sense, we know from the Remark \[rmk3.1\] that $G_{P,p}\equiv 0$, a contradiction. Hence $G_{P,p}\left( q\right) >0$ for $p\neq q$. Next let us give the full argument of Theorem \[thm1.1\]. (1)$\Rightarrow $(2): This follows from Lemma \[lem3.2\], (\[eq1.5\]) and (\[eq1.6\]). (2)$\Rightarrow $(1): This follows from the classical Krein-Rutman theorem ([@L]). Since our case is relatively simple, we provide the argument here. Define the integral operator $T$ as$$Tf\left( p\right) =\int_{M}G_{P}\left( p,q\right) f\left( q\right) d\mu \left( q\right) .$$$T$ is the inverse operator of $P$. Let$$\alpha _{1}=\sup_{f\in L^{2}\left( M\right) \backslash \left\{ 0\right\} }\frac{\int_{M}Tf\cdot fd\mu }{\left\Vert f\right\Vert _{L^{2}}^{2}}>0.$$$\alpha _{1}$ is an eigenvalue of $T$. We note all eigenfunctions of $\alpha _{1}$ does not change sign. Indeed say $T\varphi =\alpha _{1}\varphi $, $\int_{M}\varphi ^{2}d\mu =1$, we have$$\int_{M}\left( \varphi _{+}^{2}+\varphi _{-}^{2}\right) d\mu =1.$$Here $\varphi _{+}=\max \left\{ \varphi ,0\right\} $, $\varphi _{-}=\max \left\{ -\varphi ,0\right\} $. Without losing of generality, we assume $\varphi _{+}$ is not identically zero. Then$$\begin{aligned} \alpha _{1} &=&\int_{M}T\varphi \cdot \varphi d\mu \\ &=&\int_{M}T\varphi _{+}\cdot \varphi _{+}d\mu +\int_{M}T\varphi _{-}\cdot \varphi _{-}d\mu -2\int_{M}T\varphi _{+}\cdot \varphi _{-}d\mu \\ &\leq &\alpha _{1}-2\int_{M}T\varphi _{+}\cdot \varphi _{-}d\mu .\end{aligned}$$Hence $\int_{M}T\varphi _{+}\cdot \varphi _{-}d\mu =0$. Since $T\varphi _{+}>0$, we see $\varphi _{-}=0$. Hence $\varphi \geq 0$. Because $T\varphi =\alpha _{1}\varphi $ we see $\varphi \in C^{\infty }\left( M\right) $ and $\varphi >0$. It follows that $\alpha _{1}$ must be a simple eigenvalue and $P\varphi =\alpha _{1}^{-1}\varphi $, hence$$Q_{\varphi ^{\frac{4}{n-4}}g}=\frac{2}{n-4}P_{\varphi ^{\frac{4}{n-4}}g}1=\frac{2}{n-4}\varphi ^{-\frac{n+4}{n-4}}P_{g}\varphi =\frac{2}{n-4}\alpha _{1}^{-1}\varphi ^{-\frac{8}{n-4}}>0.$$ (2)$\Rightarrow $(3): Assume $p_{0}\in M$ such that $G_{P,p_{0}}>0$. For $p\in M$, define$$\Theta \left( p\right) =\min_{q\in M\backslash \left\{ p\right\} }G_{P}\left( p,q\right)$$Then we have $\Theta \left( p_{0}\right) >0$. We note that $\Theta \left( p\right) \neq 0$ for any $p\in M$. Otherwise, say $\Theta \left( p\right) =0$, then $G_{P,p}\geq 0$ and $G_{P,p}\left( q\right) =0$ for some $q\neq p$. It follows from Remark \[rmk3.1\] that $G_{P,p}=const$, a contradiction. Since $M$ is connected we see $\Theta \left( p\right) >0$ for all $p$. In another word, $G_{P}\left( p,q\right) >0$ for $p\neq q$. \[rmk3.2\]In the proof of (2)$\Rightarrow $(1), a similar argument tells us if $\beta $ is an eigenvalue of $T$, $\beta \neq \alpha _{1}$, then $\left\vert \beta \right\vert <\alpha _{1}$. It follows that when $G_{P}$ is positive, the smallest **positive** eigenvalue of $P$ must be simple and its eigenfunction must be either strictly positive or strictly negative. Moreover if $\lambda $ is a negative eigenvalue of $P$, then $\left\vert \lambda \right\vert $ is strictly bigger than the smallest positive eigenvalue. By Lemma \[lem3.2\] we know $\ker P=0$ and $G_{P}>0$. From (\[eq2.1\]) we know$$P\left( G_{L,p}^{\frac{n-4}{n-2}}-c_{n}G_{P,p}\right) =-\frac{n-4}{\left( n-2\right) ^{2}}G_{L,p}^{\frac{n-4}{n-2}}\left\vert Rc_{G_{L,p}^{\frac{4}{n-2}}g}\right\vert _{g}^{2}\leq 0.$$Hence $G_{L,p}^{\frac{n-4}{n-2}}\leq c_{n}G_{P,p}$. If for some $q\neq p$, $G_{L,p}^{\frac{n-4}{n-2}}\left( q\right) =c_{n}G_{P,p}\left( q\right) $, then $Rc_{G_{L,p}^{\frac{4}{n-2}}g}=0$, hence $\left( M,g\right) $ is conformal diffeomorphic to the standard $S^{n}$ by the argument in the proof of Lemma \[lem3.1\]. $3$ dimensional case\[sec4\] ============================ Throughout this section we assume $\left( M,g\right) $ is a smooth compact Riemannian manifold of dimension $3$. If $Y\left( g\right) >0$, then for $p\in M$, (\[eq2.1\]) becomes$$P\left( G_{L,p}^{-1}\right) =-256\pi ^{2}\delta _{p}+G_{L,p}^{-1}\left\vert Rc_{G_{L,p}^{4}g}\right\vert _{g}^{2}. \label{eq4.1}$$Note here $G_{L,p}^{-1}\in H^{2}\left( M\right) $. \[lem4.1\]Assume $Y\left( g\right) >0$, $u\in H^{2}\left( M\right) $ such that $u\geq 0$, $Pu\leq 0$ in distribution sense. If for some $p\in M$, $u\left( p\right) =0$, then either $u\equiv 0$ or $\left( M,g\right) $ is conformal diffeomorphic to the standard $S^{3}$ and $u$ is a constant multiple of $G_{P,p}$. Using the fact $G_{L,p}^{-1}\in H^{2}\left( M\right) $, it follows from ([eq4.1]{}) that$$\int_{M}G_{L,p}^{-1}Pud\mu -\int_{M}G_{L,p}^{-1}\left\vert Rc_{G_{L,p}^{4}g}\right\vert _{g}^{2}ud\mu =0.$$Note here$$\int_{M}G_{L,p}^{-1}Pud\mu =E\left( G_{L,p}^{-1},u\right) .$$Hence $\int_{M}G_{L,p}^{-1}Pud\mu =0$ and $\int_{M}G_{L,p}^{-1}\left\vert Rc_{G_{L,p}^{4}g}\right\vert _{g}^{2}ud\mu =0$. Hence $\left\vert Rc_{G_{L,p}^{4}g}\right\vert _{g}^{2}u=0$. Since $Pu$ must be a measure, we see $Pu=const\cdot \delta _{p}$. In particular $u$ is smooth on $M\backslash \left\{ p\right\} $. If $u$ is not identically zero, it follows from unique continuation property that the set $\left\{ u\neq 0\right\} $ is dense, and hence $Rc_{G_{L,p}^{4}g}=0$. Same argument as in the proof of Lemma [lem3.1]{} tells us $\left( M,g\right) $ must be conformal diffeomorphic to the standard $S^{3}$, and hence $u=const\cdot G_{P,p}$. \[prop4.1\]Assume $Y\left( g\right) >0$, $Q\geq 0$. If $u\in C^{\infty }\left( M\right) $ such that $Pu\leq 0$ and $u$ is not identically constant, then $u>0$. If the conclusion of the proposition is false, then $u\left( p\right) =\min_{M}u\leq 0$ for some $p$. Let $\lambda =-u\left( p\right) \geq 0$, then $u+\lambda \geq 0$, $u\left( p\right) +\lambda =0$ and$$P\left( u+\lambda \right) =Pu-\lambda Q\leq 0.$$It follows from the Lemma \[lem4.1\] that $u+\lambda \equiv 0$. This contradicts with the fact $u$ is not a constant function. \[cor4.1\]Assume $Y\left( g\right) >0$, $Q\geq 0$, then $\ker P\subset \left\{ \text{constant functions}\right\} $. If in addition, $Q$ is not identically zero, then $\ker P=0$ i.e. $0$ is not an eigenvalue of $P$. Assume $Pu=0$. If $u$ is not a constant function, then it follows from Proposition \[prop4.1\] that $u>0$ and $-u>0$, a contradiction. \[lem4.2\]Assume $Y\left( g\right) >0$, $Q\geq 0$ and not identically zero, then $\ker P=0$, and the Green’s function $G_{P,p}\left( q\right) =G_{P}\left( p,q\right) <0$ for $p\neq q$. Moreover if for some $p\in M$, $G_{P,p}\left( p\right) =0$, then $\left( M,g\right) $ is conformal diffeomorphic to the standard $S^{3}$. By Corollary \[cor4.1\], we know $\ker P=0$. Hence for any $f\in C^{\infty }\left( M\right) $, there exists a unique $u\in C^{\infty }\left( M\right) $ with $Pu=f$, moreover$$u\left( p\right) =\int_{M}G_{P,p}\left( q\right) f\left( q\right) d\mu \left( q\right) .$$If $f\leq 0$, it follows from the Proposition \[prop4.1\] that $u\geq 0$. Hence $G_{P,p}\leq 0$. If $G_{P,p}\left( q\right) =0$ for some $q$, since $PG_{P,p}=\delta _{p}\geq 0$, it follows from Lemma \[lem4.1\] that $\left( M,g\right) $ must be conformal diffeomorphic to the standard $S^{3}$ and $G_{P,p}$ is a constant multiple of $G_{P,q}$, this implies $p=q$. Hence $G_{P,p}<0$ on $M\backslash \left\{ p\right\} $. Now we are ready to prove Theorem \[thm1.2\]. (1)$\Rightarrow $(2): This follows from Lemma \[lem4.2\] and (\[eq1.5\]), (\[eq1.6\]). (2)$\Rightarrow $(1): This follows from Krein-Rutman theorem, or one may use the argument in the proof of Theorem \[thm1.1\]. We also remark it follows that the largest **negative** eigenvalue of $P$ must be simple and its eigenfunction must be strictly positive or strictly negative. Moreover if $\lambda $ is a positive eigenvalue of $P$, then $\lambda $ is strictly bigger than the absolute value of the largest negative eigenvalue. (3)$\Rightarrow $(2): We can assume $\left( M,g\right) $ is not conformal diffeomorphic to the standard $S^{3}$. For any $p\in M$, we let$$\Theta \left( p\right) =\max_{M}G_{P,p}.$$Then it follows from Lemma \[lem4.1\] that $\Theta \left( p\right) \neq 0$ for any $p\in M$. Since $\Theta \left( p_{0}\right) <0$ for some $p_{0}\in M$, we see $\Theta \left( p\right) <0$ for all $p\in M$. In another word, $G_{P}<0$. With all the above analysis, we can easily deduce Proposition \[prop1.2\]. Under the assumption of Proposition \[prop1.2\], it follows from Lemma [lem4.2]{} that $\ker P=0$ and $G_{P}\left( p,q\right) <0$ for $p\neq q$. From (\[eq4.1\]) we see$$P\left( G_{L,p}^{-1}+256\pi ^{2}G_{P,p}\right) =G_{L,p}^{-1}\left\vert Rc_{G_{L,p}^{4}g}\right\vert _{g}^{2}\geq 0.$$Hence $G_{L,p}^{-1}+256\pi ^{2}G_{P,p}\leq 0$. If it achieves $0$ somewhere, then $Rc_{G_{L,p}^{4}g}=0$ and hence $\left( M,g\right) $ is conformal diffeomorphic to the standard $S^{3}$. At last we want to point out based on Proposition \[prop1.2\], using the arguments in [@HY3] we have the following statement: Let$$\mathcal{M}=\left\{ g:\begin{tabular}{l} $g$ is a smooth metric with $Y\left( g\right) >0$ and there exists \\ a positive smooth function $\rho $ such that $Q_{\rho ^{2}g}>0$\end{tabular}\right\}$$be endowed with $C^{\infty }$ topology. Then 1. For every $g\in \mathcal{M}$, there exists $\rho \in C^{\infty }\left( M\right) $, $\rho >0$ such that $Q_{\rho ^{2}g}=1$. Moreover as long as $\left( M,g\right) $ is not conformal diffeomorphic to the standard $S^{3}$, the set$$\left\{ \rho \in C^{\infty }\left( M\right) :\rho >0,Q_{\rho ^{2}g}=1\right\}$$is compact in $C^{\infty }$ topology. 2. Let $\mathcal{N}$ be a path connected component of $\mathcal{M}$. If there is a metric in $\mathcal{N}$ satisfying condition NN, then every metric in $\mathcal{N}$ satisfies condition NN. Hence as long as the metric is not conformal to the standard $S^{3}$, it satisfies condition $P$. As a consequence, for any metric in $\mathcal{N}$,$$\inf \left\{ E\left( u\right) \left\Vert u^{-1}\right\Vert _{L^{6}\left( M\right) }^{2}:u\in H^{2}\left( M\right) ,u>0\right\} >-\infty$$and is always achieved. We omit the details here. $4$ dimension case revisited\[sec5\] ==================================== Throughout this section we will assume $\left( M,g\right) $ is a smooth compact Riemannian manifold of dimension $4$. In this dimension the $Q$ curvature is written as$$Q=-\frac{1}{6}\Delta R-\frac{1}{2}\left\vert Rc\right\vert ^{2}+\frac{1}{6}R^{2}. \label{eq5.1}$$The Paneitz operator can be written as$$P\varphi =\Delta ^{2}\varphi +2\func{div}\left( Rc\left( \nabla \varphi ,e_{i}\right) e_{i}\right) -\frac{2}{3}\func{div}\left( R\nabla \varphi \right) . \label{eq5.2}$$Here $e_{1},e_{2},e_{3},e_{4}$ is a local orthonormal frame with respect to $g$. $P$ satisfies$$P_{e^{2w}g}\varphi =e^{-4w}P_{g}\varphi \label{eq5.3}$$for any smooth function $w$. The $Q$ curvature transforms as$$Q_{e^{2w}g}=e^{-4w}\left( P_{g}w+Q_{g}\right) . \label{eq5.4}$$In the spirit of Proposition \[prop2.1\], we have \[prop5.1\]Assume $\left( M,g\right) $ is a $4$ dimensional smooth compact Riemannian manifold with $Y\left( g\right) >0$, $p\in M$, then we have $\left\vert Rc_{G_{L,p}^{2}g}\right\vert _{g}^{2}\in L^{1}\left( M\right) $ and$$P\left( \log G_{L,p}\right) =16\pi ^{2}\delta _{p}-\frac{1}{2}\left\vert Rc_{G_{L,p}^{2}g}\right\vert _{g}^{2}-Q \label{eq5.5}$$in distribution sense. Here $G_{L,p}$ is the Green’s function of conformal Laplacian operator $L=-6\Delta +R$ with pole at $p$. If $\rho $ is a positive smooth function on $M$, $\widetilde{g}=\rho ^{2}g$, then$$\left\vert Rc_{G_{\widetilde{L},p}^{2}\widetilde{g}}\right\vert _{\widetilde{g}}^{2}d\widetilde{\mu }=\left\vert Rc_{G_{L,p}^{2}g}\right\vert _{g}^{2}d\mu . \label{eq5.6}$$Hence to show $\left\vert Rc_{G_{L,p}^{2}g}\right\vert _{g}^{2}\in L^{1}\left( M\right) $, in view of the existence of conformal normal coordinate, we can assume $\exp _{p}$ preserves volume near $p$. Let $x_{1},x_{2},x_{3},x_{4}$ be normal coordinate at $p$, $r=\left\vert x\right\vert $, then (see [@LP])$$G_{L,p}=\frac{1}{24\pi ^{2}}\frac{1}{r^{2}}\left( 1+O^{\left( 4\right) }\left( r^{2}\right) \right) . \label{eq5.7}$$Using$$\begin{aligned} Rc_{G_{L,p}^{2}g} &=&Rc-2D^{2}\log G_{L,p}+2d\log G_{L,p}\otimes d\log G_{L,p} \label{eq5.8} \\ &&-\left( \Delta \log G_{L,p}+2\left\vert \nabla \log G_{L,p}\right\vert ^{2}\right) g, \notag\end{aligned}$$we see $\left\vert Rc_{G_{L,p}^{2}g}\right\vert _{g}=O\left( 1\right) $, hence $\left\vert Rc_{G_{L,p}^{2}g}\right\vert _{g}^{2}\in L^{1}\left( M\right) $. On the other hand, (\[eq5.5\]) means$$\int_{M}\log G_{L,p}\cdot P\varphi d\mu =16\pi ^{2}\varphi \left( p\right) -\frac{1}{2}\int_{M}\left\vert Rc_{G_{L,p}^{2}g}\right\vert _{g}^{2}\varphi d\mu -\int_{M}Q\varphi d\mu . \label{eq5.9}$$Careful check shows (\[eq5.9\]) is conformally invariant. Hence we can assume $\exp _{p}$ preserves volume near $p$. It follows from (\[eq5.7\]) that on $B_{\delta }\left( p\right) $ for $\delta >0$ small,$$P\left( \log G_{L,p}\right) =16\pi ^{2}\delta _{p}+\text{a }L^{1}\text{ function} \label{eq5.10}$$in distribution sense. On $M\backslash \left\{ p\right\} $, we have$$P\left( \log G_{L,p}\right) =G_{L,p}^{4}Q_{G_{L,p}^{2}g}-Q=-\frac{1}{2}\left\vert Rc_{G_{L,p}^{2}g}\right\vert _{g}^{2}-Q. \label{eq5.11}$$(\[eq5.5\]) follows. By integrating (\[eq5.5\]) on $M$ and observing that$$\left\vert Rc_{G_{L,p}^{2}g}\right\vert _{g}^{2}d\mu _{g}=\left\vert Rc_{G_{L,p}^{2}g}\right\vert _{G_{L,p}^{2}g}^{2}d\mu _{G_{L,p}^{2}g}$$we immediately get \[cor5.1\]Assume $Y\left( g\right) >0$, then for any $p\in M$,$$\int_{M}Qd\mu +\frac{1}{2}\int_{M}\left\vert Rc_{G_{L,p}^{2}g}\right\vert _{G_{L,p}^{2}g}^{2}d\mu _{G_{L,p}^{2}g}=16\pi ^{2}. \label{eq5.12}$$In particular, $\int_{M}Qd\mu \leq 16\pi ^{2}$ and equality holds if and only if $\left( M,g\right) $ is conformal diffeomorphic to the standard $S^{4}$. Positive mass theorem for Paneitz operator revisited\[sec6\] ============================================================ Throughout this section we will assume $\left( M,g\right) $ is a smooth compact Riemannian manifold with dimension $n>4$. In [@HuR], for locally conformally flat manifold with $Y\left( g\right) >0$ and positive Green’s function $G_{P}$, a positive mass theorem for Paneitz operator was proved by a nice calculation. Note that this result plays similar role for $Q$ curvature equation as the classical positive mass theorem for the Yamabe problem ([@LP]). It was observed that similar calculation works for $n=5,6,7$ in [@GM] and for $n=3$ in [@HY3]. Since the case $n=3$ can be covered by Lemma \[lem4.1\], we concentrate on the case $n>4$. The main aim of this section is to show the positive mass theorem for Paneitz operator follows from the formula (\[eq2.1\]). \[lem6.1\]Assume $n>4$, $Y\left( g\right) >0$, $\ker P=0$. Let $x_{1},\cdots ,x_{n}$ be a coordinate near $p$ with $x_{i}\left( p\right) =0$, $r=\left\vert x\right\vert $. If either $M$ is conformally flat near $p$ or $n=5,6,7$, then$$c_{n}G_{P,p}-G_{L,p}^{\frac{n-4}{n-2}}=\text{const}+O^{\left( 4\right) }\left( r\right) . \label{eq6.1}$$Here $c_{n}$ is the constant given by (\[eq1.13\]). First we observe that if $\rho $ is a positive smooth function on $M$, $\widetilde{g}=\rho ^{\frac{4}{n-4}}g$, then$$c_{n}G_{\widetilde{P},p}-G_{\widetilde{L},p}^{\frac{n-4}{n-2}}=\rho \left( p\right) ^{-1}\rho ^{-1}\left( c_{n}G_{P,p}-G_{L,p}^{\frac{n-4}{n-2}}\right) . \label{eq6.2}$$Hence we only need to verify (\[eq6.1\]) for a conformal metric. For the case $M$ is conformally flat near $p$, by a conformal change of metric, we can assume $g$ is Euclidean near $p$. Then under the normal coordinate at $p$ we have$$G_{P,p}=\frac{1}{2n\left( n-2\right) \left( n-4\right) \omega _{n}}\left( r^{4-n}+A+O^{\left( 4\right) }\left( r\right) \right) . \label{eq6.3}$$Here $\omega _{n}$ is the volume of unit ball in $\mathbb{R}^{n}$ and $A$ is a constant. People usually call $A$ as the mass of Paneitz operator. The Green’s function of conformal Laplacian$$G_{L,p}=\frac{1}{4n\left( n-1\right) \omega _{n}}\left( r^{2-n}+O^{\left( 4\right) }\left( r^{-1}\right) \right) . \label{eq6.4}$$It is worth pointing out one has better estimate for the Green’s function than the one in (\[eq6.3\]) and (\[eq6.4\]), but the formula we wrote above also works for $n=5,6,7$ without locally conformally flat assumption. More precisely, for $n=5,6,7$, under the conformal normal coordinate, ([eq6.3]{}) and (\[eq6.4\]) remain true (see [@GM; @LP]). It follows that$$c_{n}G_{P,p}-G_{L,p}^{\frac{n-4}{n-2}}=\left( 4n\left( n-1\right) \omega _{n}\right) ^{-\frac{n-4}{n-2}}A+O^{\left( 4\right) }\left( r\right) . \label{eq6.5}$$ To continue, we note that under the assumption of Lemma \[lem6.1\], by (\[eq2.1\]) we have$$P\left( c_{n}G_{P,p}-G_{L,p}^{\frac{n-4}{n-2}}\right) =\frac{n-4}{\left( n-2\right) ^{2}}G_{L,p}^{\frac{n-4}{n-2}}\left\vert Rc_{G_{L,p}^{\frac{4}{n-2}}g}\right\vert _{g}^{2}, \label{eq6.6}$$hence$$G_{L,p}^{\frac{n-4}{n-2}}\left\vert Rc_{G_{L,p}^{\frac{4}{n-2}}g}\right\vert _{g}^{2}=O\left( r^{-3}\right) \label{eq6.7}$$and$$\left( 4n\left( n-1\right) \omega _{n}\right) ^{-\frac{n-4}{n-2}}A=\frac{n-4}{\left( n-2\right) ^{2}}\int_{M}G_{P,p}G_{L,p}^{\frac{n-4}{n-2}}\left\vert Rc_{G_{L,p}^{\frac{4}{n-2}}g}\right\vert _{g}^{2}d\mu . \label{eq6.8}$$ If in addition we know the Green’s function $G_{P,p}>0$, then it follows from (\[eq6.8\]) that $A\geq 0$, moreover $A=0$ if and only if $\left( M,g\right) $ is conformal equivalent to the standard $S^{n}$. This proves the positive mass theorem for Paneitz operator. [HeR1]{} T. Branson. *Differential operators canonically associated to a conformal structure*. Math. Scand. **57** (1985), no. 2, 293–345. S.-Y. A. Chang, M. J. Gursky and P. C. Yang. *An equation of Monge-Ampere type in conformal geometry, and four-manifolds of positive Ricci curvature*. Ann. of Math. (2) **155** (2002), 709–787. Z. Djadli, E. Hebey and M. Ledoux. *Paneitz-type operators and applications*. Duke Math. Jour. **104** (2000), 129–169. M. J. Gursky. *The principal eigenvalue of a conformally invariant differential operator.* Comm. Math. Phys. **207** (1999), no. 1, 131–143. M. J. Gursky and A. Malchiodi. A strong maximum principle for the Paneitz operator and a nonlocal flow for the $Q$ curvature. Preprint (2014). F. B. Hang and P. Yang. *The Sobolev inequality for Paneitz operator on three manifolds.* Calculus of Variations and PDE. **21** (2004), 57–83. F. B. Hang and P. Yang. Paneitz operator for metrics near $S^{3}$. Preprint (2014). F. B. Hang and P. Yang. $Q$* curvature on a class of* $3$* manifolds.* Comm Pure Appl Math, to appear. F. B. Hang and P. Yang. $Q$ curvature on a class of manifolds with dimension at least $5$. Preprint (2014). E. Hebey and F. Robert.* Compactness and global estimates for the geometric Paneitz equation in high dimensions*. Electron Res Ann Amer Math Soc. **10** (2004), 135–141. E. Hebey and F. Robert. *Asymptotic analysis for fourth order Paneitz equations with critical growth*. Adv Calc Var. **4** (2011), no. 3, 229–275. E. Humbert and S. Raulot. *Positive mass theorem for the Paneitz-Branson operator.* Calculus of Variations and PDE. **36** (2009), 525–531. N. H. Kuiper. *On conformally-flat spaces in the large.* Ann of Math*.* **50** (1949), 916–924. P. Lax. Functional analysis. John Wiley & Sons, Inc. 2002. J. M. Lee and T. H. Parker. *The Yamabe problem*. Bull AMS. **17** (1987), no. 1, 37–91. S. Paneitz. A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds. Preprint (1983). J. Qing and D. Raske. *On positive solutions to semilinear conformally invariant equations on locally conformally flat manifolds*. Int Math Res Not. Art. id 94172 (2006).

--- author: - | [**Cristian S. Calude**]{}$^{1}$,  [**Elena Calude**]{}$^{2}$, [**Solomon Marcus**]{}$^{3}$\ $^{1}$University of Auckland, New Zealand\ [[email protected]]{}\ $^{2}$Massey University at Albany, New Zealand\ [[email protected]]{}\ $^{3}$Romanian Academy, Mathematics, Bucharest, Romania\ [[email protected]]{} title: '**Passages of Proof**' --- To Prove or Not to Prove–That Is the Question! ============================================== \ \ \ \ In this paper we propose a new perspective on the evolution and history of the idea of mathematical proof. Proofs will be studied at three levels: syntactical, semantical and pragmatical. Computer-assisted proofs will be give a special attention. Finally, in a highly speculative part, we will anticipate the evolution of proofs under the assumption that the quantum computer will materialize. We will argue that there is little ‘intrinsic’ difference between traditional and ‘unconventional’ types of proofs. Mathematical Proofs: An Evolution in Eight Stages ================================================= [*Reason*]{} and [*experiment*]{} are two ways to acquire knowledge. For a long time mathematical proofs required only reason; this might be no longer true. We can distinguish eight periods in the evolution of the idea of mathematical proof. The first period was that of pre–Greek mathematics, for instance the Babylonian one, dominated by observation, intuition and experience. The second period was started by Greeks such as Pythagoras and is characterized by the discovery of deductive mathematics, based on theorems. Pythagoras proved his theorem, but the respective statement was discovered much earlier. Deductive mathematics saw a culminating moment in Euclid’s geometry. The importance of abstract reasoning to ancient Greeks can be illustrated by citing Aristophanes’s comedy [*The Birds*]{} which includes a cameo appearance of Meton, the astronomer, who claims that he had squared the circle. Knuth [@knuth] rhetorically asked: “Where else on earth would a playwright think of including such a scene?" Examples would have been difficult to produce in 1985, but today the situation has changed. Take for example, the movie [*Pi*]{} written and directed by Darren Aronofsky Starring Sean Gullette or Auburn’s play [*Proof*]{} [@auburn] originally produced by the Manhattan Theatre Club on 23rd May 2000. In a more careful description, we observe that deductive mathematics starts with Thales and Pythagoras, while the axiomatic approach begins with Eudoxus and especially with Aristotle, who shows that a demonstrative science should be based on some non–provable principles, some common to all sciences, others specific to some of them. Aristotle also used the expression “common notions" for axioms (one of them being the famous principle of non–contradiction). Deductive thinking and axiomatic thinking are combined in Euclid’s [*Elements*]{} (who uses, like Aristotle, “common notions" for “axioms"). The great novelty brought by Euclid is the fact that, for the first time, mathematical proofs (and, through them, science in general) are built on a long distance perspective, in a step by step procedure, where you have to look permanently to previous steps and to fix your aim far away to the hypothetical subsequent steps. Euclid became, for about two thousands years, a term of reference for the axiomatic–deductive thinking, being considered the highest standard of rigour. Archimedes, in his treatise on static equilibrium, the physicists of the Middle Age (such as Jordanus de Nemore, in [*Liber de ratione ponderis*]{}, in the 13th century), B. Spinoza in [*Ethics*]{} (1677) and I. Newton in [*Principia*]{} (1687) follow Euclid’s pattern. This tradition is continued in many more recent works, not only in the field of mathematics, but also in physics, computer science, biology, linguistics, etc. However, some shortcomings of Euclid’s approach were obstacles for the development of mathematical rigour. One of them was the fact that, until Galilei, the mathematical language was essentially the ordinary language, dominated by imprecision resulting from its predominantly spontaneous use, where emotional factors and lack of care have an impact. In order to diminish this imprecision and make the mathematical language capable to face the increasing need of precision and rigour, the ordinary language had to be supplemented by an artificial component of symbols, formulas and equations: with Galilei, Descartes, Newton and Leibniz, the mathematical language became more and more a mixed language, characterized by a balance between its natural and artificial components. In this way, it was possible to pack in a convenient, heuristic way, previous concepts and results, and to refer to them in the subsequent development of mathematical inferences. To give only one example, one can imagine how difficult was to express the $n$th power of a binomial expression in the absence of a symbolic representation, i.e., using only words of the ordinary language. This was the third step in the development of mathematical proofs. The fourth step is associated with the so–called epsilon rigour, so important in mathematical analysis; it occurred in the 19th century and it is associated with names such as A. Cauchy and K. Weierstrass. So, it became possible to renounce the predominantly intuitive approach via the infinitely small quantities of various orders, under the form of functions converging in the limit to zero (not to be confused with the Leibnizian infinitely small, elucidated in the second half of the 20th century, by A. Robinson’s non–standard analysis). The epsilon rigour brought by the fourth step created the possibility to cope in a more accurate manner with processes with infinitely many steps such as limit, continuity, differentiability and integrability. The fifth period begun with the end of the 19th century, when Aristotle’s logic, underlining mathematical proofs for two thousands years, entered a crisis with the challenge of the principle of non–contradiction. This crisis was already announced by the discovery of non–Euclidean geometries, in the middle of the 19th century. Various therapies were proposed to free the mathematical proof of the dangerous effects of paradoxes (Russell–Whitehead, Hilbert, Brouwer, etc). This period covers the first three decades of the 20th century and is dominated by the optimistic view stating the possibility to arrange the whole mathematics as a formal system and to decide for any possible statement whether it is true or false. However, even during this period mathematicians were divided with respect to the acceptance of non–effective (non–constructive) entities and proofs (for example, Brouwer’s intuitionism rejects the principle of excluded middle in the case of infinite sets). Intuitionism was a signal for the further development of constructive mathematics, culminating with the algorithmic approach leading to computer science. The sixth period begins with Gödel’s incompleteness theorem (1931), for many meaning the unavoidable failure of any attempt to formalise the whole of mathematics. Aristotle’s requirement of complete lack of contradiction can be satisfied only by paying the price of incompleteness of the working formal system. Chaitin (1975) has continued this trend of results by proving that from $N$ bits of axioms one cannot prove that a program is the smallest possible if it is more than $N$ bits long; he suggested that complexity is a source of incompleteness because a formal system can capture only a tiny amount of the huge information contained in the world of mathematical truth. This principle has been proved in Calude and J" urgensen [@cj]. Hence, incompleteness is natural and inevitable rather then mysterious and esoteric. This raises the natural question (see Chaitin [@ch02]): [*How come that in spite of incompleteness, mathematicians are making so much progress?*]{} The seventh period belongs to the second half of the 20th century, when algorithmic proofs become acceptable only when their complexities were not too high. Constructiveness is no longer enough, a reasonable high complexity (cost) is mandatory. We are now living in this period. An important event of this period was the 1976 proof of the Four–Colour Problem (4CP): it marked the reconciliation of empirical–experimental mathematics with deductive mathematics, realized by the use of computer programs as pieces of a mathematical proof. Computer refers to classical von Neumann computer. At the horizon we can see the (now hypothetical) quantum computer which may modify radically the relation between empirical–experimental mathematics and deductive mathematics … With the eighth stage, proofs are no longer exclusively based on logic and deduction, but also empirical and experimental. On the other hand, in the light of the important changes brought by authors like Hilbert, already at the beginning of the 20th century, primitive terms became to have an explicit status, axioms show their dependence on physical factors and the axiomatic–deductive method displays its ludic dimension, being a play with abstract symbols. Hilbert axiomatization of geometry is essentially different from Euclid’s geometry and this fact is well pointed out by Dijkstra in [@dijkstra] where he considers that, by directing their attention towards provability, formalists circumvented the vague metaphysical notion of “truth". Dijkstra qualifies as “philosophical pollution" the mentality which pushed Gauss not to publish his ideas related to non–Euclidean geometry. Contrary to appearances, believes Dijkstra, Euclidean geometry is not a prototype of a deductive system, because it is based to a large extent on pictures (so–called definitions of points and lines, for instance) motivated by the need of geometric intuition. For Dijkstra, the claim that the Euclidean geometry is a model of deductive thinking, is a big lie. As a matter of fact, the shortcomings to which Dijkstra refers were well-known, as can be seen in Morris Kline’s book [@kline], pp. 86–88. In contemporary mathematics we are facing a change of perspective, a change of scenario, replacing the old itinerary definition–theorem–proof by another one (see, for instance, W. Thurston), based on ideas, examples and motivations. The interesting fact is that the gap created between proof and intuition by Hilbert prepared the way for a new marriage between deduction and experiment, made possible by the computational revolution, as it was shown by the latest step in the evolution of proofs. Proofs, Theorems and Truths =========================== \ What is a mathematical proof? At a first glance the answer seems obvious: a proof is a series of logical steps based on some axioms and deduction rules which reaches a desired conclusion. Every step in a proof can be checked for correctness by examining it to ensure that it is logically sound. In David Hilbert’s words: “The rules should be so clear, that if somebody gives you what they claim is a proof, there is a mechanical procedure that will check whether the proof is correct or not, whether it obeys the rules or not." By making sure that every step is correct, one can tell once and for all whether a theorem has been proved. Simple! A moment of reflection shows that the problem may not be so simple. For example, what if the “agent" (human or computer) checking a proof for correctness makes a mistake (agents are fallible)? Obviously, another agent has to check that the agent doing the checking did not make any mistakes. Some other agent will need to check that agent, and so on. Eventually one runs out of agents who could check the proof and, in principle, they could all have made a mistake! The mistake is the neighbour and the brother of proof, it is both an opponent and a stimulus. An interesting analysis, responding to Joseph L. Doob’s challenge, of various possible mistakes in the proof of the 4CT can be found in the work of Schmidt [@schmidt]. In 1976, Kenneth Appel and Wolfgang Haken proved the 4CT. They used some of Alfred Kempe’s ideas, but avoided his mistake.[^1] They showed that if there is a map which needs five colours, then a contradiction follows. If there are several five–colour maps, they have chosen one with the smallest number of countries and proved that this map must contain one of 1,936 possible configurations; they also proved that every one of these possible configurations can be reduced into a smaller configuration which also needs five colours. This is a contradiction because we assumed that we already started with the smallest five–colour map. The reduction step, i.e., the step in which one shows that the 1,936 configurations could be reduced was actually done by brute force computer search through every configuration. No human being could ever actually read the entire proof to check its correctness. For Ron Graham, “The real question is this: If no human being can ever hope to check a proof, is it really a proof?" In 1996 Robertson, Sanders, Seymour and Thomas [@rsst] offered a simpler proof involving only 633 configurations. The paper [@rsst] concludes with the following interesting comment (p. 24): “We should mention that both our programs use only integer arithmetic, and so we need not be concerned with round–off errors and similar dangers of floating point arithmetic. However, an argument can be made that our “proof“ is not a proof in the traditional sense, because it contains steps that can never be verified by humans. In particular, we have not proved the correctness of the compiler we compiled our programs on, nor have we proved the infallibility of the hardware we ran our programs on. These have to be taken on faith, and are conceivably a source of error. However, from a practical point of view, the chance of a computer error that appears consistently in exactly the same way on all runs of our programs on all the compilers under all the operating systems that our programs run on is infinitesimally small compared to the chance of a human error during the same amount of case–checking. Apart from this hypothetical possibility of a computer consistently giving an incorrect answer, the rest of our proof can be verified in the same way as traditional mathematical proofs. We concede, however, that [*verifying a computer program is much more difficult than checking a mathematical proof of the same length*]{}.”[^2] According to Vladimir Arnold, “Proofs are to mathematics what spelling (or even calligraphy) is to poetry. Mathematical works do consist of proofs, just as poems do consist of characters." These analogies point out both the necessity and the insufficiency of proofs in the development of mathematics. Indeed, spelling is the way poetry takes expression, but it is equally the tool used by the common everyday language, in most cases devoid of any poetic effect. What should be added to spelling in order to get a piece of poetry remains a mystery. A poem consists of characters, but it is much more than a meaningful concatenation of characters. Mathematics cannot be conceived in the absence of proofs. According to Foiaş [@foias], “the theorem is the brick of mathematics". Obviously, “proof" and “theorem" go together; the object of a proof is to reach a theorem, while theorems are validated by proofs. Theorems are, for the construction of mathematics, what bricks are for the construction of a building. A building is an articulation of bricks and, analogically, a mathematical work is an articulation of theorems. Motivated by a similar view, Jean Dieudonné [@dieu] defines a mathematician as a person who has proved at least one theorem. In contrast, Arnold’s analogies point out the fact that mathematics is much more than a chain of theorems and proofs, so implicitly a mathematician should be much more than the author of a theorem. Probably the best example is offered by Bernhard Riemann whose lasting fame does not come (in the first instance) from his theorems or proofs, but from his conjectures, definitions, concepts and examples (see for example, the discussion in Hersh [@hersh], pp. 50–51). Srinivasa Ramanujan is another famous example of a mathematician who produced more results than proofs. What the mathematical community seems to value most are “ideas". “The most respected mathematicians are those with strong ‘intuition’ " (Harris [@harris], p. 19). Mathematical Proofs: The Syntactic Dimension ============================================ Of course, the first thing to be discussed is Gödel’s incompleteness theorem (GIT) which says that [*every formal system which is (1) finitely specified, (2) rich enough to include the arithmetic, and (3) consistent, is incomplete.*]{} That is, there exists an arithmetical statement which (A) can be expressed in the formal system, (B) is true, but (C) is unprovable within the formal system. All conditions are necessary. Condition (1) says that there is an algorithm listing all axioms and inference rules (which could be infinite). Taking as axioms all true arithmetical statements will not do, as this set is not finitely listable. But what does it mean to be a “true arithmetical statement"? It is a statement about non-negative integers which cannot be invalidated by finding any combination of non-negative integers that contradicts it. In Alain Connes terminology (see [@cls], p. 6), a true arithmetical statement is a “primordial mathematical reality". Condition (2) says that the formal system has all the symbols and axioms used in arithmetic, the symbols for $0$ (zero), $S$ (successor), $+$ (plus), $\times$ (times), $=$ (equality) and the axioms making them work (as for example, $x +S(y) = S(x+y)$). Condition (2) cannot be satisfied if you do not have individual terms for $0, 1, 2, \dots $; for example, Tarski proved that Euclidean geometry, which refers to points, circles and lines, is complete. Finally (3) means that the formal system is free of contradictions. The essence of GIT is to distinguish between truth and provability. A closer real life analogy is the distinction between truths and judicial decisions, between what is true and what can be proved in court.[^3] How large is the set of true and unprovable statements? If we fix a formal system satisfying all three conditions in GIT, then the set of true and unprovable statements is topologically “large" (constructively, a set of second Baire category, and in some cases even “larger"), cf. Calude, J" urgensen, Zimand [@cjz]; because theorems proven in such a system have bounded complexity, the probability that an $n$-bit statement is provable tends to zero when $n$ tends to infinity (see Calude and J" urgensen [@cj]). There is a variety of reactions in interpreting GIT, ranging from pessimism to optimism or simple dismissal (as irrelevant for the practice of mathematics). For pessimists, this result can be interpreted as the final, definite failure of any attempt to formalise the whole of mathematics. For example, Hermann Weyl acknowledged that GIT has exercised a “constant drain on the enthusiasm" with which he has engaged himself in mathematics and for Stanley Jaki, GIT is a fundamental barrier in understanding the Universe. In contrast, scientists like Freeman Dyson acknowledge the limit placed by GIT on our ability to discover the truth in mathematics, but interpret this in an optimistic way, as a guarantee that mathematics will go on forever (see Barrow [@barrow], pp. 218–221). In modern times a penetrating insight into the incompleteness phenomenon has been obtained by an information–theoretic analysis pioneered by Chaitin in [@ch75]. Striking results have been obtained by studying the Chaitin’s Omega Number, $\Omega$, the halting probability of a self-delimiting universal Turing machine. This number is not only uncomputable, but also (algorithmically) random. Chaitin has proven the following important theorem: [*If $ZFC$ (Zermelo set theory with the Axiom of Choice) is arithmetically sound, that is, any theorem of arithmetic proved by $ZFC$ is *true*, then, $ZFC$ can determine the value of only finitely many bits of $\Omega$, and one can give a bound on the number of bits of $\Omega$ which $ZFC$ can determine.*]{} Robert Solovay [@solovay2k] (see more in [@cc; @crisomega; @cris; @cris2002]) has constructed [*a self-delimiting universal Turing machine such that $ZFC$, if arithmetically sound, cannot determine any single bit of its halting probability*]{} ($\Omega$). Re–phrased, the most powerful formal axiomatic system is powerless when dealing with the questions of the form “is the $m$th bit of $\Omega$ 0?" or “is the $m$th bit of $\Omega$ 1?". Chaitin has constructed an exponential Diophantine equation $F(t; x_1, \ldots ,x_n)=0$ with the following property: the infinite binary sequence whose $m$th term is 0 or 1 depending whether the equation $F(m; x_1, \ldots ,x_n)=0$ has finitely or infinitely many solutions is exactly the digits of $\Omega$, hence it is random; its infinite amount of information is algorithmically incompressible. The importance of exponential Diophantine equations comes from the fact that most problems in mathematics can be formulated in terms of these type of equations; Riemann’s Conjecture is one such example. Manin [@manin1], p. 158, noticed that “The epistemologically important point is the discovery that randomness can be defined without any recourse to physical reality … in such a way that the necessity to make an infinite search to solve a parametric series of problems leads to the technically random answers. Some people find it difficult to imagine that a rigidly determined discipline like elementary arithmetic may produce such phenomena". Last but not least, is the truth achieved through a formal proof the ultimate expression of knowledge? Many (mathematicians) will give a positive answer, but perhaps not all. For the 13th century Oxford philosopher Roger Bacon, “Argument reaches a conclusion and compels us to admit it, but it neither makes us certain nor so it annihilates doubt that the mind rests calm in the intuition of truth, unless it finds this certitude by way of experience." More recently, I. J. Schoenberg[^4] is cited by Epstein ([@hahn]) as saying that Edmund Landau kept in his desk drawer for years a manuscript proving what is now called the two constants theorem: he had the complete proof but could not believe it until his intuition was ready to accept it. Then he published it. A “proof is only one step in the direction of confidence" argued De Millo, Lipton and Perlis in a classical paper on proofs, theorems and programs [@demillo]. Written in the same spirit is Don Knuth’s warning: “Beware of bugs in the above code: I have only proved it correct, not tried it." Mathematical Proofs: The Semantic Dimension =========================================== \ The above quotation turned slogan as “more rigour, less meaning", or better still, “less rigour, more meaning" (Chaitin [@gregpccris]) points out the necessity to distinguish between the syntactic and the semantic aspects of proofs. Should proofs belong exclusively to logic, according to the tradition started by Greeks such as Pythagoras and Euclid? Or should they also be accepted as a cocktail of logical and empirical–experimental arguments, as in the proof of the 4CT (1976)? Mathematicians are now divided into those giving an affirmative answer to the first question and implicitly a negative answer to the second question and those giving a negative answer to the first question and an affirmative one to the second question. Computationally oriented mathematicians usually belong to the second category, while many other mathematicians (as, for instance, the Fields medalist William Thurston) belong to the first, so for them, the 4CT is not yet proved! Meaning is a key distinction. For mathematicians such as René Thom, Daniel Cohen and William Thurston, correctness by itself does not validate a proof; it is also necessary to “understand" it. “The mission of mathematics is understanding" says Cohen. Paul Halmos has also insisted on the “conceptual understanding". For him a “good" proof of a theorem is one that sheds light on why it is true. It is just the process of understanding which is in question with proofs like that given to the 4CT. Referring to the proof of the 4CT, Halmos says: “I do not find it easy to say what we learned from all that. … The present proof relies in effect on an Oracle, and I say down with Oracles! They are not mathematics!" In contrast with Halmos, who hopes that “100 years from now the map theorem will be … an exercise in a first–year graduate course, provable in a couple of pages by means of appropriate concepts, which will be completely familiar by then" (see [@hersh], p. 54), R. Hersh thought that the problem itself might be responsible for the way it was solved: he is cited by saying dejectedly “So it just goes to show, it wasn’t a good problem after all" (see [@casti] p. 73). We will return later to these issues. For the moment we make the following two observations. A) : Not only the hybrid proofs obtained as a combination of logical and empirical–experimental arguments might be hard/impossible to be understood in their “globality"; this happens also for some pure deductive proofs. An example is the proof of the classification of finite simple groups called by Danny Gorenstein the “Thirty Years War" (for the classification battles were fought mostly in the decades 1950–1980), a work which comprises about 10,000–15,000 pages scattered in 500 journal articles by some 100 authors.[^5] According to Knuth [@knuth] p. 18, “… program–writing is substantially more demanding than book–writing". “Why is this so? I think the main reason is that a larger attention span is needed when working on a large computer program than when doing other intellectual tasks. … Another reason is … that programming demands a significantly higher standard of accuracy. Things don’t simply have to make sense to another human being, they must make sense to a computer." Knuth compares his TeX compiler (a document of about 500 pages) with Feit and Thompson [@ft] theorem that all simple groups of odd order are cyclic. He lucidly argues that the program might not incorporate as much creativity and “daring" as the proof of the theorem, but they come even when compared on depth of details, length and paradigms involved. What distinguishes the program from the proof is the “verification": convincing a couple of (human) experts that the proof [*works in principle*]{} seems to be easier than making sure that the program [*really works*]{}. A demonstration that [*there exists a way to compile TeX*]{} is not enough! Another example, which will be discussed later in this section, is the proof of Fermat’s Last Theorem (FLT). B) : Without diminishing in any way the “understanding" component of mathematics we note that the idea of distinguishing between “good" and “bad" proofs on the light they shed on their own truth seems to be, at least to some extent, relative and subjective. Thom’s slogan ‘more rigour, less meaning’ was the main point in his controversy with Jean Dieudonné (as a representative of the Bourbaki group). Taking rigour as something that can be acquired only at the expense of meaning and conversely, taking meaning as something that can be obtained only at the expense of rigour, we oblige mathematical proof to have the status of what was called in physics a “conjugate (complimentary) pair", i.e., a couple of requirements, each of them being satisfied only at the expense of the other (see [@marcus]). Famous prototypes of conjugate pairs are (position, momentum) discovered by W. Heisenberg in quantum mechanics and (consistency, completeness) discovered by K. G" odel in logic. But similar warnings come from other directions. According to Einstein (see, for instance, [@rosen] p. 195), “in so far as the propositions of mathematics are certain, they do not refer to reality, and in so far as they refer to reality, they are not certain", hence (certainty, reality) is a conjugate pair. Obviously, reality is here understood as an empirical entity, hence mixed with all kinds of imprecision, ranging from obscurity and disorder to randomness, ambiguity and fuzziness [@marcus1]. Pythagoras’ theorem is certain, but its most empirical tests will fail. There are some genuine obstacles in our attempts to eliminate or at least to diminish the action of various sources of imprecision. Einstein implicitly calls our attention on one of them. Proof, to the extent to which it wants to be rigorous, to give us the feeling of certainty, should be mathematical; but satisfying this condition, means failing to reach reality. In other words, the price we have to pay to obtain proofs giving us the feeling of total confidence is to renounce to be directly connected to reality. There is a genuine tension between certainty and reality, they form a conjugate pair, which is the equivalent of what in the field of humanities is an oxymoronic pair. However, there is an essential difference between Gödel’s conjugate pair (consistency, completeness) and Einstein’s conjugate pair (certainty, reality). While consistency and completeness are binary logical predicates, certainty and reality are a matter of degree, exactly like the terms occurring in Thom’s conjugate pair: rigour and meaning. In last two situations there is room for manipulation and compromise. Near to the above conjugate pairs is a third one: (rigour, reality), attributed to Socrates (see [@renyi]). A price we have to pay in order to reach rigour is the replacement of the real world by a fictional one. There is no point and no line in the real world, if we take them according to their definitions in Euclid’s [*Elements*]{}. Such entities belong to a fictional/virtual universe, in the same way in which the characters of a theatrical play are purely conventional, they don’t exist as real persons. The rules of deduction used in a mathematical proof belong to a game in the style they are described in the scenario of a Hilbert formal system, which is, as a matter of fact, a machine producing demonstrative texts. A convention underlines the production of theorems and again a convention is accepted in a theatrical play. In the first case, the acceptance of the convention is required from both the author of the proof and its readers; in the second case all people involved, the author, the actors and spectators, have to agree the proposed convention. Since many proofs, if not most of them, are components of a modeling process, we have to add the unavoidable error of approximation involved in any cognitive model. The model should satisfy opposite requirements, to be as near as possible to the phenomenon modelled, in order to be relevant; to be as far as possible from the respective phenomenon, in order to useful, to make possible the existence of at least one method or tool that can be applied to the model, but not to the original (see [@marcus2]). Theorems are discovered, models are invented. Their interaction leads to many problems of adequacy, relevance and correctness, i.e., of syntactic, semantic and pragmatic nature. In the light of the situations pointed out above, we can understand some ironical comments about what a mathematician could be. It is somebody who can prove theorems, as Dieudonné claimed. But what kind of problems are solved in this way? “Any problem you want, …except those you need", said an engineer, disenchanted by his collaboration with a mathematician. Again, what is a mathematician? “It is a guy capable to give, after a long reflection, a precise, but useless answer", said another mathematician with a deep feeling of self irony. Remember the famous reflection by Goethe: “Mathematicians are like French people, they take your question, they translate it in their language and you no longer recognize it". But things are controversial even when they concern syntactic correctness. In this respect, we should distinguish two types of syntactic mistakes: benign and malign. Benign mistakes have only a local, not global effect: they can be always corrected. Malign mistakes, on the contrary, contaminate the whole approach and invalidate the claim formulated by the theorem. When various authors (including the famous probabilist J. L. Doob, see [@schmidt]) found some mistakes in the proof of the 4CT, the authors of the proof succeeded in showing that all of them were benign and more than this, [*any other possible mistake, not yet discovered, should be benign*]{}. How can we accept such arguments, when the process of global understanding of the respective proof is in question? The problem remains open. A convenient, but fragile, solution is to accept Thom’s pragmatic proposal: a theorem is validated if it has been accepted by a general agreement[^6] of the mathematical community (see [@thom1; @thom2]). The problems raised by the 4CT were discussed by many authors, starting with Tymoczko [@tymoczko] and Swart [@swart] (more recent publications are D. MacKenzie [@mc], J. Casti [@casti], A.S. Calude [@andreea]). Swart proposed the introduction of a new entity called [*agnogram*]{}, which is “a theorem–like statement that we have verified as best we could, but whose truth is not known with the kind of assurance we attach to theorems and about which we must thus remain, to some extent, agnostic." There is however the risk to give the status of agnogram to any property depending on a natural number $n$ and verified only for a large, but finite number of values of $n$. This fact would be in conflict with Swart’s desire to consider an agnogram less than a theorem, but more than a conjecture. Obviously, the 4CT is for Swart an agnogram, not a theorem. What is missing from an agnogram to be a theorem? A theorem is a statement which could be checked individually by a mathematician and confirmed also individually by at least two or three more mathematicians, each of them working independently. But already here we can observe the weakness of the criterion: how many mathematicians are to check individually and independently the status of an agnogram to give it the status of theorem? The seriousness of this objection can be appreciated by examining the case of Andrew Wiles’ proof of FLT—a challenge to mathematics since 1637 when Pierre de Fermat wrote it into the margin of one of his books. The proof is extremely intricate, quite long (over 100 printed pages[^7]), and only a handful of people in the entire world can claim to understand it.[^8] To the rest of us, it is utterly incomprehensible, and yet we all feel entitled to say that “the FLT has been proved". On which grounds? We say so because [*we believe the experts*]{} and [*we cannot tell for ourselves*]{}. Let us also note that in the first instance the original 1993 proof seemed accepted, then a gap was found, and finally it took Wiles and Richard Taylor another year to fix the error.[^9] According to Hunt [@hunt], “In no other field of science would this be good enough. If a physicist told us that light rays are bent by gravity, as Einstein did, then we would insist on experiments to back up the theory. If some biologists told us that all living creatures contain DNA in their cells, as Watson and Crick did in 1953, we wouldn’t believe them until lots of other biologists after looking into the idea agreed with them and did experiments to back it up. And if a modern biologist were to tell us that it were definitely possible to clone people, we won’t really believe them until we saw solid evidence in the form of a cloned human being. Mathematics occupies a special place, where we believe anyone who claims to have proved a theorem on the say—so of just a few people—that is, until we hear otherwise." Suppose we loosely define a religion as any discipline whose foundations rest on an element of faith, irrespective of any element of reason which may be present. Quantum mechanics, for example, would qualify as a religion under this definition. Mathematics would hold the unique position of being a branch of theology possessing a “proof" of the fact that it should be so classified. “Where else do you have absolute truth? You have it in mathematics and you have it in religion, at least for some people. But in mathematics you can really argue that this is as close to absolute truth as you can get" says Joel Spencer. Mathematical Proofs: The Pragmatic Dimension ============================================ In the second half of the 20th century, theorems together with their proofs occur with increasing frequency as components of some cognitive models, in various areas of knowledge. In such situations we are obliged to question the theorems not only with respect to their truth value, but also in respect to their adequacy and relevance within the framework of the models to which they belong. We have to evaluate the explanatory capacity of a theorem belonging to a model B, concerning the phenomenon A, to which B is referring. This is a very delicate and controversial matter, because adequacy, relevance and explanatory capacity are a matter of degree and quality, which cannot be settled by binary predicates. Moreover, there is no possibility of optimization of a cognitive model. Any model can be improved, no model is the best possible. This happens because, as we have explained before, a cognitive model B of an entity A has simultaneously the tendency to increase its similarity with A and stress its difference from A. To give only one example in this respect, we recall the famous result obtained by Chomsky [@chomsky], in the late 1950s, stating that context–free grammars are not able to generate the English language. This result was accepted by the linguistic and computer science communities until the eighties, when new arguments pointed out the weakness of Chomsky’s argument; but this weakness was not of a logical nature, it was a weakness in the way we consider the entity called “natural language". As a matter of fact, the statement “English is a context–free language" is still controversial. Mathematical proofs are “theoretical" and “practical". Theoretical proofs (formal, ideal, rigorous) are models for practical proofs (which are informal, imprecise, incomplete). “Logicians don’t tell mathematicians what to do. They make a theory out of what mathematicians actually do", says Hersh [@hersh], p. 50. According to the same author, logicians study what mathematicians do the way fluid dynamicists study water waves. Fluid dynamicists don’t tell water how to wave, so logicians don’t tell mathematicians what to do. The situation is not as simple as it appears. Logical restrictions and formal models (of proof) can play an important role in the practice of mathematics. For example, the key feature of constructive mathematics is the identification “existence = computability" (cf. Bridges [@bridges]) and a whole variety of constructive mathematics, the so–called Bishop constructive mathematics, is mathematics with intuitionistic rather than classical underlying logic. Quasi–Empirical Proofs: From Classical to Quantum ================================================= \ The use of large–scale programs, such as Mathematica, Maple or MathLab is now widespread for symbolical and numerical calculations as well as for graphics and simulations. To get a feeling of the extraordinary power of such programs one can visit, for example, the Mathematica website [http://www.wolfram.com]{}. New other systems are produced; “proofs as programs", “proof animation" or “proof engineering" are just a few examples (see [@hayashi]). In some cases an experiment conveys an aesthetic appreciation of mathematics appealing to a much broader audience (cf. [@bb1; @bbg; @crismarcus]). A significant, but simple example of the role an experiment may play in a proof is given by Beyer [@beyer]. He refers to J. North who asked for a computer demonstration that the harmonic series diverges. We quote Beyer: “His example illustrates the following principle: Suppose that one has a computer algorithm alleged to provide an approximation to some mathematical quantity. Then the algorithm should be accompanied by a theorem giving a measure of the distance between the output of the algorithm and the mathematical quantity being approximated. For the harmonic series, one would soon find that the sum was infinite." It is interesting to mention that in 1973 Beyer made together with Mike Waterman a similar attempt to compute Euler’s constant; their experiment failed, but the error was discovered later by Brent [@brent]. New types of proofs motivated by the experimental “ideology” have appeared. For example, rather than being a static object, the [*interactive proof*]{} (see Goldwasser, Micali, Rackoff [@GMR], Blum [@Blum]) is a two–party protocol in which the [*prover*]{} tries to prove a certain fact to the [*verifier*]{}. During the interactive proof the [*prover*]{} and the [*verifier*]{} exchange messages and at the end the [*verifier*]{} produces a verdict “accept" or “reject". A holographic (or probabilistic checkable) proof (see Babai [@Babai]) is still a static object but it is verified probabilistically. Errors become almost instantly apparent after a small part of the proof was checked.[^10] The transformation of a classical proof (which has to be self-contained and formal) into a holographic one requires super-linear time. The blend of logical and empirical–experimental arguments (“quasi–empirical mathematics" for Tymoczko [@tymoczko], Chaitin [@ch00; @ch02; @gregphil] or “experimental mathematics" for Bailey, Borwein [@bb], Borwein, Bailey [@bb1], Borwein, Bailey, Girgensohn [@bbg]) may lead to a new way to understand (and practice) mathematics. For example, Chaitin argued that we should introduce the Riemann hypothesis as an axiom: “I believe that elementary number theory and the rest of mathematics should be pursued more in the spirit of experimental science, and that you should be willing to adopt new principles. I believe that Euclid’s statement that an axiom is a self–evident truth is a big mistake. The Schrödinger equation certainly isn’t a self–evident truth! And the Riemann hypothesis isn’t self–evident either, but it’s very useful. A physicist would say that there is ample experimental evidence for the Riemann hypothesis and would go ahead and take it as a working assumption." Classically, there are two equivalent ways to look at the mathematical notion of proof: [*logical*]{}, as a finite sequence of sentences strictly obeying some axioms and inference rules, and [*computational*]{}, as a specific type of computation. Indeed, from a proof given as a sequence of sentences one can easily construct a Turing machine producing that sequence as the result of some finite computation and, conversely, given a machine computing a proof we can just print all sentences produced during the computation and arrange them into a sequence. This gives mathematics an immense advantage over any science: a proof is an explicit sequence of reasoning steps that can be inspected at [*leisure*]{}. [*In theory*]{}, if followed with care, such a sequence either reveals a gap or mistake, or can convince a sceptic of its conclusion, in which case the theorem [*is considered proven*]{}. The equivalence between the logical and computational proofs has stimulated the construction of programs which play the role of [*“artificial" mathematicians*]{}. The “theorem provers" have been very successful as “helpers" in proving many results, from simple theorems of Euclidean geometry to the computation of a few digits of a Chaitin Omega Number [@crisds]. “Artificial" mathematicians are far less ingenious and subtle than human mathematicians, but they surpass their human counterparts by being infinitely more patient and diligent. If a conventional proof is replaced by an “unconventional" one (that is a proof consisting of a sequence of reasoning steps obeying axioms and inference rules which depend not only on some logic, but also on the external physical medium), then the conversion from a computation to a sequence of sentences may be impossible, e.g. due to the size of the computation. An extreme, and for the time being hypothetical example, is the proof obtained as a result of a quantum computation (see Calude and Păun [@cp]). The quantum automaton would say “your conjecture is true", but (due to quantum interference) there will be no way to exhibit all trajectories followed by the quantum automaton in reaching that conclusion. The quantum automaton has the ability to check a proof, but it may fail to reveal any “trace" of the proof for the human being operating the quantum automaton. Even worse, any attempt to [*watch*]{} the inner working of the quantum automaton (e.g. by “looking" inside at any information concerning the state of the ongoing proof) may compromise forever the proof itself! We seem to go back to Bertrand Russell who said that “mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true", and even beyond by adding [*and even when it’s true we might not know why.*]{} Speculations about quantum proofs [*may not affect*]{} the essence of mathematical objects and constructions (which, many believe, have an autonomous reality quite independent of the physical reality), but they seem to [*have an impact*]{} on how we [*learn/understand mathematics,*]{} which is through the physical world. Indeed, our glimpses of mathematics are revealed only through physical objects, human brains, silicon computers, quantum automata, etc., hence, according to Deutsch [@deutsch-97], they have to obey not only the axioms and the inference rules of the theory, but the [*laws of physics*]{} as well. To complete the picture we need to take into account also the [*biological*]{} dimension. No matter how precise the rules (logical and physical) are, we need human consciousness to apply the rules and to understand them and their consequences. Mathematics is a human activity. Knowledge Versus Proof ====================== \ Are there intrinsic differences between traditional and ‘unconventional’ types of proofs? To answer this question we will consider the following interrelated questions:\ 1. Do ‘unconventional’ methods supply us with a proof in some formal language?\ 2. Do ‘unconventional’ methods supply us with a mathematical proof?\ 3. Do ‘unconventional’ methods supply us with knowledge?\ 4. Does mathematics require knowledge or proof?\ A blend of mathematical reasoning supported by some silicon or quantum computation or a classical proof of excessive length and complexity (for example, the classification of finite simple groups) are examples of “unconventional” proofs. The ultimate goal of the mathematical activity is the [*advance human understanding of mathematics*]{} (whatever this means!). The answer to the first two question is affirmative. Indeed, computations are represented in the programming language used by the computer (the ‘unconventional’ computer too), even if the whole proof cannot be globally ‘visualized’ by a human being. The proof can be checked by any other mathematician having the equipment used in the ’unconventional’ proof. A proof provides knowledge only to the extent that its syntactic dimension is balanced by the semantic one; any gap between them makes the proof devoid of knowledge and paves the way for the proof to become a ritual without meaning. Proofs generating knowledge, quite often produce much more, for example, ’insight’ (think of the insight provided by understanding the algorithm used in the proof). A misleading analogy would be to replace, in the above questions, [*‘unconventional’ methods*]{} with [*“testimony from a respected and (relevantly) competent mathematician”*]{}. Certainly, such testimony provides knowledge; it does not qualify as a mathematical proof (even less as a formalized proof), but the result is a “mathematical activity” because it advances our knowledge of mathematics. The difference between ‘unconventional’ methods and ‘relevant testimony’ can be found in the mechanisms used to produce outputs: a ‘relevant testimony’ is the gut feeling of a respected, relevant, competent mathematician, by and large based on a considerable mathematical experience, while an ‘unconventional’ method produces an objective argument. There is little ‘intrinsic’ difference between traditional and ‘unconventional’ types of proofs as i) first and foremost, [*mathematical truth*]{} cannot always be certified by proof, ii) correctness is not absolute, but almost certain, as mathematics advances by making mistakes and correcting and re–correcting them (mathematics fallibility was argued by Lakatos), iii) non–deterministic and probabilistic proofs do not allow mistakes in the applications of rules, they are just indirect forms of checking (see Pollack [@pollack], p. 210) which correspond to various degrees of rigour, iv) the explanatory component, the understanding ‘emerging’ from proofs, while extremely important from a cognitive point of view, is subjective and has no bearing on formal correctness. As Hersh noticed, mathematics like music exists by some logical, physical and biological manifestation, but “it makes sense only as a mental and a cultural activity" ([@hersh], p. 22). How do we continue to produce rigorous mathematics when more research will be performed in large computational environments where we might or might not be able to determine what the system has done or why[^11] is an open question. The blend of logical and empirical–experimental arguments are here to stay and develop. Of course, some will continue to reject this trend, but, we believe, they will have as much effect as King Canute’s royal order to the tide. There are many reasons which support this prediction. They range from economical ones (powerful computers will be more and more accessible to more and more people), social ones (skeptical oldsters are replaced naturally by youngsters born with the new technology, results and success inspire emulation) to pure mathematical (new challenging problems, wider perspective) and philosophical ones (note that incompleteness is based on the analysis of the computer’s behaviour). The picture holds marvelous promises and challenges; it does not eliminate the continued importance of extended personal interactions in training and research. Acknowledgements {#acknowledgements .unnumbered} ================ This paper is based on a talk presented at the Workshop [*Truths and Proofs*]{}, a satellite meeting of the [*Annual Conference of the Australasian Association of Philosophy (New Zealand Division)*]{}, Auckland, New Zealand, December 2001. We are most grateful to Andreea Calude, Greg Chaitin, Sergiu Rudeanu, Karl Svozil, Garry Tee, and Moshe Vardi for inspired comments and suggestions. [999]{} M. Aschbacher. The status of the classification of finite simple groups, [*Notices of the Amer. Math. Soc.*]{}51, 7 (2004), 736–740. D. Auburn. [*Proof. A Play*]{}, Faber and Faber, New York, 2001. K. Appel, W. Haken. [*Every Planar Graph is Four Colorable*]{}, Contemporary Mathematics 98, AMS, Providence, 1989. L. Babai. Probably true theorems, cry wolf? [*Notices of the Amer. Math. Soc.*]{} 41 (5) (1994), 453–454. D. H. Bailey, J. M. Borwein. Experimental mathematics: Recent developments and future outlook, in B. Engquist, W. Schmid (eds.). [*World Mathematical Year Mathematics Unlimited—2001 and Beyond*]{}, Springer-Verlag, Berlin, 2001, 51–66. J. Barrow. [*Impossibility. The Limits of Science and the Science of Limits*]{}, Oxford University Press, Oxford, 1998. J. M. Borwein, D. H. Bailey, [*The Experimental Mathematician. Plausible Reasoning in the 21st Century,*]{} A. K. Peters, Natick, Ma., 2003. J. M. Borwein, D. H. Bailey, R. Girgensohn. [*Experimentation in Mathematics. Computational Paths to Discovery*]{}, A. K. Peters, Natick, Ma., 2004. W. A. Beyer. The computer and mathematics. [*Notices of the Amer. Math. Soc.*]{} 48, 11 (2001), 1302. M. Blum. How to prove a theorem so no one else can claim it, [*Proceedings of the International Congress of Mathematicians,*]{} Berkeley, California, USA, 1986, 1444–1451. R. P. Brent. Computation of the regular continued fraction for Euler’s constant, [*Math. of Computation*]{} 31, 139 (1977), 771–777. D. S. Bridges. Constructive truth in practice, in H. G. Dales and G. Oliveri (eds.). [*Truth in Mathematics*]{}, Clarendon Press, Oxford, 1998, 53–69. A. S. Calude. The journey of the four colour theorem through time, [*The NZ Mathematics Magazine*]{} 38, 3 (2001), 27–35. C. S. Calude. [*Information and Randomness: An Algorithmic Perspective*]{}, 2nd Edition, Revised and Extended, Springer Verlag, Berlin, 2002. C. S. Calude. Chaitin $\Omega$ numbers, Solovay machines and incompleteness, [*Theoret. Comput. Sci.*]{}, 284 (2002), 269–277. C. S. Calude. Incompleteness, complexity, randomness and beyond, [*Minds and Machines: Journal for Artificial Intelligence, Philosophy and Cognitive Science*]{}, 12, 4 (2002), 503–517. C. S. Calude, G. J. Chaitin. Randomness everywhere, [*Nature*]{}, 400, 22 July (1999), 319–320. C. S. Calude, M. J. Dinneen and C.-K. Shu. Computing a glimpse of randomness, [*Experimental Mathematics*]{} 11, 2 (2002), 369–378. C. S. Calude, H. J[ü]{}rgensen. [*Is Complexity a Source of Incompleteness?*]{}, [*CDMTCS Research Report*]{} 241, 2004, 15 pp. C. Calude, H. Jürgensen, M. Zimand. Is independence an exception?, [*Appl. Math. Comput.*]{} 66 (1994), 63–76. C. S. Calude, S. Marcus. Mathematical proofs at a crossroad? in J. Karhum" aki, H. Maurer, G. Păun, G. Rozenberg (eds.). [*Theory Is Forever*]{}, Lectures Notes in Comput. Sci. 3113, Springer Verlag, Berlin, 2004, 15–28. C. S. Calude, G. Păun. [*Computing with Cells and Atoms*]{}, Taylor & Francis Publishers, London, 2001. J. L. Casti. [*Mathematical Mountaintops*]{}, Oxford University Press, Oxford, 2001. G. J. Chaitin. Randomness and mathematical proof, [*Scientific American*]{}, 232 (5) (1975), 47–52. G. J. Chaitin. [*Exploring Randomness*]{}, Springer Verlag, London, 2001. G. J. Chaitin. Computers, paradoxes and the foundations of mathematics, [*American Scientist*]{}, 90 March–April (2002), 164–171. G. J. Chaitin. Personal communication to C. S. Calude, 5 March, 2002. G. J. Chaitin. On the intelligibility of the universe and the notions of simplicity, complexity and irreducibility, [http://www.cs.auckland.ac.nz/CDMTCS/ chaitin/bonn.html]{}, September 2002. N. Chomsky. [*Syntactic Structures*]{}, Mouton, The Hague, 1957. A. Connes, A. Linchnerowicz, M. P. Sch" utzenberger. [*Triangle of Thoughts*]{}, AMS, Providence, 2001. E. W. Dijkstra. Real mathematicians don’t prove, [*EWD1012*]{}, University of Texas at Austin, 1988, [http://www.cs.utexas.edu/users/EWD/EWD1012.pdf]{}. D. Deutsch. [*The Fabric of Reality*]{}, Allen Lane, Penguin Press, 1997. R. A. De Millo, R. J. Lipton, A. J. Perlis. Social processes and proofs of theorems and programs, [*Comm. ACM*]{} 22, 5 (1979), 271–280. J. Dieudonné. [*Pour L’honneur de l’Esprit Humain*]{}, Gallimard, Paris, 1986. W. Feit, J. G. Thomson. Solvability of groups of odd order, [*Pacific J. Math.* ]{} 13 (1963), 775–1029. 01 J. E. Fenstad. Is mathematical still the science of paper, pencils and proofs?, Conference on [*Electronic Communication and Research in Europe*]{}, Darmstadt/Seeheim, 15–17 April 1998, [//academia.darmstadt.gmd.de/seeheim/thebook/finals/fenstad.html]{}. S. Goldwasser, S. Micali, C. Rackoff. The knowledge complexity of interactive proof–systems, [*SIAM Journal of Computing,*]{} 18(1) (1989), 186–208. C. Foiaş. Personal communication to S. Marcus (about 20 years ago). L. E. Hahn, B. Epstein. [*Classical Complex Analysis*]{}, Sudury, Mass., Jones and Barlettt, 1996. M. Harris. Contexts of justification, [*The Mathematical Intelligencer*]{} 23, 1 (2001), 10–22. S. Hayashi, R. Sumitomo, K. Shii. Towards the animation of proofs–testing proofs by examples, [*Theoret. Comput. Sci.*]{} 272 (2002), 177–195. R. Hunt. The philosophy of proof, [http://plus.maths.org/issue10/ features/proof4/]{}. R. Hersh. [*What Is Mathematics, Really?*]{}, Vintage, London, 1997. M. Kline. [*Mathematical Thought from Ancient to Modern Times*]{}, Oxford University Press, Oxford, Vol. 1, 1972. D. E. Knuth. Theory and practice, [*EATCS Bull.*]{} 27 (1985), 14–21. D. MacKenzie. Slaying the kraken. The sociohistory of a mathematical proof, [*Social Studies of Science*]{} 29, 2 (1999), 7–60. Yu. I. Manin. Truth, rigour, and common sense, in H. G. Dales and G. Oliveri (eds.). [*Truth in Mathematics*]{}, Clarendon Press, Oxford, 1998, 147–159. S. Marcus. No system can be improved in all respects, in G. Altmann and W. A. Koch (eds.). [*Systems. New Paradigms for the Human Sciences*]{}, Walter de Gruyter, Berlin, 1998, 143–164. S. Marcus. Imprecision, between variety and uniformity: the conjugate pairs, in J. J. Jadacki and W. Strawinski (eds.). [*The World of Signs*]{}, Poznan Studies in the Philosophy of Sciences and the Humanities 62 Rodopi, Amsterdam, 1998 59–72. S. Marcus. Metaphor as dictatorship, in J. Bernard, J. Wallmannsberger and G. Withalm (eds.). [*World of Signs. World of Things*]{}, Angewandte Semiotik 15, OGS, Wien, 1997, 87–108. R. Pollack. How to believe a machine–checked proof, in G. Sambin and J. M. Smith (eds.). [*Twenty–five Years of Constructive Type Theory*]{}, Clarendon Press, Oxford, 1998, 205–220. A. R' enyi. [*Dialogues on Mathematics*]{}, Holden Day, San Francisco, 1967. N. Robertson, D. Sanders, P. Seymour, R. Thomas. A new proof of the four–colour theorem, [*Electronic Research Announcements of the AMS*]{} 2,1 (1996), 17–25. R. R. Rosen. Complementarity in social structures, [*Journal of Social and Biological Structures*]{} 1 (1978), 191–200. R. Thom. Modern mathematics: does it exist?, in A. G. Howson (ed.). [*Developments in Mathematical Education*]{}, Cambridge University Press, 1973, 194–209. R. Thom. Topologie et linguistique, in A. Haefliger and R. Nerasimham (eds.). [*Essays on Topology and Related Topics. Memoires Dédiés à Georges de Rham*]{}, Springer Verlag, New York, 1970, 226–248. R. M. Solovay. A version of $\Omega$ for which $ZFC$ can not predict a single bit, in C. S. Calude and G. Păun (eds.). [*Finite Versus Infinite. Contributions to an Eternal Dilemma*]{}, Springer Verlag, London, 2000, 323–334. T. Tymoczko. The four–colour problem and its philosophical significance, [*J. Philosophy*]{} 2,2 (1979), 57–83. U. Schmidt. [*Überprüfung des Beweises für den Vierfarben Satz*]{}, Diplomarbeit Technische Hochschule, Aachen, 1982. E. R. Swart. The philosophical implications of the four–colour problem, [*American Mathematical Monthly*]{} 87, 9 (1980), 697–702. [^1]: In 1879 Kempe announced his ‘proof’ of the 4CT in both the magazine [*Nature*]{} and the [*American Journal of Mathematics*]{}. Eleven years later, Percy Heawood found an error in the proof which nobody had spotted, despite careful checking. [^2]: Our Italics. [^3]: The Scottish judicial system which admits three forms of verdicts, guilty, not–guilty and not–proven, comes closer to the picture described by GIT. [^4]: Landau’s son-in-law. [^5]: Still, there is a controversy in the mathematical community on whether these articles provide a complete and correct proof. For a recent account see Aschbacher [@ma]. [^6]: Perhaps “general" should be replaced here by “quasi–general". [^7]: Probabilists would argue that very long proofs can at best be viewed as only probably correct, cf. [@demillo], p. 273. In view of [@cj], the longer the statement, the lesser its chance is to be proved. [^8]: Harris [@harris] believes that no more than 5% of mathematicians have made the effort to work through the proof. Does this have anything to do with what George Hardy has noted in his famous [*Apology*]{}: “All physicists and a good many quite respectable mathematicians are contemptuous about proof."? [^9]: According to Wiles, “It was an error in a crucial part of the argument, but it was something so subtle that I’d missed it completely until that point. The error is so abstract that it can’t really be described in simple terms. Even explaining it to a mathematician would require the mathematician to spend two or three months studying that part of the manuscript in great detail." [^10]: More precisely, a traditional proof of length $l$ is checked in time a constant power of $l$ while a holographic proof requires only constant power of $\log_{2}l$. To appreciate the difference, the binary logarithm of the number of atoms in the known Universe is smaller than 300. [^11]: Metaphorically described as “relying on proof by ‘Von Neumann says’".

--- abstract: 'Determining the properties of starbursts requires spectral diagnostics of their ultraviolet radiation fields, to test whether very massive stars are present. We test several such diagnostics, using new models of line ratio behavior combining Cloudy, Starburst99 and up-to-date spectral atlases [@pauldrach01; @hillmill]. For six galaxies we obtain new measurements of [ $1.7$ ]{}/, a difficult to measure but physically simple (and therefore reliable) diagnostic. We obtain new measurements of [ $2.06$ ]{}/ in five galaxies. We find that [ $2.06$ ]{}/ and \[\]/ are generally unreliable diagnostics in starbursts. The heteronuclear and homonuclear mid–infrared line ratios (notably \[\] $15.6$  / \[\] $12.8$ ) consistently agree with each other and with [ $1.7$ ]{}/; this argues that the mid–infrared line ratios are reliable diagnostics of spectral hardness. In a sample of $27$ starbursts, \[\]/\[\] is significantly lower than model predictions for a Salpeter IMF extending to $100$ . Plausible model alterations strengthen this conclusion. By contrast, the low–mass and low–metallicity galaxies II Zw 40 and NGC 5253 show relatively high neon line ratios, compatible with a Salpeter slope extending to at least $\sim40$–$60$ . One solution for the low neon line ratios in the high–metallicity starbursts would be that they are deficient in $\ga 40$   stars compared to a Salpeter IMF. An alternative explanation, which we prefer, is that massive stars in high–metallicity starbursts spend much of their lives embedded within ultra–compact  regions that prevent the near– and mid–infrared nebular lines from forming and escaping. This hypothesis has important consequences for starburst modelling and interpretation.' author: - 'J. R. Rigby and G. H. Rieke' nocite: - '[@mrr]' - '[@pauldrach01]' - '[@kbfm; @ho3; @doherty95]' - '[@thornley]' - '[@dpj]' - '[@doherty95]' - '[@lph320]' - '[@shields]' - '[@vanzi]' - '[@vr]' - '[@chad]' - '[@fs-m82]' - '[@lph328]' - '[@seaquist]' - '[@sb]' - '[@chip]' - '[@guseva]' - '[@schmutz]' - '[@lejeune]' - '[@kurtz]' - '[@hanson]' - '[@depree]' - '[@garcia]' - '[@btk]' - '[@thb]' - '[@kj99; @vjc]' - '[@ccm]' title: 'Missing Massive Stars in Starbursts: Stellar Temperature Diagnostics and the IMF' --- INTRODUCTION {#sec:intro} ============ In the very local ($D<5 h_{100}^{-1}$) Universe, the circumnuclear regions of just four galaxies (M82, NGC 253, NGC 4945, and M83) are responsible for $\sim25\%$ of the current massive star formation [@heckman97]. In these “circumnuclear starburst galaxies”, the star formation is confined to the inner $0.2$ to $2$ kpc, in a dense, gas–rich disk where star formation rates can reach $1000$   [@robhubble]. If the starburst initial mass function (IMF) includes significant numbers of low–mass stars, then each starburst is currently building up the stellar component of its host galaxy as well. A starburst enriches and heats its interstellar medium, as well as the local intergalactic medium. Starbursts can also drive large–scale winds that eject interstellar gas, presumably casting metals into the voids and heating the gas between galaxies. Starburst galaxies thus play a number of important roles in galaxy evolution. If starbursts could be dated, then a sequence could be pieced together, charting starburst evolution from triggering to post–starburst quiescence. Starburst ages are most directly determined by understanding the population of rapidly evolving massive stars. The feedback effect of a starburst on its gas supply is transmitted through massive stellar winds and supernovae–driven superwinds. Thus, understanding the evolution of starbursts and their effects on the interstellar and intergalactic media both critically depend on understanding the populations of massive stars. Unfortunately, since starburst galaxies are too far away to count individual stars, the high-mass IMF must be determined indirectly, in ways that are model–dependent and crude. @leithconf reviews these techniques and divides them into three categories: techniques to determine a lower mass cutoff by measuring the mass–to–light ratio; to find the slope of the IMF above $\sim10$ ; and to determine an upper mass cutoff from the hardness of the ionizing radiation field. The ionizing spectrum is set by the starburst’s age, IMF, and star formation history. Consequently, the shape of the ionizing field spectrum is an important boundary condition on starburst models [@ghr-confproc]. Ionizing continua are often parameterized by an effective temperature (), as if one stellar spectral type were responsible for the flux. The UV spectrum cannot be measured directly because little ionizing continuum radiation escapes from a starburst [@lfhl]. Instead, the presence or absence of massive stars must be inferred using spectral diagnostics. Extinction in circumnuclear starbursts means that infrared diagnostics are preferred. Many line ratios have been used to estimate starburst : forbidden line ratios, mixed forbidden–recombination line ratios like $[{\hbox{{\rm O}\kern 0.1em{\sc iii}}}]/{\hbox{{\rm H}\kern 0.1em$\beta$}}$, optical lines of , the near–infrared line [ $2.06$ ]{}, and mid–infrared fine structure lines for example. Unfortunately, these diagnostics disagree by $2,000$ to $5,000$ K [@vr; @thornley; @ghr-confproc], and suffer variously from intrinsic faintness, susceptibility to shocks and reddening, dependence on nebular conditions, and uncertain atomic constants. In this paper, we use the ratio of [ $1.7$ ]{} to Brackett 10 () to diagnose the hardness of starburst ionizing fields. The faintness of the [ $1.7$ ]{} line restricts its measurement to nearby galaxies with strong emission lines. In these galaxies, the / ratio should allow estimates of  that are largely independent of reddening or nebular conditions. We then use / to assess the accuracy of diagnostics that can reach distant galaxies. Using diagnostics we find reliable, we confirm that few massive starburst galaxies have high–excitation spectra. While this may occur because the IMF is biased against high–mass stars, we propose that high–excitation spectra are scarce because the massive stars spend most of their main sequence lifetimes embedded in ultracompact  regions. OBSERVATIONS, DATA REDUCTION, AND CALIBRATION {#sec:whatwedid} ============================================= To assist in evaluating T$_{eff}$ diagnostics, we have obtained new measurements of [ $1.7$ ]{}/. This diagnostic is unaffected by metallicity, shocks, or level pumping. Regrettably, the [ $1.7$ ]{} line is very weak, less than $10\%$ the intensity of . Thus, the [ $1.7$ ]{}/ ratio can only be measured in nearby starbursts with bright lines. For our sample, we chose six nearby starburst galaxies with large measured  fluxes and, when possible, supporting observations in the literature such as mid–infrared spectra. Near–infrared spectra were obtained on the nights of 2001 April 6 and 7, using the FSPEC near–infrared spectrometer [@williams] on the Steward Observatory Bok $2.3$ m telescope. Table \[tab:obs\] lists target objects and integration times. All observations were taken with the $600~$lines mm$^{-1}$ grating, which produces effective resolutions of $R\approx 2000$ at $1.7$  and $R\approx 3000$ at $2.1$ . The slit was $2.4$ by $90$. All exposures were guided by hand using an H–band camera that images the mirrored slit. Spectra of the six targets were obtained in H–band ($1.7$ ). K–band ($2.1$) spectra were also obtained unless high–quality spectra already existed in the literature. The angular sizes of the nuclear starburst regions are small compared to the length of the FSPEC slit. For each integration, nuclear spectra were obtained at four successive positions along the slit. For the calibration stars, six spectra were taken along the slit. (For brevity, we will call each resulting two-dimensional spectrum a “frame”, and each group of frames in an integration a “set”.) Data Reduction {#sec:datared} -------------- The infrared  and  lines are faint, making the data reduction approach critical. We therefore describe it in detail. The data were reduced using `iraf`.[^1] First, dark frames were subtracted from the object frames. Each frame was then flat–fielded using a median–averaged lamp flat. The airglow and bias were removed by differencing neighboring frames. For the first and last frames of a set, the neighbor was subtracted. For each middle frame, the mean of the immediately–bracketing frames was subtracted. For the calibration stars the subtraction was simple. Over the longer integration times required for the galaxies ($\sim4$ minutes per frame), the sky background is variable. Accordingly, prior to subtraction, we scaled each galaxy frame by a constant, generally within a percent of unity, to optimize the sky cancellation. This technique of differencing neighboring two–dimensional spectra usually removes the sky emission lines accurately. For a few galaxies, the resultant sky subtraction was not adequate. In these exposures, the sky lines in a set of frames were offset along the dispersion axis by $0.005$ to $0.01$ pixels, suggesting a slight, monotonic shift in the grating tilt. To improve the cancellation, we used `onedspec.identify` and `onedspec.reidentify` to fit, for each frame, a linear shift in the position of the airglow lines with respect to a reference frame. We then used `images.imgeom.imshift` to shift the frames, using linear interpolation, to zero the offsets. This was done for H–band frames of NGC 4861, He 2–10, NGC 4102, and NGC 3504 as needed. The next step in the reduction was to combine the frames within a set. Offsets were determined by summing each frame down the dispersion axis and measuring the location of the continuum peak in the resulting one-dimensional image. Frames were magnified by a factor of six to permit fractional pixel shifts, which minimizes smearing of the data and maximizes preservation of flux during the next step of straightening. Magnified images were then remapped to make the spatial axis perpendicular to the dispersion axis. This remapping is accomplished by fitting the tilt of the dispersion axis as a cubic polynomial; this function is well–defined and does not vary with time. Straightening the spectra in this manner simplifies subsequent extraction, stacking, and wavelength calibration [@chad_reduction]. Frames were then median combined with `imcombine` using no rejection, and using scale and weight factors determined from each frame’s median continuum strength (generally within $20\%$ of unity). Frames were then de-magnified. Bad pixels were replaced with the average of their immediate two neighbors along the dispersion axis. A similar procedure created two–dimensional sky frames suitable for wavelength calibrations. The frames were straightened and median combined as before, but without weights, scales, or offsets. The resulting images have no continua, only sky lines. A high signal–to–noise 1–D sky spectrum was then created by taking, at each wavelength, the median value over all spatial positions. We then extracted the spectra of the galaxies and calibration stars. Using `apall` in `iraf`, we traced each continuum interactively using low–order Legendre polynomials. The aperture width was chosen to be where the signal at the brightest part of the continuum dropped to $30\%$ of peak. No extra background was subtracted at this point; trials with additional subtraction (to remove residual sky lines) added more noise than they eliminated. Spectra were extracted from the sky frames by using the apertures fitted for the corresponding objects. Next, the spectra were corrected for telluric absorption. To do this, each galaxy spectrum was first divided, in pixel space, by the spectrum of a calibration star. When a target observation was bracketed by calibration star observations, the average of the two stellar spectra was divided into the target spectrum. Otherwise, the calibration spectrum taken closest in time to the target was used. Calibration stars were dwarfs of spectral types F6 to G0, most within $5\degr$ of the target object. Such stars have relatively featureless intrinsic spectra at $\sim2$ , so their observed spectra reflect the variable absorption of the Earth’s atmosphere. We wavelength–calibrated the sky spectra of the galaxies, using `onedspec.identify` and tabulated vacuum wavelengths of the OH lines calculated by C. Kulesa (1996, private communication). Scatter in the wavelength calibration of the galaxies was less than $\pm0.2$ [Å]{}, and usually below $\pm0.1$ [Å]{}. Wavelength solutions of the sky spectra were transferred to the \[target/calibrator\] spectra. Due to the longer integration times, the sky spectra derived from galaxy frames give wavelength solutions more precise than those derived from sky spectra of the star frames. The wavelength solution is quite stable with time. The calibration stars are not completely featureless; their metal absorption lines produce spurious emission lines in the \[target/calibrator\] spectra. To correct for this effect, we multiplied the \[target/calibrator\] spectra by the solar spectrum, which has been corrected for atmospheric absorption [@lwsun]. The solar spectrum was first normalized and broadened to $10$ [Å]{} in H and $9$ [Å]{} in K to match our observations. Since the calibration stars are similar in spectral type to the Sun, the solar multiplication cures the final spectrum of the metal lines and the global Rayleigh–Jeans slope that the calibration star introduced. The resulting spectrum is \[target/calibrator\]$*$\[$\sun$\]. This procedure is explained more fully by Maiolino, Rieke, & Rieke (1996). At this point, NGC 4861 and NGC 4214 still showed residual sky lines. The spectra were improved by subtracting a scaled, extracted sky spectrum. For both galaxies,  and [ $1.7$ ]{} are uncontaminated by OH lines, whereas  is somewhat contaminated in NGC 4861 and seriously contaminated in NGC 4214 (the lowest redshift galaxy in our sample). Accordingly, we will consider only  in lieu of both  and  for these two galaxies. Combining Spectra {#sec:combspec} ----------------- For the H–band spectra, we observed targets for $0.5$ hr between calibrators. Because our total integration times on each galaxy were substantially longer than this, the spectra must be combined. To do this, we used `onedspec.scombine`, median combining groups of $\ge4$ images, and average combining otherwise. The [ $2.06$ ]{} and  lines were covered by different grating settings. We spliced together the two grating settings for each target by scaling the $2.085$  spectrum by a constant until it matched the flux level of the $2.15$  spectrum in the region of overlap. Scaling constants were between $1.08$ and $1.3$. We then joined the spectra using `scombine`. Subtracting the Stellar Continuum {#sec:contsub} --------------------------------- The nebular lines we seek sit atop a stellar continuum, whose absorption lines can mask or alter the emission line ratios. Before faint emission line fluxes can be measured, the stellar continuum must be removed. The lines of interest at K ( and [ $2.06$ ]{}) are strong enough that subtraction of the stellar continuum is not necessary. For H–band, we used a stellar continuum template made by combining the $15$ stars from a stellar atlas (observed with the same spectrometer: V. D. Ivanov, in preparation) which minimized the residuals when subtracted from NGC $253$ [@chad]. Thus, the template was chosen to be a good fit to a $\sim$solar metallicity starburst galaxy, and was not made by modelling the stellar populations of each galaxy in our sample. The stars in the template are of stellar types K0 to M3, with metallicities between solar and half–solar. Two of the stars are supergiants, five are bright giants, five are giants, and three are dwarfs. The resolution of the template is $130$  in H. To match the intrinsic velocity dispersion of the galaxy spectra, we convolved this template with a Gaussian kernel as necessary to lower the resolution. He 2–10 and NGC 3077 were best fit with no convolution; NGC 3504 required a template with $150$  resolution, and NGC 4102 required a $175$  template. For NGC 4214 and NGC 4861, continuum subtraction was unimportant because the continua are much weaker than the nebular emission lines. The stellar continuum template has an absorption feature at $1.7010$ . By comparison with spectra of the Sun [@lwsun] and Arcturus [@mont], we identify this feature as a blend of three nickel lines and four (weaker) iron lines. At a resolution of $130$ , the blend has an equivalent width of $\le 1.5$ [Å]{}, and dips to $94\%$ of the continuum level. Because the absorption feature is a metal line blend, its strength will depend on metallicity. Besides a simple subtraction of the stellar continuum, we also added a $20\%$ featureless continuum to the template, renormalized, and subtracted the new continuum from the galaxy spectra. This procedure crudely approximates the effect of lower metallicity. These two realizations of the continuum subtraction provide some estimate of the associated uncertainty. Continuum–subtracted H–band spectra of the galaxies and the stellar template are plotted in figure \[fig:spec\_17\]. K–band spectra are plotted in figure \[fig:spec\_20\]. Measuring Line Ratios {#sec:lineratios} --------------------- To set the continuum level, we fit a low–order Chebyshev function across each spectrum, excluding emission lines from the fit. Each line was fit by a Gaussian to measure the line fluxes listed in table \[tab:waves\]. For the noisy, non–Gaussian line profiles of NGC $4102$, we directly summed flux rather than fit Gaussians. To measure the relative strength of [ $1.7$ ]{}, we considered two ratios: [ $1.7$ ]{}/, and $[{\hbox{{\rm He}\kern 0.1em{\sc i}}~$1.7$~{\micron}}/ {\hbox{{\rm Br}\kern 0.1em$11$}}]\times[{\hbox{{\rm Br}\kern 0.1em$11$}}/ {\hbox{{\rm Br}\kern 0.1em$10$}}]_{case B}$. We assumed the value $[{\hbox{{\rm Br}\kern 0.1em$11$}}/ {\hbox{{\rm Br}\kern 0.1em$10$}}]_{case B} = 0.75$, which is appropriate for $n = 10^2$  and $T_e = 5,000$ K [@chad; @hs87]. For each galaxy, we computed both these ratios for both realizations of the continuum subtraction (with or without the $20\%$ featureless continuum), and used the mean of these four values as the [ $1.7$ ]{}/ ratio, and the standard deviation as an estimate of the uncertainty associated with the continuum subtraction. We also computed the ratio [ $2.06$ ]{}/. Our measured line ratios are listed in table \[tab-line-rats\], along with values of [ $2.06$ ]{}/ from the literature, and a weighted mean for [ $2.06$ ]{}/ that combines new and literature values. (The quoted uncertainty for the weighted mean is the error in the mean.) Because of the $1.7$  stellar absorption feature, studies that do not subtract the continuum in galaxies with weak [ $1.7$ ]{} will somewhat underestimate the [ $1.7$ ]{} line strength and therefore underestimate . To test the magnitude of this effect, in table \[tab-line-rats\] we list both continuum–subtracted and raw (un–subtracted) [ $1.7$ ]{}/ ratios. For galaxies NGC 3504 and NGC 4102, the lines are so weak relative to the stellar continuum that the [ $1.7$ ]{}/ cannot be measured without continuum subtraction. For the other galaxies in table \[tab-line-rats\], the raw and continuum–subtracted line ratios are very similar; for these galaxies (He 2–10, NGC 3077, NGC 4214, and NGC 4861), the continuum subtraction is not an important source of uncertainty. MODELLING LINE RATIO BEHAVIOR {#sec:models} ============================= Because the emission lines used to diagnose effective temperature have different excitation energies, one cannot verify that a particular diagnostic works by simply testing whether it exhibits a one-to-one correlation with another diagnostic. Instead, one must test diagnostics in light of photoionization models that, given realistic hot stellar ionizing spectra, predict line ratios appropriate to idealized nebulae. One can then ask a) whether the observed ratios populate the line ratio space permitted by models; and b) whether many observed line ratios for a particular galaxy are consistent, that is, can all be produced by one set of physical parameters. Thus, translating a nebular line ratio to a statement about stellar content is necessarily model–dependent. Past studies have run series of models in which a single main sequence star photoionizes a nebula, producing tabulated line ratios as a function of stellar . A measured galactic line ratio is then translated into an effective temperature using this tabulation [@dpj; @al; @vr; @bkl; @fs-m82]. This method has its uses: namely, to compare model inputs and assumptions, and understand what line ratios different stellar classes can produce. However, in § \[sec:specsynth\] we will argue that, especially for mid–infrared line ratios and [ $1.7$ ]{}/, starbursts are poorly approximated by single main sequence stars; to translate a line ratio into a meaningful statement about a stellar population, one must consider the flux from *all* the stars as a function of time. First, though, we will consider the insights and limitations of simple one–star models. All our models (single–star and population synthesis) are radiation–bounded thin shells created by the photoionization code Cloudy 94.00 [@hazy]. Using parameters determined by @fs-m82 for M82, we set the total hydrogen number density to $n_H = 300$  and inner radius to $R=25$ pc. This choice of radius produces line ratios within $2\%$ of the plane–parallel ($R=\infty$) case. This is because the shell is thin. Thus, the choice of radius only slightly affects the models, which are effectively plane–parallel. For single–star models, we use a constant ionization parameter of $\log U = -2.3$. We ran two sets of models, one with gas–phase abundances of solar, and the other with $1/5$ solar abundances (“the low–Z models”); neither abundance set includes depletion onto dust grains. (We address the effects of dust in § \[sec:caveats\].) Because Cloudy does not predict the intensity of [ $1.7$ ]{}, we scaled the intensity from [ $4471$]{} Å, which shares the same upper level. For Case B and T$_e=5,000$ K, the [ $1.7$ ]{} line is a factor of $7.4 \times 10^{-3}$ fainter than [ $4471$]{} Å. Models Using Individual Stars {#sec:indystars} ----------------------------- We first consider the ratio of \[\] $15.6$  to \[\] $12.8$ . For reference, it requires $22$ eV to make singly–ionized neon, and $41$ eV to make doubly–ionized neon. We took ionizing spectra from the O star models of Pauldrach, Hoffmann, & Lennon (2001), as prepared by @snc, and also the CoStar model spectra of @sk as hardwired in Cloudy. These two stellar libraries predict dramatically different line ratios. Dwarf, giant, and supergiant Pauldrach stars all produce a maximum \[\]/\[\] ratio of $10$ at $=50,000$ K. By contrast, the CoStar dwarf and giant atmospheres yield \[\]/\[\]$=40$ at $=50,000$ K. At $=35,000$ K, the predicted line ratios disagree by an order of magnitude. @fs-m82 used Pauldrach atmospheres and an earlier version of Cloudy to make their figure 8, which our Pauldrach models reproduce. The other mid–infrared line ratios also show this discrepancy. CoStar models predict ten times higher \[\]/\[\] and \[\]/\[\] ratios than Pauldrach models for most of the $25,000<{\hbox{T$_{eff}$}}<50,000$ K range; for \[\]/\[\] and \[\]/\[\], CoStar gives $2$ and $3$ times higher ratios, respectively. The near–infrared line ratios [ $1.7$ ]{}/ and [ $2.06$ ]{}/ are not sensitive to the choice of stellar atlas. It is sobering that current O star models predict such different mid–infrared line ratio strengths. On the bright side, this sensitivity suggests that mid–IR line ratios may provide astrophysical tests of O star spectral models in simple  regions. @giveon performed such a test in Galactic  regions; they find that @pauldrach01 atmospheres fit the observed \[\]/\[\] versus \[\]/\[\] relation, whereas stellar models that assume LTE do not. In another test, @pauldrach01 argued that their models successfully reproduce the observed far–UV spectra of hot stars, as opposed to other models. Finally, @snc argue that @pauldrach01 atmospheres should be more realistic than those of @sk because the latter neglect line broadening, and thus underestimate line blanketing. As a result, CoStar atmospheres have significantly higher ionizing fluxes, especially at energies exceeding the He$^+$ edge. Given these problems, and that the CoStar atmospheres predict much higher \[\]/\[\] line ratios than are observed, we will use the @pauldrach01 atmospheres in this paper. Still, that the Pauldrach spectra are better does not mean they are correct; the CoStar–Pauldrach discrepancy should serve as some warning of the current uncertainties regarding hot star spectra—a critical input to the models. We also consider Wolf–Rayet (WR) stars in simple nebulae. We use WN and WC model spectra compiled by @snc, which were generated using the code of @hillmill. For a given , these model WR stars yield much lower \[\]/\[\] ratios compared to Pauldrach O stars: at ${\hbox{T$_{eff}$}}=50,000$ K, the difference is a factor of $30$ for WN, and a factor of $10^4$ for WC stars. This is because of the very strong line blanketing found in WR stars. Since Wolf–Rayet stars can reach much hotter temperatures than main sequence stars, a ${\hbox{T$_{eff}$}}\la 140,000$ K WC star can reach $[{\hbox{{\rm Ne}\kern 0.1em{\sc iii}}}]/[{\hbox{{\rm Ne}\kern 0.1em{\sc ii}}}]\sim10$ (comparable to the ratio produced by a ${\hbox{T$_{eff}$}}=50,000$ K Pauldrach O star). Similarly, a $120,000$ K WN star can reach \[\]/\[\]$=100$. Thus, given these stellar atmospheres, only a WN star can give rise to a neon ratio between $10$ and $100$. As a result, there are mid–infrared line ratio regimes that only Wolf–Rayet stars can populate (again assuming solar metallicity.) For maximum effective temperatures of $^{MS}=50,000$ K, $^{WN}=120,000$ K, and $^{WC}=150,000$ K, for [@pauldrach01] model O stars and @hillmill WR stars, we find the following: - Main sequence O stars can only produce $[{\hbox{{\rm Ar}\kern 0.1em{\sc iii}}}]/[{\hbox{{\rm Ar}\kern 0.1em{\sc ii}}}] \le 18$, whereas WN and WC stars can reach ratios of $40$. - MS O stars can only produce \[\]/\[\]$=0.5$, while WC stars can reach $1.2$, and WN can reach $3$. - MS O stars can only produce \[\]/\[\]$=4$, whereas WC can reach $12$ and WN can reach $130$. - MS O stars and WC stars can only produce \[\]/\[\]$=2.5$, whereas WN stars can reach $14$. We have just seen that the conversion from mid–infrared line ratio to   is very different for main sequence stars than for Wolf–Rayet stars. Thus, even a modest portion of WR stars within a hot stellar population can significantly affect the line ratios. We also conclude that in a solar–metallicity starburst, if the line flux ratios exceed the maximum that main sequence stars can produce, then WR stars dominate the ionizing flux. Spectral Synthesis Models {#sec:specsynth} ------------------------- Given the influence of WR stars, we must consider the more realistic scenario of an evolving stellar population as the ionizing source. We used the spectral synthesis code Starburst99 version 4.0 [@starburst99] to create instantaneous starbursts with an initial mass function of Salpeter–slope [@salpeter] and initial stellar masses between $1$ M$_{\sun}$ and a variable upper mass cutoff, “” ($=100$, $75$, $60$, $50$, $40$, and $30$ .) ( in this paper always refers to the IMF, not the present-day mass function.) As in our single–star models, this version of Starburst99 uses O star model spectra from @pauldrach01 and Wolf–Rayet model spectra from the code of @hillmill, as prepared by @snc. We created two suites of models: the first set assumed solar metallicity in Starburst99 and Cloudy, and the “solar metallicity, high–mass loss” option, which is recommended as the default for Starburst99. (The alternative “standard mass loss” option gives qualitatively similar results.) The second set of models used a gas–phase metallicity of $1/5$ solar in Cloudy, and the “high–mass loss, Z=$1/5$ solar” and “uvlines = Magellanic” settings in Starburst99. Dust was ignored (and will be addressed in § \[sec:caveats\].) Other parameters were set as for the single–star models. Starburst99 calculated the spectral energy distribution (SED) of the burst every $0.1$ Myr for $10$ Myr after the starburst. The ionization parameter was normalized to a maximum value of $\log U = -2.3$, and scaled by the number of hydrogen–ionizing photons present in the SED. Given the SEDs as input, Cloudy calculated line ratios as a function of starburst age. Figure \[fig:models\] plots line ratios as a function of time for these simulations. Table \[tab-s99\] summarizes the spectral synthesis models with $=100$ and compares to line ratios from the single–star models. Line ratios versus time for the low–metallicity models are shown in figure \[fig:lowZmodels\]. Figure \[fig:models\] shows that in the first $2$ Myr, the mid–infrared line ratios fall from an initial plateau. By $2$ Myr, the O3 through O5 dwarfs (${\hbox{T$_{eff}$}}>44,500$ K) stars in the models have left the main sequence; by $2.5$ Myr, no O3 or O4 star of any luminosity class remains. Wolf–Rayet stars, together with the remaining main sequence stars, create a second period of relatively high line ratios from $3.5$ to $5$ Myr.[^2] While the line ratios predicted for the Wolf–Rayet phase are lower than predicted for WR–only nebulae, clearly the Wolf–Rayet stars are important: they produce a renaissance of high line ratios after the O stars have left the main sequence. It does not make sense to parameterize a mid–infrared line ratio as though the flux came from a single main sequence star; the ensemble of stars, including the Wolf–Rayets, must be considered. The models of @thornley, which otherwise used similar input spectra and nebular parameters to this work, did not include Wolf–Rayet stars. As a result, \[\]/\[\] drops monotonically with time in their figures 6 and 10, while our solar–metallicity curves (figure \[fig:models\]) are double–peaked. DIAGNOSTICS OF STELLAR  IN STARBURST GALAXIES {#sec:diags} ============================================= Approaches to Estimating ------------------------- In general, line ratios capable of indicating  also depend on metallicity, electron temperature, density, ionization parameter, and the morphology of the ionized regions. Without constraints on these other parameters,  can be difficult to determine (e.g., @morisset-apj). However, in the extreme conditions in starbursts, we expect less range in ionization parameter and morphology than in broad samples of  regions, and starburst metallicities can be constrained by other line ratios. Therefore, it is plausible that useful constraints on  can be derived for these regions. We return to this topic in § \[sec:depends\]. At optical wavelengths, the ratio of \[\] $5007$ to   is frequently used as a diagnostic since both lines are easily observed (see for example @SL96.) They are relatively close in wavelength, and Balmer ratios can be used to correct for residual extinction. However, large optical depths of interstellar extinction can make \[\]/reflect the conditions of the outer skin of starbursts only. Thus, \[\]/ may not indicate the average conditions throughout a highly–extincted starburst. Also, a ratio composed of a forbidden metal line and a hydrogen recombination line is particularly sensitive to the metallicity, electron temperature, and density of the nebular region. Additionally, \[\] can be shock excited [@cygloop]. @vp have proposed an all–forbidden line diagnostic $\eta^{\prime}$, which uses lines of \[${\hbox{{\rm O}\kern 0.1em{\sc ii}}}$\], \[${\hbox{{\rm O}\kern 0.1em{\sc iii}}}$\], \[${\hbox{{\rm S}\kern 0.1em{\sc ii}}}$\], and \[${\hbox{{\rm S}\kern 0.1em{\sc iii}}}$\] with wavelengths from $3726$ [Å]{} to $9532$ [Å]{}. This diagnostic was initially reported to work well for   regions [@kbfm], but it is sensitive to morphology and shocks [@oey]. Moreover, the diagnostic is poorly suited to starburst galaxies because it involves red lines that are seldom observed, and is extremely subject to reddening. More robust optical line diagnostics can be made by comparing strengths of helium recombination lines with recombination lines of hydrogen (Kennicutt [et al.]{} 2000; Ho, Filippenko, & Sargent 1997; Doherty [et al.]{} 1995). [ $6678$]{} Å and [ $4471$]{} Å are attractive for this purpose because their proximity to H$\alpha$ and H$\beta$, respectively, reduces reddening effects. However, these diagnostics still sample only the outer skin of the starburst, and the helium lines are weak, as discussed further in § \[sec:optical\]. Because they suffer less extinction, infrared spectral diagnostics probe more deeply into a starburst than optical ones. For example, $10$ magnitudes of extinction at $5500$ Å corresponds to only $1.1$ magnitudes at $2.2$ , and $0.8$ magnitudes at $10$ [@rl1985]. The mid–infrared fine structure lines are the most successful tools in this spectral region to estimate   [@roche; @kunze; @al; @thornley; @fs-m82]. These lines are less dependent on electron temperature than optical forbidden lines. However, their atomic constants are not well known [@f97; @f01; @vanhoof; @fe-proj-co], and they are still sensitive to metallicity and ionization parameter (see figure 10 of Thornley [et al.]{} 2000). Because of the lack of strong atomic lines, attempts to use near–infrared lines to measure  have focused on recombination lines of helium and hydrogen. For stellar ionizing sources, the hardness of the ionizing continuum determines the volume of He$^+$ relative to H$^+$ [@osterbrock]. For ${\hbox{T$_{eff}$}}> 40,000$ K, the He$^{+}$ and H$^{+}$ regions coincide; the Strömgren radii are approximately equal. For lower , the zone of ionized H extends beyond the central zone of singly–ionized He (see figures 2.4 and 2.5 of Osterbrock (1989), and figure 1 of Shields (1993)). Thus, by measuring the relative volumes of He$^+$ and H$^+$ within a nebula, one can constrain the effective temperature of the ionizing stellar source(s). The line ratio of ${\hbox{{\rm He}\kern 0.1em{\sc i}}~$2.06$~{\micron}}/ {\hbox{{\rm Br}\kern 0.1em$\gamma$}}$ has been used to estimate  in starbursts (Doyon, Puxley, & Joseph 1992; Doherty [et al.]{} 1995) and planetary nebulae (Lumsden, Puxley, & Hoare 2001a). However, the strength of the ${\hbox{{\rm He}\kern 0.1em{\sc i}}~$2.06$~{\micron}}$ $2^1P \rightarrow 2^1S$ line is not determined simply by recombination cascade, but also by the population in the $2^1$P state [@shields]. This level is pumped from the ground state by $\lambda = 584$ [Å]{} photons in the resonance transition $1^1S \rightarrow 2^1P$ [@shields; @bs]. Photoionization of hydrogen, dust absorption, or Doppler shifting can change the resonance efficiency and thus the occupation of the $2^1P$ state. The state can be further populated by collisions from the triplet states, primarily from $2^3S$ [@shields]. A small ${\hbox{{\rm He}\kern 0.1em{\sc i}}~$2.06$~{\micron}}/ {\hbox{{\rm Br}\kern 0.1em$\gamma$}}$ ratio should indicate a soft continuum where there are few $584$ Å photons and few helium recombinations. Otherwise, the ratio is likely to be a poor measure of starburst  due to the dependence on nebular dust content, electron temperature, and density, as well as on the ionizing continuum. Some of this complex behavior is seen in photoionization models (figure 1d of Shields 1993). The ${\hbox{{\rm He}\kern 0.1em{\sc i}}~$1.7$~{\micron}}/ {\hbox{{\rm Br}\kern 0.1em$10$}}$ ratio was proposed as a  diagnostic by @vanzi, and has been measured in several starburst galaxies (Vanzi [et al.]{} 1996; Vanzi & Rieke 1997; Engelbracht, Rieke, & Rieke 1998; Förster Schreiber [et al.]{} 2001) and planetary nebulae (Lumsden, Puxley, & Hoare 2001b). The [ $1.7$ ]{} line and  are close in wavelength, and A$_{1.7~\micron}$ is only one-sixth of A$_V$, making their ratio nearly reddening-independent and also allowing the photons to escape from relatively obscured regions. Unlike [ $2.06$ ]{}, the [ $1.7$ ]{} $4^3D \rightarrow 3^3P^0$ transition arises almost entirely from recombination cascade. The relevant levels are triplet states, so they cannot be pumped from the ground state, because an electron spin flip would be required [@bs]. As a result, the line ratio is insensitive to nebular conditions, and is determined almost entirely by the relative sizes of the H and He ionization zones. Figure 8 of @fs-m82 plots the behavior of the [ $1.7$ ]{}/  ratio as a function of , as predicted by photoionization models for a starburst environment ionized by hot main sequence stars. The [ $1.7$ ]{}/ ratio is small for $ < 30,000$ K because there are many more photons capable of ionizing hydrogen (ionization potential of $13.6$ eV) than neutral helium (ionization potential of $24.6$ eV). For $>30,000$ K, the ratio rapidly increases as the zone of singly–ionized helium overlaps more of the hydrogen Strömgren sphere. The ratio then saturates for $> 40,000$ K, as the He$^+$ and H$^+$ regions coincide. For $n_e = 100$ , the saturated ratio is $${\hbox{{\rm He}\kern 0.1em{\sc i}}}\ 1.7 / {\hbox{{\rm Br}\kern 0.1em$10$}}\ = 3.60~C_{1.7}~[n(He) / n(H)] , \label{eq:saturated}$$ where $n(He)/n(H)$ is the gas–phase abundance of helium (by number) relative to hydrogen, and the term C$_{1.7}$ expresses the weak dependence on electron temperature [@vanzi]. For T$_e = 10^4$ K, C$_{1.7} = 1.000$; other values are listed in table \[tab:c17\]. The helium abundance $n(He)/n(H)$ increases from the primordial value of approximately $0.08$ [@ih-he; @bono] to $0.1$ for the Milky Way. Thus, [ $1.7$ ]{}/ should saturate at a value of $0.27$ to $0.38$. The  diagnostics discussed above do not necessarily agree. For example, in the starburst galaxy He 2–10, \[\]/and \[\]/  indicate $>39,000$ K [@he2-10opt], whereas mid–infrared line ratios indicate $<37,000$ K [@roche]. The [ $2.06$ ]{}/ observed by @vr would indicate $=39,000$ K using the conversion of @dpj. At poor signal–to–noise, @vr measure [ $1.7$ ]{}/ and find $=36,000$ K. This few thousand Kelvin disagreement translates into a serious disagreement in stellar mass: a  of $36,000$ K corresponds to approximately an O8V spectral type, which from eclipsing binaries should have a mass of $\sim 22$ to $25$   [@andersen; @ostrov; @niemela; @gies]; whereas a  of $40,000$ K corresponds to an O6.5V to O7V spectral type, which should have a mass of $\sim 35$  [@gies; @niemela]. We now test  diagnostics against each other in light of the stellar synthesis models detailed above. Because the line physics of [ $1.7$ ]{}/ is simple and well–understood (see § \[sec:diags\]), we assume this diagnostic is unbiased, and thus accurately reflects the ionizing continuum, within the limitations of measurement error. Testing [ $2.06$ ]{}/ {#sec:heh_hek} --------------------- In this section we consider the galaxies for which we obtained [ $1.7$ ]{}/ measurements, as well as three galaxies with [ $1.7$ ]{}/ measurements available in the literature: NGC 253 [@chad], for which the stellar continuum was subtracted as in this work; M82 [@fs-m82], for which representative stellar spectra were subtracted; and NGC 5253 [@vr], for which the stellar continuum is weak enough to ignore. These three galaxies, together with the six galaxies for which we observed [ $1.7$ ]{}/, we term our expanded sample. We also take measurements of [ $2.06$ ]{}/ from the literature for the galaxies in the expanded sample. Figure \[fig:heh\_hek\] plots [ $2.06$ ]{}/versus [ $1.7$ ]{}/. NGC 3077, NGC 4861, NGC 4214, and He 2–10 all have [ $1.7$ ]{}/ ratios consistent with the saturated value of $\approx 0.3$, within the measurement errors and the expected variation of helium abundance. Thus, these starburst regions appear to contain massive stars ($> 39,000$ K if main sequence stars.) By contrast, NGC 253, NGC 4102, and the nucleus of M82 have [ $1.7$ ]{}/$<0.15$, and thus are inferred to have softer ionizing continua ($\lesssim 37,000$ K if main sequence stars.) NGC 3504 and the two off–nuclear regions of M82 have line ratios intermediate to these extremes. Figure \[fig:heh\_hek\] illustrates that [ $2.06$ ]{}/ does not trace [ $1.7$ ]{}/ as the models predict. The nucleus of M82 demonstrates that [ $2.06$ ]{} may be strong while [ $1.7$ ]{} is weak, contrary to the expected behavior (but expected if [ $2.06$ ]{} is pumped.) However, for most galaxies, [ $2.06$ ]{} is *too weak* for the measured [ $1.7$ ]{}. This is the first direct demonstration that [ $2.06$ ]{}/ is a poor diagnostic of  in starburst galaxies. Radiative transfer considerations have predicted that the behavior of [ $2.06$ ]{} should not be a simple function of  [@shields]. @lph320 confirm this complex behavior for planetary nebulae, though they attempt to constrain the dependence on T$_e$ and density by also considering optical  lines [@doherty95]. However, the data do not contradict the expectation that a very low [ $2.06$ ]{}/ ratio (below $\sim0.2$) indicates that the continuum is fairly soft, because there would be few ionizing photons and also few resonantly scattered photons. We further consider the reliability of the [ $2.06$ ]{}/ ratio in figure \[fig:ne\_206\], by comparing it to the mid–infrared line ratio \[\] 15.6/\[\] 12.8. Here, too, [ $2.06$ ]{}/ is too low for a given \[\]/\[\] (compared to model predictions) and there is no obvious correlation between the two ratios. An alternative interpretation of figure \[fig:ne\_206\] would be that [ $2.06$ ]{}/ is correct and \[\]/\[\] is systematically overproduced; we feel this is unlikely because, as we will demonstrate in § \[sec:midir-test\], \[\]/\[\] is *underproduced* in starburst galaxies with respect to the predictions of a Salpeter IMF extending to $100$ . Testing Optical  Indicators {#sec:optical} --------------------------- How well do optical forbidden and recombination line ratios estimate  in starbursts? Figures 9 and 10 of @kbfm show that the recombination ratios [ $5876$]{}/  and [ $6678$]{}/, as well as \[\]/, all track  well in Milky Way, LMC, and SMC  regions, where could be determined by classifying all the ionizing stars. How well do these diagnostics perform in starburst galaxies? In figure \[fig:optical\], using dereddened data from @ho3, we compare the behaviors of [ $6678$]{}/ and \[\]/  in nuclear starbursts to the predictions of Starburst99/Cloudy photoionization models. Galaxies with \[\]/$<0.5$ generally have low [ $6678$]{}/, indicating general agreement that  is low in these galaxies. At higher line ratios, there is considerable scatter. For most of the plotted galaxies, \[\]/ is systematically high for a given [ $6678$]{}/, compared to a solar–metallicity, $=100$  track. Lowering the metallicity of the model reduces but does not eliminate the disagreement between diagnostics. Only an extreme model (low metallicity, $=30$ ) can fit the data reasonably well. A possible explanation would be that \[\] in starbursts is often shock–excited by supernovae [@cygloop], which would be a rare effect in  regions and thus not affect the @kbfm plots. In particular, \[\]/ values above $\sim 2.5$ require sub–solar metallicity or excitation by shocks. Thus, figure \[fig:optical\] suggests that \[\]/is systematically high or [ $6678$]{}/ is systematically low in starburst galaxies. As a further test, figure \[fig:o3\_heh\] plots \[\]/ versus [ $1.7$ ]{}/ for our expanded sample. Overplotted are Starburst99/Cloudy models as in figure \[fig:optical\]. For low values of \[\]/, the error bars are too large to judge whether the two diagnostics correlate. As in figure \[fig:optical\], the highest \[\]/values observed require sub–solar metallicity or shock excitation of \[\]. Next, we examine the behavior of \[\]/ versus \[\]/\[\] in figure \[fig:o3\_Ne\] (omitting for now NGC 5253, II Zw 40, and NGC 55 because of their low metallicity.) Optical line ratios are from the literature, and *ISO* observations of \[\]/\[\] are from @thornley. Galaxies with \[\]/$<0.5$ generally have line ratios consistent with the overplotted Starburst99/Cloudy models. With higher \[\]/, the scatter increases. Without accurately knowing the metallicity of each galaxy in figure \[fig:o3\_Ne\], it is difficult to judge how much of the scatter in \[\]/ versus \[\]/\[\] is due to the sensitivity of \[\]/ to metallicity rather than effective temperature. According to the Cloudy models, metallicity alone cannot explain the line ratios of NGC 6240, IC 1623A, Arp 220, NGC 3690A, and NGC 7469 (and possibly NGC 972) in figure \[fig:o3\_Ne\]. Low metallicity and a upper mass cutoff of $30$  could together explain all but IC 1623A and NGC 6240. Alternatively, aperture mismatch, severe extinction, or shock excitation of \[\] could be at work. NGC 972 is not strongly centrally concentrated in optical emission line images, so the explanation may lie in aperture mismatch: the optical line ratios were measured with slitwidths of a few arcseconds, while the *ISO* neon lines were measured with a $14$ by $27$ aperture. The remaining discrepant galaxies all have very heavily obscured star formation regions, and it is likely that the discrepancy arises because the optical and mid–infrared spectra sample distinctly different regions along the line of sight. We also note that NGC 278 has extremely low \[\]/ for its measured \[\]/\[\]. Higher–spatial resolution mid–infrared spectroscopy (e.g., with SIRTF) may resolve this discrepancy. We will delay discussion of whether \[\]/\[\] is a reliable  diagnostic until § \[sec:midir-test\]. Next, we consider the optical helium and hydrogen recombination lines, which should form more accurate starburst  diagnostics than a forbidden/recombination pair like \[\]/. To reduce reddening effects, we select  lines close in wavelength to H lines. Unfortunately, the helium lines are weak: [ $6678$]{} saturates at $0.014$ of the strength of , and [ $4471$]{} saturates at $0.05$ of . As such, in the spectral atlas of @ho3, [ $6678$]{} was detected in only $108$ of $418$ galactic nuclei, and [ $4471$]{}  in only $16$ nuclei. The small sample indicates that [ $4471$]{} is only marginally detected, and we do not consider it further. Figures \[fig:optical\] and \[fig:o3\_Ne\] have already implicated \[\]/ as an unreliable  indicator for \[\]/$\ga0.5$. This makes it hard to gauge the reliability of [ $6678$]{}/ in figure \[fig:optical\]. Also, the sample sizes are too small to compare the optical recombination line ratios to \[\]/\[\], [ $1.7$ ]{}/, or [ $2.06$ ]{}/individually. Instead, we use the latter three  indicators together to test how well the optical recombination line ratios correlate with . In table \[tab-opt\], we list galaxies with measurements of at least two different  indicators, in order of increasing , as determined from \[\]/\[\], [ $1.7$ ]{}/, and [ $2.06$ ]{}/ (when $\le 0.2$), as available. Due to measurement error and uncertainty in the relative calibrations of the diagnostics, the ordering is somewhat uncertain. The published plots of the @ho3 spectra lack the dynamic range to assign upper limits to the undetected optical recombination lines. These are marked as “non det” in table \[tab-opt\]. In general, table \[tab-opt\] shows some correlation between [ $6678$]{}/ and , though with considerable scatter. Using Kendall’s $\tau$ rank correlation test on the eight galaxies with measured [ $6678$]{}/, there is only a $5\%$ chance that  and [ $6678$]{}/ are uncorrelated. Testing the Mid–Infrared Fine Structure Line Ratios {#sec:midir-test} --------------------------------------------------- In the mid–infrared, ratios of the fine structure lines \[\] $15.6$ , \[\] $12.8$ , \[\] $8.99$ , \[\] $6.99$ , \[\] $10.5$ , and \[\] $18.7$   have been used to test for the presence of hot stars in starbursts. From space, *ISO* measured these lines at low spatial resolution ($14$ by $27$ aperture for \[\]/\[\]) [@thornley; @fs-m82; @kunze]. Ground–based observations [@roche; @al] provide higher spatial resolution, but only the \[\], \[\], and \[\] transitions can be observed through the atmosphere. As a result, ground–based studies must use heteronuclear line ratios, which are less ideal than homonuclear ratios available from space because they are much more sensitive to elemental abundances and dust depletion. In table \[tab-midir\], we collect measurements of the mid–infrared line ratios and [ $1.7$ ]{}/ in starburst galaxies, including five regions within M82. M82 provides a testing ground for the accuracy of the mid–infrared line ratios as  diagnostics; *ISO* measured homonuclear line ratios in the center [@fs-m82], and this region has been mapped at 1 resolution in \[\], \[\], and \[\] [@al], identifying the nucleus and three infrared–bright regions nearby (all regions defined in the footnotes to table \[tab-midir\].) Based on the heteronuclear mid–infrared line ratios and [ $1.7$ ]{}/, we find that region W2 and the nucleus of M82 both require $<65$ , region E1 requires $<60$ , and region W1 requires $<50$ . The heteronuclear and homonuclear mid–infrared line ratios and [ $1.7$ ]{}/  within the SWS/ISO aperture require $<50$ . Models with $=100$, $75$, $70$, or $65$  do not produce the observed ratios in any of these regions. Thus, we find that the heteronuclear line ratios give consistent ages and upper mass cutoffs for individual regions near the center of M82, in agreement with [ $1.7$ ]{}/, and when averaged over the *SWS/ISO* aperture, give answers consistent with the homonuclear line ratios. We further test the mid–infrared line ratios using the five other solar–metallicity galaxies listed in table \[tab-midir\]. For NGC 4102 and NGC 6240, the constraints are poor, and the line ratios can be fit by $=40$ to $100$ . In NGC 6946, [ $2.06$ ]{}/  and \[\]/\[\] disagree unless $<65$ , but as we cautioned in § \[sec:heh\_hek\], [ $2.06$ ]{} is not a reliable diagnostic. For NGC 253, [ $1.7$ ]{}/ and the neon ratio cannot be simultaneously matched by the $=100$  model, but models with $\le 75$  can fit the ratios. For He 2–10, the line ratios require $<65$ , mostly because of low observed \[\]/\[\]. Our conclusion is that in individual regions and entire starbursts, the different heteronuclear and homonuclear mid–infrared line ratios and [ $1.7$ ]{}/  give consistent answers as to age and . This agreement supports use of the mid–infrared line ratios as diagnostics of the ionizing radiation field. The mid–infrared lines have large equivalent widths and a range of excitation energies, making them potentially powerful diagnostics. DIAGNOSING IONIZING CONDITIONS IN STARBURSTS {#sec:neon} ============================================ Mid–Infrared Line Ratio Dependencies {#sec:depends} ------------------------------------ The mid–infrared line ratios depend on several physical parameters: metallicity, ionization parameter, morphology, and the strength and shape of the ionizing continuum. To be confident in applying these ratios, we need to disentangle these various effects. We consider each parameter in turn. [**Metallicity**]{}. As metallicity decreases, the relative high–excitation line emission increases, because lower–metallicity stars have harder spectra and because lower–metallicity nebulae cool less efficiently. Another effect is that Wolf-Rayet stars require larger progenitor masses with decreasing metallicity. These effects can be seen by comparing the low–metallicity models (figure \[fig:lowZmodels\]) with the solar–metallicity models (figure \[fig:models\]). Using Starburst99 and Cloudy, we find that Z$= 0.2$ times solar models have initial mid–infrared line ratios that are $\sim 3$ times greater than solar–metallicity models; these line ratios fall more slowly with time than in solar–metallicity models. While metallicity affects the mid–infrared line ratios, metallicity can be measured and corrected for. Within galaxy samples that have similar measured metallicity, uncertainties in the metallicity should affect the mid–infrared line ratios by factors that are much smaller than the orders–of–magnitude changes in line ratio values expected due to   (as discussed in § \[sec:specsynth\]). [**Ionization parameter and morphology**]{}. The ionization parameter, as the ratio of the spectral intensity to the gas density, combines two of the fundamental parameters that determine the degree of ionization in a nebula. In a Galactic  region, the ionization parameter changes rapidly with radius because of the 1/R$^2$ falloff and absorption of UV photons by the nebula (which also alters the spectral shape.) Morphology then determines which parts of the nebula influence others. A starburst galaxy, however, is much messier than an assembly of pseudo-spherical  regions: the ISM is generally fragmented, and gas parcels are ionized by many stars. For example, in M82, it appears that the interstellar medium is highly fractionated (e.g., Seaquist, Frayer, & Bell 1998) and that the whole $\sim 450$ pc nuclear starburst and individual $\sim 20$ pc star–forming clusters can be described by a single ionization parameter (@thornley, citing the dissertation of N. Förster Schreiber.) Thus, it seems more appropriate to model a starburst as though the gas and stars are thoroughly mixed (by employing a mean UV spectrum and mean ionization parameter), rather than as a collection of spherical clouds, each with a single ionization source. This “mixed gas and stars” model is achieved in practice by assuming plane–parallel geometry and a composite ionizing spectrum. Ionization parameters (U) have been estimated in several nearby starburst galaxies by measuring the number of Lyman continuum photons and the size of the starburst region. [@thornley] summarize measurements in NGC 253, NGC 3256, and M82, which are all consistent with $\log U = -2.3$. Measurements have also been obtained for Arp 299 [@arp299]; NGC 1614 [@ngc1614]; NGC 1808 [@ngc1808]; IC 342 [@ic342]; NGC 6946 [@ngc6946]; and NGC 3049 [@ngc6946]. In addition, @hbt measure the Lyman continuum flux in fourteen nearby galaxies. Six of these galaxies have multiple measurements of $\log U$, which gives some estimate of the (often considerable) uncertainty. In figure \[fig:IP-local\] we plot the ionization parameters derived from these studies. It should be noted that each of these $U$ values is actually a *lower limit*, since we use the maximum radius of the starburst region to compute the ionization parameter. When the gas density was not measured, we assume $n_e = 300$ ; the true ionization parameter scales as $IP = IP_{300} - \log (n_e / 300~{\hbox{cm$^{-3}$}})$. Figure \[fig:IP-local\] shows that the ionization parameter used in our simulations, $\log U \le -2.3$, is a reasonable average value given the measurements available for nearby starbursts. How sensitive are the mid–infrared line ratios to the ionization parameter? Reducing $U$ in our models by a factor of $10$ lowers the \[\] $15.6$  / \[\] $12.8$  line ratio by a factor of $\sim 7$. Therefore, if the ionization parameters of starburst galaxies vary by a factor of $\sim10$ or more, this parameter could account for considerable spread in observed mid–infrared line ratios. However, there is no tendency for galaxies with small \[\]/\[\] to have low ionization parameters in figure \[fig:IP-local\], indicating that $U$ is not the dominant parameter determining this flux ratio. Comparing with the restricted range of [*U*]{} observed in starbursts, we conclude from the modeling in § \[sec:specsynth\] that  dominates variations in this line ratio in such regions. Starburst ISM morphologies are far too complex to reproduce in simulations; fortunately, parameterization of a starburst by a single, global ionization parameter and a mean ionizing spectrum is physically motivated, agrees with observations, and simplifies the problem sufficiently to allow modeling. Another test of the diagnostic usefulness of the mid–infrared line ratios is provided by studies of Galactic  regions. @mh2 found that, in compact  regions, the line ratios \[\] $15.6$ / \[\] $12.8$ , \[\] $8.99$ / \[\] $6.99$ , and \[\] $10.5$ / \[\] $18.7$ correlate very well with each other, suggesting their reliability. @morisset-apj has also demonstrated the use of these lines to estimate  and $U$ in Galactic  regions, though as demonstrated by @morisset-aa, outside constraints on ionization parameter and metallicity are usually necessary. The Spectrum of the Ionizing Radiation -------------------------------------- We now focus on using the fine structure line ratios to estimate the spectrum of the ionizing radiation in starbursts. As figure \[fig:models\] illustrates, once an instantaneous burst is older than $6$ Myr, \[\]/\[\], \[\]/\[\], and \[\]/\[\] are so low ($<0.001$) that the higher–ionization line should not be detected. \[\]/\[\] and \[\]/\[\] fall off more slowly, but still require a dynamic range exceeding $100$ to detect both lines in each ratio. Such very low line ratios are not seen in @thornley, which with $27$ \[\] $15.6$ /\[\] $12.8$   measurements is the largest sample to date of mid–infrared fine structure lines in starburst galaxies. The lowest ratio detected by @thornley is $0.05$, and $5$ galaxies have upper limits. The simplest explanation of this behavior is that massive stars continue to form at low rates after the peak of a starburst. In the @thornley sample, all but $3$ galaxies have \[\]$<$\[\]. The three outliers, with neon ratios from $1$ to $12$, are all low–mass, low–metallicity galaxies (NGC 55, NGC 5253, and II Zw 40). We will consider the higher–metallicity galaxies now, in the context of the solar–metallicity models, and defer discussion of the low–mass, low–metallicity galaxies to § \[sec:lowZspecsynth\]. In figure \[fig:models\], as  decreases, the line ratios decrease during the main sequence phase (because the ionizing spectrum softens), and the gap widens between the two phases of high line ratios (because fewer stars become Wolf–Rayets.) We now consider these models in light of the measured neon ratios of @thornley, which are overplotted in figure \[fig:ne-newplot\]. In the $=100$  model, for $46\%$ of the first $5$ Myr, the predicted \[\]/\[\] exceeds the highest line ratio measured by @thornley for a high–mass, $\sim$solar–metallicity galaxy; thus, this model poorly fits the data. A much better fit is the $Z=Z_{\sun}$, $=40$  model. For only $6\%$ of the first $5$ Myr does this model predict \[\]/\[\]$>1$; for $65\%$ of that time, it predicts neon line ratios within the range of the Thornley detections. The $=40$  model fits markedly better than the $=50$ and $30$  models. Because one–quarter of the Thornley datapoints are upper limits (excluding the three low–mass, low–metallicity galaxies), the M$_{up}=40$ model is a better fit to the Thornley data than the above percentages indicate. One draws the same conclusion from continuous star formation models, as shown in figure \[fig:continuous\]. Such models with $=100$ and $75$  predict a constant neon ratio above $1$, while the neon ratio for the $=30$  model falls below the Thornley range. The $=40$ and $50$  models predict neon line ratios within the Thornley range; the $=40$ model comes closer to the median. These results are consistent with those of § \[sec:midir-test\], which found that the heteronuclear and homonuclear mid–infrared line ratios within four regions of M82 required $<50$ to $<65$ (depending on the region), that He 2–10 required $<65$ , and that NGC 253 required $<100$ . Thus, \[\]/\[\] in the high–mass, solar–metallicity @thornley galaxies, and a concordance of line ratios in M82 and He 2–10, are all significantly lower than the predictions of a Salpeter IMF extending to $100$ . An IMF that is deficient in massive ($\ga40$ ) stars could produce the observed line ratios. Ionizing Conditions in Low Metallicity Starbursts {#sec:lowZspecsynth} ------------------------------------------------- We now discuss mid–infrared line ratios in low metallicity starbursts. Lowering the metallicity from solar elevates the mid–infrared line ratios, and fills in the valley between the MS and WR phases. [ $1.7$ ]{}/ is completely saturated until the WR stars die, by contrast to its double–peaked behavior for solar metallicity. Overplotted in figure \[fig:lowZmodels\] are the line ratios for II Zw 40, NGC 5253, and NGC 55, in order of decreasing \[\]/\[\] ratio from @thornley. All three of these galaxies have low metallicity: II Zw 40 has measured \[O/H\] $=0.20\pm0.01$ [@diazetal] and \[O/H\]$=0.19\pm0.04$ [@garnett89], \[S/H\]$=0.12\pm0.03$ [@garnett89], and \[Ne/H\]$=0.3$ [@martinhernandez], all linear and relative to solar abundance. NGC 5253 has measurements of \[O/H\]$=0.28$ (Storchi–Bergmann, Kinney, & Challis 1995) and \[Ne/H\]$=0.58$ [@martinhernandez]. NGC 55 has measured \[O/H\]$=0.25$ to $0.37$ [@websmith]. We consider the line ratios of these galaxies in light of the low–metallicity models. For NGC 55, the only mid–infrared line ratio available in the literature is \[\]/\[\]; the observed value can easily be produced by any  from $30$ to $100$ . For II Zw 40, the observed \[\]/\[\]$=12$ cannot be achieved by Starburst99/Cloudy models with solar metallicity. With the low metallicity models, we find that the observed \[\]/\[\], \[\]/\[\], and \[\]/\[\] line ratios cannot be produced at any age unless  is greater than $40$ . The [ $1.7$ ]{}/ ratio agrees that the ionizing field is rather hard, but is insensitive to . For NGC 5253, unless the burst is $<0.5$ Myr old, the measured \[\]/\[\] requires $>40$ . This constraint is strengthened if we consider the \[\]/\[\], \[\]/\[\], and [ $1.7$ ]{}/ ratios, which all predict ages within $3$ to $5$ Myr, for a broad range of  ($40$ to $100$.) The [ $1.7$ ]{}/ constraint is particularly insensitive to . If one assumes this age range, the \[\]/\[\] ratio requires $>60$ . Thus, while high–mass, solar–metallicity starburst galaxies are seen to have lower \[\]/\[\] ratios than a Salpeter IMF with $=100$ predicts, the low–metallicity galaxies II Zw 40 and NGC 5253 have the high neon ratios expected if they contain very massive stars. Caveats and Assumptions {#sec:caveats} ----------------------- How robust is the conclusion that the nebular line ratios indicate that most high–mass, solar–metallicity starbursts have soft ionizing continua? First, we have assumed that the Thornley galaxies are generally of solar metallicity. If they were more metal–poor, this would raise the predicted line ratio curves, and thus increase the discrepancy between the predicted and observed ratios. The opposite effect (weakening our constraint) occurs if the Thornley galaxies have super–solar metallicity. @thornley use the strong–line method to derive metallicities of $1.9\pm 1$ Z$_{\odot}$ for $13$ of their galaxies (excluding NGC 5253 and II Zw 40.) This result is consistent with the metallicities from optical line ratios, but we prefer the Thornley mid–infrared estimate because it should be reddening-independent. Starburst99 is not optimized for such metallicities, but we use twice–solar models nonetheless to crudely estimate whether super–solar metallicities could void our result. For $=100$, $75$, and $60$ , doubling the metallicity from solar lowers the \[\]/\[\] line ratios and increases the duration of the WR phase by $\sim0.5$ Myr, which brings the models closer to agreement with observations, but deepens the trough between the main sequence and WR phases to $100\times$ below the lowest Thornley detection. These models predict neon line ratios within the observed Thornley range for $\sim40\%$ of the first 6 Myr—little better than the solar–metallicity $=100$ model. To summarize, while uncertainties remain because metal–rich stellar evolution is not well understood, current models indicate that the low line ratios observed in starburst galaxies are unlikely to be explained away by metallicity effects. Another way to negate the conclusion would be for the high–mass, solar–metallicity starburst galaxies to have much lower ionization parameters than we assumed. For the observed neon line ratios to arise in starbursts with $=100$ , the starbursts must have $U$ about $10$ times weaker than our assumed $\log U_{max}=-2.3$. None of the 18 galaxies in figure \[fig:IP-local\] has a measured ionization parameter this low. In fact, because in our models the ionization parameter starts at $\log U = -2.3$ and falls with the ionizing flux, the ionization parameter in our models is already fairly low. (For example, 5 Myr after a solar–metallicity, $ = 100$  burst, the ionization parameter has fallen to $\log U = -3.15$.) Thus, our ionization parameter assumptions are conservative in that they tend to predict low line ratios for a given ; as a result, when comparing to observed line ratios, the models will be slightly biased toward finding high . Thus, the choice of ionization parameter is not the reason we find generally low  in starburst galaxies; the models are actually biased against finding this result. For simplicity, we have modelled star formation as an instantaneous burst. Starburst galaxies are of course more complicated. An instantaneous burst is the *most conservative* assumption of star formation history for the purpose of constraining . As illustrated in figure \[fig:continuous\], extended star formation or a series of bursts would elevate predicted line ratios above the instantaneous–burst case for most of the burst duration. As such, extended star formation would increase the discrepancy between the low ratios observed in starbursts and the high ratios predicted by high– models. Dust grains harden the ultraviolet ionizing continuum, as pointed out by @aannestad. Thus, if dust competes for the ionizing photons, this elevates the line ratios, and our conclusions are strengthened. Figure \[fig:ne-newplot\] shows this effect in Starburst99/Mappings models with and without dust. These models were created using the Starburst99/Mappings III web interface, beta test version 3q [@mappings]. That figure also shows that the two different photoionization codes Mappings and Cloudy, given the same input spectra and nebular conditions, predict very similar neon line ratios. This helps address the concern that our results depend on the reliability of photoionization codes and their input atomic constants. The other major assumptions in our work are the choice of stellar evolution tracks and hot stellar spectra. Had we used the (hard–spectrum) CoStar models, they would have increased the predicted line ratios and made the @thornley galaxies seem even more deficient in high–mass stars. Thus, our use of the softer @pauldrach01 atlas is conservative in terms of existing hot star models. However, our conclusions could be invalidated if real stars have much softer ionizing continua than @pauldrach01. We note that NGC 3077, 4214, and 4861 now have well–measured, saturated [ $1.7$ ]{}/, but no published mid–infrared spectra. Mid–infrared spectra of these galaxies should further test the trends in nebular line behavior discussed in this paper (all of these galaxies would appear to fall into the low–mass, low–metallicity category). UV and Nebular Diagnostics in Conflict? {#sec:UV} --------------------------------------- The very massive stellar populations of a number of starburst galaxies have been constrained by ultraviolet spectroscopy. In cases where the burst age is more than $\sim5$ Myr, the UV spectra cannot test for stars above $40$  because the most massive stars have already exploded as supernovae or evolved off the main sequence (e.g., @delgado99). A small number of starbursts have strong P Cygni profiles indicative of a very young burst and the presence of very massive stars. Thus, there appears to be a tendency for ultraviolet spectra of stellar populations to indicate larger  than do the nebular lines (although the galaxy samples observed in the UV and mid–infrared hardly overlap). We now consider the cases of He 2–10 and NGC 3049; the UV spectra of both these starburst galaxies show P Cygni profiles, and nebular spectra are available (mid–infrared for the former galaxy, and optical for the latter.) He 2–10 is an extremely rare case of a starburst which has available ultraviolet spectra of adequate quality to search for P Cygni line profiles as well as high–quality mid–infrared line measurements. Although He 2–10 is of low mass and metallicity globally, the abundances in its nuclear  regions are approximately solar (Kobulnicky, Kennicutt, & Pizagno 1999). Best fits to the UV spectrum require $\ge 60$  [@chandar]. From our modelling of the mid–infrared line ratios, we find $<65$ . Thus, these observations permit a discrepancy between the diagnostics, but do not require one. For NGC 3049, mid–infrared spectra are not available, but optical and UV spectra are. This galaxy is of solar (or slightly higher) metallicity in the starburst regions (Guseva, Izotov, & Thuan 2000) although it is of low mass and luminosity, and hence probably of low global metallicity. @delgado find that the P Cygni line profiles of  and   in NGC 3049 require $\ge60$ , and rule out ages younger than $2.5$ Myr and older than $4$ Myr. Further, they find that the UV diagnostics disagree with optical nebular lines as to whether massive stars are present; they fit the optical nebular lines by a $=40$ , t$=2.5$ Myr model—parameters which would not create the observed P Cygni profiles in the UV. Given these results, the authors question whether nebular line ratios can reliably indicate the presence of massive stars. We therefore re–examine the nebular line results for NGC 3049. In modelling these lines, @delgado used an older version of Starburst99 that employed pure helium WR models (Schmutz, Leitherer, & Gruenwald 1992) and Kurucz O star atmospheres prepared by Lejeune, Cuisinier, & Buser (1997). An update of Starburst99 incorporating new stellar models (@pauldrach01 and @hillmill, as packaged by @snc) became available after submission of their paper. The authors note that these new stellar models would soften the ionizing spectrum and reduce the discrepancy with the UV results, but they did not make a detailed reconciliation. Using our models, which make use of these new stellar atmospheres, we re–examine the nebular lines of NGC 3049. In @delgado, $\log U$ is fixed with time, and varies with radius ($\log U = -2.58$ at R$_{max}=100$ pc) in a spherical model. This results in a generally stronger $U$ than in our models, in which $U$ falls with time. To compare with the results of @delgado, we ran new models with $\log U$ fixed at $-2.3$. This value for $\log U$ is within the measurement uncertainties of n$_H$ and Q(H) of the value used by @delgado. This choice of slightly higher ionization parameter biases our test toward low values of  (and agreement with the results of @delgado). We consider the age range $3\le t \le 4$ Myr, as required by the UV lines [@delgado]. Over this time period,  5876/ can be fit by $40< {\hbox{M$_{up}$}}\le 100$ , and \[\] 5007/ by $40 \le {\hbox{M$_{up}$}}\le 100$ . \[\] 6584/ only requires $<75$, \[\] 6716/ and \[\] 3727/ can be fit by any  from $30$ to $100$, and \[\] 6731/ and \[\] 6300/ cannot be fit by any model. Thus, even using a high $U$ model, we do not find that low  is required. The updated stellar models remove the inconsistency between the UV and nebular lines noted by @delgado in NGC 3049. DISCUSSION ========== We have used a number of tests to show that the nebular line ratio \[\] $15.6$ /\[\] $12.8$  is a robust measure of the hot stellar population in starbursts. The line ratio is virtually unaffected by extinction, and as a homonuclear ratio involving a rare gas it is not subject to abundance variation or depletion onto dust. Where it can be compared to other reliable  indicators, the agreement is good. Since the mid–infrared neon lines vary over several orders of magnitude during a few million years of starburst evolution, measurements of moderate precision can give good  constraints. The neon ratio indicates low  in all members of a reasonably large sample of massive, high–metallicity starburst galaxies [@thornley]. We have shown that plausible modifications to the interpretive models (adding dust, lowering the metallicity, changing the ionization parameter) leave the basic constraint of low  unchanged or strengthened. The conclusion from \[\] $15.6$ /\[\] $12.8$  contrasts with the evidence for massive, hot stars from P Cygni line profiles in the ultraviolet spectra of two galaxies, He 2–10 and NGC 3049. In the first case, we find that the infrared nebular lines are consistent with the hot stellar spectrum indicated in the UV. In the second case, no mid–infrared spectrum exists, and the optical spectrum of @delgado does not conflict with the UV result. Thus, there is no overt conflict between the P Cygni lines in UV starburst spectra and the limits on  set by nebular lines. However, to account for the observed low–excitation nebular spectra of starbursts, galaxies like NGC 3049 must represent a very rare stage in starburst evolution. Can the UV wind observations, infrared nebular line results, and starburst models be reconciled, given this new constraint? Our calculations of the emission–line properties of starbursts are based on traditional synthesis modelling, as introduced by @rieke1980. Such modelling makes the assumption that newly–formed stars appear on the main sequence according to an assumed formation rate with masses given by an initial mass function. It has recently become popular to assume a Salpeter IMF, although @rieke-m82 derived a very similar IMF [*ab initio*]{} to fit the starburst properties of M82. (Both of these IMFs differ significantly from estimates of the local IMF, in that both have a substantially larger portion of massive stars). Assuming a Salpeter IMF extending to $100$ , we have shown that these models predict an early phase in starburst evolution, of duration $3$ to $4$ million years, when hot, massive stars should produce high–excitation emission lines. “Starbursts” are identified as–such up to ages of $15$ to $20$ Myr; thus, about $20\%$ of active starbursts should be in the early $<4$ Myr phase. However, the data of @thornley show no starbursts in massive, high–metallicity galaxies with the line ratios predicted for this early phase. One explanation for this discrepancy would be that the Salpeter IMF substantially overestimates the numbers of very massive stars. We have shown that the mid–infrared line ratios can be explained if the IMF cuts off at $40$ to $50$ . Parameterizing the IMF by a cutoff is an oversimplification; a substantial steepening of the IMF slope is probably a more appropriate description. One advantage of such an IMF is that it suppresses the production of oxygen, which can otherwise reach very high abundances in starbursts [@rieke-m82]. However, in addition to the indications from UV spectra that stars more massive than $40$  can form in substantial numbers in starbursts, the Arches Cluster near the center of the Milky Way has a large population of $\sim100$   stars [@figer]. (The mid–infrared line ratios in the Arches [@giveon] are consistent with a burst of age $2$–$3$ or $6$ Myr in our models, assuming twice–solar metallicity.) None of these observations can confirm the standard assumption of a Salpeter IMF extending to $100$ , and the possibility of rolloff in the IMF toward very high masses needs to be considered in detail. However, the Arches and the UV starburst results suggest it is unlikely that the lack of high–excitation emission lines can be explained entirely in terms of a substantial steepening in the IMF above $40$ – $50$ . We have therefore searched for other causes for this behavior. We believe an explanation can be found in an incorrect assumption in the standard synthesis models: that the full luminous output of newly–formed stars escapes into surrounding diffuse gas. This assumption justifies modelling starbursts as traditional low–density  regions. Instead, we suggest that the majority of massive stars in starbursts spend a substantial part of their main sequence lifetimes embedded within dense, highly–extincted regions—similar to the ultracompact  regions of the Milky Way—and are thus invisible to optical, near–infrared, and mid–infrared nebular line studies. In the solar neighborhood, it appears that about $15\%$ of the main sequence life of a massive star is spent within an ultracompact  region (Kurtz, Churchwell, & Wood 1994). Hanson, Luhman, & Rieke (1996) have detected in the near–infrared about half of a sample of radio–selected ultracompact  regions. They conclude that the detected regions typically are obscured by A$_V$ $= 30$ – $50$. Since the undetected regions in their sample should be even more heavily obscured, we take a typical case to be A$_V$ $\sim 50$. Thus, these objects would not contribute to the optical or near–infrared emission–line spectra of the Milky Way. The heavy extinction would even diminish the fluxes of the mid–infrared fine structure lines such as \[\] $12.8$  and \[\] $15.6$ by a magnitude or more. More importantly, the densities in many ultracompact  regions exceed the critical densities for these lines (e.g., $2 \times 10^5$ cm$^{-3}$ for \[\] $15.6$ ). Thus, even in the solar neighborhood, the accuracy of traditional synthesis models would be improved by assuming that massive stars contribute their bolometric luminosity to the region for their entire main sequence lifetimes, but influence the usual  indicators in emission line spectra for only $85\%$ of their lives. The correction suggested above would be small for synthesis modelling of the solar neighborhood. However, if the ultracompact  region lifetimes were significantly greater, a substantial deviation from traditional synthesis models would be expected. For nuclear starbursts in massive galaxies, the external pressure is large, due to both the high density and high temperature of the interstellar medium. As a result, the ultracompact  regions of starbursts should be small and their expansion retarded compared with Galactic ones (De Pree, Rodríguez, & Goss 1995; Garciá-Segura & Franco 1996). The gravitational field of the central star(s) should also play an important role, slowing the expansion further [@keto]. Thus, it is likely that the massive stars in nuclear starbursts spend a substantial fraction of their lifetimes embedded in high–extinction regions. It is even plausible that this phase is only terminated when these stars begin to lose mass in strong winds—the evolutionary phase seen in UV spectra of starbursts. This possibility is suggested by the failure, to date, to detect any nuclear starburst that appears younger than about 3 million years, based on either nebular line ratios or UV spectroscopy. Another indication supporting the UC hypothesis is that starburst models under–predict the observed bolometric luminosities of starbursts. Further evidence is that radio recombination lines and free–free continua in starbursts indicate substantially more extinction than indicated by the Brackett lines. For example, @chad deduced A$_V$ $\sim 50$ to the ionized gas in NGC 253 and suggested that much of this gas lies in very compact  regions. In addition, Beck, Turner, & Kovo (2000) found a substantial population of sources in young starburst galaxies whose spectra *rise* from $\lambda=6$ cm to $\lambda=2$ cm, indicating self–absorbed (optically thick to electron scattering) emission. Parsec–scale estimated sizes and large ionizing fluxes (estimated at roughly $10^3$ to $10^4$ OB stars) suggest that these sources are highly obscured young superstar clusters—similar to ultracompact  regions, but containing many more stars. Ironically, these results apply to low–metallicity, low–luminosity galaxies similar to those that emit high–excitation nebular lines. The other galaxies known to have non-AGN rising spectrum sources are NGC 5253 (Turner, Ho, & Beck 1998) and He 2–10 (Kobulnicky & Johnson 1999; Vacca, Johnson, & Conti 2002). Massive spiral starburst galaxies do not show this self–absorbed emission, though it is plausible that such sources exist but are are hidden by non-thermal emission from supernovae. There are a number of consequences for starburst modelling. Because the duration of the obscured phase may depend on the mass of the central star, it may be difficult to deduce an accurate IMF in starbursts using nebular diagnostics. The derivation of the IMF from fitting UV spectra would also be suspect, since there could be a mass/age dependence on the stars contributing to these spectra rather than their providing a snapshot of the integrated hot stellar population. In addition, by suppressing the signatures of the youngest stars, the UC stage will tend to make the duration of starbursts appear artificially short. CONCLUSION {#sec:summary} ========== We have obtained $1.7$  and $2.1$  spectra for six nearby circumnuclear starburst galaxies to measure the [ $1.7$ ]{}/and [ $2.06$ ]{}/ line ratios. Simple recombination physics and independence from nebular conditions and extinction make [ $1.7$ ]{}/ an accurate diagnostic of the hardness of the ionizing continuum (“”). The lines are too weak to be used in distant galaxies, but in nearby galaxies can test more detectable but potentially problematic diagnostics. We present models for the behavior of the  diagnostics discussed in this paper. SEDs were generated by the spectral synthesis code Starburst99, which uses the most current O star and Wolf–Rayet model spectra [@pauldrach01; @hillmill]. Although T$_{eff}$ remains a useful shorthand term, accurate models of the ionizing continuum must consider the entire population of hot main sequence and Wolf–Rayet stars. Wolf–Rayet stars maintain high line ratio values after the O stars have left the main sequence. Our updated models may aid interpretation of IRS spectra from SIRTF. We then test whether the observed line ratios are consistent with these new models. By comparison with [ $1.7$ ]{}/, we confirm that [ $2.06$ ]{}/ is a problem–ridden diagnostic, as predicted [@shields]. [ $2.06$ ]{}/also fails to correlate with the mid–infrared diagnostic \[\]/\[\]. However, we point out that a low [ $2.06$ ]{}/ ratio may indicate a soft UV continuum. NGC 4102, in which [ $1.7$ ]{}/ and [ $2.06$ ]{}/ are both low, is an example. We test \[\] 5007 Å/ as a  diagnostic in starburst galaxies. In comparison to [ $6678$]{}/, \[\]/\[\], and [ $1.7$ ]{}/, we find that \[\]/ is systematically elevated. While aperture mismatch may contribute to the poor correlation with \[\]/\[\], the other two diagnostics were observed with apertures comparable to that for \[\]/. We suggest that shock–excitation of \[\] by supernovae is the likely cause of very high \[\]/, and that this effect plus differing sensitivities to extinction may explain the considerable scatter and lack of correlation with the other  diagnostics. We attempt to test optical He and H recombination line ratios. Sample sizes in the literature are too small to compare [ $6678$]{}/ or [ $4471$]{}/ to each other or to \[\]/\[\], [ $1.7$ ]{}/, or [ $2.06$ ]{}/. Instead, we estimate  in galaxies by the latter three diagnostics, and test for a correlation with [ $6678$]{}/. We find a correlation at $2\sigma$ significance. We re-evaluate the optical nebular lines of NGC 3049, and find them consistent with the age and IMF inferred from the UV spectrum; this removes the discrepancy between the UV and nebular diagnostics noted previously using less sophisticated stellar atmospheres. We show that the mid–infrared lines and [ $1.7$ ]{}/ give consistent answers for well-studied starbursts, lending credibility to the mid–infrared lines’ use as diagnostics of starburst ionizing fields. We also demonstrate theoretically that the behavior of these lines in starbursts should be dominated by T$_{eff}$ for galaxies with similar metallicity. Having found the mid–infrared line ratios to be credible  diagnostics, we use them to address the conditions in starbursts. @thornley found low values of the \[\] $15.6$  / \[\] $12.8$  ratio in their sample of $27$ starburst galaxies. In the context of our models, this result would suggest that high–mass, solar–metallicity starbursts form fewer M$\ga40$  stars than a Salpeter IMF. Adding dust, lowering the metallicity, choosing a different stellar atlas, or choosing a more extended star formation history would strengthen this conclusion. However, another more likely possibility can account for this result. The relatively high density and temperature of the interstellar medium in nuclear starbursts should increase the duration of the ultracompact  region phase. As a result, most of the very massive stars may spend virtually their entire main sequence lifetimes embedded within dense, highly extincted regions, and thus will be nearly undetectable to conventional optical or near-to-mid– infrared spectroscopy. This situation will make it difficult to determine the high–mass IMF in starbursts. In contrast to the high–mass, solar–metallicity starbursts, in the low–mass, low–metallicity galaxies II Zw 40 and NGC 5253, high neon line ratios seem to require stars more massive than $\sim40$–$60$ . This contrast can be understood if these galaxies form stars in regions where the interstellar medium is less effective at confining ultracompact  regions, or if the lifetimes of these regions are reduced at low metallicity. We thank the Steward Observatory TAC for time allocation, telescope operator Dennis Means and the SO Kitt Peak staff, and Chad Engelbracht for his FSPEC–specific `iraf` scripts. We thank Gary Ferland for making Cloudy available to the astronomical community, and Claus Leitherer for making Starburst99 available. We also thank Doug Kelly and Lisa Kewley for modelling advice, and Luis Ho for assistance using his atlas. Don McCarthy, Ann Zabludoff, Ed Olszewski, and Dave Arnett provided comments that improved this paper. An anonymous referee provided an exceptionally helpful critique of the original version of this paper. JRR was partially supported by an NSF Graduate Research Fellowship. This work was also partially supported by the MIPS Project, under contract to the Jet Propulsion Laboratory. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. Aannestad, P. A. 1989, , 338, 162 Andersen, J. 1991, , 3, 91 Armus, L., Heckman, T. M., & Miley, G. K. 1989, , 347, 727 Achtermann, J. M., & Lacy, J. H. 1995, , 439, 163 Alonso-Herrero, A., Rieke, G. H., Rieke, M. J., & Scoville, N. Z. 2000, , 532, 845 Alonso-Herrero, A., Engelbracht, C. W., Rieke, M. J., Rieke, G. H., & Quillen, A. C. 2001, , 546, 952 Bashkin, S., & Stoner, J. O. 1975, Atomic Energy Levels and Grotrian Diagrams, Vol. 1 (Amsterdam: North–Holland Pub.) Beck, S. C., Kelly, D. M., & Lacy, J. H. 1997, , 114, 585 Beck, S. C., Turner, J. L., & Kovo, O. 2000, , 120, 244 Bono, G., Balbi, A., Cassisi, S., Vittorio, N., & Buonanno, R. 2002, , 568, 463 Boker, T., Forster-Schreiber, Natascha M., Genzel, R. 1997, , 114, 1883 Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, , 345, 245 Chandar, R., Leitherer, C., Tremonti, C., Calzetti, D. 2003, , 586, 939 Chernoff, D. A. & Weinberg, M. W. 1990, , 351, 121 Crowther, P. A., Beck, S. C., Willis, A. J., Conti, P. S., Morris, P. W., & Sutherland, R. S. 1999, , 304, 654 De Pree, C. G., Rodriguez, L. F., & Goss, W. M. 1995, Rev. Mexicana Ast. y Ast., 31, 39. Diaz, A. I., Terlevich, E., Pagel, B. E., Vilchez, J. M., & Edmunds, B. E. 1990, RMxAA, 21, 223 Doherty, R. M., Puxley, P. J., Lumsden, S. L., & Doyon, R. 1995, , 277, 57 Doyon, R., Puxley, P. J., & Joseph, R. D. 1992, , 397, 117 Engelbracht, C. W. 1996, Supplement to Data Reduction for FSpec, Univ. Arizona Internal Report Engelbracht, C. W., Rieke, M. J., Rieke, G. H., & Latter, W. B. 1996, , 467, 227 Engelbracht, C. W. 1997, Ph.D. Dissertation, Univ. Arizona Engelbracht, C. W., Rieke, M. J., & Rieke, G. H. 1998, , 505, 639 Ferland, G. J. 1997, Hazy, Univ. Kentucky Internal Report Feuchtgruber, H. [et al.]{} 1997, , 487, 962 Feuchtgruber, H., Lutz, D., Beintema, D. 2001, , 136, 221 Figer, D. F., et al. 2002, , 581, 258 Förster Schreiber, N. M., Genzel, R., Lutz, D., Kunze, D., & Sternberg, A. 2001, , 552, 544 Galavis, M. E., Mendoza, C., & Zeippen, C. J. 1997,  Supplement, 123, 159 Garciá-Segura, G., & Franco, J. 1996, , 469, 171. Garnett, D. R. 1989, , 345, 282 Genzel, R., Weitzel, L., Tacconi-Garman, L. E., Bleitz, M., Cameron, M., Krabbe, A., Lutz, D., & Sternberg, A. 1995, , 444, 129 Gies, D. R., Penny, L. R., Mayer, P., Drechsel, H., & Lorenz, R. 2002, , 574, 957 Giveon, U., Sternberg, A., Lutz, D., Feuchtgruber, H., & Pauldrach, A. W. 2002, , 566, 880 González Delgado, R., Leitherer, C. L., Stasińska, G., & Heckman, T. M. 2002, , 580, 824 González Delgado, R., García-Vargas, M. L., Goldader, J., Leitherer, C., & Pasquali, A. 1999, , 513, 707 Guseva, N. G., Izotov, Y. I., & Thuan, T. X. 2000, , 531, 776 Hameed, S. & Devereus, N. 1999, , 118, 730 Hanson, M. M., Luhman, K., L., & Rieke, G. H. 2002, , 138, 35 Heckman, T. M. 1997, in Star Formation Near and Far, ed. S. S. Holt & L. G. Mundy (Wodbury, NY: AIP Press), 271 Hillier, D. J., & Miller, D. L., 1999, , 496, 407 Ho, P. T., Beck, S. C., & Turner, J. L. 1990, , 349, 57 Ho, L. C., Filippenko, A. V., & Sargent, W. L. W. 1997, , 112, 315 Hummer, D. G., & Storey, P. J. 1987, , 224, 801 Izotov, Y. I., & Thuan, T. X. 1998, , 500, 188 Kennicutt, R. C., 1998, , 36, 189 Kennicutt, R. C., 1998, in ASP Conf. Ser. 142, The Stellar Initial Mass Function, ed. G. Gilmore & D. Howell (San Francisco: ASP), 61 Kennicutt, R. C., Bresolin, F., French, H., & Martin, P. 2000, , 537, 589 Keto, E. 2002, , 580, 980. Kewley, L., Dopita, M., Evans, I., Groves, B., Sutherland, R., Binette, L., Allen, M., & Leitherer, C. 2003, in preparation Kobulnicky, H. A., & Johnson, K. E. 1999, , 527, 154 Kobulnicky, H. A., Kennicutt, R. C., Jr., & Pizagno, J. L. 1999, , 514, 544 Krabbe, A., Sternberg, A., & Genzel, R. 1994, , 425, 72 Kunze, D. [et al.]{} 1996, , 315, L101 Kurtz, S., Churchwell, E., & Wood, D. O. S. 1994, , 91, 659. Lehnert, M. D., & Heckman, T. M. 1996, , 472, 546 Leitherer, C. 1998, in ASP Conf. Ser. 142, The Stellar Initial Mass Function, ed. G. Gilmore & D. Howell (San Francisco: ASP), 61 Leitherer, C., Ferguson, H. C., Heckman, T. M., & Lowenthal, J. D. 1995, , 454, L19 Leitherer, C. [et al.]{} 1999, , 123, 3 Lejeune, T., Cuisinier, F., & Buser, R. 1997, , 125, 229 Lester, D. F., Gaffney, N., Carr, J. S., & Joy, M. 1990, , 352, 544 Livingston, W. & Wallace L. 1991, N.S.O. Technical Report \#91-001 Lumsden, S. L., Puxley, P. J., & Doherty, R. M. 1994, , 268, 821 Lumsden, S. L., Puxley, P. J., & Hoare, M. G. 2001a, , 320, 83 Lumsden, S. L., Puxley, P. J., & Hoare, M. G. 2001b, , 328, 419 Maiolino, R., Rieke, G. H., & Rieke, M. J. 1996, , 111, 537 Mal’Kov, Y. F. 1997, Astronomy Reports, 41, 760 Martín–Hernández, N. L., Vermeij, R., van der Hulst, J. M., & Peeters, E. 2002, , 389, 286 Martín–Hernández, N. L., [et al.]{} 2002, , 381, 606 Mihalas, D. 1972, Non–LTE Model Atmospheres for B and O Stars, NCAR-TN/STR-76 Montgomery, E. F., Connes, P., Connes, J., & Edmonds, F. N. 1969, , 167, 19 Morisset, C., 2003, , in press (astro-ph/0310275) Morisset, C., Schaerer, D., Bouret, J.–C., & Martins, F. 2003, , in press (astro-ph/0310151) Nagamine, K. 2002, , 564, 73 Niemela, V. S., & Bassino, L. P. 1994, , 437, 332 Oey, M. S., Dopita, M. A., Shields, J. C., & Smith, R. C. 2000, , 128, 511 Osterbrock, D. E. 1989, Astrophysics of Gaseous Nebulae and Active Galactic Nuclei (Sausalito: University Science Books) Osterbrock, D. E., & Pogge, R. W. 1987, , 323, 108 Ostrov, P. G., Lapasset, E., & Morrell, N. I. 2000, , 356, 935 Pauldrach, A. W. A., Hoffmann, T. L., & Lennon, M. 2001, , 375, 161 Pauldrach, A. W. A., Lennon, M., Hoffmann, T. L., Sellmaier, F., Kutritzki, R.-P., & Puls, J. 1998, in ASP Conf. Ser. 131, Properties of Hot Luminous Stars, ed. I. Howarth (San Francisco: ASP), 258 Pogge, R. 1989, , 71, 433 Raymond, J. C., Hester, J. J., Cox, D., Blair, W. P., Fesen, R. A., & Gull, T. R. 1988, , 324, 869 Rieke, G. H. 2000, in Springer Proc. in Physics 88, Starburst Galaxies: Near and Far, ed. L. Tacconi & D. Lutz (Berlin: Springer), 73 Rieke, G. H., Loken, K., Rieke, M. J., & Tamblyn, P. 1993, , 412, 99 Rieke, G. H., & Lebofsky, M. J. 1985, , 288, 618 Rieke, G. H., Lebofsky, M. J., Thompson, R. I., Low, F. J., & Tokunaga, A. T. 1980, , 238, 24 Roche, P. F., Aitken, D. K., Smith, C. H., & Ward, J. M. 1991, , 248, 606 Salpeter, E. E. 1955, , 121, 161 Schaerer, D., de Koter, A., Schmutz, W., & Maeder, A. 1996a, , 310, 837 Schaerer, D., de Koter, A., Schmutz, W., & Maeder, A. 1996b, , 312, 475 Schaerer, D., & de Koter, A. 1997, , 322, 598 Schmutz, W., Leitherer, C., & Gruenwald, R. 1992, , 104, 1164 Shields, J. C. 1993, , 419, 181 Schinnerer, E., Eckart, A., Quirrenbach, A., Boker, T., Tacconi-Garman, L. E., Krabbe, A., & Sternbert, A. 1997, , 488, 174 Seaquist, E. R., Frayer, D. T., & Bell, M. B. 1998, , 507, 745 Smith, L. J., Norris, R. P. F., & Crowther, P. A. 2002, , 337, 1309 Stasińska, G. & Leitherer, C. 1996, , 107, 661 Storchi–Bergmann, T., Kinney, A. L., & Challis, P. 1995, , 98, 103 Sugai, H., & Taniguchi, Y., 1992, , 103, 1470 Thornley, M. D., Förster Schreiber, N. M., Lutz, D., Genzel, R., Spoon, H. W. W. & Kunze, D. 2000, , 539, 641 Turner, J. L., Ho, P. T. P., & Beck, S. C. 1998, , 116, 1212 Vacca, W. D., Johnson, K. E., & Conti, P. S. 2002, , 123, 772 Vacca, W. D., & Conti, P. S. 1992, , 401, 543 Vaceli, M. S., Viegas, S. M., Gruenwald, R., & de Souza, R. E. 1997, , 114, 1345 van Hoof, P. A., Beintema, D. A., Verner, D. A., Ferland, G. J. 2000, , 354, 41 Vanzi, L. & Rieke, G. H. 1997, , 479, 694 Vanzi, L., Rieke, G. H., Martin, C. L., & Shields, J. C. 1996, , 466, 150 Vílchez, J. M., & Pagel, B. E. J. 1988, , 231, 257 Webster, B. L., & Smith, M. G. 1983, , 204, 743 Williams, D. M., Thompson, C. L., Rieke, G. H., & Montgomery, E. F. 1993, Proc. SPIE, 1946, 482 Wu, H., Zou, Z. L., Xia, X. Y., & Deng, Z. G. 1998,  Supplement, 127, 521 \[fig:spec\_17\] \[fig:spec\_20\] \[fig:models\] \[fig:lowZmodels\] \[fig:heh\_hek\] \[fig:ne\_206\] \[fig:optical\] \[fig:o3\_heh\] \[fig:o3\_Ne\]  \[fig:IP-local\] \[fig:ne-newplot\] \[fig:continuous\] [^1]: IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. [^2]: The Starburst99 model for $t=3.0$ Myr and M$_{up}=100~$ predicts line ratios that are sharply discontinuous from ratios at $2.8$, $2.9$, $3.1$, and $3.2$ Myr. (The \[\]/\[\] spike is 25 times higher than the surrounding points.) The M$_{up}<100$  models and sub-solar metallicity models have no spike. Though we have been unable to pinpoint the cause from the Starburst99 output, we feel the $3$ Myr spike is spurious, not a physical effect, and we have removed it from the figures.

--- abstract: 'We evaluate quantum corrections to conductivity in an electrically gated thin film of a three-dimensional (3D) topological insulator (TI). We derive approximate analytical expressions for the low-field magnetoresistance as a function of bulk doping and bulk-surface tunneling rate. Our results reveal parameter regimes for both weak localization and weak antilocalization, and include diffusive Weyl semimetals as a special case.' author: - Ion Garate and Leonid Glazman title: 'Weak Localization and Antilocalization in Topological Insulator Thin Films with Coherent Bulk-Surface Coupling' --- Introduction and Overview ========================= The theoretical discovery[@ti] of 3D topological insulators (TIs) in 2006 precipitated an avalanche of experiments aimed at detecting the signature behavior of these unconventional solids. Since then, angle-resolved photoemission spectra[@hasan2011] have given evidence for the Dirac-like dispersion and the momentum-dependent spin texture of TI surface states, whereas local STM probes have indicated a characteristic suppression of backscattering off surface imperfections.[@stm] However, the most desired observation of a hallmark dc conduction confined to the surface layer of a 3D TI remains elusive.[@dimi2011] The main problem is conduction through the bulk: 3D TIs are narrow-gap semiconductors, rich in bulk carriers that are either thermally activated and/or donated by crystalline lattice imperfections. Along with attempts to reduce bulk charge carriers, experimentalists are developing techniques which allow to register a separate conduction channel on the surface of a 3D TI.[@kim2012] Chief among these are measurements of low-field magnetoresistance combined with electrostatic gating of thin-film samples.[@wang2011; @chen2010; @checkelsky2011; @he2011; @chen2011; @steinberg2011; @hong2012] Low-field magnetoresistance measurements unveil the interference correction $\delta\sigma$ to the Drude conductivity $\sigma_D$.[@inter] At low temperatures, $\sigma_D$ is defined by independent acts of scattering of electrons off the crystal’s imperfections, and is proportional to the classical diffusion constant $D$. When the phase relaxation length $l_\phi$ is parametrically longer than the scattering mean free path, quantum interference affects the conductivity to a measurable extent. The sign of the interference correction depends on the strength of spin-orbit interactions. For weak spin-orbit interactions ($l_{\rm so}\gg l_\phi$, where $l_{\rm so}$ is the spin-orbit scattering length), it follows that $\delta\sigma<0$. This is called weak localization (WL). In contrast, strong spin-orbit interaction ($l_{\rm so}\ll l_\phi$) leads to suppression of backscattering and thus $\delta\sigma>0$. This is called weak antilocalization (WAL). Being interference effects, WL and WAL are degraded by a magnetic field $H$ when $H\gtrsim H_\phi\equiv \Phi_0/(8\pi l_\phi^2)$, where $\Phi_0=h/e$ is the flux quantum. Yet, $\sigma_D$ is nearly immune to $H$ at such low fields. Therefore, the low-field magnetoconductivity reads $\Delta\sigma(H)\equiv\sigma(H)-\sigma(0)\simeq\delta\sigma(H)-\delta\sigma(0)$. All experiments to date report WAL in 3D TI thin films,[@kapitulnik2012] and ascribe it to the strong spin-orbit interaction in the electronic bands of these materials. For film thickness less than $l_\phi$, the measured $\Delta\sigma(H)$ agrees well with the functional form provided by 2D WAL theory, namely $$\label{eq:hikami} \Delta\sigma(H)\simeq \alpha\,(e^2/2\pi^2 \hbar) f(H_\phi/H),$$ where $f(z)\equiv \ln z-\psi(1/2+z)$, with $\psi$ and $\alpha$ being the digamma function and a number,[@hikami1980] respectively. In a system with a single conduction channel, $\alpha$ is universal and equals $1/2$. The WAL contributions add for systems which are isolated from each other. For example, having two independent parallel conduction channels yields $\alpha=1$, irrespective of the ratio of Drude conductivities of the two subsystems. The relation between $\alpha$ and the number of parallel channels is at the heart of recent magnetoresistance experiments in 3D TIs.[@checkelsky2011; @chen2011; @steinberg2011] Overall, the coefficient $\alpha$ is found to depend on the gate voltage. For some devices,[@checkelsky2011; @chen2011; @steinberg2011] it changes from $\alpha=1/2$ all the way to $\alpha=1$. A plausible interpretation for this variation is presented in Ref. \[\]. At zero or positive bias applied to the top gate, electrons from the $n$-doped bulk reach the surface states easily: the entire film acts as a single electron system, and $\alpha=1/2$. At negative bias, electrons are repeled from the top surface and, for strong enough bias, a depletion layer is formed adjacent to it. This depletion region separates the film into two subsystems: bulk carriers (combined with surface carriers from the bottom surface) on one side, and top-surface carriers on the other side. For a wide enough depletion layer, $\alpha=1$. In spite of the ongoing scrutiny on the experimental front, quantum corrections to conductivity in 3D TIs have stimulated relatively little theoretical activity. Even though the WAL contribution from TI surface states has been calculated explicitly,[@lu2011a; @tkachov2011] there are no calculations that incorporate conducting 3D bulk states. The main reason for this omission may be the prevailing view that quantum corrections originating from bulk TI states ought to be conceptually identical to those in ordinary strongly spin-orbit coupled systems, i.e. of WAL type. Recently, an objection to this viewpoint has been raised,[@lu2011] declaring that quantum well states in ultrathin TI films may contribute via WL rather than WAL. Although suggestive, the calculation of Ref. \[\] is limited to quasi-2D films and disregards the coupling between bulk and surface states, which leaves out several experiments of interest. Besides, its extrapolation to 3D bulk states has not been carried out properly. In this paper we evaluate $\Delta\sigma$ for gated 3D thin films, as a function of the bulk carrier concentration and accounting for the coupling between surface and bulk states. Our calculation applies to TI films that are thicker than the bulk mean free path, thinner than $l_\phi$, and not highly doped. In these films, bulk carriers are three-dimensional and are concentrated around the $\Gamma$ point of the electronic band structure. The resulting approximate analytical expressions for $\Delta\sigma$ (Eqs. (\[eq:magres\_bulk\]), (\[eq:res\_tot\]) and (\[eq:res\_tot5\])) are aimed at improving the interpretation of magnetoresistance measurements in TIs, in Weyl semimetals,[@burkov2011] and in some class of topologically trivial materials. Although a few of our observations resemble those developed for graphene[@mccann2006] and 2D TIs,[@tkachov2011] there are qualitative differences originating from the 3D Dirac nature of bulk carriers in 3D TIs. Altogether, the results reported here paint a richer picture than previously anticipated. On one hand, we confirm the conventional crossover between $\alpha=1/2$ and $\alpha=1$ as a function of the gate voltage: the former corresponds to the case of coherently-coupled bulk and surface electron states, while the latter indicates a single decoupled Dirac cone on the top surface along with generic WAL from the rest of the film (containing coupled bulk and bottom surface). On the other hand, less conventional results arise when the Fermi energy is close to the bulk band edge or when the Fermi energy is much larger than the bulk bandgap: in the former regime the bulk exhibits WL with $\alpha=-1$, whereas in the latter regime the bulk exhibits an anomalous WAL with $\alpha=1$. These two “unusual” bulk regimes, combined with the surface contributions, may result in a range of $\alpha$ including $\alpha<0$ and $\alpha>1$. The rest of this work is organized as follows. In Section II we evaluate quantum corrections to [*bulk*]{} conductivity. Readers not interested in technical details should read subsection IIA and quickly scan through IIB and IIC in order to get acquainted with the nomenclature; the main results of the section are collected in Section IID. The well-known message from IIA is that at low energies bulk electrons of TI films behave as massive 3D Dirac fermions with spin and valley (or orbital) degrees of freedom. The direction of spin is locked with that of momentum, and valleys are coupled to one another by the mass of the Dirac fermions. The special case in which the Dirac mass vanishes is a time- and inversion-symmetric Weyl semimetal. In Section IID we identify and count the number of “soft” Cooperon modes, which determine the magnitude and sign of $\Delta\sigma$ in the bulk. Each soft Cooperon obeys a classical difussion equation and is thus associated with a conserved physical quantity. Since charge is conserved, there is at least one soft Cooperon in (non-magnetic) bulk TIs. We find that additional soft Cooperons can emerge depending on the bulk doping concentration as well as the bulk bandgap. This realization leads to the most important results in IID, Eqs. (\[eq:res\_bulk\])-(\[eq:magres\_bulk\]), which indicate that for bulk states $\alpha$ may acquire three different universal values. On one hand, WL with $\alpha=-1$ is possible when the bulk Fermi surface is “small” (as defined in the text), because in this case the spin-momentum locking of bulk states becomes weak and the spin of electrons is nearly conserved. In contrast, WAL with $\alpha=1$ can arise for bulk TIs with particularly small bandgaps, because in such case bulk electrons can be described by a 3D analogue of graphene with two nearly decoupled valleys, each contributing $1/2$ to $\alpha$. For a more generic case, in which neither valley nor spin are approximately conserved, the quantum interference is similar to that of an ordinary film with strong spin-orbit coupling and therefore $\alpha=1/2$. Magnetic fields perpendicular to the TI film can be used to induce crossovers between different universal regimes of $\alpha$. The accessible values of $\alpha$ and the corresponding crossover fields depend on the bulk electron density. In Section III we evaluate the [*full*]{} $\Delta\sigma$ in 3D TI thin films, which comprises coupled bulk and surface contributions. Sections IIIA and IIIB cover preliminary material that is needed to derive the main results in IIIC. Section IIIA reviews the well-established fact that, in absence of magnetic order, isolated TI surface states exhibit WAL with $\alpha=1/2$ (in this paper we assume one Dirac cone per surface). Section IIIB develops a diagrammatic framework for evaluating quantum corrections to conductivity in ordinary tunnel-coupled layers. Readers who are not interested in technicalities can disregard the diagrams in the figures and concentrate on the outcome of the calculation (Eqs. (\[eq:q1\])-(\[eq:Dii\])), as well as on the subsequent discussion. One qualitative point made therein is that the crossover from weak to strong coupling (which is accompanied by a change in $\alpha$ from $1$ to $1/2$) occurs when the interlayer resistance for a square of area $l_\phi^2$ becomes smaller than the sum of the classical intralayer resistances. Section IIIC combines results from IID, IIIA and IIIB in order to figure out quantum corrections to conductivity in experimentally realized TI films. The most important results in IIIC are Eqs. (\[eq:res\_tot\]) and  (\[eq:res\_tot5\]), which describe how $\Delta\sigma$ depends on the bulk doping concentration, on the phase relaxation rate, and on the bulk-surface tunneling rate. Some special cases of these results are highlighted in Appendix \[sec:special\]. A salient conclusion is that the WL regime of isolated bulk states is generally eliminated when either one of the film surfaces is strongly coupled to bulk states, in which case the film displays $1/2\leq\alpha\leq 1$. However, WL can still be present if the TI surfaces have short phase relaxation lengths. Finally, Section IIID characterizes the electrostatics of the depletion layer and estimates the bulk-surface tunneling rate in TI films. This estimate confirms experimental indications showing that both weak and strong bulk-surface coupling are accessible by mediation of a gate voltage. Quantum Corrections to Bulk Conductivity ======================================== This section is devoted to evaluating $\delta\sigma$ for the bulk states of a 3D TI. As a byproduct, we derive $\delta\sigma$ for a time-reversal symmetric Weyl semimetal. The contribution from TI surface states will be discarded until the next section. Model ----- The bulk band structure of a 3D TI near the $\Gamma$ point can be approximated by the following ${\bf k}\cdot{\bf p}$ Hamiltonian: [@zhang2009] $$\begin{aligned} \label{eq:model_b} &{\cal H}=\sum_{\bf k}\Psi^\dagger_{\bf k} h({\bf k}) \Psi_{\bf k}\nonumber\\ & h({\bf k}) \simeq \epsilon({\bf k}){\bf 1}_4+M({\bf k}){\bf 1}_2\,\tau^z+\hbar\left(v_z k_z\sigma^z+v_\perp {\bf k}_\perp\cdot{\boldsymbol \sigma}^\perp\right)\tau^x, \end{aligned}$$ where ${\boldsymbol \tau}$ is an orbital pseudospin ($\tau^z=T,B$), ${\boldsymbol \sigma}$ is the real spin ($\sigma^z=\uparrow,\downarrow$), ${\bf k}=({\bf k}_\perp,k_z)$ is the momentum measured from the $\Gamma$ point of the Brillouin zone, ${\bf 1}_N$ is an $N\times N$ identity matrix, $\Psi=(\Psi_{T\uparrow},\Psi_{T\downarrow},\Psi_{B\uparrow},\Psi_{B\downarrow})$ is a 4-spinor, $\epsilon({\bf k})=\epsilon(-{\bf k})$ is the part of the Hamiltonian that is independent of spin/pseudospin indices, $v_z$ and $v_\perp$ are the Fermi velocities, and $M({\bf k})=M_0-M_1 k_\perp^2-M_2 k_z^2$ is the mass term (independent of spin). $M_0$, $M_1$ and $M_2$ are constants. Equation (\[eq:model\_b\]) captures the bottom of the conduction band and the top of the valence band in the vicinity of the $\Gamma$ point ($k\equiv 0$), where the bandgap is smallest. It models 3D Dirac fermions with a Dirac mass that equals half the energy gap. For the purposes of this paper we ignore $\epsilon({\bf k})$, and assume $M({\bf k})=M={\rm const}>0$ as well as spherical symmetry ($v_z=v_\perp=v$). These assumptions simplify calculations without incurring in qualitative loss of generality. For instance, the XXZ anisotropy can be modeled by promoting the diffusion constant from a scalar to a matrix. Also, the $k^2$ terms in $M({\bf k})$ can be incorporated into our final results by $M\to |M({\bf k}_F)|$, where ${\bf k}_F$ is the Fermi wave vector. Note that in absence of spherical symmetry the Fermi surface does not have a constant mass; this complication will be disregarded in the present paper. Finally, $\epsilon({\bf k})$ can be absorbed into the definition of the Fermi energy. ![Bulk energy bands of an $n$-doped 3D TI near the $\Gamma$ point, in the spherical approximation. The momentum $k$ is measured from the $\Gamma$ point. The energies $\epsilon_F$ and $M$ are measured with respect to midgap.[]{data-label="fig:bands"}](./bands.eps) The energy eigenvalues for $h({\bf k})$ in the spherical approximation are $E_{{\bf k}\pm}=\pm\sqrt{\hbar^2v^2 k^2+M^2}$, each doubly degenerate (Fig. \[fig:bands\]). The corresponding Bloch states can be written as $$\label{eq:eigen0} |\Psi_{{\bf k}\alpha}\rangle=(1/\sqrt{V})\exp(i {\bf k}\cdot{\bf r})|\alpha {\bf k}\rangle,$$ where $V$ is the volume of the TI and $\alpha\in\{1, 2, 3, 4\}$ is a band index ($1$ and $2$ denote conduction bands, while $3$ and $4$ denote valence bands). This $\alpha$ is obviously unrelated to that of Eq. (\[eq:hikami\]); from here on it will be clear from the context which one we are referring to. For concreteness we set the chemical potential in the bulk conduction band, although all results obtained below will be directly applicable to $p$-doped bulk TIs as well. The density of conduction band electrons is then $$\label{eq:n} n\simeq \frac{\left(\epsilon_F^2-M^2\right)^{3/2}}{\pi^2\hbar^3 v^3},$$ where $\epsilon_F$ is the Fermi energy measured from the middle of the bulk energy gap. Adopting the basis $\{|T\uparrow\rangle, |T\downarrow\rangle,|B\uparrow\rangle,|B\downarrow\rangle\}$, the two eigenspinors corresponding to the conduction bands near the $\Gamma$ point are $$\begin{aligned} \label{eq:eigenstates} |1 {\bf k}\rangle&=&\sqrt{\frac{E_k+M}{2 E_k}}\left(1,0,\frac{\hbar v k_z}{E_k+M},\frac{\hbar v k_+}{E_k+M}\right)\nonumber\\ |2 {\bf k}\rangle&=&\sqrt{\frac{E_k+M}{2 E_k}}\left(0,1,\frac{\hbar v k_-}{E_k+M},\frac{-\hbar v k_z}{E_k+M}\right),\end{aligned}$$ where $k_\pm=k_x\pm i k_y$ and $E_k=E_{{\bf k},+}$. Since all non-Hall dc transport properties of good conductors involve states close to the Fermi energy, we hereafter ignore valence bands. Unlike in the ${\bf k}\cdot{\bf p}$ Hamiltonians for graphene and 2D (or quasi-2D) TIs, Eq. (\[eq:model\_b\]) cannot be decomposed into two $2\times 2$ block diagonal matrices (due to $M\neq 0$). In addition, the $k_z$ band dispersion absent in 2D cannot be neglected in our case. These two features make the calculations and results of this section quite different from those of Refs. \[\]. Equation (\[eq:model\_b\]) becomes inaccurate when the chemical potential moves up in the conduction band and electron pockets away from the $\Gamma$ point begin to be populated. These additional pockets contribute to quantum interference, and the total $\delta\sigma$ depends on the scattering rate between different electron pockets. Although a realistic study of the full band structure is beyond the scope of this paper, we expect calculations based on Eq. (\[eq:model\_b\]) to provide a generic understanding of quantum corrections to conductivity in 3D Dirac materials at low-to-moderate doping concentrations. Formalism --------- In order to quantify the conductivity of a bulk TI, we begin by characterizing the simplest possible disorder potential: $V_{\rm dis}({\bf r})=V({\bf r}){\bf 1}_4$, which is time-independent (elastic) and independent of spin as well as orbital degrees of freedom. For simplicity we assume $V({\bf r})$ to be slowly-varying at the atomic scale, yet short-ranged compared to the mean free path: $V({\bf r})=V_0\delta({\bf r})$. It is due to its slow spatial variation on atomic lenghtscales that $V_{\rm dis}$ becomes an identity operator in orbital space. With such disorder realization, the Fermi-surface lifetime $\tau_0$ for the $\alpha=1,2$ eigenstates in Eq. (\[eq:eigen0\]) obeys $$\begin{aligned} \label{eq:lifetime} \frac{1}{\tau_0} &=\frac{2\pi u_0}{\hbar}\int_{{\bf k}'}\sum_{\alpha'}|\langle\alpha {\bf k}_F|\alpha' {\bf k}_F'\rangle|^2 \delta(\epsilon_F-E_{{\bf k}'\alpha'})\nonumber\\ &\simeq\frac{\pi u_0 \nu}{\hbar} \left(1+\frac{M^2}{\epsilon_F^2}\right),\end{aligned}$$ where $\int_{\bf k}\equiv\int d^3 k/(2\pi)^3$, $u_0\equiv n_i V_0^2$, $n_i$ is the density of impurities, and $\nu$ is the density of states per band and per unit volume at $\epsilon_F$. A related quantity is the transport scattering rate $\tau^{-1}$, $$\begin{aligned} \frac{1}{\tau} &\equiv \frac{2\pi u_0}{\hbar}\int_{{\bf k}'}\sum_{\alpha'} (1-\hat{{\bf k}}_F\cdot\hat{{\bf k}}_F') |\langle\alpha {\bf k}_F|\alpha' {\bf k}_F'\rangle|^2 \delta(\epsilon_F-E_{{\bf k}'\alpha'})\nonumber\\ &= \frac{2}{3\tau_0}\frac{\epsilon_F^2+2 M^2}{\epsilon_F^2+M^2}.\end{aligned}$$ The momentum-dependence of $|\alpha {\bf k}\rangle$ makes $\tau_0\neq\tau$ even for $\delta$-function impurity potentials. Throughout this work we impose $(\epsilon_F-M)\tau\gg \hbar$ or equivalently $k_F l\gg 1$, where $l=(D\tau)^{1/2}$ is the elastic mean free path, $$k_F=(\epsilon_F^2-M^2)^{1/2}/(\hbar v)$$ is the Fermi wave vector and $$D=v_F^2\tau/3=v^2\tau(1-M^2/\epsilon_F^2)/3$$ is the classical diffusion constant. Next, we consider a TI with spatial dimensions $L\times L$ in the $xy$ plane and a thickness $W$ along the $z$ direction. We take a thin film geometry with $L\gg l_\phi\gg l$ and $l_\phi\gg W\gg l$, where $l_\phi=(D\tau_\phi)^{1/2}$ is the coherence length and $\tau_\phi$ is the phase relaxation time. The conductivity of this film is $$\sigma=\sigma_D+\delta\sigma,$$ where $\sigma_D$ is the classical (Drude) part and $\delta\sigma$ is the part coming from quantum interference. On one hand, the Drude conductivity can be approximated as $$\label{eq:sd} \sigma_D\simeq\frac{e^2\hbar}{2\pi}\sum_{\alpha,\beta}\int_{\bf k} v^x_{\alpha\beta}({\bf k})\tilde{v}^x_{\beta\alpha}({\bf k}) G^R_\alpha({\bf k}) G^A_\beta({\bf k}),$$ where we have assumed a spatially uniform dc electric field and $\alpha,\beta\in\{1,2\}$. $v^x_{\alpha\beta}=\langle\alpha{\bf k}|{\bf v}\cdot\hat{x}|\beta {\bf k}\rangle$ is a matrix element for the $x$-component of the bare velocity operator ${\bf v}=v\tau^x{\boldsymbol\sigma}$, which obeys $$\label{eq:vbare} {\bf v}_{\alpha\beta}({\bf k})=\hbar v^2 ({\bf k}/E_k) \delta_{\alpha\beta}\,\,\,\mbox{(for $\alpha,\beta\in\{1,2\}$)}.$$ Disorder vertex corrections renormalize Eq. (\[eq:vbare\]) to $$\tilde{{\bf v}}_{\alpha\beta}={\bf v}_{\alpha\beta}(\tau/\tau_0),$$ see Appendix \[sec:ren\]. In addition, $$\label{eq:ds} G_\alpha^{R(A)}({\bf k})= \left[\epsilon_F-E_{{\bf k}\alpha}+(-) \frac{i\hbar}{2\tau_0}\right]^{-1}$$ is the zero-frequency retarded (advanced) Green’s function in the band eigenstate basis. Using $G^{R(A)}_1({\bf k})=G^{R(A)}_2({\bf k})\equiv G^{R(A)}({\bf k})$, Eq. (\[eq:sd\]) yields $$\sigma_D=2 e^2 \nu D.$$ ![(a) Feynman diagram for $\delta\sigma_1$, defined in the text. Filled squares denote velocity operators (including disorder vertex corrections), $C$ is the Cooperon. (b) Diagrammatic representation of the Bethe-Salpeter equation for the Cooperon. Crosses correspond to impurity scattering centers. Solid lines with arrows are disorder-averaged Green’s functions. (c) Additional Feynman diagrams that contribute to conductivity of 3D TIs even when impurity scattering is isotropic.[]{data-label="fig:cofig"}](./diag0.eps) On the other hand, the quantum correction $\delta\sigma$ can be written as $\delta\sigma\simeq\delta\sigma_1+\delta\sigma_2$, represented pictorially in Fig. \[fig:cofig\]. Following standard approximations, the expression for $\delta\sigma_1$ is $$\begin{aligned} \label{eq:ds} \delta\sigma_1 &\simeq \frac{e^2\hbar}{2\pi}\sum_{\alpha,\alpha',\beta,\beta'}\int_{\bf k} \tilde{v}^x_{\alpha\beta}({\bf k}) \tilde{v}^x_{\beta'\alpha'}(-{\bf k}) G^R_\alpha({\bf k}) G^R_{\alpha'}(-{\bf k})\nonumber\\ &~~~\times G^A_\beta({\bf k}) G^A_{\beta'}(-{\bf k})\frac{1}{W}\int \frac{d^2 Q}{(2\pi)^2} C^{\beta\beta'}_{\alpha'\alpha}({\bf k},{\bf k},{\bf Q}).\end{aligned}$$ In the second line of Eq. (\[eq:ds\]) we have exploited the condition $W\ll l_\phi$, which allows to set $Q_z=0$ everywhere. $C^{\beta\beta'}_{\alpha'\alpha}({\bf k}_1,{\bf k}_2,{\bf Q})$ are the matrix elements of the Cooperon matrix $\hat{C}$ in the band eigenstate basis. ${\bf Q}=(Q_x,Q_y)$ is the momentum of the Cooperon, whose magnitude ranges from $0$ to $\simeq (D\tau)^{-1/2}$. $\hat{C}$ obbeys the Bethe-Salpeter equation (Fig. \[fig:cofig\]b): $$\begin{aligned} \label{eq:bs} &C^{\beta\beta'}_{\alpha'\alpha}({\bf k}_1,{\bf k}_2,{\bf Q})=\Gamma^{\beta\beta'}_{\alpha'\alpha}({\bf k}_1,{\bf k}_2,{\bf Q})+ \int_{{\bf k}_3}\Gamma^{\beta\beta''}_{\alpha'\alpha''}({\bf k}_1,{\bf k}_3,{\bf Q})\nonumber\\ &~~~~~~~\times G^A_{\beta''}({\bf k}_3) G^R_{\alpha''}(-{\bf k}_3+{\bf Q}) C^{\beta''\beta'}_{\alpha''\alpha}({\bf k}_3,{\bf k}_2,{\bf Q}),\end{aligned}$$ where a sum over repeated indices is implied and $$\Gamma^{\beta\beta'}_{\alpha'\alpha}({\bf k}_1,{\bf k}_2,{\bf Q})\equiv u_0\langle\beta {\bf k}_1|\beta' -{\bf k}_2+{\bf Q}\rangle\langle\alpha' -{\bf k}_1+{\bf Q}|\alpha {\bf k}_2\rangle\nonumber$$ is the bare disorder vertex (first term on the right hand side of Fig. \[fig:cofig\]b). Equation (\[eq:bs\]) is a complicated integral equation because $C^{\beta\beta'}_{\alpha'\alpha}$ is a function of three momenta. This is unlike in simplest examples, where the Cooperon depends only on ${\bf Q}$. The difficulty originates from the momentum-dependence of $|\alpha {\bf k}\rangle$, which cannot be overlooked as it crucially determines both the magnitude and the sign of $\delta\sigma$. One procedure[@garate2009] to solve Eq. (\[eq:bs\]) starts by writing the Cooperon in the two-particle spin/orbit basis $\{|m,m'\rangle\}$, where $m\in\{T\uparrow,T\downarrow,B\uparrow,B\downarrow\}$: $$\begin{aligned} \label{eq:trans} &C^{\beta \beta'}_{\alpha' \alpha}({\bf k}_1, {\bf k}_2, {\bf Q})=\sum_{m,m',n,n'}\langle\alpha',-{\bf k}_1+{\bf Q}|m'\rangle\langle\beta {\bf k}_1|m\rangle\nonumber\\ &\times\langle n|\beta',-{\bf k}_2+{\bf Q}\rangle\langle n'|\alpha {\bf k}_2\rangle C^{m n }_{m' n'}({\bf Q}).\end{aligned}$$ We then make the ansatz that $C^{m n}_{m' n'}$ depends on ${\bf Q}$ but not on ${\bf k}_1$ and ${\bf k_2}$; the entire ${\bf k}_1$- and ${\bf k}_2$-dependence of $C^{\beta \beta'}_{\alpha' \alpha}({\bf k}_1,{\bf k}_2,{\bf Q})$ originates from the overlap matrix elements of Eq. (\[eq:trans\]). The internal consistency of this ansatz can be demonstrated by substituting Eq. (\[eq:trans\]) in Eq. (\[eq:bs\]), which produces an algebraic equation for $C^{m n}_{m' n'}$ that is more tractable than the original integral equation:$$\label{eq:C_mat} C^{m n}_{m' n'}({\bf Q})= u_0 \delta_{m n}\delta_{m' n'}+\sum_{l,l'}U^{m l}_{m' l'}({\bf Q}) C^{l n}_{l' n'}({\bf Q}),$$ where $$\label{eq:U} U^{m l}_{m' l'}({\bf Q})=u_0\int \frac{d^3 k}{(2\pi)^3} G_{m l}^A({\bf k})G^R_{m' l'}(-{\bf k}+{\bf Q})$$ and $$G^{R (A)}_{m l}({\bf k})=\sum_\alpha \langle m|\alpha {\bf k}\rangle G_\alpha^{R (A)}({\bf k}) \langle\alpha {\bf k}|l\rangle.$$ In matrix language, Eq. (\[eq:C\_mat\]) can be rewritten as $$\label{eq:C_mat2} \hat{C}=({\bf 1}_{16}-\hat{U})^{-1} \hat{C}^{(0)},$$ where $\hat{C}^{(0)}=u_0 {\bf 1}_{16}$. Once we obtain $C^{m n}_{m' n'}$, we use Eq. (\[eq:trans\]) in order to recover $C^{\beta\beta'}_{\alpha'\alpha}$. During this operation we neglect ${\bf Q}$ in the overlap matrix elements, which is a good approximation because $\delta\sigma$ is dominated by elements of $C^{m n}_{m' n'}({\bf Q})$ that are strongly peaked at $Q\simeq 0$. For $\epsilon_F$ in the conduction band, we once again limit ourselves to $\alpha,\beta,\alpha',\beta'\in\{1,2\}$ in Eq. (\[eq:ds\]). Then we can approximate ${\bf k}\simeq {\bf k}_F$ inside the Cooperon, and an integration over $|{\bf k}|$ yields $$\label{eq:ds2bis} \delta\sigma_1 \simeq -6 \frac{e^2}{\hbar^2} \nu D \tau\tau_0\frac{1}{W}\int\frac{d^2 Q}{(2\pi)^2}\overline{C}({\bf Q}),$$ where $$\label{eq:overline_c} \overline{C}({\bf Q}) \equiv \int \frac{d\Omega_{\bf k}}{4\pi}\hat{\bf k}_x^2\sum_{\alpha,\alpha'=1,2}C^{\alpha \alpha'}_{\alpha' \alpha}({\bf k}_F,{\bf k}_F,{\bf Q})$$ and $d\Omega_{\bf k}$ is the differential solid angle subtended by $\hat{{\bf k}}$. Note that $\delta\sigma_1$ depends on the lifetime $\tau_0$ of Bloch states as well as on the transport relaxation time $\tau$. As mentioned above, the difference between $\tau$ and $\tau_0$ comes from the momentum dependence of $|\alpha {\bf k}\rangle$ states. At any rate, the full correction $\delta\sigma$ depends only on $\tau$ due to the additional contribution from $\delta\sigma_2$ (see Fig. \[fig:cofig\]c and Eq. (\[eq:nasty\])). Equation (\[eq:nasty\]) can be evaluated using the same procedure as for $\delta\sigma_1$. For instance, in Appendix \[sec:ds2\] we derive $$\label{eq:ds2} \delta\sigma_2\simeq\left\{\begin{array}{ccc} 0 & {\rm if } & (\epsilon_F-M)/M\ll 1\\ -(1/3)\delta\sigma_1 & {\rm if }& (\epsilon_F-M)/M\gg 1.\\ \end{array}\right.$$ The full form of the quantum correction, $\delta\sigma_1+\delta\sigma_2$, depends only on the transport mean free path $\tau$ and has (in appropriate limits) a universal magnitude, see Eqs. (\[eq:res\_bulk\]) and  (\[eq:magres\_bulk\]). Calculations ------------ The road map to $\delta\sigma$ starts from a calculation of $\hat{U}$ in Eq. (\[eq:U\]). In Appendix \[sec:u\] we derive $$\begin{aligned} \label{eq:coeffs} U^{m l}_{m' l'} &= a\,\delta_{m l}\delta_{m' l'}+ \sum_\mu b_\mu\,\Lambda^\mu_{m' l'}\delta_{m l}\nonumber\\ &+\sum_\mu c_\mu\,\Lambda^\mu_{m l}\delta_{m' l'}+\sum_{\mu,\nu} d_{\mu\nu}\,\Lambda^\mu_{m l}\Lambda^\nu_{m' l'},\end{aligned}$$ where $\mu,\nu\in\{1,2,3,4\}$, $\Lambda^i=\sigma^i\tau^x$ for $i\in\{1,2,3\}$ and $\Lambda^4={\bf 1}_{2}\,\tau^z$. In addition, $a$, $b_\mu$, $c_\nu$ and $d_{\mu\nu}$ are ${\bf Q}$-dependent coefficients whose explicit expressions are shown in Appendix \[sec:u\]. With those, $\hat{U}$ is fully determined. The next task is to get $C^{m n}_{m' n'}({\bf Q})$ from Eq. (\[eq:C\_mat2\]). While $({\bf 1}_{16}-\hat{U}({\bf Q}))$ can be inverted numerically, it is not possible to do so analytically for $Q\neq 0$. Since we are interested in analytical expressions, we follow an approximate three-step route. First, we diagonalize $({\bf 1}_{16}-\hat{U})$ for $Q=0$, analytically. All eigenvalues can be written in the form $\Delta_g\tau_0$, where $\Delta_g$ is the “intrinsic” Cooperon gap or mass. We find that one of the eigenvalues has $\Delta_g=0$ for any $\epsilon_F$ and $M$, which is a reflection of combined time-reversal symmetry and charge conservation. As we elaborate in the next subsection, there may be additional eigenvalues with $\Delta_g\simeq 0$ when $(\epsilon_F-M)/M\ll 1$ and $(\epsilon_F-M)/M\gg 1$. Hereafter we refer to eigenvectors of $\Delta_g\simeq 0$ eigenvalues as gapless (or massless, or “soft”) modes. Because $\Delta_g\simeq 0$ eigenvalues make $\hat{C}$ large, $\delta\sigma$ is determined mainly by soft modes. Second, we extrapolate the $Q=0$ case to $Q\neq 0$ perturbatively, with the objective of finding how the eigenvalues of the gapless modes depend on $Q$. To that end $\delta \hat{U} ({\bf Q})\equiv \hat{U}({\bf 0})-\hat{U}({\bf Q})$ is written in the basis that diagonalizes $\hat{U}({\bf 0})$. The shift of $Q=0$ eigenvalues under $\delta\hat{U}({\bf Q})$ is then evaluated through standard second order perturbation theory. The need to go to second order in $\delta \hat{U}$ originates from the fact that several matrix elements of $U^{m n}_{m' n'}({\bf Q})$ are linear in $Q$ (see Appendix \[sec:u\]). When $(\epsilon_F-M)/M\ll 1$ and $(\epsilon_F-M)/M\gg 1$, perturbation theory leads to eigenvalues $(D Q^2+\Delta_g)\tau_0$. The fact that $D$ contains the transport time $\tau$ rather than the scattering time $\tau_0$ is generally crucial in order to arrive at the correct result for $\delta\sigma$. Third, we invert the diagonalized matrix, and transform its outcome to the $|m,m'\rangle$ basis by using the $Q=0$ eigenvector matrix (the change of unperturbed eigenvectors under $\delta\hat{U}({\bf Q})$ is deemed unimportant.) This yields $C^{m n}_{m' n'}({\bf Q})$. Once we have $C^{m n}_{m' n'}({\bf Q})$, we use Eq. (\[eq:trans\]) in order to extract $C^{\alpha\beta}_{\beta\alpha}({\bf k},{\bf k}',{\bf Q})$. This is then plugged in Eqs. (\[eq:ds2bis\]) and  (\[eq:nasty\]). Results ------- The diagonalization of Eq. (\[eq:C\_mat2\]) at $Q=0$ shows one genuinely gapless Cooperon mode ($\Delta_g=0$, c.f. Sec. IIC), with a spin-singlet and orbital-triplet eigenvector: $$\label{eq:g1} \left[\frac{\epsilon_F+M}{2\sqrt{\epsilon_F^2+M^2}}|T T\rangle+ \frac{\epsilon_F-M}{2\sqrt{\epsilon_F^2+M^2}}|B B\rangle\right]\left(|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle\right).$$ The fact that Eq. (\[eq:g1\]) remains gapless for any $\epsilon_F/M$ is a physical manifestation of charge conservation. This situation differs qualitatively from 2D TIs in HgTe quantum wells,[@tkachov2011] where a nonzero mass term gaps all Cooperons. The reason for the difference is that in 2D TIs the mass term acts somewhat like a Zeeman field in a 2D electron gas with Rashba spin-orbit interaction. Importantly, the diagonalization of Eq. (\[eq:C\_mat2\]) reveals two qualitatively distinct regimes of quantum interference, $(\epsilon_F-M)/M\ll 1$ and $(\epsilon_F-M)/M\gg 1$, which potentially host additional gapless Cooperon modes. As we discuss below, these additional gapless modes can change and even reverse the contribution to $\delta\sigma$ coming from Eq. (\[eq:g1\]). When $(\epsilon_F-M)/M\gg 1$, we identify a slightly gapped (soft) Cooperon mode with $$\label{eq:tau_s} \Delta_g=2 (M^2/\epsilon_F^2)\tau_0^{-1}\equiv\tau_v^{-1}\ll\tau_0^{-1},$$ whose eigenvector is a spin-singlet and an orbital-triplet: $$\label{eq:g2} \frac{1}{2}\left(|T B\rangle+|B T\rangle\right)\left(|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle\right).$$ Physically, $\tau_v^{-1}$ is the rate of “intervalley” transitions ($|T\rangle+|B\rangle \to |T\rangle-|B\rangle$) induced by the “mass term” ($M\tau^z$) in Eq. (\[eq:model\_b\]). Because both Eq. (\[eq:g1\]) and Eq. (\[eq:g2\]) are spin-singlets, their contributions to $\delta\sigma$ are of WAL type (this is proven below). Incidentally, $M=0$ is the physically relevant regime for Weyl semimetals, which have two degenerate Dirac points with linear energy dispersion along the three momenta axes. Unlike in graphene,[@mccann2006] where there are $4$ gapless Cooperon modes (in absence of atomically sharp defects and hexagonal warping), in a Weyl semimetal we obtain only $2$ gapless Cooperon modes. This difference stems from the fact that the SU(2) “valley symmetry” of graphene[@mccann2006] gets reduced to a U(1) symmetry in Weyl semimetals, due to the band dispersion along $z$. Acting somewhat like a Zeeman field would in a free electron gas, the $k_z$ dispersion generates a mass for orbital-singlet modes, which is why the nearly-gapless Cooperons in Eq. (\[eq:g1\]) and  (\[eq:g2\]) are orbital-triplets. When $(\epsilon_F-M)/M\ll 1$, there are three soft modes with gap $$\label{eq:tau_v} \Delta_g=(2/9)(1-M/\epsilon_F)^2\tau_0^{-1}\equiv\tau_s^{-1}\ll\tau_0^{-1}.$$ Physically, $\tau_s^{-1}$ is the rate of spin-flip transitions induced by the “spin-orbit term” ($v {\bf k}\cdot{\boldsymbol\sigma} \tau^x$) in Eq. (\[eq:model\_b\]). The eigenvectors for the three slightly gapped modes are $$\begin{aligned} \label{eq:trip} &(\lambda_1|T T\rangle+\lambda_2|B B\rangle)|\downarrow\downarrow\rangle\nonumber\\ & (\lambda_1|T T\rangle+\lambda_2|B B\rangle)|\uparrow\uparrow\rangle\nonumber\\ & (\lambda_3|T T\rangle +\lambda_4|B B\rangle)\left(|\uparrow\downarrow\rangle+\downarrow\uparrow\rangle\right),\end{aligned}$$ where $\lambda_1,...,\lambda_4$ are coefficients that depend only on $\epsilon_F/M$, such that $\lambda_1\simeq\lambda_3\simeq 1+O[(\epsilon_F/M-1)^2]$ and $\lambda_2\simeq\lambda_4\simeq O[(\epsilon_F/M-1)]$. Therefore, the three soft modes in Eq. (\[eq:trip\]) are all spin and orbital triplets. As will be demonstrated momentarily, their contribution to $\delta\sigma$ is of WL type. Next we determine $\overline{C}$ (c.f. Eq. (\[eq:overline\_c\])) by diagonalizing Eq. (\[eq:C\_mat\]) at $Q\neq 0$ and doing the angular integration in Eq. (\[eq:overline\_c\]). For $(\epsilon_F-M)/M\ll 1$ we obtain $$\label{eq:cav1} \overline{C}\simeq \frac{\hbar}{6\pi\nu\tau^2}\left[-\frac{1}{D Q^2+\tau_\phi^{-1}}+\frac{3}{D Q^2+\tau_\phi^{-1}+\tau_s^{-1}}\right].$$ For $(\epsilon_F-M)/M\gg 1$, we instead get $$\label{eq:cav2} \overline{C}\simeq \frac{3 \hbar}{8\pi\nu\tau^2}\left[-\frac{1}{D Q^2+\tau_\phi^{-1}}-\frac{1}{D Q^2+\tau_\phi^{-1}+\tau_v^{-1}}\right].$$ In the derivation of Eqs. (\[eq:cav1\]) and  (\[eq:cav2\]) we have included the phase relaxation time $\tau_\phi$ and exploited $D Q^2\tau_0\ll 1$. The first term in the square brackets of Eqs. (\[eq:cav1\]) and  (\[eq:cav2\]) is large at $Q\to 0$ irrespective of $\epsilon_F/M$, and originates from the spin-singlet Cooperon mode in Eq. (\[eq:g1\]). Its negative sign means that it makes a contribution towards WAL. Equation (\[eq:cav1\]) displays a competition between WL and WAL, which is no different from that found in an ordinary metal with spin-orbit interactions. WL terms originate from the three spin triplet modes of Eq. (\[eq:trip\]). WL prevails if $\tau_\phi^{-1}\gg\tau_s^{-1}$, whereas WAL rules if $\tau_{\phi}^{-1}\ll\tau_s^{-1}$. Equation (\[eq:cav2\]) unveils two different regimes of WAL. On one hand, if $\tau_\phi^{-1}\gg\tau_v^{-1}$, the spin-singlet Cooperon mode of Eq. (\[eq:g2\]) makes a contribution to $\delta\sigma$ that equals that of Eq. (\[eq:g1\]). In this limit, quantum interference can be interpreted as coming from two identical and nearly-decoupled Dirac valleys. On the other hand, if $\tau_\phi^{-1}\ll\tau_v^{-1}$, the contribution from Eq. (\[eq:g2\]) becomes relatively unimportant and the magnitude of WAL is halved. In other words, when the intervalley transition rate induced by the mass term $M\tau^z$ is fast compared to the phase relaxation rate, the two valleys contribute as one. This is quite different from graphene, where strong intervalley scattering changes WAL into WL.[@mccann2006] The underlying reason for such a qualitative difference is that in graphene a gapless valley-singlet mode is responsible for producing WL, whereas in a Weyl semimetal the valley-singlet Cooperons are strongly gapped by the $k_z$ band dispersion. Substituting Eqs. (\[eq:cav1\]) and  (\[eq:cav2\]) in Eq. (\[eq:ds2\]) and doing the $Q$-integral, we arrive at $$\begin{aligned} \label{eq:res_bulk} &\delta G\simeq\alpha\, G_q\ln(\tau_\phi/\tau)\nonumber\\ &\alpha=\left\{\begin{array}{ccc} -1 & {\rm if } & \tau_\phi\ll\tau_s\\ 1/2 & {\rm if }& \tau_\phi\gg(\tau_v,\tau_s)\\ 1 & {\rm if } & \tau_\phi\ll\tau_v, \end{array}\right.\end{aligned}$$ where $\delta G\equiv W\delta\sigma$ is the quantum interference correction to [*conductance*]{} and $$G_q\equiv e^2/(2\pi^2 \hbar)$$ is a universal conductance unit. In the derivation of Eq. (\[eq:res\_bulk\]) we have used Eq. (\[eq:ds2\]). The reason why $\alpha=1/2$ when $\tau_\phi\gg(\tau_v,\tau_s)$ is that in such regime there is only one gapless Cooperon mode (hence $|\alpha|=1/2$), which is a spin-singlet (hence $\alpha=|\alpha|$). While Eq. (\[eq:res\_bulk\]) is valid in absence of external magnetic fields, the magnetoconductance $\Delta G(H)\equiv G(H)-G(0)\simeq\delta G(H)-\delta G(0)$ can be easily obtained from Eq. (\[eq:res\_bulk\]) for $H$ perpendicular to the TI thin film. The replacement of $\int d^2 Q$ by an appropriate sum over Landau levels[@hikami1980] results in $$\begin{aligned} \label{eq:magres_bulk} &\Delta G\simeq\alpha\, G_q f(H_\phi/H)\nonumber\\ &\alpha=\left\{\begin{array}{ccc} -1 & {\rm if } & \tau_H\ll\tau_s\\ 1/2 & {\rm if } & \tau_H\gg(\tau_v,\tau_s)\\ 1 & {\rm if } & \tau_H\ll\tau_v, \end{array}\right.\end{aligned}$$ where $f(z)\equiv\ln z-\psi(1/2+z)$ with asymptotes $f(z)\propto z^{-2}$ for $z\gg 1$ and $f(z)\propto\ln(1/z)$ for $z\ll 1$, $\psi$ is the digamma function, $$\tau_H^{-1}\equiv \tau_\phi^{-1}+2 e D H/\hbar\,\,\mbox{ and }\,\, H_\phi\equiv\hbar/(4 e D \tau_\phi).$$ Three conclusions of experimental relevance can be extracted from Eqs. (\[eq:res\_bulk\]) and  (\[eq:magres\_bulk\]), which apply when highest occupied electronic states are all located near the $\Gamma$ point. First, bulk TI bands can display $\alpha=-1$ (WL) as long as the chemical potential is sufficiently close to the bottom of the bulk conduction band. Second, bulk TI bands can produce $\alpha=1$ when $\epsilon_F/M$ is sufficiently large. Third, when $(\epsilon_F-M)/M$ is neither large nor small, $\alpha=1/2$ ensues; this is the conventional result expected for ordinary conducting thin films with strong spin-orbit coupling, and is the one that has been often presumed in experiments on TI films.[@chen2010; @checkelsky2011; @wang2011; @he2011; @chen2011; @steinberg2011] At $\tau_H\simeq\tau_s$ there is a crossover between $\alpha=-1$ and $\alpha=1/2$; likewise, at $\tau_H\simeq\tau_v$ there is a crossover between $\alpha=1/2$ and $\alpha=1$. The particular expressions for $\tau_s$ and $\tau_v$ in Eqs. (\[eq:tau\_s\]) and (\[eq:tau\_v\]) rely on our assumption of $V_{\rm dis}\propto {\bf 1}_4$. Spin-orbit coupled impurities and/or atomically sharp disorder potentials would induce additional spin- and valley-flip processes, whose rates $\tau_{sf}^{-1}$ and $\tau_{vf}^{-1}$ would have to be incorporated via $\tau_s^{-1}\to\tau_s^{-1}+\tau_{sf}^{-1}$ and $\tau_v^{-1}\to\tau_v^{-1}+\tau_{vf}^{-1}$. If $\tau_{vf}^{-1}$ and $\tau_{sf}^{-1}$ are strong enough and insensitive to the value of $\epsilon_F/M$, then the only surviving regime of interference corrections is the conventional $\alpha=1/2$. The conventional $\alpha=1/2$ can be found in a wide range of parameter space at low temperatures, whereas the unconventional $\alpha=-1$ and $\alpha=1$ emerge in the relatively narrow regimes $\tau\ll\tau_H\ll\tau_s$ and $\tau\ll\tau_H\ll\tau_v$ (respectively). How accessible are these unconventional regimes? Suppose $M\simeq 150 {\rm meV}$, $v\simeq 5\times 10^5 {\rm m/s}$ and a bulk carrier density of $n\simeq 3\times 10^{18} {\rm cm}^{-3}$. This situation corresponds to having a small bulk Fermi surface. Then, it follows that $\alpha\simeq -1$ for a fairly wide range of magnetic fields ($l_H/(12 l)\ll 1$, where $l_H\equiv(D \tau_H)^{1/2}$). The limit $\alpha\to 1$ is not accesible in this regime. Instead, $\alpha\simeq 1$ should be accessible in (i) Weyl semimetals or in TIs with very narrow bandgaps, (ii) in TIs with large bandgap but $M({\bf k}_F)\simeq 0$. For the latter case it must be kept in mind that in the absence of spherical symmetry $M({\bf k}_F)$ is not constant on the Fermi surface. Suppose $M\simeq 5 {\rm meV}$ and a bulk carrier density of $\simeq 2\times 10^{18} {\rm cm}^{-3}$. Then, $\alpha\simeq 1$ in the range of fields corresponding to $l_H/(10 l)\ll 1$. For typical thin films, the requirements for $\alpha=\pm 1$ are compatible with $k_F l\gg 1$. Materials like BiTl(S$_{1-\delta}$Se$_\delta$)$_2$, where controlled changes of $\delta$ can tune $M$ from 0 to large values,[@xu2011] appear to be good candidates to observe crossovers between different regimes of magnetoresistance in Eq. (\[eq:magres\_bulk\]). Our analysis has thus far neglected surface states of the TI, which can also contribute to the measured magnetoresistance. It can be argued that surface states are unimportant and Eq. (\[eq:magres\_bulk\]) suffices in trivial insulators described by Eq. (\[eq:model\_b\]), as well as in time-reversal-invariant Weyl semimetals and in TIs with very small bulk bandgaps ($\lesssim\hbar/\tau_0$). In contrast, when the surface states of the TI are robust, Eq. (\[eq:magres\_bulk\]) is incomplete and must be generalized. Such generalization is the subject for the rest of this paper. Quantum Corrections to Conductivity from Coupled Bulk and Surface States ======================================================================== In this section we consider the combined bulk-surface contribution to $\delta\sigma$ in 3D TIs with relatively large bandgaps. We concentrate on a particular setup that consists of a TI thin film gated on one surface. The gate voltage can produce a depletion layer that spatially separates bulk and surface carriers (Fig. \[fig:dep\]), and carriers tunnel back and forth across the depletion layer. We assume the bulk-surface tunneling rate to be much smaller than the elastic scattering time on either side of the depletion layer, so that electrons scatter many times within the bulk (surface) before tunneling to the surface (bulk). This assumption is experimentally realistic, and it simplifies the microscopic theory of this section considerably. Single isolated TI surface -------------------------- As a preliminary step, we recall the expression for $\delta\sigma$ on a single TI surface that is decoupled from the bulk. Taking $\epsilon_{Fs} \tau\gg 1$, where $\epsilon_{Fs}$ is the Fermi energy measured from the Dirac point of the surface states, one arrives[@tkachov2011; @lu2011a] at $$\label{eq:res_s} \Delta G/G_q=(1/2) f(H_\phi/H)$$ for any $\tau_H$. The prefactor $1/2$ is consistent with having a gapless spin-singlet Cooperon (the spin-triplet Cooperons have large gaps due to the strong spin-momentum coupling on the surface). Two coupled 2D layers without spin-orbit coupling ------------------------------------------------- As another preliminary step, here we compute $\delta\sigma$ for two ordinary metallic 2D layers separated by a tunnel barrier. In a double layer system, the current flowing in layer $i$ can be written as ${\bf j}_i=\sum_j \sigma_{i j} {\bf E}_j$, where ${\bf E}_j$ is the electric field in layer $j$. For concreteness we take ${\bf E}_1={\bf E}_2\equiv {\bf E}$, so that the measured current is ${\bf j}={\bf j}_1+{\bf j}_2=\sigma{\bf E}$ with $\sigma=\sum_{i j}\sigma_{i j}$. Consequently, the quantum corrections to conductivity are $\delta\sigma=\sum_{i j}\delta\sigma_{i j}$. The goal of this section is to compute $\delta\sigma$ from microscopic theory. ![Diagrammatic representation for $\delta\sigma_{i j}$, where $i$ and $j$ are layer indices. For 2D layers without spin-orbit coupling, the Cooperon matrix elements are fully characterized by layer indices. The velocity operator is diagonal in the layer index; therefore, the Cooperons $C^{1 1}_{2 2}$ and $C^{2 2}_{1 1}$ do not enter in the expression for $\delta\sigma_{i j}$.[]{data-label="fig:dsij"}](./diag6.eps) ![Typical microscopic process that gives rise to $\delta\sigma_{1 2}$. It can be neglected when the intralayer disorder potentials in the two layers are uncorrelated.[]{data-label="fig:ds12"}](./diag3.eps) The interference correction $\delta\sigma_{i j}$ has the diagrammatic representation shown in Fig. \[fig:dsij\]. Because the velocity operator is diagonal in the layer index, the only Cooperons that enter in the conductivity are $C^{i j}_{j i}$, with $i,j\in\{1,2\}$. In particular $\delta\sigma_{i i}$ involves intralayer Cooperons $C^{i i}_{i i}$, whereas $\delta\sigma_{1 2}$ and $\delta\sigma_{2 1}$ involve interlayer Cooperons $C^{1 2}_{2 1}$ and $C^{2 1}_{1 2}$ (Fig. \[fig:ds12\]). Assuming that disorder potentials in the two layers are uncorrelated, $C^{i j}_{j i}=0$ for $i\neq j$. This is a reasonable assumption when electrons in the two layers scatter off different sets of impurities. Hence, we are left with $\delta\sigma=\sum_i\delta\sigma_{i i}$. From here on we simplify the notation via $C^{i i}_{i i}\equiv C_i$. When evaluating $\delta\sigma_{i i}$ we will neglect spin-orbit interactions; however, the main lessons learned in this subsection will be transferrable to the spin-orbit coupled case studied in the next subsection. In absence of interlayer coupling, a standard calculation yields $$\label{eq:dnotu} \delta\sigma_{i i}^{(0)}\simeq -4 \frac{e^2}{\hbar^2} \nu_i D_i \tau_{d i}^2\int_{\bf Q} C_i^{(0)}({\bf Q}),$$ where $\int_{\bf Q}\equiv \int d^2 Q/(2\pi)^2$, an extra factor of $2$ is due to spin degeneracy, $\tau_{d i}$ is the scattering time in layer $i$ due to elastic impurities (we assume purely s-wave scattering, so that there is no difference between the transport scattering time and the quantum lifetime), $\nu_i$ is the density of states per unit area in layer $i$ and $$C_i^{(0)} ({\bf Q})=\frac{\hbar}{2\pi \nu_i \tau_{d i}^2} \frac{1}{D_i Q^2+\tau_{\phi i}^{-1}}$$ is the Cooperon for an isolated layer. In presence of interlayer tunneling, $C_i^{(0)}$ in Eq. (\[eq:dnotu\]) is replaced by $C_i$: $$\label{eq:dd} \delta\sigma_{i i}\simeq -4 \frac{e^2}{\hbar^2} \nu_i D_i \tau_{d i}^2\int_{\bf Q}C_i({\bf Q}),$$ in whose prefactor we have neglected terms containing the ratio between the tunneling rate and the elastic scattering rate. ![Single-particle Green’s functions. (a) Dressing of Bloch states due to intralayer impurity scattering. (b) Dressing of disorder-averaged Green’s functions due to interlayer tunneling. The tunneling amplitude is regarded as a random variable.[]{data-label="fig:gfig"}](./diag.eps) In order to compute $C_i$, we recognize that the influence of interlayer coupling occurs at two different levels. On one hand, it modifies the single-particle Green’s function for each layer (Fig. \[fig:gfig\]). Because the thickness of the depletion layer typically shows microscopic variations within the same film as well as from sample to sample, the interlayer tunneling amplitude can be regarded as a random variable. Consequently, the change in the ensemble-averaged Green’s function due to tunneling can be captured via $\tau_{d i}^{-1}\to \tau_{d i}^{-1}+\tau_{t i}^{-1}$, where $$\tau_{t i}^{-1}=(2\pi/\hbar) \langle|t|^2\rangle S\, \nu_j$$ is the tunneling rate from layer $i$ onto layer $j\neq i$, $\langle|t|^2\rangle$ is the averaged square of the tunneling matrix element and $S$ is the layer area. Note that $\langle|t|^2\rangle$ scales like $S^{-1}$, so that $\tau_{t i}^{-1}$ is independent of the layer area. ![(a) Cooperon $C_i^{(0)}$ without interlayer tunneling. (b) Partially dressed Cooperon $\tilde{C}_i^{(0)}$, where tunneling is included solely in the single-particle Green’s functions. $\tilde{C}_i^{(0)}$ can be directly obtained from $C_i^{(0)}$ via $\tau_{\phi i}\to\tilde{\tau}_{\phi i}$. (c) Fully dressed Cooperon $C_i$, where tunneling is incorporated both in the single-particle Green’s function and in the particle-particle correlations.[]{data-label="fig:cfig"}](./diag5.eps) ![Typical processes not included in Fig. \[fig:cfig\], as they are subdominant for $\tau_{t i}\gg\tau_{d i}$.[]{data-label="fig:nfig"}](./diag4b.eps) On the other hand, interlayer tunneling modifies particle-particle correlations that build up Cooperons. An approximate diagrammatic expression for these correlations is shown in Fig. \[fig:cfig\]. The equation of Fig. \[fig:cfig\]c can be solved in momentum space and it yields $$\label{eq:cii} C_i =\frac{\hbar}{2\pi\nu_i\tau_{d i}^2}\frac{D_j Q^2+\tilde{\tau}_{\phi j}^{-1}}{(D_1 Q^2+\tilde{\tau}_{\phi 1}^{-1})(D_2 Q^2+\tilde{\tau}_{\phi 2}^{-1})-\tau_{t 1}^{-1}\tau_{t 2}^{-1}}$$ for $j\neq i$. In the derivation of Eq. (\[eq:cii\]) we have introduced $$\label{eq:tau_tilde} \tilde{\tau}_{\phi i}^{-1}\equiv\tau_{\phi i}^{-1}+\tau_{t i}^{-1}$$ as an effective phase relaxation rate that incorporates tunneling, and have used $$\int_{\bf k} G_i^R ({\bf k}) G^A_i(-{\bf k}+{\bf Q})\simeq \frac{2\pi \nu_i\tau_{d i} }{\hbar}\left(1-\frac{\tau_{d i}}{\tilde{\tau}_{\phi i}}-D_i Q^2 \tau_{d i}\right).\nonumber$$ Microscopic processes depicted in Fig. \[fig:cfig\] leave out those in which two consecutive tunneling events occur without any intralayer scattering in between. Likewise, they ignore electron trajectories in which a tunneling event precedes any intralayer scattering (Fig. \[fig:nfig\]). These processes are relatively unimportant if $\tau_{t i}\gg \tau_{d i}$. Not surprisingly, Eq. (\[eq:cii\]) arises in the coupled equations for the classical diffusive conductivity as well (see Appendix \[sec:cond\]). It is convenient to rewrite $C_i$ in Eq. (\[eq:cii\]) as $$\label{eq:q1} C_i=\frac{\hbar}{2\pi\nu_i D_i \tau_{d i}^2}\left[\frac{A_i}{Q^2+q_a^2}+\frac{B_i}{Q^2+q_b^2}\right],$$ where $$\label{eq:ab1} 2 q_{a (b)}^2=\frac{1}{\tilde{l}_{\phi 1}^{2}}+\frac{1}{\tilde{l}_{\phi 2}^{2}}\pm \sqrt{\left(\frac{1}{\tilde{l}_{\phi 1}^{2}}-\frac{1}{\tilde{l}_{\phi 2}^{2}}\right)^2+ \frac{4}{l_{t 1}^2 l_{t 2}^2}}$$ and $$\label{eq:ab2} A_i = 1-B_i= (\tilde{l}_{\phi j}^{-2}-q_a^2)/(q_b^2-q_a^2)\,\,\mbox{ for $j\neq i$.}$$ In Eq. (\[eq:ab1\]) we have defined $\tilde{l}_{\phi i}\equiv(D_i\tilde{\tau}_{\phi i})^{1/2}$ as an effective coherence length and $l_{t i}\equiv(D_i\tau_{t i})^{1/2}$ as the interlayer leakage length. Besides, $q_a^2 (q_b^2)$ gets the positive (negative) sign in front of the square root. Combining Eq. (\[eq:dd\]) with Eq. (\[eq:q1\]) and using $A_1+A_2=B_1+B_2=1$, we get $$\delta\sigma=\sum_i\delta\sigma_{i i}=-2 \frac{e^2}{\pi\hbar} \int_{\bf Q} \left[\frac{1}{Q^2+q_a^2}+\frac{1}{Q^2+q_b^2}\right].$$ Therefore, the low-field magnetoconductance reads $$\label{eq:Dii} \Delta\sigma=\sum_i\Delta\sigma_{i i}=-G_q\left[ f\left(\frac{H_a}{H}\right)+ f\left(\frac{H_b}{H}\right) \right],$$ where $$H_{a (b)}\equiv \hbar\, q_{a (b)}^2/(4 e).$$ In the limit of very strong tunneling ($\tau_{t i}/\tau_{\phi i}\to 0$), Eq. (\[eq:Dii\]) becomes $\Delta\sigma\simeq -G_q f(H_b/H)$, as though there was a single layer. In the limit of very weak tunneling ($\tau_{t i}/\tau_{\phi i}\to\infty$), $\Delta\sigma$ is the sum of contributions from two independent films. It is helpful to understand the weak and strong coupling regimes in terms of measurable quantities like the interlayer conductance per square, $$g_t=(2\pi e^2/\hbar)\langle|t|^2\rangle S \nu_1\nu_2=\sigma_{D i}/l_{t i}^2,$$ where $\sigma_{D i}$ is the Drude conductivity in layer $i$. For simplicity suppose that $\tau_{\phi 1}\simeq\tau_{\phi 2}\equiv\tau_\phi$. In this case the crossover from weak to strong tunneling occurs when $$\label{eq:cross} \frac{1}{g_t l_\phi^2}\lesssim \frac{1}{\sigma_{D 1}}+\frac{1}{\sigma_{D 2}}\,\,\,\mbox{(crossover condition)},$$ namely when the tunneling resistance for a square of area $l_\phi^2$ becomes smaller than the sum of the classical intralayer resistivities. Let us define $$g_c^{-1}\equiv (\sigma_{D 1}^{-1}+\sigma_{D 2}^{-1}) l_\phi^2.$$ If $g_t\ll g_c$, then $\Delta\sigma/G_q\simeq -2\ln(H/H_\phi)$ for $H\gg H_\phi$. If $g_t\gg g_c$, then $\Delta\sigma/G_q\simeq -\ln(H/H_\phi)$ for $H_\phi\ll H\ll H_\phi (g_t/g_c)$. Thus changing the interlayer conductance results in a factor-of-two change for the magnitude of the WL correction. ![Example of an interlayer scattering process that is allowed in multilayer systems. Its analog in multivalley semiconductors of Ref. \[\] is forbidden.[]{data-label="fig:fuku"}](./diag7.eps) Limits reminiscent of the above were first discussed in inversion layers of multivalley semiconductors like Si,[@fukuyama1980] where the role of the layers is played by different electron pockets in the Brillouin zone. Similarities notwithstanding, there are clear differences between our microscopic theory and that of multivalley semiconductors. On one hand, the separation in momentum between valleys of Si prevents scattering processes such as the one in Fig. \[fig:fuku\]. These processes are not only allowed in our case, but also lead to the Cooperon dressing shown in Fig \[fig:cfig\]c. On the other hand, in our case the interlayer Cooperon vanishes due to uncorrelated disorder potentials in the two spatially separated layers. That is not the case in multivalley semiconductors, where both valleys scatter off the same set of real-space impurities and intervalley Cooperons contribute crucially to $\delta\sigma$. Finally, it should be mentioned that Eqs. (\[eq:ab1\]), (\[eq:ab2\]) and (\[eq:Dii\]) coincide with those derived by G. Bergmann,[@bergmann1989] who invoked macroscopic arguments based on the diffusion equation. The microscopic theory of this subsection supports Bergmann’s results, insofar as $\tau_{t i}\gg \tau_{d i}$ and the disorder potentials in the two layers are uncorrelated. Incidentally, yet another way to arrive at the same results is unveiled in Appendix \[sec:coupled\_cooper\]; this later method will prove convenient in the upcoming subsection. 3D TI film with bulk-surface coupling ------------------------------------- We now consider a 3D TI film (Fig. \[fig:dep\]) with a gate electrode placed near its top surface. At the moment we neglect the bottom surface of the TI, which will be incorporated below. For ease of notation we use subscript “1” to refer to “bulk”, and subscript “2” to refer to “top surface”. Like in the preceding subsection we assume bulk-surface disorder correlations to be negligible, so that the quantum corrections to conductance can be written as $\delta G=\delta G_{1 1}+\delta G_{2 2}=W \delta\sigma_{1 1}+\delta\sigma_{2 2}$. $\delta G$ is approximately independent of the film thickness $W$ as long as $W\ll\tilde{l}_{\phi 1}$ , where $\tilde{l}_{\phi 1}$ was defined below Eq. (\[eq:ab2\]). The goal of this subsection is to calculate $\delta G$ from microscopic theory. Unlike in the previous subsection, here both “layers” are spin-orbit coupled. We assume that tunneling events, albeit being time-reversal invariant, conserve neither spin nor orbital degrees of freedom. Indeed, in a TI spin is not conserved for non-momentum-conserving tunneling. Similarly, the orbital degree of freedom is not conserved due to broken inversion symmetry near the surface. Let us begin with no tunneling. On one hand, there are four surface Cooperon modes: one gapless spin-singlet mode and three spin-triplet modes with large ($\sim\tau_{d 2}^{-1}$) gaps. On the other hand, there are sixteen bulk Cooperons, of which a spin-singlet mode (Eq. (\[eq:g1\])) is always gapless. In addition, four of the bulk modes (the spin-singlet of Eq. (\[eq:g2\]) and three spin-triplets of Eq. (\[eq:trip\])) can be “soft” depending on $\epsilon_F/M$. The rest of the bulk Cooperon modes have large gaps of order $\tau_{d 1}^{-1}$. Let us now turn on tunneling. Since $\tau_{t i}\gg \tau_{d i}$, we can limit ourselves to analyzing the effects of tunneling within the low-energy subspace formed by the soft Cooperons. If there are no magnetic impurities in the depletion layer, the total spin of the Cooperon is a good quantum number even in presence of tunneling. Accordingly tunneling does not mix spin-singlet modes with spin-triplet modes, and the full (dressed) Cooperons can also be classified into spin-singlets and a spin-triplet. In the regimes $\tau_{\phi 1}\ll\tau_s$ and $\tau_{\phi 1}\gg (\tau_s,\tau_v)$, tunneling dresses one soft spin-singlet Cooperon in the bulk with another soft spin-singlet Cooperon on the surface. This dressing is completed as explained in Section IIIB: first by renormalizing the phase relaxation time $\tau_{\phi i}\to \tilde{\tau}_{\phi i}$, and afterwards proceeding with the series expansion of Fig. \[fig:cfig\]c. All “blocks” appearing in this series expansion are spin-singlets. When $\tau_{\phi 1}\ll\tau_s$, the soft spin-triplet Cooperons from the bulk are dressed simply through $\tau_{\phi 1}\to\tilde{\tau}_{\phi 1}$: they do not get appreciably admixed with the spin-triplet Cooperon on the surface because the latter has a large gap. In the regime $\tau_{\phi 1}\ll\tau_v$, there are two gapless singlet Cooperons in the bulk, each of which can hybridize with the singlet gapless Cooperon on the surface. For this situation, Fig. \[fig:cfig\]c does not capture all possible processes and the calculation from the previous subsection must be generalized; this generalization is carried out in Appendix \[sec:coupled\_cooper\]. With the above considerations in mind, we combine Eqs. (\[eq:magres\_bulk\]) and  (\[eq:res\_s\]) in order to obtain the total contribution to low-field magnetoconductance: $$\begin{aligned} \label{eq:res_tot} & \frac{\Delta G}{G_q}\simeq\frac{1}{2}\left\{\begin{array}{ccc} f\left(\frac{H_a}{H}\right)+f\left(\frac{H_b}{H}\right)-3 f\left(\frac{\tilde{H}_{\phi 1}}{H}\right) &{\rm if } & \tilde{\tau}_H\ll\tau_s\\ f\left(\frac{H_a}{H}\right)+f\left(\frac{H_b}{H}\right) &{\rm if } & \tilde{\tau}_H\gg(\tau_s,\tau_v)\\ f\left(\frac{H_c}{H}\right)+f\left(\frac{H_d}{H}\right)+f\left(\frac{\tilde{H}_{\phi 1}}{H}\right) &{\rm if } &\tilde{\tau}_H\ll\tau_v, \end{array}\right.\end{aligned}$$ where $H_l=\hbar\, q_l^2/(4 e)$ for $l\in\{a,b,c,d\}$, $$\tilde{H}_{\phi 1}\equiv \hbar/(4 e D_1 \tilde{\tau}_{\phi 1}),\,\,\,{\rm and}\,\,\, \tilde{\tau}_H^{-1}\equiv \tilde{\tau}_{\phi 1}^{-1}+ 2 e D_1 H/\hbar.$$ Note that the effective phase relaxation rate increases linearly with the bulk-to-surface tunneling rate (c.f. Eq. (\[eq:tau\_tilde\])). The characteristic momenta $q_{a (b)}$ have been introduced earlier in Eq. (\[eq:ab1\]). The additional momenta $q_{c(d)}$ are identical to $q_{a (b)}$, except for $\tau_{t 2}^{-1}\to 2\,\tau_{t 2}^{-1}$. The reason for this difference is that the surface Cooperon can decay into two gapless bulk Cooperons when $\tau_{\phi 1}\ll\tau_v$. The first line of Eq. (\[eq:res\_tot\]) displays a competition between WL and WAL, and suggests that it is possible to induce a WAL-to-WL transition with a varying gate voltage. In the weak tunneling regime WL prevails, whereas in the strong tunneling regime WAL takes over. Similarly, a gate voltage can induce transitions between three different WAL coefficients: $\alpha\in(1/2,1)$ in the second line, and $\alpha\in(1/2,3/2)$ in the third line. The second line of Eq. (\[eq:res\_tot\]) describes quantum corrections as if they originated from two independent thin films with mixed bulk-surface character; indeed, universal results expected for the simplectic symmetry class are recovered when the effective phase relaxation times become the longest timescales of the problem. Some simple limiting cases of Eq. (\[eq:res\_tot\]) are discussed in Appendix \[sec:special\]. Thus far we have considered the coupling between the bulk and [*one*]{} (the top) surface of the TI film. As a consequence, Eq. (\[eq:res\_tot\]) applies to a TI film only if the phase relaxation time of the bottom surface (adjacent to the substrate) is short compared to other phase relaxation and tunneling times in the problem. This condition is likely not met in some recent experiments,[@checkelsky2011; @chen2011] which report on independent contributions from both surfaces. Partly motivated by these experiments, we now generalize Eq. (\[eq:res\_tot\]) so as to capture two surfaces, each coupled to bulk states. We consider the scenario depicted in Fig. \[fig:dep\], where the bottom surface contains bulk carriers. Since there is no depletion layer near $z=W$, we assume that the bulk-surface tunneling rate therein is strong compared to the phase relaxation rate, yet weak compared to disorder scattering rate. Hence we describe the hybrid of bottom surface and bulk states via Eq. (\[eq:res\_tot2\]), and thereafter couple this hybrid to the top surface along the lines of Eq. (\[eq:res\_tot\]). The resulting expression for $\Delta G$ can be approximated as $$\label{eq:res_tot5} \frac{\Delta G}{G_q}\simeq\frac{1}{2} f\left(\frac{H'_a}{H}\right)+\frac{1}{2}f\left(\frac{H'_b}{H}\right),$$ where $H'_{a(b)}\equiv\hbar (q'_{a(b)})^2/(4 e)$. The characteristic momenta $q'_a$ and $q'_b$ obey Eq. (\[eq:ab1\]), where “1” labels the top surface and “2” labels a hybrid between the bottom surface and the bulk. Notably, Eq. (\[eq:res\_tot5\]) implies that WL is no longer possible once the bottom surface is strongly coupled to bulk states. Instead, conventional WAL ensues with $\alpha\in(1/2,1)$. This observation not only sheds light on why current experiments see no indication for WL, but it also gives insight as to how WL could be observed in TI films. A possible strategy is to degrade the surfaces, e.g. by depositing magnetic impurities on them, and decoupling them from the bulk by double-sided gating. One may expect WL even if only the top surface is decoupled, while the (degraded) bottom surface is in contact with the bulk. In this case, Eq. (\[eq:res\_tot\]) reduces to Eq. (\[eq:magres\_bulk\]) derived for the sole bulk conduction, with the replacement $\tau_{\phi 1}^{-1}\to\tau_{\phi 1}^{-1}+\tau_{t 3}^{-1}$, where $\tau_{t 3}$ is the tunneling rate of electrons from bulk to the bottom surface. If the film is thick enough, then $\tau_{t 3}^{-1}$ may become sufficiently small to provide some dynamic range for observing WL behavior. This same strategy can also facilitate the observation of WAL with $\alpha>1$. Estimates for the bulk-surface coupling --------------------------------------- ![Schematic energy band profile for a gated 3D TI thin film. $z=0$ corresponds to the top surface of the device, immediately under the gate. $z=W$ corresponds to the bottom (ungated) surface. The vertical (blue) solid lines at $z=0, W$ are surface states. The curved solid (red) line is the bulk conduction band, and the dot-dashed (brown) curve is the bulk valence band. The chemical potential is depicted by a horizontal dashed line. $z_d$ is the thickness of the depletion layer, where neither bulk nor surface carriers are present. $\epsilon_{F s}$ is the Fermi energy of the surface states measured from the Dirac point ($\epsilon_{Fs}<0$ in this figure). $\epsilon_F$ is the Fermi energy of the bulk states, measured with respect to the midgap point. []{data-label="fig:dep"}](./dep.eps) This subsection is devoted to an approximate electrostatic and quantum mechanical analysis of the depletion layer in a TI film, which will result in quantitative estimates for the bulk-surface coupling. For a TI with an $n$-doped bulk, a negative charge per unit area $(-Q_g)$ placed at the gate repels electrons from bulk bands at $z=0$ as well as from the surface states at $z=0$. This leaves a positive net charge on the top surface, which is equivalent to a downward shift in the local chemical potential at $z=0$: $\Delta\mu_s=\epsilon_F-\epsilon_{F s}$. Recall that $\epsilon_{Fs}$ is the Fermi energy of the surface states measured from the Dirac point (for simplicity the Dirac point is assumed to be in the middle of the bandgap at $z=0$) and $\epsilon_F$ is the Fermi energy of the bulk states measured from the middle of the bandgap. Since the chemical potential deep inside the bulk must be unaffected by the gate, $\Delta\mu_s\neq 0$ implies a band bending of magnitude $\phi_s=\Delta\mu_s$ near the gated surface (Fig. \[fig:dep\]). When $\Delta\mu_s>(\epsilon_F-M)$ there are no bulk carriers left at $z=0$ and a depletion layer appears at $z\in(0,z_d)$, where $z_d$ is determined below. For each value of $Q_g$, $\Delta\mu_s$ (or equivalently $\epsilon_{F s})$ can be uniquely determined from the overall neutrality condition $Q_s+Q_d=Q_g$, where $Q_s$ is the positive net charge induced on the surface, and $Q_d$ is the positive net charge in the depletion layer. In the depletion approximation[@sze2002] one has $Q_d\simeq n z_d$, where $n$ (c.f. Eq. (\[eq:n\])) is equal to the density of charged donors in the depleted region. The electrostatic energy profile in the depleted region then obeys $$\label{eq:fi} \phi(z)=\phi_b-\frac{1}{2}\frac{e^2 n}{\kappa}(z-z_d)^2,$$ where $\phi_b\equiv\phi_s-(\epsilon_F-M)=M-\epsilon_{F s}$, $\kappa$ is the static dielectric constant and $$\label{eq:depl} z_d=\sqrt{\frac{2\kappa \phi_b}{e^2 n}}.$$ In the derivation of Eq. (\[eq:fi\]) we have assumed that the electric field vanishes at $z=z_d$, which is accurate within a screening radius. As the gate voltage is made more negative, the maximum width of the depletion layer ($z_d^{\rm max}$) is achieved when $\phi_b\simeq 2 M$. For $\phi_b>2 M$, the bulk bands get inverted at $z=0$ and $z_d$ saturates. We estimate $z_d^{\rm max}\simeq 20 {\rm nm}$ for some typical parameter values ($M=150 {\rm meV}$, $n\simeq 4\times 10^{18} {\rm cm}^{-3}$, $\kappa=50$). Once the electrostatic profile of the TI film is characterized, we can analyze the quantum mechanical tunneling of electrons across the depletion layer. The tunneling conductance per unit area is roughly $$\label{eq:kappa} g_t\sim (e^2/h) \lambda_F^{-2} \exp(-2\chi),$$ where $\lambda_F$ is the smallest between bulk and surface Fermi wavelengths, and $$\label{eq:kappa2} \chi\simeq \int_0^{z_d} dz\frac{\phi_b-\phi(z)}{\hbar v}\simeq\frac{1}{6}\frac{e^2 n z_d^3}{\kappa\,\hbar v}.$$ In Eq. (\[eq:kappa2\]) we have ignored effective mass and Fermi velocity mismatches across the depletion layer. The WKB exponent $\chi$ can be tuned by a gate voltage: as $z_d$ varies from $0$ to $z_d^{\rm max}$, $\chi$ goes from $0$ to $\simeq 6$. Drawing from the previous subsection (c.f. Eq. (\[eq:cross\])), the crossover from weak to strong bulk-surface coupling occurs when $$\label{eq:cross1} \frac{1}{g_t l_\phi^2}\lesssim\frac{1}{\sigma_{D 1} W}+\frac{1}{\sigma_{D 2}}\simeq \frac{1}{\sigma_{D 2}},$$ where in the second equality we have assumed that $\sigma_{D 1} W\gg\sigma_{D 2}$. This is a good assumption provided that (i) the bulk mean free path is of the same order as the surface mean free path, and (ii) $k_F W\gg 1$. Plugging Eq. (\[eq:kappa\]) in Eq. (\[eq:cross1\]), the latter becomes $$\label{eq:cross2} \frac{l_\phi}{\lambda_F}\gtrsim (k_{Fs} l_2)^{1/2} \exp(\chi),$$ where $k_{Fs}=|\epsilon_{F s}|/{\hbar v}$ is the Fermi wave vector for the surface states and we have used $\sigma_{D 2}\sim (e^2/h) k_{F s} l_2$. When $z_d=z_d^{\rm max}$, the right hand side of Eq. (\[eq:cross2\]) reaches $\simeq 1000$, which exceeds the typical $l_\phi/\lambda_F$ in TI thin films by at least an order of magnitude. Therefore, when the depletion layer has its maximum width, the top surface and the bulk of the TI film can be regarded as weakly coupled. This state of affairs changes rapidly when the depletion layer is made thinner by a gate voltage. For instance, when $z_d=z_d^{\rm max}/\sqrt2$, the right hand side of Eq. (\[eq:cross2\]) equals $\simeq 30$, which is comparable to the typical $l_\phi/\lambda_F$. Further slight reductions in $z_d$ can subsequently drive the film into a regime of strong bulk-surface coupling. These estimates justify the interpretation of experimental data given in e.g. Ref. \[\]. Summary and conclusions ======================= We have completed a theoretical study of low-field magnetoresistance in electrostatically gated 3D TI films. The concise analytical expressions presented here \[Eqs. (\[eq:magres\_bulk\]), (\[eq:res\_tot\]) and (\[eq:res\_tot5\])\] may shed light on the quantum magnetoresistance of TIs, Weyl semimetals, as well as some topologically trivial materials. Only magnetic fields that are perpendicular to the TI thin film have been considered in this work; for in-plane fields and small bulk bandgaps, quantum interference contributions might be masked by classical magnetoresistance anomalies.[@son2012] A number of predictions from this work have not been articulated in previous studies and await experimental confirmation. For instance, we find that TI thin films with low bulk doping may exhibit weak localization (WL) or negative magnetoresistance, instead of the often presumed weak antilocalization (WAL) or positive magnetoresistance. Admittedly, the parameter space for WL is relatively narrow, and vanishes when either surface of the TI film is strongly coupled to bulk states. However, WL may be experimentally accessible in thicker films, or in thin films where the surfaces have short phase relaxation times. Under these conditions, a gate can induce a crossover between WL and WAL. On a separate note, we find that the “universal” prefactor for WAL varies depending on the bandgap of the TI, on the bulk doping concentration, on the phase relaxation times, and on the applied gate voltage. The results from this work are applicable to conducting yet lighly doped TIs, with thicknesses ranging between the bulk transport mean free path and the bulk phase relaxation length. It may be useful to find out how the results derived here change in highly doped TIs containing additional electrons pockets away from the $\Gamma$ point. Likewise, it may be helpful to extend our results to thinner films. Other potentially interesting tasks involve investigating universal conductance fluctuations and determining the influence of electron-electron interactions in the magnetoresistance of doped TI films. This research has been financially supported by a fellowship from Yale University (I.G.), and by NSF DMR Grant No. 0906498 (L.G.). L.G. thanks Pablo Jarillo-Herrero for a discussion that initiated the present work, I.G. thanks Ewelina Hankiewicz for an informative conversation, and both authors thank Aharon Kapitulnik for bringing Ref. \[\] to their attention. Renormalized velocity operator {#sec:ren} ============================== The velocity operators appearing in the expressions for $\sigma_D$ and $\delta\sigma$ (c.f. Sec. IIB) must be renormalized with ladder diagrams containing impurity scattering. The Dyson equation for the renormalized velocity operator is (Fig. \[fig:vertex\]) $$\label{eq:vd} \tilde{{\bf v}}_{\alpha\beta}({\bf k})= {\bf v}_{\alpha\beta}({\bf k})+u_0\sum_{\alpha,\beta\in\{1,2\}}\int_{{\bf k}'}\langle\alpha {\bf k}|\alpha' {\bf k}'\rangle\langle\beta'{\bf k}'|\beta{\bf k}\rangle G^A({\bf k}') G^R({\bf k}') \tilde{{\bf v}}_{\alpha'\beta'}({\bf k}'),$$ where ${\bf v}_{\alpha\beta}({\bf k})=\delta_{\alpha\beta} \hbar v^2 {\bf k}/E_k$ is a matrix element for the bare velocity operator. We solve Eq. (\[eq:vd\]) by guessing a solution of the form $$\label{eq:guess} \tilde{{\bf v}}_{\alpha\beta}({\bf k})=\gamma_k {\bf k}\delta_{\alpha\beta},$$ where $\gamma_k$ is a scalar that depends on $|{\bf k}|$ but not $\hat{\bf k}$. Although it is [*a priori*]{} not obvious that the renormalized velocity operator should be diagonal in the band indices, substituting Eq. (\[eq:guess\]) in Eq. (\[eq:vd\]) and using Eq. (\[eq:eigenstates\]) we find that $\tilde{{\bf v}}_{\alpha\beta}({\bf k})\propto\delta_{\alpha\beta}$ is indeed appropriate provided that $$\gamma_k=\frac{\hbar v^2}{E_k}\frac{\tau}{\tau_0}.$$ Here $$\frac{\hbar}{\tau}=2\pi\nu u_0\int\frac{d\Omega_{{\bf k}'}}{4\pi}\sum_{\alpha'}|\langle\alpha {\bf k}_F|\alpha' {\bf k}'_F\rangle|^2 (1-\hat{\bf k}_F\cdot\hat{\bf k}'_F)$$ is the transport scattering time. Therefore, the final result for the renormalized velocity is $\tilde{{\bf v}}_{\alpha\beta}({\bf k})={\bf v}_{\alpha\beta}({\bf k}) (\tau/\tau_0)$. ![Impurity vertex corrections for the velocity operator []{data-label="fig:vertex"}](./vertex.eps) Evaluation of $\delta\sigma_2$ in some simple cases {#sec:ds2} =================================================== The expression for $\delta\sigma_2$ (depicted in Fig. \[fig:cofig\]c) reads $$\label{eq:nasty} \delta\sigma_2\simeq -2\frac{e^2\hbar}{2\pi}\int_{{\bf k}, {\bf k}'} \tilde{v}^x({\bf k}) \tilde{v}^x({\bf k}') G^A({\bf k}) G^A({\bf k}') G^A(-{\bf k}) G^A(-{\bf k}') G^R(-{\bf k}') G^R({\bf k})\sum_{\alpha\beta\alpha'\beta'}\Gamma^{\alpha\alpha'}_{\beta'\beta}({\bf k},-{\bf k}',0) \frac{1}{W}\int\frac{d^2 Q}{(2\pi)^2} C^{\beta\alpha}_{\alpha'\beta'}({\bf k},{\bf k}',{\bf Q}),$$ where the overall factor of two stems from the fact that the two diagrams in Fig. \[fig:cofig\]c give identical contribution, and the band indices $\alpha,\beta$ etc. are summed over $1,2$. For generic $(\epsilon_F-M)/M$, the calculation of $\delta\sigma_2$ is cumbersome. Here we focus on two simple limits that are of interest: $(\epsilon_F-M)/M\ll 1$ and $(\epsilon_F-M)/M\gg 1$. When $(\epsilon_F-M)/M\ll 1$, the momentum dependence of $|\alpha {\bf k}_F\rangle$ is negligible. Consequently, $C^{\beta\alpha}_{\alpha'\beta'}({\bf k}_F,{\bf k}_F',{\bf Q})$ and $\Gamma^{\alpha\alpha'}_{\beta'\beta}({\bf k}_F,-{\bf k}_F',0)$ become independent of ${\bf k}_F$ and ${\bf k}_F'$. Since the matrix elements of the velocity operator are odd under ${\bf k}\to -{\bf k}$ and ${\bf k}'\to -{\bf k}'$, it is clear that $$\delta\sigma_2\simeq 0.$$ The limit of $(\epsilon_F-M)/M\gg 1$ is less trivial. In this regime the Hamiltonian is approximately block diagonal both in absence and in presence of disorder, because the disorder potential we take is spin- and orbital-indpendent. Therefore we may focus on a $2\times 2$ Hamiltonian (describing a Weyl node of positive chirality), $$\label{eq:h_simple} h'({\bf k})=\hbar v {\bf k}\cdot{\boldsymbol \sigma}+V_0({\bf r}){\bf 1}_{2\times 2},$$ where ${\bf k}=k(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$. The result for $\delta\sigma$ obtained from such Hamiltonian needs to be multiplied by two at the end, as each block makes an equal contribution. The eigenstates for $h'({\bf k})$ are $|+,{\bf k}\rangle=(\cos(\theta/2),\exp(i\phi)\sin(\theta/2))^T$ and $|-,{\bf k}\rangle=(\sin(\theta/2),-\exp(i\phi)\cos(\theta/2))^T$. One significant simplification from Eq. (\[eq:h\_simple\]) is that there is only one band at the Fermi energy. This allows us to rewrite Eq. (\[eq:nasty\]) as $$\begin{aligned} \label{eq:less_ugly} \delta\sigma_2=& -4\frac{e^2\hbar^3}{2\pi} u_0\frac{\tau^2}{\tau_0^2}\left[\int\frac{dk k^2}{(2\pi)^2}\frac{k v^2}{E_k} (G^A)^2 G^R\right]^2\frac{1}{W}\int\frac{d^2 Q}{(2\pi)^2}\nonumber\\ &\times\int\frac{d\Omega_{{\bf k}}}{4\pi}\int\frac{d\Omega_{{\bf k}'}}{4\pi} \sin\theta\cos\phi\sin\theta'\cos\phi'\langle +,{\bf k}_F|+,{\bf k}_F'\rangle\langle +,-{\bf k}_F|+,-{\bf k}_F'\rangle C^{+ +}_{+ +}({\bf k}_F,{\bf k}_F',{\bf Q}),\end{aligned}$$ where the aforementioned extra factor of two has been accounted for. It is illustrative to compare Eq. (\[eq:less\_ugly\]) with its counterpart in $\delta\sigma_1$: $$\label{eq:pretty} \delta\sigma_1=-2\frac{e^2 \hbar^3}{2\pi}\frac{\tau^2}{\tau_0^2}\int\frac{dk k^2}{(2\pi)^2}\frac{k^2 v^4}{E_k^2} (G^R)^2 (G^A)^2 \frac{1}{W}\int\frac{d^2 Q}{(2\pi)^2} \sin^2\theta\cos^2\phi\, C^{+ +}_{+ +}({\bf k}_F,{\bf k}_F,{\bf Q}).$$ In Section II we detailed the steps to follow for the evaluation of Eq. (\[eq:pretty\]). Applying those same steps to Eq. (\[eq:less\_ugly\]) and using $$\int\frac{dk k^2}{2\pi^2} \frac{k^2}{E_k^2} (G^R)^2 (G^A)^2 \simeq \frac{4\pi\nu\tau_0^3}{\hbar^5 v^2}\,\,\,\mbox{ and }\,\,\,\left[\int\frac{dk k^2}{(2\pi)^2}\frac{k}{E_k} (G^A)^2 G^R\right]^2 u_0 \simeq -\frac{4\pi \nu \tau_0^3}{\hbar^5 v^2},$$ we arrive at $$\delta\sigma_2=-\frac{1}{3}\delta\sigma_1=-\frac{1}{3} G_q \ln\left(\frac{\tau_\phi}{\tau}\right).$$ Evaluation of matrix elements for $\hat{U}$ {#sec:u} =========================================== In this Appendix we calculate the coefficients entering in Eq. (\[eq:coeffs\]). These coefficients generally depend on the frequency $\Omega$ and wave vector ${\bf Q}$ of the external perturbation. Even though only $\Omega=0$ is needed for our evaluation of $\delta\sigma$, for completeness here we allow for $\Omega\neq 0$ as well. The calculation is facilitated by rewriting Eq. (\[eq:model\_b\]) as $$h({\bf k})=\sum_\mu\eta_\mu({\bf k}) \Lambda^\mu,$$ where $\mu\in\{1,2,3,4\}$, $\eta_i({\bf k})= \hbar v k_i$ and $\Lambda^i=\sigma^i\tau^x$ for $i\in\{1,2,3\}$, $\eta_4({\bf k})=M$ and $\Lambda^4={\bf 1}_{2}\,\tau^z$. Then, the finite-frequency retarded and advanced Green’s functions read $$\label{eq:gf} G_{m n}^{R(A)}({\bf k},\Omega)=\frac{\epsilon^{R(A)}\delta_{m n}^0+\sum_\mu\eta_\mu \Lambda_{m n}^\mu}{[\epsilon^{R(A)}]^2-E_k^2},$$ where $\epsilon^R\equiv\epsilon_F+i\gamma$ and $\epsilon^A\equiv\epsilon_F+\hbar\Omega-i\gamma$, with $\gamma\equiv\hbar/(2\tau_0)$ (c.f. Eq. (\[eq:lifetime\])). Substituting Eq. (\[eq:gf\]) in Eq. (\[eq:U\]), we get $$U^{m l}_{m' l'} = a\,\delta_{m l}\delta_{m' l'}+ \sum_\mu b_\mu\,\Lambda^\mu_{m' l'}\delta_{m l}+\sum_\mu c_\mu\,\Lambda^\mu_{m l}\delta_{m' l'}+\sum_{\mu\nu} d_{\mu\nu}\,\Lambda^\mu_{m l}\Lambda^\nu_{m' l'},$$ where $$\begin{aligned} \label{eq:ints} a&= u_0\int \frac{d^3 k}{(2\pi)^3}\frac{\epsilon^R(\epsilon^A+\hbar \Omega)}{[(\epsilon^R)^2-E_{-{\bf k}}^2][(\epsilon^A+\hbar \Omega)^2-E_{{\bf k}+{\bf Q}}^2]} \,\,\,\mbox{ ; }\,\,\,b_\mu=u_0\int \frac{d^3 k}{(2\pi)^3}\frac{\epsilon^R d_\mu({\bf k}+{\bf Q})}{[(\epsilon^R)^2-E_{-{\bf k}}^2][(\epsilon^A+\hbar \Omega)^2-E_{{\bf k}+{\bf Q}}^2]}\nonumber\\ c_\mu&=u_0\int \frac{d^3 k}{(2\pi)^3}\frac{(\epsilon^A+\hbar \Omega) d_\mu(-{\bf k})}{[(\epsilon^R)^2-E_{-{\bf k}}^2][(\epsilon^A+\hbar \Omega)^2-E_{{\bf k}+{\bf Q}}^2]}\,\,\,\mbox{ ; }\,\,\,d_{\mu\nu}=u_0\int \frac{d^3 k}{(2\pi)^3}\frac{d_\mu(-{\bf k}) d_\nu({\bf k}+{\bf Q})}{[(\epsilon^R)^2-E_{-{\bf k}}^2][(\epsilon^A+\hbar \Omega)^2-E_{{\bf k}+{\bf Q}}^2]},\nonumber\\\end{aligned}$$ and $\mu,\nu\in\{1,2,3,4\}$. In the diffusive transport regime, namely $(\epsilon_F-M)\gg\gamma\gg (\hbar v Q,\hbar \Omega)$, the integrals in Eq. (\[eq:ints\]) can be analytically performed and the outcome is $$\begin{aligned} \label{eq:coeffs2} a&\simeq a^{(0)}\left[1-\frac{1}{12}\left(1-\frac{M^2}{\epsilon_F^2}\right)\frac{\hbar^2 v^2 Q^2}{\gamma^2}-\frac{i\hbar \Omega}{2\gamma}\right]\nonumber\\ b_1 &=-c_1\simeq \frac{i}{6} a^{(0)}\left(1-\frac{M^2}{\epsilon_F^2}\right)\frac{\hbar v Q_x}{\gamma}\,\,\,\mbox{ ; }\,\,\,b_2=-c_2\simeq \frac{i}{6} a^{(0)}\left(1-\frac{M^2}{\epsilon_F^2}\right)\frac{\hbar v Q_y}{\gamma}\,\,\,\mbox{ ; }\,\,\,b_4=c_4=\frac{M}{\epsilon_F} a\nonumber\\ d_{1 1}&\simeq -\frac{1}{3}\left(1-\frac{M^2}{\epsilon_F^2}\right) a^{(0)}\left[1-\frac{1}{20}\left(1-\frac{M^2}{\epsilon_F^2}\right)\frac{(3 Q_x^2+Q_y^2) \hbar^2 v^2}{\gamma^2}-\frac{i\hbar \Omega}{2\gamma}\right]\nonumber\\ d_{2 2}&\simeq -\frac{1}{3}\left(1-\frac{M^2}{\epsilon_F^2}\right) a^{(0)}\left[1-\frac{1}{20}\left(1-\frac{M^2}{\epsilon_F^2}\right)\frac{(3 Q_y^2+Q_x^2) \hbar^2 v^2}{\gamma^2}-\frac{i\hbar \Omega}{2\gamma}\right]\nonumber\\ d_{3 3}&\simeq -\frac{1}{3}\left(1-\frac{M^2}{\epsilon_F^2}\right) a^{(0)}\left[1-\frac{1}{20}\left(1-\frac{M^2}{\epsilon_F^2}\right)\frac{\hbar^2 v^2 Q^2}{\gamma^2}-\frac{i\hbar \Omega}{2\gamma}\right]\nonumber\\ d_{4 4}&\simeq\frac{M^2}{\epsilon_F^2} a\nonumber\\ d_{1 2} &=d_{2 1}\simeq a^{(0)}\frac{1}{30}\left(1-\frac{M^2}{\epsilon_F^2}\right)^2\frac{\hbar^2 v^2 Q_x Q_y}{\gamma^2}\nonumber\\ d_{1 4}&=-d_{4 1}\simeq -i a^{(0)}\frac{M}{\epsilon_F}\left(1-\frac{M^2}{\epsilon_F^2}\right)\frac{\hbar v Q_x}{\gamma} \,\,\,\mbox{ ; }\,\,\,d_{2 4}=-d_{4 2}\simeq -i a^{(0)}\frac{M}{\epsilon_F}\left(1-\frac{M^2}{\epsilon_F^2}\right)\frac{\hbar v Q_y}{\gamma},\end{aligned}$$ where $a^{(0)}\equiv[2 (1+M^2/\epsilon_F^2)]^{-1}$, and the elements omitted above are zero. It is worth noting that Eq. (\[eq:coeffs2\]) can be used to investigate the dynamical spin-charge coupling in doped TIs. Since this task is not directly related to the theme of this paper, it will be pursued elsewhere. Classical conductivity of two coupled layers {#sec:cond} ============================================ In this Appendix we analyze the classical conductivity of two coupled layers. The current in layer $i$ is given by ${\bf j}_i=\sum_j \sigma_{i j} {\bf E}_j$. It is illustrative to write $\sigma_{i j}$ in terms of the diffusive density-density response, using the continuity equation $$\frac{\partial\rho_i}{\partial t}+\nabla\cdot{\bf j}+\lambda\sum_j (\rho_j-\rho_i)=0$$ along with the constitutive equation ${\bf j}_i=-D_i {\boldsymbol\nabla}\rho_i-e^2\nu_i D_i {\bf E}_i$. $\lambda$ is the interlayer tunneling rate. Thus it follows that $$\sigma_{i j}({\bf q},\omega)=-\frac{i\omega}{q^2} \chi_{i j}+\frac{\lambda}{q^2}\sum_k (\chi_{i j}-\chi_{k j}),$$ where $\chi_{i j} ({\bf q},\omega)=e^2 \nu_j D_j q^2 p_{i j}({\bf q},\omega)$ is the density-density response function and $$p_{i j}({\bf q},\omega) = \left\{\begin{array}{ccc} \tilde{p}_i^{(0)}/(1-\lambda^2 p_1^{(0)} p_2^{(0)}) & {\rm if } & i=j\\ \lambda \tilde{p}_1^{(0)} \tilde{p}_2^{(0)}/(1-\lambda^2 p_1^{(0)} p_2^{(0)}) & {\rm if } & i\neq j\end{array}\right.,$$ with $\tilde{p}_i^{(0)} \equiv(D_i q^2-i\omega+\lambda)^{-1}$. The dressed diffusion probability $p_{i i}$, derived here from the continuity equation, has identical form as Eq. (\[eq:cii\]), which was derived microscopically in Section IIIB. Here $\omega$ and ${\bf q}$ are the frequency and momentum associated with the applied electric field. A straightforward calculation shows that $\sigma_{1 2}=\sigma_{2 1}=0$ when ${\bf E}_i$ is spatially uniform (${\bf q}=0$). Equations for coupled Cooperons {#sec:coupled_cooper} =============================== In the first part of this Appendix we present an alternative derivation for the results of Section IIIB. In the second part of the Appendix we generalize the derivation to make it suitable for TI thin films with $\tau_{\phi 1}\ll\tau_v$, which contain two gapless singlet Cooperons in the bulk and one gapless singlet Cooperon on the surface. The outcome of such generalization is the third line of Eq. (\[eq:res\_tot\]). Two 2D layers without spin-orbit coupling ----------------------------------------- In this subsection we use “1” and “2” to label the two layers. The relevant Cooperon modes are then $C_{1 1}$, $C_{1 2}$, $C_{2 1}$ and $C_{2 2}$. Recognizing that Cooperons must obey a diffusion equation in absence of phase relaxation, we posit the following coupled equations: $$\begin{aligned} \label{eq:1} (D_1 Q^2 +\tau_{\phi 1}^{-1}) C_{1 1}+\lambda (C_{1 1}-C_{2 1}) &=\hbar/(2\pi\nu_1\tau_{d 1}^2)\nonumber\\ (D_2 Q^2+\tau_{\phi 2}^{-1}) C_{2 1}+\lambda (C_{2 1}-C_{1 1}) &= 0\nonumber\\ (D_2 Q^2+\tau_{\phi 2}^{-1}) C_{2 2}+\lambda(C_{2 2}-C_{1 2}) &=\hbar/(2\pi\nu_2\tau_{d 2}^2)\nonumber\\ (D_1 Q^2+\tau_{\phi 1}^{-1}) C_{1 2}+\lambda (C_{1 2}-C_{2 2})&=0,\end{aligned}$$ where $\lambda$ is the interlayer tunneling rate. Note that the source term appears only for the diagonal terms of the $2\times 2$ Cooperon matrix. The solution of Eq. (\[eq:1\]) reads $$\begin{aligned} C_{1 1}&=\frac{\hbar}{2\pi\nu_1\tau_1^2}\frac{D_2 Q^2+\tilde{\tau}_{\phi 2}^{-1}}{(D_1 Q^2+\tilde{\tau}_{\phi 1}^{-1})(D_2 Q^2+\tilde{\tau}_{\phi 2}^{-1})-\lambda^2}\nonumber\\ C_{2 2}&=\frac{\hbar}{2\pi\nu_2\tau_2^2}\frac{D_1 Q^2+\tilde{\tau}_{\phi 1}^{-1}}{(D_1 Q^2+\tilde{\tau}_{\phi 1}^{-1})(D_2 Q^2+\tilde{\tau}_{\phi 2}^{-1})-\lambda^2}\nonumber\\ C_{1 2}&=C_{2 1}=\frac{\lambda}{D_2 Q^2+\tilde{\tau}_{\phi 2}^{-1}} C_{1 1},\end{aligned}$$ where $\tilde{\tau}_{\phi i}^{-1}\equiv \tau_{\phi i}^{-1}+\lambda$. The expressions for $C_{1 1}$ and $C_{2 2}$ agree with Eq. (\[eq:cii\]). In addition, $C_{1 2}$ and $C_{2 1}$ agree with the expressions for $p_{1 2}$ and $p_{2 1}$ derived in Appendix \[sec:cond\] (where we discussed the classical diffusive conductivity). $C_{i i}$ of Eq. (\[eq:1\]) is equivalent to $C^{i i}_{i i}$ of Fig. \[fig:dsij\]. Likewise, $C_{1 2}$ and $C_{2 1}$ of Eq. (\[eq:1\]) correspond to $C^{1 1}_{2 2}$ and $C^{2 2}_{1 1}$ of Fig. \[fig:dsij\]. Although $C_{1 2}$ and $C_{2 1}$ are nonzero, they do not contribute to $\delta\sigma$ because the velocity operator is diagonal in the layer index. Therefore, we reproduce the expression of Section IIIB for $\delta\sigma$. TI film with two gapless bulk Cooperons and one gapless surface Cooperon ------------------------------------------------------------------------ In this subsection we use “1” and “3” to label the two bulk Cooperons, and “2” to label the surface Cooperon. The generalization of Eq. (\[eq:1\]) is $$\begin{aligned} \label{eq:2} (D_1 Q^2+\tau_{\phi 1}^{-1}) C_{1 1}+\lambda(C_{1 1}-C_{2 1})&=\hbar/(2\pi\nu_1\tau_{d 1}^2)\nonumber\\ (D_2 Q^2+\tau_{\phi 2}^{-1}) C_{2 1}+\lambda(2 C_{2 1}-C_{1 1}-C_{3 1})&=0\nonumber\\ (D_1 Q^2+\tau_{\phi 1}^{-1}) C_{3 1}+\lambda(C_{3 1}-C_{2 1})&=0,\end{aligned}$$ $$\begin{aligned} \label{eq:3} (D_1 Q^2+\tau_{\phi 1}^{-1}) C_{1 2}+\lambda (C_{1 2}-C_{2 2}) &= 0\nonumber\\ (D_2 Q^2+\tau_{\phi 2}^{-1}) C_{2 2}+\lambda( 2 C_{2 2}-C_{1 2}-C_{3 2})&=\hbar/(2\pi\nu_2\tau_{d 2}^2)\nonumber\\ (D_1 Q^2+\tau_{\phi 1}^{-1}) C_{3 2}+\lambda(C_{3 2}-C_{2 2})&=0\end{aligned}$$ and $$\begin{aligned} \label{eq:4} (D_1 Q^2+\tau_{\phi 1}^{-1}) C_{1 3}+\lambda(C_{1 3}-C_{2 3})&=0\nonumber\\ (D_2 Q^2+\tau_{\phi 2}^{-1}) C_{2 3}+\lambda(2 C_{2 3}-C_{1 3}-C_{3 3})&=0\nonumber\\ (D_1 Q^2+\tau_{\phi 1}^{-1}) C_{3 3}+\lambda(C_{3 3}-C_{2 3})&=\hbar/(2\pi\nu_1\tau_{d 1}^2),\end{aligned}$$ Once again in Eqs. (\[eq:2\])-(\[eq:4\]) the source term appears for the diagonal components of the $3\times 3$ Cooperon matrix. In addition, a factor of $2$ has been multiplied in front of some tunneling rates associated to surface Cooperons. The rationale behind this is that the Cooperon on the surface can decay into two bulk modes, i.e. the effective decay rate becomes $\tau_{\phi 2}^{-1}+2\lambda$. Aside from this, we have assumed a unique tunneling rate $\lambda$ between all pairs of Cooperons. The quantum correction to conductance can be written as $$\label{eq:ds_app} \delta G=2\frac{e^2}{\hbar^2}\nu_1 D_1 \tau_{d 1}^2\int_{\bf Q} (C_{1 1}+C_{3 3})+ 2\frac{e^2}{\hbar^2}\nu_2 D_2 \tau_{d 2}^2\int_{\bf Q} C_{2 2}.$$ Solving Eqs. (\[eq:2\])-(\[eq:4\]) requires some algebra. The results for the Cooperons of interest are $$\begin{aligned} \label{eq:cii_app} C_{1 1}&=C_{3 3}=\frac{\hbar}{2\pi\nu_1\tau_{d 1}^2}\frac{(D_1 Q^2+\tilde{\tau}_{\phi 1}^{-1})(D_2 Q^2+\tilde{\tau}_{\phi 2}^{-1}+\lambda)-\lambda^2}{(D_1 Q^2+\tilde{\tau}_{\phi 1}^{-1})\left[(D_1 Q^2+\tilde{\tau}_{\phi 1}^{-1})(D_2 Q^2+\tilde{\tau}_{\phi 2}^{-1}+\lambda)-2 \lambda^2\right]}\nonumber\\ C_{2 2} &=\frac{\hbar}{2\pi\nu_2\tau_{d 2}^2}\frac{D_1 Q^2+\tilde{\tau}_{\phi 1}^{-1}}{(D_1 Q^2+\tilde{\tau}_{\phi 1}^{-1})(D_2 Q^2+\tilde{\tau}_{\phi 2}^{-1}+\lambda)-2 \lambda^2},\end{aligned}$$ which are not illuminating expressions. It is better to rewrite them as $$\begin{aligned} \label{eq:ciib_app} C_{1 1} &=C_{3 3}=\frac{\hbar}{2\pi\nu_1\tau_{d 1}^2}\frac{1}{D_1}\left[\frac{X}{Q^2+q_x^2}+\frac{Y}{Q^2+q_y^2}+\frac{Z}{Q^2+q_z^2}\right]\nonumber\\ C_{2 2} &=\frac{\hbar}{2\pi\nu_2\tau_{d 2}^2} \frac{1}{D_2}\left[\frac{A}{Q^2+q_a^2}+\frac{B}{Q^2+q_b}\right],\end{aligned}$$ so that Eq. (\[eq:ds\_app\]) transforms into $$\label{eq:ds2_app} \delta G=\frac{e^2}{\pi\hbar}\int_{\bf Q} \left[2\frac{X}{Q^2+q_x^2}+2\frac{Y}{Q^2+q_y^2}+2\frac{Z}{Q^2+q_z^2}+\frac{A}{Q^2+q_a^2}+\frac{B}{Q^2+q_b^2}\right].$$ Comparing Eqs. (\[eq:cii\_app\]) and  (\[eq:ciib\_app\]), we arrive at $$\begin{aligned} \label{eq:A_app} A &=\frac{\frac{1}{D_1\tilde{\tau}_{\phi 1}}-q_a^2}{q_b^2-q_a^2}\,\,\,\mbox{ ; }\,\,\,B = 1-A\nonumber\\ X &= \frac{(D_1 q_x^2-\tilde{\tau}_{\phi 1}^{-1})(D_2 q_x^2-\tilde{\tau}_{\phi 2}^{-1}-\lambda)-\lambda^2}{D_1 D_2 (q_x^2-q_y^2)(q_x^2-q_z^2)}\nonumber\\ Y &= \frac{ D_2 q_y^2 \tilde{\tau}_{\phi 1}^{-1}-\tilde{\tau}_{\phi 1}^{-1}(\tilde{\tau}_{\phi 2}^{-1}+\lambda)+D_1 q_y^2 (-D_2 q_y^2+\tilde{\tau}_{\phi 2}^{-1}+\lambda)+\lambda^2}{D_1 D_2 (q_x^2-q_y^2)(q_y^2-q_z^2)}\nonumber\\ Z &= \frac{ D_2 q_z^2 \tilde{\tau}_{\phi 1}^{-1}-\tilde{\tau}_{\phi 1}^{-1}(\tilde{\tau}_{\phi 2}^{-1}+\lambda)+D_1 q_z^2 (-D_2 q_z^2+\tilde{\tau}_{\phi 2}^{-1}+\lambda)+\lambda^2}{D_1 D_2 (q_x^2-q_z^2)(q_z^2-q_y^2)}\end{aligned}$$ and $$\begin{aligned} \label{eq:qa_app} 2 q_{a(b)}^2 &=\frac{1}{D_1\tilde{\tau}_{\phi 1}}+\frac{1}{D_2\tilde{\tau}_{\phi 2}}+\frac{\lambda}{D_2}\pm\sqrt{\left(\frac{1}{D_1\tilde{\tau}_{\phi 1}}-\frac{1}{D_2 \tilde{\tau}_{\phi 2}}-\frac{\lambda}{D_2}\right)^2+\frac{8\lambda^2}{D_1 D_2}}\nonumber\\ q_{x (y)}^2 &=q_{a (b)}^2\,\,\,\mbox{ ; }\,\,\,q_z^2 = 1/(D_1\tilde{\tau}_{\phi 1}).\end{aligned}$$ Note that $q_{a (b)}=q_{x (y)}$, which will be important below. Also note that the expressions for $A$, $B$ and $q_{a(b)}$ are identical to the ones in Section IIIB, except for the following difference: the effective inelastic scattering rate for layer $2$ is now $\tau_{\phi 2}^{-1}+2\lambda$ instead of $\tau_{\phi 2}^{-1}+\lambda$, for the reason explained above. Although Eqs.  (\[eq:A\_app\]) and  (\[eq:qa\_app\]) look cumbersome, after substituting Eq. (\[eq:qa\_app\]) back in Eq. (\[eq:A\_app\]) we find some remarkable simplifications. In particular $$Z=1/2\,\,\,\mbox{ , }\,\,\,2 X+ A=1 \,\,\,\mbox{ and }\,\,\, 2 Y+ B =1.$$ Replacing these in Eq. (\[eq:ds2\_app\]) immediately leads to $$\delta G=\frac{e^2}{\pi\hbar}\int_{\bf Q} \left[\frac{1}{Q^2+q_a^2}+\frac{1}{Q^2+q_b^2}+\frac{1}{Q^2+q_z^2}\right].$$ In consequence, we recover the third line of Eq. (\[eq:res\_tot\]) for the low-field magnetoconductance: $$\label{eq:dg} \frac{\Delta G}{G_q}=\frac{1}{2}\left[f\left(\frac{H_a}{H}\right)+f\left(\frac{H_b}{H}\right)+f\left(\frac{H_z}{H}\right)\right],$$ where $H_a=\hbar q_a^2/(4 e)$, etc. As a reality check, let us take some simple limits. First, consider the case of no bulk-surface coupling, $\lambda\to 0$. In this case $H_a=H_z=\hbar/(4 e D_1 \tau_{\phi 1})$ and $H_b=\hbar/(4 e D_2 \tau_{\phi 2})$, which produces $$\frac{\Delta G}{G_q}=\frac{1}{2}\left[2 f\left(\frac{H_a}{H}\right)+f\left(\frac{H_b}{H}\right)\right].$$ This is indeed the result one would have expected when bulk and surface are decoupled. Second, suppose both $\tau_{\phi 1}$ and $\tau_{\phi 2}$ are infinitey large, for arbitrary tunneling rate. Then it follows that $H_b=0$, $$H_a=\frac{\hbar}{4 e}\lambda\left(\frac{1}{D_1}+\frac{2}{D_2}\right)\,\,\,\mbox{ and }\,\,\, H_z=\frac{\hbar}{4 e}\frac{\lambda}{D_1}$$ Then, $$\frac{\Delta G}{G_q}=\frac{1}{2}\left[f\left(\frac{H_a}{H}\right)+f\left(\frac{H_z}{H}\right)\right].$$ The fact that $H_b=0$ means that we recover the conventional WAL case (as we should when the phase relaxation times are infinitely long). Finally, consider the case of very strong tunneling between bulk and surface states. In this case $H_a$ and $H_z$ become very large ($\propto\lambda$), whereas $H_b$ becomes independent of $\lambda$. Consequently $$\frac{\Delta G}{G_q}=\frac{1}{2} f\left(\frac{H_b}{H}\right),$$ as if we had a single channel contributing to WAL. This seems to make sense too, because when tunneling is strong, $C_{i i}$ are strongly coupled to one another ($i=1,2,3$). Some special cases of Eq. (\[eq:res\_tot\]) {#sec:special} =========================================== In this Appendix we analyze some simple limiting cases of Eq. (\[eq:res\_tot\]), which considers a single TI surface coupled to bulk states. First, suppose that surface-bulk tunneling is strong, so that $\tau_{t i}\ll \tau_{\phi i}$ for $i=1,2$. In this case $(H_a, H_c, \tilde{H}_1)\gg (H_b,H_d)$ and thus Eq. (\[eq:res\_tot\]) turns into $$\begin{aligned} \label{eq:res_tot2} & \frac{\Delta G}{G_q}=\frac{1}{2}\left\{\begin{array}{ccc} f(H_b/H) &{\rm if } & \tilde{\tau}_H\ll\tau_s\\ f(H_b/H) &{\rm if } & \tilde{\tau}_H\gg(\tau_v,\tau_s)\\ f(H_d/H) &{\rm if } & \tilde{\tau}_H\ll\tau_v, \end{array}\right.\end{aligned}$$ where $H_b \simeq \hbar/(4 e)(1/\tau_{\phi 1}+1/\tau_{\phi 2})/(D_1+D_2)$ and $H_d\simeq \hbar/(4 e)(2/\tau_{\phi 1}+1/\tau_{\phi 2})/(2 D_1+D_2)$. For simplicity we have taken $\tau_{t 1}=\tau_{t 2}$, but this assumption can be easily relaxed. In sum, WL is [*not*]{} possible when the bulk-surface coupling is strong, and the film exhibits conventional WAL ($\alpha=1/2$) regardless of the bulk carrier concentration. Next, we consider a weak surface-bulk tunneling, so that $\tau_{t i}\gg \tau_{\phi i}$ for $i=1,2$. In this case the outcome depends on whether $D_1\tau_{\phi 1}>D_2\tau_{\phi 2}$ or $D_1\tau_{\phi 1}<D_2\tau_{\phi 2}$. Without loss of generality suppose that $D_1\tau_{\phi 1}>D_2\tau_{\phi 2}$. Then Eq. (\[eq:res\_tot\]) yields $$\begin{aligned} \label{eq:res_tot3} & \frac{\Delta G}{G_q}\simeq\frac{1}{2}\left\{\begin{array}{ccc} f(H_{\phi 2}/H)-2 f(H_{\phi 1}/H) &{\rm if } & \tilde{\tau}_H\ll\tau_s\\ f(H_{\phi 2}/H)+f(H_{\phi 1}/H) &{\rm if } & \tilde{\tau}_H\gg(\tau_v,\tau_s)\\ f(H_{\phi 2}/H)+2 f(H_{\phi 1}/H) &{\rm if } & \tilde{\tau}_H\ll\tau_v, \end{array}\right.\end{aligned}$$ where $H_{\phi i}= \hbar/(4 e D_i \tau_{\phi i})$ for $i=1,2$. When $H_{\phi 1}$ and $H_{\phi 2}$ are of the same order, the first line of Eq. (\[eq:res\_tot3\]) displays WL with $\alpha=-1/2$ and the third line exhibits WAL with $\alpha=3/2$. If instead $H_{\phi 1}\ll H_{\phi 2}$, $\Delta G$ is the same as if there were no surface states. This latter regime can be experimentally accessible by e.g. depositing magnetic impurities on the surface of the TI. Last, we consider the case $\tau_{t 1}\gg\tau_{\phi i}\gg\tau_{t 2}$ for $i=1,2$. This situation may be relevant for some thicker TI films where $\tau_{t 1}/\tau_{t 2}=W\nu_1/\nu_2\gg 1$ (for thicker films, surface states have more bulk states to decay onto). The resulting magnetoconductance is once again as though there were no surface states: $$\begin{aligned} \label{eq:res_tot5bis} & \frac{\Delta G}{G_q}=\left\{\begin{array}{ccc} -f(H_{\phi 1}/H) &{\rm if } & \tilde{\tau}_H\ll\tau_s\\ \frac{1}{2}f(H_{\phi 1}/H) &{\rm if } & \tilde{\tau}_H\gg(\tau_v,\tau_s)\\ f(H_{\phi 1}/H) &{\rm if } & \tilde{\tau}_H\ll\tau_v. \end{array}\right.\end{aligned}$$ [100]{} For reviews see e.g. J. Moore, Nature [**464**]{}, 194 (2010); M.Z. Hasan and C.L. Kane, Rev. Mod. Phys. [**82**]{}, 3045 (2010); X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. [**83**]{}, 1057 (2011). For a review see e.g. M.Z. Hasan, D. Hsieh, Y. Xia, L.A. Wray, S.-Y. Xu and C.L. Kane, arXiv:1105.0396 (2011). Z. Alpichshev, J.G. Analytis, J.-H. Chu, I.R. Fisher, Y.L Chen, Z.X. Shen, A. Fang and A. Kapitulnik, Phys. Rev. Lett. [**104**]{}, 016401 (2010); J. Seo, P. Roushan, H. Beidenkopf, Y.S. Hor, R.J. Cava and A. Yazdani, Nature [**466**]{}, 343 (2010). For a review see e.g. D. Culcer, Physica E [**44**]{}, 860 (2012) See e.g. D. Kim, S. Cho, N.P. Butch, P. Syers, K. Kirshenbaum, S. Adam, J. Paglione and M.S. Fuhrer, Nature Physics (2012), doi: 10.1038/nphysS2286; G. Zhang, H. Qin, J. Chen, X. He, L. Lu, Y. Li and K. Wu, Adv. Func. Mater. [**21**]{}, 2351 (2011). J. Chen, H.J. Qin, F. Yang, J. Liu, T. Guan, F.M. Qu, G.H. Zhang, J.R. Shi, X.C. Xie, C.L. Yang, K.H. Wu, Y.Q. Li and L. Lu, Phys. Rev. Lett. [**105**]{}, 176602 (2010). J.G. Checkelsky, Y.S. Hor, R.J. Cava and N.P. Ong, Phys. Rev. Lett, [**106**]{}, 196801 (2011). J. Wang, A.M. Silva, C.-Z. Chang, K. He, J.K. Jain, N. Samarth, X.-C. Ma, Q.-K. Xue and M.H.W. Chan, Phys. Rev. B [**83**]{}, 245438 (2011). H.-T. He, G. Wang, T. Zhang, I.-K. Sou, G.K.L. Wong, J.-N. Wang, H.-Z. Lu, S.-Q. Shen and F.-C. Zhang, Phys. Rev. Lett. [**106**]{}, 166805 (2011). J. Chen, X. Y. He, K.H. Wu, Z.Q. Ji, L. Lu, J.R. Shi, J.H. Smet and Y.Q. Li, Phys. Rev. B [**83**]{}, 241304 (2011). H. Steinberg, J.B. Laloö, V. Fatemi, J.S. Moodera and P. Jarillo-Herrero, Phys. Rev. B [**84**]{}, 233101 (2011). S.S. Hong, J.J. Cha, D. Kong and Y. Cui, Nature Communications (2012), doi:10.1038/ncomms1771. For reviews see e.g. E. Akkermans and G. Montambaux, [*Mesoscopic Physics of Electrons and Photons*]{} (Cambridge University Press, Cambridge, 2007); B.L. Altshuler, A.G. Aronov, D.E. Khmelnitskii and A.I. Larkin, [*Quantum Theory of Solids*]{} (MIR Publishers, Moscow, 1982). After submitting our work we have been informed of a new experiment that reports on the first observation of WL (negative magnetoresistance) in TI thin films; see L. Zhang, R. Hammond, M. Dolev, M. Liu, A. Palevski and A. Kapitulnik, arXiv:1205.5832 (2012). S. Hikami, A.I. Larkin and Y. Nagaoka, Progr. Theor. Phys. [**63**]{}, 707 (1980). H.-Z. Lu, J. Shi and S.-Q. Shen, Phys. Rev. Lett. [**107**]{}, 076801 (2011). G. Tkachov and E.M. Hankiewicz, Phys. Rev. B [**84**]{}, 035444 (2011). H.-Z. Lu and S.-Q. Shen, Phys. Rev. B [**84**]{}, 125138 (2011). S. Murakami, New J. Phys. [**9**]{}, 356 (2007); A.A. Burkov and L. Balents, Phys. Rev. Lett. [**107**]{}, 127205 (2011); X. Wan, A.M. Turner, A. Vishwanath and S.Y. Savrasov, Phys. Rev. B [**83**]{}, 205101 (2011); P. Hosur, S.A. Parameswaran and A. Vishwanath, Phys. Rev. Lett. [**108**]{}, 046602 (2012). E. McCann, K. Kechedzhi, V.I. Falko, H. Suzuura, T. Ando and B.L. Altshuler, Phys. Rev. Lett. [**97**]{}, 146805 (2006). H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang and S.-C. Zhang, Nature Physics [**5**]{}, 438 (2009); H.-J. Zhang, C.-X Liu, X.-L. Qi, X.-Y. Deng, X. Dai, S.-C. Zhang and Z. Fang, Phys. Rev. B [**80**]{}, 085307 (2009). I. Garate, J. Sinova, T. Jungwirth and A.H. MacDonald, Phys. Rev. B [**79**]{}, 155207 (2009). S.-Y. Xu, Y. Xia, L.A. Wray, S. Jia, F. Meier, J.H. Dil, J. Osterwalder, B. Slomski, A. Bansil, H. Lin, R.J. Cava and M.Z. Hasan, Science [**332**]{}, 560 (2011). H. Fukuyama, Supplement of the Prog. of Theor. Phys. [**69**]{}, 220 (1980); H. Fukuyama, J. Phys. Soc. Japan [**49**]{}, 649 (1980). G. Bergmann, Phys. Rev. B [**39**]{}, 11280 (1989). See e.g. S.M. Sze, [*Semiconductor Devices*]{}, 2nd ed. (John Wiley & Sons, New York, 2002). D.T. Son and B.Z. Spivak, arXiv:1206.1627 (2012).

--- abstract: 'We investigate the notion of Connes-amenability, introduced by Runde in [@Runde1], for bidual algebras and weighted semigroup algebras. We provide some simplifications to the notion of a $\sigma WC$-virtual diagonal, as introduced in [@Runde2], especially in the case of the bidual of an Arens regular Banach algebra. We apply these results to discrete, weighted, weakly cancellative semigroup algebras, showing that these behave in the same way as C$^*$-algebras with regards Connes-amenability of the bidual algebra. We also show that for each one of these cancellative semigroup algebras $l^1(S,\omega)$, we have that $l^1(S,\omega)$ is Connes-amenable (with respect to the canonical predual $c_0(S)$) if and only if $l^1(S,\omega)$ is amenable, which is in turn equivalent to $S$ being an amenable group. This latter point was first shown by Gr[ö]{}nbæk in [@Gron1], but we provide a unified proof. Finally, we consider the homological notion of injectivity, and show that here, weighted semigroup algebras do not behave like C$^*$-algebras.' author: - Matthew Daws title: '<span style="font-variant:small-caps;">Connes-amenability of bidual and weighted semigroup algebras</span>' --- *2000 Mathematics Subject Classification:* 22D15, 43A20, 46H25, 46H99 (primary), 46E15, 46M20, 47B47. Introduction ============ We first fix some notation, following [@Dales]. For a Banach space $E$, we let $E'$ be its dual space, and for $\mu\in E'$ and $x\in E$, we write ${{\langle {\mu} , {x} \rangle}} = \mu(x)$ for notational convenience. We then have the canonical map $\kappa_E:E\rightarrow E''$ defined by ${{\langle {\kappa_E(x)} , {\mu} \rangle}} = {{\langle {\mu} , {x} \rangle}}$ for $\mu\in E',x\in E$. For Banach spaces $E$ and $F$, we write ${\mathcal{B}}(E,F)$ for the Banach space of bounded linear maps between $E$ and $F$. We write ${\mathcal{B}}(E,E) = {\mathcal{B}}(E)$. For $T\in{\mathcal{B}}(E,F)$, the *adjoint* of $T$ is $T'\in{\mathcal{B}}(F',E')$, defined by ${{\langle {T'(\mu)} , {x} \rangle}} = {{\langle {\mu} , {T(x)} \rangle}}$, for $\mu\in F'$ and $x\in E$. Let ${\mathcal{A}}$ be a Banach algebra. A *Banach left ${\mathcal{A}}$-module* is a Banach space $E$ together with a bilinear map ${\mathcal{A}}\times E \rightarrow E; (a,x) \mapsto a\cdot x$, such that $\|a\cdot x\|\leq \|a\|\|x\|$ and $a\cdot(b\cdot x)= ab\cdot x$ for $a,b\in{\mathcal{A}}$ and $x\in E$. Similarly, we have the notion of a *Banach right ${\mathcal{A}}$-module* and a *Banach ${\mathcal{A}}$-bimodule*. If $E$ is a Banach ${\mathcal{A}}$-bimodule (resp. left or right module) then ${\mathcal{A}}'$ is a Banach ${\mathcal{A}}$-bimodule (resp. right or left module) with module action given by $${{\langle {a\cdot\mu} , {x} \rangle}} = {{\langle {\mu} , {x\cdot a} \rangle}} \qquad {{\langle {\mu\cdot a} , {x} \rangle}} = {{\langle {\mu} , {a\cdot x} \rangle}} \qquad (a\in{\mathcal{A}}, x\in E).$$ Notice that as ${\mathcal{A}}$ is certainly a bimodule over itself (with module action induced by the algebra product) we also have that ${\mathcal{A}}'$, ${\mathcal{A}}''$ etc. are Banach ${\mathcal{A}}$-bimodules. Given a Banach ${\mathcal{A}}$-bimodule $E$, a subspace $F$ of $E$ is a *submodule* if $a\cdot x, x\cdot a\in F$ for each $a\in{\mathcal{A}}$ and $x\in F$. For Banach ${\mathcal{A}}$-bimodules $E$ and $F$, $T\in{\mathcal{B}}(E,F)$ is an *${\mathcal{A}}$-bimodule homomorphism* when $$a \cdot T(x) = T(a\cdot x) \qquad T(x) \cdot a = T(x\cdot a) \qquad (a\in{\mathcal{A}}, x\in E).$$ A linear map $d:{\mathcal{A}} \rightarrow E$ between a Banach algebra ${\mathcal{A}}$ and a Banach ${\mathcal{A}}$-bimodule $E$ is a *derivation* if $d(ab) = a \cdot d(b) + d(a)\cdot b$ for $a,b\in{\mathcal{A}}$. For $x\in E$, we define $\delta_x : {\mathcal{A}}\rightarrow E$ by $\delta_x(a) = a\cdot x-x\cdot a$. Then $\delta_x$ is a derivation, called an *inner derivation*. A Banach algebra ${\mathcal{A}}$ is said to be *super-amenable* or *contractable* if every bounded derivation $d:{\mathcal{A}}\rightarrow E$, for every Banach ${\mathcal{A}}$-bimodule $E$, is inner. For example, a C$^*$-algebra ${\mathcal{A}}$ is super-amenable if and only if ${\mathcal{A}}$ is finite-dimensional. It is conjectured that there are no infinite-dimensional, super-amenable Banach algebras. If we restrict to derivations to $E'$ for Banach ${\mathcal{A}}$-bimodules $E$ then we arrive at the notion of *amenability*. For example, a C$^*$-algebra ${\mathcal{A}}$ is amenable if and only if ${\mathcal{A}}$ is nuclear; a group algebra $L^1(G)$ is amenable if and only if the locally compact group $G$ is amenable (which is the motivating example). See [@RundeBook] for further discussions of amenability and related notions. Let $E$ be a Banach space and $F$ a closed subspace of $E$. Then we naturally, isometrically, identify $F'$ with $E' / F^\circ$, where $$F^\circ = \{ \mu\in E' : {{\langle {\mu} , {x} \rangle}}=0 \ (x\in F) \}.$$ \[dual\_means\] Let $E$ be a Banach space and $E_*$ be a closed subspace of $E'$. Let $\pi_{E_*}: E'' \rightarrow E''/E_*^\circ$ be the quotient map, and suppose that $\pi_{E_*} \circ \kappa_E$ is an isomorphism from $E$ to $E_*'$. Then we say that $E$ is a *dual Banach space* with *predual $E_*$*. When ${\mathcal{A}}$ is a dual Banach space with predual ${\mathcal{A}}_*$ which is also a submodule of ${\mathcal{A}}'$ we say that ${\mathcal{A}}$ is a *dual Banach algebra*. [$\square$]{} For a dual Banach algebra ${\mathcal{A}}$ with predual ${\mathcal{A}}_*$, we henceforth identify ${\mathcal{A}}$ with ${\mathcal{A}}_*'$. Thus we get a weak$^*$-topology on ${\mathcal{A}}$, which we denote by $\sigma({\mathcal{A}},{\mathcal{A}}_*)$. It is a simple exercise to show that ${\mathcal{A}}$ is a dual Banach algebra if and only if ${\mathcal{A}}$ is a dual Banach space such that the algebra product is separately $\sigma({\mathcal{A}},{\mathcal{A}}_*)$-continuous (see [@Runde1]). The following lemma is standard. \[weak\_star\_cts\] Let $E$ and $F$ be dual Banach spaces with preduals $E_*$ and $F_*$ respectively, and let $T\in{\mathcal{B}}(E,F)$. Then the following are equivalent: 1. $T$ is $\sigma(E,E_*) - \sigma(F,F_*)$ continuous; 2. $T'(\kappa_{F_*}(F_*)) \subseteq \kappa_{E_*}(E_*)$; 3. there exists $S\in{\mathcal{B}}(F_*,E_*)$ such that $S'=T$. [$\square$]{} As noticed by Runde (see [@Runde1]), there are very few Banach algebras which are both dual and amenable. For von Neumann algebras, which are the motivating example of dual Banach algebras, there is a weaker notion of amenablity, called Connes-amenability, which has a natural generalisation to the case of dual Banach algebras. Let ${\mathcal{A}}$ be a dual Banach algebra with predual ${\mathcal{A}}_*$. Let $E$ be a Banach ${\mathcal{A}}$-bimodule. Then $E'$ is a *w$^*$-Banach ${\mathcal{A}}$-bimodule* if, for each $\mu\in E'$, the maps $${\mathcal{A}}\rightarrow E',\quad a\mapsto \begin{cases} a\cdot \mu, \\ \mu\cdot a \end{cases}$$ are $\sigma({\mathcal{A}},{\mathcal{A}}_*) - \sigma(E',E)$ continuous. Then $({\mathcal{A}},{\mathcal{A}}_*)$ is Connes-amenable if, for each w$^*$-Banach ${\mathcal{A}}$-bimodule $E'$, each derivation $d:{\mathcal{A}}\rightarrow E'$, which is $\sigma({\mathcal{A}},{\mathcal{A}}_*) - \sigma(E',E)$ continuous, is inner. [$\square$]{} Given a Banach algebra ${\mathcal{A}}$, we define bilinear maps ${\mathcal{A}}''\times{\mathcal{A}}'\rightarrow{\mathcal{A}}'$ and ${\mathcal{A}}'\times{\mathcal{A}}''\rightarrow{\mathcal{A}}'$ by $${{\langle {\Phi\cdot\mu} , {a} \rangle}} = {{\langle {\Phi} , {\mu\cdot a} \rangle}} \quad {{\langle {\mu\cdot\Phi} , {a} \rangle}} = {{\langle {\Phi} , {a\cdot\mu} \rangle}} \qquad (\Phi\in{\mathcal{A}}'', \mu\in{\mathcal{A}}', a\in{\mathcal{A}}).$$ We then define two bilinear maps ${\Box},{\Diamond}:{\mathcal{A}}''\times{\mathcal{A}}'' \rightarrow{\mathcal{A}}''$ by $${{\langle {\Phi{\Box}\Psi} , {\mu} \rangle}} = {{\langle {\Phi} , {\Psi\cdot\mu} \rangle}} \quad {{\langle {\Phi{\Diamond}\Psi} , {\mu} \rangle}} = {{\langle {\Psi} , {\mu\cdot\Phi} \rangle}} \qquad (\Phi,\Psi\in{\mathcal{A}}'', \mu\in{\mathcal{A}}').$$ We can check that ${\Box}$ and ${\Diamond}$ are actually algebra products, called the *first* and *second Arens products* respectively. Then $\kappa_A:{\mathcal{A}}\rightarrow{\mathcal{A}}''$ is a homomorphism with respect to either Arens product. When ${\Box}= {\Diamond}$, we say that ${\mathcal{A}}$ is *Arens regular*. In particular, when ${\mathcal{A}}$ is Arens regular, we may check that ${\mathcal{A}}''$ is a dual Banach algebra with predual ${\mathcal{A}}'$. \[ca\_facts\] Let ${\mathcal{A}}$ be an Arens regular Banach algebra. When ${\mathcal{A}}$ is amenable, ${\mathcal{A}}''$ is Connes-amenable. If ${\kappa_{{\mathcal{A}}}({\mathcal{A}})}$ is an ideal in ${\mathcal{A}}''$ and ${\mathcal{A}}''$ is Connes-amenable, then ${\mathcal{A}}$ is amenable. Let ${\mathcal{A}}$ be a C$^*$-algebra. Then ${\mathcal{A}}$ is Arens regular, and ${\mathcal{A}}''$ is Connes-amenable if and only if ${\mathcal{A}}$ is amenable. The first statements are [@Runde1 Corollary 4.3] and [@Runde1 Theorem 4.4]. The statement about C$^*$-algebras is detailed in [@RundeBook Chapter 6]. Another class of Connes-amenable dual Banach algebras is given by Runde in [@Runde4], where it is shown that $M(G)$, the measure algebra of a locally compact group $G$, is amenable if and only if $G$ is amenable. The organisation of this paper is as follows. Firstly, we study intrinsic characterisations of amenability, recalling a result of Runde from [@Runde2]. We then simplify these conditions in the case of Arens regular Banach algebras. We recall the notion of an *injective* module, and quickly note how Connes-amenability can be phrased in this language. The final section of the paper then applies these ideas to weighted semigroup algebras. We finish with some open questions. Characterisations of amenability ================================ Let $E$ and $F$ be Banach spaces, and form the algebraic tensor product $E\otimes F$. We can norm $E\otimes F$ with the *projective tensor norm*, defined as $$\|u\|_\pi = \inf\Big\{ \sum_{k=1}^n \|x_k\|\|y_k\| : u = \sum_{k=1}^n x_k \otimes y_k \Big\} \qquad (u\in E\otimes F).$$ Then the completion of $(E\otimes F, \|\cdot\|_\pi)$ is $E {{\widehat{\otimes}}}F$, the *projective tensor product* of $E$ and $F$. Let ${\mathcal{A}}$ be a Banach algebra. Then ${\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}}$ is a Banach ${\mathcal{A}}$-bimodule for the module actions given by $$a\cdot (b\otimes c) = ab \otimes c, \quad (b\otimes c) \cdot a = b\otimes ca \qquad (a\in{\mathcal{A}}, b\otimes c\in{\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}}).$$ Define $\Delta_{{\mathcal{A}}} : {\mathcal{A}} {{\widehat{\otimes}}}{\mathcal{A}} \rightarrow {\mathcal{A}}$ by $\Delta_{{\mathcal{A}}}(a\otimes b)=ab$. Then $\Delta_{{\mathcal{A}}}$ is an ${\mathcal{A}}$-bimodule homomorphism. \[when\_amen\] Let ${\mathcal{A}}$ be a Banach algebra. Then the following are equivalent: 1. ${\mathcal{A}}$ is amenable; 2. ${\mathcal{A}}$ has a *virtual diagonal*, which is a functional $M\in ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})''$ such that $a\cdot M =M\cdot a$ and $\Delta_{{\mathcal{A}}}''(M) \cdot a = \kappa_{{\mathcal{A}}}(a)$ for each $a\in{\mathcal{A}}$. [$\square$]{} Runde introduced, in [@Runde2], the following notion in order to prove a version of the above theorem for Connes-amenability. Let ${\mathcal{A}}$ be a dual Banach algebra with predual ${\mathcal{A}}_*$, and let $E$ be a Banach ${\mathcal{A}}$-bimodule. Then $x\in \sigma WC(E)$ if and only if the maps ${\mathcal{A}}\rightarrow E$, $$a \mapsto \begin{cases} a\cdot x, \\ x\cdot a \end{cases}$$ are $\sigma({\mathcal{A}},{\mathcal{A}}_*) - \sigma(E,E')$ continuous. [$\square$]{} It is clear that $\sigma WC(E)$ is a closed submodule of $E$. The ${\mathcal{A}}$-bimodule homomorphism $\Delta_{{\mathcal{A}}}$ has adjoint $\Delta'_{{\mathcal{A}}}:{\mathcal{A}}' \rightarrow ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})'$. In [@Runde2 Corollary 4.6] it is shown that $\Delta'_{{\mathcal{A}}}({\mathcal{A}}_*) \subseteq \sigma WC( ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})' )$. Consequently, we can view $\Delta_{{\mathcal{A}}}'$ as a map ${\mathcal{A}}_* \rightarrow \sigma WC( ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})' )$, and hence view $\Delta''_{{\mathcal{A}}}$ as a map $\sigma WC( ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})' )' \rightarrow {\mathcal{A}}_*' = {\mathcal{A}}$, denoted by $\tilde \Delta_{{\mathcal{A}}}$. \[Runde\_Thm\] Let ${\mathcal{A}}$ be a dual Banach algebra with predual ${\mathcal{A}}_*$. Then the following are equivalent: 1. ${\mathcal{A}}$ is Connes-amenable; 2. ${\mathcal{A}}$ has a *$\sigma WC$-virtual diagonal*, which is $M\in \sigma WC( ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})' )'$ such that $a\cdot M = M\cdot a$ and $a \tilde \Delta_{{\mathcal{A}}}(M) = a$ for each $a\in{\mathcal{A}}$. This is [@Runde2 Theorem 4.8]. In particular, we see that a Connes-amenable Banach algebra is unital (which can of course be shown in an elementary fashion, as in [@Runde1 Proposition 4.1]). Connes-amenability for biduals of algebras {#con_amen_bidual} ========================================== Recall Gantmacher’s theorem, which states that a bounded linear map $T:E\rightarrow F$ between Banach spaces $E$ and $F$ is *weakly-compact* if and only if $T''(E'') \subseteq \kappa_F(F)$. We write ${\mathcal{W}}(E,F)$ for the collection of weakly-compact operators in ${\mathcal{B}}(E,F)$. \[wsw\_cty\] Let $E$ be a dual Banach space with predual $E_*$, let $F$ be a Banach space, and let $T\in{\mathcal{B}}(E,F')$. Then the following are equivalent, and in particular each imply that $T$ is weakly-compact: 1. $T$ is $\sigma(E,E_*) - \sigma(F',F'')$ continuous; 2. $T'(F'') \subseteq \kappa_{E_*}(E_*)$; 3. there exists $S\in{\mathcal{W}}(F,E_*)$ such that $S' = T$. That (1) and (2) are equivalent is standard (compare with Lemma \[weak\_star\_cts\]). Suppose that (2) holds, so that we may define $S\in{\mathcal{B}}(F,E_*)$ by $\kappa_{E_*}\circ S = T'\circ\kappa_F$. Then, for $x\in E$ and $y\in F$, we have $${{\langle {x} , {S(y)} \rangle}} = {{\langle {T'(\kappa_F(y))} , {x} \rangle}} = {{\langle {T(x)} , {y} \rangle}},$$ so that $S'=T$. Then $S''(F'') = T'(F'') \subseteq \kappa_{E_*}(E_*)$, so that $S$ is weakly-compact, by Gantmacher’s Theorem, so that (3) holds. Conversely, if (3) holds, as $S$ is weakly-compact, we have $\kappa_{E*}(E_*) \supseteq S''(F'') = T'(F'')$, so that (2) holds. It is standard that for Banach spaces $E$ and $F$, we have $(E{{\widehat{\otimes}}}F)' = {\mathcal{B}}(F,E')$ with duality defined by $${{\langle {T} , {x\otimes y} \rangle}} = {{\langle {T(y)} , {x} \rangle}} \qquad (T\in{\mathcal{B}}(F,E'), x\otimes y\in E{{\widehat{\otimes}}}F).$$ Then we see, for $a,b,c\in{\mathcal{A}}$ and $T\in({\mathcal{A}} {{\widehat{\otimes}}}{\mathcal{A}})' = {\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')$, that ${{\langle {a\cdot T} , {b\otimes c} \rangle}} = {{\langle {T(ca)} , {b} \rangle}}$ and that ${{\langle {T\cdot a} , {b\otimes c} \rangle}} = {{\langle {T(c)} , {ab} \rangle}} = {{\langle {T(c)\cdot a} , {b} \rangle}}$ so that $$(a\cdot T)(c) = T(ca), \quad (T\cdot a)(c) = T(c)\cdot a \qquad (a,c\in{\mathcal{A}}, T:{\mathcal{A}}\rightarrow{\mathcal{A}}'). \label{eq:one}$$ Notice that we could also have defined $(E{{\widehat{\otimes}}}F)'$ to be ${\mathcal{B}}(E,F')$. This would induce a different bimodule structure on ${\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')$, and we shall see in Section \[Inj\_predual\] that our chosen convention seems more natural for the task at hand. \[first\_wap\_prop\] Let ${\mathcal{A}}$ be a dual Banach algebra with predual ${\mathcal{A}}_*$. For $T\in{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}') = ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})'$, define maps $\phi_r, \phi_l : {\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}}\rightarrow{\mathcal{A}}'$ by $$\phi_r(a\otimes b) = T'\kappa_{{\mathcal{A}}}(a) \cdot b, \quad \phi_l(a\otimes b) = a \cdot T(b) \qquad (a\otimes b\in{\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}}).$$ Then $T\in\sigma WC({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}'))$ if and only if $\phi_r$ and $\phi_l$ are weakly-compact and have ranges contained in ${\kappa_{{\mathcal{A}}_*}({\mathcal{A}}_*)}$. For $T\in{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}') = ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})'$, define $R_T,L_T : {\mathcal{A}}\rightarrow ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})'$ by $R_T(a) = a\cdot T$ and $L_T = T\cdot a$, for $a\in{\mathcal{A}}$. By definition, $T\in\sigma WC({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}'))$ if and only if $R_T$ and $L_T$ are $\sigma({\mathcal{A}},{\mathcal{A}}_*)-\sigma({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}'),({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})'')$ continuous. By Lemma \[wsw\_cty\], this is if and only if there exist $\varphi_r, \varphi_l \in {\mathcal{W}}({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}},{\mathcal{A}}_*)$ such that $\varphi_r' = R_T$ and $\varphi_l' = L_T$. For $a\otimes b\in{\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}}$ and $c\in{\mathcal{A}}$, we see that $$\begin{aligned} {{\langle {c} , {\varphi_r(a\otimes b)} \rangle}} &= {{\langle {R_T(c)} , {a\otimes b} \rangle}} = {{\langle {c\cdot T} , {a\otimes b} \rangle}} = {{\langle {T(bc)} , {a} \rangle}} \\ &= {{\langle {T'\kappa_{{\mathcal{A}}}(a)} , {bc} \rangle}} = {{\langle {T'\kappa_{{\mathcal{A}}}(a)\cdot b} , {c} \rangle}} = {{\langle {\phi_r(a\otimes b)} , {c} \rangle}}, \\ {{\langle {c} , {\varphi_l(a\otimes b)} \rangle}} &= {{\langle {L_T(c)} , {a\otimes b} \rangle}} = {{\langle {T\cdot c} , {a\otimes b} \rangle}} = {{\langle {T(b)} , {ca} \rangle}} \\ &= {{\langle {a\cdot T(b)} , {c} \rangle}} = {{\langle {\phi_l(a\otimes b)} , {c} \rangle}}.\end{aligned}$$ Thus $\kappa_{{\mathcal{A}}_*}\circ\varphi_r = \phi_r$ and $\kappa_{{\mathcal{A}}_*}\circ\varphi_l = \phi_l$. Consequently, we see that $T\in\sigma WC({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}'))$ if and only if $\phi_r$ and $\phi_l$ are weakly-compact and take values in ${\kappa_{{\mathcal{A}}_*}({\mathcal{A}}_*)}$. The following definition is [@Runde2 Definition 4.1]. Let ${\mathcal{A}}$ be a Banach algebra and let $E$ be a Banach ${\mathcal{A}}$-bimodule. An element $x\in E$ is *weakly almost periodic* if the maps $${\mathcal{A}}\rightarrow E,\quad a \mapsto \begin{cases} a\cdot x, \\ x\cdot a \end{cases}$$ are weakly-compact. The collection of weakly almost periodic elements in $E$ is denoted by ${\operatorname{WAP}}(E)$. [$\square$]{} \[wap\_to\_maps\] Let ${\mathcal{A}}$ be a Banach algebra, and let $T\in{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}') = ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})'$. Let $\phi_r,\phi_l : {\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}} \rightarrow{\mathcal{A}}'$ be as above. Then $T\in{\operatorname{WAP}}( {\mathcal{B}}({\mathcal{A}},{\mathcal{A}}') )$ if and only if $\phi_r$ and $\phi_l$ are weakly-compact. Let $R_T,L_T:{\mathcal{A}}\rightarrow{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')$ be as in the above proof. By definition, $T\in{\operatorname{WAP}}( {\mathcal{B}}({\mathcal{A}},{\mathcal{A}}') )$ if and only if $L_T$ and $R_T$ are weakly-compact. We can verify that $$\phi_r'\circ\kappa_{{\mathcal{A}}} = R_T,\ \phi_l'\circ\kappa_{{\mathcal{A}}} = L_T,\ R_T'\circ\kappa_{{\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}}} = \phi_r,\ L_T'\circ\kappa_{{\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}}} = \phi_l,$$ which completes the proof. \[unital\_dual\_sigma\] Let ${\mathcal{A}}$ be a unital, dual Banach algebra with predual ${\mathcal{A}}_*$, and let $T\in{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}') = ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})'$. The following are equivalent, and, in particular, each imply that $T$ is weakly-compact: 1. $T\in\sigma WC({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}'))$; 2. $T({\mathcal{A}}) \subseteq \kappa_{{\mathcal{A}}_*}({\mathcal{A}}_*)$, $T'(\kappa_{{\mathcal{A}}}({\mathcal{A}})) \subseteq \kappa_{{\mathcal{A}}_*}({\mathcal{A}}_*)$, and $T\in\sigma WC({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}'))$; 3. $T({\mathcal{A}}) \subseteq \kappa_{{\mathcal{A}}_*}({\mathcal{A}}_*)$, $T'(\kappa_{{\mathcal{A}}}({\mathcal{A}})) \subseteq \kappa_{{\mathcal{A}}_*}({\mathcal{A}}_*)$, and $T \in {\operatorname{WAP}}( {\mathcal{B}}({\mathcal{A}},{\mathcal{A}}') )$. Let $e_{{\mathcal{A}}}$ be the unit of ${\mathcal{A}}$, so that for $a\in{\mathcal{A}}$, we have $T(a) = \phi_l(e_{{\mathcal{A}}} \otimes a)$ and $T'\kappa_{{\mathcal{A}}}(a) = \phi_r(a \otimes e_{{\mathcal{A}}})$, which shows that (1) implies (2); clearly (2) implies (1). As ${\mathcal{A}}_*$ is an ${\mathcal{A}}$-bimodule, (2) and (3) are equivalent by an application of Lemma \[wap\_to\_maps\] and Proposition \[first\_wap\_prop\]. \[When\_Con\_Amen\] Let ${\mathcal{A}}$ be a dual Banach algebra with predual ${\mathcal{A}}_*$. Then ${\mathcal{A}}$ is Connes-amenable if and only if ${\mathcal{A}}$ is unital and there exists $M\in({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})''$ such that: 1. ${{\langle {M} , {a\cdot T-T\cdot a} \rangle}}=0$ for $a\in{\mathcal{A}}$ and $T\in\sigma WC( {\mathcal{W}}({\mathcal{A}},{\mathcal{A}}') )$; 2. $\kappa_{{\mathcal{A}}_*}' \Delta''_{{\mathcal{A}}}(M) = e_{{\mathcal{A}}}$, where $e_{{\mathcal{A}}}$ is the unit of ${\mathcal{A}}$. As $\sigma WC( ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})' )'$ is a quotient of $({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})''$, this is just a re-statement of Theorem \[Runde\_Thm\]. When ${\mathcal{A}}$ is an Arens regular Banach algebra, ${\mathcal{A}}''$ is a dual Banach algebra with canonical predual ${\mathcal{A}}'$. In this case, we can make some significant simplifications in the characterisation of when ${\mathcal{A}}''$ is Connes-amenable. For a Banach algebra ${\mathcal{A}}$, we define the map $\kappa_{{\mathcal{A}}} \otimes \kappa_{{\mathcal{A}}}: {\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}} \rightarrow {\mathcal{A}}'' {{\widehat{\otimes}}}{\mathcal{A}}''$ by $$( \kappa_{{\mathcal{A}}} \otimes \kappa_{{\mathcal{A}}} ) (a\otimes b) = \kappa_{{\mathcal{A}}}(a) \otimes \kappa_{{\mathcal{A}}}(b) \qquad (a\otimes b\in {\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}}).$$ We turn ${\mathcal{A}}'' {{\widehat{\otimes}}}{\mathcal{A}}''$ into a Banach ${\mathcal{A}}$-bimodule in the canonical way. Then $\kappa_{{\mathcal{A}}}\otimes\kappa_{{\mathcal{A}}}$ is an ${\mathcal{A}}$-bimodule homomorphism. The following is a simple verification. \[can\_proj\] Let ${\mathcal{A}}$ be a Banach algebra. The map $$\iota_{{\mathcal{A}}}:{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}') \rightarrow {\mathcal{B}}({\mathcal{A}}'',{\mathcal{A}}'''); \ T\mapsto T'',$$ is an ${\mathcal{A}}$-bimodule homomorphism which is an isometry onto its range. Furthermore, we have that $(\kappa_{{\mathcal{A}}} \otimes \kappa_{{\mathcal{A}}})' \circ \iota_{{\mathcal{A}}} = I_{{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')}$. Define $\rho_{{\mathcal{A}}}: {\mathcal{A}}''{{\widehat{\otimes}}}{\mathcal{A}}''\rightarrow ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})''$ by $${{\langle {\rho_{{\mathcal{A}}}(\tau)} , {T} \rangle}} = {{\langle {T''} , {\tau} \rangle}} \qquad (\tau\in {\mathcal{A}}''{{\widehat{\otimes}}}{\mathcal{A}}'', T\in{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')=({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})').$$ Then $\rho_{{\mathcal{A}}}$ is a norm-decreasing ${\mathcal{A}}$-bimodule homomorphism which satisfies $\rho_{{\mathcal{A}}} \circ (\kappa_{{\mathcal{A}}}\otimes\kappa_{{\mathcal{A}}}) = \kappa_{{\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}}}$. [$\square$]{} For a Banach algebra ${\mathcal{A}}$, it is clear that ${\mathcal{W}}({\mathcal{A}},{\mathcal{A}}')$ is a sub-${\mathcal{A}}$-bimodule of ${\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')=({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})'$. \[arens\_wap\] Let ${\mathcal{A}}$ be an Arens regular Banach algebra such that ${\mathcal{A}}''$ is unital, and let $T\in{\mathcal{B}}({\mathcal{A}}'',{\mathcal{A}}''') = ({\mathcal{A}}''{{\widehat{\otimes}}}{\mathcal{A}}'')'$. Then the following are equivalent: 1. $T \in \sigma WC({\mathcal{B}}({\mathcal{A}}'',{\mathcal{A}}'''))$, where we treat ${\mathcal{B}}({\mathcal{A}}'',{\mathcal{A}}''')$ as an ${\mathcal{A}}''$-bimodule; 2. $T = S''$ for some $S\in {\operatorname{WAP}}( {\mathcal{W}}({\mathcal{A}},{\mathcal{A}}') )$, where now we treat ${\mathcal{W}}({\mathcal{A}},{\mathcal{A}}')$ as an ${\mathcal{A}}$-bimodule. We apply Corollary \[unital\_dual\_sigma\] to ${\mathcal{A}}''$, so that (1) is equivalent to $T$ being weakly-compact, $T({\mathcal{A}}'')\subseteq \kappa_{{\mathcal{A}}'}({\mathcal{A}}')$, $T'(\kappa_{{\mathcal{A}}''}({\mathcal{A}}'')) \subseteq \kappa_{{\mathcal{A}}'}({\mathcal{A}}')$, and $T\in{\operatorname{WAP}}( {\mathcal{B}}({\mathcal{A}}'',{\mathcal{A}}''') )$. Thus, if (1) holds, then there exists $T_0 \in {\mathcal{W}}({\mathcal{A}}'',{\mathcal{A}}')$ such that $T = \kappa_{{\mathcal{A}}'} \circ T_0$, and there exists $T_1 \in {\mathcal{W}}({\mathcal{A}}'',{\mathcal{A}}')$ such that $T'\circ\kappa_{{\mathcal{A}}''} = \kappa_{{\mathcal{A}}'} \circ T_1$. Let $S = T_0\circ\kappa_{{\mathcal{A}}} \in {\mathcal{W}}({\mathcal{A}},{\mathcal{A}}')$. Then, for $a\in{\mathcal{A}}$ and $\Psi\in{\mathcal{A}}''$, we have $$\begin{aligned} {{\langle {S'(\Psi)} , {a} \rangle}} &= {{\langle {\Psi} , {T_0(\kappa_{{\mathcal{A}}}(a))} \rangle}} = {{\langle {T(\kappa_{{\mathcal{A}}}(a))} , {\Psi} \rangle}} = {{\langle {T'(\kappa_{{\mathcal{A}}''}(\Psi))} , {\kappa_{{\mathcal{A}}}(a)} \rangle}} \\ &= {{\langle {\kappa_{{\mathcal{A}}}(a)} , {T_1(\Psi)} \rangle}} = {{\langle {T_1(\Psi)} , {a} \rangle}},\end{aligned}$$ so that $S' = T_1$. Thus, for $\Phi,\Psi\in{\mathcal{A}}''$, we have $$\begin{aligned} {{\langle {S''(\Phi)} , {\Psi} \rangle}} &= {{\langle {\Phi} , {T_1(\Psi)} \rangle}} = {{\langle {T'(\kappa_{{\mathcal{A}}''}(\Psi))} , {\Phi} \rangle}} = {{\langle {T(\Phi)} , {\Psi} \rangle}},\end{aligned}$$ so that $S''=T$. We know that the maps $L_T,R_T: {\mathcal{A}}''\rightarrow{\mathcal{B}}({\mathcal{A}}'',{\mathcal{A}}''')$, defined by $L_T(\Phi) = T\cdot\Phi$ and $R_T(\Phi) = \Phi\cdot T$ for $\Phi\in{\mathcal{A}}''$, are weakly-compact. Define $L_S,R_S: {\mathcal{A}}\rightarrow{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')$ is an analogous manner, using $S\in{\mathcal{W}}({\mathcal{A}},{\mathcal{A}}')$. For $a\in{\mathcal{A}}$, $S\cdot a \in{\mathcal{W}}({\mathcal{A}},{\mathcal{A}}')$, so for $\Psi\in{\mathcal{A}}''$ and $b\in{\mathcal{A}}$, $${{\langle {(S\cdot a)'(\Psi)} , {b} \rangle}} = {{\langle {\Psi} , {(S\cdot a)(b)} \rangle}} = {{\langle {\Psi} , {S(b)\cdot a} \rangle}} = {{\langle {a\cdot\Psi} , {S(b)} \rangle}} = {{\langle {S'(a\cdot\Psi)} , {b} \rangle}}.$$ Thus, for $a\in{\mathcal{A}}$ and $\Phi,\Psi\in{\mathcal{A}}''$, we have that $$\begin{aligned} {{\langle { \iota_{{\mathcal{A}}}(L_S(a))(\Phi) } , {\Psi} \rangle}} &= {{\langle { (S\cdot a)''(\Phi) } , {\Psi} \rangle}} = {{\langle { \Phi } , { S'(a\cdot\Psi) } \rangle}} = {{\langle { S''(\Phi) \cdot a } , {\Psi} \rangle}},\end{aligned}$$ so that $\iota_{{\mathcal{A}}}(L_S(a))(\Phi) = S''(\Phi)\cdot a$, and hence that $\iota_{{\mathcal{A}}}(L_S(a)) = S''\cdot a = T\cdot a = T\cdot\kappa_{{\mathcal{A}}}(a) = L_T(\kappa_{{\mathcal{A}}}(a))$. Thus we have that $L_S = (\kappa_{{\mathcal{A}}}\otimes\kappa_{{\mathcal{A}}})' \circ R_T \circ \kappa_{{\mathcal{A}}}$, so that $L_S$ is weakly-compact. A similar calculation shows that $R_S$ is also weakly-compact, so that $S \in{\operatorname{WAP}}({\mathcal{W}}({\mathcal{A}},{\mathcal{A}}'))$. This shows that (1) implies (2). Conversely, if (2) holds, then $L_S$ and $R_S$ are weakly-compact. As $S$ is weakly-compact, $T({\mathcal{A}}'') = S''({\mathcal{A}}'') \subseteq \kappa_{{\mathcal{A}}'}({\mathcal{A}}')$ and $T'(\kappa_{{\mathcal{A}}''}({\mathcal{A}}'')) = S'''(\kappa_{{\mathcal{A}}''}({\mathcal{A}}'')) = \kappa_{{\mathcal{A}}'}(S'({\mathcal{A}}'')) \subseteq \kappa_{{\mathcal{A}}'}({\mathcal{A}}')$, and $T$ is weakly-compact. Thus, to show (1), we are required to show that $L_T$ and $R_T$ are weakly-compact. For $a,b\in{\mathcal{A}}$ and $\Phi\in{\mathcal{A}}'$, we have $${{\langle {(a\cdot S)'(\Phi)} , {b} \rangle}} = {{\langle {\Phi} , {S(ba)} \rangle}} = {{\langle {a \cdot S'(\Phi)} , {b} \rangle}}.$$ Then, for $\Phi,\Psi\in{\mathcal{A}}''$ and $a\in{\mathcal{A}}$, we thus have $$\begin{aligned} {{\langle {R_S'(\rho_{{\mathcal{A}}}(\Phi\otimes\Psi))} , {a} \rangle}} &= {{\langle {(a\cdot S)''} , {\Phi\otimes\Psi} \rangle}} = {{\langle {(a\cdot S)''(\Psi)} , {\Phi} \rangle}} = {{\langle {\Psi} , {a\cdot S'(\Phi)} \rangle}} \\ &= {{\langle {\Psi\cdot a} , {S'(\Phi)} \rangle}} = {{\langle {\Psi{\Box}\kappa_{{\mathcal{A}}}(a)} , {S'(\Phi)} \rangle}} = {{\langle {S'(\Phi)\cdot\Psi} , {a} \rangle}}.\end{aligned}$$ Hence we see that $R_S'(\rho_{{\mathcal{A}}}(\Phi\otimes\Psi)) = S'(\Phi)\cdot\Psi$. Let $U=R_S'\circ\rho_{{\mathcal{A}}}: {\mathcal{A}}''{{\widehat{\otimes}}}{\mathcal{A}}'' \rightarrow {\mathcal{A}}'$, so that as $R_S$ is weakly-compact, so is $U$. Then, for $\Phi,\Psi, \Gamma\in{\mathcal{A}}''$, we have that $$\begin{aligned} {{\langle {U'(\Gamma)} , {\Phi\otimes\Psi} \rangle}} &= {{\langle {\Gamma} , {S'(\Phi)\cdot\Psi} \rangle}} = {{\langle {\Psi{\Diamond}\Gamma} , {S'(\Phi)} \rangle}} = {{\langle {S''(\Psi{\Box}\Gamma)} , {\Phi} \rangle}} = {{\langle {(\Gamma\cdot S'')(\Psi)} , {\Phi} \rangle}},\end{aligned}$$ so that $U'(\Gamma) = \Gamma\cdot T$, that is, $U' = R_T$, so that $R_T$ is weakly-compact. Similarly, we can show that $L_T$ is weakly-compact, completing the proof. \[Connes\_Amen\] Let ${\mathcal{A}}$ be an Arens regular Banach algebra. Then ${\mathcal{A}}''$ is Connes-amenable if and only if ${\mathcal{A}}''$ is unital and there exists $M \in ({\mathcal{A}} {{\widehat{\otimes}}}{\mathcal{A}})''$ such that: 1. $\Delta_{{\mathcal{A}}}''(M) = e_{{\mathcal{A}}''}$, the unit of ${\mathcal{A}}''$; 2. ${{\langle {M} , {a\cdot T-T\cdot a} \rangle}}=0$ for each $a\in{\mathcal{A}}$ and each $T\in {\operatorname{WAP}}( {\mathcal{W}}({\mathcal{A}},{\mathcal{A}}') )$. By Theorem \[When\_Con\_Amen\], we wish to show that the existence of such an $M$ is equivalent to the existence of $N\in({\mathcal{A}}''{{\widehat{\otimes}}}{\mathcal{A}}'')''$ such that: (N1) : $\kappa_{{\mathcal{A}}'}'\Delta_{{\mathcal{A}}''}''(N) = e_{{\mathcal{A}}''}$; (N2) : ${{\langle {N} , {\Phi\cdot S - S\cdot\Phi} \rangle}}=0$ for each $\Phi\in{\mathcal{A}}''$ and each $S\in\sigma WC( {\mathcal{B}}({\mathcal{A}}'',{\mathcal{A}}''') )$. We can verify that $\iota_{{\mathcal{A}}} \circ \Delta_{{\mathcal{A}}}' = \Delta'_{{\mathcal{A}}''} \circ \kappa_{{\mathcal{A}}'}$, so that (N1) is equivalent to $\Delta_{{\mathcal{A}}}'' \iota'_{{\mathcal{A}}}(N) = e_{{\mathcal{A}}''}$. For $S\in\sigma WC( {\mathcal{B}}({\mathcal{A}}'',{\mathcal{A}}''') )$, we know that $S=T''$ for some $T \in {\operatorname{WAP}}( {\mathcal{W}}({\mathcal{A}},{\mathcal{A}}') )$, by Theorem \[arens\_wap\]. That is, the maps $\phi_r$ and $\phi_l$, formed using $T$ as in Proposition \[first\_wap\_prop\], are weakly-compact. Then, for $\Phi\in{\mathcal{A}}''$, $\phi_r'(\Phi), \phi_l'(\Phi) \in {\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')$, and we can check that $$\phi_r'(\Phi)(a) = \kappa_{{\mathcal{A}}}'T''(a\cdot\Phi), \quad \phi_l'(\Phi)(a) = T(a)\cdot\Phi \qquad (a\in{\mathcal{A}}).$$ Then $\phi_r'(\Phi)', \phi_l'(\Phi)'\in{\mathcal{B}}({\mathcal{A}}'',{\mathcal{A}}')$ are the maps $$\phi_r'(\Phi)'(\Psi) = \Phi\cdot T'(\Psi), \quad \phi_l'(\Phi)'(\Psi) = T'(\Phi{\Box}\Psi) \qquad (\Psi\in{\mathcal{A}}''),$$ where we remember that $T''({\mathcal{A}}'') \subseteq \kappa_{{\mathcal{A}}'}({\mathcal{A}}')$. Consequently $\phi_r'(\Phi)'', \phi_l'(\Phi)''\in{\mathcal{B}}({\mathcal{A}}'',{\mathcal{A}}''')$ are given by $$\phi_r'(\Phi)''(\Psi) = T''(\Psi{\Box}\Phi), \quad\phi_l'(\Phi)''(\Psi) = T''(\Psi) \cdot \Phi \qquad (\Psi\in{\mathcal{A}}''),$$ where ${\mathcal{A}}'''$ is an ${\mathcal{A}}''$-bimodule, as ${\mathcal{A}}''$ is Arens regular. That is, $\phi_r'(\Phi)'' = \Phi\cdot S$ and $\phi_l'(\Phi)'' = S\cdot\Phi$. Hence (N2) is equivalent to $$0 = {{\langle {N} , {\phi_r'(\Phi)'' - \phi_l'(\Phi)''} \rangle}} = {{\langle {N} , {\iota_{{\mathcal{A}}}( \phi_r'(\Phi) - \phi_l'(\Phi) )} \rangle}} = {{\langle {\iota_{{\mathcal{A}}}'(N)} , {\phi_r'(\Phi) - \phi_l'(\Phi)} \rangle}},$$ for each $\Phi\in{\mathcal{A}}''$ and $S\in\sigma WC( {\mathcal{B}}({\mathcal{A}}'',{\mathcal{A}}''') )$. That is, (N2) is equivalent to $$\phi_r''\iota_{{\mathcal{A}}}'(N) - \phi_l''\iota_{{\mathcal{A}}}'(N) = 0 \qquad ( S\in\sigma WC( {\mathcal{B}}({\mathcal{A}}'',{\mathcal{A}}''') ) ).$$ As $\phi_r$ and $\phi_l$ are weakly-compact, $\phi_r''$ and $\phi_l''$ take values in ${\kappa_{{\mathcal{A}}'}({\mathcal{A}}')}$, and so (N2) is equivalent to $$0 = {{\langle {\phi_r''\iota_{{\mathcal{A}}}'(N) - \phi_l''\iota_{{\mathcal{A}}}'(N)} , {\kappa_{{\mathcal{A}}}(a)} \rangle}} = {{\langle {\iota_{{\mathcal{A}}}'(N)} , { \phi_r'(\kappa_{{\mathcal{A}}}(a)) - \phi_l'(\kappa_{{\mathcal{A}}}(a)) } \rangle}},$$ for each $a\in{\mathcal{A}}$ and each $S\in\sigma WC( {\mathcal{B}}({\mathcal{A}}'',{\mathcal{A}}''') )$. However, $\phi_r'(\kappa_{{\mathcal{A}}}(a)) - \phi_l'(\kappa_{{\mathcal{A}}}(a)) = a\cdot T - T\cdot a$, so that (N2) is equivalent to $$0 = {{\langle {\iota_{{\mathcal{A}}}'(N)} , {a\cdot T - T \cdot a} \rangle}} \qquad (a\in{\mathcal{A}}),$$ for each $T\in{\mathcal{W}}({\mathcal{A}},{\mathcal{A}}')$ such that $\phi_r$ and $\phi_l$ are weakly-compact. Thus we have established that (N1) holds for $N$ if and only if (1) holds for $M=\iota_{{\mathcal{A}}}'(N)$, and that (N2) holds for $N$ if and only if (2) holds for $M=\iota_{{\mathcal{A}}}'(N)$, completing the proof. We immediately see that ${\mathcal{A}}$ amenable implies that ${\mathcal{A}}''$ is Connes-amenable. Furthermore, if ${\mathcal{A}}$ is itself a dual Banach algebra, then Corollary \[unital\_dual\_sigma\] shows that if ${\mathcal{A}}''$ is Connes-amenable, then ${\mathcal{A}}$ is Connes-amenable: notice that if $e_{{\mathcal{A}}''}$ is the unit of ${\mathcal{A}}''$, then $${{\langle {\kappa_{{\mathcal{A}}_*}'(e_{{\mathcal{A}}''})a} , {\mu} \rangle}} = {{\langle {e_{{\mathcal{A}}''} \cdot a} , {\kappa_{{\mathcal{A}}_*}(\mu)} \rangle}} = {{\langle {\kappa_{{\mathcal{A}}}(a)} , {\kappa_{{\mathcal{A}}_*}(\mu)} \rangle}} = {{\langle {a} , {\mu} \rangle}} \qquad (a\in{\mathcal{A}}, \mu\in{\mathcal{A}}_*),$$ so that $\kappa_{{\mathcal{A}}_*}'(e_{{\mathcal{A}}''})$ is the unit of ${\mathcal{A}}$. Injectivity of the predual module {#Inj_predual} ================================= Let ${\mathcal{A}}$ be a Banach algebra, and let $E$ and $F$ be Banach left ${\mathcal{A}}$-modules. We write ${_{{\mathcal{A}}}}{\mathcal{B}}(E,F)$ for the closed subspace of ${\mathcal{B}}(E,F)$ consisting of left ${\mathcal{A}}$-module homomorphisms, and similarly write ${\mathcal{B}}_{{\mathcal{A}}}(E,F)$ and ${_{{\mathcal{A}}}}{\mathcal{B}}_{{\mathcal{A}}}(E,F)$ for right ${\mathcal{A}}$-module and ${\mathcal{A}}$-bimodule homomorphisms, respectively. We say that $T\in{_{{\mathcal{A}}}}{\mathcal{B}}(E,F)$ is *admissible* if both the kernel and image of $T$ are closed, complemented subspaces of, respectively, $E$ and $F$. If $T$ is injective, this is equivalent to the existence of $S\in{\mathcal{B}}(F,E)$ such that $ST=I_E$. Let ${\mathcal{A}}$ be a Banach algebra, and let $E$ be a Banach left ${\mathcal{A}}$-module. Then $E$ is *injective* if, whenever $F$ and $G$ are Banach left ${\mathcal{A}}$-modules, $\theta\in{_{{\mathcal{A}}}}{\mathcal{B}}(F,G)$ is injective and admissible, and $\sigma\in{_{{\mathcal{A}}}}{\mathcal{B}}(F,E)$, there exists $\rho\in{_{{\mathcal{A}}}}{\mathcal{B}}(G,E)$ with $\rho\circ\theta = \sigma$. We say that $E$ is *left-injective* when we wish to stress that we are treating $E$ as a left module. Similar definitions hold for right modules and bimodules (written *right-injective* and *bi-injective* where necessary). Let ${\mathcal{A}}$ be a Banach algebra, let $E$ be a Banach left ${\mathcal{A}}$-module, and turn ${\mathcal{B}}({\mathcal{A}},E)$ into a left ${\mathcal{A}}$-module by setting $$(a\cdot T)(b) = T(ba) \qquad (a,b\in{\mathcal{A}}, T\in{\mathcal{B}}({\mathcal{A}},E)).$$ Then there is a canonical left ${\mathcal{A}}$-module homomorphism $\iota:E\rightarrow {\mathcal{B}}({\mathcal{A}},E)$ given by $$\iota(x)(a) = a\cdot x\qquad (a\in{\mathcal{A}},x\in E).$$ Notice that if $E$ is a closed submodule of ${\mathcal{A}}'$, then ${\mathcal{B}}({\mathcal{A}},E)$ is a closed submodule of $(A{{\widehat{\otimes}}}A)' = {\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')$, and $\iota$ is the restriction of $\Delta'_{{\mathcal{A}}}:{\mathcal{A}}'\rightarrow {\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')$ to $E$. Similarly, we turn ${\mathcal{B}}({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}},E)$ into a Banach ${\mathcal{A}}$-bimodule by $$(a\cdot T)(b\otimes c) = T(ba\otimes c), \ (T\cdot a)(b\otimes c) = T(b\otimes ac) \quad ( a,b,c\in{\mathcal{A}}, T\in{\mathcal{B}}({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}},E) ).$$ We then define (with an abuse of notation) $\iota:E\rightarrow{\mathcal{B}}({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}},E)$ by $$\iota(x)(a\otimes b) = a\cdot x \cdot b\qquad (x\in E, a\otimes b\in{\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}}),$$ so that $\iota$ is an ${\mathcal{A}}$-bimodule homomorphism. We can also turn ${\mathcal{B}}({\mathcal{A}},E)$ into a right ${\mathcal{A}}$-module by reversing the above (in particular, we need to take the other possible choice in Section \[con\_amen\_bidual\] leading to different module actions as compared to those in (\[eq:one\]).) Let ${\mathcal{A}}$ be a Banach algebra, and let $E$ be a *faithful* Banach left ${\mathcal{A}}$-module (that is, for each non-zero $x\in E$ there exists $a\in{\mathcal{A}}$ with $a\cdot x\not=0$). Then $E$ is injective if and only if there exists $\phi\in{_{{\mathcal{A}}}}{\mathcal{B}}( {\mathcal{B}}({\mathcal{A}},E), E )$ such that $\phi \circ \iota = I_E$. Similarly, if $E$ is a left and right faithful Banach ${\mathcal{A}}$-bimodule (that is, for each non-zero $x\in E$ there exists $a,b\in{\mathcal{A}}$ with $a\cdot x\not=0$ and $x\cdot b\not=0$). Then $E$ is injective if and only if there exists $\phi\in {_{{\mathcal{A}}}}{\mathcal{B}}_{{\mathcal{A}}}( {\mathcal{B}}({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}},E), E)$ such that $\phi\circ\iota = I_E$. The first claim is [@DP Proposition 1.7], and the second claim is an obvious generalisation. Again, there exists a similar characterisation for right modules. Let ${\mathcal{A}}$ be a dual Banach algebra with predual ${\mathcal{A}}_*$. It is simple to show (see [@Runde2]) that if ${\mathcal{A}}_*$ is bi-injective, then ${\mathcal{A}}$ is Connes-amenable. Helemskii showed in [@Hel2] that for a von Neumann algebra ${\mathcal{A}}$, the converse is true. However, Runde (see [@Runde2]) and Tabaldyev (see [@Tab]) have shown that $M(G)$, the measure algebra of a locally compact group $G$, while being a dual Banach algebra with predual $C_0(G)$, has that $C_0(G)$ is a left-injective $M(G)$-module only when $G$ is finite. Of course, Runde (see [@Runde4]) has shown that $M(G)$ is Connes-amenable if and only if $G$ is amenable. Similarly, it is simple to show (using a virtual diagonal) that if ${\mathcal{A}}$ is a Banach algebra with a bounded approximate identity, then ${\mathcal{A}}$ is amenable if and only if ${\mathcal{A}}'$ is bi-injective. Let $E$ and $F$ be Banach left ${\mathcal{A}}$-modules, and let $\phi:E\rightarrow F$ be a left ${\mathcal{A}}$-module homomorphism which is bounded below. Then $\phi(E)$ is a closed submodule of $F$, so that $F / \phi(E)$ is a Banach left ${\mathcal{A}}$-module. Hence we have a *short exact sequence*: $$\spreaddiagramrows{4ex} \spreaddiagramcolumns{6ex} \xymatrix{ 0 \ar[r] & E \ar@<1ex>[r]^{\phi} & F \ar@{->>}[r] \ar@<1ex>@{-->}[l]^{P} & F/\phi(E) \ar[r] & 0 }.$$ If there exists a bounded linear map $P:F\rightarrow E$ such that $P\circ\phi = I_E$, then we say that the short exact sequence is *admissible*. If, further, we may choose $P$ to be a left ${\mathcal{A}}$-module homomorphism, then the short exact sequence is said to *split*. Similar definitions hold for right modules and bimodules. Let ${\mathcal{A}}$ be a Banach algebra, let $E$ be a Banach left ${\mathcal{A}}$-module, and consider the following admissible short exact sequence: $$\spreaddiagramrows{4ex} \spreaddiagramcolumns{6ex} \xymatrix{ 0 \ar[r] & E \ar@<1ex>[r]^{\iota} & {\mathcal{B}}({\mathcal{A}},E) \ar@{->>}[r] \ar@<1ex>@{-->}[l]^{P} & {\mathcal{B}}({\mathcal{A}},E) / \iota(E) \ar[r] & 0 }.$$ Then $E$ is injective if and only if this short exact sequence splits. See, for example, [@RundeBook Section 5.3]. Let ${\mathcal{A}}$ be a unital dual Banach algebra with predual ${\mathcal{A}}_*$, and consider the following admissible short exact sequence of ${\mathcal{A}}$-bimodules: $$\spreaddiagramcolumns{3ex} \xymatrix{ 0 \ar[r] & {\mathcal{A}}_* \ar@<1ex>[r]^(.35){\Delta_{{\mathcal{A}}}'} & \sigma WC(({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})') \ar@{->>}[r] \ar@<1ex>@{-->}[l]^(0.65){P} & \sigma WC(({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})')/\Delta_{{\mathcal{A}}}'({\mathcal{A}}_*) \ar[r] & 0 }. \label{con_amen_ses}$$ Then ${\mathcal{A}}$ is Connes-amenable if and only if this short exact sequence splits. Notice that $\Delta'_{{\mathcal{A}}}$ certainly maps ${\mathcal{A}}_*$ into $\sigma WC(({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})') = \sigma WC({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}'))$, and that Corollary \[unital\_dual\_sigma\] shows that we can define $P:\sigma WC({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}'))\rightarrow{\mathcal{A}}_*$ by $P(T) = T(e_{{\mathcal{A}}})$ for $T\in \sigma WC({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}'))$. Suppose that we can choose $P$ to be an ${\mathcal{A}}$-bimodule homomorphism. Then let $M = P'(e_{{\mathcal{A}}})$, so that for $a\in{\mathcal{A}}$ and $T\in\sigma WC({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}'))$, $${{\langle {a\cdot M-M\cdot a} , {T} \rangle}} = {{\langle {e_{{\mathcal{A}}}} , {P(T\cdot a-a\cdot T)} \rangle}} = {{\langle {a-a} , {P(T)} \rangle}} = 0,$$ so that $a\cdot M - M\cdot a$. Also $\Delta''_{{\mathcal{A}}}(M) = (P\circ\Delta'_{{\mathcal{A}}})'(e_{{\mathcal{A}}}) = e_{{\mathcal{A}}}$, so that $M$ is a $\sigma WC$-virtual diagonal, and hence ${\mathcal{A}}$ is Connes-amenable by Runde’s theorem. Conversely, let $M$ be a $\sigma WC$-virtual diagonal and define $P:\sigma WC({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}'))\rightarrow{\mathcal{A}}'$ by $${{\langle {P(T)} , {a} \rangle}} = {{\langle {M} , {a\cdot T} \rangle}} \qquad (a\in{\mathcal{A}}, T\in\sigma WC({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')).$$ Let $(a_\alpha)$ be a bounded net in ${\mathcal{A}}$ which tends to $a\in{\mathcal{A}}$ in the $\sigma({\mathcal{A}},{\mathcal{A}}_*)$-topology. By definition, $a_\alpha\cdot T\rightarrow a\cdot T$ weakly, for each $T\in\sigma WC({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}'))$, so that ${{\langle {P(T)} , {a_\alpha} \rangle}} \rightarrow {{\langle {P(T)} , {a} \rangle}}$. This implies that $P$ maps into ${\mathcal{A}}_*$, as required. Then, for $\mu\in{\mathcal{A}}_*$, $${{\langle {a} , {P\Delta_{{\mathcal{A}}}'(\mu)} \rangle}} = {{\langle {M} , {a\cdot\Delta_{{\mathcal{A}}}'(\mu)} \rangle}} = {{\langle {M} , {\Delta_{{\mathcal{A}}}'(a\cdot\mu)} \rangle}} = {{\langle {e_{{\mathcal{A}}}} , {a\cdot \mu} \rangle}} = {{\langle {a} , {\mu} \rangle}} \qquad (a\in{\mathcal{A}}),$$ so that $P\Delta_{{\mathcal{A}}}'=I_{{\mathcal{A}}_*}$. Finally, we note that $$\begin{aligned} {{\langle {P(a\cdot T\cdot b)} , {c} \rangle}} &= {{\langle {M} , {ca\cdot T\cdot b} \rangle}} = {{\langle {b\cdot M} , {ca\cdot T} \rangle}}= {{\langle {M\cdot b} , {ca\cdot T} \rangle}} \\ &= {{\langle {P(T)} , {bca} \rangle}} = {{\langle {a\cdot P(T) \cdot b} , {c} \rangle}} \qquad (a,b,c\in{\mathcal{A}}, T\in\sigma WC({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}'))),\end{aligned}$$ so that $P$ is an ${\mathcal{A}}$-bimodule homomorphism, as required. Let ${\mathcal{A}}$ be an Arens regular Banach algebra. By reversing the argument Theorem \[arens\_wap\], we can show that $\Delta'_{{\mathcal{A}}}:{\mathcal{A}}'\rightarrow{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')$ actually maps into ${\operatorname{WAP}}({\mathcal{W}}({\mathcal{A}},{\mathcal{A}}'))$. Furthermore, if ${\mathcal{A}}''$ is unital, then we may define $P:{\operatorname{WAP}}({\mathcal{W}}({\mathcal{A}},{\mathcal{A}}')) \rightarrow {\mathcal{A}}'$ by $${{\langle {P(T)} , {a} \rangle}} = {{\langle {e_{{\mathcal{A}}''}} , {P(a)} \rangle}} \qquad (a\in{\mathcal{A}}, T\in{\operatorname{WAP}}({\mathcal{W}}({\mathcal{A}},{\mathcal{A}}'))).$$ Then we have that $${{\langle {P\Delta'_{{\mathcal{A}}}(\mu)} , {a} \rangle}} = {{\langle {e_{{\mathcal{A}}''}} , {a\cdot\mu} \rangle}} = {{\langle {\mu} , {a} \rangle}} \qquad (a\in{\mathcal{A}}, \mu\in{\mathcal{A}}').$$ Let ${\mathcal{A}}$ be an Arens regular Banach algebra such that ${\mathcal{A}}''$ is unital, and consider the following admissible short exact sequence of ${\mathcal{A}}$-bimodules: $$\spreaddiagramcolumns{3ex} \xymatrix{ 0 \ar[r] & {\mathcal{A}}' \ar@<1ex>[r]^(0.3){\Delta_{{\mathcal{A}}}'} & {\operatorname{WAP}}({\mathcal{W}}({\mathcal{A}},{\mathcal{A}}')) \ar@{->>}[r] \ar@<1ex>[l]^(0.7){P} & {\operatorname{WAP}}({\mathcal{W}}({\mathcal{A}},{\mathcal{A}}')) / \Delta_{{\mathcal{A}}}'({\mathcal{A}}') \ar[r] & 0 }. \label{eq:two}$$ Then ${\mathcal{A}}''$ is Connes-amenable if and only if this short exact sequence splits. This follows in the same manner as the above proof, using Theorem \[Connes\_Amen\]. Beurling algebras ================= Let $S$ be a discrete semigroup (we can extend the following definitions to locally compact semigroups, but for the questions we are interested in, the results for non-discrete groups are trivial). A *weight* on $S$ is a function $\omega: S \rightarrow \mathbb R^{>0}$ such that $$\omega(st)\leq \omega(s)\omega(t) \qquad (s,t\in S).$$ Furthermore, if $S$ is unital with unit $u_S$, then we also insist that $\omega(u_S)=1$. This last condition is simply a normalisation condition, as we can always set $\hat\omega(s) = \sup\{ \omega(st) \omega(t)^{-1} : t\in S \}$ for each $s\in S$. For $s,t\in S$, we have that $\omega(st) \leq \hat\omega(s)\omega(t)$, so that $$\hat\omega(st) = \sup\{ \omega(str)\omega(r)^{-1} : r\in S \} \leq \sup \{ \hat\omega(s)\omega(tr)\omega(r)^{-1} : r\in S \} = \hat\omega(s) \hat\omega(t).$$ Clearly $\hat\omega(u_S)=1$ and $\hat\omega(s)\leq\omega(s)$ for each $s\in S$, while $\hat\omega(s) \geq \omega(s)\omega(u_S)^{-1}$, so that $\hat\omega$ is equivalent to $\omega$. We form the Banach space $$l^1(S,\omega) = \Big\{ (a_g)_{g\in S} \subseteq\mathbb C : \|(a_g)\| := \sum_{g\in S} |a_g| \omega(g) < \infty \Big\}.$$ Then $l^1(S,\omega)$, with the convolution product, is a Banach algebra, called a *Beurling algebra*. See [@CY] and [@DL] for further information on Beurling algebras and, in particular, their second duals. It will be more convenient for us to think of $l^1(S,\omega)$ as the Banach space $l^1(S)$ together with a weighted algebra product. Indeed, for $g\in S$, let $\delta_g\in l^1(S)$ be the standard unit vector basis element which is thought of as a point-mass at $g$. Then each $x\in l^1(S)$ can be written uniquely as $x = \sum_{g\in S} x_g \delta_g$ for some family $(x_g) \subseteq \mathbb C$ such that $\|x\| = \sum_{g\in S} |x_g| <\infty$. We then define $$\delta_g \star_\omega \delta_h = \delta_g \star \delta_h = \delta_{gh} \Omega(g,h) \qquad (g,h\in S),$$ where $\Omega(g,h) = \omega(gh)\omega(g)^{-1}\omega(h)^{-1}$, and extend $\star$ to $l^1(S)$ by linearity and continuity. For example, if $\omega$ and $\hat\omega$ are equivalent weights on $S$, the define $\psi: l^1(S,\omega) \rightarrow l^1(S,\hat\omega)$ by $\psi(\delta_s) = \hat\omega(s) \omega(s)^{-1} \delta_s$. As $\omega$ and $\hat\omega$ are equivalent, $\psi$ is an isomorphism of Banach spaces. Then $\psi(\delta_s \star \delta_t) = \omega(st) \omega(s)^{-1} \omega(t)^{-1} \hat\omega(st) \omega(st)^{-1} \delta_{st} = \psi(\delta_s) \star \psi(\delta_t)$, so that $\psi$ is a homomorphism. For a set $I$, we define the space $c_0(I)$ as $$c_0(I) = \Big\{ (a_i)_{i\in I} : \forall\, \epsilon>0, |\{ i\in I : |a_i|\geq\epsilon \}|<\infty \Big\},$$ where $| \cdot |$ is the cardinality of a set. We equip $c_0(I)$ with the supremum norm; then $c_0(I)' = l^1(I)$. For $i\in I$, we let $e_i\in c_0(I)$ be the point mass at $i$, that is, ${{\langle {\delta_j} , {e_i} \rangle}} = \delta_{i,j}$, the Kronecker delta, for $\delta_j\in l^1(I)$. Then $c_0(I)$ is the closed linear span of $\{ e_i : i\in I \}$. We let $l^\infty(I)$ be the Banach space of all bounded families $(a_i)_{i\in I}$, with the supremum norm. Then $l^1(I)'=l^\infty(I)$, we can treat $c_0(I)$ as a subspace of $l^\infty(I)$, and the map $\kappa_{c_0(I)}: c_0(I)\rightarrow l^\infty(I)$ is just the inclusion map. For a semigroup $S$ and $s\in S$, we define maps $L_s,R_s:S \rightarrow S$ by $$L_s(t) = st, \quad R_s(t) = ts \qquad (t\in S).$$ If, for each $s\in S$, $L_s$ and $R_s$ are finite-to-one maps, then we say that $S$ is *weakly cancellative*. When $L_s$ and $R_s$ are injective for each $s\in S$, we say that $S$ is *cancellative*. When $S$ is abelian and cancellative, a construction going back to Grothendieck shows that $S$ is a sub-semigroup of some abelian group. However, this can fail to hold for non-abelian semigroups. Let $S$ be a weakly cancellative semigroup, let $\omega$ be a weight on $S$, and let ${\mathcal{A}} = l^1(S,\omega)$. Then $c_0(S) \subseteq l^\infty(S) = {\mathcal{A}}'$ is a sub-${\mathcal{A}}$-module of ${\mathcal{A}}'$, so that $l^1(S,\omega)$ is a dual Banach algebra with predual $c_0(S)$. For $g,h\in S$ and $a=(a_s)_{s\in S} \in l^1(S,\omega)$, we have $${{\langle {e_g \cdot \delta_h} , {a} \rangle}} = {{\langle {e_g} , {\delta_h \star a} \rangle}} = {{\langle {e_g} , {\sum_{s\in S} a_s \delta_{hs} \Omega(h,s)} \rangle}} = \sum_{\{ s\in S : hs = g\}} a_s \Omega(h,s).$$ As $S$ is weakly cancellative, there exists at most finitely many $s\in S$ such that $hs=g$, so that $e_g \cdot \delta_h$ is a member of $c_0(S)$. Thus we see that $c_0(S)$ is a right sub-${\mathcal{A}}$-module of ${\mathcal{A}}'$. The argument on the left follows in an analogous manner. Notice that the above result will hold for some semigroups $S$ which are not weakly cancellative, provided that the weight behaves in a certain way. However, it would appear that the later results do not easily generalise to the non-weakly cancellative case. Following [@DL Definition 2.2], we have the following definition. Let $I$ and $J$ be non-empty infinite sets, and let $f:I\times J\rightarrow \mathbb C$ be a function. Then *$f$ clusters on $I\times J$* if $$\lim_{n\rightarrow\infty} \lim_{m\rightarrow\infty} f(x_m,y_n) = \lim_{m\rightarrow\infty} \lim_{n\rightarrow\infty} f(x_m,y_n),$$ whenever $(x_m)\subseteq I$ and $(y_n)\subseteq J$ are sequences of distinct elements, and both iterated limits exist. Furthermore, *$f$ $0$-clusters on $I\times J$* if $f$ clusters on $I\times J$, and the iterated limits are always $0$, when they exist. [$\square$]{} From now on we shall exclude the trivial case when our (semi-)group is finite. \[B\_AR\] Let $S$ be a discrete, weakly cancellative semigroup, and let $\omega$ be a weight on $S$. Then the following are equivalent: 1. $l^1(S,\omega)$ is Arens regular; 2. for sequences of distinct elements $(g_j)$ and $(h_k)$ in $S$, we have $$\lim_{j\rightarrow\infty} \lim_{k\rightarrow\infty} \Omega(g_j,h_k) = 0,$$ whenever the iterated limit exists; 3. $\Omega$ $0$-clusters on $S\times S$. That (1) and (2) are equivalent for cancellative semigroups is [@CY Theorem 1]. Close examination of the proof shows that this holds for weakly cancellative semigroups as well. That (1) and (3) are equivalent follows by generalising the proof of [@DL Theorem 7.11], which is essentially an application of Grothendieck’s criterion for an operator to be weakly-compact. Alternatively, it follows easily that (2) and (3) are equivalent by considering the *opposite semigroup* to $S$ where we reverse the product. In [@CY] it is also shown that if $G$ is a discrete, uncountable group, then $l^1(G,\omega)$ is not Arens regular for any weight $\omega$. Furthermore, by [@CY Theorem 2], if $G$ is a non-discrete locally compact group, then $L^1(G,\omega)$ is never Arens regular. We shall consider both the Connes-amenability of $l^1(S,\omega)''$ and $l^1(S,\omega)$ (with respect to the canonical predual $c_0(S)$) as, with reference to Corollary \[unital\_dual\_sigma\] and Theorem \[arens\_wap\], the calculations should be similar. \[weak\_comp\_infty\] Let $I$ be a non-empty set, and let $X\subseteq l^\infty(I)$ be a subset. Then the following are equivalent: 1. $X$ is relatively weakly-compact; 2. $X$ is relatively sequentially weakly-compact; 3. the absolutely convex hull of $X$ is relatively weakly-compact; 4. if we define $f:I \times X \rightarrow \mathbb C$ by $f(i,x) = {{\langle {x} , {\delta_i} \rangle}}$ for $i\in I$ and $x\in X$, then $f$ clusters on $I\times X$; That (1) and (2) are equivalent is the Eberlien-Smulian theorem; that (1) and (3) are equivalent is the Krein-Smulian theorem. That (1) and (4) are equivalent is a result of Grothendieck, detailed in, for example, [@DL Theorem 2.3]. It is standard that for non-empty sets $I$ and $J$, we have that $l^1(I) {{\widehat{\otimes}}}l^1(J) = l^1(I \times J)$, where, for $i\in I$ and $j\in J$, $\delta_i \otimes \delta_j\in l^1(I){{\widehat{\otimes}}}l^1(J)$ is identified with $\delta_{(i,j)}\in l^1(I\times J)$. Thus we have $( l^1(I) {{\widehat{\otimes}}}l^1(J) )' = {\mathcal{B}}(l^1(I), l^\infty(J)) = l^1(I\times J)' = l^\infty(I\times J)$, where $T\in {\mathcal{B}}(l^1(I), l^\infty(J))$ is identified with $(T_{(i,j)}) \in l^\infty(I\times J)$, where $T_{(i,j)} = {{\langle {T(\delta_i)} , {\delta_j} \rangle}}$. **Is this paragraph used?** Let $S$ be a countable, discrete, unital semigroup, and let $\omega$ be a weight on $S$. Then $l^1(S\times S)$ is a Banach $l^1(S,\omega)$-bimodule, with module actions $$\delta_k \cdot \delta_{(g,h)} = \delta_{(kg,h)} \Omega(k,g) \quad , \quad \delta_{(g,h)} \cdot \delta_k = \delta_{(g,hk)} \Omega(h,k) \qquad (g,h,k\in S).$$ For a non-empty set $I$, the unit ball of $l^1(I)$ is the closure of the absolutely-convex hull of the set $\{ \delta_i : i\in I\}$, so that for a Banach space $E$, by the Krein-Smulian theorem, a map $T:l^1(I)\rightarrow E$ is weakly-compact if and only if the set $\{ T(\delta_i) : i\in I \}$ is relatively weakly-compact in $E$. \[wap\_c\_zero\] Let $S$ be a weakly cancellative semigroup, let $\omega$ be a weight on $S$, and let ${\mathcal{A}} = l^1(S,\omega)$. Let $T\in{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')$ be such that $T({\mathcal{A}}) \subseteq \kappa_{c_0(S)}(c_0(S))$ and $T'(\kappa_{{\mathcal{A}}}({\mathcal{A}})) \subseteq \kappa_{c_0(S)}(c_0(S))$. Then $T\in{\mathcal{W}}({\mathcal{A}},{\mathcal{A}}')$, and $T\in{\operatorname{WAP}}({\mathcal{W}}({\mathcal{A}},{\mathcal{A}}'))$ if and only if, for each sequence $(k_n)$ of distinct elements of $S$, and each sequence $(g_m,h_m)$ of distinct elements of $S\times S$ such that the repeated limits $$\begin{gathered} \lim_n \lim_m {{\langle {T(\delta_{h_m})} , {\delta_{k_ng_m}} \rangle}}, \ \lim_n \lim_m \Omega(k_n,g_m) \label{cond_one} \\ \lim_n \lim_m {{\langle {T(\delta_{h_mk_n})} , {\delta_{g_m}} \rangle}}, \ \lim_n \lim_m \Omega(h_m,k_n) \label{cond_two}\end{gathered}$$ all exist, we have that at least one repeated limit in each row is zero. That $T$ is weakly-compact follows from Gantmacher’s Theorem (compare with Corollary \[unital\_dual\_sigma\]). To show that $T\in{\operatorname{WAP}}$, by Lemma \[wap\_to\_maps\], we are required to show that the maps $\phi_r$ and $\phi_l$ are weakly-compact. We shall show that $\phi_l$ is weakly-compact if and only if one of the repeated limits in the first line (\[cond\_one\]) is zero; the proof that $\phi_r$ is related to (\[cond\_two\]) follows in a similar way. We have that $$\phi_l(\delta_{(g,h)}) = \phi_l(\delta_g \otimes \delta_h) = \delta_g \cdot T(\delta_h) \qquad (g,h\in S).$$ By Proposition \[weak\_comp\_infty\], $\phi_l$ is weakly-compact if and only if the function $$S \times (S \times S) \rightarrow \mathbb C ; \ (k,(g,h)) \mapsto {{\langle {\delta_g\cdot T(\delta_h)} , {\delta_k} \rangle}} = {{\langle {T(\delta_h)} , {\delta_{kg}} \rangle}} \Omega(k,g) \qquad (g,h,k\in S)$$ clusters on $S \times (S \times S)$. As $T$ is weakly-compact, the function $$S \times S\rightarrow\mathbb C; \quad (g,h) \mapsto {{\langle {T(\delta_g)} , {\delta_h} \rangle}} \qquad (g,h\in S)$$ does cluster on $S\times S$. Let $(k_n)$ be a sequence of distinct elements of $S$, and let $(g_m,h_m)$ be a sequence of distinct elements of $S\times S$ such that the iterated limits $$\lim_n \lim_m {{\langle {T(\delta_{h_m})} , {\delta_{k_ng_m}} \rangle}} \Omega(k_n,g_m) , \quad \lim_m \lim_n {{\langle {T(\delta_{h_m})} , {\delta_{k_ng_m}} \rangle}} \Omega(k_n,g_m) \label{eq:five}$$ exist. We now investigate when these iterated limits are equal. Suppose firstly that, by moving to a subsequence if necessary, we have that $g_m = g$ for all $m$. Further, by moving to a subsequence if necessary, we may suppose that $\lim_n \Omega(k_n,g) = \alpha$, say, and that $(k_ng)$ is a sequence of distinct elements (as $S$ is weakly cancellative). Then $$\begin{aligned} \lim_n \lim_m {{\langle {T(\delta_{h_m})} , {\delta_{k_ng_m}} \rangle}} & \Omega(k_n,g_m) = \lim_n \Omega(k_n,g) \lim_m {{\langle {T(\delta_{h_m})} , {\delta_{k_ng}} \rangle}} \\ &= \alpha \lim_n \lim_m {{\langle {T(\delta_{h_m})} , {\delta_{k_ng}} \rangle}} = \alpha \lim_m \lim_n {{\langle {T(\delta_{h_m})} , {\delta_{k_ng}} \rangle}} \\ &= \lim_m \lim_n {{\langle {T(\delta_{h_m})} , {\delta_{k_ng_m}} \rangle}} \Omega(k_n,g_m),\end{aligned}$$ where we can swap the order of taking limits, as $T$ is weakly-compact. Alternatively, if we cannot move to a subsequence such that $(g_m)$ is constant, then we may move to subsequence such that $(g_m)$ is a sequence of distinct elements, and such that the iterated limits $$\begin{gathered} \lim_m \lim_n \Omega(k_n,g_m),\quad \lim_n \lim_m \Omega(k_n,g_m), \\ \lim_m \lim_n {{\langle {T(\delta_{h_m})} , {\delta_{k_ng_m}} \rangle}},\quad \lim_n \lim_m {{\langle {T(\delta_{h_m})} , {\delta_{k_ng_m}} \rangle}}\end{gathered}$$ all exists. As $T({\mathcal{A}})\subseteq\kappa_{c_0(S)}(c_0(S))$, we have that $$\{ g\in S : |{{\langle {T(\delta_h)} , {\delta_g} \rangle}}|\geq\epsilon \} \text{ is finite } \qquad (\epsilon>0, h\in S).$$ Consequently, and using the fact that $S$ is weakly cancellative, we see that $$\lim_n {{\langle {T(\delta_{h_m})} , {\delta_{k_ng_m}} \rangle}} = 0$$ for each $m$. Hence the iterated limits in (\[eq:five\]) are equal if and only if we have that at least one repeated limit in (\[cond\_one\]) is zero. \[auto\_semi\_wap\] Let $S$ be a discrete, unital, weakly cancellative semigroup, and let $\omega$ be a weight on $S$ such that ${\mathcal{A}} = l^1(S,\omega)$ is Arens regular. Then ${\operatorname{WAP}}({\mathcal{W}}({\mathcal{A}},{\mathcal{A}}')) = {\mathcal{W}}({\mathcal{A}},{\mathcal{A}}')$. Let $T\in{\mathcal{W}}({\mathcal{A}},{\mathcal{A}}')$. We can follow the above proof through until the point at which we use the fact that $T({\mathcal{A}})\subseteq \kappa_{c_0(S)}(c_0(S))$. However, as $l^1(S,\omega)$ is Arens regular, by Theorem \[B\_AR\], we have that $$\lim_m \lim_n \Omega(k_n,g_m) = \lim_n \lim_m \Omega(k_n,g_m) = 0,$$ so that the iterated limits in (\[eq:five\]) must be $0$, implying that $\phi_l$ is weakly-compact. In a similar manner, $\phi_r$ is weakly-compact. \[when\_loneg\_cam\] Let $S$ be a discrete weakly cancellative semigroup, and let $\omega$ be a weight on $S$ such that ${\mathcal{A}} = l^1(S,\omega)$ is Arens regular and ${\mathcal{A}}''$ is unital with unit $e_{{\mathcal{A}}''}$. Then ${\mathcal{A}}''$ is Connes-amenable if and only if there exists $M\in ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})'' = l^\infty(S\times S)'$ such that: 1. ${{\langle {M} , {(f_{gh}\Omega(g,h))_{(g,h)\in S\times S}} \rangle}} = {{\langle {e_{{\mathcal{A}}''}} , {f} \rangle}}$ for each bounded family $(f_g)_{g\in S}$; 2. ${{\langle {M} , {( f(hk,g)\Omega(h,k) - f(h,kg)\Omega(k,g) )_ {(g,h)\in S\times S}} \rangle}} = 0$ for each $k\in S$, and each bounded function $f:S\times S\rightarrow\mathbb C$ which clusters on $S\times S$. We use Theorem \[Connes\_Amen\] and Proposition \[auto\_semi\_wap\]. For $f=(f_g)_{g\in S} \in l^\infty(S)$, we have $${{\langle { \Delta'_{{\mathcal{A}}}(f) } , {\delta_g\otimes \delta_h} \rangle}} = {{\langle {f} , {\delta_{gh}} \rangle}}\Omega(g,h) \qquad (g,h\in S),$$ so that $\Delta'_{{\mathcal{A}}}(f) = ( {{\langle {f} , {\delta_{gh}} \rangle}}\Omega(g,h) )_{(g,h)\in S\times S} \in l^\infty(S\times S)$. As $f\in l^\infty(S)$ was arbitrary, we have condition (1). For $T\in{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')$, we treat $T$ as being a member of $l^\infty(S\times S)$. Then $T$ is weakly-compact if and only if the family $({{\langle {T(\delta_g)} , {\delta_h} \rangle}})_{(g,h)\in S\times S}$ clusters on $S\times S$. For $k\in S$, we have $${{\langle { \delta_k\cdot T - T\cdot \delta_k } , {\delta_g\otimes \delta_h} \rangle}} = {{\langle {T(\delta_{hk})} , {\delta_g} \rangle}} \Omega(h,k) - {{\langle {T(\delta_h)} , {\delta_{kg}} \rangle}} \Omega(k,g).$$ Thus we have condition (2). Notice that if $S$ is unital with unit $u_S$, then the unit of ${\mathcal{A}}$ (and hence ${\mathcal{A}}''$) is $\delta_{u_S}$. In this case, condition (1) reduces to ${{\langle {M} , {(f_{gh}\Omega(g,h))_{(g,h)\in S\times S}} \rangle}} = f_{u_S}$. \[when\_loneg\_am\] Let $S$ be a discrete unital semigroup, let $\omega$ be a weight on $S$, and let ${\mathcal{A}}=l^1(S,\omega)$. Then ${\mathcal{A}}$ is amenable if and only if there exists $M\in ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})'' = l^\infty(S\times S)'$ such that: 1. ${{\langle {M} , {(f_{gh}\Omega(g,h))_{(g,h)\in S\times S}} \rangle}} = f_{u_S}$, where $u_S\in S$ is the unit of $S$, for each bounded family $(f_g)_{g\in S}$; 2. ${{\langle {M} , {( f(hk,g)\Omega(h,k) - f(h,kg)\Omega(k,g) )_ {(g,h)\in S\times S}} \rangle}} = 0$ for each $k\in S$, and each bounded function $f:S\times S\rightarrow\mathbb C$. This follows from Theorem \[when\_amen\] in the same way that the above follows from Theorem \[Connes\_Amen\]. Notice that condition (2) of Theorem \[when\_loneg\_am\] is strictly stronger than condition (2) of Theorem \[when\_loneg\_cam\]. \[C\_amen\_predual\] Let $S$ be a discrete, weakly cancellative semigroup, let $\omega$ be a weight on $S$, and let ${\mathcal{A}} = l^1(S,\omega)$ be unital with unit $e_{{\mathcal{A}}}$. Then ${\mathcal{A}}$ is Connes-amenable, with respect to the predual $c_0(S)$, if and only if there exists $M\in ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})'' = l^\infty(S\times S)'$ such that: 1. ${{\langle {M} , {(f_{gh}\Omega(g,h))_{(g,h)\in S\times S}} \rangle}} = {{\langle {e_{{\mathcal{A}}}} , {f} \rangle}}$ for each family $(f_g)_{g\in S} \in c_0(S)$; 2. ${{\langle {M} , {( f(hk,g)\Omega(h,k) - f(h,kg)\Omega(k,g) )_ {(g,h)\in S\times S}} \rangle}} = 0$ for each $k\in S$, and each bounded function $f:S\times S\rightarrow\mathbb C$ which satisfies the conclusions of Proposition \[wap\_c\_zero\]. We now use Theorem \[When\_Con\_Amen\]. By $f$ satisfying the conclusions of Proposition \[wap\_c\_zero\], we identify $f:S\times S\rightarrow\mathbb C$ with $T\in{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')$ by ${{\langle {T(\delta_g)} , {\delta_h} \rangle}} = f(g,h)$, for $g,h\in S$. We shall now establish when $l^1(S,\omega)$ and $l^1(S,\omega)''$ are Connes-amenable. For a discrete group $G$, a weight $\omega$ on $G$ and $h\in G$, define $J_h\in{\mathcal{B}}(l^\infty(G))$ by $$J_h(f) = \big( f_{hg} \Omega(h,g)\omega(h) \Omega(g^{-1},h^{-1}) \omega(h^{-1}) \big)_{g\in G} \qquad (f=(f_g)_{g\in G}\in l^\infty(G)).$$ Notice then that, for $f\in l^\infty(G)$, we have $$\| J_h(f) \| = \sup_g |f_{hg}| \omega(hg)\omega(g)^{-1} \omega(g^{-1}h^{-1})\omega(g^{-1})^{-1} \leq \|f\| \omega(h) \omega(h^{-1}),$$ so that $J_h$ is bounded. \[group\_omega\_amen\] Let $G$ be a discrete group, and let $\omega$ be a weight on $G$. We say that $G$ is *$\omega$-amenable* if there exists $N\in l^\infty(G)'$ such that: 1. ${{\langle {N} , {(\Omega(g,g^{-1}))_{g\in G}} \rangle}}=1$, where $\Omega$ is defined by $\omega$, and hence $(\Omega(g,g^{-1}))_{g\in G}$ is a bounded family forming an element of $l^\infty(G)$; 2. $J_h'(N)=N$ for each $h\in G$. [$\square$]{} Notice that if $\omega$ is identically $1$, then this condition reduces to the usual notion of a group being amenable (we usually require that $N$ is a *mean*, in that $N$ is a positive functional on $l^\infty(G)$, but by forming real and imaginary parts, and then positive and negative parts, we can easily generate a non-zero scalar multiple of a mean from a functional $N$ satisfying the definition above). Let $G$ be a discrete group, let $\omega$ be a weight on $G$, and let ${\mathcal{A}}=l^1(G,\omega)$. Then the following are equivalent: 1. ${\mathcal{A}}$ is Connes-amenable, with respect to the predual $c_0(G)$; 2. ${\mathcal{A}}$ is amenable; 3. $G$ is $\omega$-amenable. Furthermore, if ${\mathcal{A}}$ is Arens regular, then these conditions are equivalent to ${\mathcal{A}}''$ being Connes-amenable. It is clear that (2) implies (1). When ${\mathcal{A}}$ is Arens regular, (2) implies that ${\mathcal{A}}''$ is Connes-amenable, and ${\mathcal{A}}''$ Connes-amenable implies (1). We shall thus show that (1) implies (3), and that (3) implies (2). If (1) holds, then let $M\in l^\infty(G\times G)'$ be given as in Theorem \[C\_amen\_predual\]. Define $\phi:l^\infty(G) \rightarrow l^\infty(G\times G)$ by $${{\langle {\phi(f)} , {\delta_{(g,h)}} \rangle}} = \begin{cases} f_g & : g = h^{-1}, \\ 0 &: g\not=h^{-1}, \end{cases} \quad\qquad ( f=(f_g)_{g\in G} \in l^\infty(G) ).$$ Let $N = \phi'(M) \in l^\infty(G)'$. Then we have $$\phi( (\Omega(g,g^{-1}))_{g\in G} ) = ( \delta_{h,g^{-1}} \Omega(g,h) )_{(g,h)\in G\times G} = ( \delta_{gh,e_G} \Omega(g,h) )_{(g,h)\in G\times G},$$ where $\delta$ is the Kronecker delta, so that $${{\langle {N} , {(\Omega(g,g^{-1}))_{g\in G}} \rangle}} = \delta_{e_G,e_G} = 1,$$ by condition (1) on $M$ from Theorem \[C\_amen\_predual\]; clearly $(\delta_{e_G,g})_{g\in G} \in c_0(G)$. Fix $k\in G$ and $f\in l^\infty(G)$. Define $F:G\times G\rightarrow \mathbb C$ by $$F(h,g) = \delta_{gh,k} f_g \omega(k) \omega(hk^{-1}) \omega(h)^{-1}. \qquad (g,h\in G).$$ Then we have $|F(h,g)| \leq |f_g| |\omega(k)| |\omega(hk^{-1})| |\omega(h)|^{-1} \leq \|f\|_\infty |\omega(k)| |\omega(k^{-1})|$, so that $F$ is bounded. Let $T:{\mathcal{A}}\rightarrow{\mathcal{A}}'$ be the operator associated with $F$. For $g,h\in G$, we have that $F(h,g)\not=0$ only when $gh=k$, so that $T({\mathcal{A}})\subseteq c_0(S)$ and $T'(\kappa_{{\mathcal{A}}}({\mathcal{A}})) \subseteq c_0(S)$. Furthermore, if $(k_n)$ is a sequence of distinct elements in $G$, and $(g_m,h_m)$ is a sequence of distinct elements in $G\times G$, then $\lim_n \lim_m F(h_m,k_ng_m) = 0$. This follows, as for $n_0$ fixed, $k_{n_0}g_mh_m = k$ only if $g_mh_m = k_{n_0}^{-1} k$, so if this holds for all sufficiently large $m$, we have that $k_ng_mh_m\not=k$ for sufficiently large $m$ and $n\not=n_0$. Similarly, $\lim_n \lim_m F(h_mk_n,g_m) = 0$, so that $F$ satisfies the conditions of Proposition \[wap\_c\_zero\]. Notice that $${{\langle { \phi(J_k(f)) } , { \delta_{(g,h)} } \rangle}} = \delta_{gh,e_G} {{\langle { J_k(f) } , { \delta_g } \rangle}} = \delta_{gh,e_G} f_{kg} \omega(kg)\omega(g)^{-1} \omega(g^{-1}k^{-1})\omega(g^{-1})^{-1}.$$ Thus we have $$\begin{aligned} F(hk,g)&\Omega(h,k) - F(h,kg)\Omega(k,g) \\ &= \delta_{ghk,k} f_g \omega(k) \omega(hkk^{-1}) \omega(hk)^{-1} \Omega(h,k) - \delta_{kgh,k} f_{kg} \omega(k) \omega(hk^{-1}) \omega(h)^{-1} \Omega(k,g) \\ &= \delta_{gh,e_G} f_g - \delta_{gh,e_G} f_{kg} \omega(hk^{-1}) \omega(h)^{-1} \omega(kg) \omega(g)^{-1} \\ &= {{\langle {\phi(f) - \phi(J_k(f)) } , { \delta_{(g,h)} } \rangle}}.\end{aligned}$$ So, by condition (2) from Theorem \[C\_amen\_predual\], we have that $${{\langle {N} , { f - J_k(f) } \rangle}} = 0,$$ which, as $f$ was arbitrary, shows that $N = J_k'(N)$, as required. Now suppose that $G$ is $\omega$-amenable. We shall show that ${\mathcal{A}}$ is amenable, which completes the proof. Define $\psi : l^\infty(G\times G)\rightarrow l^\infty(G)$ by $${{\langle { \psi(F) } , { \delta_g } \rangle}} = F(g,g^{-1}) \qquad ( F\in l^\infty(G\times G), g\in G).$$ Let $N\in l^\infty(G)'$ be as in Definition \[group\_omega\_amen\], and let $M =\psi'(N)$. Then let $(f_g)_{g\in G}$ be a bounded family in $\mathbb C$, so that $${{\langle {M} , {(f_{gh}\Omega(g,h))_{(g,h)\in G\times G}} \rangle}} = {{\langle {N} , {(f_{e_G}\Omega(g,g^{-1}))_{g\in G}} \rangle}} = f_{e_G},$$ verifying condition (1) of Theorem \[when\_loneg\_am\] for $M$. Let $f:G\times G\rightarrow\mathbb C$ be a bounded function, and let $k\in G$. Then $$\begin{aligned} \psi\big( (f(hk,g) & \Omega(h,k) - f(h,kg)\Omega(k,g))_{(g,h)\in G\times G} \big) \\ &= \big( f(g^{-1}k,g)\Omega(g^{-1},k) - f(g^{-1},kg)\Omega(k,g) \big)_{g\in G}.\end{aligned}$$ Define $F:G\times G\rightarrow\mathbb C$ by $$F(g,h) = f(hk,g) \Omega(h,k) \qquad (g,h\in G),$$ so that $F$ is bounded. For $g\in G$, we have that $$\begin{aligned} &{{\langle { \psi(F)-J_k(\psi(F)) } , { \delta_g } \rangle}} \\ &= f(g^{-1}k,g) \Omega(g^{-1},k) - f((kg)^{-1}k,kg) \Omega((kg)^{-1},k) \omega(kg) \omega(g)^{-1} \omega(g^{-1}k^{-1}) \omega(g^{-1})^{-1} \\ &= f(g^{-1}k,g) \Omega(g^{-1},k) - f(g^{-1},kg) \omega(k)^{-1} \omega(kg) \omega(g)^{-1} \\ &= f(g^{-1}k,g) \Omega(g^{-1},k) - f(g^{-1},kg) \Omega(k,g).\end{aligned}$$ Consequently, using condition (2) of Definition \[group\_omega\_amen\], we have established condition (2) of Theorem \[when\_loneg\_am\] for $M$. This shows that $l^1(G,\omega)$ is amenable. If $S$ is a semigroup which is not cancellative, then it is possible for $l^1(S)$ to be unital while $S$ is not. For example, let $S$ be $(\mathbb N,\max)$ (where $\mathbb N=\{1,2,3,\ldots\}$ say) with adjoined idempotents $u$ and $v$ such that $uv=vu=1$ and $un = nu = vn = nv = n$ for $n\in\mathbb N$. Then $S$ is a weakly cancellative, commutative semigroup without a unit, but $e = \delta_u+\delta_v-\delta_1$ is easily seen to be a unit for $l^1(S)$. Indeed, $S$ is seen to be a finite semilattice of groups, so by the result of [@Gron2], $l^1(S)$ is amenable. In [@Gron1 Theorem 2.3] it is shown that if $l^1(S,\omega)$ is amenable for a cancellative, unital semigroup $S$ and some weight $\omega$, then $S$ is actually a group. We shall now show that this holds for Connes-amenability as well. For a cancellative, unital semigroup $S$, with unit $u_S$, if $g\in S$ is invertible, then $g$ has a unique inverse, denoted by $g^{-1}$. Furthermore, if $g$ has a left inverse, say $hg=u_S$, then $ghg = g = u_Sg$ so that $gh=u_S$; similarly, if $gh=u_S$ then $hg=u_S$. Let $S$ be a weakly cancellative semigroup, let $\omega$ be a weight on $S$, and let ${\mathcal{A}}=l^1(S,\omega)$. Suppose that ${\mathcal{A}}$ is Connes-amenable with respect to the predual $c_0(S)$. If $S$ is cancellative or unital, then $S$ is a group. As ${\mathcal{A}}$ is Connes-amenable, let $M\in ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})''$ be as in Theorem \[C\_amen\_predual\]. Then ${\mathcal{A}}$ is unital, with unit $e_{{\mathcal{A}}} = (a_s)_{s\in S} \in l^1(S,\omega)$ say. For now, we shall not assume that $e_{{\mathcal{A}}}$ has norm one, as the standard renorming to ensure this will not (a priori) necessarily yield an $l^1(S,\hat\omega)$ algebra for some weight $\hat\omega$. Suppose that $S$ is cancellative. Fix $h\in S$, so that $$\sum_{s\in S} a_s \delta_{sh} \Omega(s,h) = e_{{\mathcal{A}}}\star \delta_h = \delta_h = \delta_h \star e_{{\mathcal{A}}} = \sum_{s\in S} a_s \delta_{hs} \Omega(h,s).$$ In particular, for each $h\in S$ there is a unique $u_h\in S$ such that $h u_h = h$ (so that $h u_h h = h^2$ implying that $u_h h=h$), and we have that $a_{u_h} \omega(u_h)^{-1} =1$. We also see that $a_s=0$ for each $s\in S$ such that $sh\not=h$, that is, $s\not=u_h$. However, $h$ was arbitrary, so that $S$ is unital with unit $u_S$, and $e_{{\mathcal{A}}} = \omega(u_S) \delta_{u_S}$, where we can now assume that $\omega(u_S)=1$ by a renorming. Now suppose that $S$ is a unital, weakly cancellative semigroup, so that the unit of ${\mathcal{A}}$ is $\delta_{u_S}$. Suppose that $s\in S$ has no right inverse. Define $F:S\times S\rightarrow\mathbb C$ by $$F(h,sg)=0, \quad F(hs,g) = \begin{cases} \Omega(g,hs) &: gh=u_S, \\ 0 &: \text{otherwise.} \end{cases} \qquad (g,h\in S).$$ To show that this is well-defined, suppose that for $g,h,j,k\in S$, we have that $h=js$, $sg=k$ and $kj=u_S$. Then $s(gj) = kj = u_S$, so that $s$ has a right inverse, a contradiction. Then $F$ is bounded, so let $T:{\mathcal{A}}\rightarrow {\mathcal{A}}'$ be the operator associated with $F$. Then $F(a,b)\not=0$ only when $ba=s$, so as $S$ is weakly cancellative, we see that $T({\mathcal{A}})\subseteq c_0(S)$ and $T'(\kappa_{{\mathcal{A}}}({\mathcal{A}})) \subseteq c_0(S)$. Suppose that for sequences of distinct elements $(k_n)\subseteq S$ and $(g_m,h_m)\subseteq S\times S$, we have that $$\lim_n \lim_m {{\langle {T(\delta_{h_m})} , {\delta_{k_ng_m}} \rangle}} = \lim_n \lim_m F(h_m,k_ng_m) \not=0.$$ Then, for some $N>0$ and $\epsilon>0$, for each $n\geq N$, $\lim_m F(h_m,k_ng_m)\geq \epsilon$. Hence, for $n\geq N$, there exists $M_n>0$ such that if $m\geq M_n$, then $k_ng_mh_m = s$ (as otherwise $F(h_m,k_ng_m)=0$). This, however, contradicts $S$ being weakly cancellative. Similarly, if $\lim_n \lim_m {{\langle {T(\delta_{h_mk_n})} , {\delta_{g_m}} \rangle}}\not=0$, then we need $g_mh_mk_n=s$ for all $n,m$ sufficiently large, which is a contradiction. Thus $T$ satisfies all the conditions of Proposition \[wap\_c\_zero\]. Then, for $g,h\in S$, if $gh=u_S$, we have that $\Omega(h,s)\Omega(g,hs) = \omega(h)^{-1} \omega(g)^{-1} = \Omega(g,h)$, so that $$F(hs,g)\Omega(h,s) - F(h,sg)\Omega(s,g) = \begin{cases} \Omega(g,h) &: gh=u_S, \\ 0 &: \text{otherwise}. \end{cases}$$ Hence condition (2) of Theorem \[C\_amen\_predual\] implies that ${{\langle {M} , {(\delta_{gh,u_S} \Omega(g,h))_{(g,h)\in S\times S}} \rangle}}=0$, which contradicts condition (1) of this theorem. Hence every element of $S$ has a right inverse. By symmetry (or by repeating the argument on the left) we see that every element of $S$ has a left inverse, and that hence $S$ must be a group. We hence have the following theorem, which shows that weighted semigroup algebras behave like C$^*$-algebras with regards to Connes-amenability. \[main\_thm\] Let $S$ be a discrete cancellative semigroup, and let $\omega$ be a weight on $S$. The following are equivalent: 1. $l^1(S,\omega)$ is amenable; 2. $l^1(S,\omega)$ is Connes-amenable, with respect to the predual $c_0(S)$; If $l^1(S,\omega)$ is Arens regular, then these conditions are equivalent to $l^1(S,\omega)''$ being Connes-amenable. These equivalent conditions imply that $S$ is a group. [$\square$]{} This result extends the result of [@Runde3], where it is shown that $M(G)$, the *measure algebra* of a locally compact group $G$, is Connes-amenable if and only if $G$ is amenable. This follows as, for discrete groups $G$, $M(G) = l^1(G)$. Let $\omega$ be the weight on $\mathbb Z$ defined by $\omega(n) = 1+|n|$ for $n\in\mathbb Z$. By Theorem \[B\_AR\], ${\mathcal{A}} = l^1(\mathbb Z,\omega)$ is Arens regular. For $m,n\in\mathbb Z$ and $f = (a_k)_{k\in\mathbb Z}\in l^\infty(\mathbb Z)$, we have that $${{\langle { \delta_m \cdot f } , { \delta_n } \rangle}} = {{\langle {f} , {\delta_{n+m}\Omega(n,m)} \rangle}} = f_{n+m} \frac{1+|n+m|}{(1+|n|)(1+|m|)}.$$ Suppose that $M {\Box}\kappa_{{\mathcal{A}}}(\delta_m) = \kappa_{{\mathcal{A}}}(a)$ for some $m\in\mathbb Z$, $M \in l^\infty(\mathbb Z)'$ and $a\in{\mathcal{A}}$. Then ${{\langle {M} , {\delta_m\cdot f} \rangle}} = {{\langle {f} , {a} \rangle}}$ for each $f\in l^\infty(\mathbb Z)$, so by letting $f = \kappa_{c_0(\mathbb Z)}(e_k) \in c_0(\mathbb Z)$, we see that $a = \sum_{k\in\mathbb Z} a_k \delta_k$, where $a_k = {{\langle {M} , {\delta_m\cdot\kappa_{c_0(\mathbb Z)}(e_k)} \rangle}}$. However, $\delta_m\cdot\kappa_{c_0(\mathbb Z)}(e_k) \in \kappa_{c_0(\mathbb Z)}(c_0(\mathbb Z))$ for each $k\in\mathbb Z$, so if $M \in c_0(\mathbb Z)^\circ$, then $a=0$. Consequently, if $M {\Box}\kappa_{{\mathcal{A}}}(\delta_m) \in \kappa_{{\mathcal{A}}}({\mathcal{A}})$ for each $m\in\mathbb Z$ and $M \in l^\infty(\mathbb Z)'$, then $\delta_m \cdot f \in \kappa_{c_0(\mathbb Z)}(c_0(\mathbb Z))$ for each $m\in\mathbb Z$ and $f \in l^\infty(\mathbb Z)$. However, if $\mathbf{1} \in l^\infty(\mathbb Z)$ is the constant $1$ sequence, then $$\lim_n {{\langle {\delta_m\cdot\mathbf{1}} , {\delta_n} \rangle}} = \lim_n \frac{1+|n+m|}{(1+|n|)(1+|m|)} = \frac{1}{1+|m|},$$ so that $\delta_m\cdot\mathbf{1} \not\in \kappa_{c_0(\mathbb Z)}(c_0(\mathbb Z))$. We hence conclude that ${\mathcal{A}}$ is not an ideal in ${\mathcal{A}}''$, and so we cannot apply Theorem \[ca\_facts\] in this case. [$\square$]{} Unfortunately, it is not possible for $l^1(S,\omega)$ to be both amenable and Arens regular. \[Gron\_Thm\] Let $G$ be discrete group, and let $\omega$ be a weight on $G$. Then $l^1(G,\omega)$ is amenable if and only if $G$ is an amenable group, and $\sup\{ \omega(g) \omega(g^{-1}) : g\in G \} < \infty$. This is [@Gron1 Theorem 3.2]. Let $S$ be a discrete, unital semigroup, and let $\omega$ be a weight on $S$ such that ${\mathcal{A}} = l^1(S,\omega)$ is Arens regular. Let $K>0$ and $B\subseteq S$ be such that for each $g\in B$, $g$ has a right inverse $g^{-1}$ (which need not be unique), and $\omega(g)\omega(g^{-1})\leq K$. Then $B$ is finite. For $g\in B$ and $h\in S$, we have $$\omega(g)\omega(h) = \omega(g) \omega(hgg^{-1}) \leq \omega(g) \omega(hg) \omega(g^{-1}) \leq K \omega(hg),$$ so that $\Omega(h,g) \geq K^{-1}$. Suppose now that $B$ is infinite. Then we can easily construct sequences which violate condition (2) of Theorem \[B\_AR\], showing that ${\mathcal{A}}$ is not Arens regular. This contradiction shows that $B$ must be finite. Injectivity of the predual module {#injectivity-of-the-predual-module} --------------------------------- Let $S$ be a unital, weakly cancellative semigroup, let $\omega$ be a weight on $S$, and let ${\mathcal{A}}=l^1(S,\omega)$, ${\mathcal{A}}_* = c_0(S)$. Then ${\mathcal{B}}({\mathcal{A}},{\mathcal{A}}_*) = {\mathcal{B}}(l^1,c_0) = l^\infty(c_0) \subseteq l^\infty(S\times S)$, where we identify $T:{\mathcal{A}}\rightarrow{\mathcal{A}}_*$ with the bounded family $({{\langle {\delta_s} , {T(\delta_t)} \rangle}})_{(s,t)\in S\times S}$. Let $\phi : {\mathcal{B}}({\mathcal{A}},{\mathcal{A}}_*) \rightarrow {\mathcal{A}}_*$, so that $\phi$ is represented by a bounded family $(M_s)_{s\in S} \subseteq {\mathcal{B}}({\mathcal{A}},{\mathcal{A}}_*)'$ using the relation $${{\langle {\delta_s} , {\phi(T)} \rangle}} = {{\langle {M_s} , {T} \rangle}} \qquad (s\in S, T\in{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}_*)).$$ Suppose further that $\phi$ is a left ${\mathcal{A}}$-module homomorphism. Then $${{\langle {\delta_s} , {\phi(T)} \rangle}} = {{\langle {\delta_{u_s}} , {\phi(\delta_s\cdot T)} \rangle}} = {{\langle {M_{u_S}} , {\delta_s\cdot T} \rangle}} = {{\langle {M_s} , {T} \rangle}} \qquad (s\in S, T\in{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}_*)), \label{eq:three}$$ so that $M_s = M_{u_S} \cdot \delta_s$ for each $s\in S$. We see also that $\phi$ maps into $c_0(S)$ (and not just $l^\infty(S)$) if and only if $$\lim_{s\rightarrow\infty} {{\langle {M_{u_S}} , {\delta_s\cdot T} \rangle}} = 0 \qquad (T\in{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}_*)).$$ Conversely, if condition (\[eq:three\]) holds, then for $s,t\in S$ and $T\in{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}_*)$, we have that $$\begin{aligned} {{\langle {\delta_s} , {\phi(\delta_t\cdot T)} \rangle}} &= {{\langle {M_s} , {\delta_t\cdot T} \rangle}} = {{\langle {M_{u_S}} , {\delta_s\cdot\delta_t\cdot T} \rangle}} = \Omega(s,t) {{\langle {M_{st}} , {T} \rangle}} \\ &= \Omega(s,t){{\langle {\delta_{st}} , {\phi(T)} \rangle}} = {{\langle {\delta_s} , {\delta_t\cdot \phi(T)} \rangle}}.\end{aligned}$$ Hence $\phi$ is a left ${\mathcal{A}}$-module homomorphism. Notice that $c_0(S\times S) \subseteq {\mathcal{B}}({\mathcal{A}},{\mathcal{A}}_*)$, so that $c_0(S\times S)^\circ \subseteq {\mathcal{B}}({\mathcal{A}},{\mathcal{A}}_*)'$. Let $G$ be a group and $\omega$ be a weight on $G$ such that for each $\epsilon>0$, the set $\{ g\in G : \omega(g) \omega(g^{-1}) < \epsilon^{-1} \}$ is finite. Then we say that the weight $\omega$ is *strongly non-amenable*. Let $G$ be a group, and let $\omega$ be a weight on $G$ such that $\omega$ is not strongly non-amenable, and let $\phi:{\mathcal{B}}({\mathcal{A}},c_0(G))\rightarrow c_0(G)$ be a left ${\mathcal{A}}$-module homomorphism. If $\phi$ is represented by $(M_g)_{g\in G}$ as above, then $M_{u_G} \in c_0(S\times S)^\circ$. We adapt the methods of [@DP] to the weighted, discrete case. As $\omega$ is not strongly non-amenable, there exists some $K>0$ such that the set $X_K = \{ g\in G : \omega(g)\omega(g^{-1})\leq K \}$ is infinite. Let $M=M_{u_G}$, and suppose that $M\not\in c_0(G\times G)^\circ$, so that for some $g,h\in G$, we have that $\delta:={{\langle {M} , {e_{(g,h)}} \rangle}}\not=0$. We shall henceforth treat $e_{(g,h)}$ as a member of ${\mathcal{B}}({\mathcal{A}},c_0(G))$, noting that for $k\in G$, $${{\langle {\delta_s} , {(\delta_k\cdot e_{(g,h)})(\delta_t)} \rangle}} = \begin{cases} \Omega(t,k) &: s=g, t=hk^{-1}, \\ 0 &: \text{otherwise.} \end{cases}$$ We claim that we can find a sequence $(g_n)_{n\in\mathbb N}$ of distinct elements in $G$ such that $$\begin{gathered} |{{\langle {M\cdot\delta_{g_m^{-1}g_n}} , {e_{(g,h)}} \rangle}}| \leq K^{-1} 2^{-2-|m-n|} \qquad (n\not=m), \\ \omega(g_n) \omega(g_n^{-1}) \leq K \qquad (n\in\mathbb N).\end{gathered}$$ We can do this as $\phi$ must map into $c_0(G)$, so that for any $T:{\mathcal{A}}\rightarrow c_0(G)$, we have $\lim_{g\rightarrow\infty} {{\langle {M\cdot\delta_g} , {T} \rangle}}=0$. Explicitly, let $g_1\in X_K$ be arbitrary, and suppose that we have found $g_1,\ldots,g_k$. Then notice that the sets $$\begin{gathered} \{ s\in G : |{{\langle {M\cdot\delta_{s^{-1}g_n}} , {e_{(g,h)}} \rangle}}| > K^{-1} 2^{-2-|k+1-n|} : 1\leq n\leq k \}, \\ \{ s\in G : |{{\langle {M\cdot\delta_{g_m^{-1}s}} , {e_{(g,h)}} \rangle}}| > K^{-1} 2^{-2-|k+1-m|} : 1\leq m\leq k \}\end{gathered}$$ are finite, so as $X_K$ is infinite, we can certainly find some $x_{k+1}$. Then, for $x=(x_n)\in l^\infty(\mathbb N)$, define $T_x: {\mathcal{A}}\rightarrow c_0(G)$ by setting ${{\langle {\delta_g} , {T_x(\delta_{hg_n^{-1}})} \rangle}} = x_n \Omega(hg_n^{-1},g_n)$ for $n\geq 1$, and ${{\langle {\delta_s} , {T_x(\delta_t)} \rangle}}=0$ otherwise. Then clearly $T_x$ does map into $c_0(G)$, and $\|T_x\| \leq \|x\|$. Notice that for $s,t\in G$, we have $$\begin{aligned} {{\langle {\delta_s} , {T_x(\delta_t)} \rangle}} &= \begin{cases} x_n \Omega(t,g_n) &: s=g, t=hg_n^{-1}, \\ 0 &: \text{otherwise,} \end{cases} \\ &= \sum_n x_n {{\langle {\delta_s} , {(\delta_{g_n}\cdot e_{(g,h)})(\delta_t)} \rangle}}.\end{aligned}$$ Define $Q:l^\infty(\mathbb N)\rightarrow c_0(\mathbb N)$ by $${{\langle {\delta_n} , {Q(x)} \rangle}} = {{\langle {M} , {\delta_{g_n^{-1}}\cdot T_x} \rangle}} \qquad (n\in\mathbb N),$$ so that $Q$ is bounded and linear. Let $n_0\geq 1$ and let $x = e_{n_0} \in c_0(\mathbb N) \subseteq l^\infty(\mathbb N)$. Then, $T_x = \delta_{g_{n_0}} \cdot e_{(g,h)}$, so that $$\begin{aligned} {{\langle {\delta_n} , {Q(x)} \rangle}} &= {{\langle {M} , {\delta_{g_n^{-1}}\cdot T_x} \rangle}} = {{\langle {M} , {\delta_{g_n^{-1}}\cdot(\delta_{g_{n_0}}\cdot e_{(g,h)})} \rangle}} \\ &= \begin{cases} \delta\, \Omega(g_{n_0}^{-1},g_{n_0}) &: n=n_0, \\ \Omega(g_n^{-1},g_{n_0}) {{\langle {M\cdot e_{g_n^{-1}g_{n_0}}} , {e_{(g,h)}} \rangle}} &: n\not=n_0. \end{cases}\end{aligned}$$ Define $Q_1 \in{\mathcal{B}}(c_0(\mathbb N))$ by $$Q_1(x) = \big( \Omega(g_n^{-1},g_n) x_n \big)_{n\in\mathbb N} \qquad ( x=(x_n) \in c_0(\mathbb N) ).$$ Then, as each $g_n\in X_K$, $Q_1$ is an invertible operator. Let $Q_2$ be the restriction of $Q$ to $c_0(\mathbb N)$, so that $Q_2\in{\mathcal{B}}(c_0(\mathbb N))$ and $Q_2 = \delta Q_1 + \delta Q_3Q_1$ for some $Q_3\in{\mathcal{B}}(c_0(\mathbb N))$. Thus $Q_3 = \delta^{-1}Q_2Q_1^{-1} - I_{c_0(\mathbb N)}$, so that for $x\in c_0(\mathbb N)$, we have that $$\begin{aligned} \|Q_3(x)\| &= \sup_n |{{\langle {\delta_n} , {\delta^{-1}Q_2Q_1^{-1}(x) - x} \rangle}}| = \sup_n \Big| \sum_m x_m {{\langle {\delta_n} , {\delta^{-1}Q_2Q_1^{-1}(e_m) - e_m} \rangle}} \Big| \\ &= \sup_n \Big| \sum_{m\not=n} x_m \Omega(g_m^{-1},g_m)^{-1} \Omega(g_n^{-1},g_m) {{\langle {M\cdot\delta_{g_n^{-1}g_m}} , {e_{(g,h)}} \rangle}} \Big| \\ &\leq K^{-1} \sup_n \sum_{m\not=n} |x_m| 2^{-2-|m-n|} \omega(g_m)\omega(g_m^{-1}) \leq \|x\| / 2.\end{aligned}$$ Consequently $Q_3-I_{c_0(\mathbb N)}$ is invertible, so that $Q_2Q_1^{-1}$ is invertible, showing that $Q_2$ is invertible. However, this implies that $Q_2^{-1}Q:l^\infty(\mathbb N)\rightarrow c_0(\mathbb N)$ is a projection, which is a well-known contradiction, completing the proof. \[amen\_not\_inj\] Let $G$ be a countable group, let $\omega$ be a weight which is not strongly non-amenable, and let ${\mathcal{A}}=l^1(G,\omega)$. Then $c_0(G)$ is not left-injective. Suppose, towards a contradiction, that $c_0(G)$ is left-injective, so that there exists $M=M_{u_G} \in {\mathcal{B}}({\mathcal{A}},{\mathcal{A}}_*)'$ as above, with the additional condition that $$\begin{aligned} \delta_{g,h} &= {{\langle {\delta_g} , {\phi\Delta'_{{\mathcal{A}}}(e_h)} \rangle}} = {{\langle {M} , {\delta_g \cdot \Delta'_{{\mathcal{A}}}(e_h)} \rangle}} = \Omega(hg^{-1},g) {{\langle {M} , {\Delta'_{{\mathcal{A}}}(e_{hg^{-1}})} \rangle}} \\ &= \Omega(hg^{-1},g) {{\langle {M} , {\big( \delta_{st,hg^{-1}} \Omega(s,t) \big)_{(s,t)\in G\times G}} \rangle}} \qquad (g,h\in G). \end{aligned}$$ This clearly reduces to $$\delta_{g,u_G} = {{\langle {M} , {\big( \delta_{st,g} \Omega(s,t) \big)_{(s,t)\in G\times G}} \rangle}} \qquad (g\in G).$$ As $G$ is countable, we can enumerate $G$ as $G = \{ g_n : n\in\mathbb N\}$. Then, for $g_n\in G$, let $X_{g_n} = \{ g_1, \ldots, g_n \} \subseteq G$. Define $Q:l^\infty(G)\rightarrow{\mathcal{B}}({\mathcal{A}},c_0(G))$ by $${{\langle {\delta_s} , {Q(x)(\delta_t)} \rangle}} = \Omega(s,t) \sum_{g\in X_t} x_g \delta_{st,g} \qquad (s,t\in G, x\in l^\infty(G)).$$ Then, for each $t\in G$, as $X_t$ is finite, we see that $Q(x)(\delta_t)\in c_0(G)$, so $Q$ is well-defined. Clearly $Q$ is linear, and we see that for $x\in l^\infty(G)$, $$\|Q(x)\| = \sup_{s,t\in G} \Omega(s,t) \Big|\sum_{g\in X_t} x_g \delta_{st,g}\Big| \leq \sup_{s,t\in G} \sum_{\{g\in X_t : g=st\}} |x_g| = \|x\|,$$ so that $Q$ is norm-decreasing. Then, for $h\in G$, we have that $${{\langle {\delta_s} , {Q(e_h)(\delta_t)} \rangle}} = \Omega(s,t) \sum_{g\in X_t} \delta_{g,h} \delta_{st,g} = \begin{cases} {{\langle {\delta_s} , {\Delta_{{\mathcal{A}}}'(e_h)(\delta_t)} \rangle}} &: h\in X_t, \\ 0 &: h\not\in X_t. \end{cases}$$ Let $h=g_{n_0}$, so that $\{ t\in G : h\not\in X_t \} = \{ g_n\in G : h\not\in X_{g_n} \} = \{ g_1, g_2, \ldots, g_{n_0-1}\}$. We hence see that $Q(e_{g_0}) - \Delta'_{{\mathcal{A}}}(e_{g_0}) \in c_0(G\times G)$. By the preceding proposition, we hence have that $I_{c_0(G)} = \phi\circ\Delta_{{\mathcal{A}}}' = \phi\circ (Q|_{c_0(G)})$. However, this implies that $\phi\circ Q:l^\infty(G)\rightarrow c_0(G)$ is a projection onto $c_0(G)$, giving us the required contradiction. \[sg\_not\_inj\] Let $S$ be a discrete, weakly cancellative semigroup, let $\omega$ be a weight on $S$, and let ${\mathcal{A}}=l^1(S,\omega)$. When $S$ is unital, or $S$ is cancellative, $c_0(G)$ is not a bi-injective ${\mathcal{A}}$-bimodule. Suppose, towards a contradiction, that $c_0(G)$ is bi-injective. Then ${\mathcal{A}}$ is Connes-amenable, so that Theorem \[main\_thm\] implies that ${\mathcal{A}}$ is amenable, and that $S=G$ is a group. By Theorem \[Gron\_Thm\], we know that $\omega$ is not strongly non-amenable. Suppose that $G$ is countable, so that the above theorem shows that $c_0(G)$ is not left-injective, and that hence $c_0(G)$ is certainly not bi-injective, a contradiction. Suppose that $G$ is not countable. Then let $H$ be some countably infinite subgroup of $G$. Let $K = \sup\{ \omega(g)\omega(g^{-1}) : g\in G \}<\infty$, and let $g,h\in G$. Then $$\Omega(g,h) = \frac{\omega(gh)}{\omega(g)\omega(h)} = \frac{\omega(gh)}{\omega(g)\omega(g^{-1}gh)} \geq \frac{\omega(gh)}{\omega(g)\omega(g^{-1})\omega(gh)} = \frac{1}{\omega(g)\omega(g^{-1})} \geq K^{-1},$$ so that $\Omega$ is bounded below on $G\times G$, and hence on $H\times H$. Then we can find $X\subseteq G$ such that $G = \bigcup_{x\in X} Hx$ and $Hx\cap Hy = \emptyset$ for distinct $x,y\in X$. Notice that if $g\in Hx$ then $g^{-1}\in x^{-1}H$, so that $G = \bigcup_{x\in X} x^{-1}H$ as well. By the proof of Theorem \[amen\_not\_inj\], we see that $c_0(H)$ is not a left-injective $l^1(H,\omega)$-module. Suppose, towards a contradiction, that we do have some left ${\mathcal{A}}$-module homomorphism $\phi:{\mathcal{B}}(l^1(G,\omega),c_0(G)) \rightarrow c_0(G)$ with $\phi \Delta_{{\mathcal{A}}}' = I_{{\mathcal{A}}'}$. Notice that certainly ${\mathcal{B}}(l^1(G,\omega),c_0(G))$ and $c_0(G)$ are Banach left $l^1(H,\omega)$-modules, by restricting the action from $l^1(G,\omega)$. Define a map $\psi : {\mathcal{B}}(l^1(H,\omega),c_0(H)) \rightarrow {\mathcal{B}}(l^1(G,\omega),c_0(G))$ by, for $g,k\in G$, $${{\langle {\delta_g} , {\psi(T)(\delta_k)} \rangle}} = \begin{cases} \frac{\omega(s) \omega(t)}{\omega(tx) \omega(k)} {{\langle {\delta_t} , {T(\delta_s)} \rangle}} &: g=tx, k=x^{-1}s \text{ for some } x\in X, s,t\in H, \\ 0 &: \text{otherwise.} \end{cases}$$ Certainly $\psi$ is linear, while $$\|\psi(T)\| \leq \|T\| \sup_{s,t\in H, x\in X} \frac{\omega(s)\omega(t)}{\omega(tx)\omega(x^{-1}s)} \leq \|T\| \sup_{s,t\in H, x\in X} \frac{\omega(s)\omega(t)}{\omega(txx^{-1}s)} = \|T\| \sup_{s,t\in H} \Omega(t,s)^{-1},$$ so that $\psi$ is bounded. For $h,s,t\in H$, and $x\in X$, we have $$\begin{aligned} &{{\langle {\delta_{tx}} , {(\delta_h\cdot\psi(T))(\delta_{x^{-1}s})} \rangle}} = \Omega(x^{-1}s,h) {{\langle {\delta_{tx}} , {\psi(T)(\delta_{x^{-1}sh})} \rangle}} \\ &= \frac{\Omega(x^{-1}s,h) \omega(sh) \omega(t)}{\omega(tx) \omega(x^{-1}sh)} {{\langle {\delta_t} , {T(\delta_s)} \rangle}} = \frac{\omega(sh) \omega(t)}{\omega(x^{-1}s) \omega(h) \omega(tx)} {{\langle {\delta_t} , {T(\delta_s)} \rangle}} \\ &= \omega(s) \omega(x^{-1}s)^{-1} \omega(t) \omega(tx)^{-1} {{\langle {\delta_t} , {(\delta_h\cdot T)(\delta_s)} \rangle}} = {{\langle {\delta_{tx}} , {\psi(\delta_h\cdot T)(\delta_{x^{-1}s})} \rangle}}.\end{aligned}$$ Thus $\psi$ is a left $l^1(H,\omega)$-module homomorphism. For $h,s,t\in H$ and $x\in X$, we then have that $$\begin{aligned} {{\langle {\delta_{tx}} , {\psi(\Delta'_{l^1(H,\omega)}(e_h))(\delta_{x^{-1}s})} \rangle}} &= \frac{\omega(t) \omega(s)}{\omega(x^{-1}s) \omega(tx)} {{\langle {\delta_t} , {\delta_s\cdot e_h} \rangle}} = \Omega(tx,x^{-1}s) \delta_{ts,h} \\ &= {{\langle {\delta_{tx}} , {\delta_{x^{-1}s} \cdot e_h} \rangle}} = {{\langle {\delta_{tx}} , {\Delta'_{{\mathcal{A}}}(e_h)(\delta_{x^{-1}s})} \rangle}}.\end{aligned}$$ If $g,k\in G$ are such that $gk\not\in H$ then $g=tx$ and $k=y^{-1}s$ for some $s,t\in H$ and distinct $x,y\in X$. Then, for $h\in H$, we have that $gk\not=h$, so that $${{\langle {\delta_g} , {\Delta'_{{\mathcal{A}}}(e_h)(\delta_k)} \rangle}} = \Omega(g,k) \delta_{gk,h} = 0 = {{\langle {\delta_g} , {\psi(\Delta'_{l^1(H,\omega)}(e_h))(\delta_k)} \rangle}}.$$ Hence $\psi\circ \Delta'_{l^1(H,\omega)}$ is equal to $\Delta'_{{\mathcal{A}}}$ restricted to $l^1(H,\omega)$. Let $P : c_0(G) \rightarrow c_0(H)$ be the natural projection, which is obviously an $l^1(H,\omega)$-module homomorphism. Then $Q = P \circ \phi \circ \psi : {\mathcal{B}}( l^1(H,\omega), c_0(H) ) \rightarrow c_0(H)$ is a bounded left $l^1(H,\omega)$-module homomorphism, and $Q \circ \Delta'_{l^1(H,\omega)} = I_{c_o(H)}$. This contradiction completes the proof. We note that just because $\Omega$ is bounded below does not imply that $\omega$ is bounded, so that $l^1(G,\omega)$ is not necessarily isomorphic to $l^1(G)$, and hence we cannot simply apply the results of [@DP]. We have not been able to establish if $c_0(S)$ can every be a left-injective $l^1(S,\omega)$-module for some semigroup $S$ and weight $\omega$. Open questions ============== We state a few open questions of interest: 1. Let ${\mathcal{A}}$ be an Arens regular Banach algebra such that ${\mathcal{A}}''$ is Connes-amenable. Need ${\mathcal{A}}$ be amenable? 2. This is true for C$^*$-algebras. Can we find a “simple” proof? 3. Let ${\mathcal{A}}$ be a dual Banach algebra with predual ${\mathcal{A}}_*$, and suppose that ${\mathcal{A}}_*$ is bi-injective. If ${\mathcal{A}}$ necessarily a von Neumann algebra or the bidual of an Arens regular Banach algebra ${\mathcal{B}}$ such that ${\mathcal{B}}$ is an ideal in ${\mathcal{A}}$? 4. Let $S$ be a (weakly cancellative) semigroup, and let $\omega$ be a weight on $S$. Classify (up to isomorphism) the preduals of $l^1(S,\omega)$, and calculate which preduals yield a Connes-amenable Banach algebra. 5. This question was asked by Niels Gr[ø]{}nbæk. In most of our examples, it is obvious that when ${\mathcal{A}}$ is a Connes-amenable dual Banach algebra, there is ${\mathcal{B}}\subseteq{\mathcal{A}}$ which is weak$^*$-dense and amenable. Is this always true? [9]{} [<span style="font-variant:small-caps;">I.G. Craw, N.J. Young</span>, ‘Regularity of multiplication in weighted group and semigroup algebras’, *Quart. J. Math. Oxford* 25 (1974) 351–358.]{} [<span style="font-variant:small-caps;">H.G. Dales</span>, *Banach algebras and automatic continuity*, (Clarendon Press, Oxford, 2000).]{} [<span style="font-variant:small-caps;">H.G. Dales, A.T.-M. Lau</span>, ‘The second dual of Beurling algebras’, preprint.]{} [<span style="font-variant:small-caps;">H.G. Dales, M.E. Polyakov</span>, ‘Homological properties of modules over group algebras’, *Proc. Lon. Math. Soc.* 89 (2004) 390–426.]{} [<span style="font-variant:small-caps;">N. Gr[ø]{}nbæk</span>, ‘Amenability of weighted discrete convolution algebras on cancellative semigroups’, *Proc. Roy. Soc. Edinburgh Sect. A* 110 (1988) 351–360.]{} [<span style="font-variant:small-caps;">N. Gr[ø]{}nbæk</span>, ‘Amenability of discrete convolution algebras, the commutative case’, *Pacific J. Math.* 143 (1990) 243–249.]{} [<span style="font-variant:small-caps;">A.Ya. Helemskii</span>, ‘Homological essence of amenability in the sense of A. Connes: the injectivity of the predual bimodule.’, *Math. USSR-Sb.* 68 (1991) 555–566.]{} [<span style="font-variant:small-caps;">A.Ya. Helemskii</span>, *Banach and locally convex algberas*, (Oxford Science Publications, New York, 1993).]{} [<span style="font-variant:small-caps;">A.Ya. Helemskii</span>, ‘Some aspects of topological homology since 1995: a survey’, *‘Banach algebras and their applications’ in Contemp. Math.* 363 (2004) 145–179.]{} [<span style="font-variant:small-caps;">V. Runde</span>, ‘Dual Banach algebras: Connes-amenability, normal, virtual diagonals, and injectivity of the predual bimodule’, *Math. Scand.* 95 (2004) 124–144.]{} [<span style="font-variant:small-caps;">V. Runde</span>, ‘Connes-amenability and normal, virtual diagonals for measure algebras. I.’, *J. London Math. Soc.* 67 (2003) 643–656.]{} [<span style="font-variant:small-caps;">V. Runde</span>, ‘Connes-amenability and normal, virtual diagonals for measure algebras, II’, *Bull. Austral. Math. Soc.* 68 (2003) 325–328.]{} [<span style="font-variant:small-caps;">V. Runde</span>, *Lectures on amenability*, (Springer-Verlag, Berlin, 2002).]{} [<span style="font-variant:small-caps;">V. Runde</span>, ‘Amenability for dual Banach algebras’, *Studia Math.* 148 (2001) 47–66.]{} [<span style="font-variant:small-caps;">S.B. Tabaldyev</span>, ‘Noninjectivity of the predual bimodule of the measure algebra of infinite discrete groups.’, *Math. Notes* 73 (2003) 690–696.]{} St. John’s College,\ Oxford\ OX1 3JP\ United Kingdom `[email protected]`

--- abstract: 'This paper introduces a new methodology for the complexity analysis of higher-order functional programs, which is based on three components: a powerful type system for size analysis and a sound type inference procedure for it, a ticking monadic transformation and a concrete tool for constraint solving. Noticeably, the presented methodology can be fully automated, and is able to analyse a series of examples which cannot be handled by most competitor methodologies. This is possible due to various key ingredients, and in particular an abstract index language and index polymorphism at higher ranks. A prototype implementation is available.' author: - Martin Avanzini - Ugo Dal Lago bibliography: - 'references.bib' title: | Automating Sized-Type Inference\ and Complexity Analysis --- Introduction ============ One successful approach to automatic verification of termination properties of higher-order functional programs is based on *sized types* [@HPS:POPL:96]. In sized types, a type carries not only some information about the *kind* of each object, but also about its *size*, hence the name. This information is then exploited when requiring that recursive calls are done on arguments of *strictly smaller* size. Estimating the size of intermediate results is crucial for complexity analysis, but up to now, the only attempt of using sized types for complexity analysis is due to Vasconcelos [@Vasconcelos:Diss:08], and confined to space complexity. If one wants to be sound for time analysis, size types need to be further refined, e.g., by turning them into linear dependently types [@LG:LMCS:11]. Since the first inception in the seminal paper of Hughes et. al. [@HPS:POPL:96] the literature on sized typed has grown to a considerable extend. Indeed, various significantly more expressive systems have been introduced, with the main aim to improve the expressiveness in the context of termination analysis. For instance, Blanqui [@Blanqui:CSL:05] introduced a novel sized type system on top of the *calculus of algebraic construction*. Notably, it has been shown that for size indices over the successor algebra, type checking is decidable [@Blanqui:CSL:05]. The system is thus capable of expressing additive relations between sizes. In the context of termination analysis, where one would like to statically detect that a recursion parameter decreases in size, this is sufficient. In this line of research falls also more recent work of Abel and Pientka [@AP:JFP:16], where a novel sized type system for termination analysis on top of $\mathsf{F}_\omega$ is proposed. Noteworthy, this system has been integrated in the dependently typed language .[^1] As we will see, capturing only additive relations between value sizes is not enough for our purpose. On the other hand, even slight extensions to the size index language render current methods for type inference, even type checking, intractable. In this paper, we thus take a fresh look at sized-type systems, with a particular emphasis on a richer index language and feasible automation on existing constraint solving technology. Our system exhibits many similarities with the archetypal system from [@HPS:POPL:96], which itself is based on a Hindley-Milner style system. Although conceptually simple, our system is substantially more expressive than the traditional one. This is possible mainly due to the addition of one ingredient, viz, the presence of *arbitrary rank index polymorphism*. That is, functions that take functions as their argument can be polymorphic in their size annotation. Of course, our sized type system is proven a sound methodology for *size* analysis. In contrast to existing works, one can also device an inference machinery that is sound and (relative) complete. Finally, this system system is amenable to time complexity analysis by a ticking monadic transformation. A prototype implementation is available, see below for more details. More specifically, our contributions can be summarized as follows: We show that size types can be generalised so as to encompass a notion of index polymorphism, in which (higher-order subtypes of) the underlying type can be universally quantified. This allows for a more flexible treatment of higher-order functions. Noticeably, this is shown to preserve soundness (i.e. subject reduction), the minimal property one expects from such a type system. On the one hand, this is enough to be sure that types reflect the size of the underlying program. On the other hand, termination is not enforced anymore by the type system, contrarily to, e.g. [@Blanqui:CSL:05; @AP:JFP:16]. In particular, we do not require that recursive calls are made on arguments of smaller size. The polymorphic sized types system, by itself, does not guarantee any complexity-theoretic property on the typed program, except for the *size* of the output being bounded by a function on the size of the input, itself readable from the type. Complexity analysis of a program $\progone$ can however be seen as a size analysis of another program $\tprogone$ which computes not only $\progone$, but its complexity. This transformation, called the *ticking transformation*, has already been studied in similar settings [@DLR:ICFP:15]. Contrarily to many papers from the literature, we have taken care not only of constraint *inference*, but also of constraint *solving*. This has been done by building a prototype called which implements type inference and ticking, and then relies on an external tool, called , to check the generated constraints for satisfiability. borrows heavily from the advances made over the last decade in the synthesis of *polynomial interpretations*, a form of polynomial ranking function, by the rewriting community. It features also some novel aspects, most importantly, a bottom-up SCC analysis for incremental constraint solving. We thus arrive at a fully automated runtime analysis of higher-order functional programs. Noteworthy, we are able to effectively infer polynomial, not necessarily linear, bounds on the runtime of programs. Both tools are open source and available from the first authors homepage. [^2]  is able to analyse, fully automatically, a series of examples which cannot be handled by most competitor methodologies. Indeed, it is to our best knowledge up until today the only approach that can fully deal with function closures whose complexity depends on the captured environment, compare for instance the very recent work of Hoffmann et. al. [@HDW:POPL:17]. Dealing with such closures is of crucial importance, e.g., when passing partially applied functions to higher-order combinators, a feature pervasively used in functional programming. For brevity, we only give a formalisation of our system and state the central theorem here. An extended version, including all the technical details is available online [@EV]. Our Type System at a Glance {#sect:ERW} =========================== Applicative Programs and Simple Types {#sect:APST} ===================================== Sized Types and Their Soundness {#sect:STS} =============================== Conclusions {#sect:C} =========== [^1]: See <http://wiki.portal.chalmers.se/agda>. [^2]: See <https://cl-informatik.uibk.ac.at/users/zini/software>.

--- abstract: 'We study electronic structure of hole- and electron-doped Mott insulators in the two-dimensional Hubbard model to reach a unified picture for the normal state of cuprate high-$T_{\text{c}}$ superconductors. By using a cluster extension of the dynamical mean-field theory, we demonstrate that structure of coexisting zeros and poles of the single-particle Green’s function holds the key to understand Mott physics in the underdoped region. We show evidence for the emergence of non-Fermi-liquid phase caused by the topological quantum phase transition of Fermi surface by analyzing low-energy charge dynamics. The spectra calculated in a wide range of energy and momentum reproduce various anomalous properties observed in experiments for the high-$T_{\text{c}}$ cuprates. Our results reveal that the pseudogap in hole-doped cuprates has a $d$-wave-like structure only below the Fermi level, while it retains non-$d$-wave structure with a fully opened gap above the Fermi energy even in the nodal direction due to a zero surface extending over the entire Brillouin zone. In addition to the non-$d$-wave pseudogap, the present comprehensive identifications of the spectral asymmetry as to the Fermi energy, the Fermi arc, and the back-bending behavior of the dispersion, waterfall, and low-energy kink, in agreement with the experimental anomalies of the cuprates, do not support that these originate from (the precursors of) symmetry breakings such as the preformed pairing and the $d$-density wave fluctuations, but support that they are direct consequences of the proximity to the Mott insulator. Several possible experiments are further proposed to prove or disprove our zero mechanism.' author: - 'Shiro Sakai,$^{1,2}$ Yukitoshi Motome,$^2$ and Masatoshi Imada$^2$' title: | Doped high-$T_{\text{c}}$ cuprate superconductors\ elucidated in the light of zeros and poles of electronic Green’s function --- INTRODUCTION ============ Anomalous behaviors of high-$T_{\text{c}}$ cuprates observed in the normal metallic state above $T_{\text{c}}$ hold the key not only to understanding the mechanism of the superconductivity but also to a possible manifestation of an unexplored metallic phase distinguished from the Fermi liquid.[@ts99] Extraordinary electronic structure is induced by a small density of carrier doping into the Mott insulator. Angle-resolved photoemission spectroscopies (ARPES) have in fact revealed detailed anomalies of the normal-state spectra, such as momentum-dependent excitation gap (pseudogap), a truncated Fermi surface (Fermi arc), and kinks in the dispersion.[@dh03] Toward the understanding of the anomalous metals, especially the pseudogap formation, many theoretical proposals have been made so far.[@yj03] The proposals include a Cooper paring without phase coherence,[@ek95] and hidden orders or its fluctuations competing with the superconductivity, such as antiferromagnetism,[@ks90; @p97; @vt97] charge or stripe orderings,[@kb03] and $d$-density wave.[@v97; @cl01] Mechanisms attributing the origin of the pseudogap to a direct consequence of the proximity to the Mott insulator have also been proposed.[@rice05; @sp03; @sk06; @p09; @sm09] Among the theoretical efforts, recent development of the dynamical mean-field theory (DMFT) [@gk96] and its cluster extensions [@ks01; @mj05] has enabled studies on dynamics of microscopic models without any [*ad hoc*]{} approximation. In particular, studies on the two-dimensional (2D) Hubbard model using the cluster DMFT (CDMFT) have offered many useful insights into the electronic structure of cuprates, by identifying the pseudogap, Fermi arc,[@mp02; @st04; @sm09; @cc05; @kk06; @sk06; @mj06; @aa06; @hk07; @lt09; @fc09] and high-energy kink [@mj07] in the calculated spectra. The CDMFT is specifically suited and powerful for this problem because of its nonperturbative framework, namely, it is based on neither weak nor strong coupling expansions. Furthermore, it takes account of short-range spatial correlations within a cluster explicitly. These are big advantages in exploring momentum-resolved dynamics in the intermediate coupling region, which is relevant to physics of the cuprates. In particular, recent CDMFT studies on doped Mott insulators have revealed emergence of non-Fermi-liquid phases, characterized by unexpected coexistence of two singularities at the Fermi level, one characteristic to the weak-coupling and the other to the strong-coupling regions.[@sk06; @sm09; @sm09-2; @lt09; @wg09] The singularity characterizing weakly interacting metals, namely, Fermi liquids, is a pole of the single-particle Green’s function $G$. Energy dependence of its locus in the momentum space determines band dispersions, particularly, the Fermi surface at the Fermi level. On the other hand, a crucial singularity in the strong coupling region is a zero of $G$. The situations are illustrated in Fig. \[fig:zero\], which schematically shows how the real part of $G$ changes the sign in the energy-momentum space. Because Re$G$ must be positive at high energy while negative at low energy, it has to change its sign at least once between these two regions. The sign change also has to take place in the Brillouin zone at the Fermi level, say, between the zone center and the zone boundary, if the dispersion far from the Fermi level ensures positive Re$G$ in some part of momenta and negative Re$G$ in the other part. In the normal metals, the sign change occurs at the poles of $G$ (i.e., the band dispersions), where Re$G$ goes to $-\infty$ on the lower side of the band and comes back from $+\infty$ on the upper side \[Fig. \[fig:zero\](a)\]. On the other hand, what happens when correlation effects induce a gap in the band at the Fermi level? Because of the absence of poles inside the gap by definition, there is no way for Re$G$ to change its sign at the Fermi level except for getting through zeros \[Fig. \[fig:zero\](b)\].[@d03; @sp07] Re$G=0$ means the divergence of the self-energy, so that a zero of $G$ is a singular point of the self-energy. The non-Fermi liquids discovered in CDMFT show a coexistence of these two characteristics — poles and zeros of $G$ at the Fermi level.[@sk06; @sm09] In a lightly hole-doped region, the poles form a hole-pocket Fermi surface around the nodal direction \[from $\Vec{k} = (0,0)$ to $(\pi,\pi)$\], while the zeros form a surface enclosing $(\pi,\pi)$. It has been suggested that the Fermi arc emerges because of such peculiar electronic structure: The pocket loses the spectral intensity on the side closer to the zero surface, leaving the other side as an arc.[@sk06; @sm09; @sm09-2] Similar mechanisms were proposed, based on an assumed functional form of Green’s functions [@kr06; @ks06] or a weakly-coupled chain model.[@et02; @bg06] The pseudogap is also characterized by the same zero surface crossing the Fermi level. In the previous paper [@sm09] we clarified the structure of poles and zeros in the whole energy-momentum space for hole-doped and undoped Mott insulators. We thus proposed a unified picture for understanding various puzzling features of Mott physics in the light of reconstructions of pole-zero structure under progressive doping from the Mott insulator to the Fermi liquid in the overdoped region. The aim of the present paper is to show how and to what extent we can understand experimental findings within the ‘pole-zero mechanism’. We implement CDMFT calculations at zero and finite temperatures, clarifying the relationship between zero-temperature pole-zero structure and the spectra observed in experiments at finite temperatures. We find that a number of non-Fermi-liquid aspects, i.e., spectral asymmetry as to the Fermi level, back-bending or incoherent feature of the dispersion, high- and low-energy kinks, and Fermi arc or pockets in the hole- and electron-doped cuprates, are comprehensively understood within the pole-zero mechanism. In addition to a coherent picture for various experiments, our numerical results predict a distinctive feature of the pseudogap: The numerical data support a fully opened gap above the Fermi level even in the nodal direction, which is incompatible with scenarios on the basis of the $d$-wave gap in the zero-temperature limit. Our result offers a mechanism distinct from the scenarios based on the preformed pairing or $d$-density-wave order. The paper is organized as follows. In Sec. \[sec:method\] we introduce the 2D Hubbard model and present the central ideas of the CDMFT. We also describe the essence of numerical solvers for the effective cluster problem that we used in the present study. The numerical results are presented and discussed in comparison with various experiments in Sec. \[sec:result\]. From the low-energy pole-zero structure, we derive a simple interpretation of several unusual features in hole-doped cuprates (Sec. \[ssec:nond\]-\[ssec:back\]). We also find several anomalies in the band dispersion, which are consistent with ARPES data (Sec. \[ssec:edc\]-\[ssec:kink\]). In Sec. \[ssec:arc\] we discuss the results of the Fermi surface in comparison with the recent ARPES observation of the hole pocket. The comparison is further made in Sec. \[ssec:tp\] by changing the next-nearest-neighbor hopping to reach more quantitative understanding of the experimental results. In Sec. \[ssec:tp\], \[ssec:ele2\], and \[ssec:ele\], we extend our theory to electron-doped cuprates and find good agreements with ARPES data. We summarize our results and make concluding remarks in Sec. \[sec:summary\]. MODEL AND METHOD {#sec:method} ================ As a simplest model for high-$T_{\text{c}}$ cuprates we take the Hubbard Hamiltonian, $$\begin{aligned} H= \sum_{\Vec{k}\s}\e(\Vec{k})c_{\Vec{k}\s}^\dagger c_{\Vec{k}\s} -\mu \sum_{i\s}n_{i\s}+ U\sum_{i}n_{i\ua}n_{i\da}, \label{eq:hubbard}\end{aligned}$$ on a square lattice. Here $c_{\Vec{k}\s}$ $(c_{\Vec{k}\s}^\dagger)$ annihilates (creates) an electron of spin $\s$ with momentum $\Vec{k} = (k_x, k_y)$, $c_{i\s}$ $(c_{i\s}^\dagger)$ is its Fourier component at site $i$, and $n_{i\s}\equiv c_{i\s}^\dagger c_{i\s}$. $U$ represents the onsite Coulomb repulsion, $\mu$ the chemical potential, and $$\begin{aligned} \e(\Vec{k})\equiv -2t(\cos k_x + \cos k_y) -4t' \cos k_x\cos k_y, \label{eq:disp}\end{aligned}$$ where $t$ $(t')$ is the (next-)nearest-neighbor transfer integral. Based on a first principles calculation, the value of $t$ was estimated to be $\sim 0.4$eV for La$_2$CuO$_4$.[@hs90] $-t'/t$ is considered to be $\sim 0.2$ and $\sim 0.4$ for La$_{2-x}$Sr$_x$CuO$_4$ and Bi$_2$Sr$_2$CaCu$_2$O$_{8+\d}$, respectively. We adopt $U=8t$ or $12t$, which are realistic values for the cuprates and indeed reproduce the Mott insulating state for undoped case. In the CDMFT [@ks01] we map the system (\[eq:hubbard\]) onto a model consisting of an $N_{\text{c}}$-site cluster C and bath degrees of freedom B. The bath is determined in a self-consistent way to provide an $N_{\text{c}}\times N_{\text{c}}$ dynamical mean-field matrix $\hat{g}_0(i\w_n)$ at temperature $T(\equiv 1/\b)$, where $\w_n\equiv (2n+1)\pi T$. After the self-consistency loop converges, we calculate a quantity $Q^{\text{L}}$ defined on the original lattice from those on the cluster, $Q^{\text{C}}$.[@ks01] This periodization procedure is based on the Fourier transformation truncated by the cluster size $N_{\text{c}}$, $$\begin{aligned} Q^{\text{L}}(\Vec{k})=\frac{1}{N_{\text{c}}}\sum_{ij\in \text{C}} [Q^{\text{C}}]_{ij}e^{i\Vec{k}\cdot\Vec{r}_{ij}}, \label{eq:periodize}\end{aligned}$$ where $\Vec{k}$ is defined on the entire Brillouin zone of the original lattice and $\Vec{r}_{ij}$ is the real-space vector connecting two cluster sites $i$ and $j$. The truncation by a small $N_{\text{c}}$ gives a good approximation to the thermodynamic limit of $N_{\text{c}} \to \infty$ if $Q^{\text{C}}$ is well localized within the cluster. In reality it is difficult to find a quantity which is short ranged in the entire parameter range of the interaction strength and the doping concentration. Therefore we need to choose an appropriate quantity to periodize, according to situations. For example, $Q=\S$ ($\S$: self-energy) is a good choice for weakly interacting normal metals,[@ks01] but it becomes highly nonlocal and long ranged in the strong coupling region (e.g., in the Mott insulator). This is due to an appearance of zeros of $G$, i.e., poles of $\S$ in the momentum space (see APPENDIX A for further detail). On the other hand, in the Mott insulator $Q=G$ is more appropriate because it is nearly local in the strong coupling regime.[@kk06] Another choice for the periodization is the cumulant $$M=[i\w_n+\mu-\S]^{-1}.$$ It was pointed out that the cumulant periodization works well in a wide range of $U$ including the strong coupling, because it is similar in the functional form to the atomic Green function.[@sk06] Moreover, the periodization by the cumulant has an important feature, i.e., it can describe both poles and zeros of $G$ at the same time, while the periodization by using $\S$ ($G$) describes only poles (zeros). This opens up the intriguing possibility of exploring the coexistence of poles and zeros at the Fermi level, as substantiation of anomalous metals. In the present study, we adopt the cumulant periodization scheme, $Q=M$, to investigate [*metals in the vicinity of the Mott insulator*]{}. In fact, $M$ is highly local even in the doped metallic states. In APPENDIX B we demonstrate this local nature by CDMFT calculations for $N_{\text{c}}=4\times4$ cluster; we find that $M^{\text{C}}$ is nearly localized already within the inner $2\times 2$ cluster for a parameter region relevant to the present study. After obtaining the lattice cumulant $M^{\text{L}}$ through Eq. (\[eq:periodize\]), we calculate the self-energy $\S^{\text{L}}$, Green’s function $G^{\text{L}}$, and spectral function $A^{\text{L}}$ on the original lattice with $$\begin{aligned} \S^{\text{L}}(\Vec{k},\w)\equiv \left[\w+\mu-{M^{\text{L}}}^{-1}(\Vec{k},\w)\right]^{-1},\nonumber\\ G^{\text{L}}(\Vec{k},\w)\equiv \left[\w+\mu-\e(\Vec{k})-\S^{\text{L}}(\Vec{k},\w)\right]^{-1},\end{aligned}$$ and $$\begin{aligned} A^{\text{L}}(\Vec{k},\w)\equiv -\frac{1}{\pi}\text{Im}G^{\text{L}}(\Vec{k},\w).\end{aligned}$$ In the following calculations, we employ an $N_{\text{c}}=2\times 2$ cluster, and concentrate on the paramagnetic metallic solution. We numerically solve the effective cluster problem by means of the exact diagonalization (ED) method at $T=0$ and the continuous-time quantum Monte Carlo (CTQMC) method at $T>0$, as we elaborate below. We hereafter omit the superscript L in $\S^{\text{L}}$, $G^{\text{L}}$ and $A^{\text{L}}$. Exact diagonalization method {#ssec:ed} ---------------------------- Although the pseudogap state in cuprates is experimentally detectable only above $T_{\text{c}}$, it is still significant to elucidate the nature of the pseudogap state in the zero temperature limit, by assuming the paramagnetic metal for the doped Mott insulator. This is a circumstance similar to the normal metal, where the concept of the Fermi liquid justified only in the zero temperature limit in the strict sense has proven to be fruitful. For this purpose of clarifying the zero temperature limit, we employ the Lanczos ED method [@l50] for the cluster problem. In this scheme we take a large but finite value of pseudo inverse temperature $\b'$ ($=100/t$ or $200/t$ throughout the paper), which practically represents the ground state with an energy resolution corresponding to $1/\b'$. We then represent the dynamical mean field $\hat{g}_0$ with a finite number $N_{\text{B}}$ of bath degrees of freedom, which constitute together with C sites the effective Hamiltonian to be diagonalized. We take $N_{\text{B}}=8$ throughout the paper. The optimization of B sites is done by minimizing the distance function defined by $$\begin{aligned} d\equiv \sum_{ij\in \text{C}}\sum_n \left|\left[\hat{g}_0(i\w_n)\right]_{ij} -\left[\hat{g}_{0,N_{\text{B}}}(i\w_n)\right]_{ij}\right|^2 e^{-\w_n/t}, \label{eq:distance}\end{aligned}$$ where $\hat{g}_{0,N_{\text{B}}}$ is the non-interacting Green’s function for the effective Hamiltonian and we have introduced the exponential weight factor $e^{-\w_n/t}$ with $\w_n\equiv (2n+1)\pi/\b'$ to reproduce more precisely the important low-frequency part. We have examined several other types of distance functions and confirmed that qualitative feature of the results obtained in this paper does not depend on the choice. An advantage of the Lanczos method is that Green’s function is obtained as a function of real frequency $\w$ by the continued-fraction expansion, $$\begin{aligned} G(\w)&=\frac{\langle 0| c c^\dagger | 0\rangle} {\w+i\eta-a_0^>-\displaystyle{ \frac{{b_1^>}^2}{\w+i\eta-a_1^> - \displaystyle{\frac{{b_2^>}^2}{\w+i\eta-a_2^>-\cdots}}}}}\nonumber\\ &+\frac{\langle 0| c^\dagger c | 0\rangle} {\w+i\eta-a_0^<-\displaystyle{\frac{{b_1^<}^2}{\w+i\eta-a_1^< -\displaystyle{\frac{{b_2^<}^2}{\w+i\eta-a_2^<-\cdots}}}}}, \label{eq:cf}\end{aligned}$$ where $|0\rangle$ is the ground-state vector and the coefficients $a_i^{>,<}$ and $b_i^{>,<}$ are the elements of the tridiagonal matrix appearing in the Lanczos algorithm.[@gb87] We take account of up to 2000th order in the expansion. A small positive $\eta$ is introduced to satisfy the causality. In principle, $\eta$ is taken to be infinitesimal, but in practice, it is useful to consider $\eta$ as a parameter, which serves as a resolution in energy or mimics an infinite-size effect not incorporated into the ED calculation. In the following study, we pursue a further benefit of the parameter $\eta$: We use $\eta$ as a mimic of the source of incoherence in the electronic structure, such as thermal or impurity scattering. This is based on the observation that $\eta$ does not substantially change the location of poles and zeros of $G$ but changes only the sharpness of these singularities.[@sm09; @sm09-2] Indeed, as long as $\eta$ is sufficiently smaller than the typical energy scale of the system, the location of the poles and zeros is virtually determined only by $a_i^{>,<}$ and $b_i^{>,<}$ in Eq. (\[eq:cf\]), which are calculated in the self-consistency loop performed on the Matsubara-frequency axis and independent from $\eta$. Thus, $\eta$ provides an opportunity to get insight into how the electronic structure at finite temperatures evolves from the pole-zero structure at zero temperature. The effect of $\eta$ is confirmed by a direct comparison with the finite-temperature results obtained by QMC introduced below. Note that this smearing technique by $\eta$ is very useful because it is difficult for QMC to obtain the precise electronic structure at real frequencies because it requires an analytic continuation of the numerical data. We use the smearing technique in Sec. \[sec:result\] to compare the CDMFT+ED results with ARPES ones. Quantum Monte Carlo method {#ssec:ctqmc} -------------------------- In order to discuss thermal effects directly, we implement QMC calculations for the effective cluster problem. Since the scheme takes into account infinite bath degrees of freedom, the results complement the limitation in the bath size in the CDMFT+ED results as well. We adopt the algorithm based on a weak-coupling series expansion and auxiliary-field transformation. The idea was first proposed by Rombouts [*et al.*]{},[@rh99] applied to DMFT by Sakai [*et al.*]{},[@sa06] and recently formulated in a sophisticated fashion by Gull [*et al.*]{}[@gw08], which enabled a study at sufficiently low temperatures. In the QMC, we first expand the many-body partition function in the interaction part of the Hamiltonian and apply the Hubbard-Stratonovich decoupling [@rh98; @h83] to the interaction part. This decomposes the many-body partition function into a sum of single-particle systems, which we collect. While the algorithm is based on the series expansion up to infinite order, it is feasible to obtain a numerically exact result because, after numerical convergence, all the orders in the expansion are virtually taken into account.[@rh99] We implement the CTQMC sampling for the auxiliary fields, employing the updating algorithm proposed in Ref. . We typically take $2\times10^5$ steps for each QMC simulation, and after convergence in the self-consistency loop, we average over 30 data starting from $g_0$’s which are different within a statistical error bar. RESULT AND DISCUSSION {#sec:result} ===================== Non-[*d*]{}-wave pseudogap {#ssec:nond} -------------------------- First of all, we make a remark on the structure of pseudogap obtained in Fig. \[fig:fig2\](a) in Ref.  \[reproduced in Fig. \[fig:fig2\](a)\], which depicts the low-energy pole and zero surfaces calculated by CDMFT+ED for 9% hole doping to the Mott insulator at $U=8t$ and $t'=0$. The pseudogap is formed by the zero surface (red) connecting two separated pole surfaces (green). Here we define the $\Vec{k}$-dependent amplitude $\D(\Vec{k})$ of the pseudogap as the energy difference between the upper and the lower poles at each $\Vec{k}$. Then we see in Figs. \[fig:fig2\](a) and (c) that the gap in the direct transition opens in the entire Brillouin zone, though it is somewhat larger in the antinodal region \[$\D((\pi,0))=0.35t$\] than in the nodal region \[$\D(\Vec{k}_{\text{F1}}\equiv(0.55\pi,0.55\pi))=0.31t$\]. This is one of the central results of this paper, and clearly different from previous scenarios of pseudogap which assume a $d$-wave gap above $T_{\text{c}}$, such as preformed pair,[@ek95] [*d*]{}-density-wave,[@v97; @cl01] resonating valence-bond,[@rice05; @kr06] and nodal-liquid theory,[@bf98; @ft01] since the direct transition gap closes in the nodal direction in these scenarios. Nevertheless the electronic structure in Fig. \[fig:fig2\](a), which was calculated without any assumption on the gap structure for the microscopic model, is consistent with the ARPES data, as described below. To start with, it should be noted that ARPES observes only the spectra below the Fermi energy $E_{\text{F}}(\equiv 0)$ if $T$ is low, so that the gap amplitude is often estimated by symmetrizing the spectra below and above $E_{\text{F}}$.[@nd98] This is a misleading, artificial procedure because it assumes a symmetric structure of the gap as to the Fermi energy and neglects the structure above $E_{\text{F}}$ if any. Suppose we have the result in Fig. \[fig:fig2\](a) only below $E_{\text{F}}$ and symmetrize it as in the ARPES procedure to estimate the pseudogap, we end up with a $d$-wave like gap since the gap in the part below $E_{\text{F}}$ is larger in the antinodal region while it is smaller or even zero in the nodal region \[this is more clearly seen in Fig. \[fig:fig2\](c) where we plot Re$G^{-1}(\Vec{k},\w)$ against energy and momentum along symmetric lines\]. The situation is schematically shown in Fig. \[fig:fig2\](b). Our results suggest that the “$d$-wave structure" is an artifact of the symmetrizing analysis, and in reality, the pseudogap has a non-$d$-wave (namely, full-gap) structure. In APPENDIX C we confirm that the gap in the nodal direction persists for a larger cluster, by implementing an $N_\text{c}=8$ CDMFT+CTQMC calculation at a low temperature. It is interesting to examine this interpretation experimentally. Our theory predicts that a gap exists above $E_{\text{F}}$ even in the nodal direction if the paramagnetic metal persists down to zero temperature. This means that the symmetrization [@nd98] used in ARPES breaks down in the pseudogap state due to the spectral asymmetry as to the Fermi energy. This may be examined by spectroscopic or scattering probes to observe unoccupied electronic states above the Fermi level, such as the inverse photoemission spectroscopy (IPES). It is highly desired to reveal the pseudogap structure by combining both PES and IPES without any symmetrization procedure. Other possible experimental probes suited for this purpose may be electron energy loss spectroscopy, resonant inelastic X-ray scattering, and time-resolved photoemission. In addition, another possible study is ARPES on electron-doped cuprates. When we interpret the result with the electron-hole transformation, it provides information on the spectra of unoccupied states in a hole-doped system. We will discuss this in Sec. \[ssec:ele2\] and \[ssec:ele\], comparing the results with existing ARPES data on electron-doped cuprates. Spectral symmetry and asymmetry around the Fermi energy {#ssec:eh} ------------------------------------------------------- Although a thorough comparison of the calculated pole-zero structure with experiments is not possible at present because of the lack of the experimental spectra above $E_{\text{F}}$, it is still significant to make comparison with available experimental data which partly elucidated them. The scanning tunnelling microscopy (STM) [@hl04] and the ARPES [@yr08] reported such data, where they found an electron-hole asymmetry of the spectra around $E_{\text{F}}$. The asymmetry provides a clue to the mechanism of the pseudogap, especially to the relation between the preformed pairing and the pseudogap, because the pairing will lead to a symmetric spectrum as established in the Bardeen-Cooper-Schrieffer (BCS) theory.[@bc57] Figure \[fig:asym\](a) shows the low-energy part of the density of states (DOS) for $U=8t$, $t'=0$, and $n=0.94$ and for $U=8t$, $t'=-0.4t$, and $n=0.95$. The data show that the weight of the low-energy occupied states is significantly larger than that of unoccupied ones, in accord with other CDMFT studies.[@sk06; @kk06] The asymmetry is consistent with the spatially averaged electron-tunnelling spectra \[Fig. \[fig:asym\](b), reproduced from Fig. 1(c) in Ref. \] measured by the STM for lightly hole-doped cuprates, Na$_{0.12}$Ca$_{1.88}$CuO$_2$Cl$_2$ and Bi$_2$Sr$_2$CaCu$_2$O$_{8+\d}$, where the observed probability of electron extraction is considerably greater than that of injection. Moreover, the numerical data show a peak around $\w=0.3t(0.4t)$ for $t'=0(-0.4t)$, which roughly agrees with the peak injection energies (100-300meV) seen in Fig. \[fig:asym\](b). We note that the numerical data have the minimum above the Fermi level while the STM spectra have a V-shape gap with the minimum at zero bias. The V-shape gap might be attributed to the soft Coulomb gap [@es75] or soft Hubbard gap [@si09] caused by an interplay of electron correlations and randomness, as indicated by the strong charge inhomogeneity observed by the STM at surfaces.[@hl04] ARPES allows a more detailed comparison of the spectra. Yang [*et al.*]{} [@yr08] succeeded in deriving low-energy ($\w\lesssim 0.03$eV) spectra above the Fermi level by carefully analyzing the ARPES data for Bi$_2$Sr$_2$CaCu$_2$O$_{8+\d}$. This was done by using the fact that ARPES data at finite temperatures contains information of unoccupied states because of the smeared tail of the Fermi distribution function for $\w > 0$. In the superconducting state they observed electron-hole symmetric spectra both in the nodal and antinodal region, in accordance with the BCS spectral function.[@bc57] Meanwhile in the pseudogap state they found that (i) around the antinode the spectrum is nearly symmetric with intense peaks below and above $E_{\text{F}}$ separated by a gap of $\sim 0.06$eV, and that (ii) as approaching the node the peak below $E_{\text{F}}$ goes up and eventually crosses $E_{\text{F}}$ while the peak above $E_{\text{F}}$ disappears from the measured energy range. While (ii) clearly shows the asymmetry as to $E_{\text{F}}$, Yang [*et al.*]{} [@yr08] interpreted (i) as an evidence of preformed pairing. Liebsch and Tong [@lt09] obtained an asymmetry similar to (ii) with the CDMFT. Nevertheless it is still worthwhile to see if the symmetry (i) can be reproduced since it is relevant to the mechanism of the pseudogap. Figure \[fig:asym\](c) shows the normal-state spectral function $A(\Vec{k},\w)$, calculated with the CDMFT+ED using $\eta=0.03t$. We find that around the antinode, the two intense peaks reside nearly symmetric with opening of a gap. As approaching to nodal region, the gap below the Fermi level monotonically decreases and vanishes when the lower peak reaches the Fermi level \[right panel of Fig. \[fig:asym\](c); see also Fig. \[fig:fig2\](a)\]. These behaviors are consistent with the experimental observations (i) and (ii). We note that in the $t'=0$ case \[Fig. \[fig:fig2\](c)\] the gap at $(\pi,0)$ above $E_{\text{F}}$ is larger than that below $E_{\text{F}}$, whereas the spectrum is more symmetric for $t'=-0.4t$ \[Fig. \[fig:asym\](c)\], which is more appropriate for Bi$_2$Sr$_2$CaCu$_2$O$_{8+\d}$. In the nodal direction, the gap lies above $\w\simeq 0.1t$. This is again consistent with the ARPES [@yr08] which saw the region below $0.03\text{eV} \sim 0.08t$ and observed no gap in this direction. We note that this is distinct from the picture in Ref. , where the gap was assumed to close in the nodal direction. Thus we have shown that the ARPES data [@yr08] does not necessarily indicate a preformed pairing, but is rather naturally interpreted as a consequence of the dispersive zero surface. Back-bending behavior of dispersion {#ssec:back} ----------------------------------- Another important aspect captured in Fig. \[fig:fig2\](a) is the back-bending behavior of the band cutting the Fermi level. Namely, the pole surface, which monotonically goes up to $(\pi,\pi)$ in the bare dispersion, is bent back below $E_{\text{F}}$ around $(\pi,\pi)$. This is more clearly seen in Fig. \[fig:back\](a), which is an enlarged view of Fig. \[fig:fig2\](a) in a lower energy range. The back bending can be seen below $E_{\text{F}}$ around the antinodes while it is above $E_{\text{F}}$ around the nodal direction. Actually the ARPES [@kc08] observed a similar behavior. Figure \[fig:back\](b) is a reproduction of Fig. 2(d) in Ref. , where the back-bending behavior is seen only around the antinode. In the light of Fig. \[fig:back\](a) the absence of the back bending around the node will be simply because the top of the band is located at a higher energy than the maximum energy of the measurement. Note that $t'<0$ lifts (lowers) the band around the nodes (antinodes). Enhancing $-t'$ shifts the Fermi surface to $(0,0)$ as well, as we will discuss in Sec. \[ssec:tp\]. Then a better agreement with the ARPES data may be reached. In Ref.  the back-bending behavior around the antinodes was interpreted as an evidence of preformed Cooper pairs in the pseudogap state because it resembles a band dispersion in the BCS superconductors.[@bc57] However, our result implies another simple interpretation: The band is pushed down by the neighboring zero surface, which cuts the Fermi level around $(\pi,\pi)$, due to the large self-energy around it. This picture is confirmed in Fig. \[fig:fig2\](d), where we plot the energy dependence of the self-energy at the Fermi momentum $\Vec{k}_{\text{F2}}$ closest to the zero surface. We see that the real part of the self-energy is negatively large below the zero at $\w\simeq0.06t$, which should push down the band around the momentum. The result indicates that the pair formation is not necessary and the zeros of $G$ resulting directly from strong correlation effects give an alternative interpretation of the back-bending dispersions. Recently, an evidence against the scenario that the back bending is a consequence of preformed pair was further reported.[@hh10] In this ARPES observation, the back-bending momentum, i.e., the location of the pole at the lowest binding energy along momentum cuts, is clearly deviated from the Fermi momentum at above the pseudogap opening temperature. In contrast, we note that the two momenta should agree in the BCS theory and in the preformed pair scenario as well. Furthermore, the ARPES reported that the back-bending momentum along $(\pi,0)$-$(\pi,\pi)$ shifts closer to $(\pi,\pi)$ than the Fermi momentum. This indicates a zero surface around $(\pi,\pi)$ at the Fermi level and a center of the gap residing above $E_\text{F}$, in full consistency with our pole-zero structure Fig.2(a). We note that a back-bending dispersion was already observed in an early QMC study,[@ph97] where antiferromagnetic fluctuations were proposed as the origin of the pseudogap. Although our numerical data do not exclude the antiferromagnetic fluctuations from the possible mechanisms, the less-$\Vec{k}$-dependent pseudogap as well as the asymmetric location of the hole pocket (see Sec. \[ssec:arc\]) seems to oppose the mechanism; instead it rather supports that the pseudogap is a direct consequence of the proximity to the Mott insulator. Energy-distribution curve {#ssec:edc} ------------------------- Zeros of $G$ are not directly seen in spectra. Their footprints may, however, be detected through a sudden suppression of spectra due to a large Im$\S$ around the zeros. Figure \[fig:edc\](a) shows energy-distribution curves (EDC) of the spectral function along momentum cuts $(0,0)\text{-}(\pi,0)\text{-}(\pi,\pi)$, calculated by the CDMFT+ED with $\eta=0.1t$ for $t'=-0.2t$, $U=12t$, and $n=0.93$. The results exhibit a coherent peak around $(\pi,0)$ just below the Fermi level, its shift to lower energy from $(\pi,0)$ to $(\frac{\pi}{2},0)$, and the incoherent feature around $(0,0)$ and $(\pi,\pi)$. All these features are consistent with ARPES data \[Fig. 5 in Ref. , reproduced in Fig. \[fig:edc\](b)\]. This agreement supports our zero mechanism and shows that the incoherent feature can be interpreted as the effect of zeros of $G$: Since the zero surface exists just above the band around $(\pi,\pi)$ \[Fig. \[fig:fig2\](a)\] and since another one is located just below the band around $(0,0)$ \[not shown in Fig. \[fig:fig2\](a), but can refer to Fig. \[fig:rkw\_tp\](b) below\], the spectrum associated with the band is smeared around these momenta due to the large Im$\S$ around the zeros. While the suppression around $(\pi,\pi)$ is relevant to the emergence of Fermi arc,[@sk06; @sm09; @sm09-2] that around $(0,0)$ is related to the waterfall behavior discussed in the next subsection. Waterfall {#ssec:wf} --------- Recent ARPES [@kb05; @gg07; @xy07; @vk07; @mz07] observed an anomalous spectral structure, called “waterfall", at high binding energies $\sim 0.3\text{-}0.4$eV, where the band cutting the Fermi level suddenly loses the spectral weight and starts falling down to $\sim -0.7$eV with a suppressed intensity. Below the energy a strong intensity emerges again from nearly the same momentum as that the waterfall starts. This high-energy anomaly has been found quite generally in hole-doped cuprates, irrespective of the presence or absence of superconductivity, under, optimally or overdoped, and detailed compositions. Moreover similar features have been reported in other transition-metal compounds such as SVO$_3$ [@yt05] and LaNiO$_3$.[@ec09] These suggest a universality of the phenomenon in strongly-correlated metals.[@bk07] The dynamical cluster approximation (DCA) + QMC study [@mj07] for the 2D Hubbard model with $t'=0$ reported a similar structure in the spectra. Based on the similarity of the DCA results to those with a perturbative calculation incorporating antiferromagnetic spin fluctuations, the authors proposed that the waterfall results from high-energy spin fluctuations. Here we implement a CDMFT+ED study, and closely examine how the pole-zero structure underlies the waterfall phenomenon and how the energy scale of the waterfall depends on model parameters. Our analysis indicates alternative interpretation for the phenomenon. Figures \[fig:wf\](a)-(d) plot the spectral intensity calculated along the momentum cut from $(0,\frac{\pi}{8})$ to $(\frac{7\pi}{8},\pi)$ for various parameter sets. The results show nice resemblances with the ARPES result \[Fig. 1(c) in Ref. , reproduced in Fig. \[fig:wf\](e)\], i.e., (i) an abrupt change in the slope of the band around $\w=E_1$, accompanied by the simultaneous reduction of the intensity, (ii) nearly vertical dispersion with suppressed intensity in the waterfall region, and (iii) reemergence of a strong intensity around $\w=E_2$. The energy scale also agrees roughly with the experimental value, if one keeps in mind $t\sim 0.4$ eV. We investigate how $E_1$ and $E_2$ depend on $t'$, $n$, and $U$. First, comparing Figs. \[fig:wf\](a) and (b), we see that $t'<0$ reduces both $|E_1|$ and $|E_2|$. Second, comparison of Figs. \[fig:wf\](b) and (c) shows that doping also reduces the energies. This doping dependence is qualitatively consistent with the DCA results [@mj07] for 16 sites cluster at a finite temperature. Third, the decrease of $U$ from $12t$ \[Fig. \[fig:wf\](a)\] to $8t$ \[Fig. \[fig:wf\](d)\] increases the energies. These parameter dependences can be naturally explained as follows. In general, strong correlation effects make the width of coherent band (i.e., the band above $E_1$) narrower by increasing the mass of the low-energy particles. This leads to the decrease of $|E_1|$ as $U$ increases. Indeed the slope of the coherent band decreases from $1.3ta$ ($a$: the lattice constant) to $1.0ta$ as $U$ increases from $8t$ to $12t$. Meanwhile $t'$ lifts the band around the nodal direction, as will be discussed in Sec. \[ssec:tp\]. This elevates $E_1$ and $E_2$. The doping dependence seen in Figs. \[fig:wf\](b) and (c) can be qualitatively understood as a downward shift of the chemical potential with doping. As we showed in Ref. , in hole-doped Mott insulators doping causes a rigid-band-like shift of the chemical potential, though it cannot be described within the single-electron picture. In fact, the low-doping phase is not adiabatically connected with the Fermi liquid phase at higher dopings because of the intervening Lifshitz transition and the zero-surface emergence. The above interpretation suggests that the energy scale of the waterfall is not necessarily related to the antiferromagnetic spin fluctuations [@mj07] but is considered to be a rather direct consequence of electron correlation, i.e., the mass renormalization of the coherent band. The picture is corroborated by the fact that a similar behavior can be seen within the single-site DMFT.[@bk07] In view of the pole-zero structure in Fig. 1(b) in Ref. , the waterfall emerges in the energy region $-2t\lesssim\w\lesssim-t$, where many smeared pole and zero surfaces pile. The congestion of poles and zeros within the finite-cluster calculation indicates an incoherent nature of this energy-momentum region, which would result in the waterfall. Notice that the waterfall obtained in this paper has neither a gap nor dispersive features [@mj07], but a vertical structure with a suppressed intensity. This unusual structure comes out thanks to the use of the cumulant periodization (see Sec. \[sec:method\]) which can describe the strong momentum dependence of the self-energy. Below the pile we see a relatively coherent pole surface extending down to $\w\sim -3.5t$. This corresponds to the band reemerging at the high binding energy \[(iii)\]. Note that a comparison at higher binding energies is difficult because the ARPES spectra are overlapped by other Cu-$d$ or O-$p$ bands. Low-energy kink in dispersion {#ssec:kink} ----------------------------- In prior to the discovery of the waterfall, another anomaly (kink) in dispersion has been found in ARPES at lower binding energies [@bl00] and its origin has been in dispute. The kink is located at a binding energy around $0.05$eV, where the band cutting the Fermi level sharply changes its slope. In the normal state the kink is clearly seen around the nodal region while, as approaching the antinodal region, the band becomes flatter and then the kink becomes less visible. The band structure continuously changes from the nodal to antinodal region where a pseudogap exists just above the band.[@kr01; @sm03] The CDMFT+ED results in Fig. \[fig:kink\] show similar behaviors: (i) The band suddenly changes the slope at a kink around $\w=-0.1t\sim -0.04$eV in the nodal region \[Fig. \[fig:kink\](a)\], and (ii) as approaching the antinode, the band becomes flatter and the slope change becomes weaker \[Fig. \[fig:kink\](b)\]. In general, the zero surface which generates the pseudogap pushes down the dispersion near the Fermi level. This makes a quick change of the slope of the dispersion distinct from the part deeply below the Fermi level, which may be the underlying origin of the kink formation. However, the sudden change of the dispersion observed in Fig. \[fig:kink\](a) naturally requires a precursory formation of a tiny zero surface around $\w=-0.1t$. This means a coupling of the quasiparticle with some other excitations with this energy, whichever bosonic or fermionic. The energy resolution of the present cluster size is not obvious and it could be an artifact of the present small cluster calculation. Nevertheless, it is remarkable to note that the observed kink in Fig. \[fig:kink\](a) is rather universal and we see similar kinks in other part of the quasiparticle dispersions, for instance in Fig. \[fig:rkw\_tp\](b) below. This may be alternatively interpreted that, in the strongly correlated region, the quasiparticle is strongly renormalized and may couple to various intrinsic electronic modes of charge and spin origins. In high-$T_{\text{c}}$ cuprates the mechanism of the kink has been extensively discussed in the literature from the viewpoint of a coupling of electrons with some bosonic mode, such as phonon [@lb01] and magnetic ones.[@jv01; @kr01] Our result, however, implies that the kink may be a rather general phenomenon in the proximity to the Mott insulator,[@bk07] while the specific mechanism of kink formation is left for future studies. Fermi arc and hole-pocket Fermi surface {#ssec:arc} --------------------------------------- Next we shift our focus on the zero-energy electronic structure, namely, Fermi surface. The spectra in hole-doped cuprates have been extensively studied by ARPES.[@dh03] One of the most remarkable findings in these studies is the observation of truncated Fermi surfaces, called “Fermi arc".[@nd98] Several authors have discussed the Fermi arc in terms of coexisting poles and zeros of $G$.[@sm09; @kr06; @et02; @bg06; @ks06; @sk06; @sm09-2; @sp07] In this mechanism hole-pocket Fermi surfaces around the nodal directions coexist with a zero surface around $(\pi,\pi)$. Because Im$\S$ is large around the zero surface, the pockets lose the spectral intensity more on the side closer to the zero surface, leaving arc-like spectra on the opposite side. This is reproduced in Figs. \[fig:pocket\](a) and (b). Figure \[fig:pocket\](a) shows the underlying pole-zero structure calculated by the CDMFT+ED at $T=0$ without any substantial smearing to the singularities, where a clear hole-pocket Fermi surface coexists with a zero surface. When we introduce a smearing to the singularities (see Sec. \[ssec:ed\]), we obtain the spectral map, Fig. \[fig:pocket\](b), in which the arc-like structure is observed. We confirm the zero mechanism by directly calculating a spectral weight at finite temperatures by employing the CTQMC method as the solver for the CDMFT. To obtain the spectrum without through any analytic continuation procedure which inevitably suffers from a large error bar, we calculate $$\begin{aligned} \label{eq:ghb} -\b G(\Vec{k},\t=\beta/2)=\frac{\b}{2} \int_{-\infty}^{\infty}\frac{A(\Vec{k},\w)}{\cosh(\b\w/2)}d\w\end{aligned}$$ as a function of $\Vec{k}$. This quantity is an integral of the spectral weight over a width $\sim T$ around $\w=0$ and approaches $A(\Vec{k},\w=0)$ for large $\beta$, so that gives an estimate of $A(\Vec{k},\w=0)$ at low temperatures.[@footnote2] In Figs. \[fig:pocket\](c) and (d) we plot the results at $T=0.025t$. For $n=0.95$ \[Fig. \[fig:pocket\](c)\] we see an arc at a location similar to the one in Fig. \[fig:pocket\](b) while for $n=0.9$ \[Fig. \[fig:pocket\](d)\] the spectra extend to the antinodal regions. The doping evolution of the spectra is qualitatively consistent with that obtained with the CDMFT+ED in Fig. 2(c)-(e) in Ref. . The qualitative agreement between the CDMFT+ED result with broadening $\eta$ and the CDMFT+CTQMC result at finite temperatures supports that the phenomenological smearing factor $\eta$ well simulates the thermal effects. Moreover it corroborates the above-mentioned zero mechanism for the emergence of the Fermi arc. We note that the zero surface around $(\pi,\pi)$ is also consistent with the results by other cluster schemes such as DCA, which observed a strong scattering amplitude in the momentum patch around $(\pi,\pi)$ in both $N_\text{c}=4$ (Ref. ) and $N_\text{c}=8$ (Ref. ) calculations. Interestingly, recent high-resolution ARPES [@ml09] reported the existence of hole-pocket Fermi surfaces around the nodal directions in underdoped Bi$_2$Sr$_{2-x}$La$_x$CuO$_{6+\d}$. The observed hole pockets have much less intensity on the side closer to $(\pi,\pi)$ than the opposite. This is consistent with the zero mechanism in Fig. \[fig:pocket\](b). Important findings in the ARPES are that (i) the hole pockets are not located symmetrically with respect to the antiferromagnetic Brillouin zone boundary, i.e., $(\pi,0)\text{-}(0,\pi)$ line (and its symmetrically-related ones), and that (ii) the spectral intensity on the $(\pi,\pi)$ side is finite even in the nodal directions. (i) excludes several scenarios for the pseudogap, which attribute the pockets to symmetry breakings [@ks90; @v97; @cl01] because in these theories hole pockets should be centered symmetrically as to the $(\pi,0)\text{-}(0,\pi)$ line.[@ml09] Meanwhile (i) is consistent with our zero mechanism that does not assume any symmetry breaking. (ii) is at odds with the theory based on an ansatz that gives a zero intensity in the outer point of the pocket crossing the nodal direction,[@ml09] while it agrees well with our numerical data in Fig. \[fig:pocket\](b). This is more clearly seen in Fig. \[fig:pocket\](e), where we plot $A(\Vec{k},0)$ along the momentum cut in the nodal direction: For $\eta=0.01t$, in addition to the main peak at $\Vec{k}\simeq(0.56\pi,0.56\pi)$, we see the secondary peak at $\simeq (0.7\pi,0.7\pi)$, corresponding to the hole pocket structure. The second peak is, however, not visible for $\eta=0.1t$ and only the broadened main peak can be seen there. Namely, in our theory, the spectrum looks either a pocket like or an arc like depending on how the incoherence due to the zero surface is strong. The strength of the incoherence is controlled by the value of $\eta$ or temperature, or the distance between zeros and poles, as partly demonstrated in Fig. 4 in Ref. . Figure \[fig:pocket\](e) also suggests that the energy resolution required to detect the pocket in the example of the pole-zero structure as Fig. \[fig:pocket\](a) is in between $0.01t(\sim 4\text{meV})$ and $0.1t(\sim 40\text{meV})$, which roughly corresponds to the energy resolution in ARPES. This consideration on the incoherence explains why the pocket had not been detected until the recent high-resolution ARPES.[@ml09] One obvious difference between Fig. \[fig:pocket\](b) and the ARPES [@ml09] is the location of the pocket: The pocket in Fig. \[fig:pocket\](b) resides closer to $(\pi,\pi)$ than that observed in the ARPES. This may be attributed to the difference between the model parameters we used and realistic ones for Bi$_2$Sr$_{2-x}$La$_x$CuO$_{6+\d}$. In particular, we show in the next subsection that the next-nearest-neighbor transfer $t'$, which is zero in Fig. \[fig:pocket\](b), indeed shifts the pocket in the direction to $(0,0)$. Effect of the next nearest-neighbor transfer $t'$ {#ssec:tp} ------------------------------------------------- Here we systematically study the effect of $t'$ on the electronic structure in lightly hole- and electron-doped regions. Figure \[fig:arc\_tp\] shows $t'$ dependence of the integrated spectra, Eq. (\[eq:ghb\]), calculated with CDMFT+CTQMC at $T=0.05t$ for (a)-(c) $U=8t$ and $n=0.93$ and (d)-(f) $U=12t$ and $n=0.90$. We see that the arc shifts to $(0,0)$ as $-t'/t$ increases, and that at $t'=-0.4t$, which is a reasonable value for Bi$_2$Sr$_{2-x}$La$_x$CuO$_{6+\d}$, the arc resides inside of the antiferromagnetic Brillouin zone, in consistency with experiments.[@ml09] We also see that $t'$ increases the curvature of the arc. A comparison with ARPES in further detail will require a more realistic determination of the model parameters, longer-range transfer integrals and Coulomb interactions, and a calculation on a larger cluster. These remain for future researches. The above shift of the arc with $t'$ can be understood by plotting the underlying pole-zero structures. Figures \[fig:rkw\_tp\](a) and (b) compare the structures for $t'=0$ and $t'=-0.2t$ in a lightly hole-doped region. The figures show that at low energies $t'$ lifts the poles in the nodal direction while lowers them around the antinodes, as already expected from the bare dispersion Eq. (\[eq:disp\]). Then the hole pocket expands and the Fermi surface with the stronger intensity shifts to $(0,0)$ direction. We note that, as we discuss in the next subsection, the third-neighbor transfer integral, which is not taken into account in the present calculation but exists in real materials, further enhances the pocket around $(\frac{\pi}{2},\frac{\pi}{2})$. The effect of $t'$ is totally different for electron-doped cases. Figures \[fig:rkw\_tp\](c) and (d) show the structures for $t'=0$ and $-0.2t$, respectively, in an electron-doped region. The effect of $t'$ at low energies is again understood by Eq. (\[eq:disp\]): $t'$ lifts the poles in the nodal direction while lowers around the antinodes. However, the consequence on the Fermi surface completely differs from that in hole-doped cases. For the particle-hole symmetric $t'=0$ case, the electron pockets appear around the nodes for electron doping, corresponding to the hole pockets in hole-doped cases, however, for finite $t'$, the electron pockets appear around the antinodes, as shown in Fig. \[fig:rkw\_tp\](d). This is because the low-energy zero surface pushes up the dispersion around $(0,\frac{\pi}{2})$. We compare the Fermi surface structure with ARPES for the electron-doped cuprates in detail in Sec. \[ssec:ele\]. High-energy spectra in electron-doped cuprates {#ssec:ele2} ---------------------------------------------- Recently high-energy spectra of an electron-doped cuprate Nd$_{1.85}$Ce$_{0.15}$CuO$_4$ were studied by ARPES.[@iy09] To discuss the results we first emphasize that here the ARPES observes a totally different energy region from that for hole-doped cuprates. This is illustrated in Figs. \[fig:wf2\](a) and (b): ARPES observes the lower Hubbard band (LHB) in hole-doped cases while in electron-doped cases it does ingap states between the Mott gap and the pseudogap, in addition to the pseudogap structure itself. As mentioned in Sec. \[ssec:eh\], with the electron-hole transformation the ARPES result can be interpreted as that for unoccupied states in a hole-doped system. Therefore the results in Ref.  provide a precious opportunity to compare the calculated spectra of ingap states with experiments. The pseudogap and the Fermi surface are discussed in the next section. The ARPES spectra and its second derivative in $\w$ (Fig. 1 in Ref. ) are reproduced in Figs. \[fig:wf2\](c)(d) and (e)(f), respectively. Around the antinode the spectra \[Fig. \[fig:wf2\](d)\] suddenly lose the intensity at $\w \simeq -0.4$eV. The analysis on the second derivative \[Fig. \[fig:wf2\](f)\] shows that the low-energy band reaches the bottom at this energy. On the other hand, around the node the low-energy band persists down to $\w \simeq -0.7$eV \[Fig. \[fig:wf2\](c)(e)\]. Figures \[fig:wf2\](g) and (h) depict the spectra calculated with the CDMFT+ED for $t'=-0.4t$, $U=12t$, and $n=1.11$ along momentum cuts, $(0,0)\text{-}(\pi,\pi)$ and $(0,\pi)\text{-}(\pi,\pi)$, respectively. For $\w<0$ we see that the band bottom is located at $\w\simeq -0.7t$ around $(0,\pi)$ while it is located at a higher binding energy $\w\simeq -t$ around $(0,0)$. This is consistent with the ARPES data. Note that the numerical data do not show any intensity below the band bottoms, $\w < -0.7t$ ($-t$) around $(0,\pi)$ \[$(0,0)$\], which is within the gap region between LHB and the ingap states, whereas the ARPES spectra show waterfall-like structures in $-1\text{eV}\lesssim \w \lesssim -0.7\text{eV}$ ($\w \lesssim -0.4\text{eV}$) around $(0,\pi)$ \[$(0,0)$\]. The origin of this discrepancy is not clear, but one possibility is a multiband effect, which is not considered in the present single-band model. Meanwhile for $\w>0$, we see a waterfall in the dispersion at $\w\simeq t\text{-}2t$. This corresponds, with the electron-hole transformation, to the waterfall in hole-doped cases. The result predicts that this anomaly will be observed in electron-doped cuprates when the unoccupied spectra become available up to the high energy.[@footnote5] Pseudogap and Fermi pockets in electron-doped cuprates {#ssec:ele} ------------------------------------------------------ Lastly we discuss low-energy electronic structures in electron-doped cases. The pseudogap has been experimentally observed also in electron-doped cuprates,[@ot01] although the doping range is much more limited than that in hole-doped cases due to the wider antiferromagnetic region near the undoped Mott insulator. The mechanism of the pseudogap has extensively been discussed within weak-coupling theories [@kr03; @kh04], which has successfully reproduced various experimental results such as the doping evolution of the Fermi surface. In these theories the mechanism has been ascribed to the antiferromagnetic long-range correlations. On the other hand, in the strong-coupling regime the cluster-perturbation theory [@st04] and the CDMFT [@cc05; @kk06] found that the pseudogap results from nonlocal but short-ranged dynamics without any long-range order or correlations. As pointed out in Ref.  and , the gap amplitude does not scale as $J\sim t^2/U$, so that the short-ranged dynamics is not simply ascribed to the AF correlation. Because it is unknown which mechanism, weak-coupling or strong-coupling, is relevant to electron-doped cuprates, it is worthwhile to see whether and how experimental data are understood in the strong-coupling picture, especially from the perspective of zeros of $G$. The available ARPES data for the electron doped cuprates seems to have relatively poor resolutions and a clear dispersion has not been reported. Nevertheless, a recent ARPES [@iy09-2] measurement appears to have revealed the structure of the pseudogap for a family of electron-doped cuprates, $Ln_{2-x}$Ce$_x$CuO$_4$ ($Ln=$ Nd, Sm, and Eu). In Fig. 2 in Ref.  the EDC peak position jumps from $\w\sim-0.03$eV to $\w\sim-0.15$eV around antinode, which implies the presence of the pseudogap opening below the electron pocket. This highly asymmetric position of the pseudogap is a feature which cannot be seen by ARPES for the hole-doped cuprates but observable for electron-doped ones: Because in electron-doped cases the zero surface forming the pseudogap mainly extends below $E_\text{F}$ \[see Fig. \[fig:rkw\_tp\](d)\], the ARPES can observe the major part of the pseudogap, in contrast to the hole-doped cases. Around the node, the EDC peak position saturates at $\w\sim-0.05$eV for $Ln=$Eu compound, implying the pseudogap at $\omega \gtrsim -0.05$eV, while the pseudogap is not clear for $Ln=$Nd in EDC. This is consistent with the observation in Fig. \[fig:rkw\_tp\] that the pseudogap position in the nodal region is lowered for smaller $|t'/t|$ and that the Eu compound appears to have a relatively small $|t'/t|$. Namely, the main part of the pseudogap in the nodal region of the Nd compound is located in the positive energy side, while it becomes visible below $E_\text{F}$ for the Eu compound. This suggests a gap around the node, which can be interpreted as an indirect evidence of a gap opening above $E_\text{F}$ in hole-doped cases via the electron-hole transformation. These observations are qualitatively consistent with the pole-zero structure in Fig. \[fig:rkw\_tp\](d), supporting the non-$d$-wave (fully-opened) pseudogap proposed in Sec. \[ssec:nond\]. We propose to perform ARPES measurements of the electron doped compounds with a better resolution than Ref. , because the pseudogap appears to be suggested only by the jump in EDC and the detailed pseudogap structure in the momentum space is not very clear so far. The ARPES [@ar02; @iy09-2] has also revealed a characteristic evolution of the Fermi surface with doping. Figures \[fig:edope\](a)-(c) are the reproductions of the ARPES results (Fig. 3 in Ref. ) on Nd$_{2-x}$Ce$_x$CuO$_4$. The ARPES study found that (i) at small doping ($x=0.04$) strong intensity emerges around $(\pi,0)$ and its symmetrically-related points \[Fig. \[fig:edope\](a)\], (ii) as doping is increased, additional intensity develops around $(\frac{\pi}{2},\frac{\pi}{2})$ \[Fig. \[fig:edope\](b)\], and (iii) at large doping the spectra merge, forming a single large surface around $(\pi,\pi)$ \[Fig. \[fig:edope\](c)\]. In Ref.  it is also seen that the band cutting the Fermi level at around $(\frac{\pi}{2},\frac{\pi}{2})$ evolves with doping from a high to low binding energy while that around $(\pi,0)$ emerges first around the Fermi level and extends to a high binding energy. This suggests that the Fermi surface around $(\frac{\pi}{2},\frac{\pi}{2})$ is a hole pocket while the one around $(\pi,0)$ is an electron pocket. The above features (i) and (iii) were qualitatively reproduced by the cluster perturbation theory [@st04] and by CDMFT with the $\S$ [@cc05] and $G$ [@kk06] periodizations on the 2D Hubbard model at $T=0$. In Ref.  and the suppression of the intensity around the node at low dopings was understood by the presence of the hot spot, where Im$\S$ is large, in this region. In the following we clarify how the above observations (i)-(iii) can be understood in terms of the underlying zero surface. First, we show the CDMFT+CTQMC results at $T=0.05t$. Figures \[fig:edope\](d)-(f) depict the integrated low-energy spectra, Eq. (\[eq:ghb\]), for $t'=-0.4t$, $U=12t$, at $5$, $10$, and $15$% electron dopings, respectively. At $n=1.05$ a strong intensity emerges only around the antinodal points, in accord with (i). A large Fermi surface around $(\pi,\pi)$ at $n=1.15$ agrees with (iii). As to (ii), our results do not show strong intensity around $(\frac{\pi}{2},\frac{\pi}{2})$; this will be attributed to the absence of the third-neighbor transfer integral $t''$ in the present calculation, as discussed below. Second, Figs. \[fig:edope\](g)-(i) show the CDMFT+ED results for $t'=-0.4t$ and $U=12t$ at $n=1.14$, $1.17$, and $1.19$, respectively. We see that a zero surface exists at the Fermi level, surrounding $(0,0)$. For low dopings electron-pocket Fermi surfaces reside around the antinodal points. As the doping is increased, the pockets expand. Then they merge around the nodal point and change the topology into two Fermi surfaces, one around $(\pi,\pi)$ and the other around $(0,0)$. Further doping makes the inner Fermi surface annihilate in pair with the zero surface, resulting in the Fermi liquid with the single large Fermi surface around $(\pi,\pi)$.[@footnote3] Thus in electron doping there occur, at least, two phase transitions, the Lifshitz transition [@ko07] and a pole-zero annihilation transition, on the way from the Mott insulator to the Fermi liquid. The presence of the zero surface accounts for the weak spectral intensity around the nodal direction in Figs. \[fig:edope\](d) and (e). Then how is the intensity around the node in Fig. \[fig:edope\](b) and (c) explained? To answer this we consider how the electronic structure, Fig. \[fig:rkw\_tp\](d), changes with $t''$, which is not included in our calculations based on the $2\times2$ cluster. Although $|t''|$ is usually several times smaller than $|t'|$ and therefore negligible for discussing global electronic structures, it can be crucial to understand the nodal intensity. This is because around $(\frac{\pi}{2},\frac{\pi}{2})$ the pole surface at $\w\simeq -0.15t$ in Fig. \[fig:rkw\_tp\](d) moves up with $-t'/t$, as seen in comparison with Fig. \[fig:rkw\_tp\](c), and is located just below $E_{\text{F}}$ for $-t'/t=0.4$ (not shown). Since to the first approximation $t''(>0)$ elevates dispersions around $(\frac{\pi}{2},\frac{\pi}{2})$ while lowers around $(0,0)$, $(\pi,\pi)$, $(\pi,0)$, and $(0,\pi)$, according to $$\begin{aligned} \e(\Vec{k}) \rightarrow \e(\Vec{k})-2t''[\cos(2k_x)+\cos(2k_y)],\end{aligned}$$ a hole pocket can emerge around $(\frac{\pi}{2},\frac{\pi}{2})$. This hole pocket will show up just inside the zero surface in Figs. \[fig:edope\](g) and (h). Recent experimental observation [@hk09] of the Shubnikov-de Haas oscillation in Nd$_{2-x}$Ce$_x$CuO$_4$ also indicates a presence of the hole pocket around optimal doping. Therefore, to confirm the above scenario with a larger cluster calculation incorporating $t''$ is an intriguing future problem. We note some additional indications of the zero surface in experiments. In Fig. 2(c)-(e) in Ref. , which plotted the ARPES EDCs for Nd$_{2-x}$Ce$_x$CuO$_4$ along the underlying Fermi surface, a clear jump of the peak positions can be seen between the nodal and antinodal directions. This indicates the presence of the zero surface around the jump, in accord with the above picture \[see Fig. \[fig:rkw\_tp\](d)\]. In Fig. 1 in Ref. , which plotted the ARPES spectra at the Fermi level, a similar structure to Fig. \[fig:edope\](c) was found while it clearly shows a finite intensity in some regions on the $(\pi,0)\text{-}(0,\pi)$ line. This appears to be inconsistent with the zero surface on the $(\pi,0)\text{-}(0,\pi)$ line assumed in Ref.  for hole-doped cases, but compatible with our zero surface around $(0,0)$ as seen in Figs. \[fig:edope\](g) and (h). SUMMARY AND CONCLUSION {#sec:summary} ====================== In summary, we have shown that various spectral anomalies observed in the pseudogap states of hole- or electron-doped cuprates are naturally understood in terms of underlying pole-zero structure of electronic Green’s function. The pole-zero structure has been calculated for the paramagnetic metallic phase in the 2D Hubbard model with employing the CDMFT+ED at $T=0$ and the CDMFT+CTQMC at $T>0$. We have confirmed that the cumulant periodization scheme [@sk06] is suited for the parameter region of our interest, namely lightly doped Mott insulator in the intermediate to strong coupling region, since the cumulant is well localized within the cluster that we employed, as demonstrated in APPENDIX B. This corroborates the convergence and reliability of our momentum resolution in the studies on Fermi surface topology and the differentiation in the momentum space. Furthermore, the cumulant periodization is useful since it enables to study the coexistence of pole and zero surfaces. The result calculated by CDMFT+ED shows a pseudogap in the lightly doped region around the Mott insulator. The pseudogap is characterized by a low-energy zero surface, which connects the bifurcated low-energy bands \[Fig. \[fig:fig2\](a)\]. In hole-doped cases the lower band cuts the Fermi level while the upper one resides at the energy higher than and adjacent to the pseudogap above the Fermi level. This directly leads to (i) the non-$d$-wave character of the fully opened pseudogap and (ii) the spectral asymmetry around the Fermi level. Looking into the structure around $(\pi,\pi)$ we have also found that a large Re$\S$ around the zero surface causes (iii) the back-bending dispersion. All of (i), (ii), and (iii) are consistent with experiments semiquantitatively. The agreement imposes a strong constraint on theories for pseudogap: Although (ii) and (iii) have been interpreted as evidences of the preformed pairing in the literatures, (i) is in sharp contrast to $d$-wave-gap scenarios including those by preformed pair. It is compatible neither with other precursory or real $d$-wave-type symmetry breakings such as $d$-density wave nor with commensurate antiferromagnetism. We have confirmed a full gap formation at the nodal point in a larger-cluster calculation in APPENDIX C. Moreover, the non-$d$-wave gap is supported by the agreement of our result on the pseudogap structure with that observed by ARPES for electron-doped cuprates. The zero mechanism also enables a simple and unified understanding of various spectral anomalies: In hole-doped cases the same zero surface which causes the back-bending behavior around the antinode induces the incoherence around $(\pi,\pi)$, Fermi pockets and Fermi arcs, while a pile of pole and zero surfaces at a higher binding energy results in the high-energy kink (waterfall) and the incoherence around $(0,0)$. Moreover we have found a low-energy kink structure in the nodal dispersion. On the other hand, in electron-doped cases the zero surface which constitutes the pseudogap crosses the Fermi level around $(0,0)$, making electron pockets around the antinodes. All these features are consistent with experimental results. We would like to emphasize that the zero surface does not result from symmetry breakings, but is a direct consequence of the strong correlation effect, i.e., proximity to the Mott insulator. To elucidate the effect of zeros at finite temperatures, we have also implemented CDMFT+CTQMC calculations. The results qualitatively agree with those of CDMFT+ED with a smearing factor $\eta$, which confirms that the thermal scatterings broaden the effect of zeros and indeed create the characteristic spectra similar to those observed in the cuprates. To conclude, the origins of various anomalies in the electronic structure of the normal state in the high-$T_{\text{c}}$ cuprates are unified into the presence of the low-energy zero surface which persists against doping. The zero surface is also expected to cause anomalous metallic behaviors in other physical quantities, such as the specific heat, electronic resistivity, and Hall coefficient.[@ts99] Comparisons of these quantities with experiments remain for future studies. Although the microscopic mechanism creating the zero surface has not been discussed in this paper, it has emerged clearly as the proximity to the Mott insulator. The similarity in the structure of the zero surfaces between the pseudogap and the Mott gap in the undoped system \[see Fig. 1(a) in Ref. \] also implies the Mott origin of the pseudogap. At the same time, however, the pseudogap does not appear as that directly or continuously connected to the zero surface which forms the Mott gap in the undoped state. This is because, for example in the hole-doped case, the zero surface creating the fully-opened pseudogap is bounded above by the existence of low-energy excitations far below the upper Hubbard band (UHB). The reason why the ingap states are separated by the pseudogap from the main quasiparticle band is left for an important future issue. In Ref.  we proposed a scenario in which the emergence of such a zero surface in the low-energy scale and the resultant pseudogap are a remnant of the binding of doubly occupied site (doublon) and empty site (holon) drastically weakened by the screening of Coulomb repulsions by mobile carriers. This has something to do neither with any symmetry breaking nor with its precursor. Recently, it has been proposed that a gap arising from hybridization between the quasiparticle and a composite fermion excitation is responsible for the pseudogap and the related zero surface. [@yamaji10] Possible supplementary role of strong antiferromagnetic fluctuations are left for future studies. Further study on the mechanism of pseudogap will be described elsewhere. ACKNOWLEDGMENT {#acknowledgment .unnumbered} ============== S. S. thanks Masafumi Udagawa, Giorgio Sangiovanni and Alessandro Toschi, and M. I. thanks Youhei Yamaji for useful discussions. S. S. is also grateful to Karsten Held for the hospitality. This work is supported by a Grant-in-Aid for Science Research on Priority Area “Physics of New Quantum Phases in Superclean Materials" (Grant No.17071003) from MEXT, Japan. S. S. is supported by Research Fellowship of the Japan Society for the Promotion of Science for Young Scientists. The calculations are partly performed at the Supercomputer Center, ISSP, University of Tokyo. APPENDIX A: Breakdown of $\S$ periodization for the Mott insulator {#appendix-a-breakdown-of-s-periodization-for-the-mott-insulator .unnumbered} ================================================================== Here we note that a standard periodization technique, which substitutes $Q=\S$ to Eq. (\[eq:periodize\]), cannot reproduce the Mott gap for large $U$. Because Im$G=0$ in the Mott gap, Im$\S$ must be 0 or $\infty$ in the whole Brillouin zone. However, this situation does never occur in Eq. (\[eq:periodize\]) with $Q=\S$ unless $^\forall i,j \in \text{C}, {\rm Im}\S_{ij}^{\text{C}}=0$ or $^\exists (ij), \S_{ij}^{\text{C}}=\infty$ for all $\w$ inside the gap, and these conditions are never satisfied as far as both $t$ and $U$ are nonzero and finite. As a matter of fact, we see the electronic structure shown in Fig. \[fig:speriodize\](a) when we use the $\S$-periodization procedure for the same Mott insulator shown in Fig. 1(a) in Ref.  \[and reproduced in Fig. \[fig:speriodize\](b) for comparison\]. We see that zeros of $G$ exist only at the Fermi level without a dispersion, and that poles of $G$ extend to the Fermi level around $(0,0)$ from the positive $\w$ side and around $(\pi,\pi)$ from the negative $\w$ side, making the density of states half metallic. This failure of the $\S$ periodization in the Mott insulator is ascribed to the fact that $\S$ is not localized within the $2\times 2$ cluster.[@sk06] The nonlocality of $\S$ is a direct consequence of the presence of dispersive zeros of $G$, i.e., momentum-dependent divergence of $\S$. As we have discussed so far, zeros of $G$ still persist in doped Mott insulators up to a critical doping level beyond which the Fermi liquid emerges. Therefore $\S$ should be highly nonlocal also in the non-Fermi-liquid region and the $\S$ periodization again breaks down there. On the contrary, the cumulant $M$ is well localized within the $2\times 2$ cluster in this region, as we mentioned in Sec. \[sec:method\]. In APPENDIX B we give another numerical evidence for the locality of the cumulant in doped Mott insulators. APPENDIX B: Locality of cumulant in doped Mott insulators {#appendix-b-locality-of-cumulant-in-doped-mott-insulators .unnumbered} ========================================================= Here we present a CDMFT+QMC result for an $N_{\text{c}}=4\times 4$ cluster in a parameter region of doped Mott insulators. We use the Hirsch-Fye algorithm [@hf86] and calculate the cluster cumulant $M^{\text{C}}$ for $t'=0$, $U=8t$, $n=0.91$, and $T=0.1t$. Figures \[fig:cum\](a) and (b) show $\w_n$ dependence of the real and imaginary parts of the cumulant $M(\Vec{r},\w_n)$, respectively, where $\Vec{r} = (i,j)$ is the real-space vector connecting the sites $i$ and $j$ in the cluster \[the inset of Fig. \[fig:cum\](b)\]. We notice that $\Vec{r}=(0,0)$, (1,0), and (1,1) components are much larger than the other components at longer distances. This can be more clearly seen in Fig. \[fig:cum\](c), where the cumulant at the lowest Matsubara frequency, $\w_0$, is plotted against the Euclidean distance, $\sqrt{i^2+j^2}$. We find that the cumulant is well localized within the $2\times 2$ cluster, which justifies the $M$ periodization. Although the cumulant at $T=0$ might be more extended than that at $T=0.1t$, we do not have a tractable way to examine it for a cluster larger than $2\times 2$. It is worthwhile, however, to note that the finite-temperature results obtained by our CDMFT+CTQMC calculation are consistent with the CDMFT+ED results with finite $\eta$’s, as discussed in Sec. \[ssec:arc\]. This fact implies that the $M$ periodization with $2\times 2$ cluster still remains a good approximation even at $T=0$. APPENDIX C: Nodal spectra obtained by 8-site CDMFT+CTQMC {#appendix-c-nodal-spectra-obtained-by-8-site-cdmftctqmc .unnumbered} ======================================================== To demonstrate the fully-opened pseudogap in a larger cluster, we implement an 8-site cluster calculation with CDMFT+CTQMC. We calculate the spectral function $A(\Vec{k},\w)$ at $\Vec{k}=\Vec{k}_\text{node}\equiv(\pi/2,\pi/2)$, where the $d$- and $s$-wave (fully-opened) pseudogaps are most distinguishable. The spectral function is obtained from $G(\Vec{k},i\w_n)$ through the maximum-entropy method. Figure \[fig:8site\] shows $A(\Vec{k}_\text{node},\w)$ for $t'=-0.2t$, $U=8t$, and $n=0.95$. The small but clear gap just above the Fermi level at $T=0.05t$ proves the fully-opened pseudogap.[@footnote4] The gap vanishes at a higher temperature $T=0.1t$. This is why the previous DCA study [@mj06] at a high temperature $T=0.12t$ did not observe the gap. Note that, a very recent study with 8-site DCA [@lg10] at $T=0.05t$ also detected a wispy reduction of the spectral weight at $\Vec{k}_\text{node}$ at low doping though the signal was so weak that it was not explicitly analyzed in their paper. The weakness of the signal would be due to the flat momentum dependence of the self-energy in each momentum patch in the DCA. Although the suppression of the spectrum is more subtle at higher doping in their result, it is reasonable because the pseudogap temperature decreases with doping. The gap formation at the nodal point cannot be reproduced by the assumption of the $d$-wave pseudogap. We have confirmed this by employing the assumed form of the Green function employed in Ref. . T. Timusk and B. Statt, Rep. Prog. Phys. [**62**]{}, 61 (1999). A. Damascelli, Z. Hussain, and Z.-X. Shen, Rev. Mod. Phys. [**75**]{}, 473 (2003). Y. Yanase, T. Jujo, T. Nomura, H. Ikeda, T. Hotta, and K. Yamada, Phys. Rep. [**387**]{}, 1 (2003). V. J. Emery and S. A. Kivelson, Nature (London) [**374**]{}, 434 (1995). A. Kampf and J. R. Schrieffer, Phys. Rev. B [**41**]{}, 6399 (1990). D. Pines, Z. Phys. B [**103**]{}, 129 (1997). Y. M. Vilk and A.-M. S. Tremblay, J. Phys. I France [**7**]{}, 1309 (1997). S. A. Kivelson, I. P. Bindloss, E. Fradkin, V. Oganesyan, J. M. Tranquada, A. Kapitulnik, and C. Howald, Rev. Mod. Phys. [**75**]{}, 1201 (2003). C. M. Varma, Phys. Rev. B [**55**]{}, 14554 (1997). S. Chakravarty, R. B. Laughlin, D. K. Morr, and C. Nayak, Phys. Rev. B [**63**]{}, 094503 (2001). T. M. Rice, Prog. Theor. Phys. Suppl. [**160**]{}, 39 (2005). T. D. Stanescu and P. Phillips, Phys. Rev. Lett. [**91**]{}, 017002 (2003). T. D. Stanescu and G. Kotliar, Phys. Rev. B [**74**]{}, 125110 (2006); T. D. Stanescu, M. Civelli, K. Haule, and G. Kotliar, Ann. Phys. (N. Y.) [**321**]{}, 1682 (2006). P. Philips, T.-P. Choy, and R. G. Leigh, Rep. Prog. Phys. [**72**]{}, 036501 (2009). S. Sakai, Y. Motome, and M. Imada, Phys. Rev. Lett. [**102**]{}, 056404 (2009). A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. [**68**]{}, 13 (1996). G. Kotliar, S. Y. Savrasov, G. Palsson, and G. Biroli, Phys. Rev. Lett. [**87**]{}, 186401 (2001). T. Maier, M. Jarrell, T. Pruschke, and M. H. Hettler, Rev. Mod. Phys. [**77**]{}, 1027 (2005). Th. A. Maier, Th. Pruschke, and M. Jarrell, Phys. Rev. B [**66**]{}, 075102 (2002). D. Seńećhal and A.-M. S. Tremblay, Phys. Rev. Lett. [**92**]{}, 126401 (2004). M. Civelli, M. Capone, S. S. Kancharla, O. Parcollet, and G. Kotliar, Phys. Rev. Lett. [**95**]{}, 106402 (2005). B. Kyung, S. S. Kancharla, D. Seńećhal, A.-M. S. Tremblay,M. Civelli, and G. Kotliar, Phys. Rev. B [**73**]{}, 165114 (2006). A. Macridin, M. Jarrell, T. Maier, P. R. C. Kent, and E. D’Azevedo, Phys. Rev. Lett [**97**]{}, 036401 (2006). M. Aichhorn, E. Arrigoni, M. Potthoff, and W. Hanke, Phys. Rev. B [**74**]{}, 024508 (2006). K. Haule and G. Kotliar, Phys. Rev. B [**76**]{}, 104509 (2007). A. Liebsch and N.-H. Tong, Phys. Rev. B [**80**]{}, 165126 (2009). M. Ferrero, P. S. Cornaglia, L. De Leo, O. Parcollet, G. Kotliar, and A. Georges, Phys. Rev. B [**80**]{}, 064501 (2009). A. Macridin, M. Jarrell, T. Maier, and D. J. Scalapino, Phys. Rev. Lett. [**99**]{}, 237001 (2007). P. Werner, E. Gull, O. Parcollet, and A. J. Millis, Phys. Rev. B [**80**]{}, 045120 (2009); E. Gull, O. Parcollet, P. Werner, and A. J. Millis, [*ibid*]{} [**80**]{}, 245102 (2009). S. Sakai, Y. Motome, and M. Imada, Physica B [**404**]{}, 3183 (2009). I. Dzyaloshinskii, Phys. Rev. B [**68**]{}, 085113 (2003). T. D. Stanescu, P. Phillips, and T.-P. Choy, Phys. Rev. B [**75**]{}, 104503 (2007). R. M. Konik, T. M. Rice, and A. M. Tsvelik, Phys. Rev. Lett. [**96**]{}, 086407 (2006); K.-Y. Yang, T. M. Rice, and F.-C. Zhang, Phys. Rev. B [**73**]{}, 174501 (2006); K.-Y. Yang, H.-B. Yang, P. D. Johnson, T. M. Rice, and F.-C. Zhang, Europhys. Lett. [**86**]{}, 37002 (2009); K. Le Hur and T. M. Rice, Ann. Phys. [**324**]{}, 1452 (2009). E. Z. Kuchinskii and M. V. Sadovskii, JETP [**103**]{}, 415 (2006). F. H. L. Essler and A. M. Tsvelik, Phys. Rev. B [**65**]{}, 115117 (2002); [**71**]{}, 195116 (2005). C. Berthod, T. Giamarchi, S. Biermann, and A. Georges, Phys. Rev. Lett. [**97**]{}, 136401 (2006). M. S. Hybertsen, E. B. Stechel, M. Schluter, and D. R. Jennison, Phys. Rev. B [**41**]{}, 11068 (1990). C. Lanczos, J. Research NBS [**45**]{}, 255 (1950). E. R. Gagliano and C. A. Balseiro, Phys. Rev. Lett. [**59**]{}, 2999 (1987). S. M. A. Rombouts, K. Heyde, and N. Jachowicz, Phys. Rev. Lett. [**82**]{}, 4155 (1999). S. Sakai, R. Arita, and H. Aoki, Physica B [**378-380**]{}, 288 (2006); S. Sakai, R. Arita, K. Held, and H. Aoki, Phys. Rev. B [**74**]{}, 155102 (2006). E. Gull, P. Werner, O. Parcollet, and M. Troyer, Eur. Phys. Lett. [**82**]{}, 57003 (2008). S. Rombouts, K. Heyde, and N. Jachowicz, Phys. Lett. A [**242**]{}, 271 (1998). J. E. Hirsch, Phys. Rev. B [**28**]{}, 4059 (1983); [**29**]{}, 4159 (1984). L. Balents, M. P. A. Fisher, and C. Nayak, Int. J. Mod. Phys. B [**12**]{}, 1033 (1998). M. Franz and Z. Tesanovic, Phys. Rev. Lett. [**87**]{}, 257003 (2001). M. R. Norman, H. Ding, M. Randeria, J. C. Campuzano, T. Yokoya, T. Takeuchi, T. Takahashi, T. Mochiku, K. Kadowaki, P. Guptasarma, and D. G. Hinks, Nature [**392**]{}, 157 (1998). T. Hanaguri, C. Lupien. Y. Kohsaka, D.-H. Lee, M. Azuma, M. Takano, H.Takagi, and J. C. Davis, Nature [**430**]{}, 1001 (2004). H.-B. Yang, J. D. Rameau, P. D. Johnson, T. Valla, A. Tsvelik, and G. D. Gu, Nature [**456**]{}, 77 (2008). J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. [**108**]{}, 1175 (1957). A. L. Efros and B. I. Shklovskii, J. Phys. C [**8**]{}, L49 (1975). H. Shinaoka and M. Imada, Phys. Rev. Lett. [**102**]{}, 016404 (2009); J. Phys. Soc. Jpn. [**78**]{}, 094708 (2009). A. Kanigel, U. Chatterjee, M. Randeria, M. R. Norman, G. Koren, K. Kadowaki, and J. C. Campuzano, Phys. Rev. Lett. [**101**]{}, 137002 (2008). M. Hashimoto, R.-H. He, K. Tanaka, J.-P. Testaud, W. Meevasana, R. G. Moore, D. Lu, H. Yao, Y. Yoshida, H. Eisaki, T. P. Devereaux, Z. Hussain, and Z.-X. Shen, Nature Phys. NPHYS1632 (2010). R. Preuss, W. Hanke, C. Grober, and H. G. Evertz, Phys. Rev. Lett. [**79**]{}, 1122 (1997). T. Yoshida, X. J. Zhou, D. H. Lu, S. Komiya, Y. Ando, H. Eisaki, T. Kakeshita, S. Uchida, Z. Hussain, Z.-X. Shen, and A. Fujimori, J. Phys.: Condens. Matter [**19**]{}, 125209 (2007). A. A. Kordyuk, S. V. Borisenko, A. Koitzsch, J. Fink, M. Knupfer, and H. Berger, Phys. Rev. B. [**71**]{}, 214513 (2005). J. Graf, G.-H. Gweon, K. McElroy, S. Y. Zhou, C. Jozwiak, E. Rotenberg, A. Bill, T. Sasagawa, H. Eisaki, S. Uchida, H. Takagi, D.-H. Lee, and A. Lanzara, Phys. Rev. Lett. [**98**]{}, 067004 (2007). B. P. Xie, K. Yang, D. W. Shen, J. F. Zhao, H. W. Ou, J. Weil, S. Y. Gu, M. Arita, S. Qiao, H. Namatame, M. Taniguchi, N. Kaneko, H. Eisaki, K. D. Tsuei, C. M. Cheng, I. Vobornik, J. Fujii, G. Rossi, Z. Q. Yang, and D. L. Feng, Phys. Rev. Lett. [**98**]{}, 147001 (2007). T. Valla, T. E. Kidd, W.-G. Yin, G. D. Gu, P. D. Johnson, Z.-H. Pan, and A. V. Fedorov, Phys. Rev. Lett. [**98**]{}, 167003 (2007). W. Meevasana, X. J. Zhou, S. Sahrakorpi, W. S. Lee, W. L. Yang, K. Tanaka, N. Mannella, T. Yoshida, D. H. Lu, Y. L. Chen, R. H. He, Hsin Lin, S. Komiya, Y. Ando, F. Zhou, W. X. Ti, J. W. Xiong, Z. X. Zhao, T. Sasagawa, T. Kakeshita, K. Fujita, S. Uchida, H. Eisaki, A. Fujimori, Z. Hussain, R. S. Markiewicz, A. Bansil, N. Nagaosa, J. Zaanen, T. P. Devereaux, and Z.-X. Shen, Phys. Rev. B. [**75**]{}, 174506 (2007). T. Yoshida, K. Tanaka, H. Yagi, A. Ino, H. Eisaki, A. Fujimori, and Z.-X. Shen, Phys. Rev. Lett. [**95**]{}, 146404 (2005). R. Eguchi, A. Chainani, M. Taguchi, M. Matsunami, Y. Ishida, K. Horiba, Y. Senba, H. Ohashi, and S. Shin, Phys. Rev. B [**79**]{}, 115122 (2009). K. Byczuk, M. Kollar, K. Held, Y.-F. Yang, I. A. Nekrasov, Th. Pruschke, and D. Vollhardt, Nature Phys. [**3**]{}, 168 (2007). P. V. Bogdanov, A. Lanzara, S. A. Kellar, X. J. Zhou, E. D. Lu, W. J. Zheng, G. Gu, J.-I. Shimoyama, K. Kishio, H. Ikeda, R. Yoshizaki, Z. Hussain, and Z. X. Shen, Phys. Rev. Lett. [**85**]{}, 2581 (2000). A. Kaminski, M. Randeria, J. C. Campuzano, M. R. Norman, H. Fretwell, J. Mesot, T. Sato, T. Takahashi, and K. Kadowaki, Phys. Rev. Lett. [**86**]{}, 1070 (2001). T. Sato, H. Matsui, T. Takahashi, H. Ding, H.-B. Yang, S.-C.Wang, T. Fujii, T. Watanabe, A. Matsuda, T. Terashima, and K. Kadowaki, Phys. Rev. Lett. [**91**]{}, 157003 (2003). A. Lanzara, P. V. Bogdanov, X. J. Zhou, S. A. Kellar, D. L. Feng, E. D. Lu, T. Yoshida, H. Eisaki, A. Fujimori, K. Kishio, J.-I. Shimoyama, T. Nodak, S. Uchidak, Z. Hussain, and Z.-X. Shen, Nature [**412**]{}, 510 (2001). P. D. Johnson, T. Valla, A. V. Fedorov, Z. Yusof, B. O. Wells, Q. Li, A. R. Moodenbaugh, G. D. Gu, N. Koshizuka, C. Kendziora, Sha Jian, and D. G. Hinks, Phys. Rev. Lett. [**87**]{}, 177007 (2001). Note that ARPES spectra are also usually integrated around the Fermi level and that a typical width of the energy window is $\sim 10\text{meV} \sim 0.025t$, which is similar to the $T$ window used in Eq. (\[eq:ghb\]). E. Gull, P. Werner, X. Wang, M. Troyer, and A. J. Millis, Europhys. Lett. [**84**]{}, 37009 (2008). J. Meng, G. Liu, W. Zhang, L. Zhao, H. Liu, X. Jia, D. Mu, S. Liu, X. Dong, J. Zhang, W. Lu, G. Wang, Y. Zhou, Y. Zhu, X. Wang, Z. Xu, C. Chen, and X. J. Zhou, Nature [**462**]{}, 335 (2009). M. Ikeda, T. Yoshida, A. Fujimori, M. Kubota, K. Ono, Y. Kaga, T. Sasagawa, and H. Takagi, Phys. Rev. B. [**80**]{}, 184506 (2009). The spectral feature that the band bottom is at a higher energy around $(\pi,0)$ than around $(0,0)$, as well as a waterfall-like structure at $\w>0$, can also be seen in Fig. 5(b) in Ref. . Y. Onose, Y. Taguchi, K. Ishizaka, and Y. Tokura, Phys. Rev. Lett. [**87**]{}, 217001 (2001). H. Kusunose and T. M. Rice, Phys. Rev. Lett. [**91**]{}, 186407 (2003). B. Kyung, V. Hankevych, A.-M. Dare, and A.-M. S. Tremblay, Phys. Rev. Lett. [**93**]{}, 147004 (2004). M. Ikeda, T. Yoshida, A. Fujimori, M. Kubota, K. Ono, H. Das, T. Saha-Dasgupta, K. Unozawa, Y. Kaga, T. Sasagawa, and H. Takagi, Phys. Rev. B [**80**]{}, 014510 (2009). N. P. Armitage, F. Ronning, D. H. Lu, C. Kim, A. Damascelli, K. M. Shen, D. L. Feng, H. Eisaki, Z.-X. Shen, P. K. Mang, N. Kaneko, M. Greven, Y. Onose, Y. Taguchi, and Y. Tokura, Phys. Rev. Lett. [**88**]{}, 257001 (2002). The doping evolution of the Fermi surface and zero surface in hole-doped cases was discussed in Ref. . M. M. Korshunov and S. G. Ovchinnikov, Eur. Phys. J. B [**57**]{}, 271 (2007); M. M. Korshunov, E. V. Zakharova, I. A. Nekrasov, Z. V. Pchelkina, and S. G. Ovchinnikov, J. Phys.: Condens. Matter [**22**]{}, 015701 (2010). T. Helm, M. V. Kartsovnik, M. Bartkowiak, N. Bittner, M. Lambacher, A. Erb, J. Wosnitza, and R. Gross, Phys. Rev. Lett. [**103**]{}, 157002 (2009). N. Lin, E. Gull, and A. J. Millis, arXiv:1004.2999. Y. Yamaji and M. Imada, unpublished. J. E. Hirsch and R. M. Fye, Phys. Rev. Lett. [**56**]{}, 2521 (1986). The density of states also shows a gap at the corresponding energy.

--- abstract: | We prove the analogue for continuous space-time of the quenched LDP derived in Birkner, Greven and den Hollander [@BiGrdHo10] for discrete space-time. In particular, we consider a random environment given by Brownian increments, cut into pieces according to an independent continuous-time renewal process. We look at the empirical process obtained by recording both the length of and the increments in the successive pieces. For the case where the renewal time distribution has a Lebesgue density with a polynomial tail, we derive the quenched LDP for the empirical process, i.e., the LDP conditional on a typical environment. The rate function is a sum of two specific relative entropies, one for the pieces and one for the concatenation of the pieces. We also obtain a quenched LDP when the tail decays faster than algebraic. The proof uses coarse-graining and truncation arguments, involving various approximations of specific relative entropies that are not quite standard. In a companion paper we show how the quenched LDP and the techniques developed in the present paper can be applied to obtain a variational characterisation of the free energy and the phase transition line for the Brownian copolymer near a selective interface. *MSC2010:* 60F10, 60G10, 60J65, 60K37.\ *Keywords:* Brownian environment, renewal process, annealed vs. quenched, empirical process, large deviation principle, specific relative entropy.\ *Acknowledgment:* The research in this paper is supported by ERC Advanced Grant 267356 VARIS of FdH. MB is grateful for hospitality at the Mathematical Institute in Leiden during a sabbatical leave from September 2012 until February 2013, supported by ERC. author: - | M. Birkner\ F. den Hollander date: 9th December 2013 title: A quenched large deviation principle in a continuous scenario --- Introduction and main result {#intro} ============================ When we cut an i.i.d. sequence of letters into words according to an independent integer-valued renewal process, we obtain an i.i.d. sequence of words. In the *annealed* LDP for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t. the reference law of words. Birkner, Greven and den Hollander [@BiGrdHo10] considered the *quenched* LDP, i.e., conditional on a typical letter sequence. The rate function of the quenched LDP turned out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t. the reference law of letters, with the former being obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal time distribution. The goal of the present paper is to derive the analogue of the quenched LDP for the case where the i.i.d. sequence of letters is replaced by the process of Brownian increments, and the renewal process has a length distribution with a Lebesgue density that has a polynomial tail. In Section \[setting\] we define the continuous space-time setting, in Section \[LDPs\] we state both the annealed and the quenched LDP, while in Section \[disc\] we discuss these LDPs and indicate some further extensions. In Section \[proof\] we prove the quenched LDP subject to three propositions. In Sections \[props\]–\[removeass\] we give the proof of these propositions. In Section \[proofalpha1infty\] we prove the extensions. Appendix \[metrics\] recalls a few basic facts about metrics on path space, while Appendices \[entropy\]–\[contrelentr\] prove a few basic facts about specific relative entropy that are needed in the proof and that are not quite standard. Continuous space-time {#setting} --------------------- Let $X=(X_t)_{t \geq 0}$ be the standard one-dimensional Brownian motion starting from $X_0=0$. Let ${\mathscr{W}}$ denote its law on path space: the Wiener measure on $C([0,\infty))$, equipped with the $\sigma$-algebra generated by the coordinate projections. Let $T=(T_i)_{i \in {\mathbb{N}}_0}$ ($T_0=0$) be an independent continuous-time renewal process, with interarrival times $\tau_i=T_i-T_{i-1}$, $i\in{\mathbb{N}}$, whose common law $\rho=\mathscr{L}(\tau_1)$ is absolutely continuous with respect to the Lebesgue measure on $(0,\infty)$, with density $\bar{\rho}$ satisfying $$\label{ass:rhodensdecay} \lim_{x\to\infty} \frac{\log\bar{\rho}(x)}{\log x} = - \alpha, \qquad \alpha \in (1,\infty).$$ In addition, assume that $$\label{ass:rhobar.reg0} \begin{minipage}{0.85\textwidth} $\mathrm{supp}(\rho) = [s_*,\infty)$ with $0 \leq s_* < \infty$, and $\bar{\rho}$ is continuous and strictly positive on $(s_*,\infty)$, and varies regularly near $s_*$. \end{minipage}$$ (0,0) rectangle (361.35,361.35); ( 2.40, 2.40) rectangle (358.95,358.95); ( 15.61,162.33) – (345.74,162.33); (338.42,166.36) – (341.56,164.65) – (345.14,163.38) – (349.03,162.60) – (353.07,162.33) – (353.07,162.33) – (349.03,162.07) – (345.14,161.29) – (341.56,160.02) – (338.42,158.31) – (338.42,158.31) – (338.76,158.41) – (339.08,158.71) – (339.37,159.19) – (339.61,159.82) – (339.79,160.59) – (339.90,161.44) – (339.94,162.33) – (339.90,163.23) – (339.79,164.08) – (339.61,164.84) – (339.37,165.48) – (339.08,165.96) – (338.76,166.26) – (338.42,166.36) – cycle; ( 15.61,162.33) – ( 15.94,163.78) – ( 16.27,161.59) – ( 16.60,159.54) – ( 16.93,160.84) – ( 17.26,158.90) – ( 17.59,159.60) – ( 17.92,157.91) – ( 18.25,162.99) – ( 18.58,169.37) – ( 18.91,171.53) – ( 19.24,172.96) – ( 19.57,172.49) – ( 19.90,174.03) – ( 20.23,181.44) – ( 20.56,182.39) – ( 20.89,182.86) – ( 21.22,182.87) – ( 21.55,176.96) – ( 21.88,181.97) – ( 22.21,190.43) – ( 22.54,186.60) – ( 22.87,187.90) – ( 23.20,188.38) – ( 23.53,190.46) – ( 23.86,189.69) – ( 24.19,191.41) – ( 24.52,188.72) – ( 24.85,189.63) – ( 25.18,192.42) – ( 25.51,193.43) – ( 25.84,187.48) – ( 26.17,187.08) – ( 26.50,185.68) – ( 26.83,184.93) – ( 27.16,176.03) – ( 27.49,177.69) – ( 27.82,172.43) – ( 28.15,178.64) – ( 28.48,173.22) – ( 28.81,164.52) – ( 29.14,169.59) – ( 29.47,171.64) – ( 29.80,171.10) – ( 30.13,166.26) – ( 30.46,168.12) – ( 30.79,170.35) – ( 31.12,172.40) – ( 31.45,170.64) – ( 31.78,170.16) – ( 32.11,168.50) – ( 32.44,168.21) – ( 32.77,171.97) – ( 33.10,170.81) – ( 33.43,167.06) – ( 33.76,165.03) – ( 34.09,167.83) – ( 34.42,172.37) – ( 34.75,175.09) – ( 35.08,168.99) – ( 35.41,171.08) – ( 35.74,173.81) – ( 36.07,175.59) – ( 36.40,168.07) – ( 36.73,172.84) – ( 37.06,175.36) – ( 37.39,171.05) – ( 37.72,171.36) – ( 38.05,169.27) – ( 38.39,170.48) – ( 38.72,172.81) – ( 39.05,170.21) – ( 39.38,168.26) – ( 39.71,165.13) – ( 40.04,162.23) – ( 40.37,160.18) – ( 40.70,159.97) – ( 41.03,159.69) – ( 41.36,159.26) – ( 41.69,157.81) – ( 42.02,163.04) – ( 42.35,161.43) – ( 42.68,156.62) – ( 43.01,148.97) – ( 43.34,150.66) – ( 43.67,151.58) – ( 44.00,151.35) – ( 44.33,150.49) – ( 44.66,140.15) – ( 44.99,142.98) – ( 45.32,139.32) – ( 45.65,144.90) – ( 45.98,136.91) – ( 46.31,138.99) – ( 46.64,132.30) – ( 46.97,132.08) – ( 47.30,135.59) – ( 47.63,136.14) – ( 47.96,132.33) – ( 48.29,133.80) – ( 48.62,136.85) – ( 48.95,135.64) – ( 49.28,137.39) – ( 49.61,133.43) – ( 49.94,129.17) – ( 50.27,133.93) – ( 50.60,135.33) – ( 50.93,137.62) – ( 51.26,136.47) – ( 51.59,136.07) – ( 51.92,134.90) – ( 52.25,135.79) – ( 52.58,130.82) – ( 52.91,130.10) – ( 53.24,132.73) – ( 53.57,133.36) – ( 53.90,142.93) – ( 54.23,141.95) – ( 54.56,141.64) – ( 54.89,143.25) – ( 55.22,149.41) – ( 55.55,151.74) – ( 55.88,153.06) – ( 56.21,162.66) – ( 56.54,168.64) – ( 56.87,166.72) – ( 57.20,164.37) – ( 57.53,155.91) – ( 57.86,151.62) – ( 58.19,160.67) – ( 58.52,162.00) – ( 58.85,159.70) – ( 59.18,157.72) – ( 59.51,159.15) – ( 59.84,161.56) – ( 60.17,159.91) – ( 60.50,156.12) – ( 60.83,153.61) – ( 61.16,155.32) – ( 61.49,156.12) – ( 61.82,158.02) – ( 62.16,160.85) – ( 62.49,161.11) – ( 62.82,159.73) – ( 63.15,160.63) – ( 63.48,157.34) – ( 63.81,160.60) – ( 64.14,160.85) – ( 64.47,160.34) – ( 64.80,160.90) – ( 65.13,168.70) – ( 65.46,165.38) – ( 65.79,166.83) – ( 66.12,169.33) – ( 66.45,167.26) – ( 66.78,170.76) – ( 67.11,168.82) – ( 67.44,164.58) – ( 67.77,170.12) – ( 68.10,177.37) – ( 68.43,176.15) – ( 68.76,179.62) – ( 69.09,179.71) – ( 69.42,180.19) – ( 69.75,178.91) – ( 70.08,186.32) – ( 70.41,183.84) – ( 70.74,179.87) – ( 71.07,180.00) – ( 71.40,188.72) – ( 71.73,182.11) – ( 72.06,181.57) – ( 72.39,180.12) – ( 72.72,176.85) – ( 73.05,177.30) – ( 73.38,179.62) – ( 73.71,182.55) – ( 74.04,182.70) – ( 74.37,183.30) – ( 74.70,182.27) – ( 75.03,182.22) – ( 75.36,181.50) – ( 75.69,184.22) – ( 76.02,183.65) – ( 76.35,178.58) – ( 76.68,182.69) – ( 77.01,188.94) – ( 77.34,189.21) – ( 77.67,186.79) – ( 78.00,185.20) – ( 78.33,182.63) – ( 78.66,180.80) – ( 78.99,177.16) – ( 79.32,175.06) – ( 79.65,169.62) – ( 79.98,172.75) – ( 80.31,175.36) – ( 80.64,175.16) – ( 80.97,175.56) – ( 81.30,177.05) – ( 81.63,177.49) – ( 81.96,175.25) – ( 82.29,174.08) – ( 82.62,172.44) – ( 82.95,172.27) – ( 83.28,170.18) – ( 83.61,173.79) – ( 83.94,170.97) – ( 84.27,171.84) – ( 84.60,168.01) – ( 84.93,171.34) – ( 85.26,174.13) – ( 85.59,172.13) – ( 85.93,172.15) – ( 86.26,177.04) – ( 86.59,172.69) – ( 86.92,170.13) – ( 87.25,164.17) – ( 87.58,168.98) – ( 87.91,166.36) – ( 88.24,168.31) – ( 88.57,170.04) – ( 88.90,172.97) – ( 89.23,182.76) – ( 89.56,183.62) – ( 89.89,186.49) – ( 90.22,187.63) – ( 90.55,190.17) – ( 90.88,188.71) – ( 91.21,182.24) – ( 91.54,180.87) – ( 91.87,180.77) – ( 92.20,180.59) – ( 92.53,181.03) – ( 92.86,179.60) – ( 93.19,182.92) – ( 93.52,177.21) – ( 93.85,178.44) – ( 94.18,179.21) – ( 94.51,179.97) – ( 94.84,180.61) – ( 95.17,180.12) – ( 95.50,180.53) – ( 95.83,180.34) – ( 96.16,190.58) – ( 96.49,194.48) – ( 96.82,191.22) – ( 97.15,190.34) – ( 97.48,184.60) – ( 97.81,186.95) – ( 98.14,181.80) – ( 98.47,183.21) – ( 98.80,181.83) – ( 99.13,183.31) – ( 99.46,181.57) – ( 99.79,183.81) – (100.12,183.91) – (100.45,187.30) – (100.78,191.76) – (101.11,195.42) – (101.44,195.63) – (101.77,196.44) – (102.10,194.73) – (102.43,193.23) – (102.76,191.60) – (103.09,198.12) – (103.42,196.52) – (103.75,200.69) – (104.08,201.54) – (104.41,201.90) – (104.74,202.27) – (105.07,199.36) – (105.40,195.38) – (105.73,198.02) – (106.06,196.32) – (106.39,198.78) – (106.72,202.74) – (107.05,205.89) – (107.38,200.59) – (107.71,200.39) – (108.04,197.96) – (108.37,203.79) – (108.70,209.82) – (109.03,216.18) – (109.36,218.39) – (109.70,219.87) – (110.03,212.15) – (110.36,214.22) – (110.69,207.46) – (111.02,208.96) – (111.35,207.20) – (111.68,205.55) – (112.01,201.79) – (112.34,197.57) – (112.67,197.28) – (113.00,199.80) – (113.33,202.19) – (113.66,199.62) – (113.99,201.48) – (114.32,198.95) – (114.65,198.88) – (114.98,195.00) – (115.31,198.39) – (115.64,194.21) – (115.97,196.46) – (116.30,195.87) – (116.63,198.92) – (116.96,205.32) – (117.29,202.39) – (117.62,205.91) – (117.95,201.42) – (118.28,201.44) – (118.61,197.01) – (118.94,192.33) – (119.27,196.29) – (119.60,197.02) – (119.93,200.61) – (120.26,199.05) – (120.59,195.86) – (120.92,197.09) – (121.25,192.65) – (121.58,193.14) – (121.91,199.03) – (122.24,201.72) – (122.57,196.94) – (122.90,199.67) – (123.23,196.29) – (123.56,196.94) – (123.89,204.30) – (124.22,213.13) – (124.55,213.25) – (124.88,212.46) – (125.21,216.19) – (125.54,221.37) – (125.87,213.17) – (126.20,213.06) – (126.53,213.21) – (126.86,214.20) – (127.19,222.84) – (127.52,216.51) – (127.85,223.10) – (128.18,221.10) – (128.51,221.54) – (128.84,220.01) – (129.17,221.11) – (129.50,223.96) – (129.83,228.91) – (130.16,227.96) – (130.49,234.23) – (130.82,234.61) – (131.15,231.28) – (131.48,228.32) – (131.81,228.57) – (132.14,229.23) – (132.47,239.02) – (132.80,239.74) – (133.13,241.87) – (133.47,234.42) – (133.80,232.14) – (134.13,237.75) – (134.46,237.07) – (134.79,235.75) – (135.12,233.85) – (135.45,231.74) – (135.78,233.66) – (136.11,225.93) – (136.44,226.73) – (136.77,227.52) – (137.10,230.76) – (137.43,227.00) – (137.76,234.79) – (138.09,232.12) – (138.42,234.66) – (138.75,235.80) – (139.08,234.71) – (139.41,232.44) – (139.74,234.44) – (140.07,234.81) – (140.40,224.35) – (140.73,232.53) – (141.06,238.96) – (141.39,237.36) – (141.72,230.46) – (142.05,230.84) – (142.38,231.28) – (142.71,234.70) – (143.04,234.30) – (143.37,232.08) – (143.70,232.70) – (144.03,235.58) – (144.36,235.07) – (144.69,233.43) – (145.02,228.12) – (145.35,228.37) – (145.68,232.98) – (146.01,230.77) – (146.34,233.58) – (146.67,236.81) – (147.00,241.11) – (147.33,234.45) – (147.66,235.57) – (147.99,236.56) – (148.32,235.01) – (148.65,232.37) – (148.98,237.98) – (149.31,241.02) – (149.64,246.15) – (149.97,242.29) – (150.30,229.52) – (150.63,229.61) – (150.96,228.39) – (151.29,232.02) – (151.62,227.27) – (151.95,221.02) – (152.28,213.36) – (152.61,211.49) – (152.94,210.32) – (153.27,210.94) – (153.60,206.97) – (153.93,209.02) – (154.26,210.39) – (154.59,207.60) – (154.92,207.49) – (155.25,207.10) – (155.58,208.32) – (155.91,205.47) – (156.24,208.75) – (156.57,213.58) – (156.90,212.61) – (157.24,212.47) – (157.57,217.98) – (157.90,214.83) – (158.23,212.02) – (158.56,211.23) – (158.89,217.78) – (159.22,211.94) – (159.55,214.47) – (159.88,216.76) – (160.21,220.74) – (160.54,215.35) – (160.87,217.00) – (161.20,220.68) – (161.53,226.07) – (161.86,227.25) – (162.19,224.58) – (162.52,226.61) – (162.85,225.82) – (163.18,227.91) – (163.51,224.23) – (163.84,221.02) – (164.17,223.97) – (164.50,224.40) – (164.83,228.81) – (165.16,231.43) – (165.49,237.76) – (165.82,235.92) – (166.15,230.46) – (166.48,229.64) – (166.81,235.66) – (167.14,234.52) – (167.47,234.84) – (167.80,236.71) – (168.13,237.52) – (168.46,236.74) – (168.79,231.86) – (169.12,237.77) – (169.45,235.00) – (169.78,233.38) – (170.11,230.54) – (170.44,227.22) – (170.77,229.26) – (171.10,229.10) – (171.43,229.16) – (171.76,236.46) – (172.09,233.20) – (172.42,236.14) – (172.75,239.23) – (173.08,236.66) – (173.41,244.17) – (173.74,248.64) – (174.07,239.55) – (174.40,237.44) – (174.73,238.78) – (175.06,236.14) – (175.39,240.74) – (175.72,240.68) – (176.05,241.19) – (176.38,237.89) – (176.71,242.98) – (177.04,247.72) – (177.37,246.98) – (177.70,247.28) – (178.03,248.02) – (178.36,248.41) – (178.69,245.31) – (179.02,252.07) – (179.35,253.23) – (179.68,252.12) – (180.01,248.55) – (180.34,244.15) – (180.67,246.85) – (181.01,251.95) – (181.34,254.35) – (181.67,253.16) – (182.00,245.95) – (182.33,239.46) – (182.66,241.70) – (182.99,236.71) – (183.32,234.62) – (183.65,236.87) – (183.98,230.99) – (184.31,232.04) – (184.64,223.22) – (184.97,223.69) – (185.30,219.03) – (185.63,214.00) – (185.96,212.67) – (186.29,213.98) – (186.62,214.49) – (186.95,221.15) – (187.28,222.66) – (187.61,225.91) – (187.94,227.62) – (188.27,226.36) – (188.60,224.53) – (188.93,224.52) – (189.26,223.79) – (189.59,224.46) – (189.92,225.43) – (190.25,225.91) – (190.58,226.08) – (190.91,225.43) – (191.24,218.70) – (191.57,211.95) – (191.90,213.77) – (192.23,218.90) – (192.56,223.07) – (192.89,220.63) – (193.22,222.21) – (193.55,219.58) – (193.88,222.59) – (194.21,221.76) – (194.54,221.55) – (194.87,226.57) – (195.20,220.80) – (195.53,222.11) – (195.86,221.04) – (196.19,219.93) – (196.52,215.40) – (196.85,207.54) – (197.18,209.61) – (197.51,206.69) – (197.84,205.55) – (198.17,206.15) – (198.50,202.37) – (198.83,207.16) – (199.16,204.30) – (199.49,206.52) – (199.82,199.35) – (200.15,194.79) – (200.48,200.97) – (200.81,202.15) – (201.14,201.79) – (201.47,203.54) – (201.80,199.50) – (202.13,205.55) – (202.46,206.47) – (202.79,205.18) – (203.12,202.96) – (203.45,206.30) – (203.78,207.63) – (204.11,202.62) – (204.44,200.70) – (204.78,196.79) – (205.11,204.41) – (205.44,201.80) – (205.77,200.74) – (206.10,205.37) – (206.43,204.00) – (206.76,197.62) – (207.09,194.13) – (207.42,194.06) – (207.75,198.72) – (208.08,196.40) – (208.41,193.98) – (208.74,192.00) – (209.07,190.51) – (209.40,192.45) – (209.73,201.69) – (210.06,201.17) – (210.39,205.64) – (210.72,208.75) – (211.05,207.28) – (211.38,206.20) – (211.71,205.38) – (212.04,201.88) – (212.37,200.89) – (212.70,201.29) – (213.03,194.23) – (213.36,195.33) – (213.69,197.48) – (214.02,194.52) – (214.35,199.85) – (214.68,198.18) – (215.01,195.67) – (215.34,201.15) – (215.67,199.51) – (216.00,207.69) – (216.33,205.32) – (216.66,210.90) – (216.99,213.68) – (217.32,212.47) – (217.65,211.42) – (217.98,209.79) – (218.31,211.95) – (218.64,216.32) – (218.97,218.96) – (219.30,216.24) – (219.63,222.50) – (219.96,217.14) – (220.29,211.39) – (220.62,214.94) – (220.95,216.84) – (221.28,211.99) – (221.61,212.00) – (221.94,213.57) – (222.27,222.54) – (222.60,224.81) – (222.93,228.58) – (223.26,224.40) – (223.59,230.01) – (223.92,232.62) – (224.25,238.72) – (224.58,245.31) – (224.91,245.21) – (225.24,249.23) – (225.57,250.78) – (225.90,246.38) – (226.23,249.82) – (226.56,253.98) – (226.89,251.08) – (227.22,253.96) – (227.55,252.86) – (227.88,247.13) – (228.21,245.41) – (228.55,243.95) – (228.88,248.27) – (229.21,244.80) – (229.54,248.41) – (229.87,250.08) – (230.20,246.72) – (230.53,245.55) – (230.86,248.37) – (231.19,248.10) – (231.52,247.01) – (231.85,248.99) – (232.18,251.64) – (232.51,250.12) – (232.84,250.16) – (233.17,244.70) – (233.50,244.72) – (233.83,236.17) – (234.16,233.66) – (234.49,234.23) – (234.82,239.14) – (235.15,250.48) – (235.48,245.34) – (235.81,240.36) – (236.14,240.69) – (236.47,243.92) – (236.80,246.40) – (237.13,243.28) – (237.46,244.01) – (237.79,251.20) – (238.12,254.25) – (238.45,255.04) – (238.78,255.00) – (239.11,259.04) – (239.44,259.14) – (239.77,266.13) – (240.10,264.21) – (240.43,269.21) – (240.76,266.62) – (241.09,263.77) – (241.42,266.82) – (241.75,264.14) – (242.08,264.15) – (242.41,254.93) – (242.74,251.71) – (243.07,246.16) – (243.40,242.13) – (243.73,241.97) – (244.06,240.62) – (244.39,241.02) – (244.72,249.78) – (245.05,256.52) – (245.38,258.86) – (245.71,262.44) – (246.04,260.04) – (246.37,262.31) – (246.70,257.04) – (247.03,256.15) – (247.36,250.51) – (247.69,253.00) – (248.02,258.47) – (248.35,255.29) – (248.68,259.00) – (249.01,261.94) – (249.34,263.23) – (249.67,262.69) – (250.00,267.16) – (250.33,267.94) – (250.66,271.27) – (250.99,278.01) – (251.32,273.25) – (251.65,270.81) – (251.98,270.29) – (252.32,270.85) – (252.65,269.61) – (252.98,267.66) – (253.31,271.13) – (253.64,269.60) – (253.97,263.28) – (254.30,263.28) – (254.63,260.89) – (254.96,266.78) – (255.29,265.33) – (255.62,275.53) – (255.95,277.96) – (256.28,272.13) – (256.61,275.83) – (256.94,279.30) – (257.27,278.67) – (257.60,275.19) – (257.93,270.61) – (258.26,273.76) – (258.59,272.35) – (258.92,265.35) – (259.25,261.65) – (259.58,258.02) – (259.91,262.44) – (260.24,254.63) – (260.57,252.54) – (260.90,250.93) – (261.23,251.18) – (261.56,250.71) – (261.89,246.91) – (262.22,251.58) – (262.55,254.17) – (262.88,256.25) – (263.21,249.34) – (263.54,246.51) – (263.87,247.81) – (264.20,247.92) – (264.53,251.01) – (264.86,252.13) – (265.19,248.78) – (265.52,251.45) – (265.85,251.79) – (266.18,249.59) – (266.51,248.59) – (266.84,244.71) – (267.17,240.02) – (267.50,239.30) – (267.83,239.88) – (268.16,239.07) – (268.49,234.20) – (268.82,238.51) – (269.15,233.07) – (269.48,227.14) – (269.81,229.75) – (270.14,232.43) – (270.47,238.94) – (270.80,233.41) – (271.13,231.07) – (271.46,235.33) – (271.79,234.62) – (272.12,234.92) – (272.45,240.07) – (272.78,242.76) – (273.11,244.46) – (273.44,251.92) – (273.77,249.04) – (274.10,250.86) – (274.43,246.15) – (274.76,244.27) – (275.09,238.48) – (275.42,240.93) – (275.75,241.16) – (276.09,240.75) – (276.42,247.29) – (276.75,245.28) – (277.08,247.95) – (277.41,251.00) – (277.74,251.67) – (278.07,250.16) – (278.40,249.70) – (278.73,250.12) – (279.06,258.25) – (279.39,252.94) – (279.72,240.96) – (280.05,243.59) – (280.38,242.00) – (280.71,242.91) – (281.04,237.02) – (281.37,235.68) – (281.70,235.73) – (282.03,237.13) – (282.36,234.99) – (282.69,231.86) – (283.02,234.34) – (283.35,233.88) – (283.68,235.64) – (284.01,235.89) – (284.34,231.60) – (284.67,227.84) – (285.00,230.97) – (285.33,233.05) – (285.66,228.66) – (285.99,230.28) – (286.32,233.51) – (286.65,237.46) – (286.98,236.76) – (287.31,240.35) – (287.64,240.32) – (287.97,247.11) – (288.30,250.31) – (288.63,249.51) – (288.96,250.87) – (289.29,252.69) – (289.62,249.74) – (289.95,248.08) – (290.28,253.85) – (290.61,252.54) – (290.94,250.68) – (291.27,252.45) – (291.60,247.94) – (291.93,250.95) – (292.26,251.98) – (292.59,249.74) – (292.92,257.42) – (293.25,251.20) – (293.58,257.37) – (293.91,260.03) – (294.24,265.07) – (294.57,269.85) – (294.90,267.27) – (295.23,264.21) – (295.56,266.82) – (295.89,262.48) – (296.22,257.93) – (296.55,247.13) – (296.88,248.21) – (297.21,244.84) – (297.54,250.42) – (297.87,251.20) – (298.20,254.59) – (298.53,258.18) – (298.86,262.72) – (299.19,257.03) – (299.52,253.77) – (299.86,255.90) – (300.19,254.51) – (300.52,251.41) – (300.85,256.59) – (301.18,254.11) – (301.51,249.85) – (301.84,251.42) – (302.17,247.50) – (302.50,243.68) – (302.83,243.08) – (303.16,246.73) – (303.49,250.11) – (303.82,255.08) – (304.15,256.28) – (304.48,252.91) – (304.81,254.17) – (305.14,257.11) – (305.47,259.03) – (305.80,259.48) – (306.13,262.14) – (306.46,264.52) – (306.79,261.40) – (307.12,263.11) – (307.45,267.93) – (307.78,265.37) – (308.11,275.41) – (308.44,276.54) – (308.77,273.85) – (309.10,271.35) – (309.43,272.37) – (309.76,274.79) – (310.09,271.90) – (310.42,275.56) – (310.75,276.66) – (311.08,276.26) – (311.41,283.76) – (311.74,283.90) – (312.07,280.50) – (312.40,275.85) – (312.73,279.26) – (313.06,285.42) – (313.39,289.36) – (313.72,291.51) – (314.05,291.02) – (314.38,287.60) – (314.71,284.52) – (315.04,288.90) – (315.37,289.94) – (315.70,294.69) – (316.03,286.10) – (316.36,290.07) – (316.69,290.24) – (317.02,284.59) – (317.35,286.91) – (317.68,284.17) – (318.01,288.19) – (318.34,294.16) – (318.67,298.83) – (319.00,298.05) – (319.33,303.31) – (319.66,304.55) – (319.99,304.07) – (320.32,309.41) – (320.65,313.13) – (320.98,313.67) – (321.31,316.29) – (321.64,317.33) – (321.97,314.70) – (322.30,316.53) – (322.63,308.27) – (322.96,302.30) – (323.29,297.26) – (323.63,296.35) – (323.96,297.05) – (324.29,304.16) – (324.62,302.97) – (324.95,302.08) – (325.28,308.07) – (325.61,311.21) – (325.94,310.19) – (326.27,310.64) – (326.60,309.98) – (326.93,310.49) – (327.26,309.83) – (327.59,307.09) – (327.92,302.60) – (328.25,302.22) – (328.58,308.29) – (328.91,312.81) – (329.24,317.89) – (329.57,317.26) – (329.90,319.15) – (330.23,320.71) – (330.56,326.20) – (330.89,323.12) – (331.22,326.01) – (331.55,330.71) – (331.88,331.80) – (332.21,331.79) – (332.54,332.55) – (332.87,329.31) – (333.20,336.26) – (333.53,336.09) – (333.86,336.15) – (334.19,343.27) – (334.52,342.85) – (334.85,346.95) – (335.18,346.22) – (335.51,345.34) – (335.84,345.01) – (336.17,342.60) – (336.50,347.99) – (336.83,341.90) – (337.16,344.08) – (337.49,343.88) – (337.82,338.46) – (338.15,340.81) – (338.48,338.70) – (338.81,336.82) – (339.14,338.28) – (339.47,338.43) – (339.80,341.46) – (340.13,346.47) – (340.46,349.57) – (340.79,350.21) – (341.12,346.49) – (341.45,347.69) – (341.78,337.64) – (342.11,333.52) – (342.44,329.43) – (342.77,328.51) – (343.10,327.76) – (343.43,326.43) – (343.76,328.38) – (344.09,325.85) – (344.42,323.55) – (344.75,319.87) – (345.08,320.35) – (345.41,320.71) – (345.74,322.50); ( 15.61,156.83) – ( 15.61,167.84); at ( 21.55,148.83) [$\scriptstyle T_0$]{}; (131.15,156.83) – (131.15,167.84); at (137.10,148.83) [$\scriptstyle T_1$]{}; (170.77,156.83) – (170.77,167.84); at (176.71,148.83) [$\scriptstyle T_2$]{}; (213.69,156.83) – (213.69,167.84); at (219.63,148.83) [$\scriptstyle T_3$]{}; (299.52,156.83) – (299.52,167.84); at (305.47,148.83) [$\scriptstyle T_4$]{}; ( 15.61, 2.40) – ( 15.61,358.95); (131.15, 2.40) – (131.15,358.95); (170.77, 2.40) – (170.77,358.95); (213.69, 2.40) – (213.69,358.95); (299.52, 2.40) – (299.52,358.95); ( 15.61, 55.96) – ( 15.94, 57.40) – ( 16.27, 55.21) – ( 16.60, 53.16) – ( 16.93, 54.46) – ( 17.26, 52.52) – ( 17.59, 53.22) – ( 17.92, 51.53) – ( 18.25, 56.61) – ( 18.58, 62.99) – ( 18.91, 65.15) – ( 19.24, 66.58) – ( 19.57, 66.11) – ( 19.90, 67.65) – ( 20.23, 75.06) – ( 20.56, 76.02) – ( 20.89, 76.48) – ( 21.22, 76.49) – ( 21.55, 70.58) – ( 21.88, 75.59) – ( 22.21, 84.06) – ( 22.54, 80.22) – ( 22.87, 81.52) – ( 23.20, 82.00) – ( 23.53, 84.09) – ( 23.86, 83.31) – ( 24.19, 85.04) – ( 24.52, 82.34) – ( 24.85, 83.26) – ( 25.18, 86.04) – ( 25.51, 87.06) – ( 25.84, 81.11) – ( 26.17, 80.70) – ( 26.50, 79.30) – ( 26.83, 78.55) – ( 27.16, 69.65) – ( 27.49, 71.31) – ( 27.82, 66.05) – ( 28.15, 72.26) – ( 28.48, 66.84) – ( 28.81, 58.15) – ( 29.14, 63.21) – ( 29.47, 65.26) – ( 29.80, 64.72) – ( 30.13, 59.88) – ( 30.46, 61.74) – ( 30.79, 63.97) – ( 31.12, 66.02) – ( 31.45, 64.26) – ( 31.78, 63.78) – ( 32.11, 62.12) – ( 32.44, 61.83) – ( 32.77, 65.59) – ( 33.10, 64.43) – ( 33.43, 60.69) – ( 33.76, 58.66) – ( 34.09, 61.45) – ( 34.42, 65.99) – ( 34.75, 68.72) – ( 35.08, 62.61) – ( 35.41, 64.70) – ( 35.74, 67.43) – ( 36.07, 69.22) – ( 36.40, 61.70) – ( 36.73, 66.47) – ( 37.06, 68.99) – ( 37.39, 64.67) – ( 37.72, 64.98) – ( 38.05, 62.89) – ( 38.39, 64.10) – ( 38.72, 66.43) – ( 39.05, 63.83) – ( 39.38, 61.89) – ( 39.71, 58.76) – ( 40.04, 55.85) – ( 40.37, 53.80) – ( 40.70, 53.60) – ( 41.03, 53.31) – ( 41.36, 52.88) – ( 41.69, 51.43) – ( 42.02, 56.66) – ( 42.35, 55.05) – ( 42.68, 50.24) – ( 43.01, 42.59) – ( 43.34, 44.28) – ( 43.67, 45.20) – ( 44.00, 44.98) – ( 44.33, 44.12) – ( 44.66, 33.77) – ( 44.99, 36.60) – ( 45.32, 32.95) – ( 45.65, 38.52) – ( 45.98, 30.53) – ( 46.31, 32.61) – ( 46.64, 25.92) – ( 46.97, 25.70) – ( 47.30, 29.21) – ( 47.63, 29.76) – ( 47.96, 25.95) – ( 48.29, 27.42) – ( 48.62, 30.47) – ( 48.95, 29.26) – ( 49.28, 31.01) – ( 49.61, 27.05) – ( 49.94, 22.79) – ( 50.27, 27.55) – ( 50.60, 28.95) – ( 50.93, 31.24) – ( 51.26, 30.10) – ( 51.59, 29.70) – ( 51.92, 28.52) – ( 52.25, 29.41) – ( 52.58, 24.45) – ( 52.91, 23.73) – ( 53.24, 26.35) – ( 53.57, 26.98) – ( 53.90, 36.55) – ( 54.23, 35.57) – ( 54.56, 35.26) – ( 54.89, 36.87) – ( 55.22, 43.03) – ( 55.55, 45.37) – ( 55.88, 46.68) – ( 56.21, 56.28) – ( 56.54, 62.26) – ( 56.87, 60.34) – ( 57.20, 58.00) – ( 57.53, 49.53) – ( 57.86, 45.25) – ( 58.19, 54.29) – ( 58.52, 55.62) – ( 58.85, 53.32) – ( 59.18, 51.34) – ( 59.51, 52.77) – ( 59.84, 55.18) – ( 60.17, 53.53) – ( 60.50, 49.74) – ( 60.83, 47.23) – ( 61.16, 48.94) – ( 61.49, 49.74) – ( 61.82, 51.65) – ( 62.16, 54.47) – ( 62.49, 54.73) – ( 62.82, 53.35) – ( 63.15, 54.26) – ( 63.48, 50.97) – ( 63.81, 54.22) – ( 64.14, 54.47) – ( 64.47, 53.97) – ( 64.80, 54.52) – ( 65.13, 62.32) – ( 65.46, 59.00) – ( 65.79, 60.46) – ( 66.12, 62.95) – ( 66.45, 60.88) – ( 66.78, 64.38) – ( 67.11, 62.44) – ( 67.44, 58.20) – ( 67.77, 63.75) – ( 68.10, 70.99) – ( 68.43, 69.77) – ( 68.76, 73.24) – ( 69.09, 73.34) – ( 69.42, 73.81) – ( 69.75, 72.54) – ( 70.08, 79.94) – ( 70.41, 77.46) – ( 70.74, 73.50) – ( 71.07, 73.62) – ( 71.40, 82.34) – ( 71.73, 75.73) – ( 72.06, 75.19) – ( 72.39, 73.74) – ( 72.72, 70.48) – ( 73.05, 70.92) – ( 73.38, 73.24) – ( 73.71, 76.17) – ( 74.04, 76.33) – ( 74.37, 76.93) – ( 74.70, 75.89) – ( 75.03, 75.84) – ( 75.36, 75.12) – ( 75.69, 77.84) – ( 76.02, 77.27) – ( 76.35, 72.20) – ( 76.68, 76.31) – ( 77.01, 82.56) – ( 77.34, 82.84) – ( 77.67, 80.41) – ( 78.00, 78.82) – ( 78.33, 76.25) – ( 78.66, 74.42) – ( 78.99, 70.78) – ( 79.32, 68.68) – ( 79.65, 63.24) – ( 79.98, 66.37) – ( 80.31, 68.98) – ( 80.64, 68.78) – ( 80.97, 69.18) – ( 81.30, 70.67) – ( 81.63, 71.11) – ( 81.96, 68.87) – ( 82.29, 67.70) – ( 82.62, 66.06) – ( 82.95, 65.90) – ( 83.28, 63.80) – ( 83.61, 67.42) – ( 83.94, 64.59) – ( 84.27, 65.46) – ( 84.60, 61.64) – ( 84.93, 64.97) – ( 85.26, 67.75) – ( 85.59, 65.75) – ( 85.93, 65.77) – ( 86.26, 70.67) – ( 86.59, 66.31) – ( 86.92, 63.75) – ( 87.25, 57.79) – ( 87.58, 62.60) – ( 87.91, 59.99) – ( 88.24, 61.93) – ( 88.57, 63.66) – ( 88.90, 66.59) – ( 89.23, 76.39) – ( 89.56, 77.24) – ( 89.89, 80.11) – ( 90.22, 81.25) – ( 90.55, 83.79) – ( 90.88, 82.33) – ( 91.21, 75.87) – ( 91.54, 74.49) – ( 91.87, 74.39) – ( 92.20, 74.22) – ( 92.53, 74.65) – ( 92.86, 73.22) – ( 93.19, 76.54) – ( 93.52, 70.83) – ( 93.85, 72.07) – ( 94.18, 72.83) – ( 94.51, 73.59) – ( 94.84, 74.23) – ( 95.17, 73.74) – ( 95.50, 74.15) – ( 95.83, 73.96) – ( 96.16, 84.20) – ( 96.49, 88.10) – ( 96.82, 84.84) – ( 97.15, 83.96) – ( 97.48, 78.22) – ( 97.81, 80.58) – ( 98.14, 75.42) – ( 98.47, 76.84) – ( 98.80, 75.45) – ( 99.13, 76.93) – ( 99.46, 75.20) – ( 99.79, 77.43) – (100.12, 77.53) – (100.45, 80.92) – (100.78, 85.38) – (101.11, 89.04) – (101.44, 89.26) – (101.77, 90.06) – (102.10, 88.35) – (102.43, 86.86) – (102.76, 85.22) – (103.09, 91.74) – (103.42, 90.14) – (103.75, 94.31) – (104.08, 95.16) – (104.41, 95.52) – (104.74, 95.89) – (105.07, 92.98) – (105.40, 89.00) – (105.73, 91.64) – (106.06, 89.94) – (106.39, 92.40) – (106.72, 96.36) – (107.05, 99.52) – (107.38, 94.21) – (107.71, 94.02) – (108.04, 91.58) – (108.37, 97.41) – (108.70,103.44) – (109.03,109.80) – (109.36,112.02) – (109.70,113.49) – (110.03,105.77) – (110.36,107.84) – (110.69,101.08) – (111.02,102.59) – (111.35,100.82) – (111.68, 99.17) – (112.01, 95.41) – (112.34, 91.19) – (112.67, 90.90) – (113.00, 93.42) – (113.33, 95.81) – (113.66, 93.24) – (113.99, 95.10) – (114.32, 92.57) – (114.65, 92.50) – (114.98, 88.63) – (115.31, 92.01) – (115.64, 87.83) – (115.97, 90.08) – (116.30, 89.49) – (116.63, 92.54) – (116.96, 98.94) – (117.29, 96.01) – (117.62, 99.53) – (117.95, 95.04) – (118.28, 95.06) – (118.61, 90.63) – (118.94, 85.95) – (119.27, 89.91) – (119.60, 90.64) – (119.93, 94.23) – (120.26, 92.67) – (120.59, 89.48) – (120.92, 90.71) – (121.25, 86.27) – (121.58, 86.76) – (121.91, 92.65) – (122.24, 95.34) – (122.57, 90.56) – (122.90, 93.29) – (123.23, 89.91) – (123.56, 90.56) – (123.89, 97.92) – (124.22,106.75) – (124.55,106.87) – (124.88,106.09) – (125.21,109.81) – (125.54,114.99) – (125.87,106.79) – (126.20,106.68) – (126.53,106.83) – (126.86,107.82) – (127.19,116.46) – (127.52,110.13) – (127.85,116.72) – (128.18,114.72) – (128.51,115.16) – (128.84,113.63) – (129.17,114.73) – (129.50,117.59) – (129.83,122.53) – (130.16,121.58) – (130.49,127.85) – (130.82,128.23) – (131.15,124.90); at ( 73.38, 5.77) [$\scriptstyle Y^{(1)}$]{}; (131.15, 55.96) – (131.48, 53.00) – (131.81, 53.25) – (132.14, 53.91) – (132.47, 63.70) – (132.80, 64.42) – (133.13, 66.55) – (133.47, 59.10) – (133.80, 56.82) – (134.13, 62.43) – (134.46, 61.75) – (134.79, 60.43) – (135.12, 58.53) – (135.45, 56.42) – (135.78, 58.34) – (136.11, 50.61) – (136.44, 51.41) – (136.77, 52.20) – (137.10, 55.44) – (137.43, 51.68) – (137.76, 59.47) – (138.09, 56.80) – (138.42, 59.34) – (138.75, 60.48) – (139.08, 59.39) – (139.41, 57.12) – (139.74, 59.12) – (140.07, 59.49) – (140.40, 49.03) – (140.73, 57.21) – (141.06, 63.64) – (141.39, 62.04) – (141.72, 55.14) – (142.05, 55.52) – (142.38, 55.96) – (142.71, 59.38) – (143.04, 58.98) – (143.37, 56.76) – (143.70, 57.38) – (144.03, 60.26) – (144.36, 59.75) – (144.69, 58.11) – (145.02, 52.80) – (145.35, 53.05) – (145.68, 57.66) – (146.01, 55.45) – (146.34, 58.26) – (146.67, 61.49) – (147.00, 65.79) – (147.33, 59.13) – (147.66, 60.25) – (147.99, 61.24) – (148.32, 59.69) – (148.65, 57.05) – (148.98, 62.66) – (149.31, 65.70) – (149.64, 70.83) – (149.97, 66.97) – (150.30, 54.20) – (150.63, 54.29) – (150.96, 53.07) – (151.29, 56.70) – (151.62, 51.95) – (151.95, 45.70) – (152.28, 38.04) – (152.61, 36.17) – (152.94, 35.00) – (153.27, 35.62) – (153.60, 31.65) – (153.93, 33.70) – (154.26, 35.07) – (154.59, 32.28) – (154.92, 32.17) – (155.25, 31.78) – (155.58, 33.00) – (155.91, 30.15) – (156.24, 33.43) – (156.57, 38.26) – (156.90, 37.29) – (157.24, 37.15) – (157.57, 42.66) – (157.90, 39.51) – (158.23, 36.70) – (158.56, 35.91) – (158.89, 42.46) – (159.22, 36.62) – (159.55, 39.15) – (159.88, 41.44) – (160.21, 45.42) – (160.54, 40.03) – (160.87, 41.68) – (161.20, 45.36) – (161.53, 50.75) – (161.86, 51.93) – (162.19, 49.26) – (162.52, 51.29) – (162.85, 50.50) – (163.18, 52.59) – (163.51, 48.91) – (163.84, 45.70) – (164.17, 48.65) – (164.50, 49.08) – (164.83, 53.49) – (165.16, 56.11) – (165.49, 62.44) – (165.82, 60.60) – (166.15, 55.14) – (166.48, 54.32) – (166.81, 60.34) – (167.14, 59.20) – (167.47, 59.52) – (167.80, 61.39) – (168.13, 62.20) – (168.46, 61.42) – (168.79, 56.54) – (169.12, 62.45) – (169.45, 59.68) – (169.78, 58.06) – (170.11, 55.22) – (170.44, 51.90) – (170.77, 53.94); at (150.96, 5.77) [$\scriptstyle Y^{(2)}$]{}; (170.77, 55.96) – (171.10, 55.79) – (171.43, 55.86) – (171.76, 63.15) – (172.09, 59.89) – (172.42, 62.83) – (172.75, 65.93) – (173.08, 63.35) – (173.41, 70.87) – (173.74, 75.34) – (174.07, 66.24) – (174.40, 64.14) – (174.73, 65.47) – (175.06, 62.83) – (175.39, 67.44) – (175.72, 67.37) – (176.05, 67.88) – (176.38, 64.59) – (176.71, 69.68) – (177.04, 74.41) – (177.37, 73.68) – (177.70, 73.97) – (178.03, 74.72) – (178.36, 75.11) – (178.69, 72.00) – (179.02, 78.76) – (179.35, 79.93) – (179.68, 78.81) – (180.01, 75.24) – (180.34, 70.84) – (180.67, 73.54) – (181.01, 78.64) – (181.34, 81.05) – (181.67, 79.85) – (182.00, 72.64) – (182.33, 66.16) – (182.66, 68.39) – (182.99, 63.40) – (183.32, 61.31) – (183.65, 63.56) – (183.98, 57.68) – (184.31, 58.74) – (184.64, 49.91) – (184.97, 50.38) – (185.30, 45.72) – (185.63, 40.69) – (185.96, 39.36) – (186.29, 40.67) – (186.62, 41.19) – (186.95, 47.84) – (187.28, 49.35) – (187.61, 52.60) – (187.94, 54.32) – (188.27, 53.05) – (188.60, 51.22) – (188.93, 51.22) – (189.26, 50.49) – (189.59, 51.15) – (189.92, 52.13) – (190.25, 52.60) – (190.58, 52.78) – (190.91, 52.13) – (191.24, 45.39) – (191.57, 38.64) – (191.90, 40.47) – (192.23, 45.59) – (192.56, 49.76) – (192.89, 47.32) – (193.22, 48.90) – (193.55, 46.27) – (193.88, 49.28) – (194.21, 48.46) – (194.54, 48.25) – (194.87, 53.27) – (195.20, 47.49) – (195.53, 48.80) – (195.86, 47.74) – (196.19, 46.62) – (196.52, 42.10) – (196.85, 34.23) – (197.18, 36.31) – (197.51, 33.39) – (197.84, 32.25) – (198.17, 32.84) – (198.50, 29.07) – (198.83, 33.86) – (199.16, 30.99) – (199.49, 33.21) – (199.82, 26.04) – (200.15, 21.48) – (200.48, 27.66) – (200.81, 28.85) – (201.14, 28.49) – (201.47, 30.24) – (201.80, 26.19) – (202.13, 32.25) – (202.46, 33.17) – (202.79, 31.87) – (203.12, 29.65) – (203.45, 32.99) – (203.78, 34.32) – (204.11, 29.31) – (204.44, 27.40) – (204.78, 23.48) – (205.11, 31.10) – (205.44, 28.49) – (205.77, 27.43) – (206.10, 32.06) – (206.43, 30.70) – (206.76, 24.32) – (207.09, 20.83) – (207.42, 20.76) – (207.75, 25.42) – (208.08, 23.10) – (208.41, 20.68) – (208.74, 18.69) – (209.07, 17.20) – (209.40, 19.14) – (209.73, 28.38) – (210.06, 27.86) – (210.39, 32.34) – (210.72, 35.44) – (211.05, 33.98) – (211.38, 32.89) – (211.71, 32.07) – (212.04, 28.57) – (212.37, 27.59) – (212.70, 27.98) – (213.03, 20.93) – (213.36, 22.02) – (213.69, 24.18); at (192.23, 5.77) [$\scriptstyle Y^{(3)}$]{}; (213.69, 55.96) – (214.02, 53.00) – (214.35, 58.32) – (214.68, 56.65) – (215.01, 54.14) – (215.34, 59.62) – (215.67, 57.99) – (216.00, 66.17) – (216.33, 63.80) – (216.66, 69.37) – (216.99, 72.16) – (217.32, 70.94) – (217.65, 69.90) – (217.98, 68.27) – (218.31, 70.42) – (218.64, 74.79) – (218.97, 77.43) – (219.30, 74.71) – (219.63, 80.97) – (219.96, 75.62) – (220.29, 69.86) – (220.62, 73.41) – (220.95, 75.31) – (221.28, 70.46) – (221.61, 70.47) – (221.94, 72.04) – (222.27, 81.01) – (222.60, 83.28) – (222.93, 87.06) – (223.26, 82.88) – (223.59, 88.48) – (223.92, 91.10) – (224.25, 97.19) – (224.58,103.79) – (224.91,103.68) – (225.24,107.70) – (225.57,109.25) – (225.90,104.85) – (226.23,108.30) – (226.56,112.46) – (226.89,109.55) – (227.22,112.43) – (227.55,111.33) – (227.88,105.61) – (228.21,103.88) – (228.55,102.42) – (228.88,106.74) – (229.21,103.27) – (229.54,106.88) – (229.87,108.55) – (230.20,105.19) – (230.53,104.02) – (230.86,106.85) – (231.19,106.57) – (231.52,105.48) – (231.85,107.46) – (232.18,110.11) – (232.51,108.60) – (232.84,108.63) – (233.17,103.17) – (233.50,103.20) – (233.83, 94.64) – (234.16, 92.13) – (234.49, 92.71) – (234.82, 97.61) – (235.15,108.95) – (235.48,103.81) – (235.81, 98.83) – (236.14, 99.17) – (236.47,102.39) – (236.80,104.87) – (237.13,101.75) – (237.46,102.49) – (237.79,109.67) – (238.12,112.72) – (238.45,113.51) – (238.78,113.48) – (239.11,117.52) – (239.44,117.61) – (239.77,124.61) – (240.10,122.68) – (240.43,127.69) – (240.76,125.10) – (241.09,122.24) – (241.42,125.29) – (241.75,122.61) – (242.08,122.62) – (242.41,113.40) – (242.74,110.18) – (243.07,104.63) – (243.40,100.60) – (243.73,100.44) – (244.06, 99.09) – (244.39, 99.50) – (244.72,108.25) – (245.05,114.99) – (245.38,117.33) – (245.71,120.91) – (246.04,118.51) – (246.37,120.79) – (246.70,115.51) – (247.03,114.62) – (247.36,108.98) – (247.69,111.48) – (248.02,116.94) – (248.35,113.77) – (248.68,117.47) – (249.01,120.41) – (249.34,121.71) – (249.67,121.16) – (250.00,125.63) – (250.33,126.42) – (250.66,129.75) – (250.99,136.48) – (251.32,131.72) – (251.65,129.28) – (251.98,128.77) – (252.32,129.32) – (252.65,128.09) – (252.98,126.13) – (253.31,129.60) – (253.64,128.07) – (253.97,121.75) – (254.30,121.75) – (254.63,119.36) – (254.96,125.26) – (255.29,123.80) – (255.62,134.00) – (255.95,136.43) – (256.28,130.60) – (256.61,134.30) – (256.94,137.77) – (257.27,137.14) – (257.60,133.66) – (257.93,129.08) – (258.26,132.23) – (258.59,130.82) – (258.92,123.82) – (259.25,120.12) – (259.58,116.50) – (259.91,120.91) – (260.24,113.10) – (260.57,111.01) – (260.90,109.40) – (261.23,109.65) – (261.56,109.18) – (261.89,105.38) – (262.22,110.05) – (262.55,112.64) – (262.88,114.72) – (263.21,107.81) – (263.54,104.98) – (263.87,106.29) – (264.20,106.39) – (264.53,109.48) – (264.86,110.61) – (265.19,107.25) – (265.52,109.92) – (265.85,110.26) – (266.18,108.07) – (266.51,107.06) – (266.84,103.18) – (267.17, 98.50) – (267.50, 97.78) – (267.83, 98.35) – (268.16, 97.54) – (268.49, 92.68) – (268.82, 96.98) – (269.15, 91.54) – (269.48, 85.62) – (269.81, 88.22) – (270.14, 90.90) – (270.47, 97.41) – (270.80, 91.88) – (271.13, 89.54) – (271.46, 93.80) – (271.79, 93.09) – (272.12, 93.40) – (272.45, 98.54) – (272.78,101.24) – (273.11,102.93) – (273.44,110.39) – (273.77,107.51) – (274.10,109.34) – (274.43,104.62) – (274.76,102.74) – (275.09, 96.96) – (275.42, 99.40) – (275.75, 99.63) – (276.09, 99.23) – (276.42,105.76) – (276.75,103.76) – (277.08,106.42) – (277.41,109.47) – (277.74,110.14) – (278.07,108.63) – (278.40,108.18) – (278.73,108.60) – (279.06,116.72) – (279.39,111.41) – (279.72, 99.43) – (280.05,102.07) – (280.38,100.48) – (280.71,101.38) – (281.04, 95.49) – (281.37, 94.15) – (281.70, 94.20) – (282.03, 95.61) – (282.36, 93.47) – (282.69, 90.33) – (283.02, 92.81) – (283.35, 92.35) – (283.68, 94.12) – (284.01, 94.36) – (284.34, 90.07) – (284.67, 86.31) – (285.00, 89.44) – (285.33, 91.52) – (285.66, 87.13) – (285.99, 88.75) – (286.32, 91.98) – (286.65, 95.93) – (286.98, 95.23) – (287.31, 98.82) – (287.64, 98.80) – (287.97,105.58) – (288.30,108.78) – (288.63,107.98) – (288.96,109.35) – (289.29,111.17) – (289.62,108.21) – (289.95,106.55) – (290.28,112.32) – (290.61,111.02) – (290.94,109.15) – (291.27,110.93) – (291.60,106.41) – (291.93,109.42) – (292.26,110.46) – (292.59,108.21) – (292.92,115.89) – (293.25,109.67) – (293.58,115.84) – (293.91,118.50) – (294.24,123.54) – (294.57,128.32) – (294.90,125.74) – (295.23,122.69) – (295.56,125.30) – (295.89,120.96) – (296.22,116.41) – (296.55,105.60) – (296.88,106.68) – (297.21,103.31) – (297.54,108.89) – (297.87,109.68) – (298.20,113.07) – (298.53,116.66) – (298.86,121.19) – (299.19,115.50) – (299.52,112.25); at (256.61, 5.77) [$\scriptstyle Y^{(4)}$]{}; ( 15.61, 55.96) – (345.74, 55.96); (299.52, 55.96) – (299.86, 58.09) – (300.19, 56.70) – (300.52, 53.59) – (300.85, 58.77) – (301.18, 56.29) – (301.51, 52.03) – (301.84, 53.60) – (302.17, 49.68) – (302.50, 45.86) – (302.83, 45.26) – (303.16, 48.91) – (303.49, 52.29) – (303.82, 57.26) – (304.15, 58.47) – (304.48, 55.09) – (304.81, 56.36) – (305.14, 59.29) – (305.47, 61.21) – (305.80, 61.66) – (306.13, 64.32) – (306.46, 66.71) – (306.79, 63.58) – (307.12, 65.29) – (307.45, 70.12) – (307.78, 67.55) – (308.11, 77.59) – (308.44, 78.73) – (308.77, 76.03) – (309.10, 73.53) – (309.43, 74.56) – (309.76, 76.97) – (310.09, 74.08) – (310.42, 77.75) – (310.75, 78.84) – (311.08, 78.44) – (311.41, 85.95) – (311.74, 86.09) – (312.07, 82.68) – (312.40, 78.03) – (312.73, 81.44) – (313.06, 87.60) – (313.39, 91.54) – (313.72, 93.69) – (314.05, 93.20) – (314.38, 89.78) – (314.71, 86.70) – (315.04, 91.09) – (315.37, 92.12) – (315.70, 96.87) – (316.03, 88.28) – (316.36, 92.25) – (316.69, 92.42) – (317.02, 86.78) – (317.35, 89.09) – (317.68, 86.35) – (318.01, 90.37) – (318.34, 96.35) – (318.67,101.01) – (319.00,100.23) – (319.33,105.49) – (319.66,106.73) – (319.99,106.25) – (320.32,111.59) – (320.65,115.31) – (320.98,115.85) – (321.31,118.47) – (321.64,119.51) – (321.97,116.89) – (322.30,118.72) – (322.63,110.46) – (322.96,104.48) – (323.29, 99.45) – (323.63, 98.53) – (323.96, 99.24) – (324.29,106.34) – (324.62,105.15) – (324.95,104.26) – (325.28,110.25) – (325.61,113.39) – (325.94,112.37) – (326.27,112.82) – (326.60,112.17) – (326.93,112.67) – (327.26,112.01) – (327.59,109.27) – (327.92,104.78) – (328.25,104.40) – (328.58,110.47) – (328.91,114.99) – (329.24,120.07) – (329.57,119.44) – (329.90,121.34) – (330.23,122.89) – (330.56,128.39) – (330.89,125.31) – (331.22,128.19) – (331.55,132.89) – (331.88,133.98) – (332.21,133.97) – (332.54,134.73) – (332.87,131.49) – (333.20,138.44) – (333.53,138.27) – (333.86,138.33) – (334.19,145.45) – (334.52,145.03) – (334.85,149.13) – (335.18,148.40) – (335.51,147.53) – (335.84,147.20) – (336.17,144.78) – (336.50,150.17) – (336.83,144.08) – (337.16,146.26) – (337.49,146.06) – (337.82,140.65) – (338.15,142.99) – (338.48,140.88) – (338.81,139.00) – (339.14,140.47) – (339.47,140.61) – (339.80,143.65) – (340.13,148.65) – (340.46,151.75) – (340.79,152.39) – (341.12,148.67) – (341.45,149.87) – (341.78,139.82) – (342.11,135.70) – (342.44,131.61) – (342.77,130.69) – (343.10,129.94) – (343.43,128.61) – (343.76,130.56) – (344.09,128.03) – (344.42,125.73) – (344.75,122.06) – (345.08,122.53) – (345.41,122.89) – (345.74,124.68); Define the *word sequence* $Y = (Y^{(i)})_{i\in{\mathbb{N}}}$ by putting (see Fig. \[fig-wordsequence\]) $$\qquad Y^{(i)} = \Big(T_i-T_{i-1}, \big(X_{(s+T_{i-1}) \wedge T_i} -X_{T_{i-1}}\big)_{s\geq 0}\Big),$$ which takes values in the *word space* $$\label{eq:defF} F = \bigcup_{t>0} \Big(\{t\} \times \big\{ f \in C([0,\infty))\colon\, f(0)=0, f(s)=f(t) \; \text{for} \; s > t \big\}\Big)$$ equipped with a Skorohod-type metric (see Appendix \[metrics\]). Let $$Y^{N\text{-}\mathrm{per}} = \big(\,\underbrace{Y^{(1)},Y^{(2)},\dots,Y^{(N)}},\, \underbrace{Y^{(1)},Y^{(2)},\dots,Y^{(N)}},\,\dots\big)$$ denote the $N$-periodisation of $Y$, and let $$\label{RNdef} R_N = \frac1N \sum_{i=0}^{N-1} \delta_{\widetilde{\theta}^i Y^{N\text{-}\mathrm{per}}}$$ be the *empirical process of words*, where $\widetilde{\theta}$ is the left-shift acting on $F^{\mathbb{N}}$. Note that $R_N$ takes values in $\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$, the set of shift-invariant probability measures on $F^{\mathbb{N}}$. Endow $F^{\mathbb{N}}$ with the product topology and $\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ with the corresponding weak topology. When averaged over $X$ and $T$, the law of $Y$ is ($\mathscr{L}$ denotes law) $$\label{qrhoWdef} Q_{\rho,{\mathscr{W}}}=(q_{\rho,{\mathscr{W}}})^{\otimes{\mathbb{N}}} \quad \text{with} \quad q_{\rho,{\mathscr{W}}} = \int_{(0,\infty)} \rho(dt)\, \mathscr{L}\big((t, (X_{s \wedge t})_{s \geq 0})\big).$$ By the ergodic theorem, ${\mathop{\text{\rm w-lim}}}_{N\to\infty} R_N = Q_{\rho,{\mathscr{W}}}$ a.s., where ${\mathop{\text{\rm w-lim}}}$ denotes the weak limit. Large deviation principles {#LDPs} -------------------------- For definitions and properties of specific relative entropy, we refer the reader to Appendix \[entropy\]. The following theorem is standard (see e.g. Dembo and Zeitouni [@DeZe98 Section 6.5.3]). \[thm0:contaLDP\] [[**\[Annealed LDP\]**]{}]{}\ The family $\mathscr{L}(R_N)$, $N\in{\mathbb{N}}$, satisfies the LDP on $\mathcal{P}^{\mathrm{inv}} (F^{\mathbb{N}})$ with rate $N$ and with rate function $$\label{eq:Iann} I^{\mathrm{ann}}(Q)= H(Q \mid Q_{\rho,{\mathscr{W}}}),$$ the specific relative entropy of $Q$ w.r.t. $Q_{\rho,{\mathscr{W}}}$. This rate function is lower semi-continuous, has compact level sets, is affine, and has a unique zero at $Q=Q_{\rho,{\mathscr{W}}}$. To state the quenched LDP, we need to look at the reverse of cutting out words, namely, glueing words together. Let ${y}=(y^{(i)})_{i\in{\mathbb{N}}}=((t_i,f_i))_{i\in{\mathbb{N}}} \in F^{\mathbb{N}}$. Then the *concatenation* of ${y}$, written $\kappa({y}) \in C([0,\infty))$, is defined by $$\begin{aligned} &\kappa({y})(s) = f_1(t_1)+\dots+f_{i-1}(t_{i-1}) +f_i\big(s-(t_1+\cdots+t_{i-1})\big),\\ &t_1+\cdots+t_{i-1} \leq s < t_1+\cdots+t_{i}, \;\; i \in {\mathbb{N}}. \end{aligned}$$ Write $\tau_i({y})=t_i$ to denote the length of the $i$-th word. For $Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ with finite mean word length $m_Q = {\mathbb{E}}_Q[\tau_1] ={\mathbb{E}}_Q[\tau_1(Y)]$, put $$\label{eq:PsiQcont} \Psi_Q(A) = \frac{1}{m_Q} {\mathbb{E}}_Q\left[ \int_0^{\tau_1} {1}_A(\theta^s \kappa(Y)) \, ds\right], \quad A \subset C([0,\infty))\;\;\mbox{measurable},$$ where $\theta^s$ is the shift acting on $f \in C([0,\infty))$ as $\theta^s f(t) = f(s+t)-f(s)$, $t \geq 0$. Note that $\Psi_Q$ is a probability measure on $C([0,\infty))$ with stationary increments, i.e., $\Psi_Q = \Psi_Q \circ (\theta^s)^{-1}$ for all $s \ge 0$. We can think of $\Psi_Q$ as the “stationarised” version of $\kappa(Q)$. In fact, if $m_Q<\infty$, then $$\label{eq:PsiQ} \Psi_Q = {\mathop{\text{\rm w-lim}}}_{T\to\infty} \frac{1}{T} \int_0^T \kappa(Q) \circ (\theta^s)^{-1}\,ds,$$ and $\kappa(Q)$ is asymptotically mean stationary (AMS) with stationary mean $\Psi_Q$. In fact, the convergence in also holds in total variation norm (see Lemma \[lemma:PsiQ:TVlim\] in Appendix \[entropy\]). Note that $\Psi_{Q_{\rho,{\mathscr{W}}}}={\mathscr{W}}$. To state the quenched LDP, we also need to define word *truncation*. For $(t,f) \in F$ and ${{\rm tr}}> 0$, let $$[(t,f)]_{{\rm tr}}= \big(t \wedge {{\rm tr}}, (f(s \wedge {{\rm tr}})_{s\ge 0}\big)$$ be the word $(t,f)$ truncated at length ${{\rm tr}}$. Analogously, for ${y}=(y^{(i)})_{i\in{\mathbb{N}}} \in F^{\mathbb{N}}$ set $[{y}]_{{\rm tr}}=([y^{(i)}]_{{{\rm tr}}})_{i\in{\mathbb{N}}} \in F^{\mathbb{N}}$, and denote by $[Q]_{{\rm tr}}\in \mathcal{P}^{\mathrm{inv}}(F_{0,{{\rm tr}}}^{\mathbb{N}}) \subset \mathcal{P}^{\mathrm{inv}} (F^{\mathbb{N}})$ with $F_{0,{{\rm tr}}}=[F]_{{\rm tr}}$ the image measure of $Q \in \mathcal{P}^{\mathrm{inv}} (F^{\mathbb{N}})$ under the map ${y} \mapsto [{y}]_{{\rm tr}}$. \[thm0:contqLDP\] [[**\[Quenched LDP\]**]{}]{}\ Suppose that $\rho$ satisfies [(\[ass:rhodensdecay\]–\[ass:rhobar.reg0\])]{}. Then, for ${\mathscr{W}}$ a.e. $X$, the family $\mathscr{L}(R_N \mid X)$, $N\in{\mathbb{N}}$, satisfies the LDP on $\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ with rate $N$ and with deterministic rate function $I^{\mathrm{que}}(Q)$ given by $$\label{eq:Iquelimitform} I^{\mathrm{que}}(Q) = \lim_{{{\rm tr}}\to\infty} I^{\mathrm{que}}_{{\rm tr}}([Q]_{{\rm tr}}),$$ where $$\label{def:Ique.tr} I^{\mathrm{que}}_{{\rm tr}}([Q]_{{\rm tr}}) = H\big([Q]_{{\rm tr}}\mid [Q_{\rho,{\mathscr{W}}}]_{{\rm tr}}\big) + (\alpha-1) m_{[Q]_{{\rm tr}}} H\big(\Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}\big).$$ This rate function is lower semi-continuous, has compact level sets, is affine, and has a unique zero at $Q=Q_{\rho,{\mathscr{W}}}$. Theorem \[thm0:contqLDP\] is proved in Sections \[proof\]–\[removeass\]. Let $\mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}})=\{Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}}) \colon\,m_Q<\infty\}$. We will show that the limit in exists for all $Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$, and that $$\label{eq:Ique} I^{\mathrm{que}}(Q) = H(Q \mid Q_{\rho,{\mathscr{W}}}) + (\alpha-1) m_Q H(\Psi_Q \mid {\mathscr{W}}), \qquad Q \in \mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}}).$$ We will also see that $I^{\mathrm{que}}(Q)$ is the lower semi-continuous extension to $\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ of its restriction to $\mathcal{P}^{\mathrm{inv,fin}} (F^{\mathbb{N}})$. Discussion {#disc} ---------- [**0.**]{} A *heuristic* behind Theorem \[thm0:contqLDP\] is as follows. Let $$\label{RNdefind} R^N_{t_1,\dots,t_N}(X), \qquad 0<t_1<\dots<t_N<\infty,$$ denote the empirical process of $N$-tuples of words when $X$ is cut at the points $t_1, \dots,t_N$ (i.e., when $T_i=t_i$ for $i=1,\dots,N$). Fix $Q \in \mathcal{P}^{\mathrm{inv,fin}} (F^{\mathbb{N}})$ and suppose that $Q$ is shift-ergodic. The probability ${\mathbb{P}}(R_N \approx Q \mid X)$ is an integral over all $N$-tuples $t_1,\dots,t_N$ such that $R^N_{t_1,\dots,t_N}(X) \approx Q$, weighted by $\prod_{i=1}^N \bar{\rho}(t_i-t_{i-1})$ (with $t_0=0$). The fact that $R^N_{t_1,\dots,t_N}(X) \approx Q$ has three consequences: 1. The $t_1,\dots,t_N$ must cut $\approx N$ substrings out of $X$ of total length $\approx N m_Q$ that look like the concatenation of words that are $Q$-typical, i.e., that look as if generated by $\Psi_Q$ (possibly with gaps in between). This means that most of the cut-points must hit atypical pieces of $X$. We expect to have to shift $X$ by $\approx\exp[N m_Q H(\Psi_Q \mid {\mathscr{W}})]$ in order to find the first contiguous substring of length $N m_Q$ whose empirical shifts lie in a small neighbourhood of $\Psi_Q$. By (\[ass:rhodensdecay\]), the probability for the single increment $t_1-t_0$ to have the size of this shift is $\approx \exp[-N\alpha\,m_Q H(\Psi_Q \mid {\mathscr{W}})]$. 2. The “number of local perturbations” of $t_1,\dots,t_N$ preserving the property $R^N_{t_1,\dots,t_N}(X)\approx Q$ is $\approx \exp[NH_{\tau|K}(Q)]$, where $H_{\tau|K}$ stands for the *conditional specific entropy (density) of word lengths under the law $Q$*. 3. The statistics of the increments $t_1-t_0,\dots,t_N-t_{N-1}$ must be close to the distribution of word lengths under $Q$. Hence, the weight factor $\prod_{i=1}^N \bar{\rho}(t_i-t_{i-1})$ must be $\approx \exp[N {\mathbb{E}}_Q[\log\bar{\rho}(\tau_1)]]$ (at least, for $Q$-typical pieces). Since $$\label{eqnsre1} m_Q H(\Psi_Q \mid {\mathscr{W}}) - H_{\tau|K}(Q) - {\mathbb{E}}_Q[\log\bar{\rho}(\tau_1)] = H(Q \mid q_{\rho,{\mathscr{W}}}),$$ the observations made in (1)–(3) combine to explain the shape of the quenched rate function in . For further details, see [@BiGrdHo10 Section 1.5]. *Note:* We have not defined $H_{\tau|K}(Q)$ rigorously here, nor do we prove . Our proof of Theorem \[thm0:contqLDP\] uses the above heuristic only very implicitly. Rather, it starts from the discrete-time quenched LDP derived in [@BiGrdHo10] and draws out Theorem \[thm0:contqLDP\] via control of exponential functionals through a coarse-graining approximation. [**1.**]{} We can include the cases $\alpha=1$ and $\alpha=\infty$ in . \[mainthmboundarycases\] Suppose that $\rho$ satisfies [(\[ass:rhodensdecay\]–\[ass:rhobar.reg0\])]{}.\ [(a)]{} If $\alpha=1$, then the quenched LDP holds with $I^\mathrm{que}=I^\mathrm{ann}$ given by .\ [(b)]{} If $\alpha=\infty$, then the quenched LDP holds with rate function $$\label{eq:ratefctalphainfty} I^\mathrm{que}(Q) = \begin{cases} H(Q \mid Q_{\rho,{\mathscr{W}}}) & \mbox{if} \;\; \lim\limits_{{{\rm tr}}\to\infty} m_{[Q]_{{\rm tr}}} H( \Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}) = 0, \\ \infty & \mbox{otherwise}. \end{cases}$$ Theorem \[mainthmboundarycases\] is the continuous analogue of Birkner, Greven and den Hollander [@BiGrdHo10 Theorem 1.4] and is proved in Section \[proofalpha1infty\]. [**2.**]{} We can also include the case where $\bar{\rho}$ has an exponentially bounded tail: $$\label{ass:rhoexp} \bar{\rho}(t) \leq e^{-\lambda t} \mbox{ for some } \lambda >0 \mbox{ and } t \mbox{ large enough}.$$ \[thmexp\] Suppose that $\rho$ satisfies [(\[ass:rhodensdecay\]–\[ass:rhobar.reg0\])]{} and . Then, for ${\mathscr{W}}$ a.e. $X$, the family $\mathscr{L}(R_N \mid X)$, $N\in{\mathbb{N}}$, satisfies the LDP on $\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ with rate $N$ and with deterministic rate function $I^{\mathrm{que}}(Q)$ given by $$\label{eq:ratefctexptail} I^\mathrm{que}(Q) = \left\{\begin{array}{ll} H(Q \mid Q_{\rho,{\mathscr{W}}}) &\mbox{if } Q \in {{\mathcal R}}_{\mathscr{W}},\\ \infty &\mbox{otherwise}, \end{array} \right.$$ where $${{\mathcal R}}_{\mathscr{W}}= \left\{Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})\colon\, {\mathop{\text{\rm w-lim}}}_{T\to\infty} \frac{1}{T} \int_0^T \delta_{\kappa(Y)} \circ (\theta^s)^{-1}\,ds = {\mathscr{W}}\;\; \text{for $Q$-a.e.\ $Y$}\right\}.$$ Theorem \[thmexp\] is the continuous analogue of Birkner [@Bi08 Theorem 1] and is proved in Section \[proofalpha1infty\]. On the set $\mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}})$ the following holds: $$\label{eq:calRchar} \Psi_Q = {\mathscr{W}}\quad \mbox{ if and only if } \quad Q \in {{\mathcal R}}_{\mathscr{W}}.$$ The equivalence in is the continuous analogue of [@Bi08 Lemma 2] (and can be proved analogously). [**3.**]{} By applying the contraction principle we obtain the quenched LDP for single words. Let $\pi_1\colon\,F^{\mathbb{N}}\to F$ be the projection onto the first word, and let $\pi_1R_N = R_N \circ (\pi_1)^{-1}$. \[cor:marginal\] Suppose that $\rho$ satisfies [(\[ass:rhodensdecay\]–\[ass:rhobar.reg0\])]{}. For ${\mathscr{W}}$-a.e. $X$, the family $\mathscr{L}(\pi_1 R_N \mid X)$, $N\in{\mathbb{N}}$, satisfies the LDP on $\mathcal{P}(F)$ with rate $N$ and with deterministic rate function $I^\mathrm{que}_1$ given by $$\label{eq:contractedratefct} I^\mathrm{que}_1(q) = \inf\big\{ I^\mathrm{que}(Q)\colon\, Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}}),\,\pi_1 Q = q \big\}.$$ This rate function is lower semi-continuous, has compact levels sets, is convex, and has a unique zero at $q=q_{\rho,{\mathscr{W}}}$. For general $q$ it is not possible to evaluate the infimum in (\[eq:contractedratefct\]) explicitly. For $q$ with $m_q={\mathbb{E}}_q[\tau]={\mathbb{E}}_{q^{\otimes{\mathbb{N}}}}[\tau_1]=m_{q^{\otimes{\mathbb{N}}}}<\infty$ and $\Psi_{q^{\otimes{\mathbb{N}}}}={\mathscr{W}}$, we have $I^\mathrm{que}_1(q)=h(q \mid q_{\rho,{\mathscr{W}}})$, the relative entropy of $q$ w.r.t.  $q_{\rho,{\mathscr{W}}}$. [**4.**]{} We expect assumption to be redundant. In any case, it can be relaxed to (see Section \[prop1\]): $$\label{ass:rhobar.reg-mg} \begin{minipage}{0.85\textwidth} $\mathrm{supp}(\rho)= \cup_{i=1}^M [a_i,b_i] \cup [a_{M+1},\infty)$ with $M \in {\mathbb{N}}$ and $0 \leq a_1 < b_1 \leq a_2 < \cdots < b_M \leq a_{M+1}<\infty$, and $\bar{\rho}$ is continuous and strictly positive on $\cup_{i=1}^M (a_i,b_i) \cup (a_{M+1},\infty)$ and varies regularly near each of the finite endpoints of these intervals. \end{minipage}$$ [**5.**]{} It is possible to extend Theorem \[thm0:contqLDP\] to other classes of random environments, as stated in the following theorem whose proof will not be spelled out in the present paper. Theorems [\[thm0:contqLDP\]–\[thmexp\]]{} and Corollary [\[cor:marginal\]]{} carry over verbatim when the Brownian motion $X$ is replaced by a $d$-dimensional Lévy process $\bar{X}$ with the property that ${\mathbb{E}}[e^{\langle \lambda, \bar{X}_1 \rangle}] < \infty$ for all $\lambda \in {\mathbb{R}}^d$ (where $\langle\cdot\rangle$ denotes the standard inner product), ${\mathscr{W}}$ is replaced by the law of $\bar{X}$, and in the definition of $F$ in continuous paths are replaced by càdlàg paths. [**6.**]{} In the companion paper [@BidHo13b] we apply Theorem \[thm0:contqLDP\] and the techniques developed in the present paper to the Brownian copolymer. In this model a càdlàg path, representing the configuration of the polymer, is rewarded or penalised for staying above or below a linear interface, separating oil from water, according to Brownian increments representing the degrees of hydrophobicity or hydrophilicity along the polymer. The reference measure for the path can be either the Wiener measure or the law of a more general Lévy process. We derive a variational formula for the quenched free energy, from which we deduce a variational formula for the slope of the quenched critical line. This critical line separates a *localized phase* (where the copolymer stays close to the interface) from a *delocalized phase* (where the copolymer wanders away from the interface). This slope has been the object of much debate in recent years. The Brownian copolymer is the unique attractor in the limit of weak interaction for a whole universality class of discrete copolymer models. See Bolthausen and den Hollander [@BodHo97], Caravenna and Giacomin [@CaGi10], Caravenna, Giacomin and Toninelli [@CaGiTo12] for details. Proof of Theorem \[thm0:contqLDP\] {#proof} ================================== The proof proceeds via a *coarse-graining* and *truncation* argument. In Section \[cogrtrun\] we set up the coarse-graining and the truncation, and state a quenched LDP for this setting that follows from the quenched LDP in [@BiGrdHo10] and serves as the starting point of our analysis (Proposition \[thm00:contqLDP\] and Corollary \[prop:qLDPhtr\] below). In Section \[3prop\] we state three propositions (Propositions \[prop:LambdaPhilimit1tr\]–\[prop:Ique.tr.cont\] below), involving expectations of exponential functionals of the coarse-grained truncated empirical process as well as approximation properties of the associated rate function, and we use these propositions to complete the proof of Theorem \[thm0:contqLDP\] with the help of Bryc’s inverse of Varadhan’s lemma. In Section \[3lem\] we state and prove two lemmas that are used in Section \[3prop\], involving approximation estimates under the coarse-graining. The proof of the three propositions is deferred to Sections \[props\]–\[removeass\]. Preparation: coarse-graining and truncation {#cogrtrun} ------------------------------------------- ### Coarse-graining {#cogr} Suppose that, instead of the absolutely continuous $\rho$ introduced in Section \[setting\], we are given a discrete $\hat{\rho}$ with $\mathrm{supp}(\hat\rho) \subset h {\mathbb{N}}$ for some $h>0$. Let $$\label{def:Eh} E_h = \{ f \in C([0,h])\colon\,f(0)=0 \}.$$ Path pieces of length $h$ in a continuous-time scenario can act as “letters” in a discrete-time scenario, and therefore we can use the results from [@BiGrdHo10]. Note that $(E_h)^{\mathbb{N}}$ as a metric space is isomorphic to $\{f \in C([0,\infty))\colon\,f(0)=0\}$ via the obvious glueing together of path pieces into a single path, provided the latter is given a suitable metric that metrises locally uniform convergence. Similarly, we can identify $\mathcal{P}^{\mathrm{inv}}(E_h^{\mathbb{N}})$ with $$\mathcal{P}^{h\text{-}\mathrm{inv}}(C([0,\infty))) = \big\{ Q \in \mathcal{P}(C([0,\infty))) \colon\, Q = Q \circ (\theta^{h})^{-1} \big\},$$ which is the set of laws on continuous paths that are invariant under a time shift by $h$. Note that the set $$\label{def:Fh} F_h = \bigcup_{t \in h{\mathbb{N}}} \Big(\{t\} \times \big\{ f \in C([0,\infty))\colon\,f(0)=0, f(s)=f(t) \; \text{for} \; s > t \big\}\Big)$$ is isomorphic to $\widetilde{E_h} = \cup_{n \in {\mathbb{N}}} \left(E_h\right)^n$ via the map $\iota_h\colon\, F_h \to \widetilde{E_h}$ defined by $$\label{iotahdef} \iota_h\big( (nh, f)\big) = \Big( \big(f\big((\,\cdot+(i-1)h) \wedge ih\big) -f((i-1)h)\big)\Big)_{i=1,\dots,n}, \qquad (nh, f) \in F_h.$$ For $Q \in \mathcal{P}^{\mathrm{inv, fin}}(F_h^{\mathbb{N}})$, define $$\label{eq:definitionPsiQh} \Psi_{Q,h}(A) = \frac1{m_Q} {\mathbb{E}}_Q\left[ \sum_{i=0}^{\tau_1-1} {1}_A\big(\theta^i \iota_h \kappa(Y)\big)\right] = \frac1{h \, m_Q} {\mathbb{E}}_Q\left[ \int_0^{h \tau_1} {1}_A\big( \kappa(Y)(h \lfloor u/h\rfloor +s))_ {s \geq 0} \big) \, du\right]$$ for $A \subset C([0,\infty))$ measurable, where $\tau_1$ is the length of the first word (counted in letters, so that the length of the first word viewed as an element of $F_h$ is $h\tau_1$) and $\theta$ is the left-shift acting on $(E_h)^{\mathbb{N}}$. The right-most expression in can be viewed as a coarse-grained version of (\[eq:PsiQcont\]). The following coarse-grained version of the quenched LDP serves as our starting point. \[thm00:contqLDP\] Fix $h>0$. Suppose that $\mathrm{supp}(\hat\rho) \subset h{\mathbb{N}}$ and $\lim_{n\to\infty} \log \hat\rho(\{nh\})/\log n = -\alpha$ with $\alpha \in (1,\infty)$. Then, for ${\mathscr{W}}$ a.e.  $X$, the family $\mathscr{L}(R_N \mid X)$, $N\in{\mathbb{N}}$, satisfies the LDP on $\mathcal{P}^{\mathrm{inv}}((\widetilde{E_h})^{\mathbb{N}})$ with rate $N$ and with deterministic rate function given by $$\label{eq:ratefctfixedh} I^{\mathrm{que}}_h(Q) = H(Q \mid Q_{\hat\rho,{\mathscr{W}}}) + (\alpha-1) m_Q H(\Psi_{Q,h} \mid {\mathscr{W}}), \qquad Q \in \mathcal{P}^{\mathrm{inv,fin}}((\widetilde{E_h})^{\mathbb{N}}),$$ and $$I^{\mathrm{que}}_h(Q) = \lim_{{{\rm tr}}\to\infty} I^{\mathrm{que}}_h([Q]_{{\rm tr}}), \qquad Q \notin \mathcal{P}^{\mathrm{inv,fin}}((\widetilde{E_h})^{\mathbb{N}}),$$ where $Q_{\hat\rho,{\mathscr{W}}}=(q_{\hat\rho,{\mathscr{W}}})^{\otimes{\mathbb{N}}}$ with $q_{\hat\rho,{\mathscr{W}}}$ defined as in , and $\Psi_{Q,h}$ defined via . The claim follows from [@BiGrdHo10 Corollary 1.6] by using $E_h$ as letter space and observing that $\widetilde{E_h}=\iota_h(F_h)$. Note that $F_h^{\mathbb{N}}$ is a closed subspace of $F^{\mathbb{N}}$. Since $\mathrm{supp}(\hat\rho) \subset h{\mathbb{N}}$ by assumption, we have $I^{\mathrm{que}}_h(Q) \geq H(Q \mid Q_{\hat\rho,{\mathscr{W}}}) = \infty$ for any $Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ with $Q\big(F^{\mathbb{N}}\setminus F_h^{\mathbb{N}}\big)>0$. Therefore we can consider the random variable $R_N$ as taking values in $\mathcal{P}^{\mathrm{inv}}((\widetilde{E_h})^{\mathbb{N}})$, $\mathcal{P}^{\mathrm{inv}} (F_h^{\mathbb{N}})$ or $\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$, without changing the statement of Proposition \[thm00:contqLDP\]. Note that $I^{\mathrm{que}}_h$ is finite only on $\mathcal{P}^{\mathrm{inv}}(F_h^{\mathbb{N}})\subset\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$. We want to pass to the limit $h \downarrow 0$ and deduce Theorem \[thm0:contqLDP\] from Proposition \[thm00:contqLDP\]. However, an immediate application of a projective limit at the level of letters appears to be impossible. Indeed, when we replace $h$ by $h/2$, each “$h$-letter” turns into two “$(h/2)$-letters”, so the word length changes, and even diverges as $h \downarrow 0$. This does not fit well with the way the projective limit was set up in [@BiGrdHo10 Section 8], where the internal structure of the letters was allowed to become increasingly richer, but the word length had to remain the same. In some sense, the problem is that we have finite words but only infinitesimal letters (i.e., there is no fixed letter space). To remedy this, we proceed as follows. For fixed discretisation length $h>0$ we have a fixed letter space, and so Proposition \[thm00:contqLDP\] applies. We will handle the limit $h \downarrow 0$ via Bryc’s inverse of Varadhan’s lemma. This will require several intermediate steps. ### Truncation {#trun} It will be expedient to work with a *truncated* version of Proposition \[thm00:contqLDP\]. For $h>0$, let $\lceil t \rceil_h =h\lceil t/h \rceil$ for $t \in (0,\infty)$ and put $\lceil\rho\rceil_h =\rho\,\circ (\lceil\cdot\rceil_h)^{-1}$, i.e., $$\label{def:rho.h.trunc} \lceil \rho \rceil_h = \sum_{i\in{\mathbb{N}}} w_{h,i} \delta_{ih} \, \in \mathcal{P}(h{\mathbb{N}}) \subset \mathcal{P}((0,\infty)),$$ where $$w_{h,i} = \rho\big(((i-1)h,ih]\big) = \int_{(i-1)h}^{ih} \bar{\rho}(x)dx$$ is the coarse-grained version of $\rho$ from Section \[setting\]. It is easily checked that implies $$\lim_{n\to\infty} \frac{\log \lceil \rho\rceil_h(\{nh\})}{\log n} = -\alpha.$$ Write$ \mathscr{L}_{\lceil \rho \rceil_h}([R_N]_{{\rm tr}}\mid X)$ for the law of the truncated empirical process $[R_N]_{{\rm tr}}$ conditional on $X$ when the $\tau_i$’s are drawn according to $\lceil \rho \rceil_h$. \[prop:qLDPhtr\] For ${\mathscr{W}}$-a.e. $X$, the family $\mathscr{L}_{\lceil \rho \rceil_h}([R_N]_{{\rm tr}}\mid X)$, $N\in{\mathbb{N}}$, satisfies the LDP on $\mathcal{P}^\mathrm{inv}(F_h^{\mathbb{N}})$ with rate $N$ and with deterministic rate function given by $$\label{eq:Ique.h.tr} I^{\mathrm{que}}_{h,{{\rm tr}}}(Q) = H\bigl(Q \mid Q_{\lceil\rho\rceil_h,{\mathscr{W}},{{\rm tr}}}\bigr) + (\alpha-1) m_Q H(\Psi_{Q,h} \mid {\mathscr{W}})$$ with $Q_{\lceil\rho\rceil_h,{\mathscr{W}},{{\rm tr}}} = ([q_{\lceil \rho \rceil_h, {\mathscr{W}}}]_{{\rm tr}})^{\otimes {\mathbb{N}}}$. This follows from Proposition \[thm00:contqLDP\] and the contraction principle. Alternatively, it follows from the proofs of [@BiGrdHo10 Theorem 1.2 and Corollary 1.6]. Note that $I^{\mathrm{que}}_{h,{{\rm tr}}}(Q)=\infty$ when under $Q$ the word lengths are not supported on $h{\mathbb{N}}\cap (0,{{\rm tr}}]$. Application of Bryc’s inverse of Varadhan’s lemma {#3prop} ------------------------------------------------- In this section we state three propositions (Propositions \[prop:LambdaPhilimit1tr\]–\[prop:Ique.tr.cont\] below) and show that these imply Theorem \[thm0:contqLDP\]. The proof of these propositions is deferred to Sections \[props\]–\[removeass\]. ### Notations {#subsect:notations} In what follows we obtain the quenched LDP for the truncated empirical process $[R_N]_{{{\rm tr}}}$ by letting $h \downarrow 0$ in the coarse-grained and truncated empirical process $[R_{N,h}]_{{\rm tr}}$ with ${{\rm tr}}\in {\mathbb{N}}$ fixed (for a precise definition, see in Section \[prop1\]) and afterwards letting ${{\rm tr}}\to\infty$. (We assume that ${{\rm tr}}\in {\mathbb{N}}$ and $h=2^{-M}$ for some $M \in {\mathbb{N}}$, in particular, ${{\rm tr}}$ is an integer multiple of $h$.) In the coarse-graining procedure, it may happen that a very short continuous word $y=(t,f) \in F$ disappears, namely, when $0<t<h$. We remedy this by formally allowing “empty” words, i.e., by using $$\begin{aligned} \label{def:Fhat} \widehat{F} = F \cup \big\{ (0,0) \big\} = \bigcup_{t \geq 0} \Big(\{t\} \times \big\{ f \in C([0,\infty))\colon\,f(0)=0, f(s)=f(t) \; \text{for} \; s > t \big\}\Big)\end{aligned}$$ as word space instead of $F$. The metric on $F$ defined in Appendix \[metrics\] extends in the obvious way to $\widehat{F}$. Before we proceed, we impose *additional regularity assumptions* on $\bar{\rho}$ that will be required in the proof of Proposition \[prop:LambdaPhilimit1tr\]. Recall from that $\mathrm{supp}(\rho) = [s_*,\infty)$. Let $$\begin{aligned} \label{eq:Vbarrhodef} V_{\bar{\rho}}(t,h) = \sup_{v \in (0,2h)} \left| \log \frac{\int_t^{t+h} \bar{\rho}(s)\,ds}{\int_{t+v}^{t+h+v} \bar{\rho}(s)\,ds} \right|, \qquad t,h>0.\end{aligned}$$ We assume that there exist monotone sequences $(\eta_n)_{n \in {\mathbb{N}}}$ and $(A_n)_{n\in{\mathbb{N}}}$, with $\eta_n \in (0,1)$ and $A_n \subset (s_*,\infty)$ satisfying $\lim_{n\to\infty} \eta_n = 0$ and $\lim_{n\to\infty} A_n = (s_*,\infty)$, such that $(s_*,\infty) \setminus A_n$ is a (possibly empty) union of finitely many bounded intervals whose endpoints lie in $2^{-n} {\mathbb{N}}_0$, and $$\begin{aligned} \label{ass:rhobar.reg2} \sup_{t \in A_n} V_{\bar{\rho}}(t,2^{-n}) \leq \eta_n \qquad \forall\,n \in{\mathbb{N}}. \end{aligned}$$ In addition, we assume that there exists an $\eta_0 < \infty$ such that $$\begin{aligned} \label{ass:rhobar.reg1} \sup_{n \in {\mathbb{N}}} \sup_{t \in (s_*,\infty)} V_{\bar{\rho}}(t,2^{-n}) \leq \eta_0. \end{aligned}$$ These assumptions will be removed only in Section \[removeass\]. Note that – are satisfied when $\bar{\rho}$ is continuous and strictly positive on $(s_*,\infty)$ and varies regularly near $s_*$ and at $\infty$. ### Proof of Theorem \[thm0:contqLDP\] subject to (\[ass:rhobar.reg2\]–\[ass:rhobar.reg1\]) and three propositions A function $g$ on $\widehat{F}^\ell$ is Lipschitz when it satisfies $$\begin{aligned} \label{eq:g_Lipschitz} \big| g(y^{(1)},\dots,y^{(\ell)}) - g(y^{(1)}{}',\dots,y^{(\ell)}{}') \big| \leq C_g \sum_{j=1}^\ell d_F(y^{(j)},y^{(j)}{}') \quad \mbox{ for some } C_g < \infty. \end{aligned}$$ Consider the class $\mathscr{C}$ of functions $\Phi\colon\,\mathcal{P}(\widehat{F}^{\mathbb{N}}) \to {\mathbb{R}}$ of the form $$\label{eq:Phiform1} \Phi(Q) = \int_{\widehat{F}^{\ell_1}} g_1 \, d\pi_{\ell_1} Q \wedge \cdots \wedge \int_{\widehat{F}^{\ell_m}} g_m \, d\pi_{\ell_m}Q, \quad Q \in \mathcal{P}^{\mathrm{inv}}(\widehat{F}^{\mathbb{N}}),$$ where $m \in N$, $\ell_1,\dots, \ell_m \in {\mathbb{N}}$, and $g_i$ is a bounded Lipschitz function on $\widehat{F}^{\ell_i}$ for $i=1,\dots,m$. This class is well-separating and thus is sufficient for the application of Bryc’s lemma (see Dembo and Zeitouni [@DeZe98 Section 4.4]. Our first proposition identifies the exponential moments of $[R_N]_{{\rm tr}}$. \[prop:LambdaPhilimit1tr\] The families $\mathscr{L}(R_N \mid X)$, $N\in{\mathbb{N}}$, and $\mathscr{L}([R_N]_{{{\rm tr}}} \mid X)$, ${{\rm tr}}\in {\mathbb{N}}$, are exponentially tight $X$-a.s. Moreover, for $\Phi \in \mathscr{C}$, $$\label{eq:LambdaPhilimit1tr} \Lambda_{0,{{\rm tr}}}(\Phi) = \lim_{N\to\infty} \frac1N \log {\mathbb{E}}\Big[ \exp\big( N \Phi([R_N]_{{{\rm tr}}})\big) ~\Big|~ X \Big] = \lim_{h \downarrow 0} \Lambda_{h,{{\rm tr}}}(\Phi) \quad \text{exists $X$-a.s.},$$ where $\Lambda_{h,{{\rm tr}}}$ is the generalised convex transform of $I^{\mathrm{que}}_{h,{{\rm tr}}}$ given by $$\label{eq:Phihtrform} \Lambda_{h,{{\rm tr}}}(\Phi) = \sup_{Q \in \mathcal{P}^{\mathrm{inv, fin}}((\widetilde{E_h})^{\mathbb{N}})} \big\{ \Phi(Q) - I^{\mathrm{que}}_{h,{{\rm tr}}}(Q) \big\}.$$ Furthermore, for $\Phi\in \mathscr{C}$, $$\label{eq:LambdaPhilimit3} \Lambda(\Phi) = \lim_{N\to\infty} \frac1N \log {\mathbb{E}}\Big[ \exp\big( N \Phi(R_N)\big) ~\Big|~ X \Big] = \lim_{{{\rm tr}}\to\infty} \Lambda_{0,{{\rm tr}}}(\Phi) \quad \text{exists $X$-a.s.}$$ Our second proposition identifies the limit in as the generalised convex transform of $I^{\mathrm{que}}_{{{\rm tr}}}$ defined in , $$I^{\mathrm{que}}_{{{\rm tr}}}(Q) = \begin{cases} H\bigl( Q \mid Q_{\rho,{\mathscr{W}},{{\rm tr}}} \bigr) + (\alpha-1) m_Q H\left( \Psi_Q \mid {\mathscr{W}}\right) & \text{if} \; Q \in \mathcal{P}^{\mathrm{inv}}(F_{0,{{\rm tr}}}^{\mathbb{N}}), \\[1ex] \infty & \text{otherwise}, \end{cases}$$ and implies that the latter is the rate function for the truncated empirical process $[R_N]_{{\rm tr}}$. \[prop:qLDPtrunc1\] For $\Phi\in \mathscr{C}$, $$\label{eq:Lambda0tr} \Lambda_{0,{{\rm tr}}}(\Phi) = \sup_{Q \in \mathcal{P}^{\mathrm{inv}}(F_{0,{{\rm tr}}}^{\mathbb{N}})} \big\{ \Phi(Q) - I^{\mathrm{que}}_{{{\rm tr}}}(Q) \big\}.$$ Furthermore, for ${\mathscr{W}}$-a.e. $X$, the family $\mathscr{L}([R_N]_{{{\rm tr}}} \mid X)$, $N\in{\mathbb{N}}$, satisfies the LDP on $\mathcal{P}^{\mathrm{inv}}(F_{0,{{\rm tr}}}^{\mathbb{N}})$ with deterministic rate function $I^{\mathrm{que}}_{{{\rm tr}}}$. Note that the family of truncation operators $[\cdot]_{{{\rm tr}}}$ forms a projective system as the truncation level ${{\rm tr}}$ increases. Hence we immediately get from Proposition \[prop:qLDPtrunc1\] and the Dawson-Gärtner projective limit LDP (see [@DeZe98 Theorem 4.6.1]) that the family $\mathscr{L}(R_N \mid X)$, $N\in{\mathbb{N}}$, satisfies the LDP with rate function $Q \mapsto \sup_{{{\rm tr}}\in {\mathbb{N}}} I^{\mathrm{que}}_{{{\rm tr}}}([Q]_{{{\rm tr}}})$. Furthermore, since the projection can start at any initial level of truncation, we also know that the rate function is given by $Q \mapsto \sup_{{{\rm tr}}\geq n} I^{\mathrm{que}}_{{{\rm tr}}}([Q]_{{{\rm tr}}})$ for any $n\in{\mathbb{N}}$. Thus, Proposition \[prop:qLDPtrunc1\] in fact implies that the rate function is given by $$\begin{aligned} \label{eq:Ique-DGform} \tilde{I}^{\mathrm{que}}(Q) = \limsup_{{{\rm tr}}\to\infty} I^{\mathrm{que}}_{{{\rm tr}}}([Q]_{{{\rm tr}}}).\end{aligned}$$ At this point, it remains to prove that $\tilde{I}^{\mathrm{que}}$ from actually equals $I^{\mathrm{que}}$ from and has the form claimed in . This is achieved via the following proposition, note that is the continuous analogue of [@BiGrdHo10 Lemma A.1]. \[prop:Ique.tr.cont\] [(1)]{} For $Q \in \mathcal{P}^{\mathrm{inv,fin}}(F^{{\mathbb{N}}})$, $$\begin{aligned} \label{eq:lemma:Ique.tr.cont1} \lim_{{{\rm tr}}\to\infty} I^{\mathrm{que}}_{{\rm tr}}([Q]_{{\rm tr}}) = H(Q \mid Q_{\rho,{\mathscr{W}}}) + (\alpha-1) m_Q H(\Psi_Q \mid {\mathscr{W}}).\end{aligned}$$ [(2)]{} For $Q \in \mathcal{P}^{\mathrm{inv}}(F^{{\mathbb{N}}})$ with $m_Q = \infty$ and $H(Q \mid Q_{\rho, {\mathscr{W}}})<\infty$ there exists a sequence $(\widetilde{Q}_{{\rm tr}})_{{{\rm tr}}\in {\mathbb{N}}}$ in $\mathcal{P}^{\mathrm{inv,fin}}(F^{{\mathbb{N}}})$ such that ${\mathop{\text{\rm w-lim}}}_{{{\rm tr}}\to\infty} \widetilde{Q}_{{\rm tr}}= Q$ and $$\begin{aligned} \label{eq:lemma:Ique.tr.approx2} \tilde{I}^{\mathrm{que}}(\widetilde{Q}_{{\rm tr}}) \leq I^{\mathrm{que}}_{{\rm tr}}([Q]_{{\rm tr}}) + o(1), \qquad {{\rm tr}}\to \infty.\end{aligned}$$ Proposition \[prop:Ique.tr.cont\] (1) implies that for $Q \in \mathcal{P}^{\mathrm{inv,fin}} (F^{{\mathbb{N}}})$ the $\limsup$ in is a limit, i.e., it implies on $\mathcal{P}^{\mathrm{inv,fin}}(F^{{\mathbb{N}}})$ and also . To prove for $Q \in \mathcal{P}^{\mathrm{inv}}(F^{{\mathbb{N}}})$ with $m_Q = \infty$ and $H(Q \mid Q_{\rho, {\mathscr{W}}})<\infty$, consider $\widetilde{Q}_{{\rm tr}}$ as in Proposition \[prop:Ique.tr.cont\] (2). Then $$\begin{aligned} \tilde{I}^{\mathrm{que}}(Q) \leq \liminf_{{{\rm tr}}\to\infty} \tilde{I}^{\mathrm{que}}(\widetilde{Q}_{{\rm tr}}) \leq \liminf_{{{\rm tr}}\to\infty} I^{\mathrm{que}}_{{\rm tr}}([Q]_{{\rm tr}}),\end{aligned}$$ where the first inequality uses that $\tilde{I}^{\mathrm{que}}$ is lower semi-continuous (being a rate function by the Dawson-Gärtner projective limit LDP), and the second inequality is a consequence of . For $Q \in \mathcal{P}^{\mathrm{inv}}(F^{{\mathbb{N}}})$ with $H(Q \mid Q_{\rho, {\mathscr{W}}})=\infty$ we have $$\begin{aligned} \liminf_{{{\rm tr}}\to\infty} I^{\mathrm{que}}_{{\rm tr}}([Q]_{{\rm tr}}) \geq \liminf_{{{\rm tr}}\to\infty} H([Q]_{{\rm tr}}\mid [Q_{\rho,{\mathscr{W}}}]_{{\rm tr}}) = H(Q \mid Q_{\rho, {\mathscr{W}}}) = \infty,\end{aligned}$$ i.e., also in this case the $\limsup$ in is a limit and holds. It remains to prove the properties of $I^{\mathrm{que}}$ claimed in Theorem \[thm0:contqLDP\]: lower semi-continuity of $I^{\mathrm{que}}=\tilde{I}^{\mathrm{que}}$ follows from the representation via the Dawson-Gärtner projective limit LDP in ; compactness of the level sets of $I^{\mathrm{que}}$ and the fact that $Q_{\rho,{\mathscr{W}}}$ is the unique zero of $Q \mapsto I^{\mathrm{que}}(Q)$ are inherited from the corresponding properties of $I^{\mathrm{ann}}$ because $I^{\mathrm{que}} \leq I^{\mathrm{ann}}$; affineness of $Q \mapsto I^{\mathrm{que}}(Q)$ can be checked as in [@BiGrdHo10 Proof of Theorem 1.3]. [**Remark.**]{}  Theorem \[thm0:contqLDP\] together with Varadhan’s lemma implies that $$\begin{aligned} \label{eq:LambdaPhilimit2} \Lambda(\Phi) = \sup_{Q \in \mathcal{P}^{\mathrm{inv, fin}}(F^{\mathbb{N}})} \big\{ \Phi(Q) - I^{\mathrm{que}}(Q) \big\}, \qquad \Phi\in \mathscr{C}, \end{aligned}$$ and identifies $I^{\mathrm{que}}(Q)$ as the generalised convex transform $$\label{eq:Iquetrafo} I^{\mathrm{que}}(Q) = \sup_{\Phi \in \mathscr{C}} \big\{ \Phi(Q) - \Lambda(\Phi)\big\}, \qquad Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$$ (see [@DeZe98 Theorems 4.4.2 and 4.4.10]). The supremum in can also be taken over all continuous bounded functions on $\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$. Continuity of the empirical process under coarse-graining {#3lem} --------------------------------------------------------- Before embarking on the proof of Propositions \[prop:LambdaPhilimit1tr\]–\[prop:Ique.tr.cont\] in Section \[props\], we state and prove two approximation lemmas (Lemmas \[obs:dSclose1\]–\[obs:Rdiscdiff\] below) that will be needed along the way. For $N \in {\mathbb{N}}$, $0=t_0 < t_1 < \cdots < t_N$ and $\varphi \in C([0,\infty))$, let $y_\varphi = (y_\varphi^{(i)})_{i\in{\mathbb{N}}}$ with $$\label{def:yphii} y_\varphi^{(i)} = \Big(t_i-t_{i-1}, \big(\varphi((t_{i-1}+s) \wedge t_i) -\varphi(t_{i-1})\big)_{s\geq 0}\Big) \in F, \qquad i=1,\dots,N,$$ and define $$\label{eq:defRNphi} R_{N;t_1,\dots,t_N}(\varphi) = \frac1N \sum_{i=0}^{N-1} \delta_{\widetilde{\theta}^i y_\varphi^{N\text{-}\mathrm{per}}} \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}}).$$ We need a Skorohod-type distance $d_S$ on paths, which is defined in Appendix \[metrics\]. \[obs:dSclose1\] Let $i, j \in {\mathbb{N}}$, $i \leq j$, and $t, t' \in (0,\infty)$, $t<t'$, be such that $(i-1)h < t \leq ih$, $(j-1)h < t' \leq jh$. Then, for any $\varphi \in C([0,\infty))$ and $k \in {\mathbb{N}}$, $$\begin{aligned} \label{eq:dS_wishful} & d_S\big( \varphi((ih+\cdot) \wedge jh), \varphi((t+\cdot) \wedge t')\big) \notag \\ & \leq \log\tfrac{k+1}{k} + 2 \sup_{(i-1)h \leq s \leq (i+k)h} |\varphi(s)-\varphi((i-1)h)| + 2 \sup_{(j-1)h \leq s \leq jh} |\varphi(s)-\varphi((j-1)h)|.\end{aligned}$$ The same bound holds for $d_S([\varphi((ih+\cdot) \wedge jh)]_{{\rm tr}}, [\varphi((t+\cdot) \wedge t')]_{{\rm tr}})$ for any truncation length ${{\rm tr}}> 0$. Without loss of generality we may assume that $j \geq i+k$ (otherwise, employ the trivial time transform $\lambda(s)=s$ and estimate the left-hand side of by the second term in the right-hand side of ), and use the time transformation $$\begin{aligned} \lambda(s) = \begin{cases} s \, \frac{(i+k)h-t}{kh} & \text{if} \; s < kh, \\ s + ih -t & \text{if} \; s \geq kh. \end{cases}\end{aligned}$$ In that case $\lambda(s)+t=s+ih$ for $s \geq kh$ and $\gamma(\lambda)=|\log[((i+k)h-t)/kh]| \leq \log\frac{k+1}{k}$. The same argument applies to the truncated paths $[\varphi((ih+\cdot) \wedge jh)]_{{\rm tr}}$ and $[\varphi((t+\cdot) \wedge t')]_{{\rm tr}}$ (in fact, we can drop the third term in the right-hand side of when $(j-1)h>{{\rm tr}}$). \[obs:Rdiscdiff\] Let $\varphi \in C([0,\infty))$, $h>0$, $N\in{\mathbb{N}}$ and $t_0=0<t_1<\cdots<t_N$. Let $\ell \in {\mathbb{N}}$, and let $g\colon\,\widehat{F}^\ell \to {\mathbb{R}}$ be bounded Lipschitz with Lipschitz constant $C_g$. Then, for $k \in {\mathbb{N}}$ with $k \geq \ell$, $$\begin{aligned} &N \Big| \int_{\widehat{F}^{\ell}} g\, d\pi_\ell R_{N;t_1,\dots,t_N}(\varphi) - \int_{\widehat{F}^{\ell}} g\, d\pi_\ell R_{N;\lceil t_1 \rceil_h,\dots,\lceil t_N \rceil_h}(\varphi) \Big|\\ &\qquad \leq 4\ell \|g\|_\infty + C_g \ell N \bigl(2h + \log{\textstyle\frac{k+1}{k}}\bigr) + 4 C_g \ell \sum_{i=1}^N \sup_{\lceil t_i \rceil_h-h \leq s \leq \lceil t_i \rceil_h+kh} \big| \varphi(s) -\varphi({\lceil t_i \rceil_h-h}) \big|, \end{aligned}$$ where $\pi_\ell\colon\,\widehat{F}^{\mathbb{N}}\to \widehat{F}^\ell$ denotes the projection onto the first $\ell$ coordinates. The same bound holds for the truncated versions $[R_{N;t_1,\dots,t_N} (\varphi)]_{{\rm tr}}$ and $[R_{N;\lceil t_1 \rceil_h,\dots,\lceil t_N \rceil_h}(\varphi) ]_{{\rm tr}}$ for any truncation length ${{\rm tr}}>0$. For $i=1,\dots,N$, recall $y_\varphi^{(i)}$ from , i.e., $y_\varphi^{(i)}$ is the $i$-th word obtained by cutting the continuous path $\varphi$ along the time points $t_1,\ldots,t_n$, and let $$\begin{aligned} \tilde{y}_\varphi^{(i,h)} & = \Bigl(\lceil t_i \rceil_h-\lceil t_{i-1} \rceil_h, \bigl(\varphi((\lceil t_{i-1} \rceil_h+s) \wedge \lceil t_i \rceil_h) -\varphi(\lceil t_{i-1} \rceil_h)\bigr)_{s\geq 0}\Bigr), \end{aligned}$$ be the analogous quantity when the $h$-discretised time points $\lceil t_1 \rceil_h,\ldots,\lceil t_N \rceil$ are used. By Lemma \[obs:dSclose1\] we have $$\begin{aligned} d_F\bigl(y_\varphi^{(i)}, \tilde{y}_\varphi^{(i,h)}\bigr) &\leq \bigl(2h + \log{\textstyle\frac{k+1}{k}}\bigr) + 2 \sup_{\lceil t_{i-1} \rceil_h-h \leq s \leq \lceil t_{i-1} \rceil_h+kh} \big| \varphi(s) -\varphi({\lceil t_{i-1} \rceil_h}-h) \big|\\ &\qquad\qquad + 2 \sup_{\lceil t_{i} \rceil_h-h \leq s \leq \lceil t_{i} \rceil_h} \big| \varphi(s) -\varphi({\lceil t_{i} \rceil_h}-h) \big|. \end{aligned}$$ Writing $\tilde{y}^{(h)}=(\tilde{y}^{(i,h)})_{i\in{\mathbb{N}}}$ and putting, similarly as in , $$\label{eq:defRNphi_hdisc} R_{N;\lceil t_1 \rceil_h,\dots,\lceil t_N \rceil_h}(\varphi) = \frac1N \sum_{i=0}^{N-1} \delta_{\widetilde{\theta}^i(\tilde{y}^{(h)})^{N\text{-}\mathrm{per}}},$$ we see that the claim follows from in combination with Lemma \[obs:dSclose1\]. Note that possible boundary effects due to the periodisation are estimated by the term $4\ell\|g\|_\infty$. The observation about the truncated versions of the empirical process follow analogously from Lemma \[obs:dSclose1\]. Proof of Propositions \[prop:LambdaPhilimit1tr\]–\[prop:Ique.tr.cont\] {#props} ====================================================================== Proof of Proposition \[prop:LambdaPhilimit1tr\] {#prop1} ----------------------------------------------- The proof comes in 3 Steps. #### Step 1. A.s. exponential tightness of the family $\mathscr{L}(R_N \mid X)$, $N\in{\mathbb{N}}$, is standard, because the family of unconditional distributions $\mathscr{L}(R_N)$ satisfies the LDP with a rate function that has compact level sets. Indeed, let $M > 0$, and pick a compact set $K \subset \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ such that $\limsup_{N\to\infty} \tfrac1N \log {\mathbb{P}}(R_N \not\in K) \leq -2M$. Then ${\mathbb{P}}({\mathbb{P}}(R_N \not\in K \mid X) > e^{-MN}) \leq e^{MN} {\mathbb{E}}[{\mathbb{P}}(R_N \not\in K\mid X)] \leq \exp(MN -2MN +o(N))$, which is summable in $N$. Hence we have $\limsup_{N\to\infty} \tfrac1N \log {\mathbb{P}}(R_N \not\in K \mid X) \leq -M$ a.s. by the Borel-Cantelli lemma. The same argument applies to $[R_N]_{{{\rm tr}}}$ (alternatively, use the fact that $[\cdot]_{{\rm tr}}$ is a continuous map). #### Step 2a. We next verify that the limits in exist. In Step 2a we consider the case $\mathrm{supp}(\rho)=[0,\infty)$, in Step 2b the case $\mathrm{supp}(\rho)=[s_*,\infty)$ with $s_*>0$. Let ${{\rm tr}}\in {\mathbb{N}}$ and $h = 2^{-n}$. Let $Y^{(i,h)}=(\lceil T_i \rceil_h -\lceil T_{i-1} \rceil_h, (X_{(s+\lceil T_{i-1} \rceil_h) \wedge \lceil T_i \rceil_h}-X_{\lceil T_{i-1} \rceil_h})_{s\geq 0}) \in \widehat{F}$ be the $h$-discretised $i$-th word, and let $$\label{def:RNh} R_{N,h} = \frac1N \sum_{i=0}^{N-1} \delta_{\widetilde{\theta}^i (Y^{(h)})^{N\text{-}\mathrm{per}}}$$ be the $h$-discretised empirical process, where $Y^{(h)}=(Y^{(i,h)})_{i\in{\mathbb{N}}}$. Put $\ell = \ell_1 \vee \cdots \vee \ell_m$, $C_g=C_{g_1} \vee \cdots \vee C_{g_m}$. Let $$\label{eq:defDjh} D_{j,h} = \sup_{(j-1)h \leq s \leq j h} |X_s-X_{j h}|, \qquad A_{\varepsilon,k,h}(N) = \left\{ \sum_{i=1}^N \sum_{j=0}^k D_{\lceil T_i/h \rceil+j,h} \leq N \varepsilon \right\}.$$ By Lemma \[obs:Rdiscdiff\], on the event $A_{\varepsilon,k,h}(N)$ we have $$N \big| \Phi([R_N]_{{\rm tr}}) - \Phi([R_{N,h}]_{{\rm tr}}) \big| \leq 4\ell \|\Phi\|_\infty + N C_g \ell m \Big(2h+ \log{\textstyle\frac{k+1}{k}} + 4\varepsilon \Big),$$ and hence $$\begin{aligned} \label{eq:EeNPhiRNtr.ub1} {\mathbb{E}}\big[ e^{N\Phi([R_N]_{{\rm tr}})} \big| X \big] \leq & \, \exp\big[N C_g \ell m \big(2h+ \log{\textstyle\frac{k+1}{k}} + 4\varepsilon \big) + 4\ell \|\Phi\|_\infty \big]\, {\mathbb{E}}\big[ e^{N\Phi([R_{N,h}]_{{\rm tr}})} \big| X \big] \notag \\ & \, {} + e^{N \|\Phi\|_\infty} {\mathbb{P}}\big(A_{\varepsilon,k,h}(N)^c \mid X\big), \end{aligned}$$ For $\lambda >0$, estimate $$\begin{aligned} {\mathbb{P}}([A_{\varepsilon,k,h}(N)]^c | X) \leq e^{-N \lambda \varepsilon} \, {\mathbb{E}}\left[ \exp\Big[ \lambda \sum_{i=1}^N \sum_{m=0}^k D_{\lceil T_i/h \rceil+m,h} \Big] \,\, \Big| \, X \right],\end{aligned}$$ so that, by Lemma \[lem:expmomentsDsum\] in Step 4 below, $$\begin{aligned} \label{eq:limsuplogPrAN} \limsup_{N\to\infty} \frac1N \log {\mathbb{P}}\big([A_{\varepsilon,k,h}(N)]^c \mid X\big) \leq -\varepsilon \lambda + \frac12 \log \chi\big(2 k \lambda \sqrt{h}\big). \end{aligned}$$ Since $\lim_{u\downarrow 0} \chi(u)= 1$, we have, for all $\varepsilon > 0$ and $k\in{\mathbb{N}}$, $$\label{eq_Aepskh.unlikely} \limsup_{h \downarrow 0} \limsup_{N\to\infty} \frac1N \log {\mathbb{P}}\big([A_{\varepsilon,k,h}(N)]^c \mid X\big) = - \infty \quad \text{a.s.}$$ (pick $\lambda=\lambda(h)$ in (\[eq:limsuplogPrAN\]) in such a way that $\lambda\to\infty$ and $\lambda\sqrt{h}\to 0$). Next, observe that $$\begin{aligned} {\mathbb{E}}\big[ e^{N\Phi([R_N]_{{\rm tr}})} \mid X \big] &= \int\cdots\int_{0<t_1<\cdots<t_N} \bar{\rho}(t_1) dt_1\, \bar{\rho}(t_2-t_1) dt_2 \times\cdots\times \bar{\rho}(t_N-t_{N-1}) dt_N \notag \\ &\qquad \times \exp\big[ {N\Phi\big([R_{N;t_1,\dots,t_N}(X)]_{{\rm tr}}\big)} \big],\\[0.5ex] \label{eq:expPhiRNhtr} {\mathbb{E}}\big[ e^{N\Phi([R_{N,h}]_{{\rm tr}})} \mid X \big] &= \sum_{1 \leq j_1 \leq \cdots \leq j_N} w_h(j_1,\dots,j_N) \exp\big[ {N\Phi\big([R_{N;h j_1,\dots,h j_N}(X)]_{{\rm tr}}\big)} \big], \end{aligned}$$ where $$\begin{aligned} \label{eq:wh.weights} w_h(j_1,\dots,j_N) &= \int\cdots\int_{0<t_1<\cdots<t_N} \bar{\rho}(t_1) dt_1\, \bar{\rho}(t_2-t_1) dt_2 \times\cdots\times \bar{\rho}(t_N-t_{N-1}) dt_N\\ &\qquad\qquad \times \prod_{k=1}^N {1}_{(h(j_k-1), h j_k]}(t_k). \end{aligned}$$ The idea is to replace the right-hand side of by $\prod_{k=1}^N \lceil \rho \rceil_h(h(j_k-j_{k-1}))$, which is the corresponding weight for a discrete-time renewal process with waiting time distribution $\lceil \rho \rceil_h$. The rigorous implementation of this idea requires some care, since the coarse graining can produce “empty” words. For $\underline{j}=(j_1,\dots,j_N)$ appearing in the sum in , let $R(\underline{j}) = \# \{ 1 \leq i \leq N \colon j_i = j_{i-1}\}$ be the total number of repeated values and $\underline{\hat{\jmath}}=(\hat \jmath_1,\dots,\hat \jmath_M)$ with $M=M(\underline{j}) =N-R(\underline{j})$, $1 \leq \hat \jmath_1 < \cdots < \hat \jmath_M$, the unique elements of $\underline{j}$. Note that any given $\underline{\hat{\jmath}}$ with $M=\lceil (1-\varepsilon) N \rceil$ can be obtained in this way from at most ${N \choose \lceil \varepsilon N \rceil}$ different $\underline{j}$’s. In the following, we write $\eta(h)=\eta_n$ and $A(h) = A_n$ with $\eta_n$ and $A_n$ from when $h=2^{-n}$. Let us parse through the right-hand side of successively for $k=N,N-1,\dots,1$. When $j_k=j_{k-1}$, we integrate $t_k$ out over $(h(j_k-1), h j_k]$ and estimate the (multiplicative) contribution of this integral from above by $1$. When $j_k>j_{k-1}$, we replace $\bar{\rho}(t_k-t_{k-1})$ by $\bar{\rho}(t_k- h j_{k-1})$ and integrate $t_k$ out over $(h(j_k-1), h j_k]$. For $h(j_k-j_{k-1}) \in A(h)$ we can estimate the contribution of this integral from above by $e^{\eta(h)} \lceil \rho \rceil_h(h(j_k-j_{k-1}))$) by using , while for $h(j_k-j_{k-1}) \not \in A(h)$ we can estimate it by $e^{\eta_0} \lceil \rho \rceil_h(h(j_k-j_{k-1}))$ by using with $s_*=0$. Thus, for $\underline{j}$ with $R(\underline{j}) \leq \varepsilon N$ and $\#\{ 1 \leq i < N \colon h (j_i - j_{i-1}) \not\in A(h) \} \leq \varepsilon N$, we have $$\begin{aligned} \label{eq:estwhbyrhohgr} w_h(\underline{j}) \leq e^{\varepsilon \eta_0 N} e^{\eta(h) N} \prod_{i=1}^{M} \lceil \rho \rceil_h \big(h(\hat \jmath_i- \hat \jmath_{i-1})\big) \end{aligned}$$ with $M=N-R(\underline{j})$. Furthermore, $$\begin{aligned} \label{eq:estlast} \Big| N\Phi\big([R_{N;h j_1,\dots,h j_N}(X)]_{{\rm tr}}\big) - M\Phi\big([R_{M;h \hat \jmath_1,\dots,h \hat \jmath_M}(X)]_{{\rm tr}}\big) \Big| \le (N-M) \ell \| \Phi \|_\infty \le \varepsilon N \ell \| \Phi \|_\infty.\end{aligned}$$ Combining (\[eq:expPhiRNhtr\]–\[eq:estlast\]), we find $$\begin{aligned} \label{eq:EeNPhiRNhtr.ub2} &{\mathbb{E}}\big[e^{N\Phi([R_{N,h}]_{{\rm tr}})} \mid X \big] \notag \\ &\leq e^{N \| \Phi \|_\infty} \Big\{ {\mathbb{P}}\Big( R(\lceil T_1 \rceil_h,\dots, \lceil T_N \rceil_h) \geq \varepsilon N \, \Big| \, X \Big) \notag \\ &\hspace{5em} {} + {\mathbb{P}}\Big( \#\big\{ 1 \leq i < N \colon \lceil T_i \rceil_h - \lceil T_{i-1} \rceil_h \not\in A(h) \big\} \geq \varepsilon N \, \Big| \, X \Big) \Big\} \notag \\ &\quad + e^{[\varepsilon \eta_0 + \eta(h)]N} {N \choose \varepsilon N} \sum_{M=\lceil (1-\varepsilon) N \rceil}^N \sum_{1 \leq \hat{\jmath}_1 < \cdots < \hat{\jmath}_M} e^{M \Phi\big([R_{M;h \hat{\jmath}_1,\dots,h \hat{\jmath}_M}(X)]_{{\rm tr}}\big)} \prod_{k=1}^M \lceil \rho \rceil_h(h(\hat{\jmath}_k-\hat{\jmath}_{k-1})) . \end{aligned}$$ But $$\begin{aligned} \sum_{1 \leq \hat{\jmath}_1 < \cdots < \hat{\jmath}_M} e^{M \Phi\big([R_{M;h \hat{\jmath}_1,\dots,h \hat{\jmath}_M}(X)]_{{\rm tr}}\big)} \prod_{k=1}^M \lceil \rho \rceil_h(h(\hat{\jmath}_k-\hat{\jmath}_{k-1})) = {\mathbb{E}}_{\lceil \rho \rceil_h}\big[ & e^{M\Phi([R_M]_{{\rm tr}})} \mid X \big],\end{aligned}$$ where ${\mathbb{E}}_{\lceil \rho \rceil_h}$ denotes expectation w.r.t. the reference measure $Q_{\lceil \rho \rceil_h, {\mathscr{W}}}$, and so we can apply Corollary \[prop:qLDPhtr\] and Varadhan’s lemma to obtain $$\begin{aligned} \label{eq:limEeMPhi} \lim_{M\to\infty} \frac1M \log {\mathbb{E}}_{\lceil \rho \rceil_h}\big[ & e^{M\Phi([R_M]_{{\rm tr}})} \mid X \big] = \sup_{Q \in \mathcal{P}^{\mathrm{inv, fin}}(\widetilde{E_h}^{\mathbb{N}})} \big\{ \Phi(Q) - I^{\mathrm{que}}_{h,{{\rm tr}}}(Q) \big\}.\end{aligned}$$ By elementary large deviation estimates for binomials we have, for any $\varepsilon>0$, $$\begin{aligned} \label{eq:toomanywrongloops1} &\limsup_{h \downarrow 0} \limsup_{N\to\infty} \frac1N \log {\mathbb{P}}\big( R(\lceil T_1 \rceil_h,\dots, \lceil T_N \rceil_h) \geq \varepsilon N \, \big| \, X \big) = - \infty,\\ \label{eq:toomanywrongloops2} &\limsup_{h \downarrow 0} \limsup_{N\to\infty} \frac1N \log {\mathbb{P}}\Big( \#\big\{ 1 \leq i < N \colon\, \lceil T_i \rceil_h - \lceil T_{i-1} \rceil_h \not\in A(h) \big\} \geq \varepsilon N \, \Big| \, X \Big) = - \infty. \end{aligned}$$ (Note that the events in (\[eq:toomanywrongloops1\]–\[eq:toomanywrongloops2\]) are independent of $X$.) Combining , and , and noting that $\lim_{N\to\infty} \frac1N \log {N \choose \varepsilon N} = -\varepsilon\log\varepsilon - (1-\varepsilon)\log(1-\varepsilon)$, we find $$\label{eq:eNPhiRNasympt_upper0} \begin{aligned} &\limsup_{N\to\infty} \frac1N \log {\mathbb{E}}\big[ e^{N\Phi([R_N]_{{\rm tr}})} \mid X \big] \\ &\qquad \leq \bigg\{ \sup_{Q \in \mathcal{P}^{\mathrm{inv, fin}}((\widetilde{E_h})^{\mathbb{N}})} \big\{ \Phi(Q) - I^{\mathrm{que}}_{h,{{\rm tr}}}(Q) \big\} \\ &\qquad \qquad + C_g \ell m \big(2h+ \log{\textstyle\frac{k+1}{k}} + 4\varepsilon \big) + \varepsilon \eta_0 + \eta(h) + \varepsilon\log\tfrac {1}{\varepsilon} + (1-\varepsilon)\log\tfrac{1}{1-\varepsilon} \bigg\}\\ &\qquad \vee \bigg( \|\Phi\|_\infty + \limsup_{N\to\infty} \frac1N \log {\mathbb{P}}\big(A_{\varepsilon,k,h}(N)^c \mid X\big)\bigg\} \\ &\qquad \vee \bigg\{ \|\Phi\|_\infty + \limsup_{N\to\infty} \frac1N \log {\mathbb{P}}\Big( R(\lceil T_1 \rceil_h,\dots, \lceil T_N \rceil_h) \geq \varepsilon N \, \Big| \, X \Big) \bigg\} \\ &\qquad \vee \bigg\{ \|\Phi\|_\infty + \limsup_{N\to\infty} \frac1N \log {\mathbb{P}}\Big( \#\{ 1 \leq i < N \colon \lceil T_i \rceil_h - \lceil T_{i-1} \rceil_h \not\in A(h) \} \geq \varepsilon N \, \Big| \, X \Big) \bigg\}, \end{aligned}$$ and hence $$\begin{aligned} \label{eq:eNPhiRNasympt_upper} \limsup_{N\to\infty} \frac1N \log {\mathbb{E}}\big[ e^{N\Phi([R_N]_{{\rm tr}})} \big| X \big] \leq \liminf_{h\downarrow 0} \sup_{Q \in \mathcal{P}^{\mathrm{inv, fin}}(\widetilde{E_h}^{\mathbb{N}})} \big\{ \Phi(Q) - I^{\mathrm{que}}_{h,{{\rm tr}}}(Q) \big\}\end{aligned}$$ (let $h\downarrow 0$ along a suitable subsequence, followed by $\varepsilon \downarrow 0$ and $k\to\infty$, and use and (\[eq:toomanywrongloops1\]–\[eq:toomanywrongloops2\])). Analogous arguments yield $$\begin{aligned} \label{eq:eNPhiRNasympt_lower} \liminf_{N\to\infty} \frac1N \log {\mathbb{E}}\big[ e^{N\Phi([R_N]_{{\rm tr}})} \mid X \big] \geq \limsup_{h\downarrow 0} \sup_{Q \in \mathcal{P}^{\mathrm{inv, fin}}((\widetilde{E_h})^{\mathbb{N}})} \big\{ \Phi(Q) - I^{\mathrm{que}}_{h,{{\rm tr}}}(Q) \big\}.\end{aligned}$$ Indeed, we can simply restrict the sum in to $\underline{j}$’s with $j_1 < \cdots < j_N$, so that the approximation argument is in fact a little easier because we need not pass to the $\underline{\hat\jmath}$’s. Finally, combine (\[eq:eNPhiRNasympt\_upper\]–\[eq:eNPhiRNasympt\_lower\]) to obtain . #### Step 2b. Next we consider the case $\mathrm{supp}(\rho)=[s_*,\infty)$ with $s_*>0$ and indicate the changes compared to Step 2a. To some extent this case is easier than the case $s_*=0$, since for coarse-graining level $h<s_*$ no “empty” word can appear in the coarse-graining scheme. On the other hand, when implementing a replacement similar to , it can happen that an integral $\int \bar{\rho}(t_k-t_{k-1}) {1}_{(h(j_k-1),h j_k]}(t) \,dt_k$ gets mapped to $\lceil \rho \rceil_h(h(j_k-j_{k-1}))=0$ even though the true contribution of that integral to is strictly positive (namely, when $h (j_k-j_{k-1}) \leq s_* \leq h(j_k-j_{k-1}+1)$). The idea to remedy this problem is to replace $\lceil \rho \rceil_h(h(j_k-j_{k-1}))$ by a sum of “neighbouring” weights of $\lceil \rho \rceil_h$ and to suitably control the overcounting incurred by this replacement. The details are as follows. Fix $h>0$ and $s_{*,h} = \lceil s_* \rceil_h$. For $N\in N$, consider $\underline{j}=(j_1,\dots,j_N)$ as appearing in the sum in . We say that $k \in \{1,\dots,N\}$ is “problematic” when $h(j_k-j_{k-1}) \in \{ s_{*,h}-1, s_{*,h}, s_{*,h}+1\}$, and “relaxable” when $j_k-j_{k-1} \geq 2$ and $$\max_{m=-1,0,1} \left| \log \frac{\lceil \rho \rceil_h(h(j_k-j_{k-1}+m))}{\lceil \rho \rceil_h(h(j_k-j_{k-1}))} \right| \leq 2.$$ Write $K_{\text{pro}}(\underline{j}) = \{ 1 \leq k \leq N \colon\, k\; \text{problematic}\}$ and $K_{\text{rel}} (\underline{j}) = \{ 1 \leq k \leq N \colon\,k\; \text{relaxable}\}$. Try to construct an injection $f_{\text{rel}, \underline{j}}\colon\,K_{\text{pro}} \to K_{\text{rel}}$ with the property $f_{\text{rel},\underline{j}}(k) > k$ as follows: - Start with an empty “stack” ${\sf s}$. For $k=1,\dots,N$ successively: when $k$ is problematic, push $k$ on ${\sf s}$; when $k$ is relaxable and ${\sf s}$ is not empty, pop the top element, say $k'$, from ${\sf s}$ and put $f_{\text{rel},\underline{j}}(k')=k$; when $k$ is neither problematic nor relaxable, proceed with the next $k$. We say that $\underline{j}$ is “good” when the above procedure terminates with an empty stack (in particular, $f_{\text{rel},\underline{j}}(k')$ is defined for all $k' \in K_{\text{pro}}$) and $$\sum_{k \in K_{\text{pro}}} \big( f_{\text{rel},\underline{j}}(k) - k \big) \leq \varepsilon N$$ (in particular, $\# K_{\text{pro}}(\underline{j}) \leq \varepsilon N$), and also $\# \{ 1 \leq k \leq N \colon\, j_k-j_{k-1} \not\in A(h)\} \leq \varepsilon N$. For a given good $\underline{j}$, consider the set of all $\underline{\tilde\jmath}=(\tilde\jmath_1,\dots,\tilde\jmath_N)$ obtainable by setting $$\begin{aligned} \tilde\jmath_k=j_k+\Delta_k, \; \tilde\jmath_{f_{\text{rel},\underline{j}}(k)} = j_{f_{\text{rel},\underline{j}}(k)}-\Delta_k \quad \text{with}\;\: \Delta_k \in \{-1,0,1\} \quad \text{for}\;\:k \in K_{\text{pro}},\end{aligned}$$ and $\tilde\jmath_k=j_k$ for $k \not\in(K_{\text{pro}} \cup f_{\text{rel},\underline{j}}(K_{\text{pro}}))$. Note that a given good $\underline{j}$ corresponds to at most $3^{\varepsilon N}$ different $\underline{\tilde\jmath}$’s and that, for any such $\underline{\tilde\jmath}$, $$\begin{aligned} &\Big| N\Phi\big([R_{N;h j_1,\dots,h j_N}(X)]_{{\rm tr}}\big) - N\Phi\big([R_{N;h \tilde \jmath_1,\dots,h \tilde \jmath_M}(X)]_{{\rm tr}}\big) \Big| \notag \\ &\qquad \leq \ell \| \Phi \|_\infty \sum_{k \in K_{\text{pro}}} \big( f_{\text{rel},\underline{j}}(k) - k \big) \leq \varepsilon N \ell \| \Phi \|_\infty .\end{aligned}$$ With $w_h(j_1,\dots,j_N)$ defined in , we now see that (analogously to the argument prior to ) for any good $\underline{j}$, $$\begin{aligned} \label{eq:estwhbyrhohgr.c2} w_h(\underline{j}) \leq e^{\varepsilon \eta_0 N} e^{\eta(h) N} 2^{\varepsilon N} \sum_{\underline{\tilde\jmath} \; \text{corresp.\ to}\; \underline{j}} \; \prod_{i=1}^{N} \lceil \rho \rceil_h \big(h(\tilde \jmath_i - \tilde \jmath_{i-1})\big). \end{aligned}$$ Moreover, we have $$\begin{aligned} \label{eq:toomanywrongloops3} \limsup_{h \downarrow 0} \limsup_{N\to\infty} \frac1N \log {\mathbb{P}}\big( (\lceil T_1 \rceil_h, \dots, \lceil T_N \rceil_h) \; \text{not good} \, \big| \, X \big) = - \infty. \end{aligned}$$ To check , let $S_k$ be the size of the stack ${\sf s}$ in the $k$-th step of the above construction when we use $j_k=\lceil T_k \rceil_h$, and note that $(\lceil T_1 \rceil_h, \dots, \lceil T_N \rceil_h)$ is good when $\sum_{k=1}^N S_k < \varepsilon N$. A comparison of $(S_k)_{k\in{\mathbb{N}}}$ with a (reflected) random walk on ${\mathbb{N}}_0$ that draws its steps from $\{0, \pm 1 \}$, where $(+1)$-steps have a very small probability ($\leq \int_{s_*}^{s_*+2h} \bar{\rho}(t)\, dt$) and $(-1)$-steps have a very large probability ($\rho(A_h)$) when not from $0$, shows that $\limsup_{h \downarrow h} \frac1N \log {\mathbb{P}}(\sum_{k=1}^N S_k \geq \varepsilon N) = -\infty$ for every $\varepsilon >0$. We can then estimate similarly as in , to obtain for the case $s_*>0$ as well. Analogous arguments also yield the lower bound in . #### Step 3. We next verify that the limits in exist. Note that $$| \Phi(R_N) - \Phi([R_N]_{{\rm tr}})| \leq \|\Phi\|_\infty \frac1N \#\big\{ \text{loops among the first $N$ loops that are longer than ${{\rm tr}}$} \big\},$$ which can be made arbitrarily small (also on the exponential scale, via a suitable annealing argument that uses that loop lengths are i.i.d.). A similar estimate holds for $| \Phi([R_N]_{{{\rm tr}}}) - \Phi([R_N]_{{{\rm tr}}'})|$ with ${{\rm tr}}< {{\rm tr}}'$. This shows that $\Lambda_{0,{{\rm tr}}}(\Phi)$ forms a Cauchy sequence as ${{\rm tr}}\to\infty$. The arguments in Steps [2a]{} and [2b]{} can be combined to yield the same results when assumption is relaxed to assumption . Indeed, for a given coarse-graining level $h$, gives rise to finitely many types of “problematic points” that can be handled similarly as in Step [2b]{} (combined with arguments from Step [2a]{} when $a_1=0$). #### Step 4. We close by deriving the estimate on Brownian increments over randomly drawn short time intervals that was used in in Step 2. The intuitive idea is that even though there are arbitrarily large increments over short time intervals somewhere on the Brownian path, it is extremely unlikely to hit these when sampling along an independent renewal process. The proof employs a suitable annealing argument. Recall $D_{j,h}$ from (\[eq:defDjh\]). For $h>0$ fixed, the $D_{j,h}$’s are i.i.d.  and equal in law to $\sqrt{h}D_{1,1} = \sqrt{h} \sup_{0\leq s\leq 1} |X_s|$ by Brownian scaling. \[lem:expmomentsDsum\] Let $T=(T_i)_{i\in{\mathbb{N}}}$ be a continuous-time renewal process with interarrival law $\rho$ satisfying $\mathrm{supp}(\rho) \subset [h,\infty)$. For $\lambda \geq 0$ and $k \in {\mathbb{N}}_0$, define $$\begin{aligned} \xi(\lambda,h) = \limsup_{N\to\infty} \frac1N \log {\mathbb{E}}\left[ \exp\Big[ \lambda \sum_{i=1}^N \sum_{m=0}^k D_{\lceil T_i/h \rceil+m,h} \Big] \, \Big| \, \sigma(D_{j,h}, j \in {\mathbb{N}})\right],\end{aligned}$$ which is $\geq 0$ and a.s. constant by Kolmogorov’s $0$-$1$-law. Then $$\begin{aligned} \lim_{h \downarrow 0} \xi(\lambda,h) = 0 \qquad \forall\,\lambda \geq 0. \end{aligned}$$ We consider only the case $k=0$, the proof for $k\in{\mathbb{N}}$ being analogous. Abbreviate $\mathscr{G}_h=\sigma(D_{j,h}, j \in {\mathbb{N}})$, and let $$\chi(u) = {\mathbb{E}}\Big[ \exp \big[ u \, {\textstyle\sup_{\,0 \leq t \leq 1} |X_t|} \big] \Big], \quad u \in {\mathbb{R}}.$$ Note that $\chi(\cdot)$ is finite and satisfies $\lim_{u\to 0} \chi(u) = 1$. We have $$\begin{aligned} {\mathbb{E}}\bigg[ {\mathbb{E}}\bigg[ \exp\Big[ \lambda \sum_{i=1}^N D_{\lceil T_i/h \rceil,h} \Big] \, \Big| \, \mathscr{G}_h \bigg]^2 \bigg] &\leq {\mathbb{E}}\bigg[ \exp\Big[ 2\lambda \sum_{i=1}^N D_{\lceil T_i/h \rceil,h} \Big] \bigg]\\ &= {\mathbb{E}}\big[ \exp[2\lambda D_{1,h}] \big]^N = \chi\big(2\lambda \sqrt{h}\big)^N. \end{aligned}$$ Thus, for any $\epsilon>0$, $$\begin{aligned} & {\mathbb{P}}\left( {\mathbb{E}}\left[ \exp\Big[ \lambda \sum_{i=1}^N D_{\lceil T_i/h \rceil,h} \Big] \, \Big| \, \mathscr{G}_h \right]^2 \geq \big( \chi\big(2\lambda \sqrt{h}\big) + \epsilon \big)^N \right) \\ & \leq \, \big( \chi\big(2\lambda \sqrt{h}\big) + \epsilon \big)^{-N} {\mathbb{E}}\left[ {\mathbb{E}}\left[ \exp\Big[ \lambda \sum_{i=1}^N D_{\lceil T_i/h \rceil,h} \Big] \, \Big| \, \mathscr{G}_h \right]^2 \right] \leq \left( \frac{\chi\big(2\lambda \sqrt{h}\big)}{\chi\big(2\lambda \sqrt{h}\big) + \epsilon} \right)^N, \end{aligned}$$ which is summable in $N$. The Borel-Cantelli lemma therefore yields $$\limsup_{N\to\infty} \frac1N \log {\mathbb{E}}\left[ \exp\Big[ \lambda \sum_{i=1}^N D_{\lceil T_i/h \rceil,h} \Big] \, \Big| \, \mathscr{G}_h \right] \leq \frac12 \log \chi\big(2\lambda \sqrt{h}\big).$$ Proof of Proposition \[prop:qLDPtrunc1\] {#ss:prop2} ---------------------------------------- \[lemma:Iquetrregularised\] For ${{\rm tr}}\in {\mathbb{N}}$ and $Q \in \mathcal{P}^{\mathrm{inv}}(F_{0,{{\rm tr}}}^{\mathbb{N}})$, $$\begin{aligned} \label{eq:Iquetrregularised} I^{\mathrm{que}}_{{\rm tr}}(Q) = \lim_{\varepsilon \downarrow 0} \, \limsup_{h \downarrow 0} \, \inf\Big\{ I^{\mathrm{que}}_{h,{{\rm tr}}}(Q')\colon\, Q' \in B_\varepsilon(Q) \cap \mathcal{P}^{\mathrm{inv}}((\widetilde{E}_{h,{{\rm tr}}})^{\mathbb{N}}) \Big\},\end{aligned}$$ where $h \downarrow 0$ along $2^{-m}$, $m\in {\mathbb{N}}$. Note that after $\widetilde{E}_{h,{{\rm tr}}}$ is identified with a subset of $F_{0,{{\rm tr}}}$ (see ), states that $I^{\mathrm{que}}_{h,{{\rm tr}}}$ converges to $I^{\mathrm{que}}_{{\rm tr}}$ as $h \downarrow 0$ in the sense of Gamma-convergence. Note that, when restricted to $\mathcal{P}^\mathrm{inv}(F_{0,{{\rm tr}}}^{\otimes{\mathbb{N}}})$, $$\label{eq:PsiQcont1} \text{both } Q \mapsto m_Q \text{ and } Q \mapsto \Psi_Q \text{ are continuous}$$ (by dominated convergence), while this is not true when $Q$ is allowed to vary over the whole of $\mathcal{P}^\mathrm{inv}(F^{\otimes{\mathbb{N}}})$. A more general statement is the following: if ${\mathop{\text{\rm w-lim}}}_{n\to\infty} Q_n = Q$ and $\{ \mathscr{L}_{Q_n}(\tau_1)\colon\, n \in {\mathbb{N}}\}$ are uniformly integrable, then $\lim_{n\to\infty} m_{Q_n} = m_Q$ and ${\mathop{\text{\rm w-lim}}}_{n\to\infty} \Psi_{Q_n} = \Psi_Q$. In the proof we use several properties of specific relative entropy derived in Appendix \[entropy\]. Let $Q \in \mathcal{P}^{\mathrm{inv}}(F_{0,{{\rm tr}}}^{\mathbb{N}})$, and abbreviate the right-hand side of by $\widetilde{I}^\mathrm{que}_{{\rm tr}}(Q)$. Note that, by and the lower semi-continuity of $\Psi \mapsto H(\Psi \mid {\mathscr{W}})$, the map $$\mathcal{P}^\mathrm{inv}(F_{0,{{\rm tr}}}^{{\mathbb{N}}}) \ni Q' \mapsto m_{Q'} H(\Psi_{Q'} \mid {\mathscr{W}})$$ is lower semi-continuous. Hence, for any $\delta>0$, we have $m_{Q'} H(\Psi_{Q'} \mid {\mathscr{W}}) \geq m_Q H(\Psi_Q \mid {\mathscr{W}}) - \delta$ for all $Q' \in B_\varepsilon(Q) \cap \mathcal{P}^{\mathrm{inv}} (\widetilde{E}_{h,{{\rm tr}}}^{\mathbb{N}})$ when $\varepsilon$ is sufficiently small (depending on $\delta$). Combine this with in Lemma \[lemma:hregularised1\] in Appendix \[entropy\], and note that ${\mathop{\text{\rm w-lim}}}Q_{h,{{\rm tr}}} = Q_{{\rm tr}}$ as $h\downarrow0$, to obtain $\widetilde{I}^\mathrm{que}_{{\rm tr}}(Q) \geq I^{\mathrm{que}}_{{{\rm tr}}}(Q)$. For the reverse direction, we need to find $h_n>0$ with $\lim_{n\to\infty} h_n = 0$ and $Q'_n \in \mathcal{P}^{\mathrm{inv}}((\widetilde{E}_{h_n,{{\rm tr}}})^{\mathbb{N}})$ with ${\mathop{\text{\rm w-lim}}}_{n\to\infty} Q'_n = Q$ such that $\liminf_{n\to\infty} I^{\mathrm{que}}_{h_n,{{\rm tr}}} (Q'_n) \le I^{\mathrm{que}}_{{\rm tr}}(Q)$. Here a complication stems from the fact that we must ensure that both parts of $I^{\mathrm{que}}_{h_n,{{\rm tr}}}(Q'_n)$, namely, $H(Q'_n \mid Q_{\lceil \rho \rceil_{h_n},{\mathscr{W}},{{\rm tr}}})$ and $H(\Psi_{Q'_n,h_n} \mid {\mathscr{W}})$, converge simultaneously. The proof is deferred to Lemma \[lem:cg.2lev.blockapprox\] in Appendix \[entropy\]. We are now ready to give the proof of Proposition \[prop:qLDPtrunc1\]. Fix ${{\rm tr}}\in{\mathbb{N}}$. Denote the right-hand side of by $\tilde{\Lambda}_{{\rm tr}}(\Phi)$. Let $\Phi\colon\,\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}}) \to {\mathbb{R}}$ be of the form (\[eq:Phiform1\]). For every $\delta > 0$ we can find a $Q^* \in \mathcal{P}^{\mathrm{inv}}(F_{0,{{\rm tr}}}^{\mathbb{N}})$ such that $\Phi(Q^*) - I_{{\rm tr}}^{\mathrm{que}}(Q^*) \geq \tilde{\Lambda}_{{\rm tr}}(\Phi) - \delta$. For $\varepsilon>0$ sufficiently small (depending on $\delta$) we have $\big| \Phi(Q') - \Phi(Q^*)\big| \leq \delta$ for all $Q' \in B_\varepsilon(Q^*)$ and, by Lemma \[lemma:Iquetrregularised\], $$\begin{aligned} \liminf_{h\downarrow 0} \, \inf\Big\{ I^{\mathrm{que}}_{h,{{\rm tr}}}(Q')\colon\, Q' \in B_\varepsilon(Q^*) \cap \mathcal{P}^{\mathrm{inv}}((\widetilde{E_h}_{,{{\rm tr}}})^{\mathbb{N}}) \Big\} \leq I^{\mathrm{que}}_{{\rm tr}}(Q^*) + \delta.\end{aligned}$$ Thus $$\begin{aligned} \liminf_{h\downarrow 0} \, \sup \Big\{ \Phi(Q') - I^{\mathrm{que}}_{h,{{\rm tr}}}(Q')\colon\, Q' \in B_\varepsilon(Q^*) \cap \mathcal{P}^{\mathrm{inv}}((\widetilde{E_h}_{,{{\rm tr}}})^{\mathbb{N}}) \Big\} \geq \tilde{\Lambda}_{{\rm tr}}(\Phi) - 3\delta. \end{aligned}$$ Let $\delta\downarrow 0$ to obtain $\liminf_{h\downarrow 0} \Lambda_{h,{{\rm tr}}}(\Phi) = \Lambda_{0,{{\rm tr}}}(\Phi) \geq \tilde{\Lambda}_{{\rm tr}}(\Phi)$. For the reverse direction, pick for $h \in (0,1)$ a maximiser $Q^*_h \in \mathcal{P}^{\mathrm{inv}} ((\widetilde{E_h}_{,{{\rm tr}}})^{\mathbb{N}})$ of the variational expression appearing in the right-hand side of , i.e., $\Phi(Q^*_h) - I^{\mathrm{que}}_{h,{{\rm tr}}}(Q^*_h) = \Lambda_{h,{{\rm tr}}}(\Phi)$. This is possible because $\Phi-I^{\mathrm{que}}_{h,{{\rm tr}}}$ is upper semi-continuous and bounded from above, and $I^{\mathrm{que}}_{h,{{\rm tr}}}$ has compact level sets. We claim that $$\label{claim:Q*htight} \text{the family} \; \big\{ Q^*_h : h \in (0,1) \big\} \subset \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})\; \text{is tight}.$$ Assuming (\[claim:Q\*htight\]), we can choose a sequence $h(n) \downarrow 0$ such that $$\begin{aligned} &\lim_{n\to\infty} \Big[ \Phi(Q^*_{h(n)}) - I^{\mathrm{que}}_{h(n),{{\rm tr}}}(Q^*_{h(n)}) \Big] = \limsup_{h\downarrow 0} \Lambda_{h,{{\rm tr}}}(\Phi),\\ &{\mathop{\text{\rm w-lim}}}_{n\to\infty} Q^*_{h(n)} = \widetilde{Q} \;\; \text{for some} \; \widetilde{Q} \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}}). \end{aligned}$$ Then $\lim_{n\to\infty}\Phi(Q^*_{h(n)})=\Phi(\widetilde{Q})$ because $\Phi$ is continuous, and $\liminf_{n\to\infty} I^{\mathrm{que}}_{h(n),{{\rm tr}}}(Q^*_{h(n)}) \geq I^{\mathrm{que}}_{{\rm tr}}(\widetilde{Q})$ by Lemma \[lemma:Iquetrregularised\]. Hence $$\begin{aligned} \Lambda_{0,{{\rm tr}}}(\Phi) = \limsup_{h\downarrow 0} \Lambda_{h,{{\rm tr}}}(\Phi) = \lim_{n\to\infty} \Big[ \Phi(Q^*_{h(n)}) - I^{\mathrm{que}}_{h(n),{{\rm tr}}}(Q^*_{h(n)}) \Big] \leq \Phi(\widetilde{Q}) - I^{\mathrm{que}}_{{\rm tr}}(\widetilde{Q}) \leq \tilde{\Lambda}_{{\rm tr}}(\Phi).\end{aligned}$$ It remains to prove , which follows once we show that for each $N \in {\mathbb{N}}$ the family of projections $\pi_N(Q^*_h) \in \mathcal{P}^{\mathrm{inv}}(F^N)$, $h\in(0,1)$, is tight (because $F^{\mathbb{N}}$ carries the product topology; see Ethier and Kurtz [@EK86 Chapter 3, Proposition 2.4]). Let $M= \|\Phi\|_\infty+1$. Then necessarily $H( Q^*_h \mid [Q_{\lceil \rho \rceil_h,{\mathscr{W}}}]_{{\rm tr}}) \leq M$, and hence $h(\pi_N(Q^*_h) \mid \pi_N([Q_{\lceil \rho \rceil_h,{\mathscr{W}}}]_{{\rm tr}}) \leq N M$ for all $h \in (0,1)$. Since $\pi_N([Q_{\lceil \rho \rceil_h,{\mathscr{W}}}]_{{\rm tr}}) = ([q_{\lceil \rho \rceil_h,{\mathscr{W}}}]_{{\rm tr}})^{\otimes N}$ converges weakly to $\pi_N ([Q_{\rho,{\mathscr{W}}}]_{{\rm tr}}) = ([q_{\rho,{\mathscr{W}}}]_{{\rm tr}})^{\otimes N}$ as $h \downarrow 0$, the family $\{\pi_N([Q_{\lceil \rho \rceil_h,{\mathscr{W}}}]_{{\rm tr}})\colon\,h \in (0,1)\}$ is tight, and so for any $\varepsilon > 0$ we can find a compact $\mathcal{C} \subset F^N$ such that $\pi_N([Q_{\lceil \rho \rceil_h,{\mathscr{W}}}]_{{\rm tr}})(\mathcal{C}^c) \leq \exp[-(NM + \log 2)/\varepsilon]$ uniformly in $h\in(0,1)$. By a standard entropy inequality (see in Appendix \[entropy\]), for all $h\in (0,1)$ we have $$\begin{aligned} \pi_N(Q^*_h)(\mathcal{C}^c) \leq \frac{\log 2 + h\big(\pi_N(Q^*_h) \mid \pi_N([Q_{\lceil \rho \rceil_h,{\mathscr{W}}}]_{{\rm tr}})\big)} {\log\Big(1+\big(\pi_N([Q_{\lceil \rho \rceil_h,{\mathscr{W}}}]_{{\rm tr}})(\mathcal{C}^c)\big)^{-1} \Big)} \leq \frac{\log 2 + MN}{\log\big(1+\exp[(N M + \log 2)/\varepsilon]\big)} \leq \varepsilon. \end{aligned}$$ This proves the representation of the limit $\Lambda_{0,{{\rm tr}}}(\Phi)$ from . From and , plus the exponential tightness in Proposition \[prop:LambdaPhilimit1tr\], we obtain the LDP via Bryc’s inverse of Varadhan’s lemma. Proof of Proposition \[prop:Ique.tr.cont\] {#prop3} ------------------------------------------ ### Proof of part (1) We first verify , i.e., for $Q \in \mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}})$, $$\begin{aligned} \label{eq:relentrsumlim} \lim_{{{\rm tr}}\to\infty} I^{\mathrm{que}}_{{\rm tr}}([Q]_{{\rm tr}}) & = \lim_{{{\rm tr}}\to\infty} \Big[ H([Q]_{{\rm tr}}\mid [Q_{\rho,{\mathscr{W}}}]_{{\rm tr}}) + (\alpha-1) m_{[Q]_{{\rm tr}}} H(\Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}) \Big] \notag \\ & = H(Q \mid Q_{\rho,{\mathscr{W}}}) + (\alpha-1) m_Q H(\Psi_Q \mid {\mathscr{W}}).\end{aligned}$$ The proof comes in 5 Steps. [**Step 1.**]{} Note that $\lim_{{{\rm tr}}\to\infty} H([Q]_{{\rm tr}}\mid [Q_{\rho,{\mathscr{W}}}]_{{\rm tr}}) = H(Q \mid Q_{\rho,{\mathscr{W}}})$ by the projective property of word truncations, $\lim_{{{\rm tr}}\to\infty} m_{[Q]_{{\rm tr}}} = m_Q<\infty$ by dominated convergence, and $$\liminf_{{{\rm tr}}\to\infty} H(\Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}) \geq H(\Psi_Q \mid {\mathscr{W}})$$ by the lower semi-continuity of specific relative entropy together with ${\mathop{\text{\rm w-lim}}}_{{{\rm tr}}\to\infty} \Psi_{[Q]_{{\rm tr}}} = \Psi_Q$. Hence, to obtain it remains to prove that $$\begin{aligned} \label{ineq:HPsiQtr.upper} \limsup_{{{\rm tr}}\to\infty} H(\Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}) \leq H(\Psi_Q \mid {\mathscr{W}}).\end{aligned}$$ [**Step 2.**]{} \[prop:Ique.tr.cont.part1step2\] To prove , we use coarse-graining. For every $h>0$ we can identify $\widetilde{E_h}$ with $F_h \subset F$ (recall ). In order to represent $Q\in\mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}})$ by a shift-invariant law on $(F_h)^{\mathbb{N}}$, we discretise the cut-points onto a *uniformly shifted* grid of width $h$, as follows. For $t \in {\mathbb{R}}$, $h>0$ and $u\in [0,1)$, define (compare with Section \[trun\]) $$\begin{aligned} \label{eq:t.hu} \lceil t \rceil_{h,u} = \min\big\{ (k+u)h \colon k \in {\mathbb{Z}}, (k+u)h \geq t \big\} \quad \big(= \lceil t -uh \rceil_h + uh \big).\end{aligned}$$ Draw $Y=(Y^{(i)})_{i\in{\mathbb{N}}}=((\tau_i, f_i))_{i\in{\mathbb{N}}}$ from law $Q$, and let $U$ be an independent random variable with uniform distribution on $[0,1]$. Put $T_0=0$, $T_n=\tau_1+\cdots+\tau_n$, $n \in {\mathbb{N}}$, $$\tilde{T}_i = \lceil T_i \rceil_{h,U}, \quad i \in {\mathbb{N}}_0, \qquad \tilde{\tau}_i = \tilde{T}_i - \tilde{T}_{i-1}, \; \tilde{f}_i = \big(\theta^{\tilde{T}_{i-1}} \kappa(Y)\big)(\, \cdot \wedge \tilde{\tau}_i), \quad i\in{\mathbb{N}}.$$ (Note that it may happen that $\tilde{\tau}_i=0$. We can remedy this by allowing “empty words”, i.e., by formally passing to $\widehat{F}$ as in Section \[subsect:notations\].) Write $\lceil Q \rceil_h$ for the distribution of $\tilde{Y}=(\tilde{Y}^{(i)})_{i \in {\mathbb{N}}}=((\tilde{\tau}_i, \tilde{f}_i))_{i \in {\mathbb{N}}}$ obtained in this way. We view $\lceil Q \rceil_h$ as an element of $\mathcal{P}^{\mathrm{inv,fin}}((F_h)^{\mathbb{N}})$. To check the shift-invariance of $\lceil Q \rceil_h$, note that by construction an initial part of length $S_1=\tilde{T}_0-T_0 = U h$ of the content of the first word is removed (in a two-sided situation, this part would be added at the end of the zero-th word). The corresponding quantity for the second word is $S_2=\tilde{T}_1-T_1 = \lceil T_1 \rceil_{h,U} - T_1$. Observe that, for measurable $A \subset [0,h)$ and $B \subset [0,\infty)$, $${\mathbb{P}}(S_2 \in A, T_1 \in B) = \int_B {\mathbb{P}}(T_1 \in dt) \int_{[0,1]} du\, {1}_A\big( \lceil t -uh \rceil_h - (t-uh) \big) = \frac1h\, {\mathbb{P}}(T_1 \in B)\, \lambda(A),$$ i.e., $S_2$ is distributed as $U h$ and independent of $Y$, and so $(\tilde{Y}^{(i+1)})_{i\in{\mathbb{N}}}$ again has law $\lceil Q \rceil_h$. This settles the shift-invariance. The key feature of the construction of $\lceil Q \rceil_h$ is that $\kappa(\tilde{Y}) = (\theta^{U h} \kappa)(Y)$, so that $$\label{Psihrel} \Psi_{\lceil Q \rceil_h, h} = \Psi_Q,$$ and therefore $$\begin{aligned} \label{HhHrel} H(\Psi_{\lceil Q \rceil_h, h} \mid {\mathscr{W}}) = H(\Psi_Q \mid {\mathscr{W}}).\end{aligned}$$ Thus, gives us a coarse-grained version of the right-hand of . [**Step 3.**]{} If ${{\rm tr}}$ is an integer multiple of $h$, then the coarse-graining $\lceil Q \rceil_h \in \mathcal{P}^{\mathrm{inv,fin}}((F_h)^{\mathbb{N}})$ of $Q \in \mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}})$ defined in Step 2 commutes with the word length truncation $[ \cdot ]_{{\rm tr}}$, i.e., $[ \lceil Q \rceil_h ]_{{\rm tr}}= \lceil [ Q ]_{{\rm tr}}\rceil_h$. This is a deterministic property of the construction in . Indeed, fix $u \in [0,1)$ and $h$ with ${{\rm tr}}= Mh$ for some $M\in {\mathbb{N}}$, consider $t_{i-1}<t_i$ with $t_i-t_{i-1} > {{\rm tr}}$ (so that in the un-coarse-grained truncation procedure the $i$-th loop length would be replaced by ${{\rm tr}}$), let $k_{i-1}, k_i \in {\mathbb{N}}$ be such that $\lceil t_{i-1} \rceil_{h,u} = (k_{i-1}+u)h$ and $\lceil t_i \rceil_{h,u} = (k_i+u)h$. When we first truncate and then coarse-grain, the $i$-th point becomes $\lceil t_{i-1} + {{\rm tr}}\rceil_{h,u} = (k_{i-1}+M+u)h$. When we first coarse-grain and then truncate, the $i$-th point becomes $\lceil t_{i-1} \rceil_{h,u} + \big( (\lceil t_i \rceil_{h,u} - \lceil t_{i-1} \rceil_{h,u}) \wedge M h \big) = (k_{i-1}+u)h + M h$, which is the same. [**Step 4.**]{} Let $h=2^{-M}$, define $\lceil Q \rceil_h \in \mathcal{P}^{\mathrm{inv,fin}}((F_h)^{\mathbb{N}})$ as in Step 2, and write $Q'_h = \lceil Q \rceil_h \circ \iota_h^{-1}$ for the same object considered as an element of $\mathcal{P}^{\mathrm{inv,fin}}((\widetilde{E_h})^{\mathbb{N}})$ (recall (\[def:Eh\]–\[iotahdef\])). Write $\nu_h=\mathscr{L}\big((X_{\cdot \wedge h})\big)$ for the Wiener measure on $E_h$. Then $m_{Q'_h} = m_{\lceil Q \rceil_h}/h$ (the mean word length counted in $h$-letters), while $$\begin{aligned} \label{HPsihrel} H(\Psi_{Q'_h} \mid \nu_h^{\otimes {\mathbb{N}}}) = H(\Psi_{\lceil Q \rceil_h,h} \mid {\mathscr{W}}),\end{aligned}$$ by construction, and $$\begin{aligned} \label{Qtrhrel} \lceil [ Q ]_{{\rm tr}}\rceil_h = [ \lceil Q \rceil_h ]_{{\rm tr}}= [Q'_h]_{({{\rm tr}}/h)} \circ \iota_h,\end{aligned}$$ where the first equality follows from the commutation property in Step 3 and the second equality is a truncation of the words from $Q'_h$ as elements of $\widetilde{E_h}$. [**Step 5.**]{} Fix $\varepsilon>0$ and let ${{\rm tr}}_0 = {{\rm tr}}_0(Q,\varepsilon)$ be so large that $$\begin{aligned} {\mathbb{E}}_Q\big[ \big( |Y^{(1)}|-{{\rm tr}}\big)_+ \big] < \tfrac13 \varepsilon m_Q, \qquad {{\rm tr}}\geq {{\rm tr}}_0.\end{aligned}$$ Then, for $0<h<\tfrac{1}{24} \varepsilon m_Q$, we have $$\begin{aligned} \label{eq:justso} {\mathbb{E}}_{\lceil Q \rceil_h}\big[ h\big( \tfrac{|Y^{(1)}|}{h}-\tfrac{{{\rm tr}}}{h} \big)_+ \big] < \tfrac13 \varepsilon m_Q + 2h < \tfrac12 \varepsilon m_{\lceil Q \rceil_h}. \end{aligned}$$ Divide both sides of by $h$, and observe that the continuum word of length $|Y^{(1)}|$ under $\lceil Q \rceil_h$ corresponds to the discrete word of $|Y^{(1)}|/h$ $h$-letters under $Q_h'$, to obtain $$\begin{aligned} {\mathbb{E}}_{Q_h'}\big[ \big( |Y^{(1)}|-\tfrac{{{\rm tr}}}{h} \big)_+ \big] < \tfrac12 \varepsilon m_{Q_h'}.\end{aligned}$$ This estimate allows us to use Lemma \[lem:trcontinuous\] in Appendix \[entropy\], which says that for every $0<\varepsilon<\tfrac12$, $$\begin{aligned} \label{bepsest} (1-\varepsilon) \Big[ H(\Psi_{[Q_h']_{({{\rm tr}}/h)}} \mid \nu_h^{\otimes {\mathbb{N}}}) + b(\varepsilon) \Big] \leq H(\Psi_{Q_h'} \mid \nu_h^{\otimes {\mathbb{N}}})\end{aligned}$$ with $b(\varepsilon)= - 2\varepsilon + [\varepsilon \log \varepsilon + (1-\varepsilon) \log (1-\varepsilon)]/(1-\varepsilon)$. However, by (\[HPsihrel\]–\[Qtrhrel\]) we have $$H(\Psi_{[Q'_h]_{({{\rm tr}}/h)}} \mid \nu_h^{\otimes {\mathbb{N}}}) = H(\Psi_{\lceil [Q]_{{\rm tr}}\rceil_h,h} \mid {\mathscr{W}}) = H(\Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}).$$ Substitute this relation into and use (\[HhHrel\]–\[HPsihrel\]), to obtain $$(1-\varepsilon) \Big[ H(\Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}) + b(\varepsilon) \Big] \leq H(\Psi_Q \mid {\mathscr{W}}).$$ Now let $\varepsilon \downarrow 0$ and use that $\lim_{\varepsilon\downarrow 0} b(\varepsilon)=0$, to obtain . ### Proof of part (2) {#subsect:prop:Ique.tr.cont.part2} Fix $Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ with $m_Q= \infty$ and $H(Q \mid Q_{\rho,{\mathscr{W}}}) < \infty$. We construct $\widetilde{Q}_{{\rm tr}}\in \mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}})$, ${{\rm tr}}\in{\mathbb{N}}$, satisfying via a “smoothed truncation” that has the same concatenated word content as its “hard truncation” equivalent. The proof comes in 5 Steps. [**Step 1.**]{} It will we be convenient to consider the two-sided scenario, i.e., we regard $Q$ as a shift-invariant probability measure on $F^{\mathbb{Z}}$. Define $$\chi_{{\rm tr}}\colon\,F_{0,{{\rm tr}}}^{\mathbb{Z}}\times [0,1]^{\mathbb{Z}}\to F^{\mathbb{Z}}, \qquad \chi_{{\rm tr}}\colon\,\big( (f_i,\tau_i)_{i\in{\mathbb{Z}}}, (u_i)_{i\in{\mathbb{Z}}} \big) \mapsto (\tilde f_i, \tilde \tau_i)_{i\in{\mathbb{Z}}},$$ as follows. Put $t_0=0$, $t_i=t_{i-1}+\tau_i$, $t_{-i}=t_{-i+1}-\tau_{-i+1}$ for $i\in{\mathbb{N}}$, and $\varphi = \kappa\big( (f_i,\tau_i)_{i\in{\mathbb{Z}}}\big)$, set $$\begin{aligned} \tilde{t}_i = \begin{cases} t_i-u_i & \text{if} \; \tau_i={{\rm tr}},\\ t_i & \text{if} \; \tau_i <{{\rm tr}}, \end{cases}\end{aligned}$$ $\tilde\tau_i=t_i-t_{i-1}$ and $\tilde f_i(\cdot)=\varphi( (\,\cdot \wedge \tilde\tau_i)+t_{i-1})$ for $i\in{\mathbb{Z}}$. In words, the total concatenated word content remains unchanged, and if the length of the $i$-th word $\tau_i$ equals ${{\rm tr}}$, then its end-point $t_i$ is moved $u_i$ to the left. Put $\widetilde{Q}_{{\rm tr}}= ([Q]_{{\rm tr}}\otimes \mathrm{Unif}[0,1]^{\otimes {\mathbb{Z}}}) \circ \chi_{{\rm tr}}^{-1} \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{Z}})$. By construction, $\Psi_{\widetilde{Q}_{{\rm tr}}} = \Psi_{[Q]_{{\rm tr}}}$ and $m_{\widetilde{Q}_{{\rm tr}}} = m_{[Q]_{{\rm tr}}}$. In particular, $$\begin{aligned} m_{\widetilde{Q}_{{\rm tr}}} H(\Psi_{\widetilde{Q}_{{\rm tr}}} \mid {\mathscr{W}}) = m_{[Q]_{{\rm tr}}} H(\Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}) .\end{aligned}$$ [**Step 2.**]{} Write $\widetilde{Q}_{{\rm tr}}^{\mathrm{ref}} = ([q_{\rho,{\mathscr{W}}}]_{{\rm tr}}^{\otimes{\mathbb{Z}}} \otimes \mathrm{Unif} [0,1]^{\otimes {\mathbb{Z}}}) \circ \chi_{{\rm tr}}^{-1}$ for the result of the analogous operation on the reference measure $(q_{\rho,{\mathscr{W}}})^{\otimes {\mathbb{Z}}}$. We have ${\mathop{\text{\rm w-lim}}}_{{{\rm tr}}\to\infty} \widetilde{Q}_{{\rm tr}}= Q$ and ${\mathop{\text{\rm w-lim}}}_{{{\rm tr}}\to\infty} ([q_{\rho,{\mathscr{W}}}]_{{\rm tr}}^{\otimes{\mathbb{Z}}} \otimes \mathrm{Unif}[0,1]^{\otimes {\mathbb{Z}}}) \circ \chi_{{\rm tr}}^{-1} = (q_{\rho,{\mathscr{W}}})^{\otimes {\mathbb{Z}}}$, and hence $$\begin{aligned} \label{eq:string} \liminf_{{{\rm tr}}\to\infty} H( \widetilde{Q}_{{\rm tr}}\mid \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}} ) & \geq \sup_{\varepsilon > 0} \liminf_{{{\rm tr}}\to\infty} \inf_{Q' \in B_\varepsilon(Q)} H( Q' \mid \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}} ) \notag \\ & \geq H(Q \mid (q_{\rho,{\mathscr{W}}})^{\otimes {\mathbb{Z}}}) = \lim_{{{\rm tr}}\to\infty} H([Q]_{{\rm tr}}\mid [q_{\rho,{\mathscr{W}}}]_{{\rm tr}}^{\otimes {\mathbb{Z}}}),\end{aligned}$$ where we use Lemma \[lemma:hregularised1\] (2) in the second inequality. (Note: Inspection of the proof of Lemma \[lemma:hregularised1\] (2) shows that the inequality “$\leq$” in also holds for $Q$’s that are not product.) The last equality in holds because the truncations $[\,\cdot\,]_{{\rm tr}}$ form a projective family (see [@BiGrdHo10 Lemma 8.1]). As specific relative entropy can only decrease under the operation of taking image measures, we have $H( \widetilde{Q}_{{\rm tr}}\mid \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}} ) \leq H([Q]_{{\rm tr}}\mid [q_{\rho, {\mathscr{W}}}]_{{\rm tr}}^{\otimes {\mathbb{Z}}}) \leq H(Q \mid q_{\rho,{\mathscr{W}}}^{\otimes {\mathbb{Z}}})$, so $\limsup_{{{\rm tr}}\to\infty} H( \widetilde{Q}_{{\rm tr}}\mid \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}} ) \leq H(Q \mid q_{\rho,{\mathscr{W}}}^{\otimes {\mathbb{Z}}})$ and, indeed, $$\begin{aligned} \lim_{{{\rm tr}}\to\infty} H( \widetilde{Q}_{{\rm tr}}\mid \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}} ) = \lim_{{{\rm tr}}\to\infty} H([Q]_{{\rm tr}}\mid [q_{\rho,{\mathscr{W}}}]_{{\rm tr}}^{\otimes {\mathbb{Z}}}) = H(Q \mid q_{\rho,{\mathscr{W}}}^{\otimes {\mathbb{Z}}}).\end{aligned}$$ The proof of is complete once we show that $$\begin{aligned} \label{eq:HtildeQtr.bd} H( \widetilde{Q}_{{\rm tr}}\mid q_{\rho,{\mathscr{W}}}^{\otimes {\mathbb{Z}}}) \leq H( \widetilde{Q}_{{\rm tr}}\mid \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}} ) + o(1),\end{aligned}$$ since, by part (1), $$\begin{aligned} \widetilde{I}^{\mathrm{que}}(\widetilde{Q}_{{\rm tr}}) = H( \widetilde{Q}_{{\rm tr}}\mid q_{\rho,{\mathscr{W}}}^{\otimes {\mathbb{Z}}}) + m_{\widetilde{Q}_{{\rm tr}}} H(\Psi_{\widetilde{Q}_{{\rm tr}}} \mid {\mathscr{W}}).\end{aligned}$$ [**Step 3.**]{} It remains to verify . Note that $$\begin{aligned} H( \widetilde{Q}_{{\rm tr}}\mid q_{\rho,{\mathscr{W}}}^{\otimes {\mathbb{Z}}}) - H( \widetilde{Q}_{{\rm tr}}\mid \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}} ) & = \lim_{N\to\infty} \frac1N {\mathbb{E}}_{\widetilde{Q}_{{\rm tr}}}\bigg[ \log\frac{d\pi_N \widetilde{Q}_{{\rm tr}}}{d q_{\rho,{\mathscr{W}}}^{\otimes N}} - \log\frac{d\pi_N \widetilde{Q}_{{\rm tr}}}{d\pi_N \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}}} \bigg] \notag \\ & = \lim_{N\to\infty} \frac1N {\mathbb{E}}_{\widetilde{Q}_{{\rm tr}}} \bigg[ \log\frac{d\pi_N \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}}}{d q_{\rho,{\mathscr{W}}}^{\otimes N}} \bigg],\end{aligned}$$ and that, by construction, ${d\pi_N \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}}}/{dq_{\rho,{\mathscr{W}}}^{\otimes N}}$ is a function of the word lengths $\tilde{\tau}_1,\dots,\tilde{\tau}_N$ only (indeed, because of the i.i.d. property of Brownian increments it easy to see that under both laws the word contents given their lengths are the same, namely, independent pieces of Brownian paths). Write $\widetilde{R}_{{\rm tr}}^{\mathrm{ref}}$ for the law of the sequence of word lengths under $\widetilde{Q}_{{\rm tr}}^{\mathrm{ref}}$. Then we must show that $$\begin{aligned} \label{eq:ElogdRtildedrho.bd} \limsup_{{{\rm tr}}\to\infty} \lim_{N\to\infty} \frac1N {\mathbb{E}}_{\widetilde{Q}_{{\rm tr}}} \bigg[ \log\frac{d\pi_N \widetilde{R}_{{\rm tr}}^{\mathrm{ref}}}{d \rho^{\otimes N}} (\tilde\tau_1,\dots,\tilde\tau_N) \bigg] \leq 0. \end{aligned}$$ [**Step 4.**]{} Denote the density of $\pi_N \widetilde{R}_{{\rm tr}}^{\mathrm{ref}}$ with respect to Lebesgue measure on ${\mathbb{R}}_+^N$ by $\widetilde{f}_{{{\rm tr}},N}^{\mathrm{ref}}$. Consider fixed $\tilde\tau_1,\dots,\tilde\tau_N$, and decompose into maximal stretches of $\tilde\tau_i$’s with values in $({{\rm tr}}-1,{{\rm tr}}+1)$ (note that under $\chi_{{\rm tr}}$ no word can become longer than ${{\rm tr}}+1$, while when $\tilde\tau_i < {{\rm tr}}-1$ the corresponding word is not truncated, i.e., $\tilde{t}_i=t_i$). Thus, there are $0 \leq M < N$, $i'_1 \leq j'_2 < i'_2 \leq j'_2 < \cdots < i'_M \leq j'_M \leq N$ such that $\{ 1 \leq i \leq N\colon\,\tilde\tau_i \in ({{\rm tr}}-1,{{\rm tr}}+1) \} = \cup_{k=1}^M [i'_k, j'_k] \cap {\mathbb{N}}$. Observe that, by construction, $\widetilde{f}_{{{\rm tr}},N}^{\mathrm{ref}}(\tilde\tau_1,\dots,\tilde\tau_N)$ can be decomposed into a product of $\prod_{j \colon \tilde\tau_j\leq {{\rm tr}}-1} \bar{\rho} (\tilde\tau_j)$ and $M$ further factors involving the $\tilde\tau_i$’s from these stretches, where the $k$-th factor depends only on $(\tilde\tau_i\colon\, i'_k \leq i \leq j'_k)$. We claim that $$\begin{aligned} \label{eq:ftrN.ref.bd} \frac{\widetilde{f}_{{{\rm tr}},N}^{\mathrm{ref}}(\tilde\tau_1,\dots,\tilde\tau_N)}{ \prod_{j=1}^N \bar{\rho}(\tilde\tau_j)} \leq \prod_{k=1}^M \big( C_1 {{\rm tr}}^{1+\epsilon} \big)^{j'_k-i'_k+1} = \big( C_1 {{\rm tr}}^{1+\epsilon} \big)^{\# \{ 1 \leq i \leq N\colon\, \tilde\tau_i > {{\rm tr}}-1 \}}\end{aligned}$$ for some $C_1=C_1(\rho) <\infty$ and $\epsilon=\epsilon(\rho) \in [0,1]$ uniformly in ${{\rm tr}}$ for ${{\rm tr}}$ sufficiently large. To see why holds, consider for example the first stretch and assume for simplicity that $i'_1=1<j'_1$ and that we know that the $0$-th word is not truncated (i.e., $\tilde{t}_0=t_0=0$). Let $\ell \leq j'_1+1$, and pretend we know that the first $\ell-1$ words are truncated (i.e., $\tau_1=\cdots=\tau_{\ell-1}={{\rm tr}}$), while the $\ell$-th word is not ($\tau_\ell<{{\rm tr}}$). Then $\tilde\tau_1={{\rm tr}}-u_1$ and $\tilde\tau_i={{\rm tr}}-u_i+u_{i-1}$ for $2 \leq i \leq \ell-1$, and so $u_i=\sum_{j=1}^i ({{\rm tr}}-\tilde\tau_j)$ for $1 \leq i \leq \ell-1$ and $\tau_\ell =\tilde\tau_\ell-u_{\ell-1} = \tilde\tau_\ell-\sum_{j=1}^{\ell-1} ({{\rm tr}}-\tilde\tau_j)$. This case contributes to $\widetilde{f}_{{{\rm tr}},\ell}^{\mathrm{ref}}$ the term $$\begin{aligned} \label{eq:termf.ell.ref} \rho([{{\rm tr}},\infty))^{\ell-1} \bar\rho\Big(\tilde\tau_\ell- {\textstyle \sum_{j=1}^{\ell-1} ({{\rm tr}}-\tilde\tau_j)}\Big) \prod_{i=1}^{\ell-1} {1}_{[0,1]}\Big( {\textstyle \sum_{j=1}^{i} ({{\rm tr}}-\tilde\tau_j)}\Big).\end{aligned}$$ Note that, by , we have $\eqref{eq:termf.ell.ref}/\prod_{j=1}^\ell \bar{\rho}(\tilde\tau_j) \le C_2 (C_3 {{\rm tr}}^{1+\epsilon})^{\ell-1}$ for some $C_2=C_2(\rho), C_3=C_3(\rho) <\infty$ and $\epsilon=\epsilon(\rho) \in [0,1]$ uniformly in ${{\rm tr}}$ for ${{\rm tr}}$ sufficiently large. The contribution of any given stretch of length $j'_k-i'_k+1$ can be written as a sum of at most $2^{j'_k-i'_k+1}$ cases where the indices of the truncated words are specified. Each such case can be estimated by a suitable product of terms as in . Furthermore, outside the stretches the words are necessarily untruncated and thus contribute $\bar{\rho}(\tilde\tau_i)$ to $\widetilde{f}_{{{\rm tr}},N}^{\mathrm{ref}}$, which cancels with the corresponding term in $\rho^{\otimes N}$. [**Step 5.**]{} From and the shift-invariance of $\widetilde{Q}_{{\rm tr}}$ we obtain that $$\begin{aligned} \label{eq:ElogdRtildedrho.bd2} \lim_{N\to\infty} \frac1N {\mathbb{E}}_{\widetilde{Q}_{{\rm tr}}} \bigg[ \log\frac{d\pi_N \widetilde{R}_{{\rm tr}}^{\mathrm{ref}}}{d \rho^{\otimes N}} (\tilde\tau_1,\dots,\tilde\tau_N) \bigg] \leq C(1+\log {{\rm tr}}) Q(\tau_1>{{\rm tr}}-1). \end{aligned}$$ Now, $h(\mathscr{L}_Q(\tau_1) \mid \rho) \leq H(Q \mid q_{\rho,{\mathscr{W}}}^{\mathbb{N}}) < \infty$ by assumption. Because of , this implies that ${\mathbb{E}}_Q[\log(\tau_1)] < \infty$, and hence that $Q(\tau_1>{{\rm tr}}) = o(1/\log {{\rm tr}})$. Therefore implies . $\qed$ Removal of Assumptions – {#removeass} ======================== We give a brief sketch of the proof only, leaving the details to the reader. Assumptions – are satisfied when $\bar{\rho}$ satisfies and varies regularly at $\infty$ with index $\alpha$. The latter condition is stronger than . To prove the claim under alone, note that for every $\delta>0$ and $\alpha'<\alpha$ there exists a probability density $\bar{\rho}'=\bar{\rho}' (\delta,\alpha')$ such that $\bar{\rho} \leq (1+\delta)\bar{\rho}'$, $\bar{\rho}'$ varies regularly at $\infty$ with index $\alpha'$, and $\bar{\rho}'(t)dt$ converges weakly to $\bar{\rho}(t)dt$ as $\delta \downarrow 0$ and $\alpha' \uparrow \alpha$. Since the quenched LDP holds for $\bar{\rho}'$, we can proceed similarly as in [@BiGrdHo10 Sections 3.6 and 5] to get the quenched LDP for $\bar{\rho}$. More precisely, for $B \subset \mathcal{P}^\mathrm{inv}(F^{\mathbb{N}})$ we may write (recall and ) $$\begin{aligned} P(R_N \in B \mid X) &= \int_{0 \leq t_1 < \cdots < t_N < \infty} dt_1 \cdots dt_N\, \bar{\rho}(t_1)\,\bar{\rho}(t_2-t_1) \cdots \bar{\rho}(t_N-t_{N-1})\\[-2ex] &\hspace{18em} \times 1_B\big(R_{N;t_1,\ldots,t_N}(X)\big), \notag\end{aligned}$$ and estimate $\bar{\rho}(t_1) \leq (1+\delta)\bar{\rho}'(t_1)$, etc., to get $P(R_N \in B \mid X) \leq (1+\delta)^N\,P'(R_N \in B \mid X)$, where $P,P'$ have $\bar{\rho},\bar{\rho}'$ as excursion length distributions. Let $\mathcal{C} \subset \mathcal{P}^\mathrm{inv}(F^{\mathbb{N}})$ be a closed set, and let $\mathcal{C}^{(\varepsilon)}$ be its $\varepsilon$-blow-up. Then the LDP upper bound for $\bar{\rho}'$ gives $$\limsup_{N\to\infty} \frac{1}{N} \log P(R_N \in \mathcal{C}^{(\varepsilon)}\mid X) \leq \log (1+\delta) - \inf_{Q \in \mathcal{C}^{(\varepsilon)}} I^\mathrm{que}_{\bar{\rho}'}(Q) \qquad X\text{-a.s.},$$ where the lower index $\bar{\rho}'$ indicates the excursion length distribution. Let $\delta \downarrow 0$ and $\alpha' \uparrow \alpha$, and use Lemma \[lemma:hregularised1\] (2), to get $$\limsup_{N\to\infty} \frac{1}{N} \log P(R_N \in \mathcal{C}^{(\varepsilon)}\mid X) \leq - \inf_{Q \in \mathcal{C}^{(2\varepsilon)}} I^\mathrm{que}_{\bar{\rho}}(Q) \qquad X\text{-a.s.}$$ Finally, let $\varepsilon \downarrow 0$ and use the lower semi-continuity of $I^\mathrm{que}_{\bar{\rho}}$ to get the LDP upper bound for $\bar{\rho}$. An analogous argument works for the LDP lower bound: Now we pick $\alpha' > \alpha$, $\delta > 0$ and a probability density $\bar{\rho}'=\bar{\rho}' (\delta,\alpha')$ such that $\bar{\rho} \geq (1-\delta)\bar{\rho}'$, and $\bar{\rho}'$ satisfies the same conditions as above. Arguing as before, we obtain for any open $\mathcal{O} \subset \mathcal{P}^\mathrm{inv}(F^{\mathbb{N}})$, $$\liminf_{N\to\infty} \frac{1}{N} \log P(R_N \in \mathcal{C}^{(\varepsilon)}\mid X) \geq - \inf_{Q \in \mathcal{O}} I^\mathrm{que}_{\bar{\rho}}(Q) \qquad X\text{-a.s.}$$ Proof of Theorems \[mainthmboundarycases\]–\[thmexp\] {#proofalpha1infty} ===================================================== We again give a brief sketch of the proofs only, leaving many details to the reader. Theorem \[mainthmboundarycases\](a), which says that for $\alpha=1$ the quenched rate function coincides with the annealed rate function, can be proved as follows: Since the claimed LDP upper bound holds automatically by the annealed LDP, it suffices to verify the matching lower bound. For this we can argue as in the proof of the lower bound in Section \[removeass\]. For any $\alpha'>1$ and $\delta>0$ we can approximate $\bar{\rho}$ by a suitable $\bar{\rho}'=\bar{\rho}' (\delta,\alpha')$ such that $\bar{\rho} \geq (1-\delta)\bar{\rho}'$. Then, using Theorem \[thm0:contqLDP\] with $\bar{\rho}'$ and taking $\delta \downarrow 0$, $\alpha' \downarrow 1$, we see that for any open $\mathcal{O} \subset \mathcal{P}^\mathrm{inv}(F^{\mathbb{N}})$, $$\liminf_{N\to\infty} \frac{1}{N} \log P(R_N \in \mathcal{C}^{(\varepsilon)}\mid X) \geq - \inf_{Q \in \mathcal{O} \cap \mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}})} I^\mathrm{ann}(Q) \qquad X\text{-a.s.}$$ (recall ). Finally note that any $Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ with $H(Q \mid Q_{\rho,{\mathscr{W}}}) < \infty$ can be approximated by a sequence $(Q_n) \subset \mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}})$ in such a way that $H(Q_n \mid Q_{\rho,{\mathscr{W}}}) \to H(Q \mid Q_{\rho,{\mathscr{W}}})$ to obtain the claim (using for example a “smoothed truncation” operation similar to Section \[subsect:prop:Ique.tr.cont.part2\]). Theorem \[mainthmboundarycases\](b), which says that for $\alpha=\infty$ the quenched rate function coincides with the annealed rate function on the set $\{Q\in\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})\colon\,\lim_{{{\rm tr}}\to\infty} m_{[Q]_{{\rm tr}}} H( \Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}) = 0\}$ and is infinite elsewhere, follows from arguments analogous to [@BiGrdHo10 Section 7, Part (b)]: For the upper bound, we can pick arbitrarily large $\alpha'>1$ and approximate $\bar{\rho} \leq (1+\delta) \bar{\rho}'$ with the help of a suitable probability density $\bar{\rho}'$ which has decay exponent $\alpha'$. Using Theorem \[thm0:contqLDP\] with $\bar{\rho}'$ and taking $\alpha' \uparrow \infty$, $\delta \downarrow 0$, we see that the upper bound holds with the claimed form of the rate function. For the matching lower bound we can trace through the proof of the lower bound contained in Theorem \[thm0:contqLDP\] but replacing our “coarse-graining work horses” Proposition \[thm00:contqLDP\] and Corollary \[prop:qLDPhtr\] (which rely on [@BiGrdHo10 Cor. 1.6]) by versions that are suitable for $\alpha=\infty$ (which rely on [@BiGrdHo10 Thm. 1.4 (b)] instead), still using a suitable truncation approximation of the quenched rate function analogous to the one proven in Proposition \[prop:Ique.tr.cont\]. This constitutes a way of rigorously implementing the “first long string strategy” from [@BiGrdHo10 Section 4], as explained in the heuristic given in item 0 of Section \[disc\], through the coarse-graining approximation. Theorem \[thmexp\] follows from Theorem \[mainthmboundarycases\](b) via an observation that is the analogue of [@Bi08 Lemma 6]: subject to the exponential tail condition in , any $Q\in\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ with $H(Q \mid Q_{\rho,{\mathscr{W}}}) <\infty$ necessarily has $m_Q<\infty$. Because of this observation we can argue as follows. If $m_Q<\infty$, then $\lim_{{{\rm tr}}\to\infty} m_{[Q]_{{\rm tr}}} = m_Q$ and $\lim_{{{\rm tr}}\to\infty} \Psi_{[Q]_{{\rm tr}}} = \Psi_Q$ by dominated convergence (recall ), which in turn imply that $\liminf_{{{\rm tr}}\to\infty} m_{[Q]_{{\rm tr}}} H(\Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}) = m_Q H(\Psi_Q \mid {\mathscr{W}})$, as shown in Lemma \[lem:trcontinuous\] in Appendix \[entropy\]. The limit is zero if and only if $\Psi_Q = {\mathscr{W}}$, which by holds if and only if $Q \in {{\mathcal R}}_{\mathscr{W}}$. This explains the link between and . Basic facts about metrics on path space {#metrics} ======================================= We metrise $F$, defined in (and $F_h \subset F$ defined in ) as follows. Let $d_S(\phi_1, \phi_2)$ be a metric on $C([0,\infty))$ that generates Skorohod’s $J_1$-topology on $D([0,\infty)) \supset C([0,\infty))$, allowing for a certain amount of “rubber time” (see e.g.  Ethier and Kurtz [@EK86 Section 3.5 and Eqs. (5.1–5.3)]) $$\label{def:dS} d_S(\phi_1, \phi_2) = \inf_{\lambda \in \Lambda} \bigg\{ \gamma(\lambda) \vee \int\nolimits_0^\infty e^{-u} \sup_{t\geq 0} \big| \phi_1(t \wedge u) - \phi_2(\lambda(t) \wedge u) \big| \, du \bigg\},$$ where $\Lambda$ is the set of Lipschitz-continuous bijections from $[0,\infty)$ into itself and $$\gamma(\lambda) = \sup_{0 \leq s < t < \infty} \Big| \log \frac{\lambda(t)-\lambda(s)}{t-s} \Big|.$$ With $$\label{eq:metriconF} d_F(y_1,y_2) = |t_1-t_2| + d_S(\phi_1,\phi_2)$$ for $y_i=(t_i,\phi_i) \in F$, $(F,d_F)$ becomes complete and separable, and the same holds for $(F_h,d_F)$ for any $h>0$. [**Remark. **]{} We might at first be inclined to metrise $F$ in a more straightforward way than (\[eq:metriconF\]), e.g. via $$\label{eq:firstmetriconF} d^{\mathrm{first}}_F(y_1,y_2) = |t_1-t_2| + \|\phi_1-\phi_2\|_\infty, \quad y_i=(t_i,\phi_i) \in F, \: i=1,2.$$ However, if we would use Lipschitz functions with $d_F$ replaced by $d^{\mathrm{first}}_F$ in (\[eq:g\_Lipschitz\]), then in the analogue of Lemma \[obs:Rdiscdiff\] we would be forced to use terms of the form $\sup_{s \geq 0} |\varphi(s+t \wedge t') - \varphi(s+ih \wedge jh)|$ in the right-hand side. When used for $\varphi=X$ (a realisation of Brownian motion as in Proposition \[prop:LambdaPhilimit1tr\]), this would in turn force us to control the increments of the Brownian motion not only locally near the beginning and the end of each loop, but uniformly inside loops. In fact, it seems plausible that an analogue of Proposition \[prop:LambdaPhilimit1tr\] where $d_F$ is replaced by $d^{\mathrm{first}}_F$ actually fails. Furthermore, note that we cannot arrange $d_S$ in such a way that, for $\phi \in C([0,\infty))$, $h>0$, $t_1 \leq t'_1 < t_2 \leq t_2'$ with $|t'_1-t_1| \leq h$, $|t'_2-t_2| \leq h$, $$\begin{aligned} \label{eq:dS_wishful1} d_S\big( \phi((t_1+\cdot) \wedge t_2), \phi((t'_1+\cdot) \wedge t'_2)\big) \leq 2h + \sup_{t_1 \leq s \leq t'_1} |\phi(s)-\phi(t'_1)| + \sup_{t_2 \leq s \leq t'_2} |\phi(s)-\phi(t'_2)|.\end{aligned}$$ This is why in Lemma \[obs:Rdiscdiff\] we need the freedom to use an extra $k$ and to “look in a neighbourhood of the cut-points of size $kh$”. Basic facts about specific relative entropy {#entropy} =========================================== In Section \[ss:definitions\] we recall the definition of (specific) relative entropy of two probability measures. In Section \[ss:approximations\] we prove various approximation results for (specific) relative entropy, which were used heavily in Sections \[props\]. Especially the parts with $\Psi_Q$ require care because of the delicate nature of the word concatenation map $Q \mapsto \Psi_Q$. The latter is looked at in closer detail in Section \[subs:towards.Ique.tr.cont\]. Definitions {#ss:definitions} ----------- For $\mu,\nu$ probability measures on a measurable space $(S,\mathscr{S})$, $$h(\mu \mid \nu) = \begin{cases} \int_S (\log\frac{d\mu}{d\nu})\,d\mu, &\text{if} \; \mu \ll \nu, \\ \infty, & \text{otherwise,} \end{cases}$$ is the relative entropy of $\mu$ w.r.t. $\nu$. When the measurable space is a Polish space $E$ equipped with its Borel-$\sigma$-algebra, we also have the representation (see e.g. [@DeZe98 Lemma 6.2.13]) $$\begin{aligned} \label{eq:relentraslegendretransf} h(\mu \mid \nu) = \sup_{f \in C_b(E)} \Big\{ \int f\,d\mu - \log \int e^f \, d\nu \Big\} = \sup_{\scriptstyle f\colon\, E \to {\mathbb{R}}\; \atop \scriptstyle \text{bounded measurable}} \Big\{ \int f\,d\mu - \log \int e^f \, d\nu \Big\} \end{aligned}$$ (and if $\mu \ll \nu$ with a bounded and uniformly positive density, then the supremum in the right-hand side is achieved by $f=\log d\mu/d\nu$). Equation implies the entropy inequality $$\label{ineq:entropy} \mu(A)\leq \frac{\log 2 + h(\mu \mid \nu)}{\log[1+1/\nu(A)]}$$ by choosing $f=\alpha {1}_A$ and $\alpha=\log[1+1/\nu(A)]$ (see e.g. Kipnis and Landim [@KiLa99 Appendix 1, Proposition 8.2]). For $Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$, $$\begin{aligned} \label{eq:SREwrtProd} H(Q \mid (q_{\rho,{\mathscr{W}}})^{\otimes{\mathbb{N}}}) = \lim_{N\to\infty} \frac1N h\big( \pi_N Q \mid (q_{\rho,{\mathscr{W}}})^{\otimes N}\big) = \sup_{N\in{\mathbb{N}}} \frac1N h\big(\pi_N Q \mid (q_{\rho,{\mathscr{W}}})^{\otimes N}\big)\end{aligned}$$ with $\pi_N$ the projection onto the first $N$ words, is the specific relative entropy of $Q$ w.r.t. $(q_{\rho,{\mathscr{W}}})^{\otimes{\mathbb{N}}}$. Similarly, using the canonical filtration $(\mathscr{F}^C_t)_{t \ge 0}$ on $C([0,\infty))$, for a probability measure $\Psi$ on $C([0,\infty))$ with stationary increments we denote by $$\begin{aligned} \label{eq:SREwrtWM} H(\Psi \mid {\mathscr{W}}) = \lim_{t\to\infty} \frac{1}{t} h\big(\Psi_{|}{}_{\mathscr{F}^C_t} \mid {\mathscr{W}}_{|}{}_{\mathscr{F}^C_t}\big) = \sup_{t>0} \frac{1}{t} h\big(\Psi_{|}{}_{\mathscr{F}^C_t} \mid {\mathscr{W}}_{|}{}_{\mathscr{F}^C_t}\big)\end{aligned}$$ the specific relative entropy w.r.t. Wiener measure. See Appendix \[contrelentr\] for a proof of . Approximations {#ss:approximations} -------------- Let $E$ be a Polish space. Equip $\mathcal{P}(E)$ with the weak topology (suitably metrised). $E^{\mathbb{N}}$ carries the product topology, and the set of shift-invariant probability measures $\mathcal{P}^\mathrm{inv}(E^{\mathbb{N}})$ carries the weak topology (also suitably metrised). ### Blocks For $M\in{\mathbb{N}}$ and $r \in \mathcal{P}(E^M)$, denote by $r^{\otimes {\mathbb{N}}} \in \mathcal{P}(E^{\mathbb{N}})$ the law of an infinite sequence obtained by concatenating $M$-blocks drawn independently from $r$ (i.e., we identify $(E^M)^{\mathbb{N}}$ and $E^{\mathbb{N}}$ in the obvious way), and write $$\label{def:blockmeas} {\mathsf{sblock}}_M(r) = \frac1M \sum_{j=0}^{M-1} r^{\otimes {\mathbb{N}}} \circ (\theta^j)^{-1} \; \in \mathcal{P}^\mathrm{inv}(E^{\mathbb{N}})$$ for its stationary mean. \[lemma:HblockmeasQ\] For $Q = q^{\otimes {\mathbb{N}}} \in\mathcal{P}^\mathrm{inv}(E)$ and $r \in \mathcal{P}(E^M)$, $$\label{eq:HblockmeasQ} H\big({\mathsf{sblock}}_M(r) \mid Q \big) = \frac1M h\big(r \mid \pi_M Q \big).$$ Moreover, for any $R \in \mathcal{P}^\mathrm{inv}(E)$, $$\label{eq:reconstrfromblocks} {\mathop{\text{\rm w-lim}}}_{M\to\infty} {\mathsf{sblock}}_M\big(\pi_M R\big) = R.$$ This proof is standard. Equation follows from the results in Gray [@Gr09b Section 8.4, see Theorem 8.4.1] by observing that ${\mathsf{sblock}}_M(r)$ is the asymptotically mean stationary measure of $r^{\otimes{\mathbb{N}}}$. It is also contained in Föllmer[@Foe88 Lemma 4.8], or can be proved with “bare hands” by explicitly spelling out $d\pi_N {\mathsf{sblock}}_M(r)/dq^{\otimes N}$ for $N \gg M$ and using suitable concentration arguments under $q^{\otimes N}$ as $N\to\infty$. Equation is obvious from the definition of weak convergence. ### Change of reference measure \[lemma:hregularised1\] [(1)]{} Let $\nu, \nu_1,\nu_2,\ldots \in \mathcal{P}(E)$ with ${\mathop{\text{\rm w-lim}}}_{n\to\infty} \nu_n = \nu$. Then $$\begin{aligned} \label{eq:hregularised1} h(\mu \mid \nu) = \lim_{\varepsilon \downarrow 0} \limsup_{n \to \infty} \inf_{\mu' \in B_\varepsilon(\mu)} h(\mu' \mid \nu_n), \quad \mu \in \mathcal{P}(E). \end{aligned}$$ [(2)]{} Let $Q=q^{\otimes {\mathbb{N}}}, Q_1=q_1^{\otimes {\mathbb{N}}},Q_2=q_2^{\otimes {\mathbb{N}}},\ldots \in \mathcal{P}^\mathrm{inv}(E^{\mathbb{N}})$ be product measures with ${\mathop{\text{\rm w-lim}}}_{n\to\infty} Q_n$ $= Q$. Then $$\begin{aligned} \label{eq:Hregularised1} H(R \mid Q) = \lim_{\varepsilon \downarrow 0} \limsup_{n \to \infty} \inf_{R' \in B_\varepsilon(R)} H(R' \mid Q_n), \quad R \in \mathcal{P}^\mathrm{inv}(E^{\mathbb{N}}). \end{aligned}$$ \(1) Denote the term in the right-hand side of (\[eq:hregularised1\]) by $\tilde{h}(\mu)$. Let $f \in C_b(E)$, $\delta > 0$. We can find $\varepsilon_0 > 0$ and $n_0 \in {\mathbb{N}}$ such that $$\begin{aligned} \forall \, 0 < \varepsilon \leq \varepsilon_0, \, \mu' \in B_\varepsilon(\mu)\colon\,\, &\int_E f\, d\mu' \geq \int_E f\, d\mu - \frac{\delta}{2}, \\ \forall \, n \geq n_0\colon\,\, &\log \int_E e^f \, d\nu_n \leq \log \int_E e^f \, d\nu + \frac{\delta}{2}. \end{aligned}$$ Therefore, for $0 < \varepsilon \leq \varepsilon_0$ and $n \geq n_0$, $$\begin{aligned} \inf_{\mu' \in B_\varepsilon(\mu)} h(\mu' \mid \nu_n) \geq \int_E f\, d\mu' - \log \int_E e^f \, d\nu_n \geq \int_E f\, d\mu \smallskip - \log \int_E e^f \, d\nu - \delta.\end{aligned}$$ Now optimise over $f$ and take $\delta \downarrow 0$, to obtain $\tilde{h}(\mu) \geq h(\mu \mid \nu)$ via . For the reverse inequality, we may without loss of generality assume that $h(\mu \mid \nu) = \int_E \varphi \log \varphi\, d\nu < \infty$, where $\varphi=d\mu/d\nu \geq 0$ is in $L^1(\nu)$. Then for any $\delta > 0$ we can find a $\widetilde{\varphi} \geq 0$ in $C_b(E) \cap L^1(\nu)$ such that $\int_E \widetilde{\varphi}\,d\nu = 1$ and $$\begin{aligned} \int_E \big| \widetilde{\varphi} - \varphi \big| \, d\nu < \delta, \;\; \int_E \big| \widetilde{\varphi}\log\widetilde{\varphi} - \varphi\log\varphi \big| \, d\nu < \delta.\end{aligned}$$ Note that $\lim_{n\to\infty} \int_E \widetilde{\varphi}\,d\nu_n = 1$, and let $\widetilde{\varphi}_n = \widetilde{\varphi}/\int \widetilde{\varphi}\,d\nu_n$ and $\mu_n = \widetilde{\varphi}_n \nu_n$. Then, for $g \in C_b(E)$, $$\begin{aligned} \Big| \int_E g \, d\mu_n - \int_E g \,d\mu \Big| &= \Big| \frac1{\int_E \widetilde{\varphi} \, d\nu_n} \int_E g \widetilde{\varphi}\, d\nu_n - \int_E g\varphi \,d\nu \Big| \\ &\leq \Big| \frac{1}{\int_E \widetilde{\varphi}\, d\nu_n} - 1 \Big| \, \| g \widetilde{\varphi} \|_\infty + \Big| \int_E g \widetilde{\varphi}\, d\nu_n - \int_E g \widetilde{\varphi}\, d\nu \Big| + \Big| \int_E g (\widetilde{\varphi} - \varphi) \,d\nu\Big|, \end{aligned}$$ which can be made arbitrarily small by choosing $\delta$ small enough and $n$ large enough. In particular, for any $\varepsilon > 0$ we can choose $\delta$, $\widetilde{\varphi}$ and $n_0$ such that $\mu_n \in B_\varepsilon(\mu)$ for $n \geq n_0$. Hence $$\begin{aligned} \limsup_{n\to\infty} \inf_{\mu' \in B_\varepsilon(\mu)} h(\mu' \mid \nu_n) &\leq \limsup_{n\to\infty} h(\mu_n \mid \nu_n)\\ &= \limsup_{n\to\infty} \int_E \widetilde{\varphi}_n \log \widetilde{\varphi}_n \, d\nu_n = \int_E \widetilde{\varphi} \log \widetilde{\varphi} \, d\nu \leq h(\mu \mid \nu) + \delta, \end{aligned}$$ and letting $\delta \downarrow 0$ we $\tilde{h}(\mu) \leq h(\mu \mid \nu)$. \(2) Recall that for $R \in \mathcal{P}^\mathrm{inv}(E^{\mathbb{N}})$ and $Q$ a product measure, $$\lim_{N\to\infty} \frac1N h\left( \pi_N R \mid \pi_N Q\right) = H(R \mid Q) = \sup_{N\in{\mathbb{N}}} \frac1N h\left( \pi_N R \mid \pi_N Q\right).$$ Denote the expression in the right-hand side of (\[eq:Hregularised1\]) by $\tilde{H}(R)$. Fix $N\in{\mathbb{N}}$. Since for each $\varepsilon>0$, we have $B_{\varepsilon'}(R) \subset \{ R'\colon\, \pi_N R' \in B_{\varepsilon}(\pi_N R) \}$ for $\varepsilon'$ sufficiently small we also have $$\begin{aligned} \lim_{\varepsilon' \downarrow 0} \limsup_{n\to\infty} \inf_{R' \in B_{\varepsilon'}(R)} H(R' \mid Q_n) \ge \limsup_{n \to \infty} \inf_{\mu' \in B_\varepsilon(\pi_N R)} \frac1N h(\mu' \mid \pi_N Q_n). \end{aligned}$$ Let $\varepsilon \downarrow 0$ and use Part (1), to see that $\tilde{H}(R) \ge \frac1N h(\pi_N R \mid \pi_N Q)$ for any $N$. Hence also $\tilde{H}(R) \geq H(R \mid Q)$. For the reverse inequality, we may w.l.o.g. assume that $H(R \mid Q) < \infty$. Fix $\varepsilon>0$ and $\delta>0$. There is an $N \in {\mathbb{N}}$ such that $H(R \mid Q) \leq \frac1N h \big( \pi_N R \mid \pi_N Q \big) + \delta$, and since $\pi_N R \ll \pi_N Q=q^ {\otimes N}$ we can find a continuous, bounded and uniformly positive function $f_N\colon\,E^N \to [0,\infty)$ such that $\int_E f_N \, dq^{\otimes N} = 1$, $\int_E f_N \log f_N \, dq^{\otimes N} \leq h \big( \pi_N R \mid \pi_N Q \big) + N\delta$ and $\tilde{R}_N \in B_{\varepsilon/2}(R)$, where $\tilde{R}_N = {\mathsf{sblock}}_N\big(f_N\,q^{\otimes N}\big) \in \mathcal{P}^\mathrm{inv}(E^{\mathbb{N}})$ (see Lemma \[lemma:HblockmeasQ\]). By , we have $$H(\tilde{R}_N \mid Q) = \frac1N \int_E f_N \log f_N \, dq^{\otimes N}.$$ Now put $f_{N,n} = f_N/\int_E f_N\,q_n^{\otimes N}$, and define $\tilde{R}_{N,n} = {\mathsf{sblock}}_N \big(f_{N,n} \, q^{\otimes N}\big)$ as the “stationary version” of $(f_{N,n}\,q_n^{\otimes N})^{\otimes {\mathbb{N}}}$. In particular, $H(\tilde{R}_{N,n} \mid Q_n) = \frac1N \int f_{N,n} \log f_{N,n} \, dq_n^{\otimes N}$. Since $f_N$ is continuous, we have $\tilde{R}_{N,n} \in B_{\varepsilon}(R)$ and $\int_E f_{N,n} \log f_{N,n}\,dq_n^{\otimes N} \le H(R \mid Q) + 3\delta$ for $n$ large enough. Hence $$\limsup_{n\to\infty} \inf_{R' \in B_{\varepsilon}(R)} H(R' \mid Q_n) \le \limsup_{n\to\infty} H(\tilde{R}_{N,n} \mid Q_n) \le H(R \mid Q) + 4\delta.$$ Now let $\delta \downarrow 0$ followed by $\lim_{\varepsilon\downarrow 0}$ to conclude the proof. ### Existence of sharp coarse-graining approximations to the quenched rate function The following lemma was used in the proof of Lemma \[lemma:Iquetrregularised\]. \[lem:cg.2lev.blockapprox\] Let $Q \in \mathcal{P}^{\mathrm{fin}}(F^{\mathbb{N}})$ with $H( Q \mid Q_{\rho,{\mathscr{W}}}) < \infty$. There exist a sequence $(h_n)_{n\in{\mathbb{N}}}$ with $h_n>0$ and $\lim_{n\to\infty} h_n = 0$ and a sequence $(Q'_n)_{n\in{\mathbb{N}}}$ with $Q'_n \in \mathcal{P}^{\mathrm{fin}}(\widetilde{E}_{h_n}^{\mathbb{N}})$ and ${\mathop{\text{\rm w-lim}}}_{n\to\infty} Q'_n = Q$ such that $\limsup_{n\to\infty} I^{\mathrm{que}}_{h_n}(Q'_n) \leq I^{\mathrm{que}}(Q)$. The same holds with $F$ replaced by $F_{0,{{\rm tr}}}$ and $\widetilde{E}_{h_n}$ replaced by $\widetilde{E}_{h_n,{{\rm tr}}}$. Recall the definition of $\lceil Q \rceil_h$ in Step 2 of the proof of part (1) of Proposition \[prop:Ique.tr.cont\] (see page ). For any $N\in{\mathbb{N}}$, we have $$\begin{aligned} \label{eq:anyway1} h(\pi_N \lceil Q \rceil_h \mid \pi_N \lceil Q_{\rho,{\mathscr{W}}} \rceil_h) \leq h(\pi_N Q \mid \pi_N Q_{\rho,{\mathscr{W}}}) \leq N \, H( Q \mid Q_{\rho,{\mathscr{W}}}).\end{aligned}$$ The second inequality follows from . For the first inequality, use the fact that the construction of $\lceil Q \rceil_h$ can be implemented as a deterministic function of the pair of random variables $(Y,U)$, together with the fact that relative entropy can only decrease when image measures are taken. Write $\hat{\tau}_i = (\tilde{T}_i-\tilde{T}_{i-1})/h$, $i \in {\mathbb{N}}$. Since letters both under $\lceil Q_{\rho,{\mathscr{W}}} \rceil_h$ and under $Q_{\lceil \rho \rceil_h,{\mathscr{W}}}$ are constructed from a Brownian path that is independent of the word lengths, we have (recall \[def:rho.h.trunc\]) $$\begin{aligned} {1}(\hat{\tau}_1=k_1,\dots,\hat{\tau}_N=k_N) \, \frac{d\pi_N \lceil Q_{\rho,{\mathscr{W}}} \rceil_h}{d\pi_N Q_{\lceil \rho \rceil_h,{\mathscr{W}}}} = \frac{(\pi_N \lceil Q_{\rho,{\mathscr{W}}} \rceil_h)\big(\hat{\tau}_1=k_1,\dots,\hat{\tau}_N=k_N\big)} {\prod_{\ell=1}^N \lceil \rho \rceil_h(h k_\ell)}\end{aligned}$$ with $$\begin{aligned} & (\pi_N \lceil Q_{\rho,{\mathscr{W}}} \rceil_h)\big(\hat{\tau}_1=k_1,\dots,\hat{\tau}_N=k_N\big) \notag \\ & = \int_{[0,1]} du \, \int_0^\infty \bar{\rho}(t_1) dt_1 \int_{t_1}^\infty \bar{\rho}(t_2-t_1) d(t_2-t_1) \cdots \int_{t_{N-1}}^\infty \bar{\rho}((t_N-t_{N-1})) d(t_N-t_{N-1}) \notag\\ &\qquad\qquad \times \prod_{\ell=1}^N {1}_{(h(\bar{k}_\ell-1+u), h(\bar{k}_\ell+u)]}(t_\ell),\end{aligned}$$ where $\bar{k}_\ell = k_1+\cdots+k_\ell$. Thus, by (\[eq:Vbarrhodef\]–\[ass:rhobar.reg1\]), $$\begin{aligned} \label{eq:anyway2} \sup_{N \in {\mathbb{N}}} \frac1N E_{\lceil Q \rceil_h}\left[ \Big| \log \frac{d\pi_N \lceil Q_{\rho,{\mathscr{W}}} \rceil_h}{d\pi_N Q_{\lceil \rho \rceil_h,{\mathscr{W}}}}\Big|\right] \leq r_Q(h)\end{aligned}$$ with $$\begin{aligned} r_Q(h) = \eta_n \lceil Q \rceil_h(\hat\tau_1 \in \bar{A}_n) + \eta_0 \lceil Q \rceil_h(\hat\tau_1 \not\in \bar{A}_n), \qquad h=2^{-n},\end{aligned}$$ where $\bar{A}_n \subset (s_*,\infty)$ is the set obtained from $A_n$ by removing pieces of length $2^{-n}$ from its edges (i.e., $\bar{A}_n$ is the $2^{-n}$-interior of $A_n$). But $\lim_{n\to\infty} \lceil Q \rceil_{2^{-n}}(\hat\tau_1 \not\in \bar{A}_n)=0$ because $A_n$ fills up $(s_*,\infty)$ as $n\to\infty$. Since $\lim_{n\to\infty} \eta_n=0$, we get $\lim_{h \downarrow 0} r_Q(h) = 0$. Combining (\[eq:anyway1\]–\[eq:anyway2\]), we obtain that $$\begin{aligned} H(\lceil Q \rceil_h \mid Q_{\lceil \rho \rceil_h,{\mathscr{W}}}) = \sup_{N \in {\mathbb{N}}} \frac1N h(\pi_N \lceil Q \rceil_h \mid \pi_N Q_{\lceil \rho \rceil_h,{\mathscr{W}}}) \leq H( Q \mid Q_{\rho,{\mathscr{W}}}) + r_Q(h)\end{aligned}$$ and, finally, $$\begin{aligned} &\limsup_{h \downarrow 0} H(\lceil Q \rceil_h \mid Q_{\lceil \rho \rceil_h,{\mathscr{W}}}) + (\alpha-1) m_{\lceil Q \rceil_h} H(\Psi_{\lceil Q \rceil_h, h} \mid {\mathscr{W}}) \notag\\ &\qquad \leq H( Q \mid Q_{\rho,{\mathscr{W}}}) + (\alpha-1) m_Q H(\Psi_Q \mid {\mathscr{W}}).\end{aligned}$$ The truncated case, where $F$ is replaced by $F_{0,{{\rm tr}}}$, etc., can be handled analogously. ### Approximation of $\Psi_Q$ The approximation in is stronger than just weak convergence. \[lemma:PsiQ:TVlim\] For $Q \in \mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}})$, $$\begin{aligned} \label{eq:PsiQTVlimit} \lim_{T\to\infty} \sup_{A \subset C[0,\infty)\; \text{measurable}} \bigg| \Psi_Q(A) - \frac{1}{T} \int_0^T \big(\kappa(Q) \circ (\theta^s)^{-1}\big)(A) \,ds\bigg| = 0,\end{aligned}$$ i.e., the convergence in holds in total variation. Note that, by shift-invariance, $$\begin{aligned} \Psi_Q(A) = \frac{1}{N m_Q} {\mathbb{E}}_Q \left[ \int_0^{\tau_N} {1}_A\big( \theta^s \kappa(Y) \big) \, ds \right], \qquad N\in{\mathbb{N}}.\end{aligned}$$ Suppose that $Q$ is also ergodic. Then $\lim_{N\to\infty} \tau_N/N = m_Q$ $Q$-a.s.  and in $L^1(Q)$. Hence, for given $\varepsilon > 0$ we can find a $T_0(\varepsilon)$ such that, for $T \geq T_0(\varepsilon)$, $$\begin{aligned} \label{eq:Qerg.cons1} {\mathbb{E}}_Q\Big[ \Big| \frac{\tau_{N(T)}-T}{m_Q N(T)} \Big| \Big] + \Big| \frac{T}{m_Q N(T)} - 1 \Big| \leq \varepsilon,\end{aligned}$$ where $N(T) = \lceil T/m_Q \rceil$. Thus, for $T \geq T_0(\varepsilon)$ and any measurable $A \subset C[0,\infty)$, we have $$\begin{aligned} & \bigg| \Psi_Q(A) - \frac{1}{T} \int_0^T \big(\kappa(Q) \circ (\theta^s)^{-1}\big)(A) \,ds\bigg| \notag \\ & \leq \frac{1}{m_Q N(T)} \bigg| {\mathbb{E}}_Q\left[ \int_0^{\tau_{N(T)}} {1}_A\big( \theta^s \kappa(Y) \big) \, ds - \int_0^{T} {1}_A\big( \theta^s \kappa(Y) \big) \, ds \right] \bigg| \notag \\ & \qquad + \bigg| \Big(\frac{1}{m_Q N(T)}-\frac1T \Big) \int_0^{T} {1}_A\big( \theta^s \kappa(Y) \big) \, ds \bigg| \leq {\mathbb{E}}_Q\left[ \Big| \frac{\tau_{N(T)}-T}{m_Q N(T)} \Big| \right] + \Big| \frac{T}{m_Q N(T)} - 1 \Big| \leq \varepsilon,\end{aligned}$$ i.e., holds. If $Q$ is not ergodic, then use the ergodic decomposition $$Q = \int_{\mathcal{P}^{\mathrm{erg,fin}}(F^{\mathbb{N}})} Q' \, W_Q(dQ')$$ and note that $$m_Q = \int_{\mathcal{P}^{\mathrm{erg,fin}}(F^{\mathbb{N}})} m_{Q'} \, W_Q(dQ'), \quad \Psi_Q = \int_{\mathcal{P}^{\mathrm{erg,fin}}(F^{\mathbb{N}})} \frac{m_{Q'}}{m_Q} \, \Psi_{Q'} \, W_Q(dQ')$$ (see also [@BiGrdHo10 Section 6]). We can choose $N(T)$ so large that the set of $Q'$s for which holds (with $Q$ replaced by $Q'$) has $W_Q$-measure arbitrarily close to $1$. Continuity of the “letter part” of the rate function under truncation: discrete-time {#subs:towards.Ique.tr.cont} ------------------------------------------------------------------------------------ In this section we consider a discrete-time scenario as in [@BiGrdHo10]: $\rho \in \mathcal{P}({\mathbb{N}})$, $E$ is a Polish space, $\nu \in \mathcal{P}(E)$, the sequence of words $(Y^{(i)})_{i\in{\mathbb{N}}}$ with discrete lengths has reference law $q_{\rho,\nu}^{\otimes{\mathbb{N}}}$ with $q_{\rho,\nu}$ as in [@BiGrdHo10 Eq. (1.4)]. The following lemma extends [@BiGrdHo10 Lemma A.1] to Polish spaces (in [@BiGrdHo10] it was only proved and used for finite $E$, and without explicit control of the error term). Via coarse-graining, this lemma was used in the proof of Proposition \[prop:Ique.tr.cont\]. \[lem:trcontinuous\] Let $Q \in \mathcal{P}^{\mathrm{fin}}(\widetilde{E}^{\mathbb{N}})$ and $0 < \varepsilon < \tfrac12$. Let ${{\rm tr}}\in {\mathbb{N}}$ be so large that $$\begin{aligned} \label{eq:mQtr.qb} {\mathbb{E}}_Q\Big[ \big( |Y^{(1)}|-{{\rm tr}}\big)_+ \Big] < \frac{\varepsilon}{2} m_Q.\end{aligned}$$ Then $$\begin{aligned} \label{eq:HPsiQtr.qb} (1-\varepsilon) \big( H(\Psi_{[Q]_{{\rm tr}}} \mid \nu^{\otimes {\mathbb{N}}}) + b(\varepsilon) \big) \leq H(\Psi_Q \mid \nu^{\otimes {\mathbb{N}}})\end{aligned}$$ with $b(\varepsilon) = -2\varepsilon + [\varepsilon \log\varepsilon + (1-\varepsilon) \log (1-\varepsilon)]/(1-\varepsilon) $, satisfying $\lim_{\varepsilon \downarrow 0} b(\varepsilon)=0$. In particular, $$\begin{aligned} \label{eq:HPsiQtrlim.disc} \lim_{{{\rm tr}}\to\infty} H(\Psi_{[Q]_{{\rm tr}}} \mid \nu^{\otimes {\mathbb{N}}}) = H(\Psi_Q \mid \nu^{\otimes {\mathbb{N}}}). \end{aligned}$$ We can assume w.l.o.g. that $H(\Psi_Q \mid \nu^{\otimes{\mathbb{N}}}) < \infty$ for otherwise is trivial and follows from lower-semicontinuity of specific relative entropy. First, assume that $Q$ is ergodic, then $\Psi_Q$ is ergodic as well (see [@Bi08 Remark 5]). For $\Psi \in \mathcal{P}^{\mathrm{erg}}(E^{\mathbb{N}})$ and $\delta \in (0,1)$, $$\begin{aligned} \label{eq:HPsinu.typset0} H(\Psi \mid \nu^{\otimes {\mathbb{N}}}) & = \lim_{L\to\infty} -\frac{1}{L} \log \Big( \inf\big\{ \nu^{\otimes L}(B) \colon\, B \subset E^L, (\pi_L \Psi)(B) \geq 1-\delta \big\} \Big), \\ \label{eq:HPsinu.typset1} & = \lim_{L\to\infty} \sup\Big\{ -\frac{1}{L} \log \nu^{\otimes L}(B) \, \colon\, B \subset E^L, (\pi_L \Psi)(B) \geq 1-\delta \Big\}.\end{aligned}$$ This replaces the asymptotics of the covering number and its relation to specific entropy for ergodic measures on discrete shift spaces that was employed in the proof of [@BiGrdHo10 Lemma A.1], and can be deduced with bare hands from the Shannon-McMillan-Breiman theorem. Indeed, asymptotically optimal $B$’s are of the form $\{ \frac1L \log \frac{d\pi_L\Psi}{d\nu^{\otimes L}} \in H(\Psi \mid \nu^{\otimes {\mathbb{N}}}) \pm \epsilon\}$: Put $f_L = \frac{d\pi_L \Psi}{d\nu^{\otimes L}}$ and set $B_L = \{ \frac1L \log f_L > H(\Psi \mid \nu^{\otimes {\mathbb{N}}}) - \epsilon \}$. Then $(\pi_L \Psi)(B_L) \to 1$ by the Shannon-McMillan-Breiman, and $\nu^{\otimes L}(B_L) = \int_{B_L} \frac1{f_L} d\pi_L\Psi \leq \exp[-L(H(\Psi \mid \nu^{\otimes {\mathbb{N}}}) - \epsilon)]$, i.e., the right-hand side of is $\geq H(\Psi \mid \nu^{\otimes {\mathbb{N}}})$. For the reverse inequality, consider any $B \subset E^L$ with $(\pi_L\Psi)(B) \geq \tfrac12$, say. Set $B'=B \cap \{ \frac1L \log f_L < H(\Psi \mid \nu^{\otimes {\mathbb{N}}}) + \epsilon\}$. Then $\pi_L\Psi(B') \geq \tfrac13$ for $L$ large enough and $\nu^{\otimes L}(B) \geq \nu^{\otimes L}(B') \geq \exp[-L(H(\Psi \mid \nu^{\otimes {\mathbb{N}}}) + \epsilon)] \pi_L\Psi(B')$. Hence the right-hand side of is also $\leq H(\Psi \mid \nu^{\otimes {\mathbb{N}}})$. To check , fix $\varepsilon>0$. For $L$ sufficiently large, we construct a set $B_L \subset E^L$ such that $\pi_L \Psi_Q(B_L) \geq \tfrac12$ and $\nu^{\otimes L}(B_L) \leq \exp[ - L(1-\varepsilon)(b_L(\varepsilon) + H(\Psi_{[Q]_{{\rm tr}}} \mid \nu^{\otimes {\mathbb{N}}}))]$, i.e., $$\begin{aligned} -\frac1L \log \nu^{\otimes L}(B_L) \geq (1-\varepsilon) \big[ H(\Psi_{[Q]_{{\rm tr}}} \mid \nu^{\otimes {\mathbb{N}}}) + b_L(\varepsilon) \big], \end{aligned}$$ where $\lim_{L\to\infty} b_L(\varepsilon) = b(\varepsilon)$. Via applied to $\Psi=\Psi_Q$, this yields . To construct the sets $B_L$, we proceed as follows. Put $N= \lceil (1+2\varepsilon) L/m_Q \rceil$. By the ergodicity of $Q$ (see [@BiGrdHo10 Section 3.1] for analogous arguments), we can find a set $A \subset \widetilde{E}^N$ such that $$\begin{aligned} &\forall\, (y^{(1)},\dots,y^{(N)}) \in A \colon\, \notag \\ \label{eq:BL.prop.1} & \hspace{2em} | \kappa(y^{(1)},\dots,y^{(N)}) | \geq L(1+\varepsilon), \;\; |y^{(1)}| \leq {{\rm tr}}, \;\; \sum_{i=1}^N (|y^{(i)}|-{{\rm tr}})_+ < \varepsilon L, \\ \label{eq:BL.prop.2} & \hspace{2em} {\mathbb{E}}_Q\Big[ |Y^{(1)}| {1}_A(Y^{(1)},\dots,Y^{(N)}) \big] \geq (1-\varepsilon) m_Q, \end{aligned}$$ and the set $$\begin{aligned} B'_L = B'_L(A) &= \Big\{ \pi_L \big( \theta^i \kappa([y^{(1)}]_{{\rm tr}},\dots,[y^{(N)}]_{{\rm tr}})\big)\colon \,\notag\\ &\qquad (y^{(1)},\dots,y^{(N)}) \in A, i=0,1,\dots,|y^{(1)}|-1 \Big\} \subset E^L \end{aligned}$$ satisfies $$\begin{aligned} \pi_L\Psi_{[Q]_{{\rm tr}}}(B'_L) \geq \frac12, \quad \nu^{\otimes \lceil L(1-\varepsilon) \rceil}(\pi_{\lceil L(1-\varepsilon) \rceil} B'_L) \leq \exp\big[ -L(1-\varepsilon) \big(H(\Psi_{[Q]_{{\rm tr}}} \mid \nu^{\otimes {\mathbb{N}}})-2\varepsilon\big)\big].\end{aligned}$$ Here, use in (\[eq:BL.prop.1\]–\[eq:BL.prop.2\]), and note that $N\big(1-\tfrac{\varepsilon}{2}\big)m_Q \sim (1+2\varepsilon)\big(1-\tfrac{\varepsilon}{2}\big)L \geq (1+\varepsilon) L$ and $N \tfrac{\varepsilon}{2} m_Q \sim (1+2\varepsilon)\tfrac{\varepsilon}{2} L < \varepsilon L$ as $L\to\infty$. For $I \subset \{1,\dots,L\}$, $x \in E^L$ and $y \in E^{|I|}$, write $\mathsf{ins}_I(x; y) \in E^{L+|I|}$ for the word of length $L+|I|$ consisting of the letters from $y$ at index positions in $I$ and the letters from $x$ at index positions not in $I$, with the order of letters preserved within $x$ and within $y$ (the word $y$ is inserted in $x$ at the positions in $I$). Put $$\begin{aligned} B_L = \pi_L\Big( \big\{ \mathsf{ins}_I(x; y) \colon \, x \in B'_L, I\subset \{1,\dots,L\}, |I| \leq \varepsilon L, y \in E^{|I|} \big\} \Big).\end{aligned}$$ Then $\pi_L\Psi_Q(B_L) \geq \frac12$ by construction. Furthermore, for fixed $I\subset \{1,\dots,L\}$ with $|I|=k \le \varepsilon L$, $$\begin{aligned} \nu^{\otimes L} \Big( \pi_L\big( \big\{ \mathsf{ins}_I(x; y)\colon \, x \in B'_L, y \in E^k \big\} \big)\Big) = \nu^{\otimes L} \big( \pi_{L-k}(B'_L) \big) \leq \nu^{\otimes \lceil L(1-\varepsilon \rceil]}(\pi_{\lceil L(1-\varepsilon) \rceil} B'_L),\end{aligned}$$ and hence $$\begin{aligned} \nu^{\otimes L}(B_L) & \leq [\varepsilon L] {L \choose [\varepsilon L]} \exp\big[ -L(1-\varepsilon) \big(H(\Psi_{[Q]_{{\rm tr}}} \mid \nu^{\otimes {\mathbb{N}}})-2\varepsilon\big)\big] \notag \\ & = \exp\big[ - L(1-\varepsilon) \big(b_L(\varepsilon) + H(\Psi_{[Q]_{{\rm tr}}} \mid \nu^{\otimes {\mathbb{N}}})\big) \big]\end{aligned}$$ with $b_L(\varepsilon) = - \frac{1}{(1-\varepsilon) L}(\log [\varepsilon L] + \log {L \choose [\varepsilon L]}) - 2\varepsilon$, which satisfies $\lim_{\varepsilon \downarrow 0} b_L(\varepsilon)= b(\varepsilon)$. It remains to prove . Since ${\mathop{\text{\rm w-lim}}}_{{{\rm tr}}\to\infty} \Psi_{[Q]_{{\rm tr}}} = \Psi_Q$, we have $\liminf_{{{\rm tr}}\to\infty} H(\Psi_{[Q]_{{\rm tr}}} \mid \nu^{\otimes {\mathbb{N}}}) \geq H(\Psi_Q \mid \nu^{\otimes {\mathbb{N}}})$, while the reverse inequality $ \limsup_{{{\rm tr}}\to\infty} H(\Psi_{[Q]_{{\rm tr}}} \mid \nu^{\otimes {\mathbb{N}}}) \leq H(\Psi_Q \mid \nu^{\otimes {\mathbb{N}}})$ follows from (\[eq:mQtr.qb\]–\[eq:HPsiQtr.qb\]) and the fact that $\lim_{{{\rm tr}}\to\infty} {\mathbb{E}}_Q[( |Y^{(1)}|-{{\rm tr}})_+] = m_Q$ by dominated convergence. For non-ergodic $Q$, decompose as in [@BiGrdHo10 Eqs.(6.1)–(6.3)], use the above argument on each of the ergodic components, and use the fact that specific relative entropy is affine. Existence of specific relative entropy {#contrelentr} ====================================== In this section we prove . For technical reasons, we consider the two-sided scenario. The argument is standard, but the fact that time is continuous requires us to take care. Let $\Omega = \tilde{C}({\mathbb{R}})$ be the set of continuous functions $\omega\colon\, {\mathbb{R}}\to {\mathbb{R}}$ with $\omega(0)=0$, which is a Polish space e.g. via the metric $d(\omega, \omega') = \int_{\mathbb{R}}e^{-|t|} \big(|\omega(t)-\omega'(t)| \wedge 1\big) dt$. The shifts on $\Omega$ are $\theta^t \omega(\cdot) = \omega(\cdot+t)-\omega(t)$. A probability measure $\Psi$ on $\Omega$ has stationary increments when $\Psi = \Psi \circ (\theta^t)^{-1}$ for all $t \in {\mathbb{R}}$. For an interval $I \subset {\mathbb{R}}$ denote $\mathcal{F}_I = \sigma(\omega(t)-\omega(s)\colon\, s,t \in I)$. $\Psi_I$ denotes $\Psi$ restricted to $\mathcal{F}_I$. Write ${\mathscr{W}}$ for the Wiener measure on $\Omega$, i.e., the law of a (two-sided) Brownian motion. Let $\Psi \in \mathcal{P}(\Omega)$ with stationary increments be given and assume that $h(\Psi_{[0,T]} \mid {\mathscr{W}}_{[0,T]}) < \infty$ for all $T>0$. To verify , we imitate well-known arguments from the discrete-time setup (see e.g.  Ellis [@El85 Section IX.2]). For $I_1$, $I_2$ disjoint intervals in ${\mathbb{R}}$, denote by $\kappa^\Psi_{I_1, I_2}\colon\, \Omega \times \mathcal{F}_{I_2} \to [0,1]$ a regular version of the conditional law of (the increments of) $\Psi$ on $I_2$, given the increments in $I_1$, i.e., for fixed $\omega$, $\kappa^\Psi_{I_1, I_2}(\omega, \cdot)$ is a probability measure on $\mathcal{F}_{I_2}$, for fixed $A \in \mathcal{F}_{I_2}$, $\kappa^\Psi_{I_1, I_2}(\cdot, A)$ is an $\mathcal{F}_{I_1}$-measurable function, and $\kappa^\Psi_{I_1, I_2}(\omega, A)$ is a version of ${\mathbb{E}}_{\Psi}[{1}_A | \mathcal{F}_{I_1}]$. When $I_1=\emptyset$, $\kappa^\Psi_{\emptyset, I_2}(\omega, A) = \Psi_{I_2}(A)$. Similarly, define $\kappa^{\mathscr{W}}_{I_1, I_2}$ (which is simply $\kappa^{\mathscr{W}}_{I_1, I_2}(\omega, A) = {\mathscr{W}}_{I_2}(A)$ by the independence of the Brownian increments). Put $$\begin{aligned} a_{I_1,I_2} = \int_\Omega \Psi(d\omega_1) \int_\Omega \kappa^\Psi_{I_1, I_2}(\omega_1, d\omega_2) \, \log\left[ \frac{d \kappa^\Psi_{I_1, I_2}(\omega_1, \cdot)} {d \kappa^{\mathscr{W}}_{I_1, I_2}(\omega_1, \cdot)}(\omega_2)\right], \end{aligned}$$ the expected relative entropy of the conditional distribution under $\Psi$ on $\mathcal{F}_{I_2}$ given $\mathcal{F}_{I_1}$ w.r.t. Wiener measure on $\mathcal{F}_{I_2}$). We have $a_{I_1,I_2} < \infty$ for bounded intervals, because of the assumption of finite relative entropy of $\Psi$ w.r.t.  ${\mathscr{W}}$ on compact time intervals. By stationarity, $a_{I_1,I_2}=a_{t+I_1,t+I_2}$ for any $t$. Let $I_1' \subset I_1$, note that $\kappa^\Psi_{I_1, I_2}(\omega, \cdot) \ll \kappa^\Psi_{I_1', I_2} (\omega, \cdot)$ for $\Psi$-a.e. $\omega$, and $\kappa^{\mathscr{W}}_{I_1, I_2}(\omega, \cdot) = \kappa^{\mathscr{W}}_{I_1', I_2}(\omega, \cdot) = {\mathscr{W}}_{I_2}(\cdot)$. By the consistency property of conditional distributions, we have $$\begin{aligned} a_{I_1',I_2} = \int_\Omega \Psi(d\omega_1) \int_\Omega \kappa^\Psi_{I_1, I_2}(\omega_1, d\omega_2) \log \left[\frac{d \kappa^\Psi_{I_1', I_2}(\omega_1, \cdot)} {d \kappa^{\mathscr{W}}_{I_1', I_2}(\omega_1, \cdot)}(\omega_2)\right].\end{aligned}$$ Indeed, $$\int_\Omega \Psi(d\omega_1) \int_\Omega \kappa^\Psi_{I_1, I_2}(\omega_1, d\omega_2) f(\omega_1,\omega_2) = \int_\Omega \Psi(d\omega_1) \int_\Omega \kappa^\Psi_{I'_1, I_2}(\omega_1, d\omega_2) f(\omega_1,\omega_2)$$ for any function $f(\omega_1,\omega_2)$ that is $\mathcal{F}_{I_1'} \otimes \mathcal{F}_{{\mathbb{R}}}$-measurable. Hence $$\begin{aligned} &a_{I_1,I_2} - a_{I_1',I_2} \\ & = \int_\Omega \Psi(d\omega_1) \int_\Omega \kappa^\Psi_{I_1, I_2}(\omega_1, d\omega_2) \bigg( \log \left[\frac{d \kappa^\Psi_{I_1, I_2}(\omega_1, \cdot)} {d \kappa^{\mathscr{W}}_{I_1, I_2}(\omega_1, \cdot)}(\omega_2)\right] - \log \left[\frac{d \kappa^\Psi_{I_1', I_2}(\omega_1, \cdot)} {d \kappa^{\mathscr{W}}_{I_1', I_2}(\omega_1, \cdot)}(\omega_2)\right] \bigg) \notag \\ \label{eq:hdeccont1} & = \int_\Omega \Psi(d\omega_1) \int_\Omega \kappa^\Psi_{I_1, I_2}(\omega_1, d\omega_2) \, \log \left[\frac{d \kappa^\Psi_{I_1, I_2}(\omega_1, \cdot)} {d \kappa^\Psi_{I_1', I_2}(\omega_1, \cdot)}(\omega_2)\right] \geq 0\end{aligned}$$ because the inner integral is $h( \kappa^\Psi_{I_1, I_2}(\omega_1, \cdot) \mid \kappa^\Psi_{I_1', I_2} (\omega_1, \cdot)) \geq 0$. Choosing $I_1'=\emptyset$, , we get $a_{I_1, I_2} \geq a_{\emptyset, I_2} = h( \Psi_{I_2} \mid {\mathscr{W}}_{I_2})$. Observe $$\begin{aligned} \frac{d \Psi_{(0,s+t]}}{d {\mathscr{W}}_{(0,s+t]}}(\omega) = \frac{d \Psi_{(0,t]}}{d {\mathscr{W}}_{(0,t]}}(\omega) \, \frac{d \kappa^\Psi_{(0,t],(t,s+t]}(\omega, \cdot)} {d \kappa^{\mathscr{W}}_{(0,t],(t,s+t]}(\omega, \cdot)}(\omega) \quad \Psi_{(0,s+t]}-\text{a.s.},\end{aligned}$$ take logarithms and integrate w.r.t. $\Psi$ (using consistency of conditional expectation on the right-hand side), to obtain $$\begin{aligned} h\big( \Psi_{(0,s+t]} \mid {\mathscr{W}}_{(0,s+t]} \big) = h\big( \Psi_{(0,t]} \mid {\mathscr{W}}_{(0,t]} \big) + a_{(0,t], (t,s+t]} \geq h\big( \Psi_{(0,t]} \mid {\mathscr{W}}_{(0,t]} \big) + h\big( \Psi_{(0,s]} \mid {\mathscr{W}}_{(0,s]} \big). \end{aligned}$$ Thus, the function $(0,\infty) \ni t \mapsto h( \Psi_{(0,t]} \mid {\mathscr{W}}_{(0,t]})$ is super-additive, and follows from Fekete’s lemma. Under $\kappa^\Psi_{(-\infty,0], (0,h]}$, the coordinate process will be a Brownian motion with a (possibly complicated) drift process $U_t = \int_0^t u_s\, ds$, where $(u_t)_{t \geq 0}$ can be chosen adapted, and $${\mathbb{E}}_\Psi\big[ h(\kappa^\Psi_{(-\infty,0], (0,h]} \mid {\mathscr{W}}_{(0,h]}) \big] = {\mathbb{E}}_\Psi\big[{\textstyle \int_0^h} u_s^2 \,ds \big]$$ (see Föllmer [@Foe86]). [AAA]{} M. Birkner, Conditional large deviations for a sequence of words, Stoch. Proc. Appl. 118 (2008), 703–729. M. Birkner, A. Greven and F. den Hollander, Quenched large deviation principle for words in a letter sequence, Probab. Theory Relat. Fields 148 (2010), 403–456. M. Birkner and F. den Hollander, A variational view on the Brownian copolymer, manuscript in preparation. E. Bolthausen and F. den Hollander, Localization transition for a polymer near an interface, Ann. Probab. 25 (1997), 1334–1366. F. Caravenna and G. Giacomin, The weak coupling limit of disordered copolymer models, Ann. Probab. 38 (2010), 2322–2378. F. Caravenna, G. Giacomin and F.L. Toninelli, Copolymers at selective interfaces: settled issues and open problems, In: *Probability in Complex Physical Systems. In honour of Erwin Bolthausen and Jürgen Gärtner* (eds. J.-D. Deuschel, B. Gentz, W. König, M. von Renesse, M. Scheutzow, U. Schmock), Springer Proceedings in Mathematics 11, 2012, pp. 289–312. A. Dembo and O. Zeitouni, *Large Deviations Techniques and Applications* (2nd. Ed.), Springer, 1998. R. Ellis, *Entropy, Large Deviations, and Statistical Mechanics*, Springer, 1985. S. Ethier and T. Kurtz, *Markov Processes: Characterization and Convergence*, Wiley, 1986. H. Föllmer, Time reversal on Wiener space, Lecture Notes in Math. 1158, Springer, 1986, pp. 119–129. H. Föllmer, Random fields and diffusion processes, in: École d’Été de Probabilités de Saint-Flour XV–XVII, 1985–87, pp. 101–203, Lecture Notes in Math. 1362, Springer, 1988. R.M. Gray, *Entropy and Information Theory*, Springer, 1991.\ <http://ee.stanford.edu/~gray/it.html> C. Kipnis and C. Landim, *Scaling Limits of Interacting Particle Systems*, Springer, 1999.

--- abstract: 'We propose a method to electrically control electron spins in donor-based qubits in silicon. By taking advantage of the hyperfine coupling difference between a single-donor and a two-donor quantum dot, spin rotation can be driven by inducing an electric dipole between them and applying an alternating electric field generated by in-plane gates. These qubits can be coupled with exchange interaction controlled by top detuning gates. The qubit device can be fabricated deep in the silicon lattice with atomic precision by scanning tunneling probe technique. We have combined a large-scale full band atomistic tight-binding modeling approach with a time-dependent effective Hamiltonian description, providing a design with quantitative guidelines.' author: - 'Yu Wang\*' - 'Chin-Yi Chen' - Gerhard Klimeck - 'Michelle Y. Simmons' - 'Rajib Rahman\*' title: 'All-electrical control of donor-bound electron spin qubits in silicon' --- **** Introduction ============ Donor bound electrons in silicon have been demonstrated to have long spin coherence times [@Tyryshkin; @Buch; @Hsueh], making them promising candidates for solid-state qubits. Few-donor quantum dots [@Buch; @Weber_DQD] have been patterned by scanning tunneling microscopy (STM) based lithography in silicon with atomic precision [@Schofield], an excellent fabrication technique for building a scalable quantum computer. Due to different quantum confinement and hyperfine interaction compared to single donors, few-donor quantum dots provide more flexibility in addressing [@Buch; @YW_hyperfine] and engineering the exchange coupling [@YW_exchange], which are favorable attributes for multi-qubit operations. Controlling individual electron spins is of great importance for donor-based quantum computation. Manipulation of electron spins with integrated microwave antenna has been demonstrated in both donor and gate-defined quantum dot qubits with long coherence and high gate fidelity [@Muhonen; @Menno1]. However, it is challenging to use an ac magnetic field to realize local spin control for many qubits, a crucial requirement for multi-qubit operations in a scalable quantum computer architecture. We also know that a microwave antenna can introduce deleterious noise to the coherence of a qubit [@Muhonen]. An alternative way to spin control is to utilize an oscillating electric field, which has been demonstrated in quantum dot systems [@Kato; @Laird; @Vandersypen]. Here, the qubit is modulated periodically by the difference in Zeeman energy caused by either non-uniform electron g-factor, external magnetic field or hyperfine couplings [@Pingenot; @Tokura; @Petta]. To date, all-electrical control of spins without a microwave magnetic field in donor systems in silicon has not been demonstrated. =3.4in \[fig\_devices\] In this work, we propose all-electrical control of donor-based spin qubits taking advantage of the hyperfine coupling difference between single donor and few-donor quantum dots by introducing ancillary dots (The device schematic is shown in Fig. 1(a) and (b) with “P” denoting phosphorus donors). The electron and the 3 nuclear spins (of 2P + 1P atoms) in the dashed box in Fig. 1(b) define the single-qubit operation space, and the information is encoded in the electron spin. Here the ancillary dot (1P in green) creates a difference in the local hyperfine coupling between the 1P and 2P dots due to the different number of nuclei and asymmetric quantum confinement in the dashed box [@YW_hyperfine]. The (1,0)$\leftrightarrow$(0,1) [@charge_config_note] charge transition between them can be controlled by the pair of in-plane gates G1 and G2. An electric dipole can thereby be induced by biasing the system near the (1,0)–(0,1) charge degeneracy point, where the hyperfine couplings can be modulated with an ac electric field on top of the dc electric field from the in-plane gates G1 and G2 to drive the electron spin transition. In this proposed approach, the donor dots can be placed with atomic precision far from interfaces or surfaces using an STM based lithography technique, making them less prone to noise sources close to the interface [@Shamim; @Shamim2; @Schenkel; @Paik]. Using Coulomb confined states leads to higher valley and orbital states that are not accessible as they are typically at least 10 meV above the ground state. This results in well-isolated states of operation within which the qubit coherence can be boosted. Moreover, this scheme could be realized with existing circuitry in STM-patterned devices [@Weber_DQD; @Tom_readout] without introducing extra control components. The donor dots can be coupled via the exchange interaction (e.g. the (1,1)$\leftrightarrow$(2,0) transition in Fig. 1(b)) controlled by the surface detuning gates GT1 and GT2 (Fig. 1(a)), retaining the highly tunable exchange coupling [@YW_exchange] and allowing fully electrical two-qubit operations. This design can be potentially extended to a scalable quantum computer architecture by repeating the fundamental structure (Fig. 1(a) and (b)) according to the requirements of the large-scale architecture [@Fowler; @Hill]. Methods ======= In the following, we describe the spin and charge evolution in the 2P-1P system by an effective Hamiltonian with quantitative details. The effective Hamiltonian can be expressed as: $$H = H_e + H_T + H_{Z_e} + H_{Z_n} + H_{HF} , \label{eq1}$$ with the basis $\vert s,i_1i_2i_3,d\rangle$ including spin and charge information, where we denote $s=\uparrow,\downarrow$ as electron spin, $i_j=\Uparrow,\Downarrow$ (j = 1,2,3. See Fig. 2(a)) as the three nuclear spins, and $d=2P,1P$ as the specific dot site (e.g. $\vert \downarrow,\Uparrow\Uparrow\Uparrow,2P\rangle$ is the lowest energy configuration). $H_e$ reflects both the on-site energy detuning and the applied ac electric field, which is expressed as: $$H_e = \sum_{d=1P,2P}(\epsilon_d +e \vec{E_{ac}}\cdot\vec{R})\vert d\rangle\langle d\vert, \label{eq1-1}$$ where $\epsilon_d$ is the on-site detuning energy, and $d$ is the dot index (1P or 2P). $e$ is the elementary charge, $\vec{E_{ac}}$ is the ac electric field, and $\vec{R}$ is the separation between the 2P and the 1P dots. The second term represents the tunnel coupling between the (1,0) and (0,1) charge states of the 2P and the 1P dots: $$H_T = \sum_{d\neq d'} t_c\vert d\rangle\langle d'\vert, \label{eq1-2}$$ where $t_c$ denotes the tunnel coupling between the two donor dots. $d$ and $d'$ denote different dot indices. The third and the fourth term denote the Zeeman energy of the electron spin and the nuclear spins: $$H_{Z_e} = g_e\mu_B\vec{B}\cdot\vec{S}, \label{eq1-3}$$ $$H_{Z_n} = \sum_jg_n\mu_B\vec{B}\cdot\vec{I_j}, \label{eq1-4}$$ where $g_e$ and $g_n$ are the electron and nuclear g-factors respectively. $\mu_B$ and $\mu_n$ are the Bohr and the nuclear magnetons respectively. $\vec{S}$ denotes the electron spin operator, and $\vec{I_j}$ denotes the spin operator of the $j$th nucleus.The fifth term gives the hyperfine coupling between the qubit electron and the donor nuclei: $$H_{HF} = \sum_j A_j\vec{I_j}\cdot\vec{S}, \label{eq1-5}$$ where $A_j$ is the Fermi-contact hyperfine coupling between the qubit electron and the $j$th nucleus. To provide quantitative guidelines for exploiting this proposal, it is important to obtain the parameters in $H$, the tunnel coupling $t_c$, Fermi contact hyperfine couplings $A_j$, with sufficient accuracy. Here we model the system using an atomistic tight-binding approach to obtain the Stark-shifted electron wavefunctions, from which tunnel couplings $t_c$, hyperfine couplings $A_j$ and their electric field dependency can be extracted [@YW_hyperfine; @Rahman_prl; @Park]. In the tight-binding approach, the atoms are represented by $sp^3d^5s^*$ atomic orbitals with spin-orbit coupling and nearest-neighbor interactions. Each donor is represented by a Coulomb potential screened by the dielectric constant of silicon and an on-site constant potential for the Coulomb singularity, calibrated with the P donor energy spectrum [@Ahmed]. The donor ground-state wavefunction obtained from this approach agrees well with the recent STM imaging experiments [@Salfi]. This physics-based approach automatically includes silicon conduction band valley degrees of freedom, and captures valley-orbit interaction [@Rahman_gfactor] and Stark effect in donor orbitals [@Rahman_prl]. The experimentally spin relaxation times of a single P donor and few-donor dots can also be reproduced with our approach [@Hsueh]. This provides us confidence in accurately extracting the parameters $t_c$ and $A_j$ under realistic electric fields. Results and discussions ======================= =3.4in \[fig\_hyperfine\] Fig. 2 shows the modulation of the hyperfine couplings ($A_j$) between the qubit electron spin and the nuclear spins of the P donors (2P + 1P) with electric field, with j=1,2 labeling the nuclei of the 2P dots respectively and j=3 labeling the nucleus of the 1P dot. The detail of the system in the dashed box in Fig. 1(b) is depicted as Fig. 2(a), where an atom configuration of the 2P dot is shown at the bottom. An in-plane external electric field ($E_x$) between G1 and G2 applied along the direction between the 2P dot and its ancillary 1P dot can detune the system and redistribute the electron wavefunction between them, thereby controlling the hyperfine couplings of the 2P and the 1P dots, as shown in Fig. 2(b). As can be seen, $A_3\approx 0$ at $E_x-E_0 = 5kV/m$, and $A_1, A_2\approx 0$ at $E_x-E_0 = -5kV/m$, indicating that the charge states (0,1) and (1,0) can be accessed with a small electric field range ($\sim$ 10 kV/m) with an inter-dot separation R$\approx$15 nm. At $E_x \approx E_0$, the (1,0) and (0,1) charge states are degenerate, forming hybridized bonding and anti-bonding states. As observed, the hyperfine couplings ($A_j$) have a nearly linear dependence on the electric field near the (1,0)–(0,1) charge degeneracy point ($-1 kV/m <E_x - E_0 < 1 kV/m$), indicating that the hyperfine coupling difference between the 2P and the 1P dot has a linear response to the external electric field. Near the charge degeneracy point ($E_x \approx E_0$), an electric dipole transition can be driven by an ac electric field $\varepsilon_0 sin(\omega t)$ applied from G1 and G2, thereby causing the modulations of $A_j$ with time. When the ac electric field is in resonance with the qubit energy splitting (solved from the static part of $H$ in eq. (\[eq1\])), the electron spin transition can be induced by the overall local hyperfine coupling difference [@Laird; @Tosi] between the 2P and the 1P dot. The emulation of this process in a 2P-1P system is now described by solving the time evolution based on the Hamiltonian in eq. (\[eq1\]). =3.4in \[fig\_rabi\] We assume the system is initially in its lowest energy configuration and the electron is located at the 2P dot (at (1,0)), i.e. $\vert\downarrow,\Uparrow\Uparrow\Uparrow,2P\rangle$, and assume the 2P dot configuration as shown in Fig. 2(a) with the 2P in one dot $\leq$1.5 nm apart. A static magnetic field $B_0$ is applied along the separation direction $x$. One of the possible ways to prepare the qubit in this initial state is to utilize the dynamic nuclear polarization technique [[@Petta_dynamic]]{} by repeatedly and selectively loading up-spin electrons to the donor dots [[@Morello_readout]]{} and emptying the dots to dynamically drive the nuclear spins to up orientations, then depleting the dots and loading a down-spin electron onto the dots in the end. To achieve universal quantum gates, two-axis control of single qubits, i.e. Z-gate and X-gate, is needed. A Z-gate can be simply realized by applying the external static B-field. We will focus on the X-gate in the following. As shown in Fig. 3(a), at the beginning of the control manipulation, an adiabatic in-plane gate bias between G1 and G2 is applied to ramp the static dc electric field up to and to keep it at $\sim$$E_0$, which is followed by a continuous ac electric field between the same gates. The X-gate rotation (the manipulation of $\downarrow\rightarrow\uparrow$ or $\uparrow\rightarrow\downarrow$) is then driven by the ac electric field if $h\gamma = \Delta E$, where $\Delta E$ is the energy difference between the two qubit states and $\gamma$ is the frequency of the ac electric field. To be explicit, the qubit ground state is $\frac{1}{\sqrt{2}}(\vert\downarrow,\Uparrow\Uparrow\Uparrow,2P\rangle - \vert\downarrow,\Uparrow\Uparrow\Uparrow,1P\rangle)$, and the qubit excited state is a dressed state involving both electron and nuclear spins, which can be expressed as $\alpha(\vert\uparrow,\Downarrow\Uparrow\Uparrow,2P\rangle - \vert\uparrow,\Downarrow\Uparrow\Uparrow,1P\rangle) +\beta(\vert\uparrow,\Uparrow\Downarrow\Uparrow,2P\rangle$ $- \vert\uparrow,\Uparrow\Downarrow\Uparrow,1P\rangle) + \zeta(\vert\uparrow,\Uparrow\Uparrow\Downarrow,2P\rangle - \vert\uparrow,\Uparrow\Uparrow\Downarrow,1P\rangle)$. A second adiabatic gate bias is then applied after the X-gate control is finished to bring the electron back to the 2P dot for spin storage. Here, we choose 2P over 1P because the electron spin relaxation time is longer in a 2P donor cluster than a single donor dot [@Hsueh]. To achieve high-fidelity X-gate operation and make the system less prone to charge noise and relaxation, we need to make appropriate choices of the external B-field and the tunnel coupling. On the one hand, with regard to the external B-field, the qubit energy splitting $\Delta$E can be expressed as $E_{Z_e} + \delta$, where $E_{Z_e}$ is the electron Zeeman splitting, and $\delta$ includes the effects of nuclear spin Zeeman energies and hyperfine couplings, which contributes to an effective magnetic field in the order of mT. To form well-defined qubit states and suppress nuclear spin flip-flop, we need $E_{Z_e} >> \delta$ to preserve the qubit state when no ac field is applied. As a result, the external B-field is required to be in the order of 0.1 T. On the other hand, regarding the tunnel coupling, we need $\Delta$E significantly smaller than $2t_c$ in order to make the higher anti-bonding states well separated from the lower qubit states, preventing state hybridization or excitation due to environmental noise. As an example, Fig. 3(b) shows the Rabi oscillations of the qubit electron spin under the driving ac electric-field under $\Delta E<2t_c$ ($B_0$ = 0.5 T) and $\Delta E\approx 2t_c$ ($B_0$ = 1.45 T) for $R\approx$ 11.4 nm, where the magnitude of the driving ac electric field is 15 kV/m, and its frequency is $\gamma\approx$ 14 GHz which satisfies $\Delta E = h\gamma$. The ac electric field is assumed to be a single-frequency sinusoid. As shown, a full X-gate spin rotation can be achieved for $\Delta E<2t_c$, while the X-gate fidelity (defined as max($\sum\vert\langle\Psi\vert\uparrow,i_1i_2i_3,d\rangle\vert^2$)) is diminished when $\Delta E\approx 2t_c$ due to the qubit spin-up state (solid blue curve in Fig. 3(a)) is hybridized with the upper anti-bonding spin-down state (dashed green curve). As a result, $t_c$ needs to be engineered large enough. Fig. 3(c) shows $t_c$ as a function of the inter-dot separation $R$. As can be seen, $t_c$ decreases exponentially as a function of $R$, because the wavefunction overlap of the 2P and the 1P dots decreases exponentially as $R$ increases. To achieve $\Delta E<2t_c$, if we choose B = 0.1 T, $R$ needs to be larger than 15.6 nm approximately according to Fig. 3(c). If B = 0.5 T is chosen, $R$ needs to be at least 13 nm. In the following, we investigate the effect of $R$ (or $t_c$) on the qubit coherence time. Both magnetic and charge noise can lead to qubit decoherence [@Kuhlmann]. In the proposed design, magnetic noise can be suppressed to a large extent if the substrate is made of enriched Si-28. In addition, the microwave antenna that introduces magnetic noise [@Muhonen] in the traditional magnetic qubit manipulation is excluded here. As a consequence, we mainly consider the effect of the charge noise from the charge fluctuations in the nearby gates on qubit coherence. We investigate the decoherence time $T_{2}^{*}$ possibly due to different types of charge noise from a single nearby in-plane gate, e.g. G1 in Fig. 1(b). $T_{2}^{*}$ can be obtained using [@Chirolli]: $$\begin{split} \frac{1}{T_{2}^{*}} = \frac{e^2}{\hbar^2}\vert\sum_{r_i=x,y,z}\langle\Psi_{\uparrow}\vert r_i\vert \Psi_{\uparrow}\rangle-\langle\Psi_{\downarrow}\vert r_i\vert \Psi_{\downarrow}\rangle\vert^2\\\cdot\frac{S_E(\omega)}{\omega}\bigg|_{\omega\rightarrow 0}\frac{2k_BT}{\hbar}, \end{split}$$ where $e$ is the elementary charge, $\Psi_{\uparrow}$ and $\Psi_{\downarrow}$ are the electron spin-up and spin-down molecular wavefunctions solved by the atomistic tight-binding method respectively, $\omega$ is the noise frequency and $S_E(\omega)$ is the noise field spectrum. For $S_E(\omega)$, we study $1/f^{\alpha}$ noise, Johnson noise and evanescent wave Johnson noise (EWJN) [@PHuang], assuming the noise source is 65 nm (the distance between G1 and the two dot center) away from the qubit system to be consistent with Ref. [@Tom_readout]. The expressions of the noise field spectra and the parameter estimations based on experiments [@Tom_readout; @Bent_prl; @Buch_prb] are also included in the Supplementary. =3.4in \[fig\_noise\] Fig. 4(a) demonstrates the decoherence rate $1/T_{2}^{*}$ as a function of the applied dc electric field ($E_x$) due to Johnson noise from a single noise source for the case $R \approx$ 13 nm, assuming B = 0.5 T. Using the estimated parameters in the Supplementary, we find that $T_{2}^{*}$ is limited by Johnson noise. Based on our calculations, the effect of EWJN is at least 2 orders of magnitude lower than Johnson noise, and $1/f^{\alpha}$ noise is negligible (see Supplementary), thus they are not shown here. As shown, the decoherence rate $1/T_{2}^{*}$ reaches a maximum at $E_x=E_0$, where the bonding and anti-bonding states are formed and the system is most sensitive to charge noise. The left y-axis of Fig. 4(b) shows the maximum decoherence rate $1/T_{2}^{*}$ due to Johnson noise as a function of inter-dot separation $R$. As shown, $T_{2}^{*}$ can be improved by shrinking the inter-dot separation. This can be explained by the curvature of the qubit energy curve (e.g. the solid green or blue curve in Fig. 3(a))), which serves as a metric of how the qubit is prone to charge noise near $E_x = E_0$. On the right y-axis of Fig. 4(b), we plot this curvature ($a$ in its absolute value) by fitting the energy curves with a quadratic function of $E_x$ for different $R$. The curvature term $|a|$ increases with $R$ because the tunnel coupling $t_c$ decreases with $R$ (Fig. 3(c)), causing more abrupt charge transition. As can be seen, $1/T_2^{*}$ agrees with the trend of the curvature term $|a|$. Consequently, larger tunnel coupling/smaller 2P-1P separation is preferred to enhance the qubit coherence time. So far, we have investigated the decoherence time due to a single charge noise source. In a real device, there could be multiple charge noise sources (other in-plane gates, top gates, etc.), resulting in $T_2^{*}$ being degraded by 1-2 orders of magnitude eventually. Even then, using a 2P-1P spin qubit with electrical control is likely to yield devices comparable to single electron spin qubit based on single donor ($T_2^*$ = 268 $\mu$s [@Muhonen]) and single quantum dot qubit ($T_2^*$ = 120 $\mu s$ [@Menno1]) based on magnetic control, and outperform the 1P-1P charge qubit ($T_2^*$ = 0.72 $\mu$s [@Hollenberg]) in terms of qubit coherence time. Summary ======= In summary, we propose a novel approach for all-electrical control of donor-based spin qubits in silicon using full-band atomistic tight-binding modeling and time-dependent simulations based on effective spin Hamiltonian. In this design, ancillary dots are introduced to form an asymmetric 2P-1P system to create a hyperfine coupling difference between 2P and 1P, utilized to realize electron spin control with an ac electric field. We perform a quantitative analysis to optimize this design in terms of X-gate fidelity and decoherence time through external static B-field and tunnel coupling determined by inter-dot separation. We show that a long qubit coherence time can be potentially achieved. This work can serve as an alternative design to those that exploit the hyperfine difference between the donor and the interface states [@Tosi], where the qubit coherence could be affected by the proximity of the oxide interface [@Schenkel; @Paik]. To further reduce possible sources of deleterious noise in the proposed design, we would further pursue all-in-plane electrostatic and qubit control without the top surface gates in the future. Acknowledgements {#acknowledgements .unnumbered} ================ This research was conducted by the Australian Research Council Centre of Excellence for Quantum Computation and Communication Technology (project No. CE110001027), the US National Security Agency and the US Army Research Office under contract No. W911NF-08-1-0527. Computational resources on nanoHUB.org, funded by the NSF grant EEC-0228390, were used. M.Y.S. acknowledges a Laureate Fellowship. Supplementary {#supplementary .unnumbered} ============= [*Spectrum functions of charge noise fields.*]{} Following Ref. [@PHuang], we investigate three types of charge noise. The $1/f^{\alpha}$ noise field spectrum is expressed as: $$S_E(\omega) = \frac{N}{\omega^{\alpha}},$$ where $N$ is the noise field strength in $(V/m)^2$. In this work, we estimate $N$ based on Ref. [[@Kuhlmann]]{}, where the root-mean-square electric field noise is $F_{r.m.s}$ = 46 V/m. Then $N = (\sqrt{2}F_{r.m.s})^2$ = 4232 $(V/m)^2$. We assume the bandwidth starts at 0.1 Hz also in line with Ref. [[@Kuhlmann]]{}, and $\alpha$ = 1 in the calculations. The Johnson noise field spectrum is: $$S_E(\omega) = \frac{2\xi\omega\hbar^2}{1+(\omega/\omega_R)^2}/(el_0)^2,$$ where $\xi = R/R_k$, $R_k$ is the fundamental quantum resistance $h/e^2$, $R$ is the circuit resistance, and $\omega_R = 1/RC$ is the cutoff frequency. R is estimated based on Ref. [@Bent_prl]. In this work, we assume the gate length ($l_g$) is 100 nm and the gate width is 6 nm ($w$) which leads to 18 conducting modes ($M$, number of modes) [@Bent_prl]. As estimated in Ref. [@Bent_prl], the mean-free-path ($\lambda$) of such a wire is $\sim$6 nm. Hence, R in this work is calculated by $1/R = e^2/h\cdot M\cdot \lambda/(\lambda+l_g)$. $l_0$ is the distance between the qubit and the noise source, and we assume $l_0$ = 65 nm based on experimental devices [@Tom_readout]. $C$ is estimated based on Ref. [@Buch_prb], where the donor-gate capacitance is 0.6 aF for a separation $\sim$35 nm. Therefore, $C$ in this work is evaluated as $35nm/65nm\cdot 0.36 aF = 0.17 aF$. The evanescent wave Johnson noise (EWJN) field spectrum can be expressed as: $$S_E(\omega) = \frac{\hbar\omega}{8z^3\sigma},$$ where $\sigma$ is the conductivity of the gates. We extract $\sigma$ from Ref. [@Bent_prl] where the wire length ($l_w$) is 47 nm. We assume the thickness of the wire ($t_w$) to be 2 nm considering donor diffusion and segregation, then the conductivity can be expressed as $4.8\frac{e^2}{h}\frac{l_w+\lambda}{w\cdot t_w}$. =3.4in In Fig. [\[fis\]]{}, we compare the effects of the three types of charge noise stated above on the decoherence rate [$1/T_2^{*}$]{}. As can be seen and stated in the main text, [$T_{2}^{*}$]{} is limited by Johnson noise. EWJN is at least 2 orders of magnitude lower than Johnson noise. [$1/f^{\alpha}$]{} noise is negligible, which is consistent with Ref. [[@PHuang]]{}. Electronic address: [email protected]; [email protected] [100]{} A. M. Tyryshkin [*[et al.]{}*]{}, Nat. Mater. [**[11]{}**]{}, 143-147 (2012). H. Büch, S. Mahapatra, R. Rahman, A. Morello, and M.Y. Simmons. Nat. Commun. [**[4]{}**]{}, 2017 (2013). Y. Hsueh [*[et al.]{}*]{}, Phys. Rev. Lett. [**[113]{}**]{}, 246406 (2014). B. Weber [*[et al.]{}*]{}, Nat. Nanotechnol. [**[9]{}**]{}, 430-435 (2014). S. R. Schofield [*[et al.]{}*]{}, Phys. Rev. Lett. [**[91]{}**]{}, 136104 (2003). Y. Wang [*[et al.]{}*]{}, Sci. Rep. [**[6]{}**]{}, 31830 (2016). Y. Wang [*[et al.]{}*]{}, npj Quantum Information [**[9]{}**]{}, 16008 (2016). A. G. Fowler, S. J. Devitt, L. C. L. Hollenberg, Quantum Inf. Comput. [**[4]{}**]{}, 237–245 (2004). C. D. Hill [*[et al.]{}*]{}, Sci. Adv. [**[1]{}**]{}, no. 9, e1500707 (2015). J. Muhonen [*[et al.]{}*]{}, Nat. Nanotechnol. [**[9]{}**]{}, 986 (2014). M. Veldhorst [*[et al.]{}*]{}, Nat. Nanotechnol. [**[9]{}**]{}, 981 (2014). Y. Kato [*[et al.]{}*]{}, Science [**[21]{}**]{}, vol. 299, issue 5610, 1201-1204 (2003). E. A. Laird, C. Barthel, E. I. Rashba, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Phys. Rev. Lett. [**[99]{}**]{}, 246601 (2007). E. Kawakami [*[et al.]{}*]{}, Nat. Nanotechnol. [**[9]{}**]{}, 666–670 (2014). J. Pingenot [*[et al.]{}*]{}, Appl. Phys. Lett. [**[92]{}**]{}, 222502 (2008). Y. Tokura [*[et al.]{}*]{}, Phys. Rev. Lett. 96, 047202 (2006). J. R. Petta [*[et al.]{}*]{}, Science [**[309]{}**]{}, 2180 (2005). The charge configuration of the system is denoted as (M,N), where M indicates the number of electrons on the left dot, and N the right dot. S. Shamim [*[et al.]{}*]{}, Nano Lett. [**[16]{}**]{}(9), 5779-5784 (2016). S. Shamim [*[et al.]{}*]{}, Phys. Rev. Lett. [**[112]{}**]{}, 236602 (2014). T. Schenkel [*[et al.]{}*]{}, Appl. Phys. Lett. [**[88]{}**]{}, 112101 (2006). S.-Y. Paik, S.-Y. Lee, W. J. Baker, D. R. McCamey, & C. Boehme, Phys. Rev. B. [**[81]{}**]{}, 075214 (2010). T. F. Watson [*[et al.]{}*]{}, Phys. Rev. Lett. [**[115]{}**]{}, 166806 (2015). R. Rahman [*[et al.]{}*]{}, Phys. Rev. Lett. [**[99]{}**]{}, 036403 (2007). S. H. Park [*[et al.]{}*]{}, Phys. Rev. Lett. [**[103]{}**]{}, 106802 (2009). S. Ahmed [*[et al.]{}*]{}, Springer Encyclopedia of Complexity and Systems Science, Springer-Verlag GmbH, Heidelberg, pp. 5745, ISBN: 978-0-387-75888-6 (2009). J. Salfi [*[et al.]{}*]{}, Nat. Mater. [**[13]{}**]{}, 605 (2014). R. Rahman [*[et al.]{}*]{}, Phys. Rev. B [**[80]{}**]{}, 155301 (2009). G. Tosi [*[et al.]{}*]{}, Silicon quantum processor with robust long-distance qubit couplings. arXiv:1509.08538 (2015). J. R. Petta [*[et al.]{}*]{}, Phys. Rev. Lett. [**[100]{}**]{}, 067601 (2008). A. Morello [*[et al.]{}*]{}, Nature [**[467]{}**]{}, 687–691 (2010). A. V. Kuhlmann [*[et al.]{}*]{}, Nat. Phys. [**[9]{}**]{}, 570–575 (2013). L. Chirolli and G. Burkard, Adv. in Phys. 57:3, 225-285 (2008). P. Huang and X. Hu, Phys. Rev. B [**[89]{}**]{}, 195302 (2014). B. Weber, H. Ryu, Y.-H. M. Tan, G. Klimeck, and M. Y. Simmons, Phys. Rev. Lett. [**[113]{}**]{}, 246802 (2014). H. Büch, M. Fuechsle, W. Baker, M. G. House, and M. Y. Simmons, Phys. Rev. B [**[92]{}**]{}, 235309 (2015). L. C. L. Hollenberg [*[et al.]{}*]{}, Phys. Rev. B [**[69]{}**]{}, 113301 (2004).

--- abstract: 'We discuss the isospin symmetry breaking (ISB) of the valence- and sea-quark distributions between the proton and the neutron in the framework of the chiral quark model. We assume that isospin symmetry breaking is the result of mass differences between isospin multiplets and then analyze the effects of isospin symmetry breaking on the Gottfried sum rule and the NuTeV anomaly. We show that, although both flavor asymmetry in the nucleon sea and the ISB between the proton and the neutron can lead to the violation of the Gottfried sum rule, the main contribution is from the flavor asymmetry in the framework of the chiral quark model. We also find that the correction to the NuTeV anomaly is in an opposite direction, so the NuTeV anomaly cannot be removed by isospin symmetry breaking in the chiral quark model. It is remarkable that our results of ISB for both valence- and sea-quark distributions are consistent with the Martin-Roberts-Stirling-Thorne parametrization of quark distributions.' author: - Huiying Song - Xinyu Zhang - 'Bo-Qiang Ma[^1]' title: Isospin Symmetry Breaking in the Chiral Quark Model --- Introduction ============ Isospin symmetry was originally introduced to describe almost identical properties of strong interaction of the proton and the neutron by turning off their electromagnetic interaction, i.e., their charge information. This symmetry is commonly expected to be a precise symmetry [@Henley:1979ig; @Miller:1990iz], and its breaking is assumed to be negligible in the phenomenological or experimental analysis. This is, in general, true, since electromagnetic interactions are weak compared with strong interactions. However, it is possible for isospin symmetry breaking (ISB) to have important influence on some experiments, especially its effects on the parton distributions. Therefore, it is necessary to analyze it carefully. The isospin symmetry between the proton and the neutron originates from the SU(2) symmetry between $u$ and $d$ quarks, which are isospin doublets with isospin $I= 1/2$ and isospin three-components ($I_3$) 1/2 and -1/2, respectively. The isospin symmetry at parton level indicates that the $u~(d,~\bar{u},~\bar{d})$-quark distribution in the proton is equal to the $d~(u,~\bar{d},~\bar{u})$-quark distribution in the neutron. Accordingly, the ISBs of both valance-quark and sea-quark distributions are defined, respectively, as $$\begin{aligned} \delta u_{\mathrm{V}}(x)&=&u_{\mathrm{V}}^{\mathrm{p}}(x)-d_{\mathrm{V}}^{\mathrm{n}}(x),\nonumber\\ \delta d_{\mathrm{V}}(x)&=&d_{\mathrm{V}}^{\mathrm{p}}(x)-u_{\mathrm{V}}^{\mathrm{n}}(x),\nonumber\\ \delta \bar{u}(x)&=&\bar{u}^{\mathrm{p}}(x)-\bar{d}^{\mathrm{n}}(x),\nonumber\\ \delta \bar{d}(x)&=&\bar{d}^{\mathrm{p}}(x)-\bar{u}^{\mathrm{n}}(x),\end{aligned}$$ where $q_{\mathrm{V}}^{\mathrm{N}}(x)=q^{\mathrm{N}}(x)-\bar{q}^{\mathrm{N}}(x)~(q=u,~d,~\mathrm{N}=\mathrm{p},~\mathrm{n}).$ ISB at the parton level and its possible consequences for several processes were first investigated by one of us [@Ma:1991ac]. It was pointed out that both flavor asymmetry in the nucleon sea and isospin symmetry breaking between the proton and the neutron can lead to the violation of the Gottfried sum rule reported by the New Muon Collaboration [@Amaudruz:1991at; @Arneodo:1994sh]. The possibility of distinguishing these two effects was also discussed in detail [@Ma:1992gp]. In 2002, the NuTeV Collaboration [@Zeller:2001hh] extracted $\sin^{2}\theta_{\mathrm{W}}$ by measuring the ratios of neutral current to charged current $\nu$ and $\bar{\nu}$ cross sections on iron targets. The reported $\sin^{2}\theta_{\mathrm{W}}=0.2277\pm0.0013\left(\mathrm{stat}\right)\pm0.0009\left(\mathrm{syst}\right)$ has approximately 3 standard deviations above the world average value $\sin^{2}\theta_{\mathrm{W}}=0.2227\pm0.0004$ measured in other electroweak processes. This remarkable deviation is called the NuTeV anomaly and was discussed in a number of papers from various aspects, including new physics beyond the standard model [@Davidson:2001ji], the nuclear effect [@Kovalenko:2002xe], nonisoscalar targets [@Kumano:2002ra], and strange-antistrange asymmetry [@Cao:2003ny; @Ding:2004ht; @Ding:2004dv]. Moreover, the possible influence of ISB on this measurement was also studied in a series of papers [@Sather:1991je; @Rodionov:1994cg; @Davidson:1997mb; @Londergan:2003ij; @Cao:2000dj; @Gluck:2005xh; @Ding:2006ud]. However, the correction from ISB to the NuTeV anomaly is still not conclusive. The Martin-Roberts-Stirling-Thorne (MRST) group [@Martin:2003sk] provided some evidence to support the ISB effects on parton distributions of both valance and sea quarks and included ISB in the parametrization based on experimental data. They obtained the ISB of valance quarks as $$\begin{aligned} \delta u_{\mathrm{V}}=-\delta d_{\mathrm{V}}=\kappa(1-x)^{4}x^{-0.5}(x-0.0909),\end{aligned}$$ where $-0.8\leq\kappa\leq+0.65$ with a $90\%$ confidence level, and the best fit value is $\kappa=-0.2$. They also obtained the ISB of sea quarks, as can be deduced from Eqs. (28) and (29) in Ref. [@Martin:2003sk], $$\begin{aligned} \delta\bar{u}(x)=k\bar{u}^{\mathrm{p}}(x),~~~~\delta\bar{d}(x)=k\bar{d}^{\mathrm{p}}(x),\end{aligned}$$ with the best fit value $k=0.08$. In this paper, we calculate the ISB of the valance- and sea-quark distributions between the proton and the neutron in the chiral quark model and discuss some possible effects of ISB. We assume that the ISB between the proton and the neutron is entirely from the mass difference between isospin multiplets at both hadron and parton levels.[^2] In Sec. \[section2\], we compute ISB in the chiral quark model, with the constituent-quark-model results as the bare constituent-quark-distribution inputs. Then, we calculate the ISB effect on the violation of the Gottfried sum rule. In Sec. \[section3\], we discuss the ISB correction to the measurement of the weak angle and point out the significant influence on the NuTeV anomaly. In Sec. \[section4\], we provide summaries of the paper. isospin symmetry breaking in the chiral quark model {#section2} =================================================== The chiral quark model, established by Weinberg [@Weinberg:1978kz] and developed by Manohar and Georgi [@Manohar:1983md], has an apt description of its important degrees of freedom in terms of quarks, gluons, and Goldstone (GS) bosons at momentum scales relating to hadron structure. This model is successful in explaining numerous problems, including the violation of the Gottfried sum rule from the aspect of flavor asymmetry in the nucleon sea [@Eichten:1991mt; @Wakamatsu:1991tj], the proton spin crisis [@Ashman:1987hv; @Cheng:1994zn; @Song:1997bp], and the NuTeV anomaly resulting from the strange-antistrange asymmetry [@Ding:2004dv], and has been widely recognized as an effective theory of QCD at the low-energy scale. In the chiral quark model, the minor effects of the internal gluons are negligible. The valence quarks contained in the nucleon fluctuate into quarks plus GS bosons, which spontaneously break chiral symmetry. Then, the effective interaction Lagrangian is $$L=\bar{\psi}\left(iD_{\mu}+V_{\mu}\right)\gamma^{\mu}\psi+ig_{\mathrm{A}}\bar{\psi}A_{\mu}\gamma^{\mu}\gamma_{5}\psi+\cdots,$$ where $$\psi=\left(\begin{array}{c} u \\ d \\ s \\ \end{array}\right)$$ is the quark field and $D_{\mu}=\partial_{\mu}+igG_{\mu}$ is the gauge-covariant derivative of QCD. $G_{\mu}$ stands for the gluon field, $g$ stands for the strong coupling constant, and $g_{\mathrm{A}}$ stands for the axial-vector coupling constant. $V_{\mu}$ and $A_{\mu}$ are the vector and the axial-vector currents, which are defined as $$\left(\begin{array}{c} V_{\mu} \\ A_{\mu} \\ \end{array}\right)=\frac{1}{2}\left(\xi^{+}\partial_{\mu}\xi\pm\xi\partial_{\mu}\xi^{+}\right),$$ where $\xi=\mathrm{exp}(i\Pi/f)$, and $\Pi$ has the form: $$\Pi\equiv\frac{1}{\sqrt{2}}\left( \begin{array}{ccc} \frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}} & \pi^{+} & K^{+} \\ \pi^{-} & -\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}} & K^{0} \\ K^{-} & \overline{K^{0}} & \frac{-2\eta}{\sqrt{6}} \\ \end{array} \right).$$ Expanding $V_{\mu}$ and $A_{\mu}$ in powers of $\Pi/f$, one gets $V_{\mu}=0+O(\Pi/f)^{2}$ and $A_{\mu}=i\partial_{\mu}\Pi/f+O(\Pi/f)^{2}$. The pseudoscalar decay constant is $f\simeq93$ MeV. Thus, the effective interaction Lagrangian between GS bosons and quarks in the leading order becomes [@Eichten:1991mt] $$L_{\Pi q}=-\frac{g_{\mathrm{A}}}{f}\bar{\psi}\partial_{\mu}\Pi\gamma^{\mu}\gamma_{5}\psi.$$ Based on the time-ordered perturbative theory in the infinite momentum frame, all particles are on-mass-shell, and the factorization of the subprocess is automatic, so we can express the quark distributions inside a nucleon as a convolution of a constituent-quark distribution in a nucleon and the structure functions of a constituent quark. Since the $\eta$ is relatively heavy, we neglect the minor contribution from its suppressed fluctuation in this paper. Then, the light-front Fock decompositions of constituent-quark wave functions are $$|U\rangle=\sqrt{Z_{u}}|u_{0}\rangle+a_{\pi^{+}}|d\pi^{+}\rangle+\frac{a_{u\pi^{0}}}{\sqrt{2}}|u\pi^{0}\rangle+a_{K^{+}}|sK^{+}\rangle,\label{u}$$ $$|D\rangle=\sqrt{Z_{d}}|d_{0}\rangle+a_{\pi^{-}}|u\pi^{-}\rangle+\frac{a_{d\pi^{0}}}{\sqrt{2}}|d\pi^{0}\rangle+a_{K^{0}}|sK^{0}\rangle,\label{d}$$ where $Z_{u}$ and $Z_{d}$ are the renormalization constants for the bare constituent $u$ quark $|u_{0}\rangle$ and $d$ quark $|d_{0}\rangle$, respectively, and $|a_{\alpha}|^{2}$ ($\alpha=\pi, K$) are the probabilities to find GS bosons in the dressed constituent-quark states $|U\rangle$ and $|D\rangle$. In the chiral quark model, the fluctuation of a bare constituent quark into a GS boson and a recoil bare constituent quark is given as [@Suzuki:1997wv] $$q_{j}(x)=\int^{1}_{x}\frac{\textmd{d}y}{y}P_{j\alpha/i}(y)q_{i}\left(\frac{x}{y}\right),\label{q}$$ where $P_{j\alpha/i}(y)$ is the splitting function, which gives the probability of finding a constituent quark $j$ carrying the light-cone momentum fraction $y$ together with a spectator GS boson $\alpha$, $$\begin{aligned} P_{j\alpha/i}(y)=\frac{1}{8\pi^{2}}\left(\frac{g_{\mathrm{A}}\overline{m}}{f}\right)^{2}\int \textmd{d}k^{2}_{T}\frac{(m_{j}-m_{i}y)^{2}+k^{2}_{T}}{y^{2}(1-y)[m_{i}^{2}-M^{2}_{j\alpha}]^{2}}. \label{splitting}\end{aligned}$$ $m_{i}$, $m_{j}$, and $m_{\alpha}$ are the masses of the $i$- and $ j$-constituent quarks and the pseudoscalar meson $\alpha$, respectively, and $\overline{m}=(m_{i}+m_{j})/2$ is the average mass of the constituent quarks. $M^{2}_{j\alpha}=\left(m^{2}_{j}+k^{2}_{T}\right)/y+\left(m^{2}_{\alpha}+k^{2}_{T}\right)/\left(1-y\right)$ is the square of the invariant mass of the final state. We can also write the internal structure of GS bosons in the following form $$q_{k}(x)=\int\frac{\textmd{d}y_{1}}{y_{1}}\frac{\textmd{d} y_{2}}{y_{2}}V_{k/\alpha}\left(\frac{x}{y_{1}}\right)P_{\alpha j/i}\left(\frac{y_{1}}{y_{2}}\right)q_{i}\left(y_{2}\right),$$ where $V_{k/\alpha}(x)$ is the quark $k$ distribution function in $\alpha$ and satisfies the normalization $\int_{0}^{1}V_{k/\alpha}(x)dx=1$. When we take ISB into consideration, the renormalization constant $Z$ should take the form $$\begin{aligned} Z_{u}=1-\langle P_{\pi^{+}}\rangle-\frac{1}{2}\langle P_{u\pi^{0}}\rangle-\langle P_{K^{+}}\rangle,\nonumber\\ Z_{d}=1-\langle P_{\pi^{-}}\rangle-\frac{1}{2}\langle P_{d\pi^{0}}\rangle-\langle P_{K^{0}}\rangle,\end{aligned}$$ where $\langle P_{\alpha}\rangle\equiv \langle P_{j\alpha/i}\rangle=\langle P_{\alpha j/i}\rangle=\int^{1}_{0}x^{n-1}P_{j\alpha/i}(x)\mathrm{d}x$ [@Suzuki:1997wv]. It is conventional to specify the momentum cutoff function at the quark-GS-boson vertex as $$g_{\mathrm{A}}\rightarrow g_{\mathrm{A}}^{\prime}\textmd{exp}\bigg{[}\frac{m^{2}_{i}-M^{2}_{j\alpha}}{4\Lambda^{2}}\bigg{]},$$ where $g_{\mathrm{A}}^{\prime}=1$, following the large $N_{c}$ argument [@Weinberg:1990xm], and $\Lambda$ is the cutoff parameter, which is determined by the experimental data of the Gottfried sum and the constituent-quark-mass inputs for the pion. Such a form factor has the correct $t$- and $u$-channel symmetry, $P_{j \alpha /i} (y) = P_{\alpha j/i} (1-y)$. Then, one can obtain the quark-distribution functions in the proton [@Suzuki:1997wv], $$\begin{aligned} u(x)&=&Z_{u}u_{0}(x)+P_{u\pi^{-}/d}\otimes d_{0}(x)+V_{u/\pi^{+}}\otimes P_{\pi^{+}d/u}\otimes u_{0}(x)+\frac{1}{2}P_{u\pi^{0}/u}\otimes u_{0}(x)\nonumber\\&+&V_{u/K^{+}}\otimes P_{K^{+}s/u}\otimes u_{0}(x)+ \frac{1}{2}V_{u/\pi^{0}}\otimes \left[P_{\pi^{0}u/u}\otimes u_{0}(x)+P_{\pi^{0}d/d}\otimes d_{0}(x)\right],\nonumber\\ d(x)&=&Z_{d}d_{0}(x)+P_{d\pi^{+}/u}\otimes u_{0}(x)+V_{d/\pi^{-}}\otimes P_{\pi^{-}u/d }\otimes d_{0}(x)+ \frac{1}{2}P_{d\pi^{0}/d}\otimes d_{0}(x)\nonumber\\&+&V_{d/K^{0}}\otimes P_{K^{0}s/d}\otimes d_{0}(x) +\frac{1}{2}V_{d/\pi^{0}}\otimes \left[P_{\pi^{0}u/u }\otimes u_{0}(x)+P_{\pi^{0}d/d}\otimes d_{0}(x)\right],\nonumber\\ \bar{u}(x)&=&V_{\bar{u}/\pi^{-}}\otimes P_{\pi^{-}u/d}\otimes d_{0}(x)+\frac{1}{2}V_{\bar{u}/\pi^{0}}\otimes \left[P_{\pi^{0}u/u}\otimes u_{0}(x)+P_{\pi^{0}d/d}\otimes d_{0}(x)\right],\nonumber\\ \bar{d}(x)&=&V_{\bar{d}/\pi^{+}}\otimes P_{\pi^{+}d/u}\otimes u_{0}(x)+\frac{1}{2}V_{\bar{d}/\pi^{0}}\otimes \left[P_{\pi^{0}u/u}\otimes u_{0}(x)+P_{\pi^{0}d/d}\otimes d_{0}(x)\right],\end{aligned}$$ where the constituent quark-distributions $u_{0}$ and $d_{0}$ are normalized to two and one, respectively. Convolution integrals are defined as $$\begin{aligned} P_{j \alpha / i}\otimes q_i &=&\int_{x}^{1}\frac{\textmd{d}y}{y}P_{j \alpha / i}\left(y\right)q_i\left(\frac{x}{y}\right),\nonumber\\ V_{k/ \alpha}\otimes P_{\alpha j/i}\otimes q_i&=&\int_{x}^{1}\frac{\textmd{d}y_{1}}{y_{1}}\int_{y_{1}}^{1}\frac{\textmd{d}y_{2}}{y_{2}}V_{k/ \alpha}\left( \frac{x}{y_{1}}\right)P_{\alpha j/i}\left(\frac{y_{1}}{y_{2}}\right)q_{i}\left(y_{2}\right).\end{aligned}$$ In addition, $V_{k/\alpha}(x)$ follows the relationship $$\begin{aligned} &&V_{u/\pi^{+}}=V_{\bar{d}/\pi^{+}}=V_{d/\pi^{-}}=V_{\bar{u}/\pi^{-}}=2V_{u/\pi^{0}} =2V_{\bar{u}/\pi^{0}}=2V_{d/\pi^{0}}=2V_{\bar{d}/\pi^{0}} =\frac{1}{2}V_{\pi},\nonumber\\ &&V_{u/K^{+}}=V_{d/K^{0}}.\end{aligned}$$ We postulate that the bare-quark distributions are isospin-symmetric between the proton and the neutron, so we can obtain the quark distributions of the neutron by interchanging $u_{0}$ and $d_{0}$. Employing the quark distributions of the chiral quark model, we get the Gottfried sum determined by the difference between the proton and the neutron structure functions, $$\begin{aligned} S_{\mathrm{G}}&=&\int^{1}_{0}\frac{\mathrm{d}x}{x}\left[F^{\mathrm{p}}_{2}(x)-F^{\mathrm{n}}_{2}(x)\right]\nonumber\\ &=&\frac{1}{9}\int_{0}^{1}\mathrm{d}x\left[4u^{\mathrm{p}}(x)+4\bar{u}^{\mathrm{p}}(x)-4u^{\mathrm{n}}(x)-4\bar{u}^{\mathrm{n}}(x)+d^{\mathrm{p}}(x)+\bar{d}^{\mathrm{p}}(x)-d^{\mathrm{n}}(x)-\bar{d}^{\mathrm{n}}(x)\right]\nonumber\\& =&\frac{1}{3}+\int^{1}_{0}\mathrm{d}x\left\{\frac{8}{9}\left[\bar{u}^{\mathrm{p}}(x)-\bar{u}^{\mathrm{n}}(x)\right]+\frac{2}{9}\left[\bar{d}^{\mathrm{p}}(x)-\bar{d}^{\mathrm{n}}(x)\right]\right\}\nonumber\\ &=&\frac{1}{3}-\frac{8}{9}\left<P_{\pi^{-}}\right>+\frac{2}{9}\left<P_{\pi^{+}}\right>+\frac{5}{18} \left(\left<P_{u\pi^{0}}\right>-\left<P_{d\pi^{0}}\right>\right).\label{gottfried}\end{aligned}$$ We assume that the ISB is entirely from the mass difference between isospin multiplets. In this paper, we adopt $(m_{u}+m_{d})/2=330$ MeV, $m_{\pi^{\pm}}=139.6$ MeV, $m_{\pi^{0}}=135$ MeV, $m_{K^{\pm}}=493.7$ MeV, and $m_{K^{0}}=497.6$ MeV. We choose two sets of the mass difference between $u$ and $d$ quarks, namely $\delta m=4$ MeV and $\delta m=8$ MeV, respectively, in order to show the dependence on this important parameter. Based on Eq. (\[gottfried\]) and the experimental data of the Gottfried sum [@Arneodo:1994sh], one can find that the appropriate value for $\Lambda_{\pi}$ is $1500$ MeV. However, one cannot determine $\Lambda_{K}$ in the same method, because $\langle P_{K}\rangle$ in the Gottfried sum is canceled out. Usually, it is assumed that $\Lambda_{K}=\Lambda_{\pi}=1500~\mathrm{MeV}$[@Suzuki:1997wv; @Szczurek:1996tp]. However, it is implied by the SU$\left(3\right)_{f}$ symmetry breaking that $\left<P_{K}\right>$ should be smaller, and, accordingly, one should adopt a smaller $\Lambda_{K}$. In this paper, we adopt a wide range of $\Lambda_{K}$ from $900$ to $1500$ MeV. In addition, the parton distributions of mesons are the parametrization GRS98 given by Gluck-Reya-Stratmann [@Gluck:1997ww], since the parametrization is more approximate to the actual value, $$\begin{aligned} V_{\pi}(x)=0.942x^{-0.501}(1+0.632\sqrt{x})(1-x)^{0.367},\nonumber\\ V_{u/K^{+}}(x)=V_{d/K^{0}}(x)=0.541(1-x)^{0.17}V_{\pi}(x).\end{aligned}$$ We should point out that, in principle, it is possible that the parton distributions of different mesons in the same multiplet are different, and this can contribute to ISB simultaneously. However, in this paper, we simply neglect this possibility, and calculations in future can be improved if we have a better understanding of the quark structure of mesons. Moreover, we have to specify constituent-quark distributions $u_{0}$ and $d_{0}$, but there is no proper parametrization of them because they are not directly related to observable quantities in experiments. In this paper, we adopt the constituent-quark-model distributions [@Hwa:1980mv] as inputs for constituent-quark distributions. For the proton, we have $$\begin{aligned} u_{0}(x)&=&\frac{2x^{c_{1}}(1-x)^{c_{1}+c_{2}+1}}{\textmd{B}[c_{1}+1,c_{1}+c_{2}+2]},\nonumber\\ d_{0}(x)&=&\frac{x^{c_{2}}(1-x)^{2c_{1}+1}}{\textmd{B}[c_{2}+1,2c_{1}+2]},\end{aligned}$$ where $\textmd{B}[i,j]$ is the Euler beta function. Such distributions satisfy the number and the momentum sum rules $$\begin{aligned} \int^{1}_{0}u_{0}(x)\textmd{d}x=2, ~~~~ \int^{1}_{0}d_{0}(x)\textmd{d}x=1,\nonumber\\ \int_{0}^{1}xu_{0}(x)\textmd{d}x+\int_{0}^{1}xd_{0}(x)\textmd{d}x=1.\end{aligned}$$ $c_{1}=0.65$ and $c_{2}=0.35$ are adopted in the calculation, following the original choice [@Hwa:1980mv; @Kua:1999yf]. We display the ISB of the valance- and sea-quark distributions in Figs. \[uvalance\], \[dvalance\], and \[sea\], respectively. It is shown that in most regions, $x\delta u_\mathrm{V}(x)>0$ and $x\delta \bar{u}(x)>0$, and on the contrary that $x\delta d_\mathrm{V}(x)<0$ and $x\delta \bar{d}(x)<0$. Our predictions that $x\delta \bar{u}(x)>0$ and $x\delta \bar{d}(x)<0$ are consistent with the MRST parametrization [@Martin:2003sk], and, moreover, the shapes of $x\delta \bar{u}(x)$ and $x\delta \bar{d}(x)$ are similar to the best phenomenological fitting results given by the MRST group. We should point out that our results are analogous to the results calculated in the framework of the meson cloudy model by Cao and Signal [@Cao:2000dj], and the shapes and magnitudes of $x\delta \bar{u}(x)$ and $x\delta \bar{d}(x)$ are similar to the results given in the framework of the radiatively generated ISB [@Gluck:2005xh], but with different signs. It can also be found that the difference between various choices of $\Lambda_K$ is minor, but the different choices of $\delta m$ can have remarkable influence on the distributions. Especially, larger $\delta m$ can lead to larger ISB, and this is concordant with our principle that ISB results from the mass difference between isospin multiplets at both hadron and parton levels. From the figures, we can see that $\delta u_\mathrm{V}(x)$ reaches a maximum value at $x\approx 0.5$, and $\delta d_\mathrm{V}(x)$ has a minimum value at $x\approx 0.4$. It should also be noted that $\delta q_\mathrm{V}(x)$ $(q=u,d)$ must have at least one zero point due to the valance-quark-normalization conditions. We should also point out that at large $x$, $\delta u_\mathrm{V}/u_\mathrm{V} \approx -\delta d_\mathrm{V}/d_\mathrm{V}$, and this implies that the magnitudes of the ISB for $u_{\mathrm{V}}$ and $d_{\mathrm{V}}$ are almost the same, but with opposite signs. Moreover, although both flavor asymmetry in the nucleon sea and the ISB between the proton and the neutron can lead to the violation of the Gottfried sum rule, the main contribution is from the flavor asymmetry in the framework of the chiral quark model. ![The ISB of the $u_{\mathrm{V}}$-quark distribution $x\delta u_{\mathrm{V}}(x)$ versus $x$ in the chiral quark model with different inputs. The red solid line is the result with $\delta m=4$ MeV and $\Lambda_K=1500$ MeV as inputs. The blue dashed line is the result with $\delta m=8$ MeV and $\Lambda_K=1500$ MeV as inputs. The green dotted line is the result with $\delta m=4$ MeV and $\Lambda_K=900$ MeV as inputs.[]{data-label="uvalance"}](uv.eps){width="95.00000%"} ![The ISB of the $d_{\mathrm{V}}$-quark distribution $x\delta d_{\mathrm{V}}(x)$ versus $x$ in the chiral quark model with different inputs. The red solid line is the result with $\delta m=4$ MeV and $\Lambda_K=1500$ MeV as inputs. The blue dashed line is the result with $\delta m=8$ MeV and $\Lambda_K=1500$ MeV as inputs. The green dotted line is the result with $\delta m=4$ MeV and $\Lambda_K=900$ MeV as inputs.[]{data-label="dvalance"}](dv.eps){width="95.00000%"} ![The ISB of the sea-quark distributions $x\delta \bar{q}(x)$ versus $x$ in the chiral quark model. The red solid line and the blue dashed line are the behaviors of $x\delta \bar{u}(x)$, with $\delta m=4$ MeV and $\delta m=8$ MeV, respectively. The green dotted line and the orange dash-dotted line are the behaviors of $x\delta \bar{d}(x)$, with $\delta m=4$ MeV and $\delta m=8$ MeV, respectively.[]{data-label="sea"}](sea.eps){width="95.00000%"} The contribution from isospin symmetry breaking to the NuTeV anomaly {#section3} ==================================================================== The measured $\sin^2\theta_\mathrm{W}$ by the NuTeV Collaboration is closely related to the Paschos-Wolfenstein (PW) ratio [@Paschos:1972kj] $$\begin{aligned} R^{-}=\frac{\left<\sigma_{\mathrm{NC}}^{\nu \mathrm{N}}\right>-\left<\sigma_{\mathrm{NC}}^{\overline{\nu}\mathrm{N}}\right>}{\left<\sigma_{\mathrm{CC}}^{\nu \mathrm{N}}\right>-\left<\sigma_{\mathrm{CC}}^{\overline{\nu}\mathrm{N}}\right>}=\frac{1}{2}-\sin^{2}\theta_{\mathrm{W}},\label{pw}\end{aligned}$$ where $\left<\sigma_{\mathrm{NC}}^{\nu \mathrm{N}}\right>$ is the neutral-current-inclusive cross section for a neutrino on an isoscalar target. If we take the ISB between the proton and the neutron into account, we obtain $$\begin{aligned} R^{-}_{\mathrm{N}}=\frac{\left<\sigma_{\mathrm{NC}}^{\nu \mathrm{N}}\right>-\left<\sigma_{\mathrm{NC}}^{\overline{\nu}\mathrm{N}}\right>}{\left<\sigma_{\mathrm{CC}}^{\nu \mathrm{N}}\right>-\left<\sigma_{\mathrm{CC}}^{\overline{\nu}\mathrm{N}}\right>} =R^{-}+\delta R^{\mathrm{ISB}}_{\mathrm{PW}},\label{mpw}\end{aligned}$$ where $\delta R^{\mathrm{ISB}}_{\mathrm{PW}}$ is the correction from the ISB to the PW ratio and takes the form $$\begin{aligned} \delta R^{\mathrm{ISB}}_{\mathrm{PW}}=\bigg{(}\frac{1}{2}-\frac{7}{6}\sin^{2}\theta_{\mathrm{W}}\bigg{)}\frac{\int^{1}_{0}x\bigg{[}\delta u_{\mathrm{V}}(x)-\delta d_{\mathrm{V}}(x)\bigg{]}\mathrm{d}x} {\int^{1}_{0}x\bigg{[}u_{\mathrm{V}}(x)+d_{\mathrm{V}}(x)\bigg{]}\mathrm{d}x},\label{rs}\end{aligned}$$ with $u_{\mathrm{V}}(x)$ and $d_{\mathrm{V}}(x)$ standing for valance-quark distributions of the proton. We show the renormalization constant $Z$, the total momentum fraction of valance quarks $Q_{\mathrm{V}}=\int^{1}_{0}x\left[u_{\mathrm{V}}(x)+d_{\mathrm{V}}(x)\right]\mathrm{d}x$, and the correction of the ISB to the NuTeV anomaly $\Delta R^{\mathrm{ISB}}_{\mathrm{PW}}$, with different $\delta m$ and $\Lambda_{K}$ as inputs in Table \[ISB\]. It can be found that the ISB correction is of the order of magnitude of $10^{-3}$ and is more significant with a larger $\delta m$ or $\Lambda_K$. Our result is consistent with the range $-0.009\leq\Delta R_{\mathrm{PW}}^{\mathrm{ISB}}\leq+0.007$, which is derived based on the parametrization given by the MRST group [@Martin:2003sk]. We should stress that the correction is remarkable, since the NuTeV anomaly can be totally removed if $\Delta R_{\mathrm{PW}}=-0.005$, and, consequently, we should pay special attention to ISB in such problem. It is also worthwhile to point out that the correction is in an opposite direction to remove the NuTeV anomaly in the chiral quark model. Such a conclusion is the same as that given in the baryon-meson fluctuation model [@Ding:2006ud], but the value is one or 2 orders of magnitude larger. Our result of the ISB correction to the NuTeV anomaly differs from the results in Refs. [@Londergan:2003ij; @Gluck:2005xh]. ------------------ --------------------- ---------- ---------- ------------------ ----------------------------------------- $\delta m$ (MeV) $\Lambda_{K}$ (MeV) $Z_{u}$ $Z_{d}$ $Q_{\mathrm{V}}$ $\Delta R^{\mathrm{ISB}}_{\mathrm{PW}}$ $4$ $900$ $0.7497$ $0.7463$ $0.8451$ $0.0008$ $4$ $1200$ $0.7220$ $0.7185$ $0.8222$ $0.0008$ $4$ $1500$ $0.6932$ $0.6896$ $0.7985$ $0.0009$ $8$ $900$ $0.7515$ $0.7444$ $0.8455$ $0.0016$ $8$ $1200$ $0.7239$ $0.7165$ $0.8227$ $0.0017$ $8$ $1500$ $0.6953$ $0.6874$ $0.7990$ $0.0019$ ------------------ --------------------- ---------- ---------- ------------------ ----------------------------------------- : The renormalization constant, the total momentum fraction of valance quarks, and the correction of the ISB to the NuTeV anomaly in the chiral quark model.[]{data-label="ISB"} summary {#section4} ======= In this paper, we discuss the ISB of the valance-quark and the sea-quark distributions between the proton and the neutron in the framework of the chiral quark model. We assume that isospin symmetry breaking is the result of mass differences between isospin multiplets. Then, we analyze the effects of isospin symmetry breaking on the Gottfried sum rule and the NuTeV anomaly. We show that, although both flavor asymmetry in the nucleon sea and the ISB between the proton and the neutron can lead to the violation of the Gottfried sum rule, the main contribution is from the flavor asymmetry in the framework of the chiral quark model. It is remarkable that our results of ISB for both the valence-quark and sea-quark distributions are consistent with the MRST parametrization of the ISB of valance- and sea-quark distributions. Moreover, we find that the correction to the NuTeV anomaly is in an opposite direction, so the NuTeV anomaly cannot be removed by isospin symmetry breaking in the chiral quark model. However, its influence is remarkable and should be taken into careful consideration. Therefore, it is important to do more precision experiments and careful theoretical studies on isospin symmetry breaking. Acknowledgement {#acknowledgement .unnumbered} =============== This work is supported by the National Natural Science Foundation of China (grant numbers. 10721063, 10975003, 11035003), and National Fund for Fostering Talents of Basic Science (grant numbers. J0630311, J0730316). It is also supported by Principal Fund for Undergraduate Research at Peking University. [99]{} E. M. Henley and G. A. Miller, [*in Mesons in Nuclei*]{}, edited by M. Rho and D. H. Wilkinson (North-Holland Publishing Co., Amsterdam, 1979), Vol. I, p. 405. G. A. Miller, B. M. K. Nefkens, and I. Slaus, Phys. Rept.  [**194**]{}, 1 (1990). B.-Q. Ma, Phys. Lett.  B [**274**]{}, 111 (1992). P. Amaudruz [*et al.*]{} \[New Muon Collaboration\], Phys. Rev. Lett.  [**66**]{}, 2712 (1991). M. Arneodo [*et al.*]{} \[New Muon Collaboration\], Phys. Rev.  D [**50**]{}, R1 (1994). B.-Q. Ma, A. Schafer, and W. Greiner, Phys. Rev.  D [**47**]{}, 51 (1993); J. Phys. G [**20**]{}, 719 (1994). G. P. Zeller [*et al.*]{} \[NuTeV Collaboration\], Phys. Rev. Lett.  [**88**]{}, 091802 (2002); Phys. Rev.  D [**65**]{}, 111103 (2002). S. Davidson, S. Forte, P. Gambino, N. Rius, and A. Strumia, JHEP [**0202**]{}, 037 (2002); A. Kurylov, M. J. Ramsey-Musolf, and S. Su, Nucl. Phys.  B [**667**]{}, 321 (2003);\ G. J. Ding and M. L. Yan, Phys. Rev.  D [**78**]{}, 075015 (2008). S. Kovalenko, I. Schmidt, and J. J. Yang, Phys. Lett.  B [**546**]{}, 68 (2002);\ S. J. Brodsky, I. Schmidt, and J. J. Yang, Phys. Rev.  D [**70**]{}, 116003 (2004). S. Kumano, Phys. Rev.  D [**66**]{}, 111301 (2002). F. G. Cao and A. I. Signal, Phys. Lett.  B [**559**]{}, 229 (2003). Y. Ding and B.-Q. Ma, Phys. Lett.  B [**590**]{}, 216 (2004). Y. Ding, R.-G. Xu, and B.-Q. Ma, Phys. Lett.  B [**607**]{}, 101 (2005); Phys. Rev.  D [**71**]{}, 094014 (2005). E. Sather, Phys. Lett.  B [**274**]{}, 433 (1992). E. N. Rodionov, A. W. Thomas, and J. T. Londergan, Mod. Phys. Lett.  A [**9**]{}, 1799 (1994). R. M. Davidson and M. Burkardt, Phys. Lett.  B [**403**]{}, 134 (1997). J. T. Londergan and A. W. Thomas, Phys. Rev.  D [**67**]{}, 111901 (2003); Phys. Lett.  B [**558**]{}, 132 (2003). F. G. Cao and A. I. Signal, Phys. Rev.  C [**62**]{}, 015203 (2000). M. Gluck, P. Jimenez-Delgado, and E. Reya, Phys. Rev. Lett.  [**95**]{}, 022002 (2005). See, also, A. D. Martin, R. G. Roberts, W. J. Stirling and R. S. Thorne, Eur. Phys. J.  C [**39**]{}, 155 (2005). Y. Ding and B.-Q. Ma, Phys. Rev.  D [**73**]{}, 054018 (2006). A. D. Martin, R. G. Roberts, W. J. Stirling, and R. S. Thorne, Eur. Phys. J.  C [**35**]{}, 325 (2004). S. Weinberg, Physica A [**96**]{}, 327 (1979). A. Manohar and H. Georgi, Nucl. Phys.  B [**234**]{}, 189 (1984). E. J. Eichten, I. Hinchliffe, and C. Quigg, Phys. Rev.  D [**45**]{}, 2269 (1992). M. Wakamatsu, Phys. Lett.  B [**269**]{}, 394 (1991). J. Ashman [*et al.*]{} \[European Muon Collaboration\], Phys. Lett.  B [**206**]{}, 364 (1988); Nucl. Phys.  B [**328**]{}, 1 (1989). T. P. Cheng and L. F. Li, Phys. Rev. Lett.  [**74**]{}, 2872 (1995). X. Song, J. S. McCarthy, and H. J. Weber, Phys. Rev.  D [**55**]{}, 2624 (1997). K. Suzuki and W. Weise, Nucl. Phys.  A [**634**]{}, 141 (1998). S. Weinberg, Phys. Rev. Lett.  [**65**]{}, 1181 (1990). A. Szczurek, A. J. Buchmann, and A. Faessler, J. Phys. G [**22**]{}, 1741 (1996). M. Gluck, E. Reya, and M. Stratmann, Eur. Phys. J.  C [**2**]{}, 159 (1998). R. C. Hwa and M. S. Zahir, Phys. Rev.  D [**23**]{}, 2539 (1981). H. W. Kua, L. C. Kwek, and C. H. Oh, Phys. Rev.  D [**59**]{}, 074025 (1999). E. A. Paschos and L. Wolfenstein, Phys. Rev.  D [**7**]{}, 91 (1973). [^1]: Corresponding author. Email address: `[email protected]` [^2]: As mass difference between isospin multiplets, especially that between $u$ and $d$ quarks, is not entirely due to charge difference, we refer such effect as Isospin Symmetry Breaking (ISB) instead of Charge Symmetry Breaking (CSB) as called in some papers.

--- abstract: 'Waveform design is a pivotal component of [*the fully adaptive radar*]{} construct. In this paper we consider waveform design for radar space time adaptive processing (STAP), accounting for the waveform dependence of the clutter correlation matrix. Due to this dependence, in general, the joint problem of receiver filter optimization and radar waveform design becomes an intractable, non-convex optimization problem, Nevertheless, it is however shown to be individually convex either in the filter or in the waveform variables. We derive constrained versions of: a) the alternating minimization algorithm, b) proximal alternating minimization, and c) the constant modulus alternating minimization, which, at each step, iteratively optimizes either the STAP filter or the waveform independently. A fast and slow time model permits waveform design in radar STAP but the primary bottleneck is the computational complexity of the algorithms.' author: - 'Pawan Setlur, *Member, IEEE*, Muralidhar Rangaswamy, *Fellow, IEEE*[^1] [^2] [^3]' bibliography: - 'refs.bib' title: Joint Filter and Waveform Design for Radar STAP in Signal Dependent Interference --- [AFRL, Sensors Directorate Tech. Report, 2014: Approved for Public release]{} Waveform design, waveform scheduling, space time adaptive radar, Capon beamformer, constant modulus, convex optimization, alternating minimization, regularization, proximal algorithms. Introduction ============ objective of this report is to address waveform design in radar space time adaptive processing (STAP) [@klemm2002; @ward1994; @guerci2003; @Brennan1973]. An air-borne radar is assumed with an array of sensor elements observing a moving target on the ground. We will assume that the waveform design and scheduling are performed over one CPI rather than on an individual pulse repetition interval (PRI).To facilitate waveform design, we develop a STAP model considering the fast time samples along with the slow time processing. This is different from traditional STAP which generally considers the data after matched filtering [@klemm2002; @ward1994]. Nonetheless STAP research efforts have been proposed which consider inclusion of fast time samples in space time processing, see for example [@klemm2002; @madurasinghe2006; @seliktar2006] and references therein. In line with traditional STAP, we formulate the waveform design, as an minimum variance distortion-less response (MVDR) type optimization [@capon1969]. As we will see in the sequel, inclusion of the waveform increases the dimensionality of the correlation matrix. Classical Radar STAP is computationally expensive but the waveform adaptive STAP increases the complexity by several orders of magnitude. Therefore, benefits of waveform design in STAP come at the expense of increased computational complexity. The noise, clutter, and interference are modeled stochastically and are assumed to be mutually uncorrelated. Endemic to airborne STAP, clutter is persistent in most range gates resulting from ground reflections. The clutter correlation matrix is a function of the waveform causing the joint reliever filter and waveform optimization to be non-convex with no closed form solution. However, it is analytically shown here that the STAP MVDR objective is convex with respect to (w.r.t.) the receiver filter for a fixed but arbitrary waveform, and vice versa. Therefore, alternating minimization approaches arise as natural candidate solutions. As such, alternating minimization itself has a rich history in the optimization literature, possibly motivated directly from the works in [@Powell1964; @Powell1973; @Zangwill1967; @Ortega1970], with some not so recent seminal contributions [@Luo1992; @Auslender1992; @Bertsekas1999; @Grippo2000] and recent contributions (not exhaustive) [@Attouch2010; @Beck2013]. Other celebrated algorithms such as the Arimoto-Blahut algorithm to calculate channel capacity, and the expectation-maximization (EM) algorithm are all examples of the alternating minimization. Here we address the joint optimization problem via a constrained alternating minimization approach, which has the favorable property of monotonicity in successive objective evaluations. Convergence, performance guarantees and other properties pertinent to this algorithm are further addressed. Full rank correlation matrices are required in implementing the constrained alternating minimization approach. In practice, radar STAP contends with rank deficient correlation matrices due to lack of homogeneous training data. In this case, the constrained alternating minimization approach is not implementable. To addresses this issue, we consider regularization of the STAP objective via strongly convex functions resulting in the constrained proximal alternating minimization [@Setlurasilomar2014]. Proximal algorithms, originally proposed by [@Martinet1970; @Rockafeller1973] are well suited candidate techniques for constrained, large scale optimization [@Attouch2010; @Bertsekas1994; @Rockafeller1976; @Combettes2011; @Parikh2013], applicable readily to our waveform adaptive STAP problem. In fact, as we will see subsequently the constrained proximal alternating minimization results in diagonal loading solutions, and for optimization-specific interpretations, the load factors may be related to the Lipschitz constants (w.r.t. the gradient). [**Signal dependent interference: Chicken or the Egg?**]{} The fundamental problem in practical radar waveform design is analogous to the chicken or the egg problem. Signal dependent interference, i.e., clutter, can only be perfectly characterized by transmitting a signal. Herein lies the central problem. The estimated clutter properties could therefore be dependent on what was transmitted in the first place. This is especially true for frequency selective and dispersive clutter responses frequently encountered in radar operations, for example, urban terrain. Therefore, any claim of optimality is myopic. Sadly the same problem would also persist when the target impulse responses are used to shape the waveform. Unfortunately, and as famously stated by Woodward [@woodward1952information; @woodward1953theory], “…what to transmit remains substantially unanswered” [@woodward1953probability; @Benedetto2009]. We will assume like other works in the signal dependent interference waveform design [@Delong1967; @Delong1969; @Delong1970; @Kay2007; @vaidyanathan2009; @Pillai2000; @Demaio2012; @Demaio2013; @Hongbin2014], that the clutter response is known [*a priori*]{}. To a certain extent, this may be obtained via a combination of, either previous radar transmission [@cochran2009waveform], or assuming that the topography is known from ground elevation maps, synthetic aperture radar imagery [@SetlurJSTSP2014], or access to knowledge aided databases as in the DARPA’s KASSPER program [@Guerci2006]. [**Literature:**]{} The signal dependent interference waveform design problem has had a rich history [@middleton1996; @van2004detection]. Iterative approaches but not limited alternating minimization type techniques have been the subject of work in [@Delong1967; @Delong1969; @Delong1970; @Kay2007; @vaidyanathan2009; @Pillai2000; @Stoica2012; @Demaio2012; @Demaio2013; @Hongbin2014] for SISO, MIMO radars but never in radar STAP. Waveform design for STAP without considering the signal dependent interference clutter was addressed in [@pattonstap2012], where the authors premise is that the degrees of freedom from the waveform could be used in suppressing the interference and noise, while the degrees of freedom from the filter could be used exclusively for suppressing the clutter. A joint STAP waveform and STAP filter design was never considered. Further, their premise is erroneous for the following several reasons. For any radar application, but especially in STAP, obtaining range cells which are interference free or clutter free is impossible. Nonetheless assuming this was possible, then, the weight vector for exclusive clutter suppression uses [*the inverse of the clutter interference correlation matrix*]{} only, and not, as stated in [@pattonstap2012], the inverse of the (clutter+noise+interference) correlation matrix. Furthermore such a detector may have disastrous consequences, because control in the false alarm rates becomes impossible due to the self induced coloring on other range cells which are contaminated by the clutter plus interference plus noise. Other contributions in waveform design and waveform scheduling for extended targets in radar using information theoretic measures, tracking etc can be seen in [@setlurspie2012; @setlurssp2012; @bell1993information; @bell1988thesis; @Lesham2007; @romero2008information], [@paper:AussiesSensorScheduleRadar; @sira2006waveform; @sira2009waveform; @Kershaw:2004; @kershaw1997waveform; @Li2008], and the references therein. We outline some of the contributions for the signal dependent interference problem which have thus far appeared in the literature. [*Approaches different from Alternating minimization*]{}: In [@Delong1967; @Kay2007; @Pillai2000; @Stoica2012] a single sensor radar was assumed. In [@Delong1967], the authors used the symmetry property of the cross-ambiguity function to design an iterative algorithm for the signal dependent interference problem. Their algorithm cannot be modified easily for the multi-sensor framework and when noise is in general colored. The problem was addressed from a detection perspective in [@Kay2007], and lead to a waterfilling [@bell1993information] type solution. A similar waterfilling type metric albeit in the discrete time domain was obtained in [@Stoica2012], where the authors also imposed constant modulus and peak to average power ratio (PAPR) waveform constraints. An iterative algorithm was derived in [@Pillai2000], where monotonic increase in SINR was not guaranteed, and was shown that waveform could always be chosen as minimum phase. [*Alternating minimization type approaches*]{}: In [@vaidyanathan2009], a MIMO sensor framework was employed, convergence was not addressed, convexity was not proven, and no practical waveform constraints were imposed on the design. See also in this report, Section III, paragraph following Rem. \[limitpointremk\] where some of the conclusions drawn in [@vaidyanathan2009] are further discussed. Alternating minimization was used in [@Demaio2012; @Demaio2013; @Hongbin2014] but for reasons unknown, was called as sequential optimization. In [@Demaio2012; @Demaio2013], a SISO model advocating joint filter and radar code design (after matched filtering) was employed. Analysis of the convexity of the objective in the individual filter or radar code was never shown. Convergence in iterates was not proven formally, neither was it shown via simulations. The constant modulus constraint was not invoked directly but through a similarity constraint. In [@Hongbin2014], the authors used a MIMO radar framework, and relaxation techniques were employed in their iterative algorithm. Neither convergence nor convexity was demonstrated analytically. Constant modulus constraint and similarity constraints were enforced separately in the waveform design. [**Notation:**]{} The variable $N$ is used interchangeably with the number of the fast time samples, as well as, the conventional dimension of arbitrary real or complex (sub)spaces. Its meaning is readily interpreted from context. The symbol $|| \cdot||$ always denotes the $l_2$ norm. Vectors are always lowercase bold, matrices are bold uppercase, $\lambda$ is typically reserved for eigenvalues (with $\lambda_o$ being an exception it used for the spatial frequency, defined later) and $\gamma$ is strictly reserved for the Lagrange multipliers ($\gamma_{pq}$ is an exception used for the radar cross section of the $p$-th scatterer in the $q$-th clutter patch). Solutions to the optimization are denoted as $(\cdot)_o$, i.e. the subscript $o$. the complex conjugate is denoted with $(\cdot)^{\ast}$. The set of reals, complex numbers, and natural numbers are denoted as $\mathbbm{R},\mathbbm{C},\mathbbm{N}$, respectively. Other symbols are defined upon first use and are standard in the literature. [**Organization:**]{} The STAP fast time-slow time model is delineated in Section II, and in Section III, the filter and waveform optimization is derived. Some preliminary simulations are presented in section IV and the resulting conclusions are drawn in Section V. STAP Model ========== The radar consists of a calibrated air-borne linear array, comprising $M$ sensor elements, each having an identical antenna pattern. Without loss of generality, assume that the first sensor in the array is the phase center, and acts as both a transmitter and receiver, the rest of the elements are purely receivers. The first sensor is located at $\mathbf{x_r}\in\mathbbm{R}^3$ and the ground based point target at $\mathbf{x_t}\in\mathbbm{R}^3$. The radar transmits the burst of pulses: $$\label{eq1} u(t)=\sum\limits_{l=1}^{L}s(t-lT_p)\exp(j2\pi f_o(t-lT_p)),t\in[0,T)$$ where, $f_o$ is the carrier frequency, and $T_p=1/f_p$ is the inverse of the pulse repetition frequency, $f_p$. The pulse width and bandwidth are denoted as $T$, $B$, respectively. The coherent processing interval (CPI) consists of $L$ pulses, each of width equal to $T$. The geometry of the scene is shown in Fig. 1, where $\theta_t$ and $\phi_t$ denote the azimuth and elevation. The radar and target are both assumed to be moving. For the time being, we ignore the noise, clutter and interference and assume a non-fluctuating target. Then the desired target’s received signal for the $l$-th pulse, and at the $m$-th sensor element is given by $$\label{eq2} s_{ml}(t)=\rho_t s(t-lT_p-\tau_m)e^{(j2\pi (f_o+f_{dm}) (t-lT_p-\tau_m))}$$ where the target’s observed Doppler shift is denoted as $f_{dm}$, and its complex back-scattering coefficient as $\rho_t$. Assume that the array is along the local $x$ axis as shown in Fig. 1. Then, the coordinates of the $m$-th element is given by $\mathbf{x_t}+m\mathbf{d},\mathbf{d}:=[d,0,0]^T,m=0,1,2\ldots,M-1$, where $d$ is the inter-element spacing. The delay $\tau_m$ could be re-written as $$\begin{aligned} &\tau_m=||\mathbf{x_r}-\mathbf{x_t}||/c+||\mathbf{x_r}+m\mathbf{d}-\mathbf{x_t}||/c \nonumber \\ &=\dfrac{||\mathbf{x_r}-\mathbf{x_t}||}{c}+\dfrac{||\mathbf{x_r}-\mathbf{x_t}||}{c} \sqrt{1+\frac{||m\mathbf{d}||^2}{||\mathbf{x_r}-\mathbf{x_t}||^2}+\frac{2m\mathbf{d}^T(\mathbf{x_r}-\mathbf{x_t})}{||\mathbf{x_r}-\mathbf{x_t}||^2}} \nonumber \\ &\overset{(a)}{\equiv}\dfrac{||\mathbf{x_r}-\mathbf{x_t}||}{c}+\dfrac{||\mathbf{x_r}-\mathbf{x_t}||}{c} \left(1+\frac{m\mathbf{d}^T(\mathbf{x_r}-\mathbf{x_t})}{||\mathbf{x_r}-\mathbf{x_t}||^2} \right) \label{eq3} \\ &=2\dfrac{||\mathbf{x_r}-\mathbf{x_t}||}{c}+\frac{m\mathbf{d}^T(\mathbf{x_r}-\mathbf{x_t})}{c||\mathbf{x_r}-\mathbf{x_t}||}, \label{eq4}\end{aligned}$$ where in approximation (a), the term $\propto||m\mathbf{d}||^2$ was ignored, i.e. it is assumed that $d/|| \mathbf{x_r}-\mathbf{x_t}||<<1$, and then a binomial approximation was employed. From geometric manipulations, we also have: $$\frac{\mathbf{x_r}-\mathbf{x_t}}{||\mathbf{x_r}-\mathbf{x_t}||}=[\sin(\phi_t)\sin(\theta_t),\sin(\phi_t)\cos(\theta_t),\cos(\phi_t)]^T.$$ Using the above equation in , the delay $\tau_m,m=0,1,\ldots,M-1$ can be rewritten as $$\label{eq5} \tau_m=2\dfrac{||\mathbf{x_r}-\mathbf{x_t}||}{c}+\dfrac{md\sin(\phi_t)\sin(\theta_t)}{c}.$$ The Doppler shift, i.e. $f_{dm}$ is computed as $$\begin{aligned} \label{eq6} &f_{dm}=2f_o\dfrac{(\mathbf{\dot{x}_r}-\mathbf{\dot{x}_t})^T(\mathbf{x_r}-\mathbf{x_t})}{c||\mathbf{x_r}-\mathbf{x_t}||} \\ &+ f_o\dfrac{m\mathbf{d}^T}{c}\left[\dfrac{\mathbf{\dot{x}_r}-\mathbf{\dot{x}_t}}{||\mathbf{x_r}-\mathbf{x_t}||^2} -\dfrac{(\mathbf{x_r}-\mathbf{x_t})(\mathbf{\dot{x}_r}-\mathbf{\dot{x}_t})^T(\mathbf{x_r}-\mathbf{x_t})}{\|| \mathbf{x_r}-\mathbf{x_t}||^3} \right] \nonumber\end{aligned}$$ where $\mathbf{\dot{x}_{(\cdot)}}$ is the vector differential of $\mathbf{x_{(\cdot)}}$ w.r.t. time. In practice $d$ is a fraction of the wavelength, and assuming that $d/|| \mathbf{x_r}-\mathbf{x_t}||<<1$ we approximate the second term in as $0$. The Doppler shift is no longer a function of the sensor index, $m$, and is rewritten as $$\label{eq7} f_{dm}=f_d=2f_o\dfrac{(\mathbf{\dot{x}_r}-\mathbf{\dot{x}_t})^T(\mathbf{x_r}-\mathbf{x_t})}{c||\mathbf{x_r}-\mathbf{x_t}||}$$ Vector signal model ------------------- Let $s(t)$ be sampled discretely resulting in $N$ discrete time samples. Consider for now the single range gate corresponding to the time delay $\tau_t$. After a suitable alignment to a common local time (or range) reference, and invoking some standard assumptions, see also , the radar returns in $l$-th PRI written as a vector defined as $\mathbf{y}_{\mathbi{l}}\in\mathbbm{C}^{NM}$, is given by $$\begin{aligned} &\mathbf{y}_{\mathbi{l}}=\rho_t\mathbf{s}\otimes \mathbf{a}(\theta_t,\phi_t)\exp(-j2\pi f_d(l-1)T_p) \label{eq9} \\ &\mathbf{a}(\theta_t,\phi_t):=[1,e^{-j2\pi\vartheta},\ldots,e^{-j2\pi (M-1)\vartheta}]^T \in\mathbbm{C}^M \nonumber\end{aligned}$$ where $\mathbf{s}:=[s_1,s_2,\ldots,s_N]^T \in\mathbbm{C}^N $ and $\vartheta:=d\sin(\theta_t)\sin(\phi_t)/\lambda_o$ is defined as the spatial frequency. Further it is noted that in , the constant phase terms have been absorbed into $\rho_t$. Considering the $L$ pulses together, i.e. concatenating the desired target’s response for the entire CPI in a tall vector $\mathbf{y}$, is defined as $$\begin{aligned} &\mathbf{y}\in\mathbbm{C}^{NML}=[\mathbf{y_0}^T,\mathbf{y_1}^T,\ldots,\mathbf{y}_{\mathbi{L-1}}^T]^T =\rho_t \mathbf{v}(f_d) \otimes\mathbf{s}\otimes \mathbf{a}(\theta_t,\phi_t)\nonumber \\ &\mathbf{v}(f_d):=[1,e^{-j2\pi f_dT_p},\ldots,e^{-j2\pi f_d(L-1)T_p}]^T \label{eq10}.\end{aligned}$$ The vector $\mathbf{y}$ consists of both the spatial and the temporal steering vectors as in classical STAP, as well as the waveform dependency, via waveform vector $\mathbf{s}$. Due to inclusion of the fast time samples in the waveform $\mathbf{s}$, the STAP data cube is modified to reflect this change, and is depicted in Fig. \[fig2\]. At the considered range gate, the measured snapshot vector consists of the target returns and the undesired returns, i.e. clutter returns, interference and noise. The contaminated snapshot at the considered range gate is then given by $$\begin{aligned} \mathbf{\tilde{y}}=\mathbf{y}+\mathbf{y_i}+\mathbf{y_c}+\mathbf{y_n} =\mathbf{y}+\mathbf{y_{u}} \label{eq.11} \end{aligned}$$ where $\mathbf{y_i,y_c,y_n}$ are the contributions from the interference, clutter and noise, respectively, and are assumed to be statistically uncorrelated with one another. The contribution of the undesired returns are treated in detail, starting with the noise as it is the simplest. [**Noise**]{}: The noise is assumed to be zero mean, identically distributed across the sensors, across pulses, and in the fast time samples. The correlation matrix of $\mathbf{y_n}$ is denoted as $ \mathbf{R_n}\in\mathbbm{C}^{NML\times NML}$. [**Interference**]{}: The interference consists of jammers and other intentional / un-intentional sources which may be ground based, air-borne or both. Let us assume that there are $K$ interference sources. Further, since nothing is known about the jammers waveform characteristics, the waveform itself is assumed to be a stationary zero mean random process. Consider the $k$-th interference source in the $l$-th PRI, and at spatial co-ordinates $(\theta_k,\phi_k)$. Its corresponding snapshot contribution is modeled as, $$\mathbf{y}_{kl}=\boldsymbol{\alpha_{kl}}\otimes\mathbf{a}(\theta_k,\phi_k),k=1,2,\ldots,K,l=0,1,\ldots,L-1$$ where $\boldsymbol{\alpha_{kl}}=[\alpha_{kl}(0),\alpha_{kl}(1),\ldots,\alpha_{kl}(N-1)]^T\in\mathbbm{C}^{N}$ is the random discrete segment of the jammer waveform, as seen by the radar in the $l$-th PRI. Stacking $\mathbf{y}_{kl}$ for a fixed $k$ as a tall vector, we have $$\begin{aligned} \label{eq12} \mathbf{y_k}&=\boldsymbol{\alpha_k}\otimes\mathbf{a}(\theta_k,\phi_k) =[\mathbf{y}_{ko}^T,\mathbf{y}_{k1}^T,\ldots,\mathbf{y}_{kL-1}^T]^T \in\mathbbm{C}^{NML} \nonumber\\ \boldsymbol{\alpha_k}:&=[\boldsymbol{\alpha_{k0}}^T,\boldsymbol{\alpha_{k1}}^T,\ldots,\boldsymbol{\alpha_{kL-1}}^T]^T \in\mathbbm{C}^{NL} \end{aligned}$$ Using the Kronecker mixed product property, (see for e.g. [@horn1994]), the correlation matrix of $\mathbf{y}_k$ is expressed as $\mathbbm{E}\{\mathbf{y}_k\mathbf{y}_k^H \}=\mathbf{R}_{\boldsymbol{\alpha}}^k\otimes \mathbf{a}(\theta_k,\phi_k)\mathbf{a}(\theta_k,\phi_k)^H$ where, $\mathbbm{E}\{ \boldsymbol{\alpha_k} \boldsymbol{\alpha_k}^H\}:=\mathbf{R}_{\boldsymbol{\alpha}}^k$. For $K$ mutually uncorrelated interferers, the correlation matrix is $\mathbf{R_i}=\sum\limits_{k=1}^K\mathbbm{E}\{\mathbf{y}_k\mathbf{y}_k^H \}=\sum\limits_{k=1}^K \mathbf{R}_{\boldsymbol{\alpha}}^k\otimes \mathbf{a}(\theta_k,\phi_k)\mathbf{a}(\theta_k,\phi_k)^H=\sum\limits_{k=1}^K (\mathbf{I}_{NL}\otimes\mathbf{a}(\theta_k,\phi_k))\mathbf{R}_{\boldsymbol{\alpha}}^k(\mathbf{I}_{NL}\otimes\mathbf{a}(\theta_k,\phi_k)^H) $, and is simplified as $$\begin{aligned} \label{eq13} \mathbf{R_i}&= \mathbf{A}(\theta,\phi)\mathbf{R}_{\boldsymbol{\alpha}}\mathbf{A}(\theta,\phi)^H\end{aligned}$$ where $\mathbf{R}_{\boldsymbol{\alpha}}:=\mbox{Diag}\{\mathbf{R}_{\boldsymbol{\alpha}}^1,\mathbf{R}_{\boldsymbol{\alpha}}^2,\ldots,\mathbf{R}_{\boldsymbol{\alpha}}^K\} \in\mathbbm{C}^{NMLK\times NMLK}$ and $\mathbf{A}(\theta,\phi)\in\mathbbm{C}^{NML\times NMLK}\\ =[\mathbf{I}_{NL}\otimes \mathbf{a}(\theta_1,\phi_1),\mathbf{I}_{NL}\otimes \mathbf{a}(\theta_2,\phi_2),\ldots,\mathbf{I}_{NL}\otimes \mathbf{a}(\theta_K,\phi_K)]$, here $\mathbf{I}_{NL}$ the identity matrix of size $NL\times NL$, and $\mbox{Diag}\{ \cdot,\cdot,\ldots,\cdot\}$ the matrix diagonal operator which converts the matrix arguments into a bigger diagonal matrix. For example, $\mbox{Diag}\{\mathbf{A,B,C}\}=\left[ \begin{smallmatrix} \mathbf{A}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{B}&\mathbf{0} \\ \mathbf{0}&\mathbf{0}&\mathbf{C} \end{smallmatrix} \right].$ [**Clutter**]{}: The ground is a major source of clutter in air-borne radar applications and is persistent in all range gates upto the gate corresponding to the platform horizon. Other sources of clutter surely exist, such as buildings, trees, as well as other un-interesting targets, which are ignored. We therefore consider only ground clutter and treat it stochastically. Let us assume that there are $Q$ clutter patches indexed by parameter $q$. Each of these clutter patches are comprised of say $P$ scatterers. The radar return from the $p$-th scatterer in the $q$-th clutter patch is given by $$\gamma_{pq}\mathbf{v}(fc_{pq})\otimes\mathbf{s} \otimes a(\theta_{pq},\phi_{pq})$$ where $\gamma_{pq}$ is its random complex reflectivity, $fc_{pq}$ is the Doppler shift observed from the $p$-th scatterer in the $q$-th clutter patch, and $\theta_{pq},\phi_{pq}$ are the azimuth and elevation angles of this scatterer.The Doppler $fc_{pq}$ is given by, $$\label{eq14} fc_{pq}:=\dfrac{2f_o\mathbf{\dot{x}_r}^T (\mathbf{x_r}-\mathbf{x_{pq} })}{c||\mathbf{x_r}-\mathbf{x_{pq} }||}.$$ where $\mathbf{x_{pq}}$ is the location of the $p$-th scatter in the $q$-th clutter patch. Since the clutter patch is stationary, the Doppler is purely from the motion of the aircraft as seen in . The contribution from the $q$-th clutter patch to the received signal is given by $$\label{eq15} \mathbf{y}_q=\sum \limits_{p=1}^P \gamma_{pq} \mathbf{v}(fc_{pq} )\otimes\mathbf{s} \otimes a(\theta_{pq},\phi_{pq} ),$$ with corresponding correlation matrix $$\begin{aligned} \label{cluteq} &\mathbf{R}_{\boldsymbol{\gamma}}^q:=\mathbf{B_q}\mathbf{R}_{\boldsymbol{\gamma}}^{pq} \mathbf{B_q}^H\end{aligned}$$ where, $\mathbf{B_q} =[\mathbf{v}(fc_{1q})\otimes\mathbf{s} \otimes \mathbf{a}(\theta_{1q},\phi_{1q}), \mathbf{v}(fc_{2q})\otimes\mathbf{s} \otimes \mathbf{a}(\theta_{2q},\phi_{2q})\ldots,\mathbf{v}(fc_{Pq})\otimes\mathbf{s} \otimes \mathbf{a}(\theta_{Pq},\phi_{Pq})] \in\mathbbm{C}^{NML\times P}$ and $\mathbf{R}_{\boldsymbol{\gamma}}^{pq}$ is the correlation matrix of the random vector, $[\gamma_{1q},\gamma_{2q},\ldots,\gamma_{Pq}]^T$. It is readily shown that the matrix $\mathbf{B_q}$ could be simplified as, $\mathbf{B_q}:=\mathbf{\breve{B}_q}(\mathbf{I}_P\otimes\mathbf{s})$, where $\mathbf{\breve{B}_q}:=[\mathbf{v}(fc_{1q})\otimes\mathbf{A_{1q}},\mathbf{v}(fc_{2q})\otimes\mathbf{A_{2q}},\ldots, \mathbf{v}(fc_{Pq})\otimes\mathbf{A_{Pq}}]\in \mathbbm{C}^{NML\times PN}$, and the structure of the matrix $\mathbf{A_{pq}}\in\mathbbm{C}^{NM\times N}$ (straightforward but not shown here) is defined such that $\mathbf{s}\otimes\mathbf{a}(\theta_{pq},\phi_{pq} )=\mathbf{A_{pq}}\mathbf{s},p=1,\ldots,P$. Assuming that a particular scatterer from one clutter patch is uncorrelated to any other scatterer belonging to any other clutter patch, we have the net contribution of clutter $\mathbf{y_c}=\sum\limits_{q=1}^Q \mathbf{y}_q$, with corresponding correlation matrix given by $$\label{eq16} \mathbf{R_c}=\sum\limits_{q=1}^Q \mathbf{R}_{\boldsymbol{\gamma}}^q.$$ The clutter model could further be simplified by the following arguments. Assuming a large range resolution which is typically the case for radar STAP [@ward1994] the scatterers in a particular clutter patch are in the same range gate and hence are assumed to possess approximately identical Doppler shifts, i.e. $fc_{pq}\approx fc_q=\tfrac{2f_o\mathbf{\dot{x}_r}^T (\mathbf{x_r}-\mathbf{x_{q} })}{c||\mathbf{x_r}-\mathbf{x_{q} }||}$. Similarly for the far field operation, and considering scatterers in the same azimuth resolution cell, and from the large range resolution argument, we may assume $\theta_{pq}\approx \theta_q$ and $\phi_{pq}\approx \phi_{q}$, i.e. their nominal angular centers. These assumptions can now be incorporated in matrix $\mathbf{B_q}$ to simplify the clutter model, see also [@Setlurradar2013].   Waveform Design =============== The radar return at the considered range gate is processed by a filter characterized by a weight vector, $\mathbf{w}$, whose output is given by $\mathbf{w}^H\mathbf{\tilde{y}}$. Since the vector $\mathbf{s} \in\mathbbm{C}^{N}$ prominently figures in the steering vectors, the objective is to jointly obtain the desired weight vector, $\mathbf{w}$ and waveform vector, $\mathbf{s}$. It is desired that the weight vector will minimize the output power, $\mathbbm{E}\{|\mathbf{w}^H\mathbf{y_u}|^2\}=\mathbf{w}^H\mathbf{R_u}( \mathbf{s})\mathbf{w}$. Mathematically, we may formulate this problem as: $$\begin{aligned} \min_{\mathbf{w},\mathbf{s}}\;\;\;\;\; &\mathbf{w}^H\mathbf{R_u}( \mathbf{s})\mathbf{w}\label{eq17} \\ \mbox{ s. t } \;\;\;\;\;&\mathbf{w}^H(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))=\kappa \nonumber \\ \;\;\;\;\;\; & \mathbf{s}^H \mathbf{s}\leq P_o \nonumber \nonumber\end{aligned}$$ In , the first constraint is the renowned,well known Capon constraint with $\kappa\in\mathbbm{R}$, typically $\kappa=1$. An energy constraint enforced via the second constraint is to addresses hardware limitation. Before we derive the solutions to the optimization problem, it is useful to recall Lem. \[crlemma\], which is well-known, used throughout this report but not stated explicitly. This fundamental result discusses the technique to compute stationary points of a real valued function w.r.t. its complex valued argument and its conjugate. \[crlemma\] Let $f(\mathbf{x},\mathbf{x}^{\ast}):\mathbbm{C}^N\rightarrow\mathbbm{R}$. The stationary point of $f(\mathbf{x},\mathbf{x}^{\ast})=\bar{f}(\mathbf{x_r},\mathbf{x_i})$ is found from the three equivalent conditions, 1. $\nabla_{\bf x_r}\bar{f}(\mathbf{x_r},\mathbf{x_i})=\mathbf{0}$ and $\nabla_{\bf x_i}\bar{f}(\mathbf{x_r},\mathbf{x_i})=\mathbf{0}$, or 2. $\nabla_{\bf x}f(\mathbf{x},\mathbf{x}^{\ast})=\mathbf{0}$, or 3. $\nabla_{\bf x^{\ast}} f(\mathbf{x},\mathbf{x}^{\ast})=\mathbf{0}$. Here $\bar{f}:\mathbbm{R}^{N}\times\mathbbm{R}^{N}\rightarrow\mathbbm{R}$ is the real equivalent of $f(\cdot,\cdot)$, $\mathbf{x_r}=\mathrm{Re}\{\mathbf{x}\},\mathbf{x_i}=\mathrm{Im}\{\mathbf{x}\}$, where we define the gradient $\nabla_{\bf x} f(\mathbf{x},\mathbf{x}^{\ast}):=[\tfrac{\partial f( \cdot,\cdot)}{\partial x_1},\tfrac{\partial f( \cdot,\cdot)}{\partial x_2},\cdots,\tfrac{\partial f( \cdot,\cdot)}{\partial x_N} ]^T$ with $x_i$ as the $i$-th element of ${\bf x},i=1,2,\ldots N$, and $\mathbf{0}$ is a column vector of all zeros of dimension $N$. This arises from the Wirtinger calculus see [@Gesbert2007] [^4] for a recent formal proof. Optimizing w.r.t. $\mathbf{w}$ first, the solution to is well known, and expressed as $$\label{weightcomp} \mathbf{w}_{o}=\frac{\kappa \mathbf{R}_{\bf u}^{-1}(\mathbf{s})(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))}{(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))^H\mathbf{R}_{\bf u}^{-1} (\mathbf{s})(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))}$$ where $\mathbf{R_u}(\mathbf{s})=\mathbf{R_i}+\mathbf{R_c}(\mathbf{s})+\mathbf{R_n}$. We further emphasize that the weight vector is an explicit function of the waveform. Now substituting $\mathbf{w}_{o}$ back into the cost function in , the minimization is purely w.r.t. $\mathbf{s}$, and cast as, $$\begin{aligned} \label{eq18} \min_{\mathbf{s}} &\frac{\kappa^2}{(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))^H\mathbf{R}_{\bf u}^{-1}(\mathbf{s})(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))} \nonumber \\ \mbox{ s. t. } & \mathbf{s}^H \mathbf{s}\leq P_o \end{aligned}$$ A solution to is not immediate, given the dependence of $\mathbf{R_u}$ on the waveform vector $\mathbf{s}$. We consider first, the case when the clutter dependence on the waveform is ignored. Solutions to the design when clutter is considered are treated subsequently. Rayleigh-Ritz: Minimum eigenvector solution ------------------------------------------- Ignoring the dependency of $\mathbf{R_u}$ on $\mathbf{s}$, we readily see that the can be recast as a Rayleigh-Ritz optimization, whose solution is given by $$\begin{aligned} \label{eq21} \mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t)=\boldsymbol{\mu}_{\min}(\mathbf{R_u})\end{aligned}$$ where $\boldsymbol{\mu}_{\min}(\mathbf{R_u})$ is the eigenvector corresponding to the minimum eigenvalue of $\mathbf{R_u}$. This tensor equation implicitly defines the optimal $\mathbf{s}$. It is readily seen that, $\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t)=\mathbf{G}\mathbf{s}$, where $\mathbf{G}=\mathbf{v}(f_d)\otimes\mathbf{A_t}$, and $$\begin{aligned} \mathbf{A_t}=\begin{bmatrix} \mathbf{a}(\theta_t,\phi_t) &\mathbf{0} &\mathbf{0} &\cdots &\mathbf{0} \\ \mathbf{0}& \mathbf{a}(\theta_t,\phi_t) & \mathbf{0} &\cdots &\mathbf{0} \\ \mathbf{0}& \mathbf{0} &\mathbf{a}(\theta_t,\phi_t) &\vdots &\vdots \\ \vdots & \vdots &\vdots &\vdots &\vdots \end{bmatrix} \in\mathbbm{C}^{MN\times N}.\end{aligned}$$ In general, the system is over-determined, and we solve this equation approximately via least squares (LS), $$\begin{aligned} \label{lsubopt} \mathbf{\hat{s}}=(\mathbf{G}^H\mathbf{G})^{-1}\mathbf{G}^H\mu_{\min}(\mathbf{R_u}).\end{aligned}$$ Moreover from and the structure of the temporal and spatial steering vectors, as well as the orthonormality of the eigenvectors, it is readily seen that, $$\begin{aligned} \label{eqrayleigh} ||\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t)||^2&= || \mathbf{v}(f_d)||^2 || \mathbf{s}||^2 ||\mathbf{a}(\theta_t,\phi_t) ||^2=|| \mathbf{s}||^2 \nonumber \\ ||\boldsymbol{\mu}_{\min}(\mathbf{R_u})||^2&=1.\end{aligned}$$ Hence the LS solution in must be scaled to satisfy the desired energy requirements of the radar system. [**Decoupling LS:**]{} The LS solution in can be further simplified due to the following linear relation between elements of $\mathbf{v}(f_d),\mathbf{a}(\theta_t,\phi_t),\mathbf{s}$ and elements of $\boldsymbol{\mu}_{\min}(\mathbf{R_u})$, expressed as $$\begin{aligned} \label{linearrel} v_la_ms_n=\mu_h,\; l&=1,2,\ldots,L,\;m=1,2,\ldots,M,\;n=1,2,\ldots,N \nonumber \\ h&=(l-1)MN+(n-1)M+m.\end{aligned}$$ where $v_l$, $a_m$, $s_n$ are the $l$-th, $m$-th, $n$-th elements of $\mathbf{v}(f_d)$, $\mathbf{a}(\theta_t,\phi_t)$, $\mathbf{s}$, and $\mu_h$ is the $h$-th element of $\boldsymbol{\mu}_{min}(\mathbf{R_u})$, respectively. Therefore, the LS solution in decouples as $$\begin{aligned} \label{lsfinal} s_n=\frac{(\mathbf{v}(f_d)\otimes\mathbf{a}(\theta_t,\phi_t) )^H\boldsymbol{\mu}_{\bf n} }{(\mathbf{v}(f_d)\otimes\mathbf{a}(\theta_t,\phi_t) )^H(\mathbf{v}(f_d)\otimes\mathbf{a}(\theta_t,\phi_t) )},\; n=1,2,\ldots,N\end{aligned}$$ where the vector $\boldsymbol{\mu}_{\bf n}\in\mathbbm{C}^{ML}$ for a [*particular $n$*]{} consists of the $ML$ appropriate elements, $\mu_h,\; h=(l-1)MN+(n-1)M+m,\; m=1,2,\ldots,M, \;l=1,2,\ldots,L$, as highlighted in . The min. eigenvector solution is most relevant when noise and interference are considered and clutter is ignored in the waveform design [@guerci2003]. it has some nice spectral properties similar (but not identical) to water-filling [@guerci2003; @bell1993information]. Therefore this solution, although suboptimal, is a good initial waveform to interrogate the radar scene, but is unfortunately well known to suffer from poor modulus and sidelobe properties. Nonetheless, in certain exceptional cases and in the presence of clutter, this suboptimal solution is shown to be optimal, and is discussed at a later stage. The ensuing definitions and lemma proves useful subsequently. \[lemma1\] (a) If vectors $\boldsymbol{\alpha}$, $\boldsymbol{\beta}$ and $\boldsymbol{\gamma}$ consist of the eigenvalues of the square but not necessarily Hermitian matrices, $\mathbf{X}\in\mathbbm{C}^{N\times N}$, $\mathbf{Y}\in\mathbbm{C}^{M\times M}$ and $\mathbf{X}\otimes\mathbf{Y}$, respectively. Then $\boldsymbol{\gamma}=\boldsymbol{\alpha}\otimes\boldsymbol{\beta}$. (b) Also, $\mathrm{rank}(\mathbf{X}\otimes\mathbf{Y})=\mathrm{rank}(\mathbf{X})\otimes\mathrm{rank}(\mathbf{Y})$. For (a), let $\mathbf{x}_i,i=1,2,\ldots,N$ and $\mathbf{y}_j,j=1,2,\ldots,M$ are the eigenvectors corresponding to $\alpha_i,\beta_j$ i.e. the $i$-th and $j$-th eigenvalues, of $\mathbf{X,Y}$, respectively. Then, from the mixed property of the Kronecker product, $\mathbf{X}\mathbf{x}_i\otimes\mathbf{Y}\mathbf{y}_j=(\mathbf{X}\otimes\mathbf{Y})(\mathbf{x}_i\otimes\mathbf{y}_j)$ but the eigenvector relations imply that $\mathbf{X}\mathbf{x}_i=\alpha_i\mathbf{x}_i,\mathbf{Y}\mathbf{y}_j=\beta_j\mathbf{y}_j$. This implies that the $ij$-th eigenvalue of of $\mathbf{X}\otimes\mathbf{Y}$ is $\gamma_{ij}=\alpha_i\beta_j$ with associated eigenvector $\mathbf{x}_i\otimes\mathbf{y}_j$. Since the rank is equal to the number of non-zero eigenvalues for square matrices, the second follows directly from (a). Hence proved. \[mydef1\] ([*Convexity*]{}) A function $f(\mathbf{x}):\mathbbm{R}^N\rightarrow\mathbbm{R}$ is convex if : (a) $f(t \mathbf{x}_1+(1-t)\mathbf{x}_2)\leq t f(\mathbf{x}_1)+(1-t)f(\mathbf{x}_2)$ for any $t\in[0,1]$ (b) If $f(\mathbf{x})$ is first order differentiable, then it is convex if $f(\mathbf{x}_j)\geq f(\mathbf{x}_i)+ \nabla_{\mathbf{x}_i} f(\mathbf{x}_i)^T( f(\mathbf{x}_j)-f(\mathbf{x}_i) )$where in (a)(b) $\mathbf{x}_i\in\mathbbm{R}^N,i=1,2, j=1,2,j\neq i$. From our extensive simulations, we noticed that the original cost function in is [*not jointly convex*]{} in $\mathbf{w}$ and $\mathbf{s}$. Nevertheless, it is not straightforward to prove / disprove joint convexity w.r.t. both $\mathbf{w}$ and $\mathbf{s}$ analytically. Consider, then, the following propositions: \[propos1\] The objective function in is individually convex w.r.t. $\mathbf{s}$, for any fixed but arbitrary $\mathbf{w}$ Definition \[mydef1\] cannot be directly invoked as the objective $g(\mathbf{s})= \mathbf{w}^H\mathbf{R_u}( \mathbf{s})\mathbf{w}:\mathbbm{C}^N\rightarrow \mathbbm{R}$ depends on the waveform $\mathbf{s}$, which is complex. Consider the following transformation[^5], $\mathbf{s}=\mathbf{D}\bar{\mathbf{s}}$ where $\bar{\mathbf{s}}\in\mathbbm{R}^{2N}=[\mbox{Re}\{\mathbf{s}\}^T,\mbox{Im}\{\mathbf{s}\}^T] ^T$ and $\mathbf{D}=[\mathbf{I}_N, j\mathbf{I}_N]\in\mathbbm{C}^{N\times2N}$. Now, we may define an equivalent $g(\bar{\mathbf{s}}) :\mathbbm{R}^{2N}\rightarrow \mathbbm{R}$ to invoke the definition of convexity. We have to prove that, $$\begin{aligned} \label{eq19} &\mathbf{w}^H \left( \begin{aligned} &\mathbf{R_n}+\mathbf{R_i} \\ &+\sum\limits_{q=1}^Q \mathbf{\breve{B}_q}\begin{aligned} &(\mathbf{I}_P\otimes \mathbf{D}(t\bar{\mathbf{s}}_1+(1-t)\bar{\mathbf{s}}_2 ) )\mathbf{R}_{\gamma}^{pq} \\ &( \mathbf{I}_P\otimes (t\bar{\mathbf{s}}_1+(1-t)\bar{\mathbf{s}}_2 )^T\mathbf{D}^H ) \mathbf{\breve{B}_q}^H \end{aligned} \end{aligned} \right) \mathbf{w} \nonumber \\ &\leq t \mathbf{w}^H \left( \begin{aligned} &\mathbf{R_n}+\mathbf{R_i} \\ &+\sum\limits_{q=1}^Q \mathbf{\breve{B}_q} (\mathbf{I}_P\otimes \mathbf{D}\bar{\mathbf{s}}_1 )\mathbf{R}_{\gamma}^{pq} ( \mathbf{I}_P\otimes \bar{\mathbf{s}}_1^T\mathbf{D}^H ) \mathbf{\breve{B}_q}^H \end{aligned} \right) \mathbf{w} \nonumber \\ &+(1-t)\mathbf{w}^H \left( \begin{aligned} &\mathbf{R_n}+\mathbf{R_i} \\ &+\sum\limits_{q=1}^Q \mathbf{\breve{B}_q}(\mathbf{I}_P\otimes \mathbf{D}\bar{\mathbf{s}}_2 )\mathbf{R}_{\gamma}^{pq} ( \mathbf{I}_P\otimes \bar{\mathbf{s}}_2^T\mathbf{D}^H ) \mathbf{\breve{B}_q}^H \end{aligned} \right) \mathbf{w} \end{aligned}$$ where $t\in[0,1]$ and $\mathbf{\bar{s}_i}\in \mbox{dom}\{ g(\bar{\mathbf{s}})\},i=1,2$. After elementary algebra, the convexity requirement in transforms to: $$\begin{aligned} \label{eq20} \sum\limits_{q=1}^{Q} \mathbf{x}^H_{\bf q} \left(\mathbf{R}_{\gamma}^{pq}\otimes \mathbf{D}(\bar{\mathbf{s}}_1-\bar{\mathbf{s}}_2)(\bar{\mathbf{s}}_1-\bar{\mathbf{s}}_2)^T\mathbf{D}^H\right)\mathbf{x_q} \geq 0\end{aligned}$$ where $\mathbf{x_q}\in\mathbbm{C}^{NP}:=\mathbf{\breve{B}_q}^H\mathbf{w}$. In other words, it is sufficient to show that iff is true then is also true and therefore convex. We notice immediately that is a sum of Hermitian quadratic forms. Consider the matrix $\mathbf{R}_{\gamma}^{pq}\otimes \mathbf{D}(\bar{\mathbf{s}}_1-\bar{\mathbf{s}}_2)(\bar{\mathbf{s}}_1-\bar{\mathbf{s}}_2)^T\mathbf{D}^H$, we know that $\mathbf{R}_{\gamma}^{pq}\succeq \mathbf{0}$[^6], since it is a covariance matrix and by definition atleast positive semi-definite (PSD). The other matrix, i.e. $\mathbf{D}(\bar{\mathbf{s}}_1-\bar{\mathbf{s}}_2)(\bar{\mathbf{s}}_1-\bar{\mathbf{s}}_2)^T\mathbf{D}^H$ is of course rank-1 Hermitian, and is clearly PSD. From Lem. \[lemma1\], it is straightforward to show that $\mathbf{R}_{\gamma}^{pq}\otimes \mathbf{D}(\bar{\mathbf{s}}_1-\bar{\mathbf{s}}_2)(\bar{\mathbf{s}}_1-\bar{\mathbf{s}}_2)^T \mathbf{D}^H\succeq \mathbf{0},\forall q$. Then from the definition of positive semi-definiteness, each of the $Q$ Hermitian quadratic forms in is greater than zero, hence their sum is also greater than zero. \[propos2\] The objective function in is individually convex w.r.t. $\mathbf{w}$, for any fixed but arbitrary $\mathbf{s}$. Given the guaranteed positive semi-definiteness of $\mathbf{R_u}(\mathbf{s})$, the proof is straightforward to demonstrate by invoking the convexity definition on the vector consisting of the real and imaginary parts of $\mathbf{w}$. In fact, Prop. 1, Prop. 2 may be sharpened to include strong convexity, which, as we will show subsequently is desired for the solutions to exist, see the note immediately after $\eqref{eq32}$. For now, however, individual convexity is sufficient to proceed with our analysis. \[remarkstapobj\] ([*Characteristic of STAP objective*]{}) The STAP objective in has [*at most one*]{} minima for a fixed but arbitrary $\mathbf{w}\in\mathbbm{C}^{NML}$ but $\forall \mathbf{s} \in\mathbbm{C}^{N}$. Likewise, it has [*at most one*]{} minima for a fixed but arbitrary $\mathbf{s}\in\mathbbm{C}^{N}$ but $\forall \mathbf{w} \in\mathbbm{C}^{NML}$ This is concluded readily from Prop. \[propos1\], Prop. \[propos2\], i.e. the individual convexity. An illustrative example is provided in Fig. \[figlocalmin\]. \[htbp!\] ![An [*illustrative*]{} non-convex example with multiple local minima. Contours in black are characteristic of the objective. Contours in blue violate convexity in the $\mathbf{w}$, and $\mathbf{s}$ dimension individually, and are therefore [*not characteristic*]{} of the objective function.[]{data-label="figlocalmin"}](Localminima "fig:") Constrained alternating minimization ------------------------------------ Motivated from Prop. \[propos1\], and Prop. \[propos2\], we propose a constrained alternating minimization technique which is iterative. Before we present details on this technique, consider the following minimization problem, which optimizes $\mathbf{s}$, but for a fixed and arbitrary $\mathbf{w}$: $$\begin{aligned} \min\limits_{\mathbf{s}} \;\;\;\;\; &\mathbf{w}^H\mathbf{R_u}(\mathbf{s})\mathbf{w} \nonumber \\ \mbox{s. t. }\;\;\;\;\; & \mathbf{w}^H(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))=\kappa \label{eq22} \\ \;\;\;\;\;\; & \mathbf{s}^H \mathbf{s}\leq P_o. \nonumber \nonumber\end{aligned}$$ In , the objective function could be rewritten as, $$\begin{aligned} \label{eq23} \mathbf{w}^H\mathbf{R_u}(\mathbf{s})\mathbf{w}=&\mathbf{w}^H(\mathbf{R_n}+\mathbf{R_i})\mathbf{w} \\ +&\sum\limits_{q=1}^Q \mbox{Tr}\{ \mathbf{R}_{\gamma}^{pq} (\mathbf{I}_P\otimes\mathbf{s}^H)\mathbf{x_q}\mathbf{x}_{\bf q}^H (\mathbf{I}_P\otimes\mathbf{s})\} \nonumber.\end{aligned}$$ In , the trace operation is further simplified as: $$\begin{aligned} \label{eq24} \mbox{Tr}\{ &\mathbf{R}_{\gamma}^{pq} (\mathbf{I}_P\otimes\mathbf{s}^H)\mathbf{x_q}\mathbf{x}_{\bf q}^H (\mathbf{I}_P\otimes\mathbf{s})\} \nonumber \\ &=\mbox{vec}\left( \left(\mathbf{R}_{\gamma}^{pq} (\mathbf{I}_P\otimes\mathbf{s}^H)\mathbf{x_q}\mathbf{x}_{\bf q}^H \right)^T \right)^T \mbox{vec}( \mathbf{I}_P\otimes\mathbf{s}) \nonumber \\ &=\mathbf{s}^H\mathbf{H}^T ( \mathbf{R}_{\gamma}^{pq}\otimes \mathbf{x_q}\mathbf{x}_{\bf q}^H)\mathbf{H}\mathbf{s} \nonumber\\ &=\mathbf{s}^H\mathbf{Z_q}(\mathbf{w})\mathbf{s}\end{aligned}$$ where $\mbox{vec}(\mathbf{I}_P\otimes\mathbf{s})=\mathbf{H}\mathbf{s}$, with $\mathbf{H}\in \mathbbm{R}^{P^2N \times N}=[\mathbf{H_1}^T,\mathbf{H_2}^T,\ldots,\mathbf{H_P}^T]^T$. The matrix $\mathbf{H_k}\in\mathbbm{R}^{PN \times N}, k=1,2,\ldots,P$ is further decomposed into $P$, $N\times N$ matrices, and is defined such that the $k$-th $N \times N$ matrix is $\mathbf{I}_N$ and the other $(N-1)$, $N\times N$ matrices are all equal to zero matrices. \[PSDremark\] (a) At the very least, $\sum\limits_{q=1}^Q\mathbf{Z_q}\succeq \mathbf{0}$. (b) The matrix $\mathbf{Z_q}\succeq \mathbf{0}$ for $P<N$, always. (c) However, it may be positive definite, i.e. $\mathbf{Z_q}\succ\mathbf{0}$ and hence $\sum\limits_{q=1}^Q\mathbf{Z_q}\succ\mathbf{0}$ for $P\geq N$ and for $\mathbf{R}_\gamma^{pq}\succ \mathbf{0}$. We note that (a) is readily implied from Prop. \[propos1\] since a Hermitian quadratic form $\mathbf{x}^H\mathbf{B}\mathbf{x}$ is convex (strictly convex) iff $\mathbf{B}\succeq\mathbf{0}$ ( $\mathbf{B}\succ\mathbf{0}$). Since $\mathbf{R}_{\gamma}^{pq}\otimes \mathbf{x_q}\mathbf{x}_{\bf q}^H \succeq \mathbf{0}$ always ($\leq P$ non-zero eigenvalues and the rest are zeros) and that $P<N$, in other words, the transformation $\mathbf{H}^T(\mathbf{R}_{\gamma}^{pq}\otimes \mathbf{x_q}\mathbf{x}_{\bf q}^H)\mathbf{H}:\mathbbm{C}^{P^2 N\times N} \times \mathbbm{C}^{P^2 N\times N} \rightarrow \mathbbm{C}^{N\times N}$ and from the structure of $\mathbf{H}$, the result (b) is obvious. For (c), we know that $\mathrm{rank}(\mathbf{H})=N$, hence it could be shown after some tedious algebra that $\mathbf{Z_q}$ may be PD only when $P\geq N$ and that $\mathbf{R}_{\gamma}^{pq}$ is PD in the first place, also see for example [@horn1994 pg. 399]. Using and , the Lagrangian of is readily cast as, $$\begin{aligned} \label{eq25} \mathcal{L}(\mathbf{s},\gamma_1,\gamma_2)&=\mathbf{w}^H( \mathbf{R_i+R_n})\mathbf{w}+\sum\limits_{q=1}^Q\mathbf{s}^H\mathbf{Z_q}(\mathbf{w})\mathbf{s} \\ &+\mbox{Re}\{ \gamma_1^{\ast} (\mathbf{w}^H\mathbf{G}\mathbf{s}-\kappa)\}+\gamma_2\mathbf{s}^H\mathbf{I}_N\mathbf{s}-\gamma_2P_o \nonumber\end{aligned}$$ where $\gamma_1\in\mathbbm{C}$ and $\gamma_2\in\mathbbm{R}^{+}$ are the complex and real Lagrange parameters. [**Lagrange Dual:**]{} The Lagrange dual, denoted as $\mathcal{H}(\gamma_1,\gamma_2)=\inf\limits_{\mathbf{s}} \mathcal{L}( \mathbf{s},\gamma_1,\gamma_2)$. Since consists of Hermitian quadratic forms and other linear terms of $\mathbf{s}$, we have $\mathcal{H}(\gamma_1,\gamma_2)=\mathcal{L}(\mathbf{s_o}(\gamma_1,\gamma_2),\gamma_1,\gamma_2)$, where $\mathbf{s_o}(\gamma_1,\gamma_2)$ is obtained by solving the first order optimality conditions, i.e. $$\begin{aligned} \label{eq26} \frac{\partial \mathcal{L}(\mathbf{s},\gamma_1,\gamma_2)}{\partial \mathbf{s}} ={\bf 0}\end{aligned}$$ where, $\mathbf{0}$ is a column vector of size $N$ and consists of all zeros. Further, in , while taking the derivative the usual rules of complex vector differentiation apply, i.e. treat $\mathbf{s}^H$ independent of $\mathbf{s}$. The solution to is readily obtained by differentiating , and expressed as: $$\begin{aligned} \label{eq27} \mathbf{s_o}(\gamma_1,\gamma_2)=-\frac{\gamma_1}{2}\Bigl( \sum\limits_{q=1}^Q \mathbf{Z_q}( \mathbf{w})+\gamma_2\mathbf{I}_N\Bigr)^{-1}\mathbf{G}^H\mathbf{w}.\end{aligned}$$ Using , the dual $\mathcal{H}(\gamma_1,\gamma_2)$ is given by: $$\begin{aligned} \label{eq28} &\mathcal{H}( \gamma_1,\gamma_2)= \mathbf{w}^H( \mathbf{R_i+R_n})\mathbf{w} -\kappa\mbox{Re}\{ \gamma_1^\ast \}-\gamma_2P_o \nonumber \\ -&\frac{|\gamma_1|^2}{4} \mathbf{w}^H\mathbf{G} \Bigl( \sum\limits_{q=1}^Q \mathbf{Z_q}( \mathbf{w})+\gamma_2\mathbf{I}_N\Bigr)^{-1}\mathbf{G}^H\mathbf{w}.\end{aligned}$$ Equation is further simplified by decomposing, $\gamma_1=\gamma_{1r}+j\gamma_{1i}$. In which case, we notice that is quadratic in $\gamma_{1r},\gamma_{1i}$,and purely linear in $\lambda_2$. The Lagrange dual optimization is therefore, $$\begin{aligned} \label{eq29} \max \limits_{\gamma_{1r},\gamma_{1i},\gamma_2} \;\;\;\; &\mathcal{H}(\gamma_{1r},\gamma_{1i},\gamma_2) \nonumber \\ \mbox{s. t } \;\;\;\; &\gamma_2\geq0.\end{aligned}$$ Maximizing first w.r.t. $\gamma_{1r},\gamma_{1i}$, we have the solutions, $$\begin{aligned} \bar{\gamma}_{1r}=\frac{-2\kappa}{\mathbf{w}^H\mathbf{G} \Bigl( \sum\limits_{q=1}^Q \mathbf{Z_q}( \mathbf{w})+\gamma_2\mathbf{I}_N\Bigr)^{-1}\mathbf{G}^H\mathbf{w}},\;\bar{\gamma}_{1i}=0.\end{aligned}$$ Substituting the above solutions into , the Lagrange dual optimization problem and after ignoring an unnecessary additive constant, takes the form, $$\begin{aligned} \label{eq30} \max\limits_{\gamma_2} \;\; &\kappa^2 \Bigl( \mathbf{w}^H\mathbf{G} \Bigl( \sum\limits_{q=1}^Q \mathbf{Z_q}( \mathbf{w})+\gamma_2\mathbf{I}_N\Bigr)^{-1}\mathbf{G}^H\mathbf{w}\Bigr)^{-1}-\gamma_2P_o \nonumber \\ \mbox{s. t. } \;\;&\gamma_2\geq0\end{aligned}$$ The associated Lagrangian for is $$\begin{aligned} \label{lagra1} \mathcal{D}(\gamma_2,\gamma)=\frac{\kappa^2} {\mathbf{w}^H\mathbf{G}\mathbf{F}^{-1} \mathbf{G}^H\mathbf{w}} -\gamma_2P_o-\gamma_3\gamma_2\end{aligned}$$ where $\mathbf{F}:=\sum\limits_{q=1}^Q \mathbf{Z_q}( \mathbf{w})+\gamma_2\mathbf{I}_N$. The first order optimality condition for the optimization is given by: $$\begin{aligned} &\frac{\partial}{\partial \gamma_2} \bigl( \frac{\kappa^2} {\mathbf{w}^H\mathbf{G}\mathbf{F}^{-1} \mathbf{G}^H\mathbf{w}} \bigr)-P_o-\gamma_3 =0 \nonumber \\ \mbox{or } &\frac{-\kappa^2}{ ( \mathbf{w}^H\mathbf{G}\mathbf{F}^{-1} \mathbf{G}^H\mathbf{w})^{2}} \mathbf{w}^H\mathbf{G}\frac{\partial\mathbf{F}^{-1}}{\partial\gamma_2}\mathbf{G}^H\mathbf{w}-P_o -\gamma_3 =0 \nonumber \\ \mbox{or } &\frac{\kappa^2}{ ( \mathbf{w}^H\mathbf{G}\mathbf{F}^{-1} \mathbf{G}^H\mathbf{w})^{2}} \mathbf{w}^H\mathbf{G}\bigl( \mathbf{F}^{-1}\frac{\partial\mathbf{F}}{\partial\gamma_2} \mathbf{F}^{-1} \bigr)\mathbf{G}^H\mathbf{w}-P_o -\gamma_3 =0 \nonumber \end{aligned}$$ where $\gamma_3$ is the Lagrange multiplier associated with the Lagrangian , and we also have $\tfrac{\partial \mathbf{F}}{\partial \gamma_2}=\mathbf{I}_N$. The complementary slackness and constraint qualifier for i.e. $\gamma_3\gamma_2=0$ and $\gamma_2\geq0$ form the rest of the equations comprising the KKT conditions. It is now readily shown that the solution to is given by $$\begin{aligned} \label{eq31} &\bar{\gamma}_2=\max [0,\gamma_2] \\ &\gamma_2 \mbox{ solves } \gamma_2\left( \kappa^2\mathbf{w}^H\mathbf{G}\mathbf{F}^{-2}\mathbf{G}^H\mathbf{w}-P_o( \mathbf{w}^H\mathbf{G}\mathbf{F}^{-1}\mathbf{G}^H\mathbf{w})^2\right)=0 \nonumber.\end{aligned}$$ \[propos3\] The parameter $\bar{\gamma}_2=0$ solves . The spectral theorem for Hermitian matrices, allows for a decomposition, $\mathbf{F}=\mathbf{E}(\boldsymbol{\Lambda}+\gamma_2\mathbf{I}_N)\mathbf{E}^H$. The matrix $\boldsymbol{\Lambda}$ is a diagonal matrix comprising eigenvalues in descending order, whereas, $\mathbf{E}$ is unitary and whose columns are the corresponding eigenvectors of $\mathbf{F}$. For ease of exposition, denote $\mathbf{z}\in\mathbbm{C}^{N}:=\mathbf{E}^H\mathbf{G}^H\mathbf{w}$, then assume a function $f(\gamma_2):\mathbbm{R}^{+}\rightarrow\mathbbm{R}$, expressed as $$\begin{aligned} \label{lagra2} f(\gamma_2)&:=\kappa^2\mathbf{w}^H\mathbf{G}\mathbf{F}^{-2}\mathbf{G}^H\mathbf{w}-P_o( \mathbf{w}^H\mathbf{G}\mathbf{F}^{-1}\mathbf{G}^H\mathbf{w})^2 \nonumber \\ &=\sum\limits_{n=1}^N \kappa^2\frac{|z_n|^2}{(d_n+\gamma_2)^2} -P_o\left( \sum\limits_{n=1}^N \frac{|z_n|^2}{d_n+\gamma_2}\right)^2\end{aligned}$$ where $z_n,d_n$ are the $n$-th elements of $\mathbf{z}$, and the $n$-th eigenvalue in $\boldsymbol{\Lambda}$. We analyze $f(\gamma_2)$ and $\gamma_2f(\gamma_2)$ in detail. The following (behavior at $0$ and $\infty$) are readily observed \[lagra3\] $$\begin{aligned} \lim_{\gamma_2\rightarrow \infty}f(\gamma_2)&=f(\infty) =0 \label{lagra32}\\ \lim_{\gamma_2\rightarrow 0}f(\gamma_2)&=\sum\limits_{n=1}^N \kappa^2\frac{|z_n|^2}{d_n^2} -P_o\left( \sum\limits_{n=1}^N\frac{|z_n|^2}{d_n}\right)^2=f(0) \label{lagra31}\end{aligned}$$ Furthermore, it is seen that $$\label{lagra4} \begin{aligned} &\lim_{\gamma_2\rightarrow\infty}\gamma_2f(\gamma_2)=\lim_{\gamma_2\rightarrow\infty}\frac{f(\gamma_2)}{1/\gamma_2}=\lim_{\gamma_2\rightarrow\infty}\frac{\frac{\mathrm{d}f(\gamma_2)}{\mathrm{d}\gamma_2}}{(-1/\gamma_2^2)} =0 \end{aligned}$$ Moreover, consider $f(\gamma_2)=h_1(\gamma_2)-h_2(\gamma_2)=0$, where $h_1(\gamma_2)=\kappa^2\sum\limits_{n=1}^N\frac{f_n^2(\gamma_2)}{|z_n|^2},h_2(\gamma_2)=P_o (\sum\limits_{n=1}^N f_n(\gamma_2))^2$, where $f_n(\gamma_2)=\frac{|z_n|^2}{d_n+\gamma_2}$. Note that $fn(\gamma_2)\downarrow, n=1,2,\ldots,N$ and that $h_i(\gamma_2)\downarrow,i=1,2$, i.e. decreasing functions w.r.t. $\gamma_2 \in[0,\infty)$. Then equation$f(\gamma_2)=0$ implies that $$\label{lagra41} \begin{aligned} \sum\limits_{n=1}^N \kappa^2\frac{|z_n|^2}{(d_n+\gamma_2)^2} -P_o\left( \sum\limits_{n=1}^N \frac{|z_n|^2}{d_n+\gamma_2}\right)^2=0 \\ \mbox{or } \sum\limits_{n=1}^N (\frac{\kappa^2}{|z_n|^2}-P_o )f_n^2(\gamma_2)=2\sum\limits_{n_1}\sum\limits_{\substack{n_2\\ n_2\neq n_1}} f_{n_1}(\gamma_2)f_{n_2}(\gamma_2) \end{aligned}$$ where $(n_1,n_2)\in (1,2,\ldots,N)$. Recall that $d_n\neq 0 \forall n$, $d_n\geq d_{n+1}, n=1,2,\ldots, N$, and $ |z_n|\neq 0 \forall n$. A solution to for $\gamma_2 \in[0,\infty)$ is readily derived in the trivial case, for example when $f_{n_1}(\gamma_2)=f_{n_2}(\gamma_2)$, $P_o\neq \kappa^2$, and for $|z_n|$ to be some arbitrary constant for all $n$. For $P_o\geq \kappa$ it may now be shown numerically that a solution to for $\gamma_2 \in [0,\infty)$ does not exist. In fact, our extensive numerical simulations reveal that in general and assuming $P_o\geq \kappa$ and for $\gamma_{21}\leq \gamma_{22}$ $$\begin{aligned} \label{lagra51} \begin{cases} f(\gamma_{21})\geq f(\gamma_{22}) \mbox{ if } f(0)>0 \\ f(\gamma_{21})\leq f(\gamma_{22}) \mbox{ if } f(0)<0 \end{cases} \gamma_{21} \mbox{ and } \gamma_{22} \in[0,\infty).\end{aligned}$$ That is, $f(\gamma_2)$ is monotonic. From the above arguments, therefore, $\gamma_2f(\gamma_2)=0$ implies that $\gamma_2=0$. Alternatively nevertheless, a solution to may be found numerically and is computationally cheap. [**Note:**]{} [*(Inactive power constraint)*]{} It is noted that trivially $\bar{\gamma}_2=0$ [*may always*]{} be chosen as a solution with suitable choices of the free parameter $P_o$. This implies that the power constraint is always satisfied and hence is an inactive constraint in the corresponding Lagrangian. A graphical behavior of $h_i(\gamma_2),i=1,2$ and thus the behavior of $f(\gamma_2)$ is seen from Fig. \[figlagrange\]. Using Prop. \[propos3\], the waveform design solution is unique, a function of $\mathbf{w}$ and expressed as, $$\begin{aligned} \label{eq32} \mathbf{s}_o(\mathbf{w})=\frac{\kappa \Bigl( \sum\limits_{q=1}^Q \mathbf{Z_q}( \mathbf{w})\Bigr)^{-1}\mathbf{G}^H\mathbf{w} }{\mathbf{w}^H\mathbf{G} \Bigl( \sum\limits_{q=1}^Q \mathbf{Z_q}( \mathbf{w})\Bigr)^{-1}\mathbf{G}^H\mathbf{w}}.\end{aligned}$$ [**Note:**]{} ([*Strong convexity*]{}) To compute the constrained alternating minimization solutions, the respective matrices in , must be invertible, implying strong convexity individually w.r.t. $\mathbf{w}$, $\mathbf{s}$, respectively. This directly necessitates, $\lambda_{\min}\Bigl( \sum\limits_{q=1}^Q \mathbf{Z_q}( \mathbf{w})\Bigr)\neq 0$ and $\lambda_{\min} (\mathbf{R_u}(\mathbf{s}) )\neq 0$, and hence also, positive definiteness of these matrices. The alternating minimization algorithm is now succinctly stated in Table \[table1\]. \[tbp!\] ![Two cases are presented assuming $P_o\geq \kappa$. (a) Blue: $h_1(\gamma_2)$, Red: $h_2(\gamma_2)$ and therefore $f(\gamma_2)$ is decreasing, (b) Blue: $h_2(\gamma_2)$, Red: $h_1(\gamma_2)$ and therefore $f(\gamma_2)$ is increasing. The blue and red curves intersect at $\infty$.[]{data-label="figlagrange"}](lagrange_param "fig:") \[dualprop\] ([*Strong duality*]{}) The optimal value of the lagrange dual problem is given by $$\mathbf{w}^H( \mathbf{R_i+R_n})\mathbf{w} +\frac{\kappa^2}{\mathbf{w}^H\mathbf{G} \Bigl( \sum\limits_{q=1}^Q \mathbf{Z_q}( \mathbf{w})\Bigr)^{-1}\mathbf{G}^H\mathbf{w}}.$$ It is therefore trivial to show that the duality gap between and is zero. In other words, strong duality holds between the primal in and the dual in . From Slaters condition [@Boyd2004] the sufficient condition to ensure strong duality is the existence of , i.e. the inverse of $\sum\limits_{q=1}^Q \mathbf{Z_q}( \mathbf{w})$ exists (see note below), and that the solution in satisfies the power constraint. [**Note:**]{} ([*Lower bound on $Q$*]{}) Since $\mathrm{rank}( \sum\limits_{q=1}^Q \mathbf{Z_q}( \mathbf{w}))\leq \sum\limits_{q=1}^Q\mathrm{rank}( \mathbf{Z_q}( \mathbf{w}))$, [*assume the worst case*]{} $P=1$, then we have that $\mathrm{rank}(\mathbf{Z_q})=1$. Therefore for $Q$ distinct (different spatial signature and Doppler) clutter patches, $Q\geq N$ ensures invertibility of $\sum\limits_q \mathbf{Z_q}$. [|p[3.3in]{}|]{} 1. [*Initialize*]{}: Start with an initial waveform design, defined as $\mathbf{s}_o^{(0)}$, set counter $k=1$ 2. [ *Filter design*]{}: Design the optimal filter weight vector, $\mathbf{w}_{o}^{(k)}=\mathbf{w}_o(\mathbf{s}_o^{(k-1)})$, where is used to compute $\mathbf{w}_o( \cdot)$. 3. [ *Waveform design*]{}: Design the updated waveform $\mathbf{s}_o^{(k)}=\mathbf{s}_o(\mathbf{w}_o^{(k)})$, where is used to compute $\mathbf{s}_o(\cdot)$. 4. [*Check:*]{} If convergence is achieved, exit, else $k=k+1$, go back to step-2. \ ### Convergence, performance guarantees, and other properties Denote $(\mathbf{w}_k,\mathbf{s}_k)$ as the sequence of iterates of the algorithm in Table \[table1\] and define $g(\mathbf{w}_k,\mathbf{s}_k):=\mathbf{w}_k\mathbf{R_u}(\mathbf{s}_k)\mathbf{w}_k^H$, then for $k=1,2,\ldots$ $$\begin{aligned} \label{eq33} \cdots g(\mathbf{w}_k,\mathbf{s}_{k-1} )\geq g(\mathbf{w}_k,\mathbf{s}_k ) \geq g( \mathbf{w}_{k+1},\mathbf{s}_k)\cdots.\end{aligned}$$ Moreover, since at least $\mathbf{R_u}(\mathbf{s})\succeq \mathbf{0}$, i.e. PSD $\forall\mathbf{s}$, we have that $g(\mathbf{w},\mathbf{s})\geq0,\forall\mathbf{w}$. Therefore each of the individual terms in are [*lower bounded*]{} by zero, in other words $g(\mathbf{w}_{k_1},\mathbf{s}_{k_2})\geq0,k_1=k,\mbox{ or }k+1$ and $k_2=k,\mbox{ or }k+1$, for $k=1,2,\ldots$ . \[propos4\] Iff the iterates $(\mathbf{w}_k,\mathbf{s}_k)$ of the constrained alternating minimization exist, then $\lim\limits_{k\rightarrow\infty}\mathbf{g ( \mathbf{w}_k,\mathbf{s}_k)}$ is finite. The non-increasing property in , and since each term in is lower bounded, straightforward application of the monotone convergence theorem to the sequence, $\{\mathbf{g ( \mathbf{w}_k,\mathbf{s}_k)}\}$,completes the proof. We note that convergence to a finite limit as evidenced from Prop. \[propos4\] is indeed dependent on the constraints via the existence of the iterates $(\mathbf{w}_k,\mathbf{s}_k)$. This however does not imply convergence of the sequence $\{(\mathbf{w}_k,\mathbf{s}_k) \}$, for which, consider the following. \[remark1\] The alternating minimization is a special case of the block Gauss-Siedel and block co-ordinate descent (BCD) algorithm with block size equal to two [@Grippo2000; @Luo1992]. \[deflimit\] ([*Convergence in $\mathbbm{R}^{N}$*]{}) A sequence $\{ \mathbf{x}_k\} \in\mathbbm{R}^{N} ,k=1,2,\ldots$ is said to converge to $\tilde{\mathbf{x}}$, a limit point, if, $\forall \epsilon>0,\;\; \exists K\in{\mathbbm{N}}:\;||\mathbf{x}_k-\tilde{\mathbf{x}}|| \leq\epsilon,\; k>K$. \[altminlemma\] (Constrained alternating minimization lemma) Assume that a function $g(\mathbf{z}):\mathbbm{R}^{2N}\rightarrow\mathbbm{R},\mathbf{z}=[\mathbf{x}^T,\mathbf{y}^T]^T$ is continuously differentiable over a closed nonempty convex set, $\mathcal{A}=\mathcal{A}_1\times\mathcal{A}_2$. Also, suppose the solution to the constrained optimization problems, $\min \limits_{\mathbf{x}\in\mathcal{A}_1} g(\mathbf{x},\mathbf{y} )$ and $\min \limits_{\mathbf{y}\in\mathcal{A}_2} g(\mathbf{x},\mathbf{y} )$ are uniquely attained. Let $\{ \mathbf{z}_k \}$ be the sequence generated by this algorithm, then every limit point of this sequence is also a stationary point. The proof in [@Bertsekas1999 Prop. 2.7.1] follows immediately to the alternating minimization assuming two blocks. Also see [@Grippo2000], where the convergence of the two block BCD was analyzed. The above Lem. \[altminlemma\] discusses convergence of the constrained alternating minimization.This lemma can be applied by decomposing our problem into its real equivalent along-with real and imaginary decomposition of $\mathbf{w},\mathbf{s}$, and assuming the our constraint set $\mathcal{A}=\mathcal{A}_1\times\mathcal{A}_2$ is closed convex and the minimizers are unique. The necessary condition of a unique minimizer [@Zangwill1967] at each step is not obvious, but [@Powell1973] showed that in the absence of this assumption the algorithm cycles endlessly around a particular objective value [@Bertsekas1999]. Further the algorithm provides limit points which are not stationary points [@Grippo2000]. To discuss the characteristics of the limits points at convergence, consider the remark, presented next. \[limitpointremk\] ([*Characterizing the solutions at convergence*]{}) If $(\mathbf{w}_\star,\mathbf{s}_\star)$ are the limit points of the sequence $\{(\mathbf{w}_k,\mathbf{s}_k)\}$. Then, $(\mathbf{w}_\star,\mathbf{s}_\star)$ is a local minima, i.e. by definition $g(\mathbf{w}_\star,\mathbf{s}_\star)\leq g(\mathbf{w},\mathbf{s})$,$\exists \epsilon>0$ with $(\mathbf{w},\mathbf{s}):\,\sqrt{|| \mathbf{w}-\mathbf{w}_\star||^2+|| \mathbf{s}-\mathbf{s}_\star||^2}\leq \epsilon$. Further, $(\mathbf{w}_\star,\mathbf{s}_\star):g(\mathbf{w}_\star,\mathbf{s}_\star)\leq g(\mathbf{w}_\star,\mathbf{s}),\,\forall \mathbf{s}\in\mathcal{A}_2\mbox{ and } g(\mathbf{w}_\star,\mathbf{s}_\star)\leq g(\mathbf{w},\mathbf{s}_\star),\,\forall \mathbf{w}\in\mathcal{A}_{1}$. The first statement in Rem. \[limitpointremk\] directly results from from the stationarity condition as given in Lem. \[altminlemma\] and also since the objective is non-convex. The second statement in Rem. \[limitpointremk\] arises from the individual convexity in $\mathbf{w}$ and $\mathbf{s}$ as shown in Prop. \[propos1\], Prop. \[propos2\]. We note readily from Rem. \[remarkstapobj\], that unfortunately there is nothing special or strong about $(\mathbf{w}_\star,\mathbf{s}_\star)$ except the fact that they are local minima. It is well known that global extrema (minima or maxima) are attained only when the objective is either convex or concave. For a problem similar to ours and where the alternating minimization was applied, see [@vaidyanathan2009 pg.3537] the authors state that their algorithm produces limit points which are stronger than local maxima, in our opinion this conclusion is suspect. They further claim that their algorithm produces global extrema in their filter design and waveform dimensions individually, which leads us to believe that their objective is concave, although this was never proved in [@vaidyanathan2009]. In our opinion, Rem. \[remarkstapobj\] is also relevant to their objective by replacing minima by maxima, and hence we do not believe that the limit points produced by their algorithm are stronger than local extrema. To derive the upper and lower bounds on $g(\mathbf{w}_k,\mathbf{s}_k)-g(\mathbf{w}_{k+1},\mathbf{s}_k)$, the following well known lemmas are useful. \[lemma2\] For any Hermitian matrix, $\mathbf{A}\in\mathbbm{C}^{N\times N}$ and any arbitrary vector $\mathbf{x}\in\mathbbm{C}^{N\times N}$ , we always have $\lambda_{\min}(\mathbf{A})||\mathbf{x}||^2\leq \mathbf{x}^H\mathbf{Ax}\leq\lambda_{\max}(\mathbf{A})||\mathbf{x}||^2$, where $\lambda_{\min}(\mathbf{A})$ and $\lambda_{\max}(\mathbf{A})$ are the min. and max. eigenvalues of matrix $\mathbf{A}$, respectively. The proof can be seen in [@horn1994], and is in fact fundamental to the Rayleigh-Ritz theorem. \[lemma3\] For any two Hermitian matrices, $\mathbf{A,B}$, both in $\mathbbm{C}^{N\times N}$, $$\begin{aligned} \sum\limits_{i=1}^N \lambda_i(\mathbf{A}) \lambda_{N-i+1}(\mathbf{B})\leq \mathrm{Tr}\{ \mathbf{AB}\}\leq\sum\limits_{i=1}^N \lambda_i(\mathbf{A})\lambda_i(\mathbf{B})\end{aligned}$$ where $\lambda_i(\cdot)\geq \lambda_{i+1}(\cdot)$, $i=1,2,\ldots,N$. See [@Lasserre1995 Lemma. II. I] for a proof. Consider $g(\mathbf{w}_k,\mathbf{s}_k)$, we have $$\begin{aligned} \label{eq34} g(\mathbf{w}_k,\mathbf{s}_k)&=\mathbf{w}_k^H\mathbf{R_u}(\mathbf{s}_k)\mathbf{w}_k \nonumber \\ &=(\mathbf{w}_k-\mathbf{w}_{k+1} +\mathbf{w}_{k+1} )^H \mathbf{R_u}(\mathbf{s}_k)(\mathbf{w}_k-\mathbf{w}_{k+1}+\mathbf{w}_{k+1} ) \nonumber\\ &=(\mathbf{w}_k-\mathbf{w}_{k+1} )^H \mathbf{R_u}(\mathbf{s}_k)(\mathbf{w}_k-\mathbf{w}_{k+1}) \\ &+ \mathbf{w}_{k+1} ^H \mathbf{R_u}(\mathbf{s}_k)\mathbf{w}_{k+1}+\textrm{Re}\{ (\mathbf{w}_k-\mathbf{w}_{k+1} )^H \mathbf{R_u}( \mathbf{s}_k) \mathbf{w}_{k+1} \} \nonumber\end{aligned}$$ Moreover since the square root decomposition exists i.e., $\mathbf{R_u}( \cdot)=\mathbf{R}_{\bf u}^{1/2}( \cdot) \mathbf{R}_{\bf u}^{1/2}(\cdot)$, then application of the Cauchy-Schwartz inequality produces, $$\begin{aligned} \label{eq35} &\textrm{Re}\{ (\mathbf{w}_k-\mathbf{w}_{k+1} )^H \mathbf{R_u}( \mathbf{s}_k) \mathbf{w}_{k+1} \} \leq \\ &\sqrt{(\mathbf{w}_k-\mathbf{w}_{k+1} )^H \mathbf{R_u}(\mathbf{s}_k)(\mathbf{w}_k-\mathbf{w}_{k+1}) } \sqrt{\mathbf{w}_{k+1} ^H \mathbf{R_u}(\mathbf{s}_k)\mathbf{w}_{k+1}} \nonumber\end{aligned}$$ Using in and since $\mathbf{R_u}(\cdot)$ is PSD, we can show that $ g(\mathbf{w}_k,\mathbf{s}_k)-g(\mathbf{w}_{k+1},\mathbf{s}_k)\leq (\mathbf{w}_k-\mathbf{w}_{k+1} )^H \mathbf{R_u}(\mathbf{s}_k)(\mathbf{w}_k-\mathbf{w}_{k+1}) $. Further using , we have the following upper and lower bounds $$\begin{aligned} \label{eq36} 0&\leq g(\mathbf{w}_k,\mathbf{s}_k)-g(\mathbf{w}_{k+1},\mathbf{s}_k) \nonumber \\ &\leq(\mathbf{w}_k-\mathbf{w}_{k+1} )^H \mathbf{R_u}(\mathbf{s}_k)(\mathbf{w}_k-\mathbf{w}_{k+1})\end{aligned}$$ We notice immediately, that at convergence $(\mathbf{w}_k-\mathbf{w}_{k+1} )^H \mathbf{R_u}(\mathbf{s}_k)(\mathbf{w}_k-\mathbf{w}_{k+1})\rightarrow 0$ since $\mathbf{w}_k\rightarrow\mathbf{w}_{k+1}$. Other bounds as in can be readily derived. From Lem. \[lemma2\], we can show that $$\begin{aligned} \label{eq37} &\leq \lambda_{\min}(\mathbf{R_u}(\mathbf{s}_k) )|| \mathbf{w}_k||^2-\lambda_{\max}(\mathbf{R_u}(\mathbf{s}_k) )|| \mathbf{w}_{k+1}||^2\nonumber \\ &g(\mathbf{w}_k,\mathbf{s}_k)-g(\mathbf{w}_{k+1},\mathbf{s}_k) \\ &\leq \lambda_{\max}(\mathbf{R_u}(\mathbf{s}_k) )|| \mathbf{w}_k||^2-\lambda_{\min}(\mathbf{R_u}(\mathbf{s}_k) )|| \mathbf{w}_{k+1}||^2 \nonumber.\end{aligned}$$ Consider the following. \[lemma4\] If $\mathbf{x}$, $\mathbf{y}$ are arbitrary but distinct complex vectors of size $N$ and let $\mathbf{A}:=\mathbf{xx}^H-\mathbf{yy}^H$, then, (a) matrix $\mathbf{A}$ has exactly two real non-zero eigenvalues, the rest $N-2$ eigenvalues are all zeros, (b) of the two real and non-zero eigenvalues one is always positive and the other is always negative, and (c) if the $\mathbf{x}$, $\mathbf{y}$ are not distinct, i.e. $\mathbf{y}=\beta \mathbf{x}$, $\beta\in\mathbbm{C}$, then there exists only one non-zero eigenvalue, $(|1-|\beta|^2|) ||\mathbf{x}||^2$and the rest $N-1$ eigenvalues are purely zeroes. First of all we notice $\mathbf{A}$ is Hermitian and hence its eigenvalues are real. The proof for (a) is obvious given the fact that $\mathbf{A}$ is a sum of two distinct outer products. In other words, $\mathrm{rank}(\mathbf{A})=2$, for all $\mathbf{y}\neq\beta\mathbf{x}$ . Now we know that $$\begin{aligned} \mathrm{Tr}\{ \mathbf{A}\}&=\lambda_1+\lambda_2=\mathbf{x}^H\mathbf{x}-\mathbf{y}^H\mathbf{y} \\ \mathrm{Tr}\{\mathbf{AA}^H\}&=\lambda_1^2+\lambda_2^2=||\mathbf{x}||^4+||\mathbf{y}||^4-2|\mathbf{x}^H\mathbf{y}|^2\end{aligned}$$ where $\lambda_i,i=1,2$ are the two non zero eigenvalues of $\mathbf{A}$. The above set of equations can be reduced to a quadratic in any one eigenvalue. It can be shown that the only two possible solutions are then $$\label{eq38} \begin{aligned} \lambda_1&=\frac{||\mathbf{x}||^2-||\mathbf{y}||^2}{2} \left( 1+ \sqrt{ 1-4\frac{|\mathbf{x}^H\mathbf{y}|^2 -||\mathbf{x}||^2||\mathbf{y}||^2}{(||\mathbf{x}||^2-||\mathbf{y}||^2)^2} } \right) \\ \lambda_2&=\frac{||\mathbf{x}||^2-||\mathbf{y}||^2}{2} \left( 1- \sqrt{ 1-4\frac{|\mathbf{x}^H\mathbf{y}|^2 -||\mathbf{x}||^2||\mathbf{y}||^2}{(||\mathbf{x}||^2-||\mathbf{y}||^2)^2} } \right) \end{aligned}$$ Since $\lambda_i,i=1,2$ are purely real we have, $1-4\frac{|\mathbf{x}^H\mathbf{y}|^2 -||\mathbf{x}||^2||\mathbf{y}||^2}{(||\mathbf{x}||^2-||\mathbf{y}||^2)^2} \geq0$ and from Cauchy Schwarz inequality, we also have that $|\mathbf{x}^H\mathbf{y}|^2 -||\mathbf{x}||^2||\mathbf{y}||^2\leq0 $. Using these two facts, consider two specific cases, both of which are shown easily from elementary algebra, $$\label{eq39} \begin{cases} \lambda_1>0,\lambda_2 <0, & \mbox{ if } ||\mathbf{x}||^2-||\mathbf{y}||^2 \geq0 \\ \lambda_1<0,\lambda_2>0, & \mbox{ if } ||\mathbf{x}||^2-||\mathbf{y}||^2 <0 \end{cases}.$$ When $||\mathbf{x}||^2-||\mathbf{y}||^2=0$, it is easily seen that $\lambda_1=\sqrt{||\mathbf{x}||^2||\mathbf{y}||^2-|\mathbf{x}^H\mathbf{y}|^2 } >0$, $\lambda_2=-\lambda_1<0$. We also note immediately from that when, $\mathbf{y}=\beta \mathbf{x}$, $\lambda_1 =(1-|\beta|^2)||\mathbf{x}||^2$, $\lambda_2=0$. This completes the proof. It is readily shown that $g(\mathbf{w}_k,\mathbf{s}_k)-g(\mathbf{w}_{k+1},\mathbf{s}_k)=\mathrm{Tr}\{\mathbf{R_u}(\mathbf{s}_k) ( \mathbf{w}_k\mathbf{w}_k^H-\mathbf{w}_{k+1} \mathbf{w}_{k+1}^H) \}$. Therefore, from Lem. \[lemma3\], and Lem. \[lemma4\], we have, $$\begin{aligned} \label{eq40} &\leq \lambda_{\max}\big( \mathbf{R_u}( \mathbf{s}_k) \big)\lambda_{-}(\mathbf{w}_k\mathbf{w}_k^H-\mathbf{w}_{k+1} \mathbf{w}_{k+1}^H) \nonumber\\ &+\lambda_{\min}\big( \mathbf{R_u}( \mathbf{s}_k) \big)\lambda_{+}(\mathbf{w}_k\mathbf{w}_k^H-\mathbf{w}_{k+1} \mathbf{w}_{k+1}^H) \nonumber \nonumber \\ &g(\mathbf{w}_k,\mathbf{s}_k)-g(\mathbf{w}_{k+1},\mathbf{s}_k) \\ &\leq \lambda_{\max}\big( \mathbf{R_u}( \mathbf{s}_k) \big)\lambda_{+}(\mathbf{w}_k\mathbf{w}_k^H-\mathbf{w}_{k+1} \mathbf{w}_{k+1}^H) \nonumber\\ &+\lambda_{\min}\big( \mathbf{R_u}( \mathbf{s}_k) \big)\lambda_{-}(\mathbf{w}_k\mathbf{w}_k^H-\mathbf{w}_{k+1} \mathbf{w}_{k+1}^H) \nonumber\end{aligned}$$ It is not immediately evident from the analysis which set of bounds in , , are tight, hence combining them we have $$\begin{aligned} &\max\left\{ \begin{aligned} g_{lb}^1( \mathbf{R_u}( \mathbf{s}_k), \mathbf{w}_k, \mathbf{w}_{k+1}), \;\; &g_{lb}^2( \mathbf{R_u}( \mathbf{s}_k), \mathbf{w}_k, \mathbf{w}_{k+1}), \\ &g_{lb}^3( \mathbf{R_u}( \mathbf{s}_k), \mathbf{w}_k, \mathbf{w}_{k+1}) \end{aligned}\right\} \\ &\leq g(\mathbf{w}_k,\mathbf{s}_k)-g(\mathbf{w}_{k+1},\mathbf{s}_k) \leq \\ &\min\left\{ \begin{aligned} g_{ub}^1( \mathbf{R_u}( \mathbf{s}_k), \mathbf{w}_k, \mathbf{w}_{k+1}), \;\; &g_{ub}^2( \mathbf{R_u}( \mathbf{s}_k), \mathbf{w}_k, \mathbf{w}_{k+1}), \\ &g_{ub}^3( \mathbf{R_u}( \mathbf{s}_k), \mathbf{w}_k, \mathbf{w}_{k+1}) \end{aligned}\right\} \\\end{aligned}$$ where $g_{lb}^i (\mathbf{R_u}( \mathbf{s}_k), \mathbf{w}_k, \mathbf{w}_{k+1})$, $g_{ub}^i (\mathbf{R_u}( \mathbf{s}_k), \mathbf{w}_k, \mathbf{w}_{k+1})$, $i=1,2,3$ are the lower and upper bounds as given in -, , for $i=1,2,3$, respectively. Similar upper and lower bounds can be readily derived for the other corresponding terms, $g(\mathbf{w}_{k+1},\mathbf{s}_k)-g(\mathbf{w}_{k+1},\mathbf{s}_{k+1})$ using analysis presented thus far, and is not the focus now. Let us however denote these corresponding lower and upper bounds to be $h_{lb}^i (\mathbf{R_u}( \mathbf{s}_k), \mathbf{w}_k, \mathbf{w}_{k+1})$, $ h_{ub}^i (\mathbf{R_u}( \mathbf{s}_k), \mathbf{w}_k, \mathbf{w}_{k+1})$, $i=1,2,3$. Constrained proximal alternating minimization --------------------------------------------- The proximal version of the constrained alternating minimization is iterative, and for the filter design step, optimizes at the $k$-th iteration, $$\begin{aligned} \label{eq41} \min_{\mathbf{w} }\;\;\;\;\; &\mathbf{w}^H\mathbf{R_u}( \mathbf{s}_{k-1})\mathbf{w}+\frac{\alpha_{k-1}}{2} || \mathbf{w}-\mathbf{w}_{k-1} ||^2\\ \mbox{ s. t } \;\;\;\;\;&\mathbf{w}^H(\mathbf{v}(f_d)\otimes\mathbf{s}_{k-1}\otimes\mathbf{a}(\theta_t,\phi_t))=\kappa \nonumber $$ where $\alpha_{k-1} \in \mathbbm{R}^{+}$ can be seen as a weight attached to the regularizer / penalizer $|| \mathbf{w}-\mathbf{w}_{k-1} ||^2$. This parameter can be interpreted as follows, if it is small, it encourages the optimizer to look for viable solutions in the vicinity of $\mathbf{w}_{k-1}$. However, if large, it penalizes the optimizer heavily for focusing even slightly in the immediate vicinity of $\mathbf{w}_{k-1}$. In a similar spirit, the proximal version of the constrained alternating minimization for the waveform design step at the $k$-th iteration optimizes, $$\begin{aligned} \label{eq42} \min\limits_{\mathbf{s}} \;\;\;\;\; &\mathbf{w}^H_{k}\mathbf{R_u}(\mathbf{s})\mathbf{w}_{k} +\frac{\beta_{k-1}}{2} ||\mathbf{s}-\mathbf{s}_{k-1} ||^2\nonumber \\ \mbox{s. t. }\;\;\;\;\; & \mathbf{w}^H_{k}(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))=\kappa \\ \;\;\;\;\;\; & \mathbf{s}^H \mathbf{s}\leq P_o \nonumber \nonumber\end{aligned}$$ where $\beta_{k-1}\in\mathbbm{R}^{+}$ is the weight attached to the regularizer $ ||\mathbf{s}-\mathbf{s}_{k-1} ||^2$. Bounds on $\alpha_{k-1},\beta_{k-1}$ relating it to the Lipschitz constants are deferred to forthcoming analysis. A graphical example comparing the constrained alternating minimization and the proximal constrained alternating minimization is shown in Fig. \[amcamfig\].  \[remark2\] The objective functions in , are still individually convex in $\mathbf{w}$, $\mathbf{s}$, respectively. The regularizer terms $|| \mathbf{w}-\mathbf{w}_{k-1} ||^2$ and $ ||\mathbf{s}-\mathbf{s}_{k-1} ||^2$ are strongly convex, and $\nabla^2_{\mathbf{w}} ( || \mathbf{w}-\mathbf{w}_{k-1} ||^2)=\mathbf{I}\succ \mathbf{0}$, $\nabla^2_{\mathbf{s}} ( || \mathbf{s}-\mathbf{s}_{k-1} ||^2)=\mathbf{I} \succ \mathbf{0}$, and therefore do not alter the individual convexity of $\mathbf{w}^H\mathbf{R_u}( \mathbf{s}_{k-1})\mathbf{w}$ and $\mathbf{w}^H_{k}\mathbf{R_u}(\mathbf{s})\mathbf{w}_{k}$, w.r.t. $\mathbf{w}$, $\mathbf{s}$, respectively. The solutions to , can be cast in terms of the proximal operator as $$\begin{aligned} \mathbf{w}_{k}=&\mathrm{prox}_{(\alpha_{k-1},\mathbf{w} )} \big( g(\mathbf{w},\mathbf{s}_{k-1}) ;\mathbf{w}_{k-1} \big) \label{eq43} \\ &\mbox{ s. t } \; \mathbf{w}^H(\mathbf{v}(f_d)\otimes\mathbf{s}_{k-1}\otimes\mathbf{a}(\theta_t,\phi_t))=\kappa \nonumber \\ \nonumber \\ \mathbf{s}_{k}=&\mathrm{prox}_{(\beta_{k-1},\mathbf{s} )} \big( g(\mathbf{w}_{k},\mathbf{s}) ;\mathbf{s}_{k-1} \big) \label{eq44} \\ &\mbox{s. t. } \; \mathbf{w}^H_{k}(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))=\kappa \nonumber \\ & \;\;\;\;\;\;\;\;\mathbf{s}^H \mathbf{s}\leq P_o \nonumber \end{aligned}$$ where, for a general $f(\mathbf{x}):\mathbbm{C}^N\rightarrow \mathbbm{R}$, the proximal operator is defined as $$\begin{aligned} \label{eq45} \mathrm{prox}_{(\alpha,\mathbf{x})} \big(f(\mathbf{x});\mathbf{y} \big):= \operatorname*{arg\,min}\limits_{\mathbf{x} } \; \mathbf{f} (\mathbf{x}) +\frac{\alpha}{2} ||\mathbf{x}-\mathbf{y} ||^2.\end{aligned}$$ The proximal operator has a rich history in the literature, and well documented properties, see for example [@Parikh2013; @Rockafeller1973; @Rockafeller1976; @Bertsekas1994]. A useful and interesting fact of this operator is that iff $\mathbf{x}_{o}$ minimizes $f(\mathbf{x}$) then $\mathbf{x}_{o}=\mathrm{prox}_{(\alpha,\mathbf{x})} (f(\mathbf{x});\mathbf{x}_{o})$, a proof is seen in [@Parikh2013]. [**Trust region interpretation**]{}. The objective now is to relate the unconstrained proximal minimization as in to a well known technique in numerical optimization. A generalized trust region subproblem can be formulated for $\mathbf{f}(\mathbf{x}):\mathbbm{C}^N\rightarrow \mathbbm{R}$ [@More93] $$\begin{aligned} \label{trustreg} \min \limits_{\mathbf{x}} \;\;\; &f( \mathbf{x}) \nonumber \\ \mbox{s. t. } \;\;\; & || \mathbf{U}\mathbf{x}-\mathbf{v}||^2 \leq \delta\end{aligned}$$ where $\mathbf{U}$, $\mathbf{v}$ are a general nonsingular matrix, and a vector, both characterizing the trust region. The positive scalar $\delta$ may be interpreted as a parameter which specifies the extent of the trust region. For $\mathbf{U}=\mathbf{I}$ and $\mathbf{v}=\mathbf{y}$, the proximal minimization as in and the trust region problem in are equivalent for specific values of $\alpha$ and $\delta$. In particular every solution of is a solution to for a particular $\delta$. In the same spirit, every solution to is an unconstrained minimizer to $f(\cdot)$ or a solution to for a particular $\alpha$, see also [@Rockafeller1976; @Parikh2013]. The proximal optimizations problems, , can be cast as equivalent constrained trust region subproblems, where for the $k$-th iteration, the trust region is characterized by the previous iteration, $\mathbf{w}_{k-1}$, $\mathbf{s}_{k-1}$, respectively. [**Closed form:**]{} A closed form solution to is readily derived, expressed as in $$\label{eq46} \begin{aligned} \mathbf{w}_{k}&=\big( \mathbf{R}_{\bf u}(\mathbf{s}_{k-1})+\frac{\alpha_{k-1}}{2}\mathbf{I} \big)^{-1} \big( \frac{\alpha_{k-1}}{2}\mathbf{w}_{k-1}-\frac{\gamma_4^{\ast}}{2} \big(\mathbf{v}(f_d)\otimes\mathbf{s}_{k-1}\otimes\mathbf{a}(\theta_t,\phi_t) \big) \big) \\ \gamma_4&=\frac{\alpha_{k-1} \mathbf{w}_{k-1}^H \big( \mathbf{R}_{\bf u}(\mathbf{s}_{k-1})+\dfrac{\alpha_{k-1}}{2}\mathbf{I} \big)^{-1} \big(\mathbf{v}(f_d)\otimes\mathbf{s}_{k-1}\otimes\mathbf{a}(\theta_t,\phi_t) \big)-2\kappa }{\big(\mathbf{v}(f_d)\otimes\mathbf{s}_{k-1}\otimes\mathbf{a}(\theta_t,\phi_t) \big)^H \big( \mathbf{R}_{\bf u}(\mathbf{s}_{k-1})+\dfrac{\alpha_{k-1}}{2}\mathbf{I} \big)^{-1} \big(\mathbf{v}(f_d)\otimes\mathbf{s}_{k-1}\otimes\mathbf{a}(\theta_t,\phi_t) \big)} \end{aligned}$$ where $\gamma_4$ is the Lagrange parameter associated with . The solution to is also in closed form and the procedure to obtain it is similar to that used in deriving . Assuming that the Lagrange parameters for are $\gamma_5=\gamma_{5r}+j\gamma_{5i},\,\gamma_6\in\mathbbm{R}^{+}$, the solution is expressed in , $$\label{eq47} \mathbf{s}_{k}=\big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}+\gamma_6\mathbf{I}\big)^{-1} \big(\frac{\beta_{k-1}}{2}\mathbf{s}_{k-1} -\frac{\gamma_5}{2}\mathbf{G}^H\mathbf{w}_{k}\big)$$ where, $$\begin{aligned} \gamma_{5r}&=2\frac{\frac{\beta_{k-1}} {2}\mathrm{Re} \left\{\mathbf{w}_{k}^H\mathbf{G} \big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}+\gamma_6\mathbf{I}\big)^{-1} \mathbf{s}_{k-1}\right\}-\kappa} {\mathbf{w}_{k}^H\mathbf{G} \big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}+\gamma_6\mathbf{I}\big)^{-1} \mathbf{G}^H\mathbf{w}_{k} } \\ \gamma_{5i}&=\frac{\beta_{k-1} \mathrm{Im} \left\{\mathbf{w}_{k}^H\mathbf{G} \big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}+\gamma_6\mathbf{I}\big)^{-1} \mathbf{s}_{k-1}\right\} }{\mathbf{w}_{k}^H\mathbf{G} \big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}+\gamma_6\mathbf{I}\big)^{-1} \mathbf{G}^H\mathbf{w}_{k} }. \end{aligned}$$ The Lagrange parameter $\gamma_6$ is obtained by solving, the following $$\begin{aligned} \label{eq48} \gamma_6 r(\gamma_6)=0,\;\gamma_6\geq0\end{aligned}$$ obtained from the complementary slackness constraint on the Lagrange dual and where, $$\begin{aligned} r(\gamma_6)&=(P_o-\frac{\beta_{k-1}^2}{4} a_k) \bigg( \mathbf{w}_{k}^H\mathbf{G} \big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}+\gamma_6\mathbf{I}\big)^{-1} \mathbf{G}^H\mathbf{w}_{k} \bigg)^2 \\ -&2\big( b_i\frac{db_i}{d\gamma_6} +(b_r-\kappa) \frac{db_r}{d\gamma_6}\big) \mathbf{w}_{k}^H\mathbf{G} \big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}+\gamma_6\mathbf{I}\big)^{-1} \mathbf{G}^H\mathbf{w}_{k} \\ -&(b_{i}^2+(b_r-\kappa)^2) \mathbf{w}_{k}^H\mathbf{G} \big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}+\gamma_6\mathbf{I}\big)^{-2} \mathbf{G}^H\mathbf{w}_{k}. \end{aligned}$$ Where we also define $$\begin{aligned} a_{k}=&\mathbf{s}_{k-1}^H\big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}+\gamma_6\mathbf{I}\big)^{-1} \mathbf{s}_{k-1} \\ b_{r}&=\frac{\beta_{k-1}}{2}\mathrm{Re}\left\{ \mathbf{w}_{k}^H\mathbf{G} \big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}+\gamma_6\mathbf{I}\big)^{-1} \mathbf{s}_{k-1}\right\} \\ b_{i}&=\frac{\beta_{k-1}}{2}\mathrm{Im}\left\{ \mathbf{w}_{k}^H\mathbf{G} \big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}+\gamma_6\mathbf{I}\big)^{-1} \mathbf{s}_{k-1}\right\} \end{aligned}$$ Further, since the derivative, $\mathrm{Re}\{ \cdot\},\mathrm{Im}\{ \cdot\}$ are all linear we also have $$\begin{aligned} \dfrac{db_r}{d\gamma_6}&=-\frac{\beta_{k-1}}{2} \mathrm{Re}\left\{ \mathbf{w}_{k}^H\mathbf{G} \big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}+\gamma_6\mathbf{I}\big)^{-2} \mathbf{s}_{k-1}\right\} \\ \dfrac{db_i}{d\gamma_6}&=-\frac{\beta_{k-1}}{2} \mathrm{Im}\left\{ \mathbf{w}_{k}^H\mathbf{G} \big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}+\gamma_6\mathbf{I}\big)^{-2} \mathbf{s}_{k-1}\right\}. \end{aligned}$$ \[propos5\] In general $r(\gamma_6)$ is not monotone and there exist one or more zero crossings excluding $\gamma_6=\infty$. However in our extensive numerical simulations, and assuming $P_o>>\kappa^2, \gamma_6=0$ solves . It is readily seen that $\lim \limits_{\gamma_6\rightarrow 0} r(\gamma_6)=r(0)\neq 0,\lim \limits_{\gamma_6\rightarrow \infty} r(\gamma_6)=0,\lim \limits_{\gamma_6\rightarrow \infty} \gamma_6 r(\gamma_6)=0$. Nevertheless unlike Prop. \[propos3\], Rem. \[propos5\] is not straightforward to demonstrate analytically, however can be shown numerically. See Section IV for some demonstrative examples not specific to the radar problem. The value of $\gamma_6=0$ is substituted in to obtain the final waveform solution $\mathbf{s}_{k}(\cdot)$. \[remstrongdual\] ([*Strong duality*]{}) The primal problem, and its associated dual have zero duality gap. This is straightforward but tedious to show. However we provide the optimal values attained by the primal as well as the dual, given below, $$\begin{aligned} \label{dualopt2} \mathbf{w}_k^H(\mathbf{R_i+R_n })\mathbf{w}_k&+\mathbf{s}_k^{\ast H} \big( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I}\big)^{-1} \mathbf{s}_k^{\ast} \nonumber \\ &+\frac{\beta_{k-1} || \mathbf{s}_{k-1}||^2}{2} -\beta_{k-1}\mbox{Re}\{ \mathbf{s}_{k}^{\ast H} \mathbf{s}_{k-1}\}\end{aligned}$$ where using , Prop. \[propos5\], $$\begin{aligned} \mathbf{s}_{k}^{\ast}&=(\sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I})^{-1} ( \frac{\beta_{k-1}}{2}\mathbf{s}_{k-1} -\frac{\gamma_5}{2}\mathbf{G}^H\mathbf{w}_{k}) \\ \gamma_5&=\frac{\beta_{k-1}\mathbf{w}_{k}^H\mathbf{G} ( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I})^{-1} \mathbf{s}_{k-1}-2\kappa} {\mathbf{w}_{k}^H\mathbf{G} ( \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}_{k})+\frac{\beta_{k-1}}{2}\mathbf{I})^{-1} \mathbf{G}^H\mathbf{w}_{k} }. \end{aligned}$$ This is not surprising since it is similar to Rem. \[dualprop\]. However, in this case the condition on the existence of the matrix is irrelevant, since the inverse in always exists. Hence Slater’s condition now is a simple constraint qualifier (the power constraint) which must be satisfied as in Rem. \[dualprop\]. [**Interpretation with specific ranges of $\alpha_{k-1},\,\beta_{k-1}$ and related to the Lipschitz constants**]{}. Some definitions and lemmas are useful for future discussions and are expressed below \[mydef2\] ([*Lipschitz continuous gradient*]{}) A function $f(\mathbf{\bar{x}}) :\mathbbm{R}^N\rightarrow\mathbbm{R}$ has a Lipschitz constant (and trivially real positive), $\mathtt{L}$, when $||\nabla_{\mathbf{\bar{x}}}f(\mathbf{\bar{x}})-\nabla_{\mathbf{\bar{y}}}f(\mathbf{\bar{y}})|| \leq\mathtt{L} || \mathbf{\bar{x}-\bar{y}}||$, and $\forall \mathbf{\bar{x}}$, $\mathbf{\bar{y}}\in \mathbbm{R}^N$. [**Note:**]{} ([*upper bound on Hessian*]{} ) If $f(\mathbf{\bar{x}})$ has a Lipschitz continuous gradient, with constant $\mathtt{L}$, then using Taylor’s theorem, it can be proved that $\nabla_{\mathbf{\bar{x}}}^2 f(\mathbf{\bar{x}})\preceq \mathtt{L}\mathbf{I}$. \[remark3\] The Lipschitz constant for $f(\mathbf{\bar{x}})=\mathbf{\bar{x}}^T\mathbf{\bar{B}}\mathbf{\bar{x}}$ is the maximum eigenvalue of $\mathbf{\bar{B}}$, i.e. $\lambda_{\max}(\mathbf{\bar{B}})$, where $\mathbf{\bar{B}}\in\mathbbm{R}^{N\times N}, \; \mathbf{\bar{x}}\in\mathbbm{R}^N$. This is readily seen since $\nabla_{\mathbf{\bar{x}}}\mathbf{\bar{x}}^T\mathbf{\bar{B}\bar{x}}=\mathbf{\bar{B}\bar{x}}$. Further since the induced (by an arbitrary $\mathbf{\bar{z}} \in \mathbbm{R}^N$) spectral norm (notation: $||| \cdot |||$) is defined as $$\begin{aligned} |||\mathbf{\bar{B}}|||:=\sup\limits_{\mathbf{\bar{z}}} \{ \frac{||\bf \bar{B}\bar{z}||}{||\bf \bar{z}||} : \mathbf{\bar{z}}\in\mathbbm{R}^N,\mathbf{\bar{z}}\neq\mathbf{0} \},||{\bf \bar{B}\bar{z}}||=\sqrt{ \mathbf{\bar{z}}^T\mathbf{\bar{B}}^T\mathbf{\bar{B}\bar{z}}}\end{aligned}$$ but we know from Lem. \[lemma2\] that $\mathbf{\bar{z}}^T\mathbf{\bar{B}\bar{z}}\leq \lambda_{\max}( \mathbf{\bar{B}})|| \bf \bar{z}||^2$ and that eigenvalues of $\mathbf{\bar{B}}$ and $\mathbf{\bar{B}}^T$ are identical. This further implies that $\mathbf{\bar{z}}^T\mathbf{\bar{B}}^T\mathbf{\bar{B}\bar{z}}\leq \lambda_{\max}^2( \mathbf{\bar{B}})|| \bf \bar{z}||^2$. Therefore from Definition \[mydef2\], it is readily seen that the Lipschitz constant is the maximum eigenvalue of $\mathbf{\bar{B}}$. \[lemma5\] ([*Descent lemma*]{}) If $f(\mathbf{\bar{x}}) :\mathbbm{R}^N\rightarrow\mathbbm{R}$ is continuously differentiable and has a Lipschitz continuous gradient described by constant $\mathtt{L}$, then $f(\mathbf{\bar{x}})\leq f(\mathbf{\bar{y}})+\nabla_{\mathbf{\bar{y}}}f(\mathbf{\bar{y}})^T(\mathbf{\bar{x}}-\mathbf{\bar{y}}) +\frac{\mathtt{L}}{2}||\mathbf{\bar{x}}-\mathbf{\bar{y}}||^2$. See [@Bertsekas1999 Prop. A.24] and also [@Beck2013 Lem2.2] relevant in general for the BCD. Consider an arbitrary $g(\mathbf{x}):= \mathbf{x}^H\mathbf{Bx}$, and $\mathbf{B}=\mathbf{B}^H$, $\mathbf{x}\in\mathbbm{C}^N$. Since $g(\mathbf{x}):\mathbbm{C}^N\rightarrow\mathbbm{R}$, a real equivalent of $g(\mathbf{x})$ could be defined as $\bar{g}(\bar{\mathbf{x}}):=\bar{\mathbf{x}}^T\bar{\mathbf{B} }\bar{\mathbf{x}}$ where $$\bar{\mathbf{B}}:= \left[ \begin{matrix} \mbox{Re}\{ \mathbf{B} \} & -\mbox{Im} \{\mathbf{B} \} \\ \mbox{Im} \{\mathbf{B} \} & \mbox{Re}\{ \mathbf{B} \} \end{matrix} \right] \in\mathbbm{R}^{2N\times 2N}, \;\; \bar{\mathbf{x}}=[\mbox{Re} \{ \mathbf{x} \}^T \mbox{Im} \{ \mathbf{x} \}^T ]^T \in \mathbbm{R}^{2N}.$$ \[lemma6\] The matrix $\bar{\mathbf{B}} :=\left[ \begin{smallmatrix} \mathrm{Re}\{ \mathbf{B} \} & -\mathrm{Im} \{\mathbf{B} \} \\ \mathrm{Im} \{\mathbf{B} \} & \mathrm{Re}\{ \mathbf{B} \} \end{smallmatrix} \right]\in\mathbbm{R}^{2N\times 2N}$ and $ \left[ \begin{smallmatrix} \mathbf{B} & \mathbf{0} \\ \mathbf{0} & \mathbf{B}^{\ast} \end{smallmatrix} \right] \in\mathbbm{C}^{2N\times 2N} $ have identical eigenvalues, $\tilde{\lambda}_i,i=1,2,\ldots,2N$. Moreover, if $\mathbf{B}$ is Hermitian, then $\tilde{\lambda}_i \in \mathbbm{R}^{+},i=1,2,\ldots,2N$ are equal to twice the multiplicity of the eigenvalues of $\mathbf{B}\in\mathbbm{C}^{N\times N}$. Owing to the complex to real-real isomorphism, it can be shown after algebraic manipulations that $$\begin{aligned} \label{eq49} \left[ \begin{matrix} \mathbf{B} & \mathbf{0} \\ \mathbf{0} & \mathbf{B}^{\ast} \end{matrix} \right] =\mathbf{P}^{H}\bar{\mathbf{B}}\mathbf{P}, \;\; \mathbf{P}=\frac{1}{\sqrt{2}}\left[ \begin{matrix} j\mathbf{I} & \mathbf{I} \\ \mathbf{I} & j\mathbf{I} \end{matrix} \right], \;\; \mathbf{P}^H=\mathbf{P}^{-1}.\end{aligned}$$ That is indicates that $\bar{\mathbf{B}}$ and $\left[ \begin{smallmatrix} \mathbf{B} & \mathbf{0} \\ \mathbf{0} & \mathbf{B}^{\ast} \end{smallmatrix} \right]$ are unitary equivalent. Therefore they share the same eigenvalues. Furthermore if $\mathbf{B}$ is Hermitian its eigenvalues are purely real, and hence trivially, the eigenvalues of $\mathbf{B}$, $ \mathbf{B}^{\ast}$ are identical, and their eigenvectors are complex conjugates of one another. Hence the block diagonal matrix has identical eigenvalues as $\mathbf{B}$ but with multiplicity two. Consider the objective in , . Define $\bar{g}( \mathbf{\bar{w}},\mathbf{\bar{s}}_{k-1}),\bar{g}(\mathbf{\bar{w}}_{k-1},\mathbf{\bar{s}}_{k-1})$ as the real equivalents of $g( \mathbf{w},\mathbf{s}_{k-1}),g (\mathbf{w}_{k-1},\mathbf{s}_{k-1})$, respectively for the filter design objective as in . In addition, denote $\mathtt{L}_{1k-1}$ as the Lipschitz constant associated with $\bar{g}(\bar{\mathbf{w}}_{k-1},\mathbf{\bar{s}}_{k-1})$. Similarly using the same notation and for the objective in the waveform design objective as in consider the real equivalents, $\bar{g}( \mathbf{\bar{s}},\mathbf{\bar{w}}_{k}),\bar{g}(\mathbf{\bar{s}}_{k-1},\mathbf{\bar{w}}_{k})$ and the Lipschitz constant denoted as $\mathtt{L}_{2k-1}$. Then the following inequalities can now be shown. $$\label{eq50} \begin{aligned} &\bar{g}( \mathbf{\bar{w}})+\frac{\mathtt{L}_{1k-1}}{2}||\mathbf{\bar{w}}_{k-1}-\mathbf{\bar{w}} ||^2 \geq \bar{g}(\mathbf{\bar{w}}_{k-1}) \\ +&\nabla \bar{g}(\mathbf{\bar{w}}_{k-1})^T(\mathbf{\bar{w}}-\mathbf{\bar{w}}_{k-1}) +\frac{\mathtt{L}_{1k-1}}{2}||\mathbf{\bar{w}}_{k-1}-\mathbf{\bar{w}}||^2 \geq \bar{g}(\mathbf{\bar{w}}) \end{aligned}$$ $$\label{eq51} \begin{aligned} &\bar{g}( \mathbf{\bar{s}})+\frac{\mathtt{L}_{2k-1}}{2}||\mathbf{\bar{s}}_{k-1}-\mathbf{\bar{s}} ||^2 \geq \bar{g}(\mathbf{\bar{s}}_{k-1}) \\ +&\nabla \bar{g}(\mathbf{\bar{s}}_{k-1})^T(\mathbf{\bar{s}}-\mathbf{\bar{s}}_{k-1}) +\frac{\mathtt{L}_{2k-1}}{2}||\mathbf{\bar{s}}_{k-1}-\mathbf{\bar{s}}||^2 \geq \bar{g}(\mathbf{\bar{s}}) \end{aligned}$$ where in , the known’s $\mathbf{\bar{s}}_{k-1}$ and in , the known’s $\mathbf{\bar{w}}_{k}$ are respectively treated as constants, therefore suppressed in notation for brevity. We further note that , are tight, i.e. for $\mathbf{\bar{w}}_{k}=\mathbf{\bar{w}}_{k-1}$, $\mathbf{\bar{s}}_{k}=\mathbf{\bar{s}}_{k-1}$ the inequalities are strict equality’s. The Lipschitz constants, $\mathtt{L}_{1k-1}$, $\mathtt{L}_{2k-1}$ are readily derived using Lem. \[lemma6\]. \[remark4\] It is readily seen that if $\alpha_{k-1}\geq \mathtt{L}_{1k-1}$ and $\beta_{2k-1}\geq \mathtt{L}_{2k-1}$ the inequalities in , are valid by replacing $\mathtt{L}_{1k-1},\mathtt{L}_{2k-1}$ with $\alpha_{k-1},\beta_{k-1}$, respectively. The term in the first inequalities of , are the proximal minimization objectives with $\alpha_{k-1}=\mathtt{L}_{1k-1},\beta_{k-1}=\mathtt{L}_{2k-1}$. The inequalities of , are obtained from first applying the convexity Def. \[mydef1\](b) (first order definition) and then subsequently adding the respective terms $\tfrac{\mathtt{L}_{1k-1}}{2}||\mathbf{\bar{w}}_{k-1}-\mathbf{\bar{w}} ||^2$, $ \tfrac{\mathtt{L}_{2k-1}}{2}||\mathbf{\bar{s}}_{k-1}-\mathbf{\bar{s}}||^2$ and then using Lem. \[lemma5\], the descent lemma. Additionally, it is recalled that the functions associated with the second inequalities of , are the (unconstrained) objectives which are minimized by the gradient descent with step size $\mathtt{L}_{1k-1}$, $\mathtt{L}_{2k-1}$, respectively. That is, the new iterations are then $\mathbf{\bar{w}}_{k}=\mathbf{\bar{w}}_{k-1}-\frac{1}{\mathtt{L}_{1k-1}} \nabla_{\mathbf{\bar{w}}}\bar{g}( \mathbf{\bar{w}})$, and $\mathbf{\bar{s}}_{k}=\mathbf{\bar{s}}_{k-1}-\frac{1}{\mathtt{L}_{2k-1}} \nabla_{\mathbf{\bar{s}}}\bar{g}( \mathbf{\bar{s}})$. Therefore from , and Rem. \[remark4\] [*we note that the proximal objective, the gradient descent objective are all surrogate albeit tight upper bounds on the true objective $\forall \alpha_{k-1}\geq\mathtt{L}_{1k-1}$ and $\forall \beta_{k-1}\geq\mathtt{L}_{2k-1}$*]{}. This interpretation is graphically depicted in Fig. \[fig3\] for the filter design objective as in but for $\alpha_{k-1}=\mathtt{L}_{1k-1}$. A similar graphic interpretation is obvious for the waveform design stage and is therefore not shown. [**Tikhonov interpretation**]{} This interpretation is immediate from , . In fact from , , the quadratic regularizers $||\mathbf{w}-\mathbf{w}_{k-1} ||^2, || \mathbf{s}-\mathbf{s}_{k-1} ||^2$ are essentially Tikhonov regularization terms. Geometrically they are spheres centered at $\mathbf{w}_{k-1}$, $\mathbf{s}_{k-1}$ and encourage the current iterates to be in the vicinity of the previous iterates. Furthermore, since in the limit, the regularizer terms only decrease, this may be also seen as a vanishing Tikhonov regularization problem [@Parikh2013] for each iteration in both the waveform and the filter vectors. [**Proximal minimization: A training data starved STAP solution**]{} The regularization in , leads to [*diagonally loaded*]{} solutions , when compared to the constrained alternating minimization solutions as in and . In particular, the diagonal loading serves two important purposes, [*firstly it offers a numerically stable solution by conditioning . Secondly and more importantly, it permits a weight vector solution when $\mathrm{rank}( \mathbf{R_u}(\mathbf{s}))\leq NML$*]{}. Practical STAP contends with rank deficient correlation matrices due to lack of sufficient training data from neighboring range cells due to outlier contamination or heterogeneity in the data. The solution in ameliorates over the training data starved STAP scenarios. So far, we have considered the algorithms for waveform design without enforcing constraints such as const. modulus or sidelobe constraints. The minimum eigenvector solution belongs to this class of unconstrained waveform design. We will revisit this design by considering and Lem. \[lemma2\]. \[remark5\] The min. eigenvector solution in is still optimal in the presence of clutter, provided $\mathbf{R_i}+\mathbf{R_n}$ and $\mathbf{R_c}(\mathbf{s})$ share the same eigenvector corresponding to their min.eigenvalues, but with $\lambda_{\min}(\mathbf{R_c}(\mathbf{s}))=0$, always. This is readily seen since the optimization in , ignoring the constraint for now could be recast as $\max \limits_\mathbf{s} (\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))^H\mathbf{R}_{\bf u}^{-1}(\mathbf{s})(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))$. Now using Woodbury’s identity [@Kayest1998], we have $$\label{eqeigmin} \begin{aligned} &(\mathbf{R_i}+\mathbf{R_n}+\mathbf{R_u}(\mathbf{s}))^{-1} =(\mathbf{R_i}+\mathbf{R_n})^{-1} \\ -&(\mathbf{R_i}+\mathbf{R_n})^{-1}\mathbf{R_c}(\mathbf{s} ) \bigl( \mathbf{I}+ (\mathbf{R_i}+\mathbf{R_n})^{-1} \mathbf{R_c}(\mathbf{s})\bigr) ^{-1}(\mathbf{R_i}+\mathbf{R_n})^{-1}. \end{aligned}$$ Further using the eigenvector relations, $(\mathbf{R_i}+\mathbf{R_n})(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))=\lambda_{\min}(\mathbf{R_i}+\mathbf{R_n})(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))$ and $\mathbf{R_c} (\mathbf{s}) (\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))=\lambda_{\min}(\mathbf{R_c}( \mathbf{s}))(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))=\mathbf{0}$ in , it is readily seen that $(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))^H(\mathbf{R_i}+\mathbf{R_n}+\mathbf{R_u}(\mathbf{s}))^{-1} )(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))=\lambda_{\min}^{-1}( \mathbf{R_i}+\mathbf{R_n})$. The simplest example where Rem. \[remark5\] is satisfied is when the noise correlation matrix is scaled identity (may not be practical for narrowband radar), clutter correlation matrix is low rank. In STAP and for ideal scenarios, insights to the clutter rank are obtained by the Brennan’s rule [@guerci2003; @klemm2002; @ward1994]. A high clutter rank prevails due to the practical effects such as, the intrinsic clutter motion,velocity misalignment and crabbing, mutual coupling and antennae element mismatches as well as clutter ambiguities in Doppler resulting in aliasing [@ward1994]. \[tbp!\] ![Upper bounds on the objective for the proximal algorithm w.r.t. the filter design. A similar graphical interpretation for the waveform design but with $\mathtt{L}_{2k-1}$ is also easy depicted but not shown here.[]{data-label="fig3"}](Lipschitz_upper "fig:") Constant modulus alternating minimization ----------------------------------------- So far, the optimization problems had no specific constraints (except the power/energy constraint) on the waveform, constant modulus is a desirable property to have in a waveform [@Setlurradar2014]. The optimum weight vector is unchanged by introducing the const. modulus constraint, and is identical to for the constrained alternating minimization [^7]. Since the optimization w.r.t. weight vector is unchanged, we only treat the optimization for $\mathbf{s}$ but with the const. mod. constraint for a fixed but arbitrary $\mathbf{w}$, formulated below $$\begin{aligned} \min\limits_{\mathbf{s}} \;\;\;\;\; &\mathbf{w}^H\mathbf{R_u}(\mathbf{s})\mathbf{w} \nonumber \\ \mbox{s. t. }\;\;\;\;\; & \mathbf{w}^H(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))=\kappa \label{eq52} \\ \;\;\;\;\;\; & |s_i|=\rho,i=1,2,\ldots,N. \nonumber \nonumber\end{aligned}$$ where $s_i$ is the $i$-th component in $\mathbf{s}$. Unlike say , notice that in , constraining the power of the waveform is unnecessary since $\rho$ is fixed but could be chosen arbitrarily to scale up / down the waveforms energy to satisfy hardware limitations. Therefore, the last $N$ constraints in implicitly impose the power requirements, but more importantly also impose the constant modulus constraint. The Lagrangian of is expressed as $$\begin{aligned} \label{lagraconmod} \mathcal{L}(\mathbf{s}, \gamma_7,\boldsymbol{\gamma}_5)&=\mathbf{w}^H\mathbf{R_u}(\mathbf{s})\mathbf{w} +\mbox{Re}\{ \gamma_7^{\ast} (\mathbf{w}^H\mathbf{Qs}-\kappa)\} \nonumber \\ &+\mathbf{s}^H\mathbf{D}_{\gamma}\mathbf{s} -\rho\mathbf{1}^T\boldsymbol{\gamma}_8\end{aligned}$$ where the Lagrange parameter, $\gamma_7\in\mathbbm{C}$, and the Lagrange parameter vector $\boldsymbol{\gamma}_8=[\gamma_{8_1},\gamma_{8_2},\ldots,\gamma_{8_N}]^T\in\mathbbm{R}^{N}$ are for the Capon constraint and the $N$ const. mod. constraints, respectively. Furthermore in , define $\mathbf{D}_{\gamma}= \left[ \begin{smallmatrix} \gamma_{8_1}& & \\ & \ddots & \\ & &\gamma_{8_N} \end{smallmatrix} \right]$, i.e. a diagonal matrix. The KKT conditions are expressed as \[kkt1\] $$\begin{aligned} \mathbf{s}_o(\mathbf{w})&=\frac{\kappa \bigl( \sum\limits_{q=1}^Q \mathbf{Z_q}( \mathbf{w})+\mathbf{D}_{\gamma}\bigr)^{-1}\mathbf{G}^H\mathbf{w} }{\mathbf{w}^H\mathbf{G} \bigl( \sum\limits_{q=1}^Q \mathbf{Z_q}( \mathbf{w}) +\mathbf{D}_{\gamma}\bigr)^{-1}\mathbf{G}^H\mathbf{w}} \\ |s_{oi}(\mathbf{w})|&=\rho,i=1,2,\ldots,N.\end{aligned}$$ The waveform which simultaneously satisfies (a)(b) is the solution. Moreover, note that (a)(b) are $2N$ non-linear equations with $2N$ unknowns. The first $N$ unknowns are $s_{oi}(\mathbf{w}),i=1,2\ldots,N$ and the next $N$ unknowns are the Lagrange parameters $\gamma_{8_i}$. Unfortunately, is not in closed form but can be solved numerically for the $N$ parameters, $\gamma_{8_i},i=1,2,\ldots,N$ via a numerical root finder. Nonetheless we note that $\gamma_{8_i}\in (-\infty,\infty)$ and a reasonable initialization point is not forthcoming for the numerical root finding. [**Eliminating the constant modulus constraints**]{} Instead of solving the $2N$ non-linear equations as in (a)(b), we take an alternative approach. One may reformulate the optimization by eliminating the last $N$ constraints, by imposing a structure on $\mathbf{s}$, namely, $s_i=\rho\exp(j\alpha_i)$. Other structures exists but from our experience, complex exponentials are the easiest to manipulate. The new optimization problem is now w.r.t. $\boldsymbol{\alpha}=[\alpha_1,\alpha_2,\ldots,\alpha_{N}]^T\in\mathbbm{R}^N$, expressed as $$\begin{aligned} \min\limits_{\boldsymbol{\alpha}} \;\;\;\;\; &\mathbf{w}^H\mathbf{R_u}(\mathbf{s})\mathbf{w} \nonumber \\ \mbox{s. t. }\;\;\;\;\; & \mathbf{w}^H(\mathbf{v}(f_d)\otimes\mathbf{s}\otimes\mathbf{a}(\theta_t,\phi_t))=\kappa \label{eq53}\end{aligned}$$ where in, $\mathbf{s}=\rho[\exp(j\alpha_1),\exp(j\alpha_2),\ldots,\exp(j\alpha_{N})]^T$ and $\alpha_i \in [0,2\pi),i=1,2,\ldots, N$. The Lagrangian corresponding to is $$\begin{aligned} \label{eq54} \mathcal{L}(\boldsymbol{\alpha},\gamma_9)=\mathbf{w}^H\mathbf{R_u}(\mathbf{s})\mathbf{w} +\mbox{Re}\{ \gamma_9^{\ast}(\mathbf{w}^H\mathbf{Gs}-\kappa)\}.\end{aligned}$$ The KKT’s are expressed as, $\tfrac{\partial\mathcal{L}(\boldsymbol{\alpha},\gamma_9)}{\partial \boldsymbol{\alpha}}=\mathbf{0}$ and $\mathbf{w}^H\mathbf{Gs}=\kappa$. Noting that $\boldsymbol{\alpha}$ is purely real, we have $$\label{eq56} \begin{aligned} \frac{\partial\mathcal{L}(\boldsymbol{\alpha},\gamma_9)}{\partial \boldsymbol{\alpha}}=-&j\sum\limits_{q=1}^Q\mathbf{Z_q}\mathbf{s}\odot \mathbf{s}^{\ast}+j\sum\limits_{q=1}^Q\mathbf{Z}_{\bf q}^{\ast}\mathbf{s}^{\ast}\odot \mathbf{s} \\ +&\mbox{Im}\{ \gamma_9^{\ast}(\mathbf{w}^H\mathbf{G} )^T\odot \mathbf{s}\} =\mathbf{0}. \end{aligned}$$ The above equation can be simplified as, $\mbox{Im}\{ \sum\limits_{q=1}^Q\mathbf{Z}_{\bf q}^{\ast}\mathbf{s}^{\ast}\odot \mathbf{s} -\frac{\gamma_9^{\ast}}{2}(\mathbf{w}^H\mathbf{G})^T\odot \mathbf{s}\}$. Using this in , and taking the complex conjugate, while absorbing the negative sign into the constant $\gamma_9$[^8], we have the KKTs in final form expressed as \[eq57\] $$\begin{aligned} \mbox{Im}\left\{ \left(\sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}) \mathbf{s}_o+\frac{\gamma_9}{2} \mathbf{G}^H \mathbf{w} \right)\odot \mathbf{s}_o^* \right\}&=\mathbf{0} \\ \mathbf{w}^H\mathbf{G}\mathbf{s}_o&=\kappa\end{aligned}$$ where $\mathbf{0}$ is a column vector of all zeros and of dimension $N$. The optimal solution, $\mathbf{s}_o$, is a function of the optimal $\boldsymbol{\alpha}_o$. This relationship although evident from is not explicitly stressed in for notational succinctness. Define $\mathbf{Z}_{\bf Q}:=\sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w})$ and let $z_{ij},i=1,2,\ldots,N,j=1,2\ldots,N$ be the $ij$-th element of $\mathbf{Z}_{\bf Q}$. Noting that $\mathbf{Z}_{\bf Q}$ is Hermitian, we also have $\mbox{Im}\{z_{ii}\}=0,\,\forall i$, $z_{ji}=z_{ij}^{\ast}$. \[propos6\] The Lagrange parameter $\gamma_9=0$ solves . For any $z\in\mathbbm{C}$, and any $\theta\in [0,2\pi]$, we have $\mbox{Im}\{ z\exp(j\theta)\}=\mbox{Re}\{z\}\sin(\theta)+\mbox{Im}\{z\}\cos(\theta)$. Using this and the fact that $\mathbf{Z}_{\bf Q}=\mathbf{Z}_{\bf Q}^H$, the $i$-th equation in (a) can be simplified as $$\label{eq58} \begin{aligned} &2\rho \big( \sum\limits_{j=1, j\neq i}^{N} \mbox{Re}\{ z_{ij}\}\sin(\alpha_{j}^o-\alpha_{i}^o )+\mbox{Im}\{z_{ij} \} \cos( \alpha_{j}^o-\alpha_{i}^o )\big)\\ &=\mbox{Im}\{ \gamma_9u_i\exp(-j\alpha_{i}^o)\}, i=1,2,\ldots,N \end{aligned}$$ where $u_i$ is the $i$-th element of $\mathbf{u}=\mathbf{G}^H\mathbf{w}$ Adding the $N$ equations in , it easily seen that $\sum \limits_{i=1}^N\mbox{Im}\{ \gamma_9u_i\exp(-j\alpha_i^o\}=0$ but we know from (b) that $\rho\sum \limits_{i=1}^N u_i\exp(-j\alpha_i^o)=\kappa$, where $\kappa\in\mathbbm{R}$. Therefore this implies that $\mbox{Im}\{\gamma_9\}=0$ or in other words, $\gamma_9 $ is purely real. Substituting this back into (a) and following the same arguments as before, this is possible if trivially $\rho=0$ or $\gamma_9=0$, the former is false since $\rho=0$ does not solve (b), therefore the latter must be true. [**Interpretation of $\gamma_9=0$**]{}. With $\gamma_9=0$, from (a) we have that $$\label{finalkkts_cm} \begin{aligned} \mbox{Im}\left\{ \sum\limits_{q=1}^Q\mathbf{Z_q}(\mathbf{w}) \mathbf{s}_o \right\}&=\mathbf{0} \\ \mathbf{w}^H\mathbf{G}\mathbf{s}_o&=\kappa. \end{aligned}$$ The first equation in does not depend on $\rho$, but the second does. Therefore $\gamma_9=0$ does not imply that the constraint in is inactive. Rather, this implies that the KKTs enforce the Capon constraint in for the constant modulus waveform by varying the [*unspecified*]{} modulus parameter $\rho$. The result in Prop. \[propos6\] has some very interesting consequences. Using $\gamma_9=0$, the $N$ equations in and therefore (a), can be rewritten as a some linear matrix equation $\bar{\mathbf{Z}}_{\bf Q}\mathbf{p}_{\boldsymbol{\alpha}_{\bf o}}=\mathbf{0}$, where $\bar{\mathbf{Z}}_{\bf Q}\in\mathbbm{R}^{N\times\binom{N}{2}}$ and the vector $\mathbf{p}_{\boldsymbol{\alpha}_{\bf o}}=[\sin(\alpha_{2}^o-\alpha_{1}^o) ,\sin(\alpha_{3}^o-\alpha_{1}^o) \ldots, \sin(\alpha_{N}^o-\alpha_{N-1}^o), \cos(\alpha_{2}^o-\alpha_{1}^o),\ldots,\cos(\alpha_{N}^o-\alpha_{N-1}^o) ]^T$ i.e. has $\binom{N}{2}$ components consisting of sines and cosines of all possible differences of $\alpha_i^o-\alpha_j^o,\forall i, \forall j\neq i$. In other words, $\mathbf{p}_{\boldsymbol{\alpha}^{\bf o}}\in \mbox{null}\big( \bar{\mathbf{Z}}_{\bf Q}\big)$. The rank of $\bar{\mathbf{Z}}_{\bf Q}$ is not easy to calculate here but its maximum value is $N$. Therefore from the rank-nullity theorem, $\dim(\mbox{null}(\bar{\mathbf{Z}}_{\bf Q}))\geq N(N-2)$. Clearly there could exist multiple vectors which are in this null space but we are not certain if this translates to multiple solutions of $\boldsymbol{\alpha}_o$ from this linear equation alone. Nonetheless, if multiple solutions exist to this linear equation, they must also satisfy (b) to be considered as a solution to . In any case the optimal solution(s) are in, $\mathcal{C}_{\boldsymbol{\alpha}^{\bf o} }\subset\mathbbm{R}^{N}$, with $$\begin{aligned} \label{eq59} \mathcal{C}_{\boldsymbol{\alpha}^{\bf o} }=\{\boldsymbol{\alpha}^o:\mathbf{p}_{\boldsymbol{\alpha}_{\bf o}} \in \mbox{null}( \bar{\mathbf{Z}}_{\bf Q}),\sum\limits_{i=1}^N u_i^{\ast}\exp(j \alpha_i^o)=\frac{\kappa}{\rho} \}. \end{aligned}$$ It remains to be seen if $\mathcal{C}_{\boldsymbol{\alpha}^{\bf o} }$ is singleton, or comprises many elements, but we are optimistic that it would not turn out to be empty. Practical Considerations: Classical STAP v.s Waveform adaptive STAP ------------------------------------------------------------------- Here we addresses practical considerations on the fast time-slow time model in STAP which aids in the waveform design and compare this with the classical model in STAP (slow time). [**Hardware**]{} The fast-time slow-time model in STAP does not necessitate newer hardware nor does it require any modifications to the existing hardware. It does however assume that the current state-of-art permits arbitrary waveform generation and adaptive transmitting capabilities [@cochran2009waveform]. [**Computational complexity**]{} The inclusion of the waveform causes the correlation matrices to have larger dimension. Inverting large matrices are computationally prohibitive. Classical STAP requires inverting a complex $ML\times ML$ matrix which has a complexity of $O((ML)^{2.373})$-$O((ML)^{3})$ [@VirginiaWilliams2012]. Waveform adaptive STAP requires inverting complex $NML\times NML $ complex matrices which has a computational complexity of $O((NML)^{2.373})$-$O((NML)^{3})$ [@VirginiaWilliams2012]. [**Training data**]{} Due to the larger dimensions of the correlation matrices by inclusion of the waveform, it suddenly appears, albeit deceivingly, that more training data (from more neighboring range cells) are needed to estimate the correlation matrices. This is not true since inclusion of waveform simply includes the fast time samples. Hence the fast-time slow-time model uses the raw data prior to pulse compression or matched filtering, hence the training data requirements is identical to that required in the classical STAP case. Note that we are not interested in resolving targets within the pulse duration but rather outside it. \ \ \ ![Constrained alternating minimization: objective costs vs. iterations for 3 random, independent waveform initializations (inset: for 25 random initializations).[]{data-label="fig4"}](AMmono) ![Convergence of non const. mod initial waveform to a con. mod Con. waveform:[]{data-label="fig6"}](FJ_CM_noCMinit) Simulations {#sec:simulations} =========== First we will addresses simulations not specific to radar. Simulations supporting: Prop. \[propos3\] and Rem. \[propos5\] -------------------------------------------------------------- We ran simulations with random $z_n$ and random $d_n$ to analyze $f(\gamma_2)$ and $\gamma_2f(\gamma_2)$ numerically. In our extensive simulations we chose $z_n$ from complex normal distributions with different means and different variances. Since $d_n>0$ for all $n$, we used uniform distributions with different supports on the positive real axis excluding zero. We show only two representative simulation results for the monotonically increasing and decreasing cases in Fig. \[lagrasupport1\](a)(c), respectively. The corresponding function $\gamma_2f(\gamma_2)$ are also shown in Fig. \[lagrasupport1\](b)(d) for the two cases. Simulations for supporting Rem. \[propos5\] is presented next. Some parameters specifying the function $r(\gamma_6)$ were simulated randomly with the identical distributions used as in generating Fig. \[lagrasupport2\]. The parameter $\kappa=2,P_o=10$ was used in generating Fig. \[lagrasupport2\](a), the function $\gamma_6r(\gamma_6)$ is also shown in Fig. \[lagrasupport2\](b). As such, it is noted that $P_o=10$ is a a contrived example, typical radar applications will require $P_o$ to be in several hundred KW or several Hundred MW. The zero crossing is the intersection of the dashed line (black) with the blue curve in Fig. \[lagrasupport2\](a). Now using $P_o=20$ and keeping the other parameters fixed we obtain Fig. \[lagrasupport2\](c) which shows that $r(\gamma_6)$ is monotonic decreasing whose limit at $\infty$ is 0. [**Radar Specific simulations:**]{} Here onward, some parameters are common to all the simulation examples and are stated now. The simulation parameters are in SI units unless mentioned otherwise. To reduce computation complexity while inverting large matrices and computing their eigen-decompositions, we considered the number of, sensors, waveform transmissions, and fast time samples in the waveform as $M=5,L=32, N=5$, respectively. The carrier frequency was chosen to be 1GHz, and the radar bandwidth was 50MHz. The element spacing $d=\lambda_o/2$. Constrained alternating minimization ------------------------------------ The noise correlation matrix was assumed to have a correlation function given by $\exp(-|0.005n|),\; n=0,1,\ldots,NML$. Two interference sources were considered at $(\theta=0.3941,\phi=0.3)$ and at $(-0.4941,0.3)$. Both these interference sources had identical discrete correlation functions given by $0.2^{|n|},\; n=\pm 0, \pm 1,\ldots$. To simulate clutter we considered two clutter patches, consisting of five scatters each. The clutter correlation functions corresponding to the two patches were $\exp(-0.2|p|) \mbox{ and } \exp(-0.1|p|), \;p=\pm 0,\pm 1,\ldots,\pm P$. The rest of the parameters are identical to those used in [@Setlurradar2013]. In Fig. \[fig4\], the STAP beamformer objective vs. iterations are shown for 3 independent, random waveform initializations but the [*inset*]{} shows 25 independent initializations or trials. The alternating minimization was initialized with waveforms whose fast time samples are chosen independently from a standard complex Gaussian distribution. The algorithm was terminated as soon as the current waveform iterate invalidated the set power constraint. From the figure and its inset it is clear that the STAP beamformer output is non-increasing thereby validating the monotonicity property of this algorithm. More importantly from Fig. \[fig4\], we see that the final objective value and the iterations to reach it for each trial are different from one another, attributed to the joint non-convexity of the objective w.r.t. $\mathbf{w}$ and $\mathbf{s}$. Sensitivity to the random initialization is therefore duly noted. Constrained proximal alternating minimization --------------------------------------------- All the simulations parameters are identical to the previous case. The constrained alternating minimization was initialized with random waveforms as in Fig. \[fig4\], immediately followed by its proximal counterpart. The termination of the former algorithm was identical to the previous case, then, the latter was run for 200 iterations. Three representative trials are shown in Fig. \[fig5\](a)(b), for the constrained alternating minimization and its proximal counterpart. In Fig. \[fig5\](b), the dashed black lines are the final objective values obtained from the min. eigenvector waveform having the same energy as its proximal counterpart. For the three trials and not surprisingly, the proximal objective value, for all practical purposes, is identical to that obtained from the waveform derived from as evidenced from the [*inset*]{}. Therefore validating the implementation of both the constrained as well as its proximal counterpart. From Fig. \[fig5\](b) and unlike Fig. \[fig4\], three accumulation points w.r.t. the objective are clearly visible for the three trials indicating [*strong convergence*]{}. Constant modulus ---------------- The constant modulus algorithm was implemented numerically via the KKTs (i.e. ) and using the results from Prop. \[propos6\]. The simulation parameters are identical to the two previous scenarios. In Fig. \[fig6\], the modulus of the fast time waveform samples vs. iterations are shown for the constant modulus alternating minimization algorithm. As seen from this figure, the algorithm was initialized with a non-constant modulus waveform. For this random initialization, convergence to a constant modulus is achieved in three iterations or less. We have however encountered cases where the algorithm has not converged for several iterations. Nevertheless this problem was not encountered when the algorithm was initialized with a random constant modulus waveform. Thus in practice, it is advocated that this algorithm be initialized with an arbitrary constant modulus waveform, viz. a chirp, rectangular pulse, etc.. The ratio of the final objective for the constant modulus algorithm to the objective for the non-constant modulus waveform design using the constrained alternating minimization is seen in Fig. \[fig7\](a)(b) for 200 random waveform initializations. After convergence, not unexpectedly, the constant modulus objective is more than the non-constant modulus objective. This trend is readily observed from Fig. \[fig7\](a)(b) for the 200 trials. This is to be expected since constant modulus waveforms are a subset of their non-constant modulus counterparts. In particular, the amplitude is constrained temporally in the constant modulus design, while the phase is allowed to be optimized. Whereas, the phase and amplitude are both optimized the non-constant modulus design. From these figures we can see that on one end, this ratio is as much as 10dB, and on the other it is almost 0dB. Nonetheless on the average, the non-const. modulus waveforms have lower objective values than objective values derived from the const. modulus waveforms. Oracle sample support requirements ---------------------------------- The ideal SINR is $\tfrac{\rho_t^2|\mathbf{w}^H_{o}(\mathbf{v}(f_d)\otimes\mathbf{s}_{o}\otimes\mathbf{a}(\theta_t,\phi_t))|^2}{\mathbf{w}^H_{o}\mathbf{R_u}(\mathbf{s}_{o})\mathbf{w}_{o}}$ where $\mathbf{w}_o,\mathbf{s}_o$ are obtained after optimization. Using the estimated covariance matrix, say the sample covariance matrix, the definition of the estimated SINR is $\tfrac{\rho_t^2|\mathbf{w}^H_{est}(\mathbf{v}(f_d)\otimes\mathbf{s}_{est}\otimes\mathbf{a}(\theta_t,\phi_t))|^2}{\mathbf{w}^H_{est}\hat{\mathbf{R}}_{\bf u}(\mathbf{s}_{est})\mathbf{w}_{est}}$, where $\hat{\mathbf{R}}_{\bf u}(\cdot)$ is the estimated sample covariance matrix, and $\mathbf{w}_{est},\mathbf{s}_{est}$ are the optimized weight and waveform vectors by using the estimated covariance in the optimization instead. A true SINR loss can be computed by using the estimated i.e. $\hat{\mathbf{R}}_{\boldsymbol{\gamma}}^{pq}$ in and running the optimization algorithm for each Monte Carlo trial, resulting in an estimated $\mathbf{s}_{est}$. This is computationally heavy on our current resources, therefore not reported here. However, we will assume that an oracle has provided the optimal waveform to be transmitted. Then the oracle loss of SINR due to the estimated covariance is a random variable, captured by, $$\begin{aligned} SINR_{\mathrm{loss}}=\frac{\mathbf{w}^H_{o}\mathbf{R_u}(\mathbf{s}_{o})\mathbf{w}_{o}}{\mathbf{w}^H_{est}\hat{\mathbf{R}}_{\bf u}(\mathbf{s}_{o})\mathbf{w}_{est}} .\end{aligned}$$ Random data is now generated from zero mean multivariate complex Gaussian distributions to compute the sample covariance matrices, i.e. $\hat{\mathbf{R}}_{\bf i}, \hat{\mathbf{R}}_{\bf n}$ and $\hat{\mathbf{R}}_{\boldsymbol{\gamma}}^{pq}$. Two hundred Monte Carlo trials were run with differing sample supports. The mean and standard deviation of the oracle $SINR_{\mathrm{loss}}$ are shown in Fig. \[fig8\](a)(b). Not surprisingly the RMB rule is followed perfectly. For the same sample support, the standard deviation is a few orders less than the mean. Adapted patterns ---------------- The adapted pattern for the waveform dependent STAP objective function is expressed as $$\begin{aligned} \mathcal{P}(f_d,\theta)=\lvert\mathbf{w}_o^H ( \mathbf{v}(f_d)\otimes\mathbf{s}_o \otimes \mathbf{a}(\theta,\phi) ) \rvert^2,\;\mbox{ for a fixed }\phi. \label{eq60}\end{aligned}$$ The adapted pattern in is a function of angle, Doppler, the optimal weight and the waveform vectors, $\mathbf{w}_o,\mathbf{s}_o$, respectively. Two examples are shown in Fig. \[fig9\](a)(b). Two interferers at $(\theta=-0.2,\phi=\pi/3)$ and at $(-0.2,\pi/3)$ were chosen. We modeled the clutter discretely from all azimuth angles from $-\pi/2\mbox{ to } \pi/2$ in discrete increments of $-0.005\pi/2$ radians. The clutter patches were fixed at an elevation angle of $\pi/4$ radians. The target was assumed to be at $\theta_t=0.7,\phi_t=\pi/4$ with normalized Doppler equal to 0.31 and $\theta_t=0,\phi_t=\pi/4$ with normalized Doppler equal to -0.4 in Fig\[fig9\](a)(b), respectively. The adapted patterns in Fig. \[fig9\] are identical (upto a scaling) to those obtained from the classical STAP adapted pattern. This is not a surprise but is rather reassuring since the waveform in affects all the Doppler frequencies and the azimuths identically. Moreover, we can always consider $\mathbf{s}_o \otimes \mathbf{a}(\theta,\phi)$ as a new /modified spatial steering vector. Hence as expected the inclusion of the optimal waveform will not alter the shape of the classical STAP adapted pattern. Detection --------- Here, we investigate the impact of detection using the optimized waveforms and randomly selected waveforms. The detection test for the presence of a target at a particular range cell is cast as a binary hypothesis test, $$\label{eq61} \begin{aligned} &\mathcal{H}_0 \,\, : \mathbf{w}^H\bar{\mathbf{y}}=\mathbf{w}^H\mathbf{y_u} \,\,\, &\mathcal{H}_1\,\, : \mathbf{w}^H\bar{\mathbf{y}}=\mathbf{w}^H\mathbf{y}+\mathbf{w}^H\mathbf{y_u} \end{aligned}$$ where $\mathbf{y},$ $\mathbf{y_u}$ have been been defined in , . Assuming that $\mathbf{y_u}$ is complex normal distributed, the test in is readily evaluated. The weight vector is obtained after the optimization. The ROC curves for SINRs 0dB, 3dB and 6dB are shown in Fig. \[fig10\](a)(b) for the non const. modulus and const. modulus design, respectively. For generating Fig. \[fig10\](a), a random waveform was used having the same energy as that obtained after the alternating minimization algorithm. The waveform samples were drawn independently from a complex Gaussian distribution. In Fig. \[fig10\](b), a chirp waveform was used having the same bandwidth and energy as its optimized constant modulus counterpart. From these figures and as expected, from a detection standpoint, an optimized waveform performs much better than transmitting an un-optimized waveform. Realistic STAP waveform design ------------------------------ We consider a scenario frequently encountered in STAP, the sample covariance matrix is rank deficient due to the paucity of training data. The simulation parameters are identical to those used as in Fig. \[fig4\], except that we considered ground clutter from all azimuths in $[-\pi/2,\pi.2]$, similar to those used in generating Fig. \[fig9\]. Furthermore, we constrained the rank of the resulting correlation matrices to be 30, equal to the numerical rank of the clutter correlation matrix for generating Fig. \[fig11\]. The alternating minimization is first used for 20 iterations assuming an arbitrary diagonal loading factor equal to 100. After termination of this algorithm, the proximal algorithm was employed for 50 iterations. The results are shown in Fig. \[fig11\](a)(b). It is noted that in practice the ‘true’ min. eigenvector cannot be computed due to the rank deficiency. Interestingly nonetheless, the designed waveforms after the proximal optimization result in a STAP objective value which is close to that obtained from the waveform estimated from the ’true’ min. eigenvector. However, extensive simulations for the rank deficient STAP are needed to verify if this behavior is seen for other classes of noise plus interference, and clutter correlation matrices. Conclusions =========== Waveform design in STAP was the focus of this report assuming the dependence of the clutter response on the transmitted waveform. Our preliminary simulations indicate that the objective function was jointly non-convex in the weight and waveform vectors. However, we showed analytically that the objective function is individually convex in the waveform and the weight vector. This motivated a constrained alternating minimization technique which iteratively optimizes one vector while keeping the other fixed. A constrained proximal alternating minimization technique was propose to handle rank deficient STAP correlation matrices. To addresses practical design constraints we incorporated constant modulus constraints in our alternating minimization formulation. Simulations were chosen to demonstrate the monotonic decrease of the MVDR objective function using this alternating minimization algorithm. Preliminary simulations were presented to validate the theory. Acknowledgment {#acknowledgment .unnumbered} ============== This work was sponsored by US AFOSR under project 13RY10COR. All views and opinions expressed here are the authors own and does not constitute endorsement from the Department of Defense or the USAF. [^1]: P. Setlur is affiliated with the Wright State Research Inst., and as a research contractor with the US AFRL, WPAFB, OH, email:[email protected]. [^2]: M. Rangaswamy is with Sensors Directorate, U.S. AFRL, WPAFB, OH, email:[email protected]. [^3]: [Approved for Public Release No.: 88ABW-2014-3392]{}. [^4]: also see refs. Brandwood, and A. van den Bos in [@Gesbert2007] [^5]: Ideally one must decompose the function into real and imaginary components (as accomplished subsequently), but due to Hermitian symmetry, real valued-ness e.t.c., we take this shortcut, here, instead [^6]: Here $\succeq$ is the Löwner partial order [@horn1994] [^7]: The analysis of the proximal constrained alternating minimization with the const. mod. constraint is omitted, but can be readily derived from the analysis of its non-proximal counterpart, presented here. [^8]: new $\gamma_9$=old $-\gamma_9$.

--- abstract: 'We construct integral bases for the $SO(3)$-TQFT-modules of surfaces in genus one and two at roots of unity of prime order and show that the corresponding mapping class group representations preserve a unimodular Hermitian form over a ring of algebraic integers. For higher genus surfaces the Hermitian form sometimes must be non-unimodular. In one such case, genus $3$ and $p=5,$ we still give an explicit basis.' address: - | Department of Mathematics\ Louisiana State University\ Baton Rouge, LA 70803\ USA - | Institut de Mathématiques de Jussieu (UMR 7586 du CNRS)\ Université Paris 7 (Denis Diderot)\ Case 7012\ 2, place Jussieu\ 75251 Paris Cedex 05\ FRANCE - | Department of Mathematics\ Louisiana State University\ Baton Rouge, LA 70803\ USA author: - 'Patrick M. Gilmer' - Gregor Masbaum - Paul van Wamelen date: 'July 8, 2002' title: Integral bases for TQFT Modules and unimodular representations of mapping class groups --- Introduction ============ Integrality properties of Witten-Reshetikhin-Turaev quantum invariants of $3$-manifolds have been studied intensively in the last several years. H. Murakami [@Mu1; @Mu2] showed that the $SU(2)$- and $SO(3)$-invariants at a root of unity $q$ of prime order are algebraic integers. This was reproved in [@MR] and generalised to all classical Lie types in [@MW; @TY] and then to all Lie types in [@Le]. These integrality properties are crucial for establishing the relationship of the invariants with the Casson invariant [@Mu1; @Mu2] and with the perturbative invariants or Ohtsuki series [@OhCambridge; @Oh1; @Le]. Quantum invariants fit into Topological Quantum Field Theories (TQFT). This means in particular that there are representations of mapping class groups associated with them. (Actually the representations are usually only projective-linear; equivalently, one has to consider certain central extensions of mapping class groups here.) If a $3$-manifold $M$ is presented as a Heegaard splitting where two handlebodies are glued together by a diffeomorphism $\varphi$ along their boundary, the quantum invariant of $M$ can be recovered from the representation of $\varphi$ on the TQFT-vector space $V({{\Sigma}})$ associated to the boundary surface ${{\Sigma}}$. The TQFT-representations are finite-dimensional and can be defined over a finite extension of the cyclotomic number field $\mathbb{Q}(q)$, where the quantum parameter $q$ is a root of unity. They also preserve a non-degenerate Hermitian form $\langle \ , \ \rangle_{{\Sigma}}$ on $V({{\Sigma}})$ (which may or may not be unitary; this usually depends on the choice of the embedding of the cyclotomic field into $\mathbb{C}$). A quite striking result was recently announced by Andersen [@An] who proved that in the $SU(n)$ case the representations are asymptotically faithful (here asymptotically means letting the order of $q$ go to infinity). At a fixed root of unity, they are certainly not faithful, as Dehn twists are always represented by matrices of finite order. Roberts [@R] showed that the representations are irreducible in the $SU(2)$-case if the order of $q$ is prime. An interesting question is to determine the image of the mapping class group in the TQFT-representations. For the $SU(2)$ and $SO(3)$-theories, the first author proved that in the genus one case the image is a finite group [@G1]. However in higher genus, the image is not finite [@Fu]; in fact, it contains elements of infinite order [@Madeira]. One might hope that this image is equal to the linear transformations which are automorphisms of some (yet to be found) structure, just as a linear transformation of the homology group $H_1({{\Sigma}};\mathbb{Z})$ is represented by a mapping class if and only if it preserves the intersection form. In this paper we are concerned with integrality properties of the TQFT representations. For simplicity, we restrict ourselves to the $SO(3)$ case; specifically, we use a variant of the $V_p$-theories of [@BHMV2] with $p$ an odd prime. Here, $p$ is the order of the root of unity $q$. Let ${{\mathcal{O}}}$ denote the ring of algebraic integers in the cyclotomic ground field. The main idea to obtain an integral structure on the TQFT already appears in [@G]. Namely, we define an ${{\mathcal{O}}}$-submodule ${{\mathcal{S}_p}}({{\Sigma}})$ of the TQFT-vector space ${{V_p}}({{\Sigma}})$ as the ${{\mathcal{O}}}$-span of vectors represented by connected $3$-manifolds with boundary ${{\Sigma}}$. The point of this definition is that the submodule ${{\mathcal{S}_p}}({{\Sigma}})$ is clearly preserved under the mapping class group. It was shown in [@G] that ${{\mathcal{S}_p}}({{\Sigma}})$ is always a free finitely generated ${{\mathcal{O}}}$-module. One can also rescale the Hermitian form $\langle \ , \ \rangle_{{\Sigma}}$ on ${{V_p}}({{\Sigma}})$ to obtain a non-degenerate ${{\mathcal{O}}}$-valued form $(\ ,\ )_{{\Sigma}}$ on ${{\mathcal{S}_p}}({{\Sigma}})$. This relies on the integrality results for the invariants mentioned above. The form $(\ ,\ )_{{\Sigma}}$ is again preserved by the mapping class group. In particular, the image of the mapping class group in the TQFT-representation $V_p({{\Sigma}})$ lies in the subgroup preserving a lattice defined over ${{\mathcal{O}}}$ and a non-degenerate Hermitian form on it. In what sense is ${{\mathcal{S}_p}}$ a TQFT defined over ${{\mathcal{O}}}$? For instance one might hope that the form $(\ ,\ )_{{\Sigma}}$ was unimodular. Here, we show that this is indeed the case in genus one and two. This is a consequence of our main result which is to describe explicit bases of ${{\mathcal{S}_p}}({{\Sigma}})$ in genus one and two. In fact, we will describe two quite different bases in genus one. The first basis is given in Theorem \[firstbasis\]. It is $$\{ \omega, t(\omega) , t^2(\omega) , \ldots , t^{d-1}(\omega)\}$$ where $\omega$ is the element appearing in the surgery axiom of the ${{V_p}}$-theory and $t$ is the twist map. It is easy to see that these elements lie in ${{\mathcal{S}_p}}(S^1\times S^1)$ and we use a Vandermonde matrix argument to show that they form a basis. A crucial step is to show that the Hermitian form $(\ ,\ )_{S^1\times S^1}$ is unimodular with respect to this basis. The second basis given in Theorem \[secondbasis\] is of a quite different nature. It is $$\{ 1, v ,v^2 , \ldots , v^{d-1}\}$$ where $v=(z+2)/(1+A)$ (here $z$ is represented by the core of the solid torus). We call it the $v$-basis. This time, it is not even obvious [*a priori*]{} that its elements lie in ${{\mathcal{S}_p}}(S^1\times S^1)$. The proof involves two different arguments. One is to show that the ${{\mathcal{O}}}$-span of the $v$-basis is stable under the twist map. This is shown in Section \[Kauf\]. In fact, we prove it in the more general context where the skein variable $A$ is an indeterminate rather than a root of unity. The second ingredient is to express $\omega$ in the $v$-basis and thereby relate the $v$-basis to the first basis. This is done in Section \[2nd\]. The $v$-basis lends itself nicely to finding bases in higher genus. In Section \[g2\], we describe a basis of ${{\mathcal{S}_p}}(\Sigma)$ in genus two consisting of $v$-colored links in a genus two handlebody. These links are described by arrangements of curves in a twice punctured disk. Again, the unimodularity of the Hermitian form with respect to this basis is a crucial step in the argument. In principle this method can be used to study ${{\mathcal{S}_p}}(\Sigma)$ in higher genus as well. It turns out, however, that the Hermitian form $(\ ,\ )_{{{\Sigma}}}$ is not always unimodular. For example, a simple argument given in Section \[non-uni\] shows that it cannot be unimodular for surfaces of genus 3 and 5, assuming $p \equiv 5 \pmod{8}.$ In this paper we will not attempt to deal with the higher genus case in general. We only give in Section \[g3\] a basis of ${{\mathcal{S}_p}}(\Sigma)$ for a surface of genus three when $p=5.$ Although in this case the Hermitian form is not unimodular, it is nearly so. This allows us to find a basis easily in this one case. Note that our definition of ${{\mathcal{S}_p}}({{\Sigma}})$ is analogous to the construction of integral modular categories in [@MW]; in both cases one constructs integral structures by considering the span, over the subring of algebraic integers of the coefficient field, of the morphisms of the geometrically defined category (tangles in the case of [@MW], $3$-dimensional cobordisms in the case at hand). It might be that this is not always enough: It is conceivable that one might be able to enlarge ${{\mathcal{S}_p}}({{\Sigma}})$ in some way to make the form always unimodular; however this enlargement would not be generated by $3$-cobordisms anymore. We conclude the paper by showing how the ${{\mathcal{S}_p}}$-theory defined over ${{\mathcal{O}}}$ can be used to prove a divisibility result for the Kauffman bracket of links in $S^3$. This generalizes a result of Cochran and Melvin [@CM] for zero framed links (see also [@OhCambridge; @KS]). [*Notational conventions.*]{} Throughout the paper, $p\geq 3$ will be an odd integer, and we put $d=(p-1)/2$. From Section \[elem\] onwards, $p$ is supposed to be prime. The twist map on the Kauffman Bracket module of a solid torus {#Kauf} ============================================================= In this section we define a sequence of submodules $K(n)$ of the Kauffman Bracket skein module of the solid torus $S^1 \times D^2$ and show that they are preserved under the twist map. We use the notations of [@BHMV1]. Suppose $R$ is a commutative ring with identity and an invertible element $A.$ The universal example is $R={{\mathbb{Z}}}[A,A^{-1}]$ which we also denote by ${{\mathbb{Z}}}[A^{\pm}]$. Recall that the Kauffman bracket skein module $K(M,R)$ of a $3$-manifold $M$ is the free $R$ module generated by isotopy classes of banded links in $M$ modulo the submodule generated by the Kauffman relations. We let $z$ denote the skein element of $K(S^1 \times D^2,R)$ given by the banded link $S^1\times J,$ where $J$ is a small arc in the interior of $D^2.$ As is well known, $K(S^1 \times D^2,R)$ is a free $R$-module on the nonnegative powers of $z,$ where $z^n$ means $n$ parallel copies of $z$. This also makes $K(S^1 \times D^2,R)$ into an $R$-algebra isomorphic to the polynomial ring $ R [z].$ Let $t:K (S^1 \times D^2,R) \rightarrow K (S^1 \times D^2,R)$ denote the twist map induced by a full right handed twist on the solid torus. It is well known (see [*e.g.*]{} [@BHMV1]) that there is a basis $ \{e_i\}_{i\ge0}$ of eigenvectors for the twist map. It is defined recursively by $$\label{ei} e_0=1, \quad e_1=z, \quad e_i= z e_{i-1}- e_{i-2} ~.$$ The eigenvalues are given by $$\label{mui}t(e_i)=\mu_i e_i, \text{ where } \mu_i=(-1)^i A^{i^2+2i} ~.$$ [*Let ${K}(n)$ denote the ${{\mathbb{Z}}}[A^{\pm}]$-submodule of $K (S^1 \times D^2, {{\mathbb{Z}}}[A^{\pm},\frac 1 {1+A}])$ generated by $\{1,v,v^2,\ldots,v^n\}$, where $$v= \frac {z+2}{1+A}~.$$* ]{} \[tK\] The twist map $t$ sends $K(n)$ to itself. Consider the basis $ \{ (z+2)^i\}_{i\ge0}$ of $K (S^1 \times D^2,{{\mathbb{Z}}}[A^{\pm}]).$ The following Lemma gives the change of basis formulas. \[changeofbasis\] For each $n \ge 1,$ $$(z+2)^{n-1}= \sum_{k=1}^n \binom{2n}{n-k} \frac k n e_{k-1}$$ $$e_{n-1}= \sum_{i=1}^n (-1)^{n-i} \binom{n+i-1}{n-i} (z+2)^{i-1}$$ Prove each separately by induction on $n$ using the recursion formula (\[ei\]). [*It follows that $\binom{2n}{n-k} \frac k n \in {{\mathbb{Z}}},$ which can also be seen directly:*]{} $$\binom{2n}{n-k} \frac k n= \binom{2n}{n-k} (1-\frac {n-k} n) = \binom{2n}{n-k}- 2\binom{2n-1}{n-k-1}~.$$ It is enough to show Theorem \[tK\] for the endomorphism $-At$ in place of $t$. Let us compute $-At$ in the basis $(z+2)^n.$ Note that $-At(e_{i-1})=(-A)^{i^2} e_{i-1}$. $$\begin{aligned} -At\left((z+2)^{n-1}\right) & = \sum_{k=1}^n \binom{2n}{n-k} \frac k n (-A) t(e_{k-1}), \\ &= \sum_{k=1}^n \binom{2n}{n-k} \frac k n (-A)^{k^2} \sum_{i=1}^k (-1)^{k-i} \binom {k+i-1}{k-i} (z+2)^{i-1},\\ &= \sum_{i=1}^n (-1)^i \left( \frac 1 n \sum_{k=i}^n k \binom {2n}{n-k} \binom {k+i-1}{k-i} A^{k^2} \right)(z+2)^{i-1},\\ &= \sum_{i=1}^n (-1)^{i} S_{1,i,n}(A) (z+2)^{i-1}, \end{aligned}$$ Here, for $ m \ge 1,$ we define $$S_{m,i,n}(A)= \frac 1 n \sum_{k=i}^n k^m \binom {2n} {n-k} \binom {k+i-1}{k-i} A^{k^2}\in {{\mathbb{Z}}}[A].$$ \[at -1\] $$S_{1,i,n}(-1) = \begin{cases} (-1)^n, &i=n\\ 0 &i \ne n \end{cases}$$ If we put $A=-1,$ then all $\mu_i=1$ and hence $-At$ is the identity. The following formula is a very special case of a transformation formula for terminating hypergeometric series due to Bailey [@B Formula 4.3.1]. This was pointed out to us by Krattenthaler’s [HYP]{} package, [@hyp]. \[recurs\] $$S_{m,i,n}= i^2 S_{m-2,i,n} + 2i (2i+1) S_{m-2,i+1,n}$$ $$\begin{aligned} S_{m,i,n} &=\frac 1 n \sum_{k=i}^n k^{m-2} \left( (k+i) (k-i) +i^2 \right) \binom {2n} {n-k} \binom {k+i-1}{k-i} A^{k^2} \\ &= i^2 S_{m-2,i,n} + \frac 1 n \sum_{k=i+1}^n k^{m-2} (k+i) (k-i) \binom {2n} {n-k} \binom {k+i-1}{k-i} A^{k^2} \\ & \text{ (the term with $k=i$ is zero) }\\ &= i^2 S_{m-2,i,n} + \frac 1 n 2 i (2i+1)\sum_{k=i+1}^n k^{m-2} \binom {2n} {n-k} \binom {k+i}{k-i-1} A^{k^2} \\ &= i^2 S_{m-2,i,n} + 2i (2i+1) S_{m-2,i+1,n}\end{aligned}$$ Here we use the simple identity: $$(k+i)(k-i) \binom {k+i-1}{k-i} = 2i (2i+1) \binom {k+i}{k-i-1}$$ \[eq2.7\] \[divis\] $S_{1,i,n}(A)$ is divisible by $(1+A)^{n-i}$ in ${{\mathbb{Z}}}[A]$ for $i \le n.$ It suffices to show: $$\label{eq2} \left[ \left( \frac {d} {dA} \right) ^{k} S_{1,i,n}(A) \right]_{A=-1} =0$$ for all $k=0,1,\ldots , n-i-1$. Note that $\frac {d} {dA} S_{m,i,n}= A^{-1} S_{m+2,i,n} \in {{\mathbb{Z}}}[A].$ Thus $$\left( \frac {d} {dA} \right) ^k S_{1,i,n} \in \text{ Span}_{{{\mathbb{Z}}}[A^{\pm}]}\{ S_{m,i,n} \,|\, m \text{ odd}, \, 1 \le m \le 2n-2i-1\}$$ for all $k$ in the required range. But using Lemma \[recurs\] one may decrease $m$ at the cost of increasing $i,$ and see that $$\text{ Span}_{{{\mathbb{Z}}}[A^{\pm}]}\{ S_{m,i,n} \,| \, m \text{ odd},\, 1 \le m \le 2n-2i-1\} \subseteq \text{ Span}_{{{\mathbb{Z}}}[A^{\pm}]}\{ S_{1,j,n} \,|\, j < n\}.$$ By Lemma \[at -1\], $S_{1,j,n}(-1) =0,$ for $ j < n,$ and (\[eq2\]) follows. [*Proof of Theorem \[tK\].*]{} We have $$t(v^{n-1}) = - A^{-1} \sum_{i=1}^n s_{i,n}(A) v^{i-1},$$ where $s_{i,n}(A)= (-1)^i (1+A)^{i-n}S_{1,i,n}(A)$ lies in ${{\mathbb{Z}}}[A]$ by Proposition \[eq2.7\]. [*Theorem \[tK\] remains valid if we take $A$ to be a root of unity, other than $-1,$ rather than an indeterminant.* ]{} [*Let $\tilde {K}(n)$ be defined as $K(n)$ but with $v=({z+2})/({1+A})$ replaced with $\tilde v= ({z+2})/({1-A^2})$. Then a similar argument shows that $\tilde {K}(n)$ is stable under $t^2$, the square of the twist map. (To see this, one should replace $-At$ with $A^2 t^2$ in the above and express everything in terms of $q=A^2$. This leads to polynomials $\tilde S_{m,i,n}(q)$ defined similarly as the $ S_{m,i,n}(A)$ except that $A$ is replaced with $q$ and an extra factor of $(-1)^k$ is inserted in the sum. The remainder of the argument is the same.)* ]{} The $SO(3)$-TQFTs ================= Let $p\geq 3$ be an odd integer. (In this section, $p$ need not be prime.) We consider a variation of the $2+1$ dimensional cobordism category considered in [@BHMV2] whose objects are closed oriented surfaces (with extra structure) with a (possibly empty) collection of banded points ($=$ small oriented arcs) colored by integers in the range $[0,p-2].$ The morphisms are (equivalence classes of) oriented 3-dimensional manifolds (with extra structure) with $p$-admissibly colored banded trivalent graphs. (Two morphisms are considered equivalent if they are related by a homeomorphism respecting the boundary identifications.) For the definition of $p$-admissibility in the $p$-odd case see [@BHMV2 Theorem 1.15]; see also Section \[g2\]. The variation consists of replacing the $p_1$-structures of [@BHMV2] with structures put forward by Walker [@W] and Turaev [@Tu]. Surfaces are equipped with a Lagrangian subspace of their first homology. We use homology with rational coefficients when considering Lagrangian subspaces. Cobordisms are equipped with integer weights, as well as Lagrangian subspaces for the target and source. This is also described in [@G]. We will denote this category by $\mathcal{C}.$ We call the objects of this category e-surfaces, and call the morphisms 3-e-manifolds. The procedure of [@BHMV2] defines a TQFT-functor ${{V_p}}$ on $\mathcal{C}$ over a commutative ring $R$ containing $p^{-1}$, a primitive 2pth root of unity $A$ and a solution of $\kappa^2= A^{ -6- p(p+1)/2 }.$ The number $\kappa$ here plays the role of $\kappa^3$ in [@BHMV2]. Here we use the term TQFT slightly loosely as the tensor product axiom does not hold unless only even colors are used in the cobordism category. The even colors correspond to irreducible representations of $SU(2)$ which lift to $SO(3)$. Therefore the ${{V_p}}$-theory for odd $p$ is considered a $SO(3)$ variant of the Witten-Reshetikhin-Turaev $SU(2)$-TQFT. For us it is convenient to use odd colors as well as even colors. However, if we insist that only even colors be used in coloring the banded points on the surfaces, then we do obtain an honest TQFT with the tensor product axiom, but we are still allowed us to use the language of odd colors to describe states. This will be useful in Sections \[g2\] and \[g3\]. If $M$ is a 3-e-manifold viewed as morphism from ${{\Sigma}}$ to ${{\Sigma}}'$ in $\mathcal{C}$, we denote the associated endomorphism from ${{V_p}}({{\Sigma}})$ to ${{V_p}}({{\Sigma}}')$ by ${{Z_p}}(M)$. (It is denoted by $(Z_p)_M$ in [@BHMV2]). If $M$ is a closed 3-e-manifold viewed as morphism from $\emptyset$ to $\emptyset,$ ${{Z_p}}(M)$ induces multiplication by a scalar from $R= V(\emptyset).$ This scalar is denoted by $\langle M\rangle.$ If $M$ is a 3-e-manifold viewed as morphism from $\emptyset$ to $\Sigma,$ let $[M]$ denote ${{Z_p}}(M)(1) \in V(\Sigma).$ ($[M]$ is denoted by $Z_p(M)$ in [@BHMV2]). We call such an element $[M]$ a [*vacuum state.*]{} If $M$ is connected, $[M]$ is called a connected vacuum state. The modules ${{V_p}}({{\Sigma}})$ are always free over $R$. They also carry a nonsingular Hermitian form [@BHMV2] : $$\langle \ ,\ \rangle _{\Sigma}: {{V_p}}(\Sigma) \times {{V_p}}(\Sigma) \rightarrow R$$ given by $$\langle [N_1],[N_2]\rangle _{\Sigma}= \langle N_1 \cup_{\Sigma} -N_2\rangle~.$$ Here $-N_2$ is the 3-e-manifold obtained by reversing the orientation, multiplying the weight by $-1, $ and leaving the Lagrangian on the boundary alone. If $\Sigma$ is an e-surface with no colored points, and $H$ is a handlebody (weighted zero) with boundary $\Sigma,$ then ${{V_p}}(\Sigma)$ has a specified isomorphism to a quotient of the skein module $K(H,R)$ [@BHMV2 p. 891]. In fact if $H$ is a subset of $S^3$ then two skeins represent the same element if and only if they are equal as “maps of outsides” in Lickorish’s phrase [@L]. Let $S^1 \times S^1$ denote an e-surface of genus one with no colored points. Let $d$ denote $(p-1)/ 2.$ It turns out that $d$ is the dimension or rank of ${{V_p}}(S^1 \times S^1).$ In fact, the module ${{V_p}}(S^1 \times S^1) $ is isomorphic as an $R$-module to the quotient of $K(S^1 \times D^2,R)=R[z]$ by the ideal generated by $e_d- e_{d-1}\in R[z].$ It follows [@BHMV1 p.696] that $e_{p-1}=0$ in ${{V_p}}(S^1 \times S^1) $ and $e_{d+i}= e_{d-1-i}.$ Thus the module ${{V_p}}(S^1 \times S^1) $ has indeed rank $d$ with the basis $\{e_0,e_1,\ldots ,e_{d-1}\}$. Note that this basis is the same, up to reordering, as the even basis $\{e_0,e_2,\ldots, e_{p-3}\}.$ Let $\Sigma_g$ denote an e-surface of genus $g$ with no colored points on the boundary. The rank of the free module ${{V_p}}(\Sigma_g)$ is given by the formula [@BHMV2 Cor. 1.16] $$\text{rank}\left({{V_p}}(\Sigma_g)\right) = \left( \frac p 4\right)^{g-1} \sum_{j=1}^{d} \left( \sin \frac {2 \pi j}{p} \right)^{2-2g} .$$ This is the same as $2^{-g}$ times the dimension of $V_{2p}({{\Sigma}}_g)$ (this fact comes from a tensor product formula, see [@BHMV2 Thm. 1.5]). Note that $V_{2p}({{\Sigma}}_g)$ is an $SU(2)$-TQFT module, with dimension given by the $SU(2)$ Verlinde formula at level $p-2$ (where the colors are again the set of integers in the range $[0,p-2]$). In genus 2, we have $$\text{rank}\left({{V_p}}(\Sigma_2)\right)= \frac {d (d+1)(2d+1)}{6}$$ as will be seen by an explicit counting argument in Section \[g2\]. Some facts from elementary number theory {#elem} ======================================== In the remainder of this paper, we assume $p$ is an odd prime. We continue to use the notation $d=(p-1)/2$. In this section, we collect some notation and a few elementary number-theoretical facts. To be specific we pick particular values for $A$ and $\kappa$. We put $A= \zeta_{2p}$ where $\zeta_n = e^{2 \pi i/{n}},$ and also use the notation[^1] $$q=A^2.$$ We may then take $\kappa = A^{-3}(-i)^{({p+1})/{2}}.$ Note that $\zeta_{2p}\in {{\mathbb{Z}}}[\zeta_p]$. Thus the coefficient ring is $R ={{\mathbb{Z}}}[\zeta_p, \frac 1 p]$ if $p \equiv -1 \pmod{4},$ and $R ={{\mathbb{Z}}}[\zeta_{p},i, \frac 1 p]={{\mathbb{Z}}}[\zeta_{4p}, \frac 1 p]$ if $p \equiv 1 \pmod{4}.$ Of course, the coefficient ring remains unchanged if $A$ is replaced by another primitive $2p$-th root of unity, and $\kappa$ is changed accordingly. We let $\eta$ denote $\langle S^3\rangle$, the invariant of $S^3$ with weight zero, and put $\mathcal{D}= \eta^{-1}$. Then using equations on [@BHMV2 p.897] $$\label{D} \mathcal{D} = \frac{i\ \sqrt{p} }{q-q^{-1}} = \frac{i^{\frac {p+1} 2} }{q-q^{-1}} \left(\frac 1 2 \sum _{m=1}^{2p}(-1)^m A^{m^2}\right)$$ In particular $$\label{DD} \mathcal{D}^2= \frac {-p}{ ({q-q^{-1}})^2}~.$$ We denote by $\mathcal{O} $ the ring of integers in $R.$ Note that $\mathcal{O}={{\mathbb{Z}}}[\zeta_p]$ if $p \equiv -1 \pmod{4},$ and $\mathcal{O} ={{\mathbb{Z}}}[\zeta_{p},i]={{\mathbb{Z}}}[\zeta_{4p}]$ if $p \equiv 1 \pmod{4}.$ The following notation will be useful. If $x,y$ are elements of ${{\mathcal{O}}}$ (or, more generally, of its quotient field), we write $x\sim y$ if there exists a unit $u\in {{\mathcal{O}}}$ such that $x=uy$. \[num\] - $1-A$ is a unit in ${{\mathcal{O}}}$, and $1-q\sim 1+A$. - One has $\mathcal{D}\in {{\mathcal{O}}}$. Moreover, $\mathcal{D}\sim (1-q)^{({p-3})/2}=(1-q)^{d-1}$. - The quantum integers $[n]=(q^n-q^{-n})/(q-q^{-1})$ are units for $1\leq n\leq p-1$. - If $0\leq i,j \leq d-1$ and $i\neq j$, then the twist coefficients $\mu_i$ (see (\[mui\])) satisfy $\mu_i-\mu_j\sim 1-q$. - Put $\lambda_i=-q^{i+1}-q^{-i-1}$. If $0\leq i \leq d-1$, then $\lambda_0-\lambda_i\sim (1-q)^2$. The fact that $1-A$ is a unit follows easily from the fact that $A$ is a zero of the $2p$-th cyclotomic polynomial $1-X+X^2-\ldots +X^{p-1}$. This proves (i). It is well-known that $p\sim (1-q)^{p-1}$ (see [*e.g.*]{} [@MR Lemma 3.1]). Together with Formulas (\[D\]) and (\[DD\]), this shows (ii). Observing that $[n]\sim 1+q^2+\ldots +q^{2n-2}$, (iii) is also shown in [@MR Lemma 3.1]. For (iv), observe that $\mu_i=\mu_{p-2-i}$ so that the set of $\mu_i$ in question is equal to the set of $\mu_{2i}=q^{ 2 i^2+2i}$ for $i=0,1,\ldots d-1$. These powers of $q$ are all distinct, which implies $$\mu_i-\mu_j \sim 1-q^n \sim 1-q$$ for some $0<n<p$. This proves (iv). Similarly, (v) follows from $\lambda_0-\lambda_i\sim(1-q^{i+2})(1-q^i)$. [*It is well-known that $1-q$ is a prime in ${{\mathbb{Z}}}[q]={{\mathbb{Z}}}[\zeta_{p}]$. But if $p \equiv 1 \pmod{4},$ then $1-q$ is not a prime in ${{\mathcal{O}}}={{\mathbb{Z}}}[\zeta_{4p}]$ (it splits as a product of two conjugate prime ideals).* ]{} Associated Integral Cobordism Functors ====================================== In [@G], a cobordism functor from a restricted cobordism category to the category of free finitely generated $\mathcal{O}$-modules is described. Let $\mathcal{C''}$ denote the subcategory of $\mathcal{C}$ defined by considering only nonempty connected surfaces and connected morphisms between such surfaces. This represents a further restriction of $\mathcal{C}$ than that considered in [@G], but it suffices for our purposes. [*If $\Sigma$ is a connected e-surface, define ${{\mathcal{S}_p}}(\Sigma)$ to be the $\mathcal{O}$-submodule of ${{V_p}}(\Sigma)$ generated by connected vacuum states. If $N: \Sigma \rightarrow \Sigma'$ is a morphism of $\mathcal{C''}$ then ${{Z_p}}(N)$ sends $ [M] \in {{\mathcal{S}_p}}(\Sigma)$ to $[M \cup_{\Sigma}N] \in {{\mathcal{S}_p}}(\Sigma').$ In this way we get a functor from $\mathcal{C''}$ to the category of finitely generated $\mathcal{O}$-modules. We also rescale the Hermitian form on ${{V_p}}(\Sigma)$ to obtain an $\mathcal{O}$-valued Hermitian form $$(\ ,\ )_{\Sigma}: {{\mathcal{S}_p}}(\Sigma) \otimes_{{\mathcal{O}}}{{\mathcal{S}_p}}(\Sigma) \rightarrow \mathcal{O},$$ defined by $$([N_1],[N_2])_{\Sigma}= \mathcal{D} \langle [N_1],[N_2]\rangle_{\Sigma} = \mathcal{D} \langle N_1 \cup_{\Sigma} -N_2\rangle .$$* ]{}** This form takes values in $\mathcal{O}$ by the integrality result for closed 3-e-manifolds [@Mu2; @MR]. These theorems are also used in proving that ${{\mathcal{S}_p}}(\Sigma)$ is finitely generated [@G]. [*Over a Dedekind domain such as ${{\mathcal{O}}}$, a finitely generated torsion-free module is always projective, but it need not be free. (The typical examples are non-principal ideals in ${{\mathcal{O}}}$.) Somewhat surprisingly, however, it turns out that the modules ${{\mathcal{S}_p}}(\Sigma)$ are always free. This is proved in [@G]. We will not actually make use of this fact in genus $1$ and $2$: freeness will follow from the construction of explicit bases.* ]{} [*A Hermitian form on a projective ${{\mathcal{O}}}$-module $S$ is called [*non-degenerate*]{} (or [*non-singular*]{}) if its adjoint map $S\rightarrow S^*$ is injective. It is called [*unimodular*]{} if the adjoint map is an isomorphism.* ]{} Note that if $S$ is free and $M$ is the matrix of the Hermitian form in some basis, then the form is non-degenerate (resp. unimodular) if $\det M$ is non-zero (resp. a unit in ${{\mathcal{O}}}$). In our situation, the form $ (\ ,\ )_{\Sigma}$ is always non-degenerate (since the original form $\langle \ ,\ \rangle_{{\Sigma}}$ on ${{V_p}}({{\Sigma}})$ is). We will show that $ (\ ,\ )_{\Sigma}$ is unimodular in genus $1$ and $2$. There is a standard basis $\{u_\sigma\}$ of ${{V_p}}(\Sigma_g)$ given by p-admissible [ *even*]{} colorings $\sigma$ of the graph (-.5,-.2)(5,1) (0,.5)[.35]{} (1,.5)[.35]{} (.35,.5)(.65,.5) (1.35,.5)(1.65,.5) (3.35,.5)(3.65,.5) (4,.5)[.35]{} (2,.45)[$\cdots$]{} (where there are $g$ loops) embedded in a 3-e-handlebody $H_g$ of genus $g$ with boundary the e-surface ${{\Sigma}}_g$ (see [@BHMV2 4.11]). One may actually use any trivalent graph in $H_g$ to which $H_g$ deformation retracts. (In the case $g=1$, this is the same as the basis given by the elements $e_i$.) These basis elements lie in ${{\mathcal{S}_p}}(\Sigma_g)$ because the denominators appearing in the Jones-Wenzl idempotents needed to expand colored graphs into skein elements are invertible in ${{\mathcal{O}}}$ (see [@MR]). Warning: the $u_\sigma$ do [*not*]{} generate ${{\mathcal{S}_p}}(\Sigma_g)$ over ${{\mathcal{O}}}$. \[detb\] The elements $u_\sigma$ are orthogonal for the form $(\ ,\ )_{\Sigma_g}.$ Moreover, one has $$(u_\sigma,u_\sigma)_{\Sigma_g}\sim {{\mathcal{D}}}^g\sim (1-q)^{(d-1)g}.$$ By [@BHMV2 Theorem 4.11] one has that $\langle u_\sigma,u_\sigma\rangle_{\Sigma_g}$ is equal to $\eta^{1-g}={{\mathcal{D}}}^{g-1}$ times a product of non-zero quantum integers or their inverses, which are units in ${{\mathbb{Z}}}[q]$ by Lemma \[num\]. Since the form $(\ ,\ )_{\Sigma_g} $ is just a rescaling of the form $\langle \ ,\ \rangle_{\Sigma_g} $, the result follows. One of the reasons to study the form $(\ ,\ )_{\Sigma_g} $ is that it is preserved by the TQFT-action of the mapping class group. More precisely, let $\tilde{\Gamma}(\Sigma)$ denote the central extension of the mapping class group $\Gamma(\Sigma)$ of $\Sigma$ realized by the subcategory of $C''$ consisting of e-manifolds homeomorphic to $\Sigma \times I$ such that the colored graph is given by $I$ times the colored banded points of $\Sigma.$ This homeomorphism need not respect the boundary identification at $\Sigma \times \{1\},$ but should respect the boundary identification at $\Sigma \times \{0\}.$ In fact considering this boundary identification at $\Sigma \times \{1\},$ defines the quotient homomorphism from $\tilde{\Gamma}(\Sigma)$ to $\Gamma(\Sigma),$ which has kernel ${{\mathbb{Z}}}$ given by the integral weights on $\Sigma \times I$ with standard boundary identifications. The group $\tilde \Gamma(\Sigma)$ is isomorphic to the signature extension (see [*e.g.*]{} Atiyah [@At], Turaev [@Tu].) This extension can be described nicely using skein theory [@MR1]. The group $\tilde \Gamma(\Sigma)$ acts on ${{V_p}}(\Sigma)$ preserving the $\mathcal{O}$-lattice ${{\mathcal{S}_p}}(\Sigma)$ and the ${{\mathcal{O}}}$-valued Hermitian form $ (\ ,\ )_{\Sigma}$. This follows from the definition of ${{\mathcal{S}_p}}(\Sigma)$ and the fact that the group $\tilde \Gamma(\Sigma)$ preserves the original Hermitian form $ \langle\ ,\ \rangle_{\Sigma}$. The module ${{\mathcal{S}_p}}(\Sigma)$ can be described using the notion of ‘mixed graph’. Recall the element $$\omega= {{\mathcal{D}}}^{-1} \sum_{i=0}^{d-1} \langle e_i\rangle e_i \in K(S^1 \times D^2,R).$$ Here $\langle e_i\rangle=(-1)^i [i+1]$. It plays an important role in the surgery axioms of the ${{V_p}}$-theory. By a [*mixed graph*]{} in a weighted 3-manifold $M, $ we mean a trivalent banded graph in $M$ whose simple closed curve components may possibly be colored $\omega$ or by integer colors in the range $[ 0,p-2]$ and whose other edges are colored p-admissibly by integers in the range $[ 0,p-2]$. A mixed graph can be expanded multilinearly into a $R$-linear combination of colored graphs. The result should be thought of as a superposition of e-morphisms. If the graph is a link and every component is colored $\omega,$ we say the link is $\omega$-colored. A mixed graph in a handlebody $H$ specifies an element in ${{V_p}}(\partial H).$ \[ml\] A mixed graph in a connected 3-e-manifold $M$ with boundary $\Sigma$ represents an element of ${{\mathcal{S}_p}}(\Sigma).$ If $H$ is a 3-e-handlebody with boundary e-surface $\Sigma$ then ${{\mathcal{S}_p}}(\Sigma)$ is generated over $\mathcal{O}$ by elements specified by mixed graphs in $H.$ The first statement follows from the fact that ${{V_p}}$ satisfies the surgery axiom (S2) [@BHMV2 p 889]. The second statement follows from the fact that any connected 3-manifold with boundary $\Sigma$ can be obtained by a sequence of 2-surgeries to $H$ [@BHMV2 Proof of Lemma p. 891]. \[5.7\] [*Suppose that we know that some collection $T$ of elements of ${{V_p}}(\Sigma)$ lie in the $\mathcal{O}$-lattice ${{\mathcal{S}_p}}(\Sigma).$ Then $\text{Span}_\mathcal{O}(T)$ is a $\mathcal{O}$-sublattice of ${{\mathcal{S}_p}}(\Sigma).$ This sublattice might not be invariant under $\tilde{\Gamma}(\Sigma)$. Let $G=\{g_i\} \in \tilde \Gamma(\Sigma)$ be a finite set of elements whose image in $\Gamma(\Sigma)$ generate. The sequence of submodules of ${{\mathcal{S}_p}}(\Sigma)$: $\text{Span}_\mathcal{O}(T),$ $\text{Span}_\mathcal{O}(T\cup G(T)),$ $\text{Span}_\mathcal{O}(T\cup G(T) \cup G(G(T))),$ , $ \ldots $ etc. must stabilize in an $\mathcal{O}$-sublattice of ${{\mathcal{S}_p}}(\Sigma)$ which is invariant under the mapping class group. This procedure is well suited to computer investigation. The basis given in Section \[2nd\] was originally found by this procedure. We used the computer program Kant [@D] starting with $T= \{e_0,e_1,\ldots, e_{d-1}\}$ in ${{\mathcal{S}_p}}(S^1\times S^1).$* ]{} First integral basis in genus 1 {#1st} =============================== By a slight abuse of notation, we let $\omega$ denote the element in ${{\mathcal{S}_p}}(S^1 \times S^1)$ given by coloring the core of $S^1 \times D^2$ with $\omega.$ Let $t$ also denote the induced map on ${{V_p}}(S^1 \times S^1)$ given by giving $S^1 \times D^2$ a full right handed twist. Note that $t^n(\omega)\in {{\mathcal{S}_p}}(S^1 \times S^1)$ for all $n$. \[firstbasis\] $\{ \omega, t(\omega) , t^2(\omega) , \ldots , t^{d-1}(\omega)\}$ is a basis for the module ${{\mathcal{S}_p}}(S^1 \times S^1).$ The form $( \ ,\ )_{S^1 \times S^1}$ is unimodular. Note that it follows in particular that the ${{\mathcal{O}}}$-span of $\{ \omega, t(\omega) , t^2(\omega) , \ldots , t^{d-1}(\omega)\}$ is stable under the action of the mapping class group $\tilde{\Gamma}( S^1 \times S^1) $. Recall that $\mu_i=(-1)^{i} A^{i^2+2i}$ denotes the eigenvalue of $e_i$ under the twist map $t$. We have that $$t^j (\omega)= {{\mathcal{D}}}^{-1} \sum_{i=0}^{d-1} \langle e_i\rangle \mu_i^j e_i~.$$ Note that $\langle e_i\rangle=(-1)^i [i+1]$ is a unit by Lemma \[num\](iii). The matrix $W$ which expresses $\{ \omega, t(\omega) , t^2(\omega) , \ldots , t^{d-1}(\omega)\}$ in terms of $\{e_0,e_1,\ldots e_{d-1}\}$ has as determinant a unit (the product of the $\langle e_i\rangle$) times ${{\mathcal{D}}}^{-d}$ times the determinant of the Vandermonde matrix $[ \mu_i^j ]$ where $0 \le i, j\le d-1.$ Moreover by Lemma \[num\](iv) $$\det [ \mu_i^j ] = \pm \prod_{i<j} (\mu_i-\mu_j)\sim (1-q)^{{d(d-1)}/2}~.$$ As ${{\mathcal{D}}}\sim (1-q)^{d-1},$ we conclude that $$\label{detW} \det W\sim (1-q)^{- {d(d-1)}/2}~.$$ In particular, this determinant is non-zero, hence the $t^j(\omega)$ are linearly independent. Let $\mathcal {W}$ denote the $\mathcal{O}$-module spanned by the $t^j(\omega)$. Clearly $\mathcal W \subset {{\mathcal{S}_p}}(S^1 \times S^1)$. Now we know by Proposition \[detb\] that $(e_i,e_i)\sim (1-q)^{d-1}$ (here we simply write $(\ ,\ )$ for the Hermitian form $(\ ,\ )_{S^1 \times S^1} $). Therefore the matrix for $(\ ,\ )$ with respect to the orthogonal basis $\{e_0,e_1,\ldots, e_{d-1}\}$ has determinant $(1-q)^{d(d-1)}$. By (\[detW\]) it follows that the matrix for $(\ ,\ )$ with respect to $\{ \omega, t(\omega) , t^2(\omega) , \ldots , t^{d-1}(\omega)\}$ has unit determinant. (Here we use that $\overline{1-q}=1-q^{-1}\sim 1-q$.) In other words, the form $(\ ,\ )$ restricted to $\mathcal {W}$ is unimodular. But then $\mathcal {W}$ must be equal to ${{\mathcal{S}_p}}(S^1 \times S^1)$. This completes the proof. \[wc\] If $H$ is a 3-e-handlebody with boundary the e-surface of $\Sigma$ and $\Sigma$ has no colored points in the boundary, then $ {{\mathcal{S}_p}}(\Sigma)$ is generated over $\mathcal{O}$ by elements represented by $\omega$-colored banded links in $H.$ By the above theorem, each $e_i$ (in particular $e_1=z$) can be expressed as an $\mathcal{O}$-linear combinations of the elements $t^j(\omega)$. Therefore every mixed graph can be written as an $\mathcal{O}$-linear combination of $\omega$-colored banded links in $H.$ The result now follows from Theorem \[ml\]. Second integral basis in genus 1 {#2nd} ================================ Consider $K(d-1)$ in the notation of Section \[Kauf\], now taking $A= \zeta_{2p}.$ Let $\mathcal{V}$ denote its image in ${{V_p}}(S^1 \times S^1).$ In other words $\mathcal{V}$ is the $\mathcal{O}$-submodule of ${{V_p}}(S^1 \times S^1)$ generated by $\{ 1, v ,v^2 , \ldots , v^{d-1}\},$ where $v=(z+2)/(1+A)$. \[secondbasis\] One has $\mathcal{V}={{\mathcal{S}_p}}(S^1 \times S^1).$ In particular, $\{ 1, v ,v^2 , \ldots , v^{d-1}\}$ is a basis for the free module ${{\mathcal{S}_p}}(S^1 \times S^1).$ We refer to this basis as the $v$-basis of ${{\mathcal{S}_p}}(S^1 \times S^1).$ We originally found it by the procedure outlined in Remark \[5.7\]. Since $\tilde{\Gamma}(\Sigma)$ preserves ${{\mathcal{S}_p}}(S^1 \times S^1)$, we have the following Corollary. $\mathcal{V}=\text{\em Span}_{{\mathcal{O}}}\{1,v,v^2,\ldots,v^{d-1}\}$ is stable under the action of the mapping class group $\tilde{\Gamma}(S^1 \times S^1)$. [*The mapping class group $\tilde{\Gamma}(S^1 \times S^1)$ is a central extension of $SL(2,{{\mathbb{Z}}})$. Its image in $GL({{V_p}}(S^1 \times S^1))$ is generated by $\kappa $ times the identity matrix (the central generator acts as multiplication by $\kappa$), the twist map $t$, and the so-called $S$-matrix. The entries of the $S$-matrix in the $e_i$-basis are well-known. One can therefore write down its entries in the $v$-basis (using the change of basis formulas in Lemma \[changeofbasis\]). The fact that these entries lie in ${{\mathcal{O}}}$ is by no means obvious. We originally proved this fact using some identities involving binomial coefficients. The argument is similar to the proof that the $v$-basis is stable under the twist map $t$ given in Section \[Kauf\], but considerably more complicated. We found proofs of these identities using Zeilberger’s algorithm together with some identities from [@B] as above. In particular the [Gosper]{} command in the Mathematica package “Fast Zeilberger” (V 2.61) by Peter Paule and Markus Schorn, [@Zeil] was used. As the proof we give below is much simpler, we omit the details of this computation.* ]{} One has $\omega \in {{\mathcal{V}}}~.$ Let $\lambda_i= -q^{i+1}- q^{-i-1}$. Recall [@BHMV1] that $e_i$ is an eigenvector with eigenvalue $\lambda_i$ for the endomorphism $c$ of $K (S^1 \times D^2,{{\mathbb{Z}}}[A^{\pm}]) $ given by sending a skein in $S^1 \times D^2$ to the skein circled by a meridian. Let $\langle\ ,\ \rangle_H$ be the Hopf pairing ([*i.e.*]{} the symmetric bilinear form on ${{V_p}}(S^1 \times S^1)$ which sends two elements $x,y$ to the bracket of the zero-framed Hopf link with one component cabled by $x$, and the other component cabled by $y$). Then $$\label{omip} \langle \omega,e_i\rangle_H = \begin{cases} \langle\omega\rangle={{\mathcal{D}}}, &\text{if } i=0\\ \ \ \ 0 &\text{if } 1\leq i\leq d-1 \end{cases}$$ Note that $\langle z-\lambda_i,e_i\rangle_H=0$ for $1\leq i\leq d-1$, and $\langle z-\lambda_i,e_0\rangle_H=\lambda_0-\lambda_i$ (since $\langle z,e_0\rangle = \langle z\rangle =-q-q^{-1}=\lambda_0$). It follows that $$\label{omi} \omega={{\mathcal{D}}}\prod_{i=1}^{d-1} \frac {z-\lambda_i} {\lambda_0-\lambda_i}$$ since the pairing $\langle\ ,\ \rangle_H$ is non-degenerate. (Note the similarity with the polynomials $Q_n$ of [@BHMV1].) Since ${{\mathcal{D}}}\sim (1-q)^{d-1}$ and $\lambda_0-\lambda_i\sim (1-q)^2$ by Lemma \[num\], it follows that $$\label{omi2} \omega \sim \prod_{i=1}^{d-1} \frac {z-\lambda_i} {1-q}\sim \prod_{i=1}^{d-1} \frac {z-\lambda_i} {1+A}$$ (where $\sim$ means equality up to multiplication by a unit). Now $$\begin{aligned} z-\lambda_i &= (z+2)-(2 +\lambda_i)\\ &= (z+2)-(1-q^{i+1})(1-q^{-i-1})\\ &= (z+2)+ u_i (1+A)^2\end{aligned}$$ where $u_i \in {{\mathcal{O}}}.$ It follows that $$(z-\lambda_i)/(1+A) \in \text{Span}_{{\mathcal{O}}}\{1,v\}~,$$ and so (\[omi2\]) implies $\omega \in {{\mathcal{V}}}~,$ proving the lemma. By Theorem \[tK\], $K(n)$ hence ${{\mathcal{V}}}$ is stable under the twist map $t$. It follows that $$\mathcal{W}=\text{Span}_{{\mathcal{O}}}\{\omega, t(\omega),\ldots, t^{d-1}(\omega)\} \subseteq \mathcal{V}~.$$ Now recall from the proof of Theorem \[firstbasis\] that the matrix $W$ which expresses $\{\omega, t(\omega),\ldots, t^{d-1}(\omega)\}$ in terms of $\{e_0,e_1,\ldots,e_{d-1}\}$ has determinant $\det W \sim (1-q)^{-d(d-1)/2}$. Remembering $v=(z+2)/(1+A)$ and $1+A\sim 1-q$, it is easy to see that the same is true for the matrix which expresses $\{ 1,v,\ldots,v^{d-1}\}$ in terms of $\{e_0,e_1,\ldots e_{d-1}\}$. Since $\mathcal{W}\subset\mathcal{V}$, it follows that actually $\mathcal{W}= \mathcal{V}$. By Theorem \[firstbasis\] we conclude $\mathcal {V}={{\mathcal{S}_p}}(S^1 \times S^1)$. This completes the proof. By a $v$-colored banded link in a 3-manifold, we mean a banded link whose components are colored $v.$ As before this should be interpreted as the linear combination (superposition) of the colored banded links that one obtains by expanding multilinearly. We note that $i$ parallel strands colored $v$ is the same as one strand colored $v^i.$ \[v\] If $H$ is a 3-e-handlebody with boundary the e-surface $\Sigma$ and $\Sigma$ has no colored points in the boundary, then ${{\mathcal{S}_p}}(\Sigma)$ is generated over $\mathcal{O}$ by elements represented by $v$-colored banded links in $H.$ *The matrix of the Hermitian form $(\ ,\ )_{S^1\times S^1}$ in the $v$-basis is easily computed. One has for $0\leq i,j\leq d-1$ $$\begin{aligned} (v^i,v^j)_{S^1\times S^1}&={{\mathcal{D}}}\langle v^i,v^j\rangle_{S^1\times S^1} = \langle v^{i+j},\omega \rangle_H ={(1+A)^{-(i+j)}} \langle (z+2)^{i+j},\omega \rangle_H\\ &=\frac {1} {(1+A)^{i+j}} \binom{2i+2j+2} {i+j} \frac 1 {i+j+1} \langle e_0,\omega \rangle_H\\ &=\frac {{{\mathcal{D}}}} {(1+A)^{i+j}} \binom{2i+2j+2} {i+j} \frac 1 {i+j+1}\end{aligned}$$ Here we have used Lemma \[changeofbasis\] to express $(z+2)^{i+j}$ in terms of the $e_n$, and then retained only the $e_0$ term. Indeed, the others are annihilated by the Hopf pairing with $\omega$ since $0\leq i+j\leq 2d-2$ (see (\[omip\]) and remember that $e_{d+i}=e_{d-1-i}$ in ${{V_p}}(S^1\times S^1)$).* It is instructive to check directly that the expression above lies in ${{\mathcal{O}}}$ (use that $p$ divides the binomial coefficient $\binom{2i+2j+2} {i+j}$ if $d\leq i+j\leq 2d-2$). Integral basis in genus 2 {#g2} ========================= Let ${{\Sigma}}_2$ be a closed surface of genus $2$. In this section, we describe a basis for the module ${{\mathcal{S}_p}}({{\Sigma}}_2)$ and show that the Hermitian form $(\ ,\ )_{{{\Sigma}}_2}$ is unimodular. Let $H_2$ be a regular neighborhood of the hand cuff graph ${ \psset{unit=.5cm} \begin{pspicture}[.4](-.5,0)(1.5,1) \pscircle(0,.5){.35} \pscircle(1,.5){.35} \psline(.35,.5)(.65,.5) \end{pspicture} }$ in $\mathbb{R}^3$. Then $H_2$ is a genus $2$ handlebody and by Corollary \[v\], ${{\mathcal{S}_p}}({{\Sigma}}_2)$ is spanned by $v$-colored banded links in $H_2$. We think of $H_2$ as $P_2 \times I$ where $P_2$ is a disk with two holes. The skein module $K(H_2,R)$ is free on the set of isotopy classes of collections of nonintersecting essential simple closed curves in $P_2$. We refer to these isotopy classes as arrangements of curves. Such arrangements can be indexed by 3-tuples of nonnegative integers. Let $C_{\alpha,\beta,\gamma}$ denote the arrangement with $\gamma$ parallel curves going around both holes, and within them $\alpha$ parallel curves going around the left hole, and $\beta$ parallel curves going around the right hole. See Figure \[fig0\] for an example. (-.6,-.8)(2.6,.8) (0,0)[.35]{}(2,0)[.35]{} (-.05,-.05)(.05,.05) (-.05,.05)(.05,-.05) (1.95,-.05)(2.05,.05) (1.95,.05)(2.05,-.05) (0,0)[.45]{}(2,0)[.45]{} (2,0)[.55]{} (0,0)[.7]{}[45]{}[315]{}(2,0)[.7]{}[225]{}[135]{} (.5,.49)(.6,.43)(.8,.4)(1.2,.4)(1.4,.43)(1.5,.49) (.5,-.49)(.6,-.43)(.8,-.4)(1.2,-.4)(1.4,-.43)(1.5,-.49) \[thg2\] Let $C_{\alpha,\beta,\gamma}(v)$ be the element of ${{\mathcal{S}_p}}({{\Sigma}}_2)$ obtained by coloring each curve of $C_{\alpha,\beta,\gamma}$ by $v=(z+2)/(1+A)$. Then the set $$\{ C_{\alpha,\beta,\gamma}(v)\,|\, 0\le \gamma \le d-1, \quad 0\le \alpha,\beta \le d-1-\gamma \}$$ is a basis of ${{\mathcal{S}_p}}({{\Sigma}}_2).$ Moreover, the Hermitian form $(\ ,\ )_{{{\Sigma}}_2}$ is unimodular. Note that $C_{\alpha,\beta,\gamma}(v)$ lies in ${{\mathcal{S}_p}}({{\Sigma}}_2)$ because $v$ lies in ${{\mathcal{S}_p}}(S^1\times S^1)$ by Theorem \[secondbasis\]. Let us first describe a basis of ${{V_p}}({{\Sigma}}_2)$ consisting of elements represented by colorings of the hand cuff graph (-.5,0)(1.5,1) (0,.5)[.35]{} (1,.5)[.35]{} (.35,.5)(.65,.5) . Let $G(i,j,k)$ be the element defined by the colored graph (-.5,0)(2,1) (0,.5)[.35]{} (1.5,.5)[.35]{} (.35,.5)(1.15,.5) (-.7,.5)[$i$]{} (2,.5)[$j$]{} (.6,.7)[$k$]{} For this element to exist, $k$ must be even. Then the coloring is $p$-admissible if and only if $\frac k 2 \leq i,j\leq p-2-\frac k 2$ (see [@BHMV2 Thm 1.15]). The standard basis of ${{V_p}}({{\Sigma}}_2)$ would be to take the $p$-admissible $G(i,j,k)$ with both $i$ and $j$ even. It is also possible to impose that one or both of $i,j$ be odd [@BHMV2 Thm 4.14]. We will need a different basis where $i,j$ are allowed to be both even and odd, but $\leq d-1$. This is given in the following Lemma. \[lemG\] The $G(i,j,k)$ with $k$ even in the range $[0,p-3]$, and both $i$ and $j$ in the range $[\frac k 2, d-1]$ (but not necessarily even), form a basis of ${{V_p}}({{\Sigma}}_2)$. [ *Let $ \mathcal{G}$ be the basis described in the above Lemma. Let $ \mathcal{G}_k$ be the subset of elements of $ \mathcal{G}$ with middle arc colored $k$. The cardinality of $ \mathcal{G}_k$ is $(d-\frac k 2)^2.$ Thus we see directly that the cardinality of this basis is $ \sum_{j=0}^{d-1} (d-j)^2= \sum_{j=1}^{d} j^2= d (d+1)(2d+1)/6 .$* ]{} Lemma \[lemG\] could be proved using the methods of [@BHMV2]. Here we give a different, more direct proof. For $i$ in the range $[0,p-2]$ we let $i'=p-2-i$. We claim that $$G(i,j,k)\sim G(i',j,k)\sim G(i,j',k)\sim G(i',j',k)$$ (where $\sim$ means equality up to multiplication by a unit in ${{\mathcal{O}}}$). It is enough to prove that $G(i,j,k)\sim G(i',j,k).$ This is done in Figure \[fig1\]. Note that if $i$ is even and $> d-1$ then $i'$ is odd and $\leq d-1$. Thus the basis of where all $i,j$ are even may be replaced by the basis of Lemma \[lemG\]. (-2,-1.5)(3.5,1.5) (-.1,-.1)(.1,.1) (-.1,.1)(.1,-.1) (0,0)[1.5]{} (1.5,0)(3,0) (-2,.1)[$i$]{} (2,.3)[$k$]{}     $=$      (-2,-1.5)(3.5,1.5) (-.1,-.1)(.1,.1) (-.1,.1)(.1,-.1) (0,0)[1]{} (-.6,.25)[$\tilde p$]{} (0,0)[1.5]{} (1.5,0)(3,0) (-2,.1)[$i$]{} (2,.3)[$k$]{} \ $ = c_1$ (-1.5,-1.5)(3.5,1.5) (-.1,-.1)(.1,.1) (-.1,.1)(.1,-.1) (0,0)[1]{}[270]{}[90]{} (.35,0)[$\tilde p$]{} (0,0)[1.5]{}[270]{}[90]{} (0,0)[1.25]{}[118]{}[242]{} (0,1.5)(-.3,1.4)(-.6,1.1) (0,1)(-.3,1.02)(-.6,1.1) (0,-1.5)(-.3,-1.4)(-.6,-1.1) (0,-1)(-.3,-1.02)(-.6,-1.1) (1.5,0)(3,0) (-1.8,.1)[$i'$]{} (2,.3)[$k$]{} (1.1,1.4)[$i$]{} (1.3,-1.3)[$i$]{}    $ = c_1 c_2$    (-1.5,-1.5)(3.5,1.5) (-.1,-.1)(.1,.1) (-.1,.1)(.1,-.1) (0,0)[1.5]{} (1.5,0)(3,0) (-2,.1)[$i'$]{} (2,.3)[$k$]{} Let $ \mathcal{A}(v)$ be the set of the $v$-colored elements $C_{\alpha, \beta,\gamma}(v)$ claimed to be a basis in Theorem \[thg2\], and let $ \mathcal{A}$ be the set of the uncolored ([*i.e.*]{} colored by $z=e_1$) elements $C_{\alpha, \beta,\gamma}$ (in the same range for $\alpha, \beta,\gamma$). The set $ \mathcal{A}$ is a basis of ${{V_p}}(\Sigma_2)$. Moreover, the basis change from $ \mathcal{G}$ to $\mathcal{A}$ has determinant $\pm 1$. Using the Wenzl recursion formula for the idempotents of the Temperley-Lieb algebra, one can expand the elements of $\mathcal{A}$ as $\mathcal{O}$-linear combinations of elements of the graph basis $\mathcal{G}.$ In fact, in the expansion of $C_{\alpha,\beta,\gamma}$, only those $G(i,j,k)$ occur where $i\leq \alpha +\gamma$, $j\leq \beta +\gamma$, and $k\leq 2\gamma$; moreover, $G(\alpha +\gamma,\beta +\gamma, 2\gamma)$ occurs with coefficient one. We can find orderings of $\mathcal{A}$ and $\mathcal{G}$ so that the matrix which expresses $\mathcal{A}$ in terms of $\mathcal{G}$ is triangular with ones on the diagonal (use the lexicographical orderings where $\gamma$ resp. $k$ is counted first). This implies the Lemma. Let $ r=d (d+1)(2d+1)/{6}$ be the rank of ${{V_p}}(\Sigma_2)$. By Proposition \[detb\], the matrix for $(\ ,\ )_{\Sigma_2} $ with respect to the orthogonal basis $\mathcal{G}$ has determinant $\sim (1-q)^{2 (d-1) r}$. The preceding Lemma shows that the same holds true for the matrix for $(\ ,\ )_{\Sigma_2} $ with respect to $\mathcal{A}$. Let $N$ denote the sum over $ \mathcal{A}$ of the number of curves appearing in each arrangement. The change of basis matrix for writing $\mathcal{A}(v)$ in terms of $\mathcal{A}$ is again triangular and has determinant $\sim (1-q)^{-N}.$ Thus the matrix for $(\ ,\ )_{\Sigma_2} $ with respect to $\mathcal{A}(v)$ has determinant $\sim (1-q)^{2 (d-1) r -2N}$. The following Lemma \[lemf\] shows that this determinant is a unit. As in the genus one case (see the proof of Theorem \[firstbasis\]), we conclude that $\mathcal{A}(v)$ is a basis for ${{\mathcal{S}_p}}({{\Sigma}}_2)$ and that the form $(\ ,\ )_{\Sigma_2} $ is unimodular on ${{\mathcal{S}_p}}({{\Sigma}}_2)$. \[lemf\] $N=(d-1)r$. To count $N, $ we write ${{\mathcal{A}}}= \cup_{ 0\le \gamma \le d-1} {{\mathcal{A}}}_\gamma,$ where $${{\mathcal{A}}}_\gamma= \{ C_{\alpha,\beta,\gamma} | 0 \le \alpha,\beta \le d-\gamma-1 \}~.$$ Note that $|{{\mathcal{A}}}_\gamma|= (d-\gamma)^2.$ The total number of curves appearing in ${{\mathcal{A}}}_\gamma$ is $$\gamma (d-\gamma)^2 + \sum_{\alpha=0}^{d-\gamma-1}\sum_{\beta=0}^{d-\gamma-1} (\alpha +\beta) = (d-1) (d-\gamma)^2~.$$ Thus each ${{\mathcal{A}}}_\gamma$ contributes $d-1$ times its cardinality to the count. As $\sum_{\gamma =0}^{d-1} |{{\mathcal{A}}}_\gamma| =r,$ we see that $N=(d-1) r.$ This completes the proof of Theorem \[thg2\]. Non-Unimodularity {#non-uni} ================== Even without knowing an explicit basis of ${{\mathcal{S}_p}}({{\Sigma}}_g)$, it is possible to see that the form $(\ ,\ )_{{{\Sigma}}_g}$ is sometimes not unimodular. \[9.1\] If $p \equiv 1 \pmod{4}$ and both the genus $g$ and the rank of ${{V_p}}({{\Sigma}}_g)$ are odd, then the form $(\ ,\ )_{{{\Sigma}}_g}$ is not unimodular on ${{\mathcal{S}_p}}({{\Sigma}}_g)$. For example, if $g=3$ and $p=5$ then the rank is $15$ and the form $(\ ,\ )_{{{\Sigma}}_3}$ is not unimodular on $\mathcal{S}_5({{\Sigma}}_3)$. [*We used Mathematica [@Wo] to calculate the rank of $V_{p}(\Sigma_g)$ for small $g$ using the formula [@BHMV2 1.16(ii)]. We found that: $$\begin{aligned} \text{rank} \left( V_{4k+1}(\Sigma_3)\right) &= (1 /{45} )(3 \ k+32 \ k^2+120 \ k^3+200 \ k^4+192 \ k^5+128 \ k^6) \\ \begin{split} \text{rank} \left(V_{4k+1}(\Sigma_5)\right) &= ( 1/ {14175} ) (45 \ k + 864 \ k^2 + 6892 \ k^3 + 30184 \ k^4\\ & + 83760 \ k^5 + 172512 \ k^6 + 304896 \ k^7 + 458112 \ k^8\\ & + 542720 \ k^9 + 487424 \ k^{10} + 294912 \ k^{11} + 98304 \ k^{12}) \end{split} \end{aligned}$$ Thus the rank of $V_p({{\Sigma}}_3)$ is odd if $p \equiv 5 \pmod{8},$ and the form $(\ ,\ )_{{{\Sigma}}_3}$ is not unimodular in this case. Similarly $(\ ,\ )_{{{\Sigma}}_5}$ is not unimodular if $p \equiv 5 \pmod{8}. $* ]{} The argument relies on the following result of [@G]. Assume $p \equiv 1 \pmod{4}$ and recall that ${{\mathcal{O}}}={{\mathbb{Z}}}[\zeta_{4p}]$ in this case. Put ${{\mathcal{O}}}^+={{\mathbb{Z}}}[\zeta_{p}]\subset {{\mathcal{O}}}$. Let the Lagrangian assigned to $\Sigma_g$ be the kernel of the map on the first homology induced by the inclusion of $ \Sigma_g$ to $H_g$ and assign $H_g$ the weight zero. Then $H_g$ is an [ *even*]{} (in the sense of [@G]) morphism from $\emptyset$ to $\Sigma_g.$ Note that the quantum integers $[n]$ for $1\le n \le p-1$ are units in ${{\mathcal{O}}}^+.$ [@G] If $p \equiv 1 \pmod{4}$ then ${{\mathcal{S}_p}}({{\Sigma}}_g)\simeq {{\mathcal{S}^+_p}}({{\Sigma}}_g)\otimes {{\mathcal{O}}}$ where ${{\mathcal{S}^+_p}}({{\Sigma}}_g)\subset {{\mathcal{S}_p}}({{\Sigma}}_g)$ is a free ${{\mathcal{O}}}^+$-module. Moreover, one has $\mathcal {G}\subset{{\mathcal{S}^+_p}}({{\Sigma}}_g)$, where $\mathcal {G}$ is the colored graph basis of ${{V_p}}({{\Sigma}}_g)$ (see Proposition \[detb\]). The matrix of $(\ ,\ )_{{{\Sigma}}_g}$ with respect to $\mathcal {G}$ has determinant ${{\mathcal{D}}}^{gr}\sim (1-q)^{(d-1)gr}$ where $r$ denotes the rank of ${{V_p}}({{\Sigma}}_g)$. Let $\mathcal {B}$ be a basis of the free ${{\mathcal{O}}}^+$-module ${{\mathcal{S}^+_p}}({{\Sigma}}_g)$, and let $D$ be the determinant of the matrix expressing $\mathcal {B}$ in terms of $\mathcal {G}$. The matrix of $(\ ,\ )_{{{\Sigma}}_g}$ with respect to the basis $\mathcal {B}$ has determinant $\sim \Delta$, where $$\label{Delta} \Delta = D \overline D (1-q)^{(d-1)gr}~.$$ If the form is unimodular, $\Delta$ must be a unit in ${{\mathcal{O}}}$, and since $\Delta$ lies in ${{\mathcal{O}}}^+$, it must be a unit in ${{\mathcal{O}}}^+$. But $1-q$ is a self-conjugate prime in ${{\mathcal{O}}}^+={{\mathbb{Z}}}[q]={{\mathbb{Z}}}[\zeta_{p}]$, and since $D^{-1}$ lies in ${{\mathcal{O}}}^+$ as well, $\Delta$ can be a unit only if $(d-1)gr$ is even. Thus one of $g$ and $r$ must be even (since $d-1=(p-3)/2$ is odd in our situation). This completes the proof. \[9.4\] [*The use of the ${{\mathcal{O}}}^+$-module ${{\mathcal{S}^+_p}}({{\Sigma}}_g)$ can in general not be avoided in this argument. Here is why. Recall that $1-q$ splits in ${{\mathcal{O}}}={{\mathbb{Z}}}[\zeta_{4p}]$ as the product of two conjugate prime ideals $\mathfrak{p}$ and $\overline{\mathfrak{p}}$. If $\mathfrak p$ is principal (this happens for example if $p=5$), then there exists $D\in {{\mathcal{O}}}$ such that the number $\Delta$ defined as in (\[Delta\]) is a unit even when $(d-1)gr$ is odd. Of course, such a $D$ does not exist in ${{\mathcal{O}}}^+$.*]{} [*If we assign extra structure to ${{\Sigma}}_g$ and $H_g$ as described above in the proof of 9.1, then ${{\mathcal{O}}}^+$ linear combinations of banded links in $H$ represent elements in ${{\mathcal{S}^+_p}}({{\Sigma}}).$ Moreover the bases described in Sections \[1st\], \[2nd\], \[g2\] for ${{\mathcal{S}_p}}(S^1 \times S^1), $ and ${{\mathcal{S}_p}}({{\Sigma}}_2)$ are actually bases for ${{\mathcal{S}^+_p}}(S^1 \times S^1), $ and ${{\mathcal{S}^+_p}}({{\Sigma}}_2).$ There are also plus versions of Theorem \[ml\] and Corollaries \[wc\] and \[v\].*]{} [*When restricted to ${{\mathcal{S}^+_p}}(\Sigma_g)$, the Hermitian form $(\ ,\ )_{\Sigma_g}$ does not take values in ${{\mathcal{O}}}^+,$ if $g$ is odd. This follows from the proof of Proposition \[detb\], since ${{\mathcal{D}}}\not\in {{\mathcal{O}}}^+$. In the next section, we will use the sesquilinear form $$(\ ,\ )^+_{\Sigma_g}: {{\mathcal{S}^+_p}}(\Sigma_g) \times {{\mathcal{S}^+_p}}(\Sigma_g) \rightarrow \mathcal{O}^+$$ obtained by multiplying the form $(\ ,\ )_{\Sigma_g}$ by $i^{\varepsilon (g)},$ where ${\varepsilon (g)}$ is zero or one accordingly as $g$ is even, or odd. This form takes values in ${{\mathcal{O}}}^+$ since $i {{\mathcal{D}}}\in {{\mathcal{O}}}^+$.* ]{} Genus three at the prime five {#g3} ============================= In genus $g\geq 3$, one can also try to find a set of banded links in a handlebody so that one obtains a basis of ${{\mathcal{S}_p}}({{\Sigma}}_g)$ by cabling each curve component with $v=(z+2)/(1+A)$. This is suggested by Corollary \[v\] and the fact that ${{\mathcal{S}_p}}({{\Sigma}}_g)$ is a free ${{\mathcal{O}}}$-module [@G]. In fact, we now find such a set of links giving a basis for $\mathcal{S}^+_5({{\Sigma}}_3)$ (and therefore also for $\mathcal{S}_5({{\Sigma}}_3)$) by adapting the above procedures. These links are described by arrangements of curves in a thrice punctured disk. Although the Hermitian form and the related ${{\mathcal{O}}}^+$-valued sesquilinear form are not unimodular, in this particular situation they are nearly so, and this is essential for our argument. It seems more difficult to find an explicit collection of banded links with this property for $\mathcal{S}_p({{\Sigma}}_3)$ for $p>5,$ and for $\mathcal{S}_p({{\Sigma}}_g)$ for $g>4.$ We plan to return to this question elsewhere. We think of the handlebody $H_3$ as $P_3 \times I$ where $P_3$ is a disk with three holes. We give $H_3$ weight zero. We equip ${{\Sigma}}_3$ with the Lagrangian given by the kernel of the map induced on the first homology by the inclusion of ${{\Sigma}}_3$ in $P_3 \times I.$ Consider the set of 15 arrangements of curves in $P_3$ $$\begin{aligned} {{\mathcal{A}}}=\{A_\emptyset,&A_1,A_2,A_3, A_1A_2,A_2A_3,A_3A_1, A_1A_2A_3,\\ &A_{12},A_{23},A_{13}, A_{12}A_3, A_{23}A_1, A_{31}A_2, A_{123}\}~.\end{aligned}$$ Here, $A_\emptyset$ is the empty arrangement, $A_i$ (resp. $A_{ij}$, resp. $A_{123}$) is a curve of the shape pictured in Figure \[fig3\] around just the $i$-th hole (resp. around both the $i$-th and $j$-th hole, resp. around all three holes), and the multiplicative notation $A_\alpha A_\beta$ means disjoint union of $A_\alpha$ and $A_\beta$. See Figure \[fig3\] for two examples. Note that the total number of curves in ${{\mathcal{A}}}$ is $22$. \[thg3\] The set ${{\mathcal{A}}}(v)=\{A_\emptyset(v)=A_\emptyset,A_1(v),A_2(v),\ldots\}$ consisting of the curve arrangements in ${{\mathcal{A}}}$ colored $v$ is a basis of $\mathcal{S}^+_5({{\Sigma}}_3), $ and thus also a basis for $\mathcal{S}_5({{\Sigma}}_3).$ Note that it follows in particular that the ${{\mathcal{O}}}^+$-span of ${{\mathcal{A}}}(v)$ is stable under the action of the index two subgroup of even morphisms in the mapping class group $\tilde{\Gamma}({{\Sigma}}_2)$. (-.6,-2.8)(2.6,.8) (-.05,-.05)(.05,.05) (-.05,.05)(.05,-.05) (1.95,-.05)(2.05,.05) (1.95,.05)(2.05,-.05) (0,0)[.6]{} (2,0)[.6]{} (1,-1.73)[.6]{} (.95,-1.78)(1.05,-1.68) (.95,-1.68)(1.05,-1.78) (1,-.577)(.51,-.29) (1,-.577)(1.49,-.29) (1,-.577)(1,-1.13) (.1,-.1)[${1}$]{} (2.1,-.1)[${2}$]{} (1.1,-1.83)[$3$]{}          (-.6,-2.8)(2.6,.8) (.1,-.1)[${1}$]{} (2.1,-.1)[${2}$]{} (1.1,-1.83)[$3$]{} (1.6,-2.2)[$A_{123}$]{} (-.05,-.05)(.05,.05) (-.05,.05)(.05,-.05) (1.95,-.05)(2.05,.05) (1.95,.05)(2.05,-.05) (0,0)[.6]{}[350]{}[310]{} (2,0)[.6]{}[230]{}[190]{} (1,-1.73)[.6]{}[110]{}[70]{} (.6,-.1)(1,-.33) (1.4,-.1)(1,-.33) (.385,-.475)(.8,-.7) (1.615,-.475)(1.2,-.7) (1.2,-.7)(1.2,-1.175) (.8,-.7)(.8,-1.175) (.95,-1.78)(1.05,-1.68) (.95,-1.68)(1.05,-1.78)          (-.6,-2.8)(2.6,.8) (.1,-.1)[${1}$]{} (2.1,-.1)[${2}$]{} (1.1,-1.83)[$3$]{} (1.6,-2.2)[$A_{3}$]{} (2.3,-.8)[$A_{12}$]{} (-.05,-.05)(.05,.05) (-.05,.05)(.05,-.05) (1.95,-.05)(2.05,.05) (1.95,.05)(2.05,-.05) (.95,-1.78)(1.05,-1.68) (.95,-1.68)(1.05,-1.78) (0,0)[.6]{}[20]{}[340]{} (2,0)[.6]{}[200]{}[160]{} (.56,.2)(1.44,.2) (.56,-.2)(1.44,-.2) (1,-1.73)[.6]{} The set ${{\mathcal{A}}}$ (where its elements are considered as planar banded links in $H_3$) is a basis of $V_5({{\Sigma}}_3)$. By the proof of Lemma \[lemG\], we can find a graph basis $\mathcal{G}$ for $V_5({{\Sigma}}_3)$ by 5-admissible colorings of the graph $G$ in Figure \[fig3\], where the loops are colored zero or one and the non-loop edges are colored zero or two. If a non-loop edge is colored two, then the loop at the end of the edge must be colored one. Also, the number of non-loop edges colored two must be zero, two, or three. This summarises 5-admissibility in this case. There are 15 such colorings. Again using Wenzl’s recursion formula, there is a triangular change of basis with ones on the diagonal from the basis $\mathcal{G}$ to the set $\mathcal{A}$ which is therefore also a basis of $V_5({{\Sigma}}_3)$. Recall that ${{\mathcal{A}}}(v)$ consists of the $15$ elements of ${{\mathcal{S}^+_5}}(\Sigma_3)$ obtained by replacing each of the 22 curves in ${{\mathcal{A}}}$ with $v=(z+2)/(1+A)$. Again there is a triangular change of basis matrix from ${{\mathcal{A}}}$ to ${{\mathcal{A}}}(v)$. Therefore the elements of ${{\mathcal{A}}}(v)$ span $V_5({{\Sigma}}_3)$ and hence are linearly independent over ${{\mathcal{O}}}^+$. Consider the inclusion $$\label{strict} \text{Span}_{{{\mathcal{O}}}^+}{{\mathcal{A}}}(v) \subset {{\mathcal{S}^+_5}}(\Sigma_3)~.$$ The matrix for $(\ ,\ )^+_{\Sigma}$ with respect to $\mathcal{A }(v)$ has determinant $\sim (1-q)^{3\cdot 15-2 \cdot 22}=1-q.$ Since $1-q$ is a prime in ${{\mathcal{O}}}^+ $ and ${{\mathcal{S}^+_5}}(\Sigma_3)$ is also a free ${{\mathcal{O}}}^+$-module, we conclude that the inclusion (\[strict\]) cannot be strict. Thus $\mathcal{A }(v)$ is a basis for ${{\mathcal{S}^+_5}}(\Sigma_3).$ [*Theorem \[thg3\] remains true if we replace $v$ by $\omega$ throughout. The same proof works.*]{} [*As in Remark \[9.4\], it is crucial for this argument to use ${{\mathcal{S}^+_5}}({{\Sigma}}_3)$ rather than ${{\mathcal{S}_5}}({{\Sigma}}_3)$, since for $p=5$ there exists $a\in {{\mathcal{O}}}={{\mathbb{Z}}}[\zeta_{20}]$ such that $1-q=a\overline{a}$.*]{} [*Kerler has announced in [@Ke] a construction of integral bases for the Reshetikhin-Turaev $SO(3)$ TQFT at the prime $p=5$ for any genus.*]{} A divisibility result for the Kauffman bracket ============================================== In this final section, we let $A$ again be an indeterminant. The fact that $v=(z+2)/(1+ \zeta_{2p} )$ lies in ${{\mathcal{S}_p}}(S^1\times S^1)$ for all odd primes $p$ has the following application to the Kauffman bracket $\langle\ \rangle$ of banded links in $S^3$. \[th11\] Let $L$ be a banded link in $S^3$ with $\mu$ components. Let $L(z+2)$ denote this link colored $z+2.$ Then the Kauffman Bracket $\langle L(z+2) \rangle \in {{\mathbb{Z}}}[A^{\pm}]$ is divisible by $(1+A)^{\mu}$. Here the Kauffman bracket is normalized so that the bracket of the empty link is $\langle \emptyset\rangle=1$. Note that $$\langle L(z+2)\rangle = \sum_{L'\subset L} 2^{\mu-\mu(L')} \langle L'\rangle~,$$ where the sum is over all sublinks $L'$ of $L$, and $\mu(L')$ denotes the number of components of $L'$. When we evaluate the Kauffman bracket $\langle J \rangle$ of a banded link $J$ in $S^3$ at $A=\zeta_{2p}$, we obtain the quantum invariant of the pair $(S^3,J)$ (where $S^3$ is given the weight zero) in the normalization $$\langle J \rangle\vert_{A=\zeta_{2p}}=I_p(S^3,J)=\frac {\langle (S^3,J) \rangle} {\langle S^3 \rangle}={{\mathcal{D}}}{\langle (S^3,J) \rangle} ~.$$ This normalization $I_p(M,J)$ of the quantum invariant is precisely the one which is always an algebraic integer [@Mu1; @MR] and which is at the basis of the integral cobordism functors ${{\mathcal{S}_p}}$. Let $f(A)$ denote the Kauffman bracket $\langle L(z+2)\rangle \in {{\mathbb{Z}}}[A^\pm].$ Since $v= (z+2)/(1+ \zeta_{2p} )\in S_p(S^1 \times S^1),$ we have $I_p(S^3,L(v)) \in {{\mathbb{Z}}}[\zeta_{2p}],$ for every odd prime $p.$ Thus $$\label{divisi} f(\zeta_{2p}) = I_p(S^3,L(z+2)) \in (1+\zeta_{2p})^\mu {{\mathbb{Z}}}[\zeta_{2p}],$$ for every odd prime $p.$ Now recall the following elementary Lemma (see [@Mu1 Lemma 5.5] and note that $-\zeta_{2p}$ is a primitive $p$-th root). Suppose $f(A) \in {{{\mathbb{Z}}}} [A^\pm].$ Let $f^{(k)}(A)$ denote the $k$-th derivative of $f(A)$. Assume $0\leq \mu<p$ where $p$ is prime. Then $f(\zeta_{2p})\in {{{\mathbb{Z}}}} [\zeta_{p} ]$ is divisible by $(1+\zeta_{2p})^\mu$ if and only if $f^{(k)}(-1)\equiv 0\pmod{p}$ for every $0\le k< \mu.$ By this lemma, (\[divisi\]) implies $f^{(k)}(-1)\equiv 0\pmod{p}$ for each $0\le k< \mu,$ provided $p$ is larger than $\mu.$ Since there are infinitely many such primes, it follows that $f^{(k)}(-1)=0$ for each $0\le k< \mu.$ But this means that $ (1+A)^{\mu}$ divides $f(A).$ \[Kcor\] If $L$ is as in the theorem, then the Kauffman Bracket $\langle L(z+[2]) \rangle \in {{\mathbb{Z}}}[A^{\pm}]$ is also divisible by $(1+A)^{\mu}$. This follows immediately from the fact that $2-[2]=2-A^2-A^{-2}=(1-A^2)(1-A^{-2})$ is divisible by $1+A.$ [*In a similar way, Theorem \[tK\] remains true if we replace $v$ with $\hat v= (z+ [2])/(1+A)$ in the definition of $K(n).$ Similarly Theorems \[secondbasis\], \[thg2\], \[thg3\] remain true if we replace $v$ by $\hat v.$* ]{} [*Theorem \[th11\] can also be proved by computing the Kauffman bracket from the (framed) Kontsevich integral via an appropriate weight system. Actually this proof is an adaptation of an argument going back to Kricker and Spence [@KS Proof of Thm. 2], but they only considered algebraically split links. Previously Ohtsuki [@OhCambridge Prop. 3.4] had obtained a stronger divisiblity result for $\langle L(z+[2]) \rangle$ using quantum groups, for algebraically split links satisfying some extra conditions. Later, Cochran and Melvin generalised the Kontsevich integral argument, and their result [@CM Theorem 2.5] contains Corollary \[Kcor\] for zero-framed links. (The results of [@OhCambridge; @KS; @CM] are stated in terms of the Jones polynomial, but it is well-known that the Jones polynomial and the Kauffman bracket are equivalent.) However, the restriction to zero framing is not really necessary (although a small additional argument is needed). We will not give details of this alternative proof here, as the techniques are completely different from the ones in the present paper.* ]{} [BHMV2]{} Asymptotic Faithfulness of the quantum $SU(n)$ representations of the mapping class groups. arXiv:math.QA/0204084 On framings of $3$-manifolds. [*Topology*]{} [**29**]{} (1990) 1-7. . Cambridge University Press, Cambridge, 1935. Three manifold invariants derived from the Kauffman bracket, [*Topology*]{} [**31**]{} (1992), 685-699 Topological quantum field theories derived from the Kauffman bracket, [*Topology*]{} [**34**]{} (1995), 883-927 Quantum cyclotomic orders of 3-manifolds. [*Topology*]{} [**40**]{} (2001), no. 1, 95-125. , [*J. Symbolic Comp,*]{} [**24**]{} (1997), 267-283 On the TQFT representations of the mapping class groups. [*Pacific J. Math.*]{} [**188**]{} (1999), no. 2, 251–274. On the Witten-Reshetikhin-Turaev representations of mapping class groups. [*Proc. AMS*]{} [**127**]{} (1999) 2483-2488. , arXiv:math.QA/0105059. HYP and HYPQ - Mathematica packages for the manipulation of binomial sums and hypergeometric series, respectively q-binomial sums and basic hypergeometric series, [ *J. Symbol. Comput.* ]{}[**20**]{} (1995), 737-744 , arXiv:math.GT/0110007. Ohtsuki’s invariants are of finite type. [*J. Knot Theory Ramifications*]{} [**6**]{} (1997) 583-597. Quantum invariants of 3-manifolds: integrality, splitting, and perturbative expansion. arXiv:math.QA/0004099. An element of infinite order in TQFT-representations of mapping class groups. [*Contemp. Math.*]{} [**233**]{} (1999) 137-139. (1993), no. 2, 171–194. On central extensions of mapping class groups. [*Math. Ann.*]{} [**302**]{}, 131-150 (1995) , Math. Proc. Cambridge Philos. Soc. [**121**]{} (1997) no. 3, 443–454 $3$-valent graphs and the Kauffman bracket. [*Pacific J. Math.*]{} [**164**]{}, (1994) 361-381. Integral modular categories and integrality of quantum invariants at roots of unity of prime order. [*J. reine angew. Math. (Crelle’s Journal)*]{} [**505**]{} (1998) 209-235. (1994), no. 2, 253–281 (1995), no. 2, 237–249 A polynomial invariant of integral homology $3$-spheres. [*Math. Proc. Camb. Phil. Soc.*]{} [**117**]{} (1995) 83-112. A polynomial invariant of rational homology $3$-spheres. [*Invent. Math.*]{} [**123**]{} (1996) 241-257. Irreducibility of some quantum representations of mapping class groups. [*J. Knot Theory Ramifications*]{} [**10**]{} (2001), no. 5, 763–767. The $PSU(n)$ invariants of $3$-manifolds are polynomials. [*J. Knot Theory Ramifications*]{} [**8**]{} (1999) 521-532 de Gruyter (1994) A Mathematica version of Zeilberger’s algorithm for proving binomial coefficient identities. (1995) 673–698. Preprint 1991 Wolfram Media, Inc., Champaign, IL; Cambridge University Press, Cambridge. (1999) [^1]: Warning: In many places ([*e.g.*]{} in [@MR]), $q$ denotes $A^4$ rather than $A^2.$

--- abstract: 'For one parameter subgroup action on a finite volume homogeneous space, we consider the set of points admitting divergent on average trajectories. We show that the Hausdorff dimension of this set is strictly less than the manifold dimension of the homogeneous space. As a corollary we know that the Hausdorff dimension of the set of points admitting divergent trajectories is not full, which proves a conjecture of Y. Cheung [@c11].' address: - 'Department of Mathematics, University of York, Heslington, York, YO10 5DD, United Kingdom' - 'Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, PR China' author: - Lifan Guan - Ronggang Shi title: Hausdorff dimension of divergent trajectories on homogeneous space --- [^1] [^2] Introduction ============ Let $G$ be a connected Lie group, $\Gamma$ be a lattice of $G$[^3] and $F=\{f_t: t\in {\mathbb{R}}\}$ be a one parameter subgroup of $G$. The action of $F$ on the homogeneous space $G/\Gamma$ by left translation defines a flow. In this paper we consider the dynamics of the semiflow given by the action of $F^+ {{\, \stackrel{\mathrm{def}}{=}\, }}\{ f_t: t\ge 0 \}$. For $x\in G/\Gamma$ we say the trajectory $F^+x{{\, \stackrel{\mathrm{def}}{=}\, }}\{f_t x: t\ge 0 \}$ is *divergent* if $f_t x$ leaves any fixed compact subset of $G/\Gamma$ provided $t$ is sufficiently large. We say $F^+x$ is *divergent on average* if for any characteristic function $\mathbbm {1}_K$ of a compact subset $K$ of $G/\Gamma $ one has $$\lim\limits_{T\to \infty}\frac{1}{T}\int_0^T \mathbbm{1}_K (f_t x){\; \mathrm{d}}t= 0.$$ Clearly, if the trajectory $F^+x$ is divergent, then it is divergent on average. The aim of this paper is to understand the set of divergent points $${\mathfrak {D}}'(F^+, G/\Gamma){{\, \stackrel{\mathrm{def}}{=}\, }}\{x\in {G/\Gamma}: F^+x \text{ is divergent}\},$$ and the set of divergent on average points $${\mathfrak {D}}(F^+, G/\Gamma){{\, \stackrel{\mathrm{def}}{=}\, }}\{x\in {G/\Gamma}: F^+x\text{ is divergent on average}\},$$ in terms of their Hausdorff dimensions. Here the Hausdorff dimension is defined by attaching ${G/\Gamma}$ with a Riemannian metric. It is well-known that different choices of Riemannian metrics will not affect the Hausdorff dimension of subsets of ${G/\Gamma}$. Indeed, specific Riemannian metric will be used later for the sake of convenience. According to the work of Margulis [@Ma] and Dani [@d84][@d86], it is well-known that if $F$ is ${\mathrm{Ad}}$-unipotent then the space $G/\Gamma$ admits no divergent on average trajectories of $F^+$. In other words, the set ${\mathfrak {D}}(F^+, G/\Gamma)$, hence the set ${\mathfrak {D}}'(F^+, G/\Gamma)$, is empty. On the other hand, the set ${\mathfrak {D}}(F^+, G/\Gamma)$ can be complicated when $F$ is ${\mathrm{Ad}}$-diagonalizable. For example, it was proved by Cheung in [@c11] that the Hausdorff dimension of ${\mathfrak {D}}'(F^+, {\mathrm{SL}}_3(\mathbb R)/{\mathrm{SL}}_3(\mathbb Z))$ with $F=\{{\mathrm{diag}}(e^t, e^t, e^{-2t}): t\in {\mathbb{R}}\}$ is equal to $7\frac{1}{3}$. Based on his results, Cheung raised the following conjecture in [@c11]. Let $\Gamma$ be a lattice of a connected Lie group $G$ and let $F=\{f_t: t\in {\mathbb{R}}\}$ be a one parameter subgroup of $G$. Then the Hausdorff dimension of ${\mathfrak {D}}'(F^+, G/\Gamma)$ is strictly less than the manifold dimension of $G/\Gamma$. The conjecture is known to be true in the following cases where $G$ is a semisimple Lie group without compact factors and $F$ is ${\mathrm{Ad}}$-diagonalizable: 1. $G$ is of rank one [@Dani]. 2. $G=\prod_{i=1}^{n}{\mathrm{SO}}(n,1)$, $\Gamma=\prod_{i=1}^{n}\Gamma_i$ with each $\Gamma_i$ lattice in ${\mathrm{SO}}(n,1)$ and $F< G$ the is diagonal embedding of any one parameter real split torus $A$ of ${\mathrm{SO}}(n,1)$ [@c07][@yang]. 3. ${G/\Gamma}={\mathrm{SL}}_{m+n}({\mathbb{R}})/{\mathrm{SL}}_{m+n}({\mathbb{Z}})$ and $F=F_{n,m}=\{{\mathrm{diag}}(e^{nt},\ldots, e^{nt}, e^{-mt}, \ldots, e^{-mt}): t\in {\mathbb{R}}\}$ with $m,n\ge 1$ [@c11][@cc16][@KKLM]. Indeed, for all the cases listed above, the Hausdorff dimension of the corresponding ${\mathfrak {D}}'(F^+, G/\Gamma)$ have been determined. There are evidences that a stronger version of this conjecture is true. It was proved by Einsiedler-Kadyrov in [@ek12] that the Hausdorff dimension of ${\mathfrak {D}}(F^+, {\mathrm{SL}}_3(\mathbb R)/{\mathrm{SL}}_3(\mathbb Z))$ is at most $7\frac{1}{3}$ when $F=F_{1,2}$ as in (3). Using the contraction property of the height function introduced in [@emm98], it was proved by Kadyrov, Kleinbock, Lindenstrauss and Margulis in [@KKLM] that for any $m,n\ge 1$, the Hausdorff dimension of ${\mathfrak {D}}(F^+, {\mathrm{SL}}_{m+n}(\mathbb R)/{\mathrm{SL}}_{m+n}(\mathbb Z))$ is at most $\dim G- \frac{mn}{m+n}$ when $F=F_{n,m}$ as in (3). See also [@elmv][@k][@kp][@lsst][@dfsu][@weiss] for related results. Now we state the main result of this paper, from which Cheung’s conjecture follows. \[t-main\] Let $\Gamma$ be a lattice of a connected Lie group $G$ and let $F=\{f_t: t\in {\mathbb{R}}\}$ be a one parameter subgroup of $G$. Then the Hausdorff dimension of ${\mathfrak {D}}(F^+, G/\Gamma)$ is strictly less than the manifold dimension of $G/\Gamma$. We will reduce the proof of Theorem \[t-main\] to the special case where $G$ is a semisimple linear group. Recall that a connected semisimple Lie group $G$ contained in ${\mathrm{SL}}_k({\mathbb{R}})$ has a natural structure of real algebraic group. So terminologies of algebraic groups have natural meanings for $G$ and are independent of the embeddings of $G$ into ${\mathrm{SL}}_k({\mathbb{R}})$. In particular, the one parameter group $F$ has the following real Jordan decomposition which is a special case of [@borel Theorem 4.4]. \[l-RJD\] Let $G\le {\mathrm{SL}}_k({\mathbb{R}})$ be a connected semisimple Lie group. For any one parameter subgroup $F=\{f_t: t\in {\mathbb{R}}\}$, there are uniquely determined one parameter subgroups $K_F=\{k_t:t\in {\mathbb{R}}\}$, $A_F=\{a_t:t\in {\mathbb{R}}\}$ and $U_F=\{u_t:t\in {\mathbb{R}}\}$ with the following properties: - $f_t=k_ta_tu_t$. - $K_F$ is bounded, $A_F$ is ${\mathbb{R}}$-diagonalizable and $U_F$ is unipotent. - All the elements of $K_F$, $A_F$ and $U_F$ commute with each other. The subgroups $K_F, A_F$ and $ U_F$ are called compact, diagonal and unipotent parts of $F$, respectively. In §\[sec;2\] we will reduce the proof of Theorem \[t-main\] to its following special case which contains the main unknown situations. \[thm;general\] Let $ G \le {\mathrm{SL}}_k({\mathbb{R}})$ be a connected center-free semisimple Lie group without compact factors. Let $F=\{f_t: t\in {\mathbb{R}}\}$ be a one parameter subgroup of $G$ such that the compact part $K_F$ is trivial but the diagonal part $A_F$ is nontrivial. We assume the followings hold: - $G=\prod_{i=1}^m G_i$ is a direct product of connected normal subgroups $G_i$. - $\Gamma =\prod_{i=1}^m \Gamma_i$ where each $\Gamma_i$ is a nonuniform irreducible lattice of $G_i$. - The group $A_F$ has nontrivial projection to each $G_i$. Then the Hausdorff dimension of $ {\mathfrak {D}}(F^+, G/\Gamma) $ is strictly less than the manifold dimension of $G/\Gamma$. The proof of Theorem \[thm;general\] is from §\[sec;3\] till the end of the paper. Indeed, the upper bound of the Hausdorff dimension in the setting of Theorem \[thm;general\] can be explicitly calculated and we will make this point clear during the proof. Our main tool will be the Eskin-Margulis height function (abbreviated as EM height function) introduced in [@em]. If $F$ is diagonalizable, i.e. $F=A_F$, Theorem \[t-main\] can be established using the strategy developed in [@KKLM] and the contraction property of the proved in [@s]. But when $F$ has nontrivial unipotent parts, i.e. $U_F$ is nontrivial, essential new ideas are needed. The following example contains the main difficulties we need to handle in the proof of Theorem \[thm;general\]: $ G={\mathrm{SL}}_4({\mathbb{R}})\times {\mathrm{SL}}_4({\mathbb{R}}), \Gamma ={\mathrm{SL}}_4({\mathbb{Z}}[\sqrt 2]) $ which embeds in $G$ diagonally via Galois conjugates, and $$f_t = \left( \begin{array}{cccc} e^t & 0 & 0 & 0\\ 0 & e^{-t} & 0 & 0\\ 0 & 0 & 1 & t \\ 0 & 0 & 0 & 1 \end{array} \right)\times \left( \begin{array}{cccc} 1 & t & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right).$$ There are two main difficulties. One is caused by the unipotent part of $f_t$ in the first ${\mathrm{SL}}_4({\mathbb{R}})$ factor, and the other is caused by the unipotent part of $f_t$ in the second ${\mathrm{SL}}_4({\mathbb{R}})$ factor. To overcome these difficulties, we will prove a uniform contraction property for a family of one parameter subgroups in §\[sec;3\] and §\[sec;height\] with respect to the EM height function. Then the last two sections are devoted to the proof of Theorem \[thm;general\]. Proof of Theorem \[t-main\] {#sec;2} =========================== In this section we prove Theorems \[t-main\] assuming Theorem \[thm;general\]. Let $G, \Gamma, F$ be as in Theorem \[t-main\]. We choose and fix a Euclidean norm $\|\cdot\|$ on the Lie algebra ${\mathfrak{g}}$ of $G$, which induces a right invariant Riemannian metric ${\mathrm{dist}}(\cdot, \cdot)$ on $G$. Moreover, this metric naturally induce a metric on $G/\Gamma$, also denoted by $``{\mathrm{dist}}"$, as follows: $${\mathrm{dist}}(g\Gamma, h\Gamma )=\inf_{\gamma\in \Gamma}{\mathrm{dist}}(g\gamma, h)\quad \mbox{where }g, h\in G .$$ Let ${\mathfrak{r}}$ be the maximal amenable ideal of the Lie algebra ${\mathfrak{g}}$ of $G$, i.e. the largest ideal whose analytic subgroup is amenable. The adjoint action of $G$ on $\mathfrak s={\mathfrak{g}}/{\mathfrak{r}}$ defines a homomorphism $\pi:G\mapsto \mathrm{Aut}(\mathfrak s)$. Let $S$ be the connected component of $\mathrm{Aut}(\mathfrak s)$. It follows from the Levi decomposition of $G$ that $\pi(G)=S$ and $S$ is a center-free semisimple Lie group without compact factors. It is known that $\Gamma\cap {\mathrm{Ker}}(\pi)$ is a cocompact lattice in ${\mathrm{Ker}}(\pi)$ and $\pi(\Gamma)$ is a lattice in $S$, see e.g. [@BQ Lemma 6.1]. Therefore, [the induced map $\overline {\pi}: G/\Gamma \mapsto S/\pi(\Gamma)$ is proper]{}, and consequently $$\begin{aligned} \label{eq;equal} \overline{\pi}({\mathfrak {D}}(F^+, G/\Gamma))={\mathfrak {D}}(\pi(F^+), S/\pi(\Gamma)). \end{aligned}$$ Let $\varphi: \mathfrak s\to \mathfrak g$ be an embedding of Lie algebras such that $\mathrm{d}\pi\circ \varphi$ is the identity map. It follows from (\[eq;equal\]) that for any $x\in G/\Gamma$ and any $v\in \mathfrak s$ $$\exp(v)\overline{\pi}(x)\in {\mathfrak {D}}(\pi(F^+), S/\pi(\Gamma))$$ if and only if $$\exp(\varphi(v))\exp (v')x\in {\mathfrak {D}}(F^+, G/\Gamma) \mbox{\quad for all }v'\in {\mathfrak{r}}.$$ By Marstrand’s product theorem, the Hausdorff dimension of the product of two sets of Euclidean spaces is bounded from above by the sum of the Hausdorff dimension of one set and the packing dimension of the other, e.g. [@bp Theorem 3.2.1]. So to prove Theorem \[t-main\] it suffices to give a nontrivial upper bound of the Hausdorff dimension of ${\mathfrak {D}}(\pi(F^+), S/\pi(\Gamma)) $. We summarize what we have obtained as follows. \[prop;more\] Theorem \[t-main\] is equivalent to its special case where $G$ is a center-free semisimple Lie group without compact factors. According to Lemma \[prop;more\], it suffices to prove the theorem under the additional assumption that $G$ is a center-free semisimple Lie group without compact factors. Under this assumption, the adjoint representation ${\mathrm{Ad}}: G\to {\mathrm{SL}}({\mathfrak{g}}) $ is a closed embedding. According to the real Jordan decomposition in Lemma \[l-RJD\], the compact part $K_{F}$ does not affect the divergence on average property of the trajectories. So we assume without loss of generality that $K_{F}$ is trivial. There exist finitely many connected semisimple subgroups $G_i$ such that $G= \prod_i G_i$ and $\Gamma_i {{\, \stackrel{\mathrm{def}}{=}\, }}\Gamma \cap G_i$ is an irreducible lattice of $G_i$ for each $i$. It follows that $\prod_i \Gamma_i$ is a finite index subgroup of $\Gamma$ and the natural quotient map $G/\prod_i \Gamma_i \rightarrow {G/\Gamma}$ is proper. So we assume moreover that $\Gamma= \prod_i \Gamma_i$. Denote by $\pi_j $ the projection of $G$ to $G/ G_j=\prod_{i\neq j}G_i$ and denote by $\overline{\pi}_j$ the induced map from $G/\Gamma$ to $\pi_j(G)/\pi_j(\Gamma)=\prod_{i\neq j} G_i/\Gamma_i$. Here if $G=G_j$ we interpret $\prod_{i\neq j}G_i$ as a trivial group and $\prod_{i\neq j} G_i/\Gamma_i$ as a singe point set. If $G_j/\Gamma_j$ is compact or the projection of $A_F$ to $G_j$ contains only the neutral element, then $${\mathfrak {D}}(F^+, G/\Gamma) = \overline{\pi}_j^{-1}\Big({\mathfrak {D}}\big(\pi_j(F^+), \prod_{i\neq j}G_i/\Gamma_i\big)\Big).$$ So either ${\mathfrak {D}}(F^+, G/\Gamma)$ is an empty set or we finally can reduce the problem to the setting of Theorem \[thm;general\] where each $\Gamma_i$ is a nonuniform lattice and the projection of $A_F$ to each $G_i $ is nontrivial. This completes the proof. Preliminary on linear representations {#sec;3} ===================================== From this section, we start the proof of Theorem \[thm;general\]. At the beginning of each section we will set up some notation that will be used later. Let $G$ and $ F$ be as in Theorem \[thm;general\]. Let $A_F=\{a_t:t\in {\mathbb{R}}\}$ and $U_F=\{u_t: t\in {\mathbb{R}}\}$ be the diagonal and unipotent parts of $F$. Let $H$ be the unique connected normal subgroup of $G$ such that $A_F\le H $ and the projection of $A_F$ to each simple factor of $H$ is nontrivial. Since $A_F$ is nontrivial and $G$ is center-free, $H$ is (nontrivial) product of some simple factors of $G$. Hence $H$ is a semisimple Lie group without compact factors. Let $S$ be the product of simple factors of $G$ not contained in $H$. Then $S$ is also semisimple normal subgroup of $G$ that commutes with $H$. Moreover, $G=HS$ and $H\cap S=1_G$, where $1_G$ is the neutral element of $G$. In this section we prove a couple of auxiliary results for a finite dimensional linear representation $\rho: G\rightarrow {\mathrm{GL}}(V)$ on a (nonzero real) normed vector space $V$. These results will be used in the next section to prove the uniform contracting property of EM height function. We will use $\|\cdot\|$ to denote the norm on $V$. For $\lambda\in {\mathbb{R}}$, we denote the $\lambda$-Lyapunov subspace of $A_F$ by $$V^{\lambda}=\{v\in V: \rho (a_t) v=e^{\lambda t} v \}.$$ Recall that if $V^\lambda\neq \{0 \}$, then $\lambda$ is a called an Lyapunov exponent of $(\rho, V)$. Since $U_F$ commutes with $A_F$, every Lyapunov subspace $V^{\lambda}$ is $U_F$-invariant. As $A_F$ is ${\mathbb{R}}$-diagonalizable, the space $V$ can be decomposed as $V^+\oplus V^0\oplus V^-$ where $$V^+=\oplus_{\lambda>0}V^{\lambda}\quad\mbox{and }\quad V^-=\oplus_{\lambda <0}V^{\lambda}.$$ Now we consider the adjoint representation of $G$ on the Lie algebra ${\mathfrak{g}}$ of $G$. It is easily checked that ${\mathfrak{g}}^+, {\mathfrak{g}}^-$ and $ {\mathfrak{g}}^c{{\, \stackrel{\mathrm{def}}{=}\, }}{\mathfrak{g}}^0$ are subalgebras of ${\mathfrak{g}}$. The connected subgroup $G^+$ (resp. $ G^-$) with Lie algebras ${\mathfrak{g}}^+$ (resp. ${\mathfrak{g}}^-$) is called unstable (resp. stable) horospherical subgroup of $a_1$. We denote the connected component of the centralizer of $a_1$ in $G$ by $G^c$ whose Lie algebra is ${\mathfrak{g}}^c$. Let $d, d^c, d^-$ be the manifold dimensions of $G^+$, $G^c$ and $G^-$, respectively. It follows from the nontriviality of $A_F$ that $d>0$. For $r\ge 0$ we let $B_r^{G}=\{h\in G: {\mathrm{dist}}(h, 1_G)<r \}$, $B_r^{\pm }=\{h\in G^{\pm}: {\mathrm{dist}}(h, 1_G)<r \}$ and $B_r^c=\{h\in G^c: {\mathrm{dist}}(h, 1_G)<r \}$. By rescaling the Riemannian metric if necessary, we may assume that: 1. the product map $B_1^-\times B_1^c\times B_1^+\to G$ is a diffeomorhism onto its image, 2. and the logarithm map is well-defined on $B_1^{G}$ and is a diffeomorphism onto its image. According to (1), it is safe to identity the product $B_1^-\times B_1^c\times B_1^+$ with $B_1^-B_1^cB_1^+$ and we will mainly use the later notation for sake of convenience. The same statement as (2) also holds for $B_1^{\pm }$ and $B_1^c$. We fix a Haar measure $\mu$ on $G^+$ normalized with $\mu(B_1^+)=1$. Since the metric $``{\mathrm{dist}}"$ is right invariant, any open ball of radius $r$ in $G^+$ has the form $B_r^+ h$ $(h\in G^+)$ and there exits $C_0\ge 1$ such that $$\begin{aligned} \label{eq;c+} C_0^{-1}r^d\le \mu( B_r^+ h)=\mu{(B_r^+)}\le C_0 r^d\quad \mbox{for all } 0\le r\le 1 . \end{aligned}$$ For $g, h\in G$ we let $g^h= h^{-1}g h$. For $z\in G^c$, let $F _z = \{f_t ^z : t\in {\mathbb{R}}\}$ and $F _z ^+= \{f_t ^z : t \ge 0 \}$. Note that $f_t ^z = a_t u_t ^z $ and $$\begin{aligned} \label{eq;compact} \{ u_1 ^z : z\in B_1^c \} \quad \mbox{is relatively compact. } \end{aligned}$$ \[l-v+-\] Let $\rho: G\to {\mathrm{GL}}(V)$ be a representation on a finite dimensional normed vector space $V$. Let $\lambda$ be a Lyapunov exponent of $(\rho, V)$. For any $0<\delta<1$, there exists $T_\delta >0$ such that, for all $t\ge T_\delta, z\in B_1^c$ and unit vector $v\in V^{\lambda}$ we have $$\label{ine-v+-} e^{(1-\delta)\lambda t}\le \|\rho(f_{t} ^z )v\| \le e^{(1+\delta)\lambda t}.$$ For all $v\in V^\lambda$ with $ \|v\|=1$ we have $ \|\rho(a_t)v\|= e^{\lambda t}$. On the other hand, in view of (\[eq;compact\]), there exists $C>0$ and $n\in {\mathbb{N}}$[^4] such that $$\|\rho (u_{ t} ^z ) \|\le C (|t|+1)^n$$ for all $z \in B^c_1$ and $t\in {\mathbb{R}}$. Therefore, for any unit vector $v\in V^\lambda, z\in B_1^c$ and sufficiently large $t$, $$\begin{aligned} \|\rho(f_{ t} ^z )v\|&\ge \|\rho(u_{ -t} ^z )\|^{-1} \|\rho(a_t)v\|\ge C^{-1}(|t|+1)^{-n}e^{\lambda t}\ge e^{(1-\delta)\lambda t}, \\ \|\rho(f_{ t} ^z )v\|&\le \|\rho(u_{ t} ^z )\| \|\rho(a_t)v\|\le C(|t|+1)^ne^{\lambda t}\le e^{(1+\delta)\lambda t}. \end{aligned}$$ From now on till the end of this section, we assume that $\rho: G\to {\mathrm{GL}}(V)$ is a representation on a finite dimensional normed vector space $V$ which has no nonzero $H$-invariant vectors. As any two norms on $V$ are equivalent, we also assume that the norm is Euclidean without loss of generality. Recall that a nonzero $H$-invariant subspace $V'$ of $V$ is said to be $H$-irreducible if $V'$ contains no $H$-invariant subspaces besides $\{0 \}$ and itself. The complete reducibility of representations of $H$ implies that there exists a unique decomposition (called $H$-isotropic decomposition) $$\begin{aligned} \label{eq;complicate} V=V_1 \oplus \cdots \oplus V_m\end{aligned}$$ such that irreducible sub-representations of $H$ in the same $V_i$ are isomorphic but irreducible sub-representations in different $V_i$ are non-isomorphic. Since $S$ commutes with $H$, each $V_i$ is $S$-invariant, and hence $G $-invariant. Each $V_i$ is called an $H$-isotropic subspace of $V$. Let $\lambda_i$ be the top Lyapunov exponent of $A_F$ in $(\rho, V_i)$, i.e., $$\begin{aligned} \lambda_i= \max\{ \lambda\in \mathbb R: V_i^{\lambda}\neq \{ 0\} \}.\end{aligned}$$ Since the projection of $A_F$ to each simple factor of $H$ is nontrivial, every $\lambda_i$ is positive. Let $\lambda$ be the minimum of top Lyapunov exponents in each $V_i$, i.e. $$\begin{aligned} \label{eq;top} \lambda= \min\{ \lambda_i: 1\le i\le m \}>0.\end{aligned}$$ Let $\pi_i: V_i\to V^{\lambda_i}_i$ be the $A_F$-equivariant projection. \[lem;sharp\] For all $v\in V_i{\smallsetminus}\{0\}$, the map $$\begin{aligned} \label{eq;varphi} \varphi_v: G^+ \mapsto {\mathbb{R}}\quad \mbox{where}\quad \varphi_v(h)=\|\pi_i(\rho(h)v)\|^2 \end{aligned}$$ is not identically zero. Suppose $\varphi_v$ is identically zero. Then $\rho(G^+) v\subset V'_i$ where $V'_i\subset V_i$ is the $A_F$-invariant complimentary subspace of $V^{\lambda_i}_i$. This implies that $\rho(G^-G^cG^+) v\subset V'_i$. Since $G^-G^cG^+$ contains an open dense subset of $G$, see e.g. [@mt94 Proposition 2.7], we moreover have that $\rho(G)v\subset V'_i$. This is impossible since the intersection of $V^{\lambda_i}_i$ with each $H$-invariant subspace of $V_i$ is nonzero. This contradiction completes the proof. \[l-c-alpha\] For all $v\in V_i{\smallsetminus}\{0\}$ and $r\ge 0$, let $$E(v,r)=\{h\in B_1^+ : \|\pi_i(\rho(h)v)\|\le r \}.$$ Then there exists $\theta_i>0$ such that $$\label{constant-c-alpha} C_i{{\, \stackrel{\mathrm{def}}{=}\, }}\sup_{\|v\|=1,v\in V_i}r^{-\theta_i}\mu(E(v,r))<\infty.$$ In particular, $ \mu(E(v,0))=0$. Since $G^+$ is a unipotent group, it is simply connected and by [@cg Theorem 1.2.10 (a)] there is an isomorphism of affine varieties ${\mathbb{R}}^{d}\to G^+$ such that the Lebesgue measure of ${\mathbb{R}}^d$ corresponds to the Haar measure $\mu$. During the proof, we will identify the group $G^+$ with ${\mathbb{R}}^{d}$ for convenience. By Lemma \[lem;sharp\], for every nonzero $v\in V_i$ the map $\varphi_v$ in (\[eq;varphi\]) is a nonzero polynomial map. So $\varphi_v|_{B_1^+}$ is nonzero. Note that the degrees of $\varphi_v$ ($v\in V_i$) are uniformly bounded from above. Therefore, the $(C, \alpha)$-good property of polynomials in [@bkm §3] implies that there exist positive constants $C$ and $ \alpha$ such that $$\begin{aligned} \label{eq;1} \mu(E(v, r))\le C \left (\frac{r^2}{ \sup_{h\in B_1^+}\varphi_v(h)}\right)^\alpha \end{aligned}$$ for all nonzero $v\in V_i$. Since the set of unit vectors of $V_i$ is compact, $$\begin{aligned} \label{eq;2} \inf_{\|v\|=1, v\in V_i} \sup_{h\in B_1^+}\varphi_v(h)>0. \end{aligned}$$ So (\[constant-c-alpha\]) follows from (\[eq;1\]) and (\[eq;2\]) by taking $\theta_i=2\alpha$. \[rem-3\] According to [@bkm Lemma 3.2] we have $\alpha=\frac{1}{dl}$ where $d$ is the manifold dimension of $G^+$ and $l$ is a uniform upper bound of the degree of $\varphi_v \ (v\in V_i)$. So the constant $\theta_i$ can be calculated explicitly. \[l-contraction\] Let $\theta_0= \min_{ 1\le i\le m} \theta_i$ where $\theta_i>0$ so that Lemma \[l-c-alpha\] holds and let $\lambda$ be as in (\[eq;top\]). Then for any $0<\delta<\theta < \theta_0$, there exists $T_{\theta, \delta}>0$ such that for all $t\ge T_{\theta, \delta}$, $z\in B_1^c$ and $v\in V$ with $\|v\|=1$, we have $$\label{ine-integration} \int_{B_1^+} \|\rho(f_{t} ^z h)v\|^{- \theta }d\mu(h)\le e^{-(\theta-\delta) \lambda t}.$$ Without loss of generality, we assume further that the Euclidean norm $\|\cdot\|$ on $V$ satisfies the following properties: - Lyapunov subspaces of $A_F$ are orthogonal to each other. - $H$-isotropic subspaces $V_i\ (1\le i\le m)$ are orthogonal to each other. Let $$\label{constant-c-r} R_i=\sup_{v\in V_i, \|v\|=1, h\in B^+_1}\|\pi_i(\rho(h)v)\|\quad \mbox{and } \quad R=\max\{ R_i: 1\le i\le m \} .$$ Let $C=\max\{C_i: 1\le i\le m \}$ where $C_i$ is given in . Let $\theta'=\max\{\theta_i: 1\le i\le m \}$. According to Lemma \[l-v+-\], there exists $T_{\frac{\delta}{{2\theta}}}>0$ such that (\[ine-v+-\]) holds for any $t\ge T_{\frac{\delta }{{2\theta}}}$, any nonzero $v\in V^{\lambda_i} \ (1\le i \le m)$ and any $z\in B_1^c$, i.e., $$\begin{aligned} \|\rho(f_t ^z )v\|^{- \theta }\le e^{ -(1-{\frac{\delta}{{2\theta}}}) \theta \lambda_i t} \|v\|^{- \theta } \le e^{-(\theta-\frac{\delta}{2} ) \lambda t} \|v\|^{- \theta }. \end{aligned}$$ This inequality and the assumption of the norm implies that for all nonzero $v\in V_i$ and $t\ge T_{\frac{\delta}{2\theta}}$ $$\label{ine-f-t} \|\rho(f_{ t} ^z h)v\|^{ -\theta }\le e^{-(\theta-\frac{\delta}{2})\lambda t }\|\pi_i(\rho(h)v)\|^{- \theta },$$ where $\frac{1}{0}$ is interpreted as $\infty$. Let $T_{\theta, \delta}\ge T_{\frac{\delta}{2\theta}}$ be a large enough real number so that $t\ge T_{\theta, \delta }$ implies $$\label{constant-t2} \frac{(2m)^{\theta'} CR^{\theta'-\theta}}{1-2^{\theta-\theta_0}} e^{-(\theta-\frac{\delta}{2})\lambda t }\le e^{-(\theta-\delta)\lambda t }.$$ Let $v$ be a unit vector of $V$. We write $v=v_1+\cdots+v_m$ where $v_i\in V_i$. Since we assume different $V_i$ are orthogonal to each other, there exists an integer $i\in [1, m]$ such that $m\|v_i\|\ge \|v\|=1$. There is a disjoint union decomposition of $B_1^+$ as $$E(v_i,0)\cup\left ( \cup_{n\ge 0} E^{+}(v_i,2^{-n}R_i)\right),$$ where $$E^{+}(v_i,2^{-n}R_i)= E(v_i,2^{-n}R_i){\smallsetminus}E(v_i,2^{-n-1}R_i).$$ Since $\mu (E(v_i, 0))=0$, for any $ z\in B_1^c$ and $t\ge T_{\theta, \delta}$ we have $$\begin{aligned} \int_{B_1^+}\|\rho(f_{ t} ^z h)v\|^{- \theta }d\mu(h) &\le \sum_{n=0}^{\infty}\int_{E^{+}(v_i,2^{-n}R_i)} \|\rho(f_{ t} ^z h)v_i\|^{-\theta }d\mu(h) \\ \text{(by \eqref{ine-f-t})}\quad &\le e^{-(\theta-\frac{\delta}{2}) \lambda t}\sum_{n=0}^{\infty}\int_{E^{+}(v_i,2^{-n}R_i)}\|\pi_i(\rho(h)v_i)\|^{- \theta }d\mu(h) \\ \text{(by \eqref{constant-c-alpha})} \quad & \le e^{-(\theta-\frac{\delta}{2})\lambda t} \sum_{n=0}^{\infty} C_i 2^{\theta}(2^{-n}R_i)^{\theta_i-\theta} \|v_i\|^{-\theta_i} \\ &\le \frac{m ^{\theta'}2^{\theta'} CR^{\theta'-\theta}}{1-2^{\theta-\theta_0}} e^{-(\theta-\frac{\delta}{2}) \lambda t} \\ \text{(by \eqref{constant-t2})} \quad &\le e^{-(\theta-\delta) \lambda t}. \end{aligned}$$ Eskin-Margulis height function {#sec;height} ============================== Let the notation be as in Theorem \[thm;general\]. In this section, we will establish a uniform contraction property of the EM height function on $G/\Gamma $ with respect to a family of one parameter groups $F_z \ (z\in B_1^c)$. Recall that $G/\Gamma=\prod_{i=1}^m G_i/\Gamma_i$ where each $G_i/\Gamma_i$ is a nonuniform irreducible quotient of a semisimple Lie group without compact factors. Since we assume the projection of $A_F$ to each $G_i$ is nontrivial, we have $ H=\prod_{i=1}^m H_i$, [where ]{} $H_i=G_i\cap H$ [ is a connected normal subgroup of ]{}$G_i $ with positive dimension. Let us recall the definition of the EM height function from [@em]. The EM height function is constructed on each $G_i/\Gamma_i$ using a finite set $\Delta_i$ of $\Gamma_i$-rational parabolic subgroups of $G_i$. Recall that a parabolic subgroup $P$ of $G_i$ is $\Gamma_i$-rational if the unipotent radical of $P$ intersects $\Gamma_i$ in a lattice. If the rank of $G_i$ is bigger than one, then Margulis’ arithmeticity theorem implies that there is a ${\mathbb{Q}}$-structure on $G_i$ such that $\Gamma_i$ is commensurable with $G_i({\mathbb{Z}})$. In this case the set $\Delta_i$ consists of standard ${\mathbb{Q}}$-rational maximal parabolic subgroups of $G_i$ with respect to a fixed ${\mathbb{Q}}$-split torus and fixed positive roots. So the irreducibility of $\Gamma_i$ implies that no conjugates of $H_i$ is contained in any $P\in \Delta_i$. The same conclusion holds in the case where $G_i$ has rank one. The reason is that in this case $H_i=G_i$ and $\Delta_i=\{P \}$ where $P$ is a maximal parabolic subgroup defined over ${\mathbb{R}}$. For each $P_{i, j}\in \Delta_i$, there exists a representation $\rho_{i,j}: G_i\to {\mathrm{GL}}(V_{i,j}) $ on a normed vector space and a nonzero vector $w_{i,j}\in V_{i,j}$ such that the stabilizer of ${\mathbb{R}}w_{i,j} $ is $P_{i,j}$. We consider $\rho_{i,j}$ as a representation of $G$ so that $\rho(G_s)$ is the identity linear map if $s\neq i$. Let $V_{i,j}^{H}$ be the $H$-invariant subspace of $V_{i,j}$ consisting of $H$-invariant vectors. Let $\pi_{i,j}$ be the projection of $V_{i,j}$ to the $H$-invariant subspace $V_{i,j}' $ complementary to $V_{i,j}^{H}$. Since no conjugates of $H_i$ is contained in $P_{i,j}$ and $G_i=K_{i} P_{i,j}$ for some maximal compact subgroup $K_i$ of $G_i$, there exists $C\ge 1 $ such that $$\|v\|\le C \|\pi_{i,j} (v)\|$$ for all $v\in \rho_{i,j}(G) w_{i,j}$. Note that $V_{i,j}'$ is $G$-invariant and it has no nonzero $H$-invariant vectors. Therefore, Lemma \[l-contraction\] implies the following lemma which corresponds to [**Condition A**]{} in [@em]. \[lem;crutial\] For each pair of index $i,j$ there exist positive constants $\theta_0^{i,j}$ and $\lambda^{i,j}$ such that for any $0<\delta<\theta<\theta_0^{i,j}$, any nonzero $v\in\rho_{i,j}( G) w_{i,j}$ and any $z\in B^c_1$ one has $$\begin{aligned} \label{eq;kao} \int_{B _1^+}\|\rho_{i, j} (f_t ^z h) v \|^{-\theta}{\; \mathrm{d}}h \le e^{-(\theta-\delta)t\lambda^{i,j} } \|v\|^{-\theta} \end{aligned}$$ provided $t\ge T_{\theta, \delta }^{i,j}$ where $T_{\theta, \delta}^{i,j}>0$ is a constant depending on $\theta$ and $\delta $. We assume without loss of generality that for all $V_{i,j} $ the norm $\|\cdot\|$ is Euclidean and $V_{i,j}^H$ and $V_{i,j}'$ are orthogonal to each other. According to Lemma \[l-contraction\], for each representation $\rho_{i,j}|_{V_{i,j}'}$, there exist positive constants $\theta^{i,j}_0$ and $ \lambda^{i,j}$ with the following properties: for any $0<\delta<\theta<\theta^{i,j}_0$ there exists $T_{ \theta, \delta}>0$ such that for any $t\ge T_{\theta, \delta}, z\in B_1^c$ and any nonzero $v\in \rho_{i,j}( G) w_{i,j}$, one has $$\begin{aligned} \int_{B _1^+}\|\rho_{i, j} (f_t ^z h) v \|^{-\theta}{\; \mathrm{d}}h& \le \int_{B _1^+}\|\rho_{i, j} (f_t ^z h) \pi_{i,j}(v ) \|^{-\theta}{\; \mathrm{d}}h \\ &\le e^{-(\theta-\delta)t\lambda^{i,j} } \|\pi_{i,j}(v)\|^{-\theta} \\ &\le C^{\theta} e^{-(\theta-\delta)t\lambda^{i,j} } \|v\|^{-\theta}.\end{aligned}$$ It is not hard to see from above estimate that (\[eq;kao\]) holds for sufficiently large $t$. Besides $\rho_{i,j}$, the EM height function is constructed using positive constants $c_{i,j}$ and $ q_{i,j}$ which are combinatorial data determined by the root system, see [@em (3.22),(3.28)]. Let $$\begin{aligned} \label{eq;simple} u _{i,j}(g\Gamma)= \max _{\gamma\in \Gamma } \frac{1}{\|\rho_{i,j} (g\gamma){w_{i,j}} \|^{1/c_{i,j} q_{i,j}}} $$ [where ]{} $g\in G$.[^5] Let $$\label{eq;alpha1} \begin{split} \theta_1=\max\{\theta>0: \frac{\theta}{q_{ij}c_{i,j}}\le \theta_0^{i,j} \quad \mbox{for all }i,j\} \quad \mbox{ and }\quad \alpha_1=\min_{ i,j } \{ \frac{\theta_1}{q_{ij}c_{i,j} }\lambda^{i,j}\}, \end{split}$$ where $\theta_0^{i,j}$ and $\lambda^{i,j}$ are constants given by Lemma \[lem;crutial\]. We call $\alpha_1$ a contraction rate for the dynamical system $(G/\Gamma, F^+)$. We will see in next sections that $\alpha_1$ plays an important role in bounding the Hausdorff dimension of ${\mathfrak {D}}(F^+,{G/\Gamma})$. We believe that optimal $\alpha_1$ is possible to give the sharp bound of the dimension. By Remark \[rem-3\], the constant $\theta_{i,j}$ can be explicitly calculated, so are the constants $\theta_1$ and $\alpha_1$. Consequently, it will be clear in the proof in the next sections that the upper bound of the dimension we obtain can also be explicitly calculated, although not optimal. \[lem;key\] For every $\alpha< \alpha_1$, there exist $0<\theta<\theta_1$ and $T>0$ such that for all $t\ge T$ and $\epsilon $ sufficiently small depending on $t$, the EM height function $$\begin{aligned} \label{eq;u} u: G/\Gamma \to (0, \infty)\quad \mbox{defined by } u(x)=\sum_{i,j}(\epsilon\, u_{i,j}(x))^{\theta} \end{aligned}$$ satisfies the following properties: 1. $u (x)\to \infty$ if and only if $x\to \infty $ in $G/\Gamma$. 2. For any compact subset $K$ of $G$, there exists $C\ge 1$ such that $u(hx)\le C u(x)$ for all $h\in K$ and $x\in G/\Gamma$. 3. There exists $b>0$ depending on $t$ such that for all $z\in B_1^c$ and $x\in G/\Gamma$ one has $$\begin{aligned} \label{eq;tech} \int_{B_1^+} u(f_t ^z hx) {\; \mathrm{d}}\mu(h)< e^{-\alpha t} u(x)+b. \end{aligned}$$ 4. There exists $\ell \ge 1$ such that if $u(x)\ge \ell $, then for all $z\in B_1^c$ $$\begin{aligned} \label{eq;contract} \int_{B_1^+} u(f_t ^z hx) {\; \mathrm{d}}\mu(h)< e^{-\alpha t} u(x). \end{aligned}$$ It follows from the corresponding results for each $G_i/\Gamma_i$ proved in [@em] that the first two conclusions hold for any choice of $\theta$ and $\epsilon$. Note that (4) is a direct corollary of (3). Now we prove (3). Let $n$ the cardinality of the indices $i,j$ appeared in the definition of $u$. We fix $\delta >0$ sufficiently small such that $$ \alpha+\delta +\frac{\delta \lambda^{i,j}}{c_{i,j} q_{i,j}}<\alpha_1\quad \forall \ i,j.$$ According to the definitions of $\theta_1$, $\alpha_1$ and the choice of $\delta $ above, there exists $\theta>0$ such that $$\theta<\theta_1\quad \mbox{and } \quad\frac{(\theta-\delta) \lambda^{i,j}}{c_{i,j} q_{i,j}}\ge \alpha +\delta\quad \forall\ i,j.$$ Let $\delta_{i,j}=\delta/ c_{i,j} q_{i,j}$, $\theta_{i,j} =\theta/ c_{i,j} q_{i,j}$, then according to Lemma \[lem;crutial\] there exists $T^{i,j}>0$ such that for $t\ge T^{i,j}$ one has (\[eq;kao\]) holds with $\delta=\delta_{i,j}$ and $\theta=\theta_{i,j}$. We will show that Lemma \[lem;key\] holds for $T=\frac{\log 2}{\delta }+\max_{i,j} T^{i,j}$. Now we fix $0<\epsilon<1$, $x=g\Gamma\in G/\Gamma, t\ge T$ and $i, j$. According to the definition of $u_{i,j}(x)$, there exists $\gamma\in \Gamma$ such that $u_{i, j}(x)=\frac{1}{\|\rho(g \gamma) w_{i,j} \|^{1/c_{i,j}q_{i,j}}}$. For any $h\in B_1^+$ and $z\in B_1^c$, if $u_{i,j}(f_t^zhx)=\frac{1}{\|\rho(f_thg \gamma) w_{i,j} \|^{1/c_{i,j}q_{i,j}}}$, then we can use (\[eq;kao\]). Otherwise, there exist $0<\kappa<1$, $b>0$ and $C'\ge 1$ where $b$ and $C'$ depend on $t$ such that $$(\epsilon u_{i,j}(f_t^zhx))^\theta\le C' \epsilon^\kappa (\epsilon u(x))^\theta+\frac{b}{n}.$$ These facts are proved in [@em §3.2]. In summary, we have $$\begin{aligned} \int_ {B_1^+} ( \epsilon u_{i, j}(f_t h x) )^\theta{\; \mathrm{d}}h & \le \epsilon^\theta \int_{B_1^+ } \frac{1}{\|\rho_{i,j}(f_t h g \gamma) w_{i,j} \|^{\theta /c_{i,j}q_{i,j}}} {\; \mathrm{d}}h + \epsilon ^\kappa C' u(x)+\frac{b}{n} \\ &\le e^{-(\theta-\delta) t \lambda^{i,j} /c_{i,j} q_{i,j} } ( \epsilon u_{i,j}(x))^\theta + \epsilon ^\kappa C' u(x)+\frac{b}{n}\\ &\le e^{-(\alpha +\delta)t}( \epsilon u_{i,j}(x))^\theta + \epsilon ^\kappa C' u(x)+\frac{b}{n}. \end{aligned}$$ Therefore, we have $$\int_ {B_1^+} u(f_t h x) {\; \mathrm{d}}h \le e^{-(\alpha +\delta)t} u(x) + n\epsilon ^\kappa C' u(x)+b.$$ We choose $\epsilon $ sufficiently small so that $n\epsilon ^\kappa C' \le e^{-(\alpha+\delta) t}$, then (\[eq;tech\]) holds. Applications of the uniform contraction property {#sec;first} ================================================ In this section we will introduce and study some auxiliary sets closely related to ${\mathfrak {D}}(F^+, {G/\Gamma})$ using the uniform contraction property of the EM height function established in Lemma \[lem;key\]. To be specific, we will prove some covering results for these auxiliary sets in Proposition \[l-main2\] and these covering results will play an important role in bounding the Hausdorff dimension of ${\mathfrak {D}}(F^+, {G/\Gamma})$. Let $\alpha_1$ be a contraction rate of the dynamical system $(G/\Gamma, F^+)$ given by (\[eq;alpha1\]) and let $\lambda$ be the top Lyapunov exponent of $A_F$ in the representation $({\mathrm{Ad}}, \mathfrak g)$. We fix $\alpha<\alpha_1,t >0$ and a EM height function $u: G/\Gamma \to (0, \infty)$ so that Lemma \[lem;key\] holds. Let $\ell \ge 1$ so that (\[eq;contract\]) holds for all $z\in B_1^c$ if $u(x)\ge \ell$. By Lemma \[lem;key\] (3), there exists $C\ge 1$ such that $$\begin{aligned} \label{eq;c} C^{-1} u(x)\le u(f_s hx)\le C u(x) \quad \mbox{for all } 0\le s\le t, h\in B_2^G \mbox{ and } x\in G/\Gamma. \end{aligned}$$ We also fix an auxiliary $\delta>0$ (which will go to zero finally) and assume that $t$ is sufficiently large so that according to Lemma \[l-v+-\] [ for all ]{} $r\le 1, z\in B_1^c$ $$\begin{aligned} B_{e^{-(\lambda+\delta) t} r}^+ \subset f_{-t} ^z B_r^+f_{t} ^z &\subset B_{r/4}^+; \label{l-t4}\\ B_{e^{-\delta t} r}^c\subset f_{-t} ^z B_r^cf_{t} ^z &\subset B_{e^{\delta t} r}^c;\label{eq;B0}\\ \label{eq;details} 2&< e^{\delta (\alpha+1) t/2}. \end{aligned}$$ Note that the logarithm map from the metric space $(B_1^+, {\mathrm{dist}})$ to the Lie algebra $\mathfrak g^+$ (with the fixed Euclidean structure) is a bi-Lipschitz homeomorphism to its image. Therefore $(B_1^+, {\mathrm{dist}})$ is Besicovitch, see [@mat], namely, for any subset $D$ of $B_1^+$ and a covering of $D$ by balls centered at $D$, there is a finite sub-covering such that each element of $D$ is covered by at most $E'$ times. Therefore, there exists $E\ge E' $ such that for all $0<r\le 1$, the set $B^+_{1/2}$ can be covered by no more than $E r^{-d}$ open balls of radius $r$, where $d=\dim G^+$. We use $|I|$ to denote the cardinality of a finite set $I$. The following is the main result of this section. \[l-main2\] Let $x\in G/\Gamma$. There exists $0<\sigma<1$ and $E_0\ge 1$ such that for $ z\in B_1^c$ and $N\in {\mathbb{N}}$, the set $$\label{def-d+} {\mathfrak {D}}_x(z, N, \sigma, C^2\ell ){{\, \stackrel{\mathrm{def}}{=}\, }}\{h\in B_{1/2}^+ : |\{1\le n\le N : u (f^z_{ nt} hx)\ge C^2\ell \}|\ge \sigma N\}$$ can be covered by no more than $E_0e^{(d\lambda-{\alpha}+\delta( d+\alpha)) tN}$ open balls of radius $e^{-(\lambda+\delta) t N}$ in $B_1^+$. The rest of this section is devoted to show that Proposition \[l-main2\] holds for $$\begin{aligned} \label{eq;sigma} \sigma& =\frac{(1-\delta/2)\alpha t +\log C}{\alpha t +\log C} $$ In the rest of this section we fix $ z\in B_1^c$ and $N\in {\mathbb{N}}$. We begin with the following simple observation. \[lem;add\] If $B\subset G^+$ is a ball of radius $e^{-(\lambda+\delta) t N}$ centered at ${\mathfrak {D}}_x({ z }, N, \sigma , C^2\ell )$, then $B \subset {\mathfrak {D}}_x({ z }, N, \sigma, C\ell )$. Let $h_0$ be the center of $B$ and $h\in B$. It suffices to show that for all $1\le n\le N$ if $u (f_{nt}^z h_0x)\ge C^2\ell $ then $u (f_{nt}^z h x)\ge C\ell $. By (\[l-t4\]) we have $${\mathrm{dist}}(f_{nt}^z h_0, f_{nt}^zh )={\mathrm{dist}}(1_G, f_{nt}^zhh_0^{-1} f_{-nt}^z)< 1.$$ By (\[eq;c\]) $$u (f_{nt}h x)=u (f_{nt}hh_0^{-1} f_{-nt}\cdot f_{nt}h_0 x)\ge C ^{-1}u (f_{nt}h_0 x)\ge C^{-1}\cdot C^2\ell = C\ell.$$ For a subset $I\subset \{ 1, \ldots, N \}$, we let $$\label{def-d+i} {\mathfrak {D}}_x( z , I, C\ell )=\{h \in B_{1/2}^+ : u (f_{ nt} ^z hx)\ge C \ell \mbox { for all } n\in I \}.$$ Since $ {\mathfrak {D}}_x({ z }, N, \sigma, C\ell )=\bigcup_{|I| \ge \sigma N} {\mathfrak {D}}_x({ z }, I, C\ell ) $, one has $$\begin{aligned} \label{eq;sum} \mu({\mathfrak {D}}_x({ z }, N, \sigma, C\ell ))\le \sum_{|I| \ge \sigma N} \mu ({\mathfrak {D}}_x({ z }, I, C\ell )).\end{aligned}$$ The following lemma will play an important role in the proof of Proposition \[l-main2\]. \[l-int-ine\] Suppose that $I\subset \{1, \ldots, N \}$ and $|I| \ge \sigma N$. Then $$\begin{aligned} \label{eq;con} \mu ({\mathfrak {D}}_x(z, I, C\ell ) )\le {C^2} u (x) e^{-(1-\delta/2)\alpha tN }. \end{aligned}$$ We fix $I$ as in the statement of Lemma \[l-int-ine\]. Our strategy is to estimate the measure of ${\mathfrak {D}}_x({ z }, I, C\ell )$ by relating it to a subset coming from random walks on $G/\Gamma$ with alphabet $f_t ^z B_1^+$. Let $p =\sup I$ and for $1\le k\le p$ let $$Z_k=\{(h_1,\ldots,h_{k })\in (B_1^+)^{k }: u (f ^z _{ t} h_n\ldots f ^z _{t} h_1x) \ge \ell \ \forall\ n\in (I\cap[1,k])\}.$$ Define $\eta :(B_1^+)^p \rightarrow G^+$ by $$\begin{aligned} \label{eq;h} \eta(h_1,\ldots, h_{p })= \tilde{h}_{p } \cdots \tilde{h}_{1}, \text{ where } \tilde{h}_{n}= f ^z _{-(n-1)t}h_n f ^z _{(n-1)t}.\end{aligned}$$ We remark here that the image of $\eta$ is contained in $B_2^+$ by (\[l-t4\]). The following two lemmas are needed in the proof of Lemma \[l-int-ine\]. \[lem;contain\] For all $h\in {\mathfrak {D}}_x({ z }, I, C\ell )$ one has $\eta^{-1}(h)\subset Z_p$. Suppose that $\eta(h_1, \ldots, h_p)=h$ where $h_i\in B_1^+$. Then for all $n\le p$ $${\mathrm{dist}}(f_{nt}^zh, f_{t}^zh_n\cdots f_t^z h_1)={\mathrm{dist}}(f_{nt}^z\tilde h_{p }\cdots \tilde h_{n+1}f_{-nt}^z, 1_G)< 2,$$ where we use (\[l-t4\]), (\[eq;h\]) and the right invariance of ${\mathrm{dist}}(\cdot, \cdot)$. Therefore by (\[eq;c\]) we have for $n\in I$ $$\begin{aligned} u (f_{t}^zh_n\cdots f_t^z h_1x)\ge C^{-1}u (f_{nt}^zhx)\ge \ell .\end{aligned}$$ So $(h_1, \ldots, h_p)\in Z_p$ and the proof is complete. Let $\widetilde \mu_{ n}$ be the Radon measure on $G^+$ defined by $$\begin{aligned} \label{eq;def} \int_{G^+}\varphi(h) {\; \mathrm{d}}\widetilde \mu_{ n}(h)=\int_{B_1^+} \varphi(f_{-nt} ^z hf_{nt} ^z ) ){\; \mathrm{d}}h\end{aligned}$$ for all $\varphi\in C_c(G^+)$. For any positive integer $n$ let $\mu_{n}=\widetilde \mu_{ n-1}*\cdots *\widetilde \mu_{1}*\widetilde \mu_{0}$ be the measure on $G^+$ defined by the $n$ convolutions. Clearly, $\mu_n$ is absolutely continuous with respect to $\mu$ and $\mu_p$ is the pushforward of $(\mu|_{B_1^+})^{\otimes p}$ by the map $\eta$. The following lemma shows that $\mu_n$ has density bigger than or equal to one at every $h\in B_{1/2}^+$. \[l-rd-conv\] For all $n\le N $ and $ h\in B_{{1}/{2}}^+$ we have $ \frac{d\mu_{n}}{d\mu}(h)\ge 1 . $ The conclusion is clear if $n=1$. Now we assume $n>1$ and let $$\nu= \widetilde \mu_{ n-1}* \widetilde \mu_{n-2}* \cdots * \widetilde \mu_{ 1}.$$ It follows from (\[l-t4\]) and (\[eq;def\]) that for $k>0$ the probability measure $\widetilde \mu_{ k}$ is supported on $B_{1/4^k}^+$. Since the metric on $G^+$ is right invariant, the measure $\nu$ is supported on $B_{1/2}^+$. Suppose $\nu=\varphi {\; \mathrm{d}}\mu$, then $\mu_{ n}=\nu * \widetilde \mu_0 =\varphi* \mathbbm{1}_{B_1^+} {\; \mathrm{d}}\mu$. So for any $h\in B_{1/2}^+$, we have $$\varphi* \mathbbm{1}_{B_1^+}(h)=\int_{G^+}\varphi(h_1)\mathbbm{1}_{B_1^+}(h_1^{-1}h) d\mu(h_1)\ge \int_{B_{1/2}^+}\varphi(h_1)d\mu(h_1) =1.$$ Now we are ready to prove Lemma \[l-int-ine\]. By Lemmas \[lem;contain\] and \[l-rd-conv\], $$\begin{aligned} \label{eq;temp} \mu ({\mathfrak {D}}_x(z, I, C\ell ) )\le \mu_\ell({\mathfrak {D}}_x(z, I, C\ell ) )\le \mu_p(Z_p). \end{aligned}$$ Now we are left to estimate $\mu_p(Z_p)$. For $1\le k \le p$ let $$\notag s(k )=\int _{Z_k } u (f_{ t} ^z h_{ k }\cdots f_{ t} ^z h_1x)d\mu^{\otimes k }(h_1, \cdots, h_{ k}).$$ Let $$\begin{aligned} \label{ine-iterated int} s(p+1)=\int_{Z_{p}}\left[ \int_{B_1^+} u (f_{ t} ^z h_{p+1 }f_t^z h_{p }\cdots f_{ t} ^z h_1x) {\; \mathrm{d}}\mu(h_{p+1} )\right ]d\mu^{\otimes p }(h_1, \cdots, h_{p}). \end{aligned}$$ Then for every $1< k\le p+1$, $$\begin{aligned} \notag s(k )&\le \int_{Z_{k-1}}\left[ \int_{B_1^+} u (f_{ t} ^z h_{k }f_t^z h_{k-1}\cdots f_{ t} ^z h_1x) {\; \mathrm{d}}\mu(h_k )\right ]d\mu^{\otimes (k -1)}(h_1, \cdots, h_{k -1}).\end{aligned}$$ If $k -1\in I$, then $s(k )\le e^{-\alpha t}s(k -1)$ by (\[eq;contract\]). If $k -1\not \in I$, then by (\[eq;c\]) we have $s(k )\le C s(k -1)$. We apply this estimate to $k=p+1, p, \cdots, 2$, then we have $$\begin{aligned} s(p+1)\le C^{(N-|I|)}e^{-|I| \alpha t} \int_{B_1^+} u (f_t hx{\; \mathrm{d}}\mu (h))\le C^{1+(1-\sigma) N }e^{-\sigma\alpha t N }u (x) . \end{aligned}$$ The choice of $\sigma$ in (\[eq;sigma\]) implies that $$\begin{aligned} \label{eq;temp2} s(p+1)\le C e^{-(1-\delta/2) \alpha t N}u (x). \end{aligned}$$ On the other hand, in view of (\[ine-iterated int\]), (\[eq;c\]) and the fact $p=\sup I $ we have $$\begin{aligned} \label{eq;che} s(p +1)\ge C^{-1} s(p)\ge C^{-1}\ell \cdot \mu_p(Z_p). \end{aligned}$$ Therefore, (\[eq;con\]) follows from (\[eq;temp\]), (\[eq;temp2\]) and (\[eq;che\]) and the observation $\ell \ge 1$. As before we fix $ z$ and $ N$ as in the statement. Let $\sigma$ be as in (\[eq;sigma\]). Since $(B_1^+, {\mathrm{dist}})$ is Besicovitch, there exists a covering ${\mathfrak{U}}$ of ${\mathfrak {D}}_x({ z }, N, \sigma, C^2\ell )$ by open balls of radius $e^{-{(\lambda+\delta) t N}}$ centered at ${\mathfrak {D}}_x({ z }, N, \sigma, C^2\ell )$ such that each element of ${\mathfrak {D}}_x({ z }, N, \sigma, C^2 \ell )$ is covered by at most $E$ times. By Lemma \[lem;add\], each $B\in {\mathfrak{U}}$ is contained in ${\mathfrak {D}}_x({ z }, N, \sigma, C\ell )$, so in view of (\[eq;c+\]) $$\begin{aligned} \label{eq;tmd} \mu( {\mathfrak {D}}_x({ z }, N, \sigma, C\ell ))\ge\frac{ |{\mathfrak{U}}|}{E} \mu(B^+_{e^{-(\lambda+\delta) t N}})\ge \frac{ |{\mathfrak{U}}|}{C_0E} e^{-(\lambda+\delta) d t N} . \end{aligned}$$ On the other hand, since there are $2^N$ subsets $I\subset \{1, \ldots, N \}$, by (\[eq;sum\]), (\[eq;details\]) and Lemma \[l-int-ine\], we have $$\label{mu-dx+1} \mu\big({\mathfrak {D}}_x({ z }, N, \sigma, C\ell )\big) \le C^2 2^Ne^{-(1-\delta/2) \alpha tN}u (x)\le e^{-(1-\delta)\alpha tN}u (x).$$ By (\[eq;tmd\]) and (\[mu-dx+1\]), $$|\mathfrak U| \le u(x) C_0C^2 E\cdot e^{ ( d\lambda -\alpha+\delta (d+\alpha)) t N}.$$ The conclusion now follows by taking $ E_0=u(x)C_0 C^2 E. $ Upper bound of Hausdorff dimension ================================== In this section, we finish the proof of Theorem \[thm;general\]. We will use the same notation as in §\[sec;first\] prior to Proposition \[l-main2\]. For $(z,h)\in B_1^c B_1^+ , \ell '>0 $ and $N\in {\mathbb{N}}$, let $I_{N, \ell'}(z,h)$ denote the set of $n\in \{1,\ldots,N\}$ satisfying $ u(f_{nt}zhx)\ge \ell ' $. For $x\in {G/\Gamma}$, let $$\label{def-d+0} {\mathfrak {D}}^{0}_x(F^+, N, \sigma, \ell')=\{(z,h)\in B_{1/2}^c B_{1/2}^+ : |I_{N, \ell '}(z,h)|\ge \sigma N\}.$$ \[l-main\] Let $x\in {G/\Gamma}$. Then there exist $0<\sigma<1$ and $E_2\ge 1$ such that for any $N\in {\mathbb{N}}$ the set ${\mathfrak {D}}^{0}_x(F^+, N, \sigma, C^4 \ell )$ can be covered by no more than $E_2e^{(d^c \lambda+d \lambda -\alpha+\delta (d^c+d+\alpha)) tN}$ open balls of radius $e^{-(\lambda+\delta) t N}$ in $G^cG^+$. Let $0<\sigma<1$ and $E_0\ge 1 $ so that Proposition \[l-main2\] holds. We fix $N\in {\mathbb{N}}$. We claim that: for $W=B_{e^{-(\lambda+\delta) tN}}^c\cdot z\subset B_{1}^c$, we have $$\label{claim-w} \Big( {\mathfrak {D}}_x^{0}(F^+,N,\sigma, C^4\ell )\cap (W B_{1}^+)\Big)\subset \Big(W {\mathfrak {D}}_x(z , N, \sigma, C^2\ell )\Big).$$ Let $(z_1,h_1)\in W B_{1}^+$. Suppose that $1\le n\le N$ and $ u(f_{nt}z_1h_1x)\ge C^4\ell $. In view of (\[eq;B0\]) and (\[eq;c\]) we have $$\begin{split} u(f_{ nt}^zh_1x)=u(z^{-1}f_{nt}zh_1x)\ge C^{-1} u(f_{nt}zh_1x) \\ =C^{-1}u(f_{nt}(zz_1^{-1})f_{-nt}\cdot f_{nt}z_1h_1x)\ge C^{-2}u(f_{nt}z_1h_1x)\ge C^2\ell . \end{split}$$ In other words, we have proved that if $n \in I_{N, C^4\ell }(z_1,h_1)$, then $u(f_{nt}^z h_1 x)\ge C^2\ell $. Therefore, if $(z_1, h_1)$ belongs to the left hand side of then it also belongs to the right hand side. Since $(B_1^c, \mathrm{dist})$ is also Besicovitch, there exists $E_1\ge 1$ such that for all $0<r\le 1$, $B_{1/2}^c$ can be covered by no more than $E_1 r^{-d^c }$ open balls of radius $r$. We fix a cover ${\mathfrak{U}}^c$ of $B_{1/2}^c$ that consists of open balls of radius $e^{-(\lambda+\delta ) Nt}$ with $|{\mathfrak{U}}^c|\le E_1 e^{d^c(\lambda+\delta) Nt}$. We assume each element of ${\mathfrak{U}}^c$ has nonempty intersection with $B_{1/2}^c$, then it is contained in $B_1^c$ in view of (\[eq;details\]). Let $W_z\in {\mathfrak{U}}^c$ be a ball centered at $z\in B_1^c$. Proposition \[l-main2\] implies that there exists a covering ${\mathfrak{U}}_{ z}$ of ${\mathfrak {D}}_x(z , N, \sigma, C^2\ell ) $ by open balls of radius $e^{-(\lambda+\delta)t N}$ such that $$\begin{aligned} |{\mathfrak{U}}_{ z}|\le E_0 e^{d\lambda-\alpha+\delta(d+\alpha)}.\end{aligned}$$ In view of claim , the following class of sets $$\begin{aligned} \label{eq;W} \{W_z B: W_z\in {\mathfrak{U}}^c, B\in {\mathfrak{U}}_{ z}\}\end{aligned}$$ forms an open cover of ${\mathfrak {D}}^{0}_x(F^+, N, \sigma, C^4 \ell)$. It is easily checked that there exists $E_1'\ge 1$ not depending on $N$ such that each element $W_zB$ of (\[eq;W\]) can be covered by $E_1'$ open balls of radius $e^{-(\lambda+\delta)Nt }$ in $G^cG^+$. Therefore the lemma holds with $E_2=E_0E_1E_1'$. \[thm;simple\] For any $x\in G/\Gamma$, the Hausdorff dimension of $ {\mathfrak {D}}_x^{0}{{\, \stackrel{\mathrm{def}}{=}\, }}\{(z,h)\in B_{1/2}^c B_{1/2}^+: zh x \in {\mathfrak {D}}(F^+, G/\Gamma) \}$ is at most $d^c+d -\frac{\alpha_1}{\lambda }$. For each $\alpha<\alpha_1$ and $0<\delta<1$ we first choose $t>0$, a height function $u$ and $\ell, C \ge 1$ so that Lemma \[lem;key\], (\[eq;c\]), (\[l-t4\]), (\[eq;B0\]) and (\[eq;details\]) hold. Then there exists $0<\sigma<1$ and $E_2\ge 1 $ so that Lemma \[l-main\] holds. It follows from Lemma \[lem;key\] (1)(2) and the definition of ${\mathfrak {D}}(F^+, G/\Gamma)$ that $${\mathfrak {D}}_x^{0}\subset \bigcup_{M\ge 1} W_M\quad \mbox{where}\quad W_{M}= \bigcap _{N\ge M} {\mathfrak {D}}^{0}_x(F^+, N, \sigma, C^4 \ell ).$$ Recall that for any metric space $S$, $$\dim_H S=\inf\left\{s>0: \inf_{\{B_i\}} \sum \rho(B_i)^s=0\right\},$$ where the latter $``\inf"$ is taken over all the countable coverings $\{B_i\}$ of $S$ that consist of open metric balls. Then in view of Lemma \[l-main\], we have $$\begin{aligned} \dim_HW_M&\le \liminf_{N\to \infty} \frac{[d^c\lambda+d\lambda -\alpha+\delta(d+d^c+\alpha)]t N+ \log E_2 }{\lambda t N}\\ & = d^c+d -\frac{\alpha}{\lambda}+\delta \frac{d+d^c+\alpha}{\lambda}.\end{aligned}$$ Therefore $$\dim_H {\mathfrak {D}}_x^{0}\le d^c+d -\frac{\alpha}{\lambda}+\delta \frac{d+d^c+\alpha}{\lambda}.$$ The conclusion follows by first letting $\delta\to 0$ and then letting $\alpha\to \alpha_1$. \[lem;stable\] If $x\in \mathfrak D(F^+, G/\Gamma)$ and $h\in G^-$, then $hx \in \mathfrak D(F^+, G/\Gamma) $. Note that by Lemma \[l-v+-\], $${\mathrm{dist}}(f_t hx, f_t x)\le {\mathrm{dist}}(f_t h f_{-t} , 1_G)\to 0$$ as $t\to \infty$. Therefore the lemma holds. We will show that $$\dim_H {\mathfrak {D}}(F^+, G/\Gamma)\le d^-+d^c+d -\frac{\alpha_1}{\lambda}.$$ In view of the local nature of Hausdorff dimension and the definition of the metric on $G/\Gamma$, it suffices to prove that for any $x\in G/\Gamma$ $$\dim_H \{g\in B_{r}^G: gx \in {\mathfrak {D}}(F^+, G/\Gamma) \}\le d^-+d^c+d -\frac{\alpha_1}{\lambda}.$$ where $r<1$ so that $B_{r}^G\subset B^-_1 B_{1/2}^cB_{1/2}^+$. By Lemma \[lem;stable\], $$\{g\in B_{r}^G: gx \in {\mathfrak {D}}(F^+, G/\Gamma) \}\subset B^-_1{\mathfrak {D}}_x^0,$$ whose Hausdorff dimension is bounded from above by $\dim_H{\mathfrak {D}}_x^0+d^-$.[^6] In view of Theorem \[thm;simple\], the Hausdorff dimension of ${\mathfrak {D}}(F^+, G/\Gamma)$ is less than $d+d^c+d^--\frac{\alpha_1}{\lambda}$ which is strictly less than the manifold dimension of $G/\Gamma$. It is worth to mention that, if $F=A_F$, then the contraction property of the Benoist-Quint height function proved in [@s] will allow us to prove a stronger result. Namely, we can get a nontrivial upper bound of the Hausdorff dimension of the intersection of ${\mathfrak {D}}(F^+, {G/\Gamma})$ and orbits of the so-called $(F^+, \Gamma)$-expanding subgroups introduced in [@KW]. But unfortunately, we are not able to prove a uniform contracting property for the Benoist-Quint height functions even in the example mentioned at the end of the introduction due to the existence of the unipotent part in the second ${\mathrm{SL}}_4({\mathbb{R}})$ factor. [99]{} Y. Benoist and J.-F. Quint, [*Random walks on finite volume homogeneous spaces*]{}, Invent. math. [**187**]{} (2012), 37–59. V. Bernik, D. Kleinbock and G.A. Margulis, [*Khintchine type theorems on manifolds: The convergence case for standard multiplicative versions*]{}, Internat. Math. Res. Notices [**9**]{} (2001), 453–486. A. Borel, [*Linear algebraic groups. Second edition*]{}, Graduate Texts in Mathematics, 126. Springer-Verlag, New York, 1991. C. Bishop and Y. Peres, [*Fractals in probability and analysis*]{}, Cambridge Studies in Advanced Mathematics 162, Cambridge University Press, Cambridge, 2017. Y. Cheung, [*Hausdorff dimension of the set of points on divergent trajectories of a homogeneous flow on a product space*]{}, Ergodic Theory Dynam. Systems [**27**]{} (2007), 65–85. Y. Cheung, [*Hausdorff dimension of the set of singular pairs*]{}, Ann. of Math. [**173**]{} (2011), 127–167. Y. Cheung and N. Chevallier, [*Hausdorff dimension of singular vectors*]{}, Duke Math. J. [**165**]{} (2016), no. 12, 2273–2329. L. Corwin and F. Greenleaf, [*Representations of nilpotent Lie groups and their applica- tions, Part I: Basic theory and examples*]{}, Cambridge Studies in Advanced Mathematics 18, Cambridge University Press, Cambridge, 1990. S.G. Dani, [*On orbits of unipotent flows on homogeneous spaces*]{}, Ergodic Theory Dynam. Systems [**4**]{} (1984), no. 1, 25–34. S.G. Dani, [*On orbits of unipotent flows on homogeneous spaces II*]{}, Ergodic Theory Dynam. Systems [**6**]{} (1986), no. 2, 167–182. S.G. Dani, [*Divergent trajectories of flows on homogeneous spaces and Diophantine approximation*]{}, J. Reine Angew. Math. [**359**]{} (1985), 55–89. T. Das, L. Fishman, D. Simmons and M. Urbanski, [*A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation*]{}, C. R. Math. Acad. Sci. Paris [**355**]{} (2017), no. 8, 835–846. M. Einsiedler and S. Kadyrov,[*Entropy and escape of mass for ${\mathrm{SL}}_3({\mathbb{R}})/{\mathrm{SL}}_3({\mathbb{Z}})$*]{}, Israel Journal of Mathematics [**190**]{} (2012), 253–288. M. Einsiedler, E. Lindenstrauss, P. Michel and A. Venkatesh, [*The distribution of closed geodesics on the modular surface, and Duke’s theorem*]{}, Enseign. Math. (2) [**58**]{} (2012), no. 3, 249–313. A. Eskin, G.A. Margulis, , [*Recurrence properties of random walks on finite volume homogeneous manifolds*]{}, In: Random Walks and Geometry, pp. 431–444. de Gruiter, Berlin (2004). A. Eskin, G.A. Margulis and S. Mozes, [*Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture*]{}, Ann. of Math. [**147**]{} (1998), 93–141. S. Kadyrov, [*Entropy and escape of mass for Hilbert modular spaces*]{}, J. Lie Theory [**22**]{} (2012), no. 3, 701–722. S. Kadyrov, D. Kleinbock, E. Lindenstrauss and G.A. Margulis, [*Singular systems of linear forms and non-escape of mass in the space of lattices*]{}, J. Anal. Math. [**133**]{} (2017), 253–277. S. Kadyrov and A. Pohl, [*Amount of failure of upper-semicontinuity of entropy in noncompact rank one situations, and Hausdorff dimension*]{}, Ergodic Theory Dynam. Systems [**37**]{} (2017), no. 2, 539–563. D. Kleinbock and B. Weiss, [*Modified Schmidt games and a conjectue of Margulis*]{}, J. Mordern Dynamics [**7**]{} (2013), no. 3, 429–460. L. Liao, R. Shi, O.N. Solan and N. Tamam, [*Hausdorff dimension of weighted singular vectors*]{}, http://arxiv.org/abs/1605.01287. G.A. Margulis, [*On the action of unipotent groups in the space of lattices, In Lie Groups and their representations*]{}, Proc. of Summer School in Group Representations, Bolyai Janos Math. Soc., Akademai Kiado, Budapest, 1971, pp. 365–370, Halsted, New York, 1975. G.A. Margulis and G.M. Tomanov, [*Invariant measures for actions of unipotent groups over local fields on homogeneous spaces*]{}, Invent. Math. [**116**]{} (1994), no. 1, 347–392. P. Mattila, [*Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability*]{}, Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. R. Shi, [*Pointwise equidistribution for one parameter diagonalizable group action on homogeneous space*]{}, http://arxiv.org/abs/1405.2067. L. Yang, [*Hausdorff dimension of divergent diagonal geodesics on product of finite–volume hyperbolic spaces*]{}, Ergodic Theory and Dynamical Systems, to appear. B. Weiss, [*Divergent trajectories on noncompact parameter spaces*]{}, Geom. Funct. Anal., [**14**]{} (2004), no. 1, 94–149. [^1]: L. G. is supported by EPSRC Programme Grant EP/J018260/1 [^2]: [^3]: A discrete subgroup $\Gamma<G$ is called a lattice if there exists a finite $G$-invariant measure on the homogeneous space ${G/\Gamma}$. [^4]: Here ${\mathbb{N}}=\{1, 2, 3, \ldots \}$. [^5]: Although only the product $c_{i,j}q_{i,j}$ is used in this paper, the constants $c_{i,j}$ and $q_{i,j}$ are given by different combinatorial data and we use both of them for the consistency with [@em]. [^6]: Here we are using Marstrand’s product theorem again.

--- abstract: 'Weyl semimetals possess unique electrodynamic properties due to a combination of strongly anisotropic and gyrotropic bulk conductivity, surface conductivity, and surface dipole layer. We explore the potential of popular tip-enhanced optical spectroscopy techniques for studies of bulk and surface topological electron states in these materials. Anomalous dispersion, extreme anisotropy, and the optical Hall effect for surface polaritons launched by a nanotip provides information about Weyl node position and separation in the Brillouin zone, the value of the Fermi momentum, and the matrix elements of the optical transitions involving both bulk and surface electron states.' author: - Qianfan Chen - Maria Erukhimova - Mikhail Tokman - Alexey Belyanin date: title: Extreme anisotropy and gyrotropy of surface polaritons in Weyl semimetals --- A number of recent studies have suggested that Weyl semimetals (WSMs) should have highly unusual optical response originated from unique topological properties of their bulk and surface electron states; see e.g. [@kargarian2015; @hofmann2016; @tabert2016-2; @ukhtary2017; @felser2017; @kotov2018; @andolina2018; @zyuzin2018; @rostami2018; @chen2019; @narang2019; @ma2019; @moore2019] and references therein. Their optical response can be used to provide detailed spectroscopic information about their electronic structure which could be difficult to obtain by any other means. Furthermore, inversion or time reversal symmetry breaking inherent to WSMs makes their optical response strongly anisotropic and/or gyrotropic, enables strong optical nonlinearity, creates anomalous dispersion of normal electromagnetic modes, breaks Lorentz reciprocity, and leads to many other optical phenomena of potential use in new generations of the optoelectronic devices. In a recent paper [@chen2019], we investigated general optical properties of Type I WSMs. Starting from a class of microscopic Hamiltonians for WSMs with two separated Weyl nodes ([@burkov2011; @okugawa2014]), we obtained both bulk and surface electron states, derived bulk and surface conductivity tensors, and described the properties of electromagnetic eigenmodes. Here we focus on one of the most popular and convenient ways to study the properties of novel materials by optical means: a tip-based optical spectroscopy, in which a tip brought in close proximity to the material surface is illuminated with laser light and the linear or nonlinear scattered signal is collected. Strong near-field enhancement at the tip apex may overcompensate the decrease in the volume of the material where light-matter interaction occurs [@raschke2016; @raschke2019]. This technique can provide information about surface states and carrier dynamics with about 10 nm spatial and 1 fs time resolution [@raschke2019]. Even more importantly in the context of this paper, nanoscale concentration of the incident light at the tip apex relaxes the optical selection and momentum matching rules. In particular, it allows one to launch various kinds of surface polariton modes which provide valuable information about the properties of both bulk and surface electron states. We use the microscopic model of the optical response of Type I WSMs developed in [@chen2019] to predict and describe theoretically the properties of surface polaritons (SPs) launched by a nanotip. We show extreme anisotropy and gyrotropy in SP radiation pattern originated from Weyl node separation and determined mainly by highly anisotropic surface current and surface dipole layer. We demonstrate anomalous dispersion and extreme sensitivity of SP anisotropy to the frequency of light and Fermi momentum, which makes them a sensitive diagnostics of Fermi arc surface states and may form the basis of efficient light modulators and switches. ![A sketch of tip-enabled SP excitation on the WSM surface. Radiation pattern of SPs is indicated in green for a particular combination of the excitation frequency and Fermi momentum, and for Weyl nodes located along the $k_x$ axis in the Brillouin zone. []{data-label="fig1"}](fig1.pdf) Figure 1 shows one possible schematic of SP excitation with a gold nanotip. Here the tip apex of $\sim 10$ nm radius is brought to a distance of $\sim 10$ nm from the WSM surface $z = 0$ in order to get access to large SP wavevectors $\sim 10^6$ cm$^{-1}$; see the SP dispersion curves in Fig. 2. A laser beam either illuminates the apex directly (e.g. [@basov]) or excites SPs on the surface of a gold tip via grating, as indicated in the figure [@raschke2016; @raschke2019]. In the latter case, gold surface plasmon-polaritons propagate to the apex, experiencing strong adiabatic amplification of the field intensity as they reach the apex [@stockman; @raschke2016]. Either way, excitation of SPs on a WSM surface is concentrated under the tip within a spot of $\sim 10$ nm. In the linear excitation regime, the frequency spectrum of SPs coincides with the spectrum of an incident laser pulse, whereas the spatial spectrum is extremely broadband, with a cutoff around $10^7$ cm$^{-1}$. The SPs propagate away from the tip, forming a strongly anisotropic radiation pattern which depends on the Weyl node position and separation and the Fermi momentum. They can be detected (converted into an outgoing EM wave) with another tip, a grating, a notch, etc. For the most sensitive diagnostics of the electronic structure of WSMs, the frequency of the probing light should be of the order of $\omega \sim v_F b$, where $2b$ is the distance between Weyl nodes in momentum space along $k_x$; see the electron bandstructure plot in Fig. 1 of [@chen2019]. In all numerical examples in the paper we assume for definiteness that $ \hbar v_F b = 100$ meV, so the incident laser light should be in the mid-infrared range. However, the formalism presented in the paper is general and does not depend on the choice of incident frequencies as long as the latter are low enough, so that the interband transitions to electron states in remote bands can be neglected. The remote states have a trivial topology and they are not of interest to this study. The Hamiltonian of a WSM with two separated Weyl nodes breaks time-reversal symmetry, which is expected for WSMs with magnetic ordering, e.g. pyrochlore iridates [@wan2011], ferromagnetic spinels [@xu2011], and Heusler compounds [@liu2018]. As we showed in [@chen2019], the tensors of both bulk and surface conductivity for Type I WSMs with time-reversal symmetry breaking have a structure corresponding to a biaxial-anisotropic and gyrotropic medium: $$\label{BulkConductivity} \sigma _{mn}^{B,S}(\omega) = \begin{pmatrix} \sigma _{xx}^{B,S} & 0 & 0\\ 0 & \sigma _{yy}^{B,S} & \sigma _{yz}^{B,S}\\ 0 & \sigma _{zy}^{B,S} & \sigma _{zz}^{B,S}\\ \end{pmatrix}$$ where the Weyl points are on the $k_x$ axis, $ \sigma _{zy}^{B,S}=- \sigma _{yz}^{B,S}$, and superscripts $B$ and $S$ denote bulk and surface conductivity elements, respectively. We add background dielectric constant $\varepsilon_b$ due to transitions to remote bulk bands, assuming it to be isotropic and dispersionless at low frequencies, so that the total bulk dielectric tensor is $\varepsilon_{mn}(\omega) = \varepsilon_b \delta_{mn} + 4 \pi i \sigma _{mn}^{B}/\omega$. The surface conductivity is due to optical transitions between different electron surface states (often called “Fermi arc states”, although they exist for all momentum states within $k_x^2 + k_y^2 \leq b^2$, not only on the Fermi arc) and between surface and bulk states. It gives rise to the surface current and surface dipole layer. Note a peculiar and most likely unique electrodynamics of WSM surface modes: they are supported by a highly anisotropic and gyrotropic surface current and surface dipole layer sitting on top of a highly anisotropic and gyrotropic bulk WSM material. The surface polaritons excited on WSM surfaces parallel to the $x$-axis (assuming that the Weyl points are located along $k_x$) can be supported by both bulk and surface electron states. However, in the quasielectrostatic approximation $ck \gg \omega$ the SPs are highly localized and the surface states make a dominant contribution to the SP dispersion and radiation pattern [@chen2019]. Here $k$ is the magnitude of the SP wavevector in the $z = 0$ plane. Below we outline the calculation of the SP dispersion, energy flux, and radiation patterns generated by the tip. The detailed derivation is in the Supplemental Material below. We model the nanotip-induced excitation source of SPs as an external point dipole, $$\mathbf{p}^{e}\left(\mathbf{r},z,t\right)=\mathrm{Re}\left[\mathbf{p}\delta\left(\mathbf{r}\right)\delta\left(z\right)e^{-i\omega t}\right]$$ where $\mathbf{r}=\left(x,y\right)$. The point source approximation is valid if the tip apex radius and its distance to the surface are smaller than the exponential extent of the excitation field. Our case is borderline as these scales are actually of the same order, but we will still assume a point source for simplicity. One can always generalize the analysis for any spatial distribution of the excitation specific to a given experiment. The corresponding external current is $ \mathbf{j}_{\omega}^{e}\left(\mathbf{r}\right)=-i\omega\mathbf{p}\delta\left(\mathbf{r}\right)$. Within the quasielectrostatic approximation the electric field of SPs can be defined through the scalar potential: $ \mathbf{E}=-\nabla\Phi$, where $$\Phi(\mathbf{r},z,t)=\mathrm{Re}\left[\Phi_{\omega}\left(\mathbf{r},z\right)e^{-i\omega t}\right].$$ Outside the surface, $\Phi_{\omega}$ is described by the Poisson equation at $z>0$ (in the air or an ambient medium), $\nabla^{2}\Phi_{\omega}=0$, and Gauss’s law in the bulk WSM at $z<0$: $$\frac{\partial}{\partial x}\left(\varepsilon_{xx}E_{x}\right)+\frac{\partial}{\partial y}\left(\varepsilon_{yy}E_{y}+\varepsilon_{yz}E_{z}\right)+\frac{\partial}{\partial z}\left(\varepsilon_{zz}E_{z}+\varepsilon_{zy}E_{y}\right)=0.$$ We assume that the medium above the surface is described by an isotropic dielectric constant $\varepsilon_{up}$. Then, the boundary conditions yield $$\varepsilon_{up}E_{z}\left(z=+0\right)-D_{z}\left(z=-0\right)=4\pi\rho^{S}=-i\frac{4\pi}{\omega}\left(\frac{\partial}{\partial x}j_{x}^{S}+\frac{\partial}{\partial y}j_{y}^{S}\right)$$ where $\rho^{S}$ is the surface charge due to surface electron states and an external source; $j_{x}^{S},$ $j_{y}^{S}$ are the components of the total surface current that are connected with the surface charge by the in-plane continuity equation. The total surface current, $\mathbf{j}_{\omega}^{S}\left(\mathbf{r}\right)=\mathbf{j}_{\omega}^{l}\left(\mathbf{r}\right)+\mathbf{j}_{\omega}^{e}\left(\mathbf{r}\right) $, is the sum of the current $\mathbf{j}_{\omega}^{l}\left(\mathbf{r}\right)$ representing the linear response to $\Phi_{\omega}$ and the current $\mathbf{j}_{\omega}^{e}\left(\mathbf{r}\right)$ induced by the external dipole source. All currents and charges are on the surface so that we drop the index $S$. The equations for the scalar potential can be solved by expansion over spatial harmonics in the $(x,y)$ plane: $$\mathbf{j}_{\omega}^{e,l}\left(\mathbf{r}\right)=\intop\int\mathbf{j}_{\omega\mathbf{k}}^{e,l}e^{i\mathbf{k\cdot r}}d^{2}k,$$ $$\Phi_{\omega}\left(\mathbf{r},z\right)=\intop\int\Phi_{\omega\mathbf{k}}\left(z\right)e^{i\mathbf{k\cdot r}}d^{2}k.$$ Here $ \mathbf{j}_{\omega\mathbf{k}}^{l}=\hat{\sigma}^{S}\cdot\mathbf{E}_{\omega}\left(z=-0\right)$, $\mathbf{E}_{\omega}\left(z=-0\right)=-i\mathbf{k}\Phi_{\omega\mathbf{k}}(z=-0)$. A surface dipole layer is formed at the boundary between the two media. Its dipole moment is oriented along the normal to the surface, with the space-time Fourier components related to the z-component of the surface current density: $$d_{z\mathbf{k}}=\frac{i}{\omega}\left[\sigma_{zy}^{S}E_{y}\left(z=-0\right)+\sigma_{zz}^{S}E_{z}\left(z=-0\right)\right].$$ The sum of an external and induced dipole creates a jump in the scalar potential $\Phi\left(z\right)$, $$\Phi_{\omega\mathbf{k}}\left(z=+0\right)-\Phi_{\omega\mathbf{k}}\left(z=-0\right)=4\pi d_{z\mathbf{k}}+\frac{1}{\pi} \mathbf{p}\cdot\mathbf{z_{0}}.$$ The solution for the SP field evanescent in $\pm z$ direction is $$\Phi_{\omega\mathbf{k}}(z>0)=\phi_{\omega\mathbf{k}}^{up}e^{-\kappa_{up}z},\,\,\,\,\,\,\,\,\Phi_{\omega\mathbf{k}}(z<0)=\phi_{\omega\mathbf{k}}^{W}e^{\kappa_{W}z}.$$ Here the spatial harmonics of the potential satisfy algebraic equations $$\label{pot1} \varepsilon_{up}\kappa_{up}\phi_{\omega\mathbf{k}}^{up}+\left[\kappa_{W}\left(\varepsilon_{zz}+\frac{4\pi}{\omega}k_{y}\sigma_{yz}^{S}\right)+ \frac{4\pi \sigma_{yz}^B}{\omega} k_{y}+i\frac{4\pi}{\omega}\left(k_{x}^{2}\sigma_{xx}^{S}+k_{y}^{2}\sigma_{yy}^{S}\right)\right]\phi_{\omega\mathbf{k}}^{W}=\frac{4\pi}{\omega}\mathbf{k}\cdot\mathbf{j}_{\omega\mathbf{k}}^{e},$$ $$\label{pot2} \phi_{\omega\mathbf{k}}^{up}+\left(i\frac{4\pi}{\omega}\kappa_{W}\sigma_{zz}^{S}-\frac{4\pi}{\omega}k_{y}\sigma_{zy}^{S}-1\right)\phi_{\omega\mathbf{k}}^{W}=\frac{1}{\pi}\mathbf{p}\cdot\mathbf{z_{0}}$$ and the decay constants $\kappa_{up,W}$ can be found from $k^{2}-\kappa_{up}^{2}=0$, $\varepsilon_{xx} k^{2}\cos^2\phi + \varepsilon_{yy}k^{2} \sin^2\phi-\varepsilon_{zz}\kappa_{W}^{2}=0$, where $k_{x}=k\cos\phi$, $k_{y}=k\sin\phi$. In the absence of an external dipole, Eqs. (\[pot1\],\[pot2\]) give the dispersion equation for SPs $\mathscr{D}\left(\omega,\phi,k\right)= 0$ (see [@supp; @chen2019] for an explicit expression). The space-dependent expressions for the scalar potential on both sides of the surface are obtained by taking the Fourier transform from $(k, \phi)$ to $(x,y) = (r\cos\theta, r\sin\theta)$. The 2D integrals in momentum space are calculated by series expansion in terms of Bessel functions and using the integral identity for Bessel functions derived in the Supplemental Material. In the far-field zone of the tip, the scalar potential scales with distance as $\displaystyle \frac{\exp\left[ik_{\omega}(\theta)r \right]}{\sqrt{r}}$. A very cumbersome expression for $k_{\omega}(\theta)$ is derived in the Supplemental Material. Figures 2a,b show the polar plots of the real part of the in-plane SP wavenumber $k_{\omega}(\theta)$ for several values of frequency and Fermi momentum. ![Polar plot of the real part of the in-plane SP wavenumber $k_{\omega}(\theta)$ for (a) several values of frequency at a given Fermi momentum $\hbar v_F k_F = 50$ meV and (b) several values of the Fermi momentum at a given frequency $\hbar \omega = 80$ meV. []{data-label="fig2"}](Fig2.pdf) These plots and all plots below were calculated for a vertical dipole orientation. In this case the excitation itself is isotropic in the plane (no $\theta$ dependence) and therefore all anisotropy comes from the properties of topological bulk and surface electron states. The conductivity tensors used in all plots were calculated assuming strongly disordered samples with high SP decay rate $\gamma = 10$ meV. In this case SPs have a low Q-factor: the imaginary part of the wave vector is only a few times lower than the real part. Obviously, in higher quality samples one should expect longer-lived SP excitations with longer propagating lengths, at least at frequencies lower than the Fermi energy-dependent interband transition cutoff determined by the Pauli blocking. We also assumed the background bulk dielectric constant $\varepsilon_b = 10$. If the contribution of surface conductivity were ignored and only the bulk carriers were taken into account, the SPs would have no dispersion at all: their frequency would depend only on the propagation angle but not on the magnitude of the wave vector [@chen2019; @supp]. Moreover, bulk electron states would support surface EM modes only below the plasma resonance, when the real part of the diagonal components of the bulk dielectric tensor is negative enough. For $\hbar v_F k_F = 50$ meV in Fig. 2a, the plasma resonance is around 50 meV [@chen2019]. SP modes plotted in Fig. 2a show a very strong dispersion in every direction and exist way beyond 50 meV. Therefore they are supported by “Fermi arc” surface electron states via surface current sheet and surface dipole that they create in response to the field, with bulk WSM serving mainly as a dielectric substrate. That is why the surface polaritons is a more appropriate term for these surface modes than surface plasmon-polaritons that would exist at low frequencies below plasma resonance. Note strong anisotropy of the wavevector and its extreme sensitivity to the relative values of frequency, Fermi momentum, and Weyl node separation in momentum space. Note also that all plots are symmetric with respect to the $y$-axis, which is perpendicular to the gyrotropy axis $x$. Similar behavior is found in the Poynting flux radiation patterns in Fig. 3. It can be interpreted as the realization of the optical Hall effect. Indeed the symmetry properties of the optical response of the system are determined by the polar symmetry axis vector $\mathbf{a} = \mathbf{n} \times \mathbf{b}$, where the axial gyrotropy vector $\mathbf{b} \parallel \hat{x}_0$ and the polar vector $\mathbf{n} \parallel \hat{z}_0$ is the normal to the surface, so that $\mathbf{a} \parallel \hat{y}_0$. This is in analogy with the Hall effect in which the current direction is determined by the cross product of the axial gyrotropy vector of the magnetic field and the polar vector of the electric field. To calculate the Poynting flux in a SP wave, we need to go beyond electrostatic approximation. Following the perturbation method detailed in [@ryan2018], we use the Maxwell’s equation $\nabla\times\mathbf{B}\left(\omega,\mathbf{r},z\right)=\frac{1}{c}\frac{\partial}{\partial t}\mathbf{D}\left(\omega,\mathbf{r},z\right)$ in each half-space to calculate the magnetic field from the electric field obtained in the electrostatic approximation. Then the time-averaged Poynting flux $\mathbf{S}\left(\mathbf{r},z\right)=\mathrm{Re}[\frac{c}{8\pi}\left(\mathbf{E\times B^{\ast}}\right)]$ can be calculated in each half-space. After integrating over $dz$, i.e. $S_{r}\left(r,\theta\right)=\int_{-\infty}^{\infty}S_{r}\left(\mathbf{r},z\right)dz$ we obtain the the time-averaged surface Poynting flux in the far field zone. The derivation and explicit expression for the Poynting flux can be found in the Supplemental Material. ![Polar plot of the in-plane Poynting vector integrated over the vertical $z$-direction, for (a) several values of frequency at a given Fermi momentum $\hbar v_F k_F = 50$ meV and (b) several values of the Fermi momentum at a given frequency $\hbar \omega = 80$ meV. The magnitudes of the Poynting flux are multiplied by different numerical factors indicated in the figure, in order to fit to one plot. []{data-label="fig3"}](Fig3.pdf) Figures 3a,b show the radiation pattern of the SPs, namely polar plots of the SP Poynting vector integrated over the vertical $z$-direction, for several values of frequency and Fermi momentum. The numerical values for the SP Poynting flux density in the plots were calculated at a distance of 250 $\mu$m from the tip and assuming that the excitation is created by the pump field of magnitude $10^6$ V/cm localized within (10 nm)$^3$. Such fields are far below damage threshold; for example, in experiments reported in [@raschke2019] the pump field under the tip was estimated at $5\times 10^7$ V/cm. Only $1/r$ divergence of the in-plane Poynting vector was included. The actual SP attenuation length is determined by the material quality and is likely to be much shorter than 250 $\mu$m. The energy flow of SPs is highly anisotropic and strongly frequency and Fermi momentum-dependent. There is again extreme sensitivity of the radiation pattern to the relative values of frequency, Fermi momentum, and Weyl node separation in momentum space. Furthermore, all plots are symmetric with respect to the $y$ axis and with increasing frequency the SP flux is mainly directed along $\hat{y}_0$. This is the manifestation of the optical Hall effect induced by Weyl node separation, as discussed above. Note an enhancement in the SP flux at low frequencies in Fig. 3a, related to intraband transitions and Drude-like enhancement of the conductivity, especially its $\sigma_{yy}^S$ element related to free-carrier motion of surface electron states with dispersion $E = \hbar v_F k_y$ [@chen2019]. Note also strong enhancement of the Poynting flux at high frequencies around 100 meV due to an increase in the wavenumber $k_{\omega}(\theta)$ and magnitude of the conductivity tensor associated with interband transitions; see the conductivity spectra in [@chen2019]. Since the surface states exist only at electron momenta $k_x^2 + k_y^2 < b^2$, at frequencies higher than 200 meV (or for high enough Fermi momenta $k_F > b$) the surface conductivity approaches zero whereas the bulk dielectric tensor approaches its background value. Therefore, there will be no SP modes supported by topological states in this limit $\omega \gg v_F b$, although other kinds of surface polariton modes could still exist due to e.g. phonon resonances. In conclusion, we showed that spectroscopy of surface polaritons can be a powerful diagnostics of topological electron states in WSMs. Anomalous dispersion and extreme anisotropy and gyrotropy of SPs launched by a nanotip provides information about Weyl node position and separation, the value of the Fermi momentum, and the matrix elements of the optical transitions involving both bulk and surface electron states. Although the quantitative results in this paper are valid only for magnetic WSMs with time-reversal symmetry breaking, one can still make some qualitative conclusions regarding the optical response of WSMs with inversion symmetry breaking. In particular, one can still expect strong anisotropy of SP propagation, related to the position and orientation of Weyl node pairs in the Brillouin zone. There will be strong dispersion of SPs associated with the presence of Fermi arc surface states. The relative enhancement or suppression of SPs associated with the Fermi edge and interband transitions will be present. The low-frequency response related to massless free carriers will be similar. This work has been supported by the Air Force Office for Scientific Research through Grant No. FA9550-17-1-0341. M.T. acknowledges the support from RFBR Grant No. 17-02-00387. M.E. acknowledges the support from the Project No. 0035-2018-0006 of the Presidium of the Russian Academy of Sciences. Supplemental Material {#supplemental-material .unnumbered} ===================== On WSM surfaces parallel to the $x$-axis (assuming that the Weyl points are located along $k_x$) the SPPs polaritons can be supported by both bulk and surface electron states. However, in the quasielectrostatic approximation $ck \gg \omega$ the SPPs are highly localized and the surface states make a dominant contribution to the SPP dispersion and radiation pattern [@chen2019]. Here $k$ is the magnitude of the SPP wavevector in the $z = 0$ plane. We model the nanotip-induced excitation source of SPPs as an external point dipole, $$\mathbf{p}^{e}\left(\mathbf{r},z,t\right)=\mathrm{Re}\left[\mathbf{p}\delta\left(\mathbf{r}\right)\delta\left(z\right)e^{-i\omega t}\right]$$ where $\mathbf{r}=\left(x,y\right)$. The corresponding external current is $ \mathbf{j}_{\omega}^{e}\left(\mathbf{r}\right)=-i\omega\mathbf{p}\delta\left(\mathbf{r}\right)$. Within the quasielectrostatic approximation the electric field of SPPs can be defined through the scalar potential: $ \mathbf{E}=-\nabla\Phi$, where $$\Phi(\mathbf{r},z,t)=\mathrm{Re}\left[\Phi_{\omega}\left(\mathbf{r},z\right)e^{-i\omega t}\right].$$ Outside the surface, $\Phi_{\omega}$ is described by the Poisson equation at $z>0$ (in the air or an ambient medium): $$\label{poisson} \nabla^{2}\Phi_{\omega}=0,$$ and Gauss’s law inside the WSM at $z<0$: $$\label{gauss} \nabla\cdot\mathbf{D}_{\omega}=0,$$ which can be expanded in components as $$\frac{\partial}{\partial x}\left(\varepsilon_{xx}E_{x}\right)+\frac{\partial}{\partial y}\left(\varepsilon_{yy}E_{y}+\varepsilon_{yz}E_{z}\right)+\frac{\partial}{\partial z}\left(\varepsilon_{zz}E_{z}+\varepsilon_{zy}E_{y}\right)=0.$$ We assume that the medium above the surface is described by an isotropic dielectric constant $\varepsilon_{up}$. Then, the boundary conditions yield $$\varepsilon_{up}E_{z}\left(z=+0\right)-D_{z}\left(z=-0\right)=4\pi\rho^{S}=-i\frac{4\pi}{\omega}\left(\frac{\partial}{\partial x}j_{x}^{S}+\frac{\partial}{\partial y}j_{y}^{S}\right)$$ where $\rho^{S}$ is the surface charge due to surface electron states and an external source; $j_{x}^{S},$ $j_{y}^{S}$ are the components of the total surface current that are connected with the surface charge by the in-plane continuity equation. A surface dipole layer is formed at the boundary between the two media. Its dipole moment is oriented along the normal to the surface, $$\mathbf{d}=\mathrm{Re}\left[\mathbf{d}_{\omega}\left(\mathbf{r}\right)e^{-i\omega t}\right],$$ $$\mathbf{d}_{\omega}\left(\mathbf{r}\right)=\mathbf{z_{0}}\intop\int d_{z\mathbf{k}}e^{i\mathbf{k\cdot r}}d^{2}k,$$ where the space-time Fourier components can be related to the z-component of the surface current density, $$d_{z\mathbf{k}}=\frac{i}{\omega}\left[\sigma_{zy}^{S}E_{y}\left(z=-0\right)+\sigma_{zz}^{S}E_{z}\left(z=-0\right)\right].$$ The sum of an external and induced dipole creates a jump in the scalar potential $\Phi\left(z\right)$, $$\Phi_{\omega\mathbf{k}}\left(z=+0\right)-\Phi_{\omega\mathbf{k}}\left(z=-0\right)=4\pi\left[d_{z\mathbf{k}}+\frac{1}{4\pi^2} \mathbf{p}\cdot\mathbf{z_{0}}\right].$$ The total surface current, $\mathbf{j}_{\omega}^{S}\left(\mathbf{r}\right)=\mathbf{j}_{\omega}^{l}\left(\mathbf{r}\right)+\mathbf{j}_{\omega}^{e}\left(\mathbf{r}\right) $, is the sum of the current $\mathbf{j}_{\omega}^{l}\left(\mathbf{r}\right)$ representing the linear response to $\Phi_{\omega}$ and the current $\mathbf{j}_{\omega}^{e}\left(\mathbf{r}\right)$ induced by the external dipole source. All currents and charges are on the surface so that we drop the index $S$. The equations for the scalar potential can be solved by expansion over spatial harmonics in the $(x,y)$ plane: $$\mathbf{j}_{\omega}^{e,l}\left(\mathbf{r}\right)=\intop\int\mathbf{j}_{\omega\mathbf{k}}^{e,l}e^{i\mathbf{k\cdot r}}d^{2}k,$$ $$\label{phi} \Phi_{\omega}\left(\mathbf{r},z\right)=\intop\int\Phi_{\omega\mathbf{k}}\left(z\right)e^{i\mathbf{k\cdot r}}d^{2}k.$$ Here $ \mathbf{j}_{\omega\mathbf{k}}^{l}=\hat{\sigma}^{S}\cdot\mathbf{E}_{\omega}\left(z=-0\right)$, $\mathbf{E}_{\omega}\left(z=-0\right)=-i\mathbf{k}\Phi_{\omega\mathbf{k}}(z=-0)$. The inverse transformation is $$\mathbf{j}_{\omega\mathbf{k}}^{e,l}=\frac{1}{\left(2\pi\right)^{2}}\intop\int\mathbf{j}_{\omega}^{e,l}\left(\mathbf{r}\right)e^{-i\mathbf{k\cdot r}}d^{2}r.$$ The solution for the SPP field evanescent in $\pm z$ direction is $$\label{phi2} \Phi_{\omega\mathbf{k}}(z>0)=\phi_{\omega\mathbf{k}}^{up}e^{-\kappa_{up}z},\,\,\,\,\,\,\,\,\Phi_{\omega\mathbf{k}}(z<0)=\phi_{\omega\mathbf{k}}^{W}e^{\kappa_{W}z}.$$ Here the spatial harmonics of the potential satisfy algebraic equations $$\label{pot1} \varepsilon_{up}\kappa_{up}\phi_{\omega\mathbf{k}}^{up}+\left[\kappa_{W}\left(\varepsilon_{zz}+\frac{4\pi}{\omega}k_{y}\sigma_{yz}^{S}\right)+gk_{y}+i\frac{4\pi}{\omega}\left(k_{x}^{2}\sigma_{xx}^{S}+k_{y}^{2}\sigma_{yy}^{S}\right)\right]\phi_{\omega\mathbf{k}}^{W}=\frac{4\pi}{\omega}\mathbf{k}\cdot\mathbf{j}_{\omega\mathbf{k}}^{e},$$ $$\label{pot2} \phi_{\omega\mathbf{k}}^{up}+\left(i\frac{4\pi}{\omega}\kappa_{W}\sigma_{zz}^{S}-\frac{4\pi}{\omega}k_{y}\sigma_{zy}^{S}-1\right)\phi_{\omega\mathbf{k}}^{W}=\frac{1}{\pi}\mathbf{p}\cdot\mathbf{z_{0}}$$ where $g = \frac{4\pi \sigma_{yz}^B}{\omega}$ and the decay constants $\kappa_{up,W}$ can be found from Eqs. (\[poisson\]), (\[gauss\]): $k^{2}-\kappa_{up}^{2}=0$, $\varepsilon_{xx} k^{2}\cos^2\phi + \varepsilon_{yy}k^{2} \sin^2\phi-\varepsilon_{zz}\kappa_{W}^{2}=0$, where $k_{x}=k\cos\phi$, $k_{y}=k\sin\phi$. This formalism allows one to add spatial dispersion of the conductivity $\hat{\sigma}^{S}\left(\omega,\mathbf{k}\right)$ and $\hat{\varepsilon}\left(\omega,\mathbf{k}\right)$ if needed, but we will ignore it below. In the absence of an external dipole, Eqs. (\[pot1\],\[pot2\]) give the dispersion equation for SPPs derived in [@chen2019], $$\label{disp} \mathscr{D}\left(\omega,\phi,k\right)=D\left(\omega,\phi\right)-k\Sigma\left(\omega,\phi\right) = 0,$$ where $$\Sigma\left(\omega,\phi\right) = \frac{4\pi}{\omega}\bigg[\sqrt{\frac{\varepsilon_{xx}\cos^{2}\phi+\varepsilon_{yy}\sin^{2}\phi}{\varepsilon_{zz}}}\left(in_{up}^{2}\sigma_{zz}^{S}-\sigma_{yz}^{S}\sin\phi\right)-n_{up}^{2}\sigma_{yz}^{S}\sin\phi-i\left(\sigma_{xx}^{S}\cos^{2}\phi+\sigma_{yy}^{S}\sin^{2}\phi\right)\bigg],$$ and $$D\left(\omega,\phi\right)=n_{up}^{2}+\varepsilon_{zz}\sqrt{\frac{\varepsilon_{xx}\cos^{2}\phi+\varepsilon_{yy}\sin^{2}\phi}{\varepsilon_{zz}}}+g\sin\phi.$$ Note that $\Sigma = 0$ if the surface terms are neglected. Therefore, $D\left(\omega,\phi\right) = 0$ is the dispersion equation of SPPs supported by bulk electron states only. Such modes would have no dispersion since $D\left(\omega,\phi\right)$ does not depend on the SPP wavenumber. Moreover, bulk states would support surface modes only below the plasma resonance, when the real part of the diagonal components of the bulk dielectric tensor is negative enough. For $\hbar v_F k_F = 50$ meV, the plasma resonance is around 50 meV [@chen2019]. SPP modes plotted in Fig. 2 of the main paper show a very strong dispersion in every direction and exist way beyond 50 meV. Therefore they are supported by surface electron states, with bulk WSM serving mainly as a dielectric substrate. Including an external source, Eqs. (\[pot1\]), (\[pot2\]) give the Fourier amplitudes of the scalar potential in both half-spaces: $$\begin{aligned} \label{phi-up} \phi_{\omega\mathbf{k}}^{up} & =\frac{\frac{4\pi}{\omega}\mathbf{k}\cdot\mathbf{j}_{\omega\mathbf{k}}^{e}\left(\frac{4\pi}{\omega}\sin\phi\sigma_{zy}^{S}+\frac{1}{k}-i\frac{4\pi}{\omega}\sqrt{\frac{\varepsilon_{xx}\cos^{2}\phi+\varepsilon_{yy}\sin^{2}\phi}{\varepsilon_{zz}}}\sigma_{zz}^{S}\right)}{\mathscr{D}\left(\omega,\phi,k\right)}+\frac{1}{\pi}\left(\mathbf{p}\cdot\mathbf{z_{0}}\right)\nonumber \\ & \times\frac{\varepsilon_{zz}\sqrt{\frac{\varepsilon_{xx}\cos^{2}\phi+\varepsilon_{yy}\sin^{2}\phi}{\varepsilon_{zz}}}+g\sin\phi+\frac{4\pi}{\omega}k\bigg[\sqrt{\frac{\varepsilon_{xx}\cos^{2}\phi+\varepsilon_{yy}\sin^{2}\phi}{\varepsilon_{zz}}}\sigma_{yz}^{S}\sin\phi+i\left(\sigma_{xx}^{S}\cos^{2}\phi+\sigma_{yy}^{S}\sin^{2}\phi\right)\bigg]}{\mathscr{D}\left(\omega,\phi,k\right)},\end{aligned}$$ $$\label{phi-w} \phi_{\omega\mathbf{k}}^{W}=\frac{\frac{4\pi}{\omega k}\mathbf{k}\cdot\mathbf{j}_{\omega\mathbf{k}}^{e}-\frac{1}{\pi}\left(\mathbf{p}\cdot\mathbf{z_{0}}\right)n_{up}^{2}}{\mathscr{D}\left(\omega,\phi,k\right)}.$$ Then the spatial field distributions on both sides of the interface can be obtained from Eqs. (\[phi2\]), (\[phi-up\]), (\[phi-w\]) by Fourier transform Eq. (\[phi\]). We will perform integration only in the case of a vertical external Hertz dipole, i.e. $\mathbf{p}=p\mathbf{z_{0}}$, when $\mathbf{k}\cdot\mathbf{p}=0$ and the source is isotropic in plane of the interface. Therefore, all anisotropy in the SPP propagation comes from the properties of topological electron states. The Fourier integral in polar coordinates $(k,\phi)$ in momentum space can be written as $$\begin{aligned} \label{phi1} \Phi_{\omega}^{\left(+\right)} \equiv \Phi_{\omega}\left(\mathbf{r},z=+0\right) & =\frac{p}{\pi}\intop\int d^{2}ke^{i\mathbf{k\cdot r}}\frac{H\left(\omega,\phi,k\right)}{\mathscr{D}\left(\omega,\phi,k\right)}\nonumber \\ & \approx -\frac{p}{\pi}\int_{0}^{2\pi}d\phi\frac{1}{\Sigma\left(\omega,\phi\right)}\int_{0}^{\infty}\frac{e^{ikr\cos\left(\phi-\theta\right)}H\left(\omega,\phi,k\right)}{k-k_{\omega}\left(\phi\right)-i\eta_{\omega}\left(\phi\right)}kdk, \end{aligned}$$ where $(r,\theta)$ are polar coordinates in real 2D space and we introduced the shortcut notation $$H\left(\omega,\phi,k\right)=D\left(\omega,\phi\right)-n_{up}^{2}+\frac{4\pi}{\omega}k\bigg[\sqrt{\frac{\varepsilon_{xx}\cos^{2}\phi+\varepsilon_{yy}\sin^{2}\phi}{\varepsilon_{zz}}}\sigma_{yz}^{S}\sin\phi+i\left(\sigma_{xx}^{S}\cos^{2}\phi+\sigma_{yy}^{S}\sin^{2}\phi\right)\bigg].$$ In the second line of Eq. (\[phi1\]) we also introduced the solution to the dispersion equation for SPPs, Eq. (\[disp\]) in terms of the real and imaginary parts of the SPP wave number, $k_{\omega}\left(\phi\right)$ and $\eta_{\omega}\left(\phi\right)$. We will also assume for simplicity that the SPP dissipation is sufficiently weak so that the real part of the solution can be found from $$\mathrm{Re}\mathscr{D}\left(\omega,\phi,k_{\omega}\left(\phi\right)\right)\approx 0,$$ whereas the imaginary part of the SPP wavenumber can be calculated as $$\eta=-\frac{\mathrm{Im}\mathscr{D}\left(\omega,\phi,k_{\omega}\right)}{\left[\frac{\partial\mathrm{Re}\mathscr{D}\left(\omega,\phi,k_{\omega}\right)}{\partial k}\right]_{k=k_{\omega}\left(\phi\right)}}.$$ The explicit expression for the SPP wavenumber is $$k_{\omega}\left(\phi\right)=\mathrm{Re}\left[\frac{\omega\left(n_{up}^{2}+\varepsilon_{zz}\sqrt{\frac{\varepsilon_{xx}\cos^{2}\phi+\varepsilon_{yy}\sin^{2}\phi}{\varepsilon_{zz}}}+g\sin\phi\right)}{4\pi\left[\sqrt{\frac{\varepsilon_{xx}\cos^{2}\phi+\varepsilon_{yy}\sin^{2}\phi}{\varepsilon_{zz}}}\left(in_{up}^{2}\sigma_{zz}^{S}-\sigma_{yz}^{S}\sin\phi\right)-n_{up}^{2}\sigma_{yz}^{S}\sin\phi-i\left(\sigma_{xx}^{S}\cos^{2}\phi+\sigma_{yy}^{S}\sin^{2}\phi\right)\right]}\right].$$ To calculate the integrals in Eq. (\[phi1\]), we use the known expansion of the exponent in terms of Bessel functions, $$e^{iz\cos\alpha} = J_0(z) + 2 \sum_{n=1}^{\infty} i^n J_n(z) \cos(n\alpha),$$ which gives $$\begin{aligned} \label{phi2} \Phi_{\omega}^{\left(+\right)} & =-\frac{p}{\pi}\int_{0}^{2\pi}d\phi\frac{1}{\Sigma\left(\phi\right)}\int_{0}^{\infty}\frac{\left\{ J_{0}\left(kr\right)+2\sum_{n=1}^{\infty}i^{n}J_{n}\left(kr\right)\cos\left[n\left(\phi-\theta\right)\right]\right\} H\left(\omega,\phi,k\right)}{k-k_{\omega}\left(\phi\right)-i\eta\left(\phi\right)}kdk. \end{aligned}$$ This integral can be calculated analytically in the far zone of the source dipole, i.e. at large $kr \gg 1$. In this case the Bessel Functions in Eq. (\[phi2\]) oscillate much faster than other $k$-dependent terms in the numerator, so we can take $H\left(\omega,\phi,k\right)$ out of the integral over $dk$ and replace $k$ with $k_{\omega}(\phi)$ in its argument. After that, the integral over $dk$ can be evaluated using the following integral identity for Bessel functions: $$\label{iden} \int_0^{\infty} \frac{k^nJ_n(kr)}{k^2-k_{\omega}^2 - i0} kdk = \left(k_{\omega}\right)^n\frac{i\pi}{2} \left( J_n(k_{\omega} r) + iY_n(k_{\omega} r) \right)$$ Equation (\[iden\]) can be derived by applying the operator $\left(\frac{1}{r} \frac{d}{dr} \right)^m$ to both sides of the known Hankel transformation [@piessens] $$\label{hankel} \int_0^{\infty} \frac{J_0(kr)}{k^2-k_{\omega}^2 - i0} kdk = \frac{i\pi}{2} \left( J_0(k_{\omega} r) + iY_0(k_{\omega} r) \right)$$ and using the recurrent formula $$\label{rec} \left(\frac{1}{z} \frac{d}{dz} \right)^m \left[ z^{-\nu} \mathfrak{G}_{\nu}(z) \right] = (-1)^m z^{-\nu - m} \mathfrak{G}_{\nu+m}(z),$$ where $\mathfrak{G}_{\nu}(z) = J_{\nu}(z), Y_{\nu}(z)$ [@abram]. Applying Eq. (\[iden\]) to the integral over $dk$ in Eq. (\[phi2\]) yields $$\begin{aligned} \Phi_{\omega}^{\left(+\right)} & =-i p\int_{0}^{2\pi}d\phi\frac{k_{\omega}\left(\phi\right)}{\Sigma\left(\phi\right)}H\left[\omega,\phi,k_{\omega}\left(\phi\right)\right]\times\nonumber \\ & \left\{ \left(J_{0}\left[k_{\omega}\left(\phi\right)r\right]+iY_{0}\left[k_{\omega}\left(\phi\right)r\right]\right)+2\sum_{n=1}^{\infty}i^{n}\cos\left[n\left(\phi-\theta\right)\right]\left(J_{n}\left[k_{\omega}\left(\phi\right)r\right]+iY_{n}\left[k_{\omega}\left(\phi\right)r\right]\right)\right\} \nonumber \\ & \approx-i p\sqrt{\frac{2}{\pi r}}\int_{0}^{2\pi}e^{ik_{\omega}\left(\phi\right)r}\left\{ e^{-i\frac{\pi}{4}}+2\sum_{n=1}^{\infty}i^{n}e^{-i\left(\frac{n\pi}{2}+\frac{\pi}{4}\right)}\cos\left[n\left(\phi-\theta\right)\right]\right\} \nonumber \\ & \times\frac{\sqrt{k_{\omega}\left(\phi\right)}}{\Sigma\left(\phi\right)}H\left[\omega,\phi,k_{\omega}\left(\phi\right)\right]d\phi. \nonumber \end{aligned}$$ In the last approximate equality we also took an advantage of the fact that in the far zone, namely when the Bessel functions argument $z \gg \left| n^2 - \frac{\pi}{4} \right|$, one can use their asymptotic values [@abram] $$J_n(z) \approx \sqrt{\frac{2}{\pi z}} \cos\left(z - \frac{n\pi}{2} - \frac{\pi}{4} \right), \; Y_n(z) \approx \sqrt{\frac{2}{\pi z}} \sin\left(z - \frac{n\pi}{2} - \frac{\pi}{4} \right). \nonumber$$ Then the integral over $\phi$ can be evaluated by using the delta-function identity: $$\begin{aligned} \label{phiplus} \Phi_{\omega}^{\left(+\right)} & =-\frac{p}{\sqrt{\pi}}\sqrt{\frac{2}{r}}\int_{0}^{2\pi}e^{i\left[k_{\omega}\left(\phi\right)r+\frac{\pi}{4}\right]}\frac{\sqrt{k_{\omega}\left(\phi\right)}}{\Sigma\left(\phi\right)}H\left[\omega,\phi,k_{\omega}\left(\phi\right)\right]\sum_{n=-\infty}^{\infty}\cos\left[n\left(\phi-\theta\right)\right]d\phi\nonumber \\ & =- 2 \sqrt{\pi} p \sqrt{\frac{2}{r}}\int_{0}^{2\pi}e^{i\left[k_{\omega}\left(\phi\right)r+\frac{\pi}{4}\right]}\frac{\sqrt{k_{\omega}\left(\phi\right)}}{\Sigma\left(\phi\right)}H\left[\omega,\phi,k_{\omega}\left(\phi\right)\right]\delta\left(\phi-\theta\right)d\phi\nonumber \\ & = - \frac{2 \sqrt{\pi} p}{\Sigma\left(\theta\right)}\sqrt{\frac{2k_{\omega}\left(\theta\right)}{r}}H\left[\omega,\theta,k_{\omega}\left(\theta\right)\right]\exp\left[ik_{\omega}\left(\theta\right)r+i\frac{\pi}{4}\right].\end{aligned}$$ Applying the same procedure, we derive the spatial distribution for the scalar potential just below the surface, i.e. inside the WSM: $$\label{phiminus} \Phi_{\omega}^{\left(-\right)}\equiv\Phi_{\omega}\left(\mathbf{r},z=-0\right) =\frac{2\sqrt{\pi} pn_{up}^{2}}{\Sigma\left(\theta\right)}\sqrt{\frac{2k_{\omega}\left(\theta\right)}{r}}\exp\left[ik_{\omega}\left(\theta\right)r+i\frac{\pi}{4}\right].$$ To calculate the Poynting flux in a SPP wave, we need to go beyond electrostatic approximation. Following the perturbation method detailed in [@ryan2018], we use the Maxwell’s equation $\nabla\times\mathbf{B}\left(\omega,\mathbf{r},z\right)=\frac{1}{c}\frac{\partial}{\partial t}\mathbf{D}\left(\omega,\mathbf{r},z\right)$ in each half-space to calculate the magnetic field from the electric field obtained in the electrostatic approximation: $$\frac{1}{r}\frac{\partial B_{z}}{\partial\theta}-\frac{\partial B_{\theta}}{\partial z}=i\frac{n_{up}^{2}\omega}{c}\frac{\partial}{\partial r}\Phi_{\omega}^{\left(+\right)}e^{-\kappa_{+}z}\,\,\,\,\,\left(z>0\right),$$ $$\frac{1}{r}\frac{\partial B_{z}}{\partial\theta}-\frac{\partial B_{\theta}}{\partial z}=i\frac{\omega}{c}\left[\left(\varepsilon_{xx}\cos^{2}\theta+\varepsilon_{yy}\sin^{2}\theta\right)\frac{\partial}{\partial r}+i\kappa_{-}g\sin\theta\right]\Phi_{\omega}^{\left(-\right)}e^{\kappa_{-}z}\,\,\,\,\,\left(z<0\right),$$ where $\kappa_{+}^{2}=k_{\omega}^{2}\left(\theta\right)-n_{up}^{2}\frac{\omega^{2}}{c^{2}}$, $\kappa_{-}^{2}=$$\frac{\varepsilon_{yy}}{\varepsilon_{zz}}\left[k_{\omega}^{2}\left(\theta\right)-\left(\varepsilon_{zz}-\frac{g^{2}}{\varepsilon_{yy}}\right)\frac{\omega^{2}}{c^{2}}\right].$ In the quasielectrostatic approximation and far field zone, i.e. $c\rightarrow\infty$ and $r\rightarrow\infty$, we have $\kappa_{+}\approx k_{\omega}\left(\theta\right)$, $\kappa_{-}\approx\left|\mathrm{Re}\left[\sqrt{\frac{\varepsilon_{yy}}{\varepsilon_{zz}}}\right]\right|k_{\omega}\left(\theta\right)$, $\frac{\partial}{\partial r}\approx ik_{\omega}\left(\theta\right)$. Furthermore, one can neglect the term $\frac{1}{r}\frac{\partial B_{z}}{\partial\theta}$ in the far field zone. Then we get $$B_{\theta}\left(z>0\right)=-\frac{n_{up}^{2}\omega}{c}\Phi_{\omega}^{\left(+\right)}e^{-k_{\omega}\left(\theta\right)z},$$ $$B_{\theta}\left(z<0\right)=\frac{\omega}{c}\left(\frac{\varepsilon_{xx}\cos^{2}\theta+\varepsilon_{yy}\sin^{2}\theta}{\left|\mathrm{Re}\left[\sqrt{\frac{\varepsilon_{yy}}{\varepsilon_{zz}}}\right]\right|}+g\sin\theta\right)\Phi_{\omega}^{\left(-\right)}e^{\left|\mathrm{Re}\left[\sqrt{\frac{\varepsilon_{yy}}{\varepsilon_{zz}}}\right]\right|k_{\omega}\left(\theta\right)z},$$ $$E_{z}\left(z>0\right)=k_{\omega}\left(\theta\right)\Phi_{\omega}^{\left(+\right)}e^{-k_{\omega}\left(\theta\right)z},$$ $$E_{z}\left(z<0\right)=-\left|\mathrm{Re}\left[\sqrt{\frac{\varepsilon_{yy}}{\varepsilon_{zz}}}\right]\right|k_{\omega}\left(\theta\right)\Phi_{\omega}^{\left(-\right)}e^{\left|\mathrm{Re}\left[\sqrt{\frac{\varepsilon_{yy}}{\varepsilon_{zz}}}\right]\right|k_{\omega}\left(\theta\right)z}.$$ Therefore the time-averaged Poynting flux $\mathbf{S}\left(\mathbf{r},z\right)=\mathrm{Re}[\frac{c}{8\pi}\left(\mathbf{E\times B^{\ast}}\right)]$ is $$S_{r}\left(\mathbf{r},z>0\right)=\frac{n_{up}^{2}\omega}{8\pi}k_{\omega}\left(\theta\right)\left|\Phi_{\omega}^{\left(+\right)}\right|^{2}e^{-2k_{\omega}\left(\theta\right)z},$$ $$S_{r}\left(\mathbf{r},z<0\right)=\frac{\omega}{8\pi}k_{\omega}\left(\theta\right)\mathrm{Re}\left[\varepsilon_{xx}^{\ast}\cos^{2}\theta+\varepsilon_{yy}^{\ast}\sin^{2}\theta+g^{\ast}\left|\mathrm{Re}\left[\sqrt{\frac{\varepsilon_{yy}}{\varepsilon_{zz}}}\right]\right|\sin\theta\right]\left|\Phi_{\omega}^{\left(-\right)}\right|^{2}e^{2\left|\mathrm{Re}\left[\sqrt{\frac{\varepsilon_{yy}}{\varepsilon_{zz}}}\right]\right|k_{\omega}\left(\theta\right)z}.$$ After integrating over $dz$, i.e. $S_{r}\left(r,\theta\right)=\int_{-\infty}^{\infty}S_{r}\left(\mathbf{r},z\right)dz$ we finally obtain the total in-plane energy flux in the far field zone: $$\begin{aligned} S_{r}\left(r,\theta\right) & =\frac{\omega}{16\pi}\left[n_{up}^{2}\left|\Phi_{\omega}^{\left(+\right)}\right|^{2}+\mathrm{Re}\left(\frac{\varepsilon_{xx}^{\ast}\cos^{2}\theta+\varepsilon_{yy}^{\ast}\sin^{2}\theta}{\left|\mathrm{Re}\left[\sqrt{\frac{\varepsilon_{yy}}{\varepsilon_{zz}}}\right]\right|}+g^{\ast}\sin\theta\right)\left|\Phi_{\omega}^{\left(-\right)}\right|^{2}\right]\nonumber \\ & =\frac{2\pi^{2}\omega p^{2}n_{up}^{2}k_{\omega}\left(\theta\right)}{\left|\Sigma\left(\theta\right)\right|^{2}r}\left[\left|H\left[\omega,\theta,k_{\omega}\left(\theta\right)\right]\right|^{2}+n_{up}^{2}\mathrm{Re}\left(\frac{\varepsilon_{xx}\cos^{2}\theta+\varepsilon_{yy}\sin^{2}\theta}{\left|\mathrm{Re}\left[\sqrt{\frac{\varepsilon_{yy}}{\varepsilon_{zz}}}\right]\right|}+g\sin\theta\right)\right]\end{aligned}$$ [24]{} M. Kargarian, M. Randeria, and N. Trivedi, Sci. Rep. 5, 12683 (2015). C. J. Tabert and J. P. Carbotte, Phys. Rev. B 93, 085442 (2016). J. Hofmann and S. Das Sarma, Phys. Rev. B 91, 241108 (2015). M. S. Ukhtary, A. R. T. Nugraha, and R. Saito, J. Phys. Soc. Jpn. 86, 104703 (2017). S. Kimura, H. Yokoyama, H. Watanabe, J. Sichelschmidt, V. Sus, M. Schmidt, and C. Felser, Phys. Rev. B 96, 075119 (2017). O. V. Kotov and Yu. E. Lozovik, Phys. Rev. B 98, 195446 (2018). G. M. Andolina, F. M. D. Pellegrino, F. H. L. Koppens, and M. Polini, Phys. Rev. B 97, 125431 (2018). K. Halterman, M. Alidoust, and A. Zyuzin, Phys. Rev. B 98, 085109 (2018). H. Rostami and M. Polini, Phys. Rev. B 97, 195151 (2018). Q. Chen, A. R. Kutayiah, I. Oladyshkin, M. Tokman, and A. Belyanin, Phys. Rev. B 99, 075137 (2019). C. A. C. Garcia, J. Coulter, and P. Narang, https://arxiv.org/abs/1907.04348. J. Ma, Q. Gu, Y. Liu, J. Lai, P. Yu, X. Zhuo, Z. Liu, J.-H. Chen, J. Feng, and D. Sun, Nat. Mater. 18, 476 (2019). J. Moore, Nat. Sci. Rev. 6, 206 (2019). A. A. Burkov and L. Balents, Phys. Rev. Lett. 107, 127205 (2011). R. Okugawa and S. Murakami, Phys. Rev. B 89, 235315 (2014). V. Kravtsov, R. Ulbricht, J. M. Atkin, and M. B. Raschke, Nat. Nanotechnol. 11, 459 (2016). T. Jiang, V. Kravtsov, M. Tokman, A. Belyanin, and M. B. Raschke, Nat. Nanotechnol. https://doi.org/10.1038/s41565-019-0515-x (2019). Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, G. Dominguez, M. Thiemens, M. M. Fogler, A. H. Castro-Neto, C. N. Lau, F. Keilmann, D. N. Basov, Nature 487, 82 (2012). M. I. Stockman, Phys. Rev. Lett. 93, 137404 (2004). X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Phys. Rev. B 83, 205101 (2011). G. Xu, H. Weng, Z. Wang, X. Dai, and Z. Fang, Phys. Rev. Lett. 107, 186806 (2011). E. Liu, Y. Sun, N. Kumar, L. Muechler, A. Sun, L. Jiao, S.-Y. Yang, D. Liu, A. Liang, Q. Xu, J. Kroder, V. Sus, H. Borrmann, C. Shekhar, Z.Wang, C. Xi,W.Wang,W. Schnelle, S. Wirth, Y. Chen, S. T. B. Goennenwein, and C. Felser, Nat. Phys. 14, 1125 (2018). A. R. Kutayiah, M. Tokman, Y. Wang, and A. Belyanin, Phys. Rev. B 98, 115410 (2018). R. Piessens, “The Hankel Transform.” The Transforms and Applications Handbook: Second Edition, Ed. A. D. Poularikas, Boca Raton, CRC Press LLC (2000). M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series - 55, Washington, D.C. (1972).

--- abstract: 'Game coloring is a well-studied two-player game in which each player properly colors one vertex of a graph at a time until all the vertices are colored. An “eternal” version of game coloring is introduced in this paper in which the vertices are colored and re-colored from a color set over a sequence of rounds. In a given round, each vertex is colored, or re-colored, once, so that a proper coloring is maintained. Player 1 wants to maintain a proper coloring forever, while player 2 wants to force the coloring process to fail. The eternal game chromatic number of a graph $G$ is defined to be the minimum number of colors needed in the color set so that player 1 can always win the game on $G$. We consider several variations of this new game and show its behavior on some elementary classes of graphs.' author: - | William F. Klostermeyer\ School of Computing\ University of North Florida\ Jacksonville, FL 32224-2669\ [[email protected]]{} - | Hannah Mendoza\ Wake Forest University\ Winston Salem, NC 27109\ [[email protected]]{} title: '**The Eternal Game Chromatic Number of a Graph**' --- Introduction ============ The [*[eternal graph coloring problem]{}*]{} was introduced by Klostermeyer in [@WFK] and is defined as follows. Let $G=(V, E)$ be a finite, undirected graph with $n$ vertices. Let $f_0: V \rightarrow Z^+$ be a proper vertex coloring of $G$. An infinite sequence of vertex [*requests*]{} $R=r_1, r_2, \ldots$ must be handled as follows. After $r_i$ is revealed, the color assigned to $r_i$, $f_{i-1}(r_i)$, must be changed, yielding a proper coloring $f_i$. We may formalize this problem as a two-player game: player 1 chooses an initial proper coloring of $G$ with at most $k$ colors; the players then alternate turns with player 2 (the [*adversary*]{}) choosing a vertex and player 1 changing the color of that vertex while still maintaining a proper coloring of $G$ with at most $k$ colors. Player 2 wins if player 1 has no valid move at some turn and player 1 wins otherwise. The smallest positive integer $k$ for which player 1 can win the game on graph $G$, for any possible sequence of moves by player 2, is the [*eternal chromatic number*]{} of $G$, is denoted $\chi^\infty(G)$. In other words, how many colors are needed to ensure player 1 can win the game? The game may be viewed as starting from some initial proper coloring, with player 1 maintaining $\chi^\infty(G)$ independent sets, dynamically moving vertices from one set to another as requests from player 2 are made so that each set remains an independent set at all times. The [*vertex coloring game*]{} was introduced in 1981 by Brams (see [@Gard]) and rediscovered later by Bodlaender [@Bod]. It is a two-player game played according to the following rules:\ $\bullet$ Alice and Bob properly color the vertices of a graph $G$ with a fixed set of $k$ colors (the colors are assumed to be integers in the range $1, \ldots, k$).\ $\bullet$ Alice and Bob take turns, coloring properly any uncolored vertex (in the standard version of the game, Alice begins).\ $\bullet$ If a vertex $v$ is impossible to color properly (that is, for any color in the color set, $v$ has a neighbor colored with that color), then Bob wins.\ $\bullet$ If all vertices in the graph are colored properly, then Alice wins.\ The game chromatic number of $G$, denoted by $\chi _{g}(G)$, is the minimum number of colors needed in the color set to guarantee that Alice wins the vertex coloring game on $G$. The purpose of this paper is to define a new graph coloring game which extends game coloring to an “eternal” model of graph coloring that is played by two players over a series of rounds. We define several versions of this game, present some basic results, and pose some questions. Eternal Game Chromatic Number ============================= We define the [*eternal vertex coloring game*]{} on finite, undirected graph $G$ as follows. $\bullet$ Alice and Bob properly color the vertices of a graph G with a fixed set of $k$ colors (the colors are assumed to be integers in the range $1, \ldots, k$).\ $\bullet$ Alice and Bob take turns, choosing and coloring (or re-coloring) properly one vertex at a time, with Alice going first:\ $\bullet$ if a vertex $v$ is uncolored and is chosen by Alice or Bob, then it must be properly colored\ $\bullet$ if a vertex $v$ is colored color $c$ and is chosen by Alice or Bob, then it must be assigned a color other than $c$ which is not used on one of its neighbors (this new color is chosen by the player whose turn it is)\ $\bullet$ all vertices must be chosen $t$ times prior to any vertex being chosen $t+1$ times, for every integer $t \geq 1$\ $\bullet$ If a vertex $v$ is chosen and is impossible to color properly (that is, for every color in the color set other than its current color, $v$ has a neighbor colored with that color), then Bob wins.\ $\bullet$ If Bob cannot win the game, then Alice wins.\ In other words, the colorings take place over a series of [*rounds*]{}: each vertex is colored, or re-colored, once during each round. The number of rounds is infinite if Alice wins. We note that if $n$ is odd, the second round, and all even numbered rounds, have Bob going first. We shall sometimes refer to this as the A-game. Let $\chi_g^\infty(G)$ be the minimum number of colors needed in the initial color set to guarantee Alice to win the eternal vertex coloring game on $G$. We call $\chi_g^\infty$ the [*eternal game chromatic number*]{}. As an example, trivially $\chi_g^\infty(K_2)=3$. Not so trivially, $\chi_g^\infty(P_3)=3$. To see this, let $P_3=uvw$. Alice colors $v$ red, Bob colors $u$ blue, and Alice colors $w$ blue. There are now two cases for Bob’s second turn: Case 1) If Bob colors $v$ green, then Alice colors $w$ red, Bob colors $u$ red, and we are back to a 2-coloring of $G$ and we can start the process over with Alice moving next.\ Case 2) If Bob colors $u$ green, then Alice colors $w$ green and Bob colors $v$ blue. Again, we are back to a 2-coloring of $G$ and we can start the process over with Alice moving next. $\Box$\ Of course, for all $G$, $\chi_g(G) \leq \chi_g^\infty(G)$. \[game-conj\]For every graph $G$, $\chi_g(G) < \chi_g^\infty(G)$. One might suspect that $\chi^\infty(G) \leq \chi_g^\infty(G)$ for all $G$. However this is not always the case. From [@WFK] we know that $\chi^\infty(K_{1, 7}) = 4$, but below we show that $\chi_g^\infty(K_{1, 7}) > 4$. On the other hand, $\chi^\infty(P_3) = 4$ and $\chi_g^\infty(P_3) =3$. Obviously $\chi^\infty(K_n) = \chi_g^\infty(K_n)=n+1$. Similarly, $\chi^\infty(K_n-e) = \chi_g^\infty(K_n-e)=n+1$. We sometimes call the previous game the A-game, since Alice starts. We can also define the following variations on the A-game:\ $\bullet$ the B-game: same as the A-game except that Bob goes first.\ $\bullet$ the A’-game: the same as the A-game except that Alice goes first in each round (so Alice may have two consecutive turns in the case $n$ is odd).\ $\bullet$ the B’-game: same as the A’-game except that Bob goes first in each round.\ Note that $P_3$ is an example in which the A-game and B’-game differ, since $\chi_g^\infty(P_3)=3$, but four colors are needed if Bob goes first. To see the latter, let $P_3=uvw$ and suppose Alice can win with three colors. Bob colors $v$ red, Alice colors $u$ green, and Bob colors $w$ blue. Bob then attempts to re-color $v$ and but cannot properly color it with the available colors, resulting in a win for Alice. In fact, $P_3$ is an example in which the A-game and B-game differ. Let $P_3=uvw$ and suppose Alice can win the B-game with three colors. Bob colors $v$ red, Alice colors $u$ green, and Bob colors $w$ blue. Alice must now color $w$ green (same as coloring $u$ blue) else she loses the game. Bob now colors $u$ blue, and Alice loses the game on her next turn. However in general, we suspect there is little difference – perhaps one additional color in some cases – in the various games defined in this section. In the next section, we consider the A-game. In Section \[alt\] we consider variations on the game and in Section \[quest\], we propose two more. Results ======= \[thm1\] Let $G$ be a graph with maximum degree $\Delta(G)$. Then $\chi_g^\infty(G) \leq \Delta(G)+2$. [*Proof:*]{} The standard proof technique showing that $\chi(G) \leq \Delta(G)+1$ can be applied to show that $\chi_g^\infty(G) \leq \Delta(G)+2$. $\Box$ \[path\] Let $P_n$ be a path with $n \geq 4$. Then $\chi_g^\infty(P_n) =4$. [*Proof:*]{} That four colors suffice follows from Theorem \[thm1\]. To see that four colors are necessary, suppose to the contrary we can color $P_4$ with three colors; the argument is similar when $n > 4$ (and, in fact, easier in the odd cases). Let $P_4 = uvwx$. There are two cases depending on which vertex Alice colors first in round 1 (up to symmetry). If she colors $u$ with, say, color 1, then Bob colors $w$ with color 2, Alice colors $x$ with color 1 (or color 3, but color 1 is a better choice), and Bob colors $v$ with color 2. On the other hand, if Alice colors $v$ with color 1 first, then Bob colors $x$ with color 2, then Alice colors $u$ with color 3 (or 2, it doesn’t matter), and Bob color $w$ with color 3. In either case, there is a vertex with all three colors in its closed neighborhood. Let us suppose that vertex is $w$, Alice must then change the color of a neighbor of $w$. But then Bob changes the color of the other vertex in $w$’s neighborhood. This then will force Bob to win when $w$ is re-colored. $\Box$ Any cycle is an example with $\chi_g^\infty(G) = \Delta(G)+2$. One can verify that $\chi_g^\infty(P_n)=4$ when $n > 3$. For comparison sake, it is known that for every graph $G$, $\chi (G)\leq \chi _{g}(G)\leq \Delta (G)+1$, see [@faig]. We ask whether or not it is always true that for any cubic graph $G$, $\chi_g^\infty(G)=5$? The graph $K_{1, n}$ is called a [*star*]{}. When $n > 1$, the unique vertex of degree greater than one is called the *center*. We note that $\chi^\infty(K_{1, n})=4$ when $n>2$ (see [@WFK]) and $\chi_g(K_{1,n})=2$ when $n \geq 1$ (see [@des]). The eternal game chromatic number of stars, however, is quite different, as we show next. Recall that above we showed that $\chi_g^\infty(K_{1, 2})=3$. \[lb\] When $n$ is odd $\chi_g^\infty(K_{1, n}) \geq \lceil \frac{n}{2} \rceil +2$. When $n \geq 4$ is even $\chi_g^\infty(K_{1, n}) \geq \frac{n}{2} +3$. [*Proof:*]{} We first explain the case when $n$ is even. Bob’s strategy in round 1 is to color at least $n/2$ of the leaves using distinct colors for each – or as many distinct colors as possible depending on the number of available colors (note that Alice may try to use one or more of these colors on other leaves). He then will force an additional color by choosing the center vertex on the first turn in round 2. It is easy to see that Bob will win the game unless at least $\frac{n}{2} +2$ colors are available. We can do one better than that by a more careful analysis which we now perform which will show that $\frac{n}{2} +3$ colors are needed. Suppose to the contrary that only $\frac{n}{2} +2$ colors are available. There are two cases. [**[Case 1.]{}**]{} Suppose during round 1, Alice does not color the center vertex. Then Bob and Alice use $\lceil \frac{n}{2} \rceil+ 1$ colors on the leaves, a different color on the center vertex, and then on Bob’s first move during round 2, he picks the center vertex and forces an additional color.\ [**[Case 2.]{}**]{} Suppose during round 1, Alice does color the center vertex (and we can assume without loss of generality that she does so on her first move). If Alice colors all leaves the same color, at the end of round 1, at most $\lceil \frac{n}{2} \rceil+ 1$ colors are used. Bob starts round 2 and plays a new color on the center vertex. He can then ensure that all $\lceil \frac{n}{2} \rceil+ 2$ are used on vertices by the end of round 2. In round 3, Alice cannot play the center vertex first, else an extra color is needed, so she plays a leaf, changing it from $c$ to $c'$. Bob then executes a “mirroring" strategy, changing a leaf with color $c'$ to $c$. At the end of the round, all $\lceil \frac{n}{2} \rceil+ 2$ are used on vertices. Bob then forces a new color on the first move of round 4. Now suppose $n$ is odd and thus Alice will go first on each round. We are claiming in this case that $\chi_g^\infty(K_{1, n}) \geq \lceil \frac{n}{2} \rceil +2$. Suppose to the contrary that only $\lceil \frac{n}{2} \rceil+ 1$ colors are available. Bob will again try to use as many different colors as possible on the leaves. There are two cases. [**[Case 1.]{}**]{} Suppose Bob colors the leaves with $\frac{n}{2} $ distinct colors in round 1 (which occurs if Alice colors the central vertex in round 1). Then the total number of colors used in round 1 is $\frac{n}{2} +1$. If Alice colors the center vertex on the first move in round 2, then an additional color is needed and we are done. So suppose Alice colors a leaf on her first move in round 2. Whatever move she makes, Bob mirrors it on his next move (e.g., if she re-colors a vertex from $c$ to $c'$, Bob will re-color a vertex from $c'$ to $c$.). Eventually, Bob can re-color the center vertex on his last turn in round 2, forcing the use of an additional color. [**[Case 2.]{}**]{} Suppose Bob colors the leaves with $\frac{n}{2} -1$ distinct colors in round 1 (which occurs if Alice does not color the central vertex in round 1 and she uses at least one of the same colors as Bob on one or more leaves). Then each of these colors Bob uses is distinct from the color Alice uses on the first leaf. Bob then uses color $\frac{n}{2} +1$ on the last move of round 1. The remainder proceeds as in Case 1. $\Box$ We next show the lower bound from the previous result is tight. \[star-lower\] When $n$ is odd $\chi_g^\infty(K_{1, n}) \leq \lceil \frac{n}{2} \rceil +2$. When $n \geq 4$ is even $\chi_g^\infty(K_{1, n}) \leq \frac{n}{2} +3$. [*Proof:*]{} We assume $n > 2$ as the result is easy to see when $n \leq 2$. Let $c(v)$ denote the color assigned to $v$ at any given time (initially null) and in each case we shall assume the number of colors specified in the statement of the theorem are available. First suppose $n$ is odd, which means Alice goes first on each round. Suppose without loss of generality that the coloring of the graph done in round 1 uses $\lceil \frac{n}{2} \rceil+1$ colors, because Alice’s strategy is to color leaves with color 1 on each of her turns. Thus, $c(v) = 1$ for $\lceil \frac{n}{2} \rceil$ of the leaves. We may assume that Bob uses a distinct color other than 1 on each of his turns on each of the remaining vertices. Alice’s strategy in the next round is to change any vertex not colored 1 to 1, and then Bob will do one of two things on his first turn of round 2. [**[Case 1)]{}**]{} If Bob decides to choose the center vertex, he must choose a $\lceil \frac{n}{2} \rceil+2^{nd}$ color. Then, Alice chooses another vertex $v$ with $c(v) \neq 1$ and re-colors it to 1, and Bob (in order to maximize the number of colors used) either (a) mirrors her actions by choosing a vertex $v$ with $c(v) = 1$ and changing it to the color (which is not equal to 1) that was the previous color on the vertex Alice chose or (b) he could also change it to the previous color of the center vertex. He continues to ensure that he does not use the same color twice during the round. When Alice runs out of vertices that are not colored 1, she chooses one of the remaining two vertices (both of color 1) to change to any of the colors Bob used on another leaf during round 2. On Bob’s final turn of the round, there are two colors of the $\lceil \frac{n}{2} \rceil + 2$ used thus far that have not yet been used during round 2. This is because Bob had one fewer turn on the leaves than Alice (so he could not counteract her actions each time), and there is already one more color in the set than necessary for the initial coloring. Thus, Bob picks one of the remaining colors, and the other is free for the same process to repeat in the following round without needing to force the use of any more colors. Therefore, $\lceil \frac{n}{2} \rceil + 2$ colors suffice.\ [**[Case 2)]{}**]{} If Bob selects a leaf to re-color, then in order to maximize the number of colors used on the leaves, he will use a mirroring strategy to counteract Alice’s turns (i.e, if Alice re-colors a vertex from $c$ to $c'$ , he will do the opposite). This can continue on the leaves until one remains of color 1 and it is Alice’s turn. She will pick a color which Bob used to re-color another leaf. Bob then chooses the center vertex and uses a $\lceil \frac{n}{2} \rceil+2^{nd}$ color. This leaves the initial color of the center vertex free for the same process to repeat on the next round. $\lceil \frac{n}{2} \rceil + 2$ colors suffice.\ Now suppose $n$ is even, which means Alice goes first on each odd-numbered round. Again, Alice’s general strategy is to color as many leaves as possible with the same color. Since she can color at least $\frac{n}{2}$ of them the same color, at most $\frac{n}{2} +1$ different colors are used on the leaves at the end of any given round, plus possibly an additional color for the center vertex. Since the center vertex needs to change colors in each round, we claim that $\frac{n}{2} +3$ colors suffice. To see this, note that at some point in a round in which Bob goes first, Bob can change the color of a leaf, thus (temporarily) there may be $\frac{n}{2} +2$ different colors on leaves. However, on Alice’s next move, she can choose a leaf to re-color that reduces the number of different colors on the leaves to $\frac{n}{2} +1$. $\Box$ It is easy to verify that $\chi_g^\infty(K_{2, 2})=4$.\ It is known that $\chi_g(K_{n, n})=3$ when $n$ and $m$ are both larger than 1, see [@des]. \[comp\] $\chi_g^\infty(K_{n, n}) \leq 5$ when $n \geq 5$. [*Proof:*]{} Let the vertex parts of $K_{n, n}$ be $A=\{a_1, \ldots, a_n\}$ and $B=\{b_1, \ldots, b_n\}$. Alice’s strategy is to ensure that at the end of each round, there are at least two distinct colors used on $A$ and at least two distinct colors used on $B$. This then allows vertices in each part to switch back and forth between those two colors each time they are re-colored. This is easily seen to be possible when $n \geq 5$. We note that in total, five colors are needed, since Bob can force a third color onto either $A$ or $B$ whilst Alice is trying to ensure there are at least two colors on both $A$ and $B$. $\Box$ This demonstrates that $\chi_g^\infty$ does not have hereditary properties, since $K_{1, n}$ is an induced subgraph of $K_{n, n}$. $\chi_g^\infty(K_{n, n}) \geq 4$ when $n \geq 3$. [*Proof*]{}: Suppose to the contrary that three colors suffice. Let the vertex parts be $A=\{a_1, \ldots, a_n\}$ and $B=\{b_1, \ldots, b_n\}$. Observe that Bob can ensure that at the end of round 1, vertices in $A$ use two distinct colors – colors 1 and 2 – and each vertex in $B$ is color 3 (without loss of generality). On Alice’s first move in round 2, she cannot re-color any vertex in $B$, as this would require a fourth color. If she re-colors a vertex in $A$ so that there are still two distinct colors in $A$, Bob will be unable to move on his turn if he chooses a vertex in $B$. So suppose Alice re-colors, say, $a_1$ so that all the vertices in $A$ are the same color, say color 2. Bob now re-colors a vertex in $B$ with color 1, so that $B$ now contains vertices of two distinct colors. Alice now needs a fourth color if she chooses a vertex in $A$, so she chooses a vertex in $B$. Bob now chooses a vertex in $A$ and needs a fourth color, winning the game. We note that if $n \geq 4$, Bob can ensure that at the end of round 1, vertices in $A$ use two distinct colors 1 and 2 and vertices in $B$ use two distinct colors. Therefore, Bob can win the game more quickly in round 2. We discuss this issue further in the last section of the paper. $\Box$ $\chi_g^\infty(K_{n, n}) \geq 5$ when $n \geq 5$. [*Proof*]{}: Suppose to the contrary that four colors suffice. Let the vertex parts be $A=\{a_1, \ldots, a_n\}$ and $B=\{b_1, \ldots, b_n\}$. Let the color set be $1, 2, 3, 4$ and let $c(v)$ denote the color assigned to $v$ at any given time (initially null). First suppose that at the end of round 1, there is only one color used on $A$, say color 1. In this case, Bob can ensure that at least three colors are used on $B$. If Alice chooses a vertex in $A$ for her first move in round 2, a fifth color must be used. Hence she must choose a vertex in $B$. In other to avoid a fifth color being played on $A$ on Bob’s first move in round 2, Alice must change the color of a vertex $v \in B$ such that $v$ is the only vertex in $B$ with that color, say color 4, which she changes to 3. Note that Bob can have assured that there are at least two vertices in $B$ of color 2 and at least two vertices of color 3 at the end of round 1. Bob now changes a vertex in $A$ to color 4, and Alice must use a fifth color on her next turn. We now claim that in fact Bob can force there to be only one color on $B$. Assume without loss of generality that Alice colors $a_1$ first with color 1. Then Bob chooses $a_2$ and colors it 2. If Alice chooses $b_1$ and colors it 3, and then Bob colors $a_3$ with color 4, then all vertices in $B$ will have to be colored with 3 in round 1. If, on the other hand, Alice choose $a_3$ and colors it 1 or 2 (note that if she colored it with color 3, then Bob would color $a_4$ with 4 and there would be no colors available for $B$), then Bob colors $a_4$ with 4, again forcing all of $B$ to be the same color. $\Box$ \[k44\] $\chi_g^\infty(K_{4, 4}) = 4$. [*Proof*]{}: We show that $\chi_g^\infty(K_{4, 4}) \leq 4$. If Alice can force there to be at least two distinct colors used on $A$ and two distinct colors used on $B$ at the end of round 1, she can successfully proceed with 4 colors, as described above. Suppose this is not the case and that only 1 color, say color 1, is used on $A$ during round 1. In order for Bob to do this, Alice can ensure that only colors 2 and 3 are used on $B$ during round 1. On her first move in round 2, Alice re-colors a vertex in $A$ with color 4. This then forces the vertices in $B$ to swap between colors 2 and 4 and color 4 to be used on the remainder of $A$. The process can be repeated in subsequent rounds. $\Box$ $\chi_g^\infty(K_{3, 3}) = 4$. [*Proof*]{}: Similar to Proposition \[k44\]. $\Box$ Let $G$ be a connected graph other than $K_1, K_2$, or $P_3$. Then $\chi_g^\infty(G) > 3$. [*Proof:*]{} We can assume $G$ has at least four vertices, since $\chi_g^\infty(K_3) > 3$. We also assume that $\Delta(G) > 2$, else $G$ is a cycle and thus four colors are needed. Of course, $G$ may be a tree, but not necessarily so. Let $v$ be a vertex of degree at least three, with $x, y, z$ three of the neighbors of $v$. Suppose to the contrary that 3 colors suffice. Our general strategy is as follows. Assume without loss of generality that $c(v)=1$. Bob can force at least two distinct colors, other than color 1, be used on $\{x, y, z\}$ in round 1. If Bob moves first in round 2, he can force a fourth color by choosing $v$. Otherwise, Bob can still force that, at any time, it is either Alice’s turn and there are two distinct colors on $\{x, y, z\}$ or it is his turn and there are either (i) two distinct colors on $\{x, y, z\}$ (ii) there is only one distinct color on $\{x, y, z\}$ but at least one of these three vertices has yet to be re-colored. Hence a fourth color will be needed unless there is a vertex $w \neq v$ that is also adjacent to each of $x, y, z$ and Alice manages to color $w$ with color 2, whilst $c(v)=1$. This then forces $x, y, z$ to all have color 3. If Bob gets to color one of these five vertices first in round 2, obviously he forces a fourth color. So assume Alice colors one of these first. Then it must be that she re-colors $v$ with color 2 (or equivalently $w$ with color 1). But then Bob can re-color $w$ with color 1 (equivalently $w$ with color 2) and then a fourth color will be forced on one of $x, y, z$. $\Box$ \(a) $\chi_g^\infty(K_{2, n}) = 4$ if $n \in \{2, 3\}$.\ (b) $\chi_g^\infty(K_{2, n}) \geq n/2 + 3$ if $n > 3$ is even.\ (c) $\chi_g^\infty(K_{2, n}) \geq \lfloor \frac{n}{2} \rfloor + 3$ if $n > 3$ is odd. [*Proof*]{}: Part (a) is not difficult to verify by hand. For parts (b) and (c), Let the vertex parts be $A=\{a_1, a_2\}$ and $B=\{b_1, \ldots, b_n\}$. Let us first suppose $n$ is even. Bob’s basic strategy in both (b) and (c) will be to make sure both vertices in $A$ have the same color and then use as many different colors on $B$ as possible (which will be $n/2 + 1$). For part (b), observe that Bob can ensure both vertices in $A$ have the same color after round 1 (and, in fact, after all subsequent rounds). Therefore, Bob can ensure $n/2 + 2$ colors are used in round 1 in total ($n/2 + 1$ on $B$ and one on $A$). An additional color will be needed on $A$ in round 2, regardless of when a vertex in $A$ in chosen (since if Alice chooses to play a vertex $v \in B$ first in round 2, she must change its color to that of another vertex $u \in B$; then on Bob’s turn he can change $u$’s color to the original color of $v$). For part (c), the argument is similar to (b), except that Bob cannot necessarily force both vertices in $A$ to be the same color after each round (otherwise, Bob can force $\lfloor \frac{n}{2} \rfloor + 2$ colors be used in round 1 and an additional color in round 2). So suppose the vertices in $A$ have two different colors at the end of some round. One can easily verify in this case that Bob can ensure that at least $\lfloor \frac{n}{2} \rfloor + 1$ colors are used on $B$. $\Box$ Alternate Games {#alt} =============== An alternate version of game chromatic number called [*game chromatic number II*]{} was introduced in [@chen]. In this version, Bob can only use the colors that have been already used on the graph, unless he is forced to use a new color to guarantee that the graph is colored properly. The number of colors needed for $G$ in this game is denoted by $\chi_g^{*}(G)$. It was shown in [@chen] that $\chi_g^{*}(T) \leq 3$ for any tree $T$, as compared to the bound of 4 for $\chi_g(T)$, see [@faig]. We define the [*eternal vertex coloring game II*]{} in the obvious manner and use $\chi_g^{\infty*}(G)$ to denote the number of colors needed when this game is played eternally on $G$: that is, the game is played over a series of rounds and in each round Bob can only use colors that have already been introduced unless he is forced to use a new color. Of course, $\chi_g^{\infty*}(G) \leq \chi_g^{\infty}(G)$. $K_3$ is an example where $\chi_g^{\infty*}(G) < 2\chi_g^*(G)$ and $P_4$ is an example where $\chi_g^{\infty*}(P_4) = 4 = 2\chi_g^*(P_4)$. We shall consider stars again and begin by noting that $\chi_g^{\infty*}(K_{1, 3})=3$ and $\chi_g^{\infty*}(K_{1, 4})=4$. Interestingly, we shall see in next result that it may take Bob several rounds to force the number of colors needed. The next result is also a good example of the power Bob gets when the number of vertices is odd. Then (a) if $n$ is odd then $\chi_g^{\infty*}(K_{1, n})=3$ and (b) if $n \geq 4$ is even then $\chi_g^{\infty*}(K_{1, n}) = \frac{n}{2}+2$. \[Note: does this proof need work?\] [*Proof:*]{} Part (a) is easy to see, noting that Alice can choose the center vertex of the star on the first turn in each round, coloring it with the smallest color possible. For part(b), note that two colors are used in round 1. Bob can then force colors 3 and 4 to be used during round 2: he does so by choosing a leaf on his first turn and the center vertex on his second turn. Subsequent to that, for the next several rounds, Bob can force an additional color to be used on either each round or every other round by coloring the leaves with as many different colors as possible during each round, eventually forcing the center vertex to use a new color – it may need to wait until a round in which Bob goes first for each new color to be introduced (as in round 2). Bob can eventually force the use of $\frac{n}{2}$ different colors on the leaves, once that many colors have brought into play. At which point the game is the same as the A-game and thus the result follows as in Theorem \[lb\]. $\Box$ One could also consider a more restrictive version of this game in which Bob must color whatever vertex he chooses with the smallest color possible (as opposed to being able to use any color, once a color becomes allowable). We call this the [*Greedy Coloring Game*]{}. Denote the number of colors needed for this game as $\chi^2_g(G) $ when only one round is played and $\chi_g^{\infty2}(G)$ when the game is played eternally. It is then natural to ask if there exist any graphs where this restriction changes the number of colors needed versus $\chi_g^{\infty*}(G)$. The answer is “yes", as we show next. $\chi_g^{\infty2}(K_{1, n})=3$ when $n$ is odd and $\chi_g^{\infty2}(K_{1, n})=4$ when $n \geq 4$ is even. [*Proof:*]{} First consider $K_{1, n}$ when $n > 3$ is odd and thus Alice goes first in each round. In this case, only 3 colors are needed. Alice chooses the center vertex of the star first in each round, giving it color 1 in odd numbered rounds and all other vertices getting color 2) and color 3 in even numbered rounds (and all other vertices getting color 1). When $n \geq 4$ is even, we show $K_{1, n}=4$. That fours colors are necessary is easy to see, so we show that four colors suffice. On round 1, Alice plays color 1 on the center and then color 2 gets played on all the leaves. In round 2, Bob goes first. If he plays the center, he will use color 3 and then color 1 gets played on all the leaves (which is no advantage to Bob). So assume Bob plays a 3 on a leaf. Then Alice plays 1 on a leaf, Bob plays 4 on the center, and 1’s gets played on the remaining leaves. Alice, starting the next round, re-colors the lead that is color 3 with color 2. Then either remaining leaves get color 2 and the center gets color 1 (which puts us back in a previous configuration), or Bob plays 3 on the center on some turn in this round and the remainder of the leaves get color 1. However, this coloring completes the round with only two different colors, 1 and 3, on the vertices, which is essentially the same at this point as the configuration in which only 1 and 2 are used. $\Box$ For every graph $G$, $\chi^2_g(G) < \chi_g^{\infty2}(G)$. One could further restrict the rules so as to force Alice to also choose smallest color for each vertex she chooses. We call this the [*Very Greedy Coloring Game*]{}. Denote the number of colors needed for this game as $\chi^3_g(G)$ when only one round is played and $\chi_g^{\infty3 }(G)$ when the game is played eternally. Recall that the [*coloring number*]{} of $G$, denoted $col(G)$, is the smallest integer $k$ such that every subgraph of $G$ has a vertex of degree less than $k$. Though intuitively, there seems to be some relationship between $col(G)$ and $\chi^3_g(G)$, in fact they are not the same: $col(K_{n, n})=n+1$ and $\chi^3_g(K_{n, n})=2$; whereas $col(P_4)=2$ and $\chi^3_g(P_4)=3$. It is clear that $\chi_g^{\infty2}(G) \leq \chi_g^{\infty3}(G)$ and $\chi_g^{\infty2}(G) \leq \chi_g^{\infty*}(G) \leq \chi_g^{\infty}(G)$ for all $G$. Is it true that $\chi_g^{\infty3}(G) \leq \chi_g^{\infty*}(G)$ for all graphs $G$? When $n$ is odd, $\chi_g^{\infty3 }(K_{1, n})=3$, when $n \geq 4$ is even, $\chi_g^{\infty3 }(K_{1, n})=4$. [*Proof:*]{} The case when $n$ is odd is trivial. So suppose $n \geq 4$ is even. It is easy to see that $\chi_g^{\infty3 }(K_{1, n}) \geq 4$, as Bob can force color 4 to be used during round 2. In order to see that four colors suffice, the key observation is that Alice can maintain the invariant that, at the end of each round and prior to the re-coloring of the center vertex during each round, the leaves are colored with at most two distinct colors. We strengthen this invariant as follows: (a) no leaf is ever color 4 and (b) if the center is color 1 after a round and the leaves have two distinct colors, then the color that appears on fewer leaves appears at most twice. Note that in order for Bob to force a fifth color, it would have to be the case that there are three distinct colors on the leaves. Of course, after the center vertex is re-colored during a round, it may be possible that three distinct colors are used on the leaves for a short time, until the end of the round at which point we claim that our invariant will be true. We now prove that the invariant can be maintained, from which the theorem follows. It is trivially true during and after round 1: Alice ensures this by coloring the center first and then all the leaves must get color 2. Now let us move past round 1. If all the leaves are the same color after some round, then they all get changed to the same color during the next round until the center vertex is re-colored, at which point the remaining leaves may get changed to a second color – however, by Alice choosing the center vertex as soon as possible, our invariant is assured. We note that if there are two distinct colors on the leaves at the end of the round in this case, one of these colors on the leaves must be color 1. Now suppose the leaves are colored with two colors after some round. It is easy to see that we never use color 4 on a leaf: since when assigning the color to a leaf, one only needs to consider its current color and the color of the center; hence one of colors 1, 2, 3, will be available to assign to each leaf. First suppose the center vertex is some color $c > 1$ and some leaf is color 1. If the center is re-colored first (which is Alice’s preference if she goes first), it cannot get color 1 (since some of the leaves have color 1), which means that each leaf will be re-colored color 1. If the center vertex is not re-colored first (meaning Bob goes first), then the leaves that are re-colored prior to the center getting re-colored get color 1, the center is re-colored some color other than 1, and the remaining leaves get color 1. Thus the invariant is maintained. Now suppose the center vertex is some color $c > 1$ and no leaf is color 1. Then it must be the case that either all the leaves have the same color, which we addressed earlier, or else some leaf is color 4, which we know cannot happen. Next suppose the center vertex is color 1. If the center is re-colored first (which is Alice’s preference if she goes first), it gets some color which is either 2, 3, 4 (since the leaves use at most two different colors). Therefore the leaves all get color 1 during this round. If the center is not re-colored first (meaning Bob goes first), there are two cases.\ [**[Case 1.]{}**]{} Suppose Bob first chooses a color that appears on at most two leaves. Then Alice simply “mirrors” Bob’s moves during the round, choosing another leaf of the same color Bob just chose. This guarantees the invariant is maintained.\ [**[Case 2.]{}**]{} Suppose Bob first chooses a color that appears on more than two leaves. Note that this may create a situation where there are three distinct colors on the leaves – otherwise Bob has changed the color of a leaf to the color that exists on another leaf and Alice can simply chooses the center vertex on her next move and we will be guaranteed the invariant is maintained. (Or alternatively, Alice can essentially duplicate what Bob just did, choosing a leaf of the same color Bob just chose and this will also allow the invariant to be preserved). So suppose we now have three colors on the leaves – the only way this occurs is if the leaves were colored with colors 2 and 4 before the round started. Since no leaves are color, 4, we are done. $\Box$ Can we differentiate between $\chi_g^{\infty2}(G)$ and $\chi_g^{\infty3 }(G)$? That is, does there exist a graph $G$ such that $\chi_g^{\infty2}(G) \neq \chi_g^{\infty3}(G)$? We observe that $\chi^3_g(T) \leq 3$ for any tree $T$, as the proof that $\chi_g^{*}(T) \leq 3$ applies to $\chi^3_g(T)$ (and thus also for $\chi^2_g(T)$). Is there an integer $c$ such that for all trees $T$ with $\chi_g^{\infty3}(T) \leq c$? Likewise $\chi_g^{\infty2}(T)$. Caterpillars are one class of trees that are easy to analyze, as it is not hard to show that $\chi_g^{\infty3}(T) \leq 6$ for any caterpillar $T$. \(a) For every graph $G$, $\chi^3_g(G) < \chi_g^{\infty3}(G)$. (b) For every graph $G$, $\chi^2_g(G) < \chi_g^{\infty2}(G)$. (c) For every graph $G$, $\chi^*_g(G) < \chi_g^{\infty*}(G)$. Characterize for each of these models discussed in this section the graphs needing three or four colors. Let $G$ be a graph with subgraph, or induced subgraph, $H$. Is it necessarily true that $\chi_g^{\infty2}(H) \leq \chi_g^{\infty2}(G)$? Is it necessarily true that $\chi_g^{\infty3}(H) \leq \chi_g^{\infty3}(G)$ ? Further Questions {#quest} ================= What is the computational complexity of each of the eternal coloring games? Characterize the connected graphs (or the trees) with $\chi_g^\infty(G) = 4$. In seems an interesting future direction to consider $\chi_g^{\infty2}$, $\chi_g^{\infty3}$, and $\chi_g^{\infty}$ as well as $\chi_g^{2}$, $\chi_g^{3}$, and $\chi_g^{*}$ for various classes of graphs. For example, it is easy to show that for a grid graph $G$ with a sufficiently large number of rows and columns, $5 \leq \chi_g^\infty(G) \leq 6$. Determine a tight bound on $\chi_g^{\infty2}$, $\chi_g^{\infty3}$, and $\chi_g^{\infty}$ for large grid graphs. We conjecture at most five colors suffice for the greedy game and probably also for the very greedy game. Is it true that if $\chi_g^\infty(G)=k$ then Alice can win the A-game on $G$ with a color set of $k+1$ colors? In the following question, we note the proof of Theorem \[lb\], in which it appears to take Bob four rounds to win on a star. Suppose the A-game is played on some graph $G$ with $k < \chi_g^\infty(G)$ colors and Bob wins the game. Is it necessarily the case that Bob can win the game prior to the beginning of round 3? What is the maximum number of rounds it takes Bob to win to any graph? Are there any graphs $G$ with $\chi^\infty(G) = \chi_g^\infty(G)$ and $\chi^\infty(G) < \Delta(G)+2$? Can we characterize the graphs with $\chi(G) + 1 = \chi_g^\infty(G)$? Likewise for the other graph coloring games introduced in this paper. We raise an issue regarding the definition of the eternal vertex coloring game (and some of its variants). Suppose Bob chooses vertex $v$ to re-color (in round 2 or later) and Bob has no legal move on this vertex, however Bob does have a legal move on vertex $u \neq v$. As we have defined the game now, Bob wins. But should we force Bob to choose some other vertex, such as $u$, where he has a legal move? It may be worth exploring this version of the game. One way to do this is to consider the alternate game discussed in Section \[alt\]. Two obvious other variations come to mind: $\bullet$ (Strong Eternal Game Coloring) We require that Bob must choose a vertex on his turn that can be re-colored properly, if one exists (as opposed to being able to choose any vertex); or $\bullet$ (Ordered Eternal Game Coloring) The vertices must be chosen in the same order (pre-determined or fixed after round 1) each round. We know that $\chi^\infty(C_5)=\chi_g^\infty(C_5)=4$. However, if one follows the Strong Eternal Game Coloring rules, three colors suffice. To see this, a 3-coloring results from round 1. On the first move in round 2, Bob must choose a vertex that can be re-colored (a vertex adjacent to two vertices of the same color), as opposed to a vertex adjacent to two different colors (which would have necessitated the use of a fourth color). It is not hard to see that round 2 can be successfully completed with three colors and the subsequent rounds can continue in a similar manner. [99]{} H. Bodlaender, On the complexity of some coloring games. [*Lecture Notes in Computer Science*]{}  **484** (1991) 30–40. G. Chen, R. Schelp, and W. Shreve, A new game chromatic number, [*Europ. J. Combinatorics*]{} **18** (1997), 1–9. C. Destacamento, A. Rodriguez, and L. Aquino-Ruivivar, The game chromatic number of some classes of graphs, De La Salle University, March 2014. U. Faigle, W. Kern, H. Kierstead, W. Trotter On the game chromatic number of some classes of graphs. [*Ars Combinatoria*]{} **35** (1993), 143–150. M. Gardner, Mathematical Games, [*Scientific American*]{} **23** (1981). W. Klostermeyer, Eternal Graph Coloring. [*Bulletin of the Inst. for Combinatorics and its Applications*]{} **70** (2014), 69-76.

--- abstract: 'In this paper we study the ratio between the number of $p$-elements and the order of a Sylow $p$-subgroup of a finite group $G$. As well known, this ratio is a positive integer and we conjecture that, for every group $G$, it is at least the $(1-\frac{1}{p})$-th power of the number of Sylow $p$-subgroups of $G$. We prove this conjecture if $G$ is $p$-solvable. Moreover, we prove that the conjecture is true in its generality if a somewhat similar condition holds for every almost simple group.' address: 'Dipartimento di Matematica e Informatica “U. Dini”,Università degli Studi di Firenze, viale Morgagni 67/a, 50134 Firenze, Italy.' author: - Pietro Gheri title: 'On the number of $p$-elements in a finite group' --- Introduction ============ Let $G$ be a finite group and $p$ be a prime dividing the order of $G$. Moreover, let $$\mathfrak{U}_p(G) = \bigcup_{P \in Syl_p(G)} P,$$ be the set of $p$-elements of $G$. A celebrated theorem of F.G. Frobenius ([@frobenius:sylow]) states that if $P$ is a Sylow $p$-subgroup of $G$, then $|P|$ divides $|\mathfrak{U}_p(G)|$. We will call the positive integer $|\mathfrak{U}_p(G)|/|P|$ the *$p$-Frobenius ratio of $G$*. The number of $p$-elements of a finite group is a fundamental invariant in finite group theory. Several different proofs of Frobenius’ theorem have been given (see, for example, [@isaacs:frobenius] and [@speyer:frobenius]). Moreover in [@steinberg:endom Theorem 15.2] it is proven that the $p$-Frobenius ratio in a finite group of Lie type is equal to the size of a Sylow $p$-subgroup. Nevertheless, it is still unknown if the Frobenius ratio has a combinatorial meaning. It is clear that the $p$-Frobenius ratio is $1$ if and only if $G$ contains a normal Sylow $p$-subgroup. In [@miller:theoryapplications], it is proven with a nice and easy argument that if the $p$-Frobenius ratio is not $1$, then it must be greater or equal than $p$. In this paper, we focus on the search for “good” bounds for the $p$-Frobenius ratio in terms of the number $n_p(G)$ of Sylow $p$-subgroups of $G$. Of course, a trivial upper bound is obtained when every pair of Sylow p-subgroups of G has trivial intersection, so that, given a Sylow $p$-subgroup $P$ of $G$, $$\frac{|\mathfrak{U}_p(G)|}{|P|} \leq n_p(G)- \frac{n_p(G)-1}{|P|} \leq n_p(G).$$ It is not hard to find examples of sequences of groups that show that a lower bound on the $p$-Frobenius ratio cannot be linear in $n_p(G)$. We state the following conjecture. Let $G$ be a finite group, $p$ be a prime dividing $|G|$ and $P$ a Sylow $p$-subgroup of $G$. Then $$\label{conj pfrobratio} \frac{|\mathfrak{U}_p(G)|}{|P|} \geq n_p(G)^{1-\frac{1}{p}}.$$ We will show in Example \[tight frob ratio\] that this bound is “asymptotically tight”. We show that Conjecture \[conj pfrobratio\] is true for $p$-solvable groups. Namely, we prove the following. \[bound on Omega psolv\] Let $G$ be a finite $p$-solvable group and $P$ be a Sylow $p$-subgroup of $G$. If $n_p(G)$ denotes the number of Sylow $p$-subgroups in $G$, then $$\label{bound p el} \frac{|\mathfrak{U}_p(G)|}{|P|} \geq n_p(G)^{\frac{p-1}{p}}.$$ Inspired by the proof of Theorem \[bound on Omega psolv\], we show that a sufficient condition for Conjecture \[conj pfrobratio\] to be true in general is that $$\label{conj lambda} \left( \prod_{x \in P} \lambda_G(x) \right)^{1/|P|} \leq n_p(G)^{\frac{1}{p}},$$ where for every $p$-element $x$ of $P$, $\lambda_G(x)$ denotes the number of Sylow $p$-subgroups of $G$ containing $x$. For this condition we give a reduction to almost simple groups. \[red alm simp\] Inequality (\[conj lambda\]) holds for every finite group if and only if it holds for every finite almost simple group. One of the most important tools used here is the so-called Wielandt’s subnormalizer (see Definition \[subnormalizer\]), which is related to the number of $p$-elements (see Lemma \[form Omega subnor\]). This connection is mainly due to the works of C. Casolo on subnormalizers ([@casolo:subnor], [@casolo:subnorsolv]). Another fundamental tool for the proof of our result is a theorem by G. Navarro and N. Rizo, concerning the number of fixed points in a coprime action of a $p$-group. Throughout the paper $G$ will be a finite group and $p$ a prime dividing $|G|$. Also, for all $x \in G$ we denote with $x^G$ the conjugacy class of $x$ in $G$. The p-solvable case =================== In this section we prove Theorem \[bound on Omega psolv\]. First of all we introduce the we introduce the concept of subnormalizer, whose definition (see [@lennox:subnormal pag. 238]) is inspired by the celebrated Wielandt’s subnormality criterion, which says that a subgroup $H$ of $G$ is subnormal in $G$ if and only if $H$ is subnormal in $\langle H, g \rangle$ for every $g \in G$. \[subnormalizer\] Let $H$ be a subgroup of $G$. The *subnormalizer of $H$ in $G$* is the set $$S_G(H) = \lbrace g \in G \ | \ H {\unlhd \unlhd \ }\langle H,g \rangle \rbrace.$$ where ${\unlhd \unlhd \ }$ means “is subnormal in”. A useful link between subnormalizers and the number of p-elements in a finite group (see Lemma \[form Omega subnor\]) is established by using a beautiful theorem by C. Casolo In order to state this theorem we introduce some notation. Let $H$ be a $p$-subgroup of $G$ and $P$ be a Sylow $p$-subgroup of $G$. We write $\lambda_G(H)$ for the number of Sylow $p$-subgroups of $G$ containing $H$ and $\alpha_G(H)$ for the number of $G$-conjugates of $H$ contained in $P$ (note that this number does not depend on the Sylow subgroup $P$ we are considering). When $H = \langle x \rangle $ is a cyclic subgroup, we simply write $S_G(x)$ and $\lambda_G(x)$, in place of $S_G(\langle x \rangle)$ and $\lambda_G(\langle x \rangle)$. In a similar fashion, we write $\alpha_G(x)$ for the number of $G$-conjugates of the element $x$ contained in $P$. We thus have that $$\label{alpha x} \alpha_G(x)=\alpha_G \left( \langle x \rangle \right) |N_G(\langle x \rangle)|/|C_G(x)|.$$ We can now state the aforementioned theorem by C. Casolo. \[form subnor theorem\] Let $H$ be a $p$-subgroup of $G$. Then the following holds and $P$ be a Sylow $p$-subgroup of $G$. - $$|S_G(H)|=\lambda_G(H) |N_G(P)| = \alpha_G(H) |N_G(H)|.$$ - If $G$ is $p$-solvable and $\mathcal{M}$ is the set of all $p'$-factors in a given normal $\lbrace p,p' \rbrace$-series of $G$, then $$|S_G(H)|= |P| \prod_{U/V \in \mathcal{M}} \left| C_{U/V} \left( HV/V \right) \right|.$$ A first easy application of this result is a formula that expresses the number of $p$-elements in $G$ in terms of the orders of the subnormalizers of the cyclic subgroups of a Sylow $p$-subgroup of $G$. \[form Omega subnor\] Let $P$ be a Sylow $p$-subgroup of $G$. We have $$|\mathfrak{U}_p(G)|=\sum_{x \in P} \frac{|G|}{|S_G(x)|}.$$ In the sum $$\sum_{x \in P} |x^G|$$ every class of $p$-elements is involved and its contribution is repeated as many times as the cardinality $|x^G \cap P|=\alpha_G(x)$. Hence $$|\mathfrak{U}_p(G)| = \sum_{x \in P} \frac{|x^G|}{\alpha_G(x)}=\sum_{x \in P} \frac{|G|}{\alpha_G(x)|C_G(x)|}=\sum_{x \in P} \frac{|G|}{|S_G(x)|},$$ by part *a)* of Theorem \[form subnor theorem\] and formula (\[alpha x\]). We now turn to the proof of Theorem \[bound on Omega psolv\]. Another fundamental tool that we are going to use in the proof is the following formula proved by Navarro and Rizo. \[nav riz\] Suppose that $P$ is a $p$-group acting on a $p'$-group $G$. Then $$|C_G(P)| = \left( \prod_{x \in P} \frac{|C_G(x)|}{|C_G(x^p)|^{1/p}} \right)^{\frac{p}{(p-1)|P|}}.$$ We can now prove Theorem \[bound on Omega psolv\]. By Lemma \[form Omega subnor\] we have that the $p$-Frobenius ratio of $G$ is the arithmetic mean of the ratios $$\frac{|G|}{|S_G(x)|}$$ when $x$ runs across $P$. By the Arithmetic-Geometric Mean Inequality, we get $$\label{FrobRatio arit geom} \frac{|\mathfrak{U}_p(G)|}{|P|} = \frac{1}{|P|} \left( \sum_{x \in P} \frac{|G|}{|S_G(x)|} \right) \geq \left( \prod_{x \in P} \frac{|G|}{|S_G(x)|} \right)^{1/|P|}.$$ Since $G$ is $p$-solvable we can take a normal $\lbrace p,p' \rbrace$-series, whose set of $p'$-factors we call $\mathcal{M}$. Then, by part *b)* of Theorem \[form subnor theorem\], we have for all $x \in P$ $$\frac{|G|}{|S_G(x)|} = \frac{|G|/|P|}{\prod_{U/V \in \mathcal{M}}|C_{U/V}(Vx)|} = \prod_{U/V \in \mathcal{M}} \frac{|U/V|}{|C_{U/V}(Vx)|}.$$ We insert this last term in (\[FrobRatio arit geom\]) and swap the products to get $$\begin{aligned} \frac{|\mathfrak{U}_p(G)|}{|P|} & \geq & \left( \prod_{U/V \in \mathcal{M}} \left( \prod_{x \in P} \frac{|U/V|}{|C_{U/V}(xV)|} \right) \right)^{1/|P|} \\ & = & \left( \prod_{U/V \in \mathcal{M}} \left| \frac{U}{V} \right|^{|P|} \left( \prod_{x \in P} \frac{1}{|C_{U/V}(xV)|} \right) \right)^{1/|P|}.\end{aligned}$$ Now for all $U/V \in \mathcal{M}$, $P$ is a $p$-group that acts on the $p'$-group $U/V$. We can then apply Theorem \[nav riz\] and use the trivial inequality $|C_{U/V}((xV)^p)| \leq |U/V|$, so that we have $$\begin{aligned} \prod_{x \in P} \frac{1}{|C_{U/V}(xV)|} & = & \left( \prod_{x \in P} \frac{1}{|C_{U/V}((xV)^p)|^{1/p}} \right) \frac{1}{|C_{U/V}(P)|^{|P|(p-1)/p}} \\ & \geq & \left( \frac{1}{|U/V|^{|P|/p}}\right)\frac{1}{|C_{U/V}(P)|^{|P|(p-1)/p}}\end{aligned}$$ and so $$\frac{|\mathfrak{U}_p(G)|}{|P|} \geq \left( \prod_{U/V \in \mathcal{M}} \frac{|U/V|}{|C_{U/V}(P)|} \right)^{\frac{p-1}{p}} = \left( \frac{|G|}{|S_G(P)|} \right)^{1-\frac{1}{p}},$$ again by part *b)* of Theorem \[form subnor theorem\]. Finally, we observe that for a Sylow $p$-subgroup $S_G(P)=N_G(P)$, so that $$\frac{|\mathfrak{U}_p(G)|}{|P|} \geq \left( \frac{|G|}{|S_G(P)|} \right)^{1-\frac{1}{p}} = \left( \frac{|G|}{|N_G(P)|} \right)^{1-\frac{1}{p}} = n_p(G)^{1-\frac{1}{p}}.$$ It is worth mentioning that the bound in Theorem \[bound on Omega psolv\] is asymptotically tight in the sense specified by the following example. \[tight frob ratio\] Let $p$ be a prime and, for $n$ a positive integer, let $P$ be an elementary abelian group of order $p^n$. Moreover set $\mathcal{M}$ to be the set of the maximal subgroups of $P$. Choose a prime $q$ such that $p$ divides $q-1$. Then for any $M \in \mathcal{M}$ we have that $P/M \simeq C_p$ acts fixed point freely as a group of automorphisms on a cyclic group $\langle a_M \rangle \simeq C_q$. We denote the image of the generator $a_M$ under this action by $a_M^{xM}$, for every $xM \in P/M$. Since $\bigcap_{M \in \mathcal{M}} M =1$, it follows that $P$ acts faithfully on the direct product $N$ of the groups $\langle a_M \rangle$. To be more explicit the following map $$\begin{aligned} P & \rightarrow Aut(N) \\ x & \mapsto \phi_x,\end{aligned}$$ where $\phi_x(a_M)=a_M^{xM}$, for all $M \in \mathcal{M}$ is an injective homomorphism. We consider the semidirect product $G_n = N \rtimes P$. The normalizer of $P$ in $G_n$ is $C_N(P)P=P$, hence the number of Sylow $p$-subgroups of $G_n$ is $$\label{npgn} n_p(G_n)= |N| = q^{|\mathcal{M}|}=q^{\frac{p^n-1}{p-1}}.$$ In order to count the number of $p$-elements in $G_n$ we use the equality $$| \mathfrak{U}_p(G_n) | = \sum_{x \in P} \frac{n_p(G_n)}{\lambda_{G_n}(x)},$$ which follows from Lemma \[form Omega subnor\] and part *a)* of Theorem \[form subnor theorem\]. We thus have to compute $\lambda_{G_n}(x)$, for $x \in P \setminus \lbrace 1 \rbrace$. Using again part *a)* of Theorem \[form subnor theorem\], we have $$\lambda_{G_n}(x)= \frac{\alpha_{G_n}(x)n_p(G_n)}{|x^{G_n}|}.$$ Now since $P$ is abelian and $G_n$ has a normal $p$-complement, we have $\alpha_{G_n}(x)=1$ and $|x^{G_n}|=|N|/|C_N(x)|$, so that $\lambda_{G_n}(x)=|C_N(x)|$. Given $x \in P \setminus \lbrace 1 \rbrace$, the centralizer of $x$ in $N$ is generated by those $a_M$ such that $a_M^{xM}=a_M$. Since $P/M$ acts fixed point freely on $\langle a_M \rangle$, this holds if and only if $x \in M$, hence $$C_N(x)= \langle a_M \ | \ x \in M \rangle.$$ The number of maximal subgroups in $P$ containing a fixed nontrivial element is $\frac{p^{n-1}-1}{p-1}$, and so $$|C_N(x)|=q^{\frac{p^{n-1}-1}{p-1}}.$$ We can then calculate the $p$-Frobenius ratio of $G_n$ $$\begin{split} \frac{|\mathfrak{U}_p(G_n)|}{|P|} &= \frac{1}{|P|} \sum_{x \in P} \frac{n_p(G_n)}{\lambda_{G_n}(x)} \\ & = \frac{1}{p^n} \left( 1 + (p^n-1) \frac{q^\frac{p^{n}-1}{p-1}}{q^\frac{p^{n-1}-1}{p-1}} \right) \\ & = \frac{1}{p^n} + \frac{p^n-1}{p^n} q^{p^{n-1}}. \end{split}$$ By (\[npgn\]) we have $$n_p(G_n)^{\frac{p-1}{p}}= q^{\frac{p^n-1}{p}}.$$ We can now compare the two members of the inequality stated by Theorem \[bound on Omega psolv\]. By considering the limit $$\lim_{n \rightarrow \infty} \frac{|\mathfrak{U}_p(G_n)|/|P|}{n_p(G_n)^{\frac{p-1}{p}}} = \lim_{n \rightarrow \infty} \left( \frac{1}{p^n q^{\frac{p^n-1}{p}}} + \frac{p^n-1}{p^n} \frac{q^{p^{n-1}}}{q^{\frac{p^n-1}{p}}} \right) = q^{1/p},$$ we see that the $p$-Frobenius ratio of $G_n$ and the $\left( 1-\frac{1}{p} \right)$-th power of the number of Sylow $p$-subgroups have the same asymptotic behaviour. The general case ================ In this section we explain why inequality (\[conj lambda\]) is sufficient for establishing Conjecture \[conj pfrobratio\] and we prove Theorem \[red alm simp\]. Let $P$ a Sylow $p$-subgroup of $G$. Since Lemma \[form Omega subnor\] is true for any group, by applying the Arithmetic-Geometric Mean Inequality as in \[FrobRatio arit geom\], we get $$\frac{|\mathfrak{U}_p(G)|}{|P|} \geq \left( \prod_{x \in P} \frac{|G|}{|S_G(x)|} \right)^{1/|P|}$$ and, recalling Theorem \[form subnor theorem\], we can write $$\frac{|\mathfrak{U}_p(G)|}{|P|} \geq \left( \prod_{x \in P} \frac{n_p(G)}{\lambda_G(x)} \right)^{1/|P|}.$$ A sufficient condition for (\[bound p el\]) is then $$\left( \prod_{x \in P} \frac{n_p(G)}{\lambda_G(x)} \right)^{1/|P|} \geq n_p(G)^{\frac{p-1}{p}},$$ that is $$\left( \prod_{x \in P} \lambda_G(x) \right)^{1/|P|} \leq n_p(G)^{\frac{1}{p}},$$ which is inequality (\[conj lambda\]). In [@gheri:degnil] it is proven that if $x$ is a $p$-element of $G$ which is not contained in the $O_p(G)$, then $\lambda_G(x)$ is at most $n_p(G)/(p+1)$. Focusing on a single element, this is the best one can get. Inequality (\[conj lambda\]), if true, would give a better bound on average, as it states that the geometric mean of the number of Sylow $p$-subgroups containing an element of a Sylow $p$-subgroup is at most the $p$-th root of the total number of Sylow $p$-subgroups. The bound (\[conj lambda\]), if true, is best possible in a strict sense. If we compute the terms of inequality (\[conj lambda\]) for the groups $G_n$ defined in Example \[tight frob ratio\], an equality occurs. \[Op 1\] For the proof of Theorem \[red alm simp\], we can assume $O_p(G)=1$. This is because if $N$ is a normal $p$-subgroup of $G$, then, for all $x \in \mathfrak{U}_p(G)$, we have that $\lambda_G(x)=\lambda_{G/N}(xN)$, and so (\[conj lambda\]) holds for $G$ if and only if it holds for $G/N$. First of all, we can reduce (\[conj lambda\]) to nonsolvable groups all of whose proper quotients are solvable (see Proposition \[reduction conj mns\]). We need some technical lemmas, the first of whom is proved in [@gheri:unsesto Lemma 3.3]. \[sp x cresce sui sottogruppi\] Let $H$ be a subgroup of $G$ and $x \in H$ be a $p$-element. Then $$\frac{\lambda_G(x)}{n_p(G)} \leq \frac{\lambda_H(x)}{n_p(H)}.$$ Moreover, if $H \unlhd G$, then the equality holds. \[SNx\] Let $N$ be a normal subgroup of $G$, $P$ be a Sylow $p$-subgroup of $G$ and $x$ an element of $P$. Assume that $G=NP$. Then $$|S_G(x)|=|S_N(x)| |NP/N|$$ If $x \in N$, the thesis follows from the fact that $|S_G(x)|/|G|=|S_N(x)|/|N|$, which can be easily derived from part *a)* of Theorem \[form subnor theorem\]. We work by induction on $m$, where $|NP/N|=p^m$. If $m=0$, then $G=N$ and there is nothing to prove. Suppose that $m > 0$ and $G \neq N \langle x \rangle$. Let $M$ be a maximal subgroup of $G$ containing $N \langle x \rangle$. Then, as $M \unlhd G$, by inductive hypothesis $$|S_G(x)|=|S_M(x)|p=|S_N(x)|p^{m-1}p.$$ Finally, if $G=N \langle x \rangle$, we observe that, given $a \in N$ and $t$ a positive integer, $ax^t \in S_G(x)$ if and only if $\langle x \rangle$ is subnormal in $\langle x, ax^t \rangle = \langle x, a \rangle $, that is if and only if $a \in S_G(x)$. It follows that $$|S_G(x)|=|S_N(x)| |NP/N|.$$ \[lambda np\] Let $N$ be a normal subgroup of $G$, $P$ be a Sylow $p$-subgroup of $G$ and $x \in P$. Then $$\lambda_G(x)= \lambda_{\frac{G}{N}}(Nx) \lambda_{NP}(x).$$ First of all we show that given a $p$-element $x$, the value $\lambda_{NP}(x)$ is independent of the particular Sylow $p$-subgroup $P$ containing $x$. By Theorem \[form subnor theorem\] and Lemma \[SNx\], we have $$\lambda_{NP}(x)= \frac{|S_{NP}(x)|}{|N_{NP}(P)|} =\frac{|S_{N}(x)|}{|N_{NP}(P)|} \left| \frac{NP}{N} \right| = \frac{|S_{N}(x)|}{|N|} |n_p(NP)| .$$ If $Q$ is another Sylow $p$-subgroup such that $x \in Q$, then of course $n_p(NP)=n_p(NQ)$, since $NP$ and $NQ$ are conjugated in $G$. Moreover $|S_N(x)|$ depends only on $N$ and $x$. Now let $\Delta_G^x$ be the set of the Sylow $p$-subgroups of $G$ containing $x$. We define the map $$\begin{split} \Delta_G^x & \rightarrow \Delta_{\frac{G}{N}}^{xN} \\ Q & \mapsto NQ/N. \end{split}$$ For all $\tilde{Q} \in \Delta_G^x$, the fiber of $N\tilde{Q}/N \in \Delta_{\frac{G}{N}}^{xN}$ is the set of Sylow $p$-subgroups $Q$ of $G$ containing $x$ and such that $NQ=N\tilde{Q}$, that is, $\Delta_{N \tilde{Q}}^{x}$. Since we proved that $\lambda_{N\tilde{Q}}(x)$ is independent of $\tilde{Q}$, we have $$\lambda_G(x)= | \Delta_{G}^{x}| = \left| \Delta_{\frac{G}{N}}^{xN}\right| \left| \Delta_{NP}^{x} \right| = \lambda_{\frac{G}{N}}(xN) \lambda_{NP}(x).$$ \[reduction conj mns\] A counterexample of minimal order to inequality (\[conj lambda\]) is a nonsolvable group having a unique minimal normal subgroup $M$, which is nonsolvable, and such that $G=MP$, where $P$ is a Sylow $p$-subgroup of $G$. Let $G$ be a counterexample of minimal order to inequality \[conj lambda\] and let $P$ be a Sylow $p$-subgroup of $G$. By Remark \[Op 1\] we have that $O_p(G)=1$. By Theorem \[bound on Omega psolv\], $G$ is nonsolvable. We show that every proper quotient of $G$ is solvable. Let $M$ be a minimal normal subgroup of $G$. By Lemma \[lambda np\], we have $$\begin{aligned} \prod_{x \in P} \lambda_G(x) & = & \prod_{x \in P} \lambda_{\frac{G}{M}}(Mx) \lambda_{MP}(x) \\ &=& \left( \prod_{xM \in \frac{PM}{M}} \lambda_{\frac{G}{M}}(Mx) \right)^{|P \cap M|} \left( \prod_{x \in P} \lambda_{MP}(x) \right). \end{aligned}$$ By the minimality of $G$, we have $$\prod_{xM \in \frac{PM}{M}} \lambda_{\frac{G}{M}}(Mx) \leq n_p \left( G/M \right)^{\frac{|PM/M|}{p}}.$$ If $MP<G$, we can again assume that the inequality is true for $MP$ and so $$\prod_{x \in P} \lambda_G(x) \leq n_p\left( \frac{G}{M} \right)^{\frac{|PM/M|}{p}|P \cap M|} n_p(MP)^\frac{|P|}{p}=n_p(G)^\frac{|P|}{p}.$$ Since this is true for every minimal normal subgroup of $G$, the usual subdirect product argument gives that $G$ has a unique minimal normal subgroup $M$, which is nonsolvable and such that $G=MP$. Referring to the notation of Lemma \[reduction conj mns\], $M$ is the direct product of simple groups permuted by $P$. The next easy lemma loosely bounds the number of $\langle x \rangle$-invariant Sylow $p$-subgroups of $M$, where $x \in P$, in terms of the action of $\langle x \rangle$ on the direct factors of $M$. \[num orb\] Let $M \unlhd G$ be a direct product of $k$ copies of a group $L$, $M=L_1 \times \dots \times L_k$, let $x \in G$ be an element that permutes the factors $L_i$ of $M$ and let $p$ be a prime number. Then the number of $\langle x \rangle$-invariant Sylow $p$-subgroups of $M$ is at most $n_p(L)^s$, where $s$ is the number of orbits of $\langle x \rangle$ on $\Delta= \lbrace L_1, \dots, L_k \rbrace$. A Sylow $p$-subgroup $Q$ of $M$ is the direct product of $k$ Sylow $p$-subgroups of $L$, $Q = Q_1 \times \dots \times Q_k$. Suppose that $Q$ is normalized by $x$. If $L_i=L_j^{x^r}$, for $r \in \mathbb{Z}$, then $Q_i=Q_j^{x^r}$ and so one has at most $n_p(L)$ choices for each $\langle x \rangle$-orbit in $\Delta$. We can now prove Theorem \[red alm simp\] Suppose that inequality (\[conj lambda\]) is true for all finite almost simple groups and let $G$ be a counterexemple of minimal order. By Proposition \[reduction conj mns\], $G=MP$ where $P$ is a Sylow $p$-subgroup of $G$ and $M$ is the unique minimal normal subgroup $$M = L_1 \times \dots \times L_k, \ L_i \simeq L, \ \forall i \in \lbrace 1, \dots , k \rbrace.$$ for some nonabelian simple group $L$. Moreover $P$ acts transitively on the set $\Delta = \lbrace L_1, \dots , L_k \rbrace$. Since we are assuming the result for almost simple groups, we have $k>1$. Let $Q=P \cap M$. For any subgroup $Q \leq X \leq P$ we set $m_X$ to be the ratio $$\label{mX} m_X=\frac{|N_M(X)|}{|N_M(P)|}= \frac{n_p(G)}{n_p(MX)}.$$ The last equality holds since $n_p(G)=[MP:N_{MP}(P)]=[M:N_M(P)]$ and $n_p(MX)=[MX:N_{MX}(X)]=[M:N_M(X)]$. Moreover observe that if $g \in N_M(P)$ and $x\in X$ then $$x^g=x[x,g] \in X(M \cap P)=XQ=X,$$ and so $N_M(P) \leq N_M(X)$. With a slight abuse of notation, we denote with $\lambda_M(x)$ the number of Sylow $p$-subgroups in $M$ normalized by $x$ even for $x \notin M$. It is easy to check that $\lambda_M(x)=\lambda_{M\langle x \rangle}(x)$. Let $H_0=N_P(L_1)$ be the stabilizer of $L_1$ in the action of $P$ on $\Delta$. Since $P$ is transitive on $\Delta$, $H_0 \neq P$. Choose now a maximal subgroup $H$ of $P$ containing $H_0$. Since $H$ is normal in $P$ and the stabilizers of the subgroups $L_i$ are all conjugated in $P$, we have that $H$ contains all of them. It follows that every element $x \in P \setminus H$ has at most $k/p$ orbits on $\Delta$ and so by Lemma \[num orb\] $$\label{orbits} \lambda_M(x) \leq n_p(L)^{\frac{k}{p}}.$$ We now consider separately the elements inside and outside $H$. As for the elements inside $H$, since $MH$ is normal in $G$, using Lemma \[sp x cresce sui sottogruppi\], we get $$\begin{split} \prod_{x \in H} \lambda_G(x) &= \prod_{x \in H} \frac{n_p(G)}{n_p(MH)}\lambda_{MH}(x) \\ &= \prod_{x \in H} m_H \lambda_{MH}(x)=(m_H)^{|H|} \prod_{x \in H} \lambda_{MH}(x). \end{split}$$ Since $H$ is a Sylow $p$-subgroup of $MH$ and inequality (\[conj lambda\]) holds for $MH<G$ we get $$\label{in H} \begin{split} \prod_{x \in H} \lambda_G(x) & = (m_H)^{|H|} \prod_{x \in H} \lambda_{MH}(x) \\ & \leq (m_H)^{|H|} n_p(MH)^\frac{|H|}{p}=m_H^{\frac{p-1}{p}|H|} n_p(G)^\frac{|H|}{p}, \end{split}$$ where we applied (\[mX\]). Now we turn our attention on elements in $P \setminus H$. Let $\mathcal{T}$ be a set of representatives for the right cosets of $Q$ in $P$ that are not contained in $H$. The cardinality of $\mathcal{T}$ is then $$|\mathcal{T}|=[P:Q]-[H:Q]=\frac{|P|-|H|}{|Q|}=(p-1) \frac{|H|}{|Q|}.$$ We have, by Lemma \[sp x cresce sui sottogruppi\] $$\begin{aligned} \prod_{x \in P \setminus H} \lambda_G(x) & = & \prod_{x \in \mathcal{T}} \prod_{g \in Q} \lambda_G(gx) \leq \prod_{x \in \mathcal{T}} \left( \prod_{g \in Q} m_{Q\langle gx \rangle} \lambda_{M\langle gx \rangle}(gx) \right) \\ &=& \prod_{x \in \mathcal{T}} \left( m_{Q\langle x \rangle}^{|Q|} \prod_{g \in Q} \lambda_{M}(gx) \right).\end{aligned}$$ Now the elements $gx$ in the previous product are not in $H$ and so by (\[orbits\]) $$\label{prod lambda outside H} \begin{split} \prod_{x \in P \setminus H} \lambda_G(x) & \leq \left( \prod_{x \in \mathcal{T}} m_{Q\langle x \rangle} \right)^{|Q|} n_p(L)^{\frac{k}{p}|Q||\mathcal{T}|}\\ & = \left( \prod_{x \in \mathcal{T}} m_{Q\langle x \rangle} \right)^{|Q|} n_p(M)^{\frac{p-1}{p}|H|}. \end{split}$$ We now want to evaluate the product $\prod_{x \in \mathcal{T}} m_{Q\langle x \rangle}$. In the following we use the bar notation for the quotients modulo $Q$. If $R=N_M(Q)$ we have a coprime action of $\bar{P}$ on the $p'$-group $\bar{R}$. We apply Theorem \[nav riz\] to this action and get $$|C_{\bar{R}}(\bar{P})|^{|\bar{P}|\frac{p-1}{p}} = \prod_{x \in \bar{P}} \frac{|C_{\bar{R}}(x)|}{|C_{\bar{R}}(x^p)|^{1/p}}.$$ Separating the elements inside $\bar{H}$ and those outside $\bar{H}$ and applying twice Theorem \[nav riz\], we get $$\begin{split} |C_{\bar{R}}(\bar{P})|^{|\bar{P}|\frac{p-1}{p}} & = \left( \prod_{x \in \bar{H}} \frac{|C_{\bar{R}}(x)|}{|C_{\bar{R}}(x^p)|^{1/p}} \right) \left( \prod_{x \in \bar{P} \setminus \bar{H}} \frac{|C_{\bar{R}}(x)|}{|C_{\bar{R}}(x^p)|^{1/p}} \right) \\ & = |C_{\bar{R}}(\bar{H})|^{|\bar{H}|\frac{p-1}{p}} \left( \prod_{x \in \bar{P} \setminus \bar{H}} \frac{|C_{\bar{R}}(x)|}{|C_{\bar{R}}(x^p)|^{1/p}} \right). \end{split}$$ Using the bound $|C_{\bar{R}}(x^p)| \leq |\bar{R}|$ and the fact that $|\bar{P} \setminus \bar{H}|=(p-1) \left| \bar{H}\right|$, we have $$\label{prodCRx} \begin{split} \prod_{x \in \bar{P} \setminus \bar{H}} |C_{\bar{R}}(x)| & = |C_{\bar{R}}(\bar{P})^{|\bar{P}|\frac{p-1}{p}} |C_{\bar{R}}(\bar{H})|^{-|\bar{H}|\frac{p-1}{p}} \left( \prod_{x \in \bar{P} \setminus \bar{H}} |C_{\bar{R}}(x^p)|^{1/p} \right) \\ &\leq |C_{\bar{R}}(\bar{P})|^{|\bar{P}|\frac{p-1}{p}} |C_{\bar{R}}(\bar{H})|^{-|\bar{H}|\frac{p-1}{p}} |\bar{R}|^{|\bar{H}|\frac{p-1}{p}}. \end{split}$$ We now observe that if $X$ is a subgroup of $P$ containing $Q$, then $Q=X \cap M$, so that $N_M(X) \leq N_M(Q)=R$ and we have $$\label{NNX CRX} \overline{N_M(X)}=C_{\bar{R}}(\bar{X}).$$ and since $Q \leq N_M(X)$ $$m_X=\frac{|N_M(X)|}{|N_M(P)|} = \frac{\left| \overline{N_M(X)} \right|}{\left| \overline{N_M(P)} \right|} = \frac{|C_{\bar{R}}(\bar{X})|}{|C_{\bar{R}}(\bar{P})|}$$ Hence, for all $x \in \mathcal{T}$, $$m_{Q\langle x \rangle} = \frac{\left| C_{\bar{R}}(xQ) \right|}{\left| C_{\bar{R}}(\bar{P}) \right|}.$$ Going back to our product and using (\[prodCRx\]), we then get $$\begin{split} \prod_{x \in \mathcal{T}} m_{Q\langle x \rangle} & = \prod_{x \in \mathcal{T}} \frac{|C_{\bar{R}}(xQ)|}{|C_{\bar{R}}(\bar{P})|} =\prod_{x \in \bar{P} \setminus \bar{H}} \frac{|C_{\bar{R}}(x)|}{|C_{\bar{R}}(\bar{P})|} \\ & \leq |C_{\bar{R}}(\bar{P})|^{-|\bar{P}|\frac{p-1}{p}} |C_{\bar{R}}(\bar{P})|^{|\bar{P}|\frac{p-1}{p}} |C_{\bar{R}}(\bar{H})|^{-|\bar{H}|\frac{p-1}{p}} |\bar{R}|^{|\bar{H}|\frac{p-1}{p}} \\ & = |C_{\bar{R}}(\bar{H})|^{-|\bar{H}|\frac{p-1}{p}} |\bar{R}|^{|\bar{H}|\frac{p-1}{p}}. \end{split}$$ We now remove the bar notation using the definition of $R$ and (\[NNX CRX\]), $$\begin{split} \prod_{x \in \mathcal{T}} m_{Q\langle x \rangle} & = \left( \frac{|N_M(H)|}{|Q|} \right)^{-\frac{|H|}{|Q|}\frac{p-1}{p}} \left( \frac{|N_M(Q)|}{|Q|} \right)^{\frac{|H|}{|Q|}\frac{p-1}{p}} \\ & = \left( \frac{|N_M(Q)|}{|N_M(H)|} \right)^{\frac{|H|}{|Q|}\frac{p-1}{p}}. \end{split}$$ Using this bound in inequality (\[prod lambda outside H\]) we get $$\begin{split} \prod_{x \in P \setminus H} \lambda_G(x) & \leq \left( \prod_{x \in \mathcal{T}} m_{Q\langle x \rangle} \right)^{|Q|} n_p(M)^{\frac{p-1}{p}|H|} \\ & \leq \left( \frac{|N_M(Q)|}{|N_M(H)|} \right)^{\frac{p-1}{p}|H|} n_p(M)^{\frac{p-1}{p}|H|} \\ & \leq \left( \frac{|N_M(Q)|}{|N_M(H)|} n_p(M)\right)^{\frac{p-1}{p}|H|}, \end{split}$$ and since $Q$ is a Sylow $p$-subgroup of $M$, $$\label{not in H} \begin{split} \prod_{x \in P \setminus H} \lambda_G(x) \leq \left( \frac{|N_M(Q)|}{|N_M(H)|} \frac{|M|}{|N_M(Q)|}\right)^{\frac{p-1}{p}|H|} = \left( \frac{|M|}{|N_M(H)|}\right)^{\frac{p-1}{p}|H|} \end{split}$$ By combining (\[in H\]) and (\[not in H\]) we obtain $$\begin{split} \prod_{x \in P} \lambda_G(x) & = \left( \prod_{x \in H} \lambda_G(x) \right)\left( \prod_{x \in P \setminus H} \lambda_G(x) \right) \\ & \leq \left( m_H^{\frac{p-1}{p}|H|} n_p(G)^\frac{|H|}{p} \right) \left( \frac{|M|}{|N_M(H)|}\right)^{\frac{p-1}{p}|H|} \\ & = n_p(G)^\frac{|H|}{p} \left( m_H \frac{|M|}{|N_M(H)|}\right)^{\frac{p-1}{p}|H|}. \end{split}$$ Finally, recalling (\[mX\]) and the fact that $|H|=|P|/p$ we get $$\begin{split} \prod_{x \in P} \lambda_G(x) & \leq n_p(G)^\frac{|H|}{p} \left( \frac{|N_M(H)|}{|N_M(P)|} \frac{|M|}{|N_M(H)|}\right)^{\frac{p-1}{p}|H|} \\ &= n_p(G)^\frac{|H|}{p} n_p(G)^{\frac{p-1}{p}|H|} = n_p(G)^\frac{|P|}{p} \end{split}$$ which is against the fact that $G$ is a counterexample. Acknowledgements {#acknowledgements .unnumbered} ---------------- This article is part of the author’s PhD thesis, which was written under the great supervision of Carlo Casolo, whose contribution to this work was essential. Thanks are also due to Francesco Fumagalli and Silvio Dolfi for his valuable comments and suggestions. This work was partially funded by the Istituto Nazionale di Alta Matematica “Francesco Severi" (Indam). [99]{} C. Casolo, *On the subnormalizer of a $p$-subgroup*, J. Pure Appl. Algebra 77 (1992), no.3. C. Casolo, *Subnormalizers in finite groups*, Comm. Algebra 18 (1990), no. 11. G. Frobenius, *Verallgemeinerung des Sylowschen Satzes*, Sitzungsberichte der Preussischen Akademie, Berlin (1895). P. Gheri, *Subnormalizers and the degree of nilpotence in finite groups*, Proc. Amer. Math. Soc., DOI: https://doi.org/10.1090/proc/15080 (to appear in print). P. Gheri, *Subnormalizers and solvability in finite groups*, preprint, (2020). I.M. Isaacs and G.R. Robinson, *On a theorem of Frobenius: solutions of $x^n=1$ in finite groups*, Amer. Math. Monthly 99 (1992), no. 4. J.C. Lennox and S.E. Stonehewer, *Subnormal subgroups of groups*, Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1987. G.A. Miller, H.F. Blichfeldt and L.E. Dickson, *Theory and applications of finite groups*, Dover Publications, Inc., New York 1961. G. Navarro and N. Rizo, *A Brauer-Wielandt formula (with an application to character tables)*, Proc. Amer. Math. Soc. 144 (2016), no. 10. Speyer, David E.(1-MI) A counting proof of a theorem of Frobenius. (English summary) Amer. Math. Monthly 124 (2017), no. 4. R. Steinberg, *Endomorphisms of linear algebraic groups*, Memoirs of the American Mathematical Society, No. 80 American Mathematical Society, Providence, R.I. 1968.

--- abstract: 'At present, Babcock-Leighton flux transport solar dynamo models appear as the most promising model for explaining diverse observational aspects of the sunspot cycle. The success of these flux transport dynamo models is largely dependent upon a single-cell meridional circulation with a deep equatorward component at the base of the Sun’s convection zone. However, recent observations suggest that the meridional flow may in fact be very shallow (confined to the top 10% of the Sun) and more complex than previously thought. Taken together these observations raise serious concerns on the validity of the flux transport paradigm. By accounting for the turbulent pumping of magnetic flux as evidenced in magnetohydrodynamic simulations of solar convection, we demonstrate that flux transport dynamo models can generate solar-like magnetic cycles even if the meridional flow is shallow. Solar-like periodic reversals is recovered even when meridional circulation is altogether absent, however, in this case the solar surface magnetic field dynamics does not extend all the way to the polar regions. Very importantly, our results demonstrate that the Parker-Yoshimura sign rule for dynamo wave propagation can be circumvented in Babcock-Leighton dynamo models by the latitudinal component of turbulent pumping – which can generate equatorward propagating sunspot belts in the absence of a deep, equatorward meridional flow. We also show that variations in turbulent pumping coefficients can modulate the solar cycle amplitude and periodicity. Our results suggest the viability of an alternate magnetic flux transport paradigm – mediated via turbulent pumping – for sustaining solar-stellar dynamo action.' author: - Soumitra Hazra and Dibyendu Nandy title: 'A Proposed Paradigm for Solar Cycle Dynamics Mediated via Turbulent Pumping of Magnetic Flux in Babcock-Leighton type Solar Dynamos' --- Introduction ============ The cycle of sunspots involves the generation and recycling of the Sun’s toroidal and poloidal magnetic field components. The magnetohydrodynamic (MHD) dynamo mechanism that achieves this is sustained by the energy of solar internal plasma motions such as differential rotation, turbulent convection and meridional circulation. The toroidal field is generated through stretching of the poloidal component by differential rotation [@park55] and is believed to be stored and amplified at the overshoot layer [@moren92] beneath the base of the solar convection zone (SCZ). Strong toroidal flux tubes are unstable to magnetic buoyancy and erupt through the surface producing sunspots, which are strongly magnetized and have a systematic tilt [@hale08; @hale19]. The poloidal field is believed to be regenerated through a combination of helical turbulent convection (traditionally known as the mean-field $\alpha$-effect; [@park55]) in the main body of the SCZ and the redistribution of the magnetic flux of tilted bipolar sunspot pairs (the Babcock-Leighton process; [@bab61; @leigh69]). Despite early, pioneering attempts to self-consistently model the interactions of turbulent plasma flows and magnetic fields in the context of the solar cycle [@gilm83; @glat85] such full MHD simulations are still not successful in yielding solutions that can match solar cycle observations. This task is indeed difficult, for the range of density and pressure scale heights, scale of turbulence and high Reynolds number that characterize the SCZ is difficult to capture even in the most powerful supercomputers. An alternative approach to modelling the solar cycle is based on solving the magnetic induction equation in the SCZ with observed plasma flows as inputs and with additional physics gleaned from simulations of convection and flux tube dynamics. These so called flux transport dynamo models have shown great promise in recent years in addressing a wide variety of solar cycle problems [@char10; @ossen03]. In particular, solar dynamo models based on the Babcock-Leighton mechanism for poloidal field generation have been more successful in explaining diverse observational features of the solar cycle [@dikp99; @nandy02; @chat04; @chou04; @guer07; @nandy11; @chou12; @haz14; @pass14]. Recent observations also strongly favor the Babcock-Leighton mechanism as a major source for poloidal field generation [@dasi10; @munoz13]. In this scenario, the poloidal field generation is essentially predominantly confined to near-surface layers. For the dynamo to function efficiently, the toroidal field that presumably resides deep in the interior has to reach the near-surface layers for the Babcock-Leighton poloidal source to be effective. This is achieved by the buoyant transport of magnetic flux from the Sun’s interior to its surface (through sunspot eruptions). Subsequent to this the poloidal field so generated at near-surface layers must be transported back to the solar interior, where differential rotation can generate the toroidal field. The deep meridional flow assumed in such models (See Fig. 1, left-hemisphere) plays a significant role in this flux transport process and is thought to govern the period of the sunspot cycle [@char20; @hatha03; @yeat08; @ghaz14]. Moreover, a fundamentally crucial role attributed to the deep equatorward meridional flow is that it allows the Parker-Yoshimura sign rule [@park55; @yosh75] to be overcome, which would otherwise result in poleward propagating dynamo waves in contradiction to observations that the sunspot belt migrates equatorwards with the progress of the cycle [@chou95; @ghaz14; @pass15; @belus15]. While the poleward meridional flow at the solar surface is well observed (Hathaway & Rightmire 2010; 2011) the internal meridional flow profile has remained largely unconstrained. A recent study utilizing solar supergranules [@hatha12] suggests that the meridional flow is confined to within the top 10% of the Sun (Fig. 1, right-hemisphere) – much shallower than previously thought. Independent studies utilizing helioseismic inversions are also indicative that the equatorward meridional counterflow may be located at shallow depths [@mitra07; @zhao13]. The latter also infer the flow to be multi-cellular and more complex. These studies motivate exploring alternative paradigms for flux transport dynamics in Babcock-Leighton type models of the solar cycle which are crucially dependent on meridional circulation linking the two segregated dynamo source regions in the SCZ. This leads us to consider the role of turbulent pumping. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --  ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- Magnetoconvection simulations supported by theoretical considerations have established that turbulent pumping preferentially transports magnetic fields vertically downwards [@brand96; @tobias01; @ossen02; @dorch01; @kapla06; @pip09; @racine11; @rogach11; @aug15; @warn16; @sim16] – likely mediated via strong downward convective plumes which are particularly effective on weak magnetic fields (such as the poloidal component). In strong rotation regimes, there is also a significant latitudinal component of turbulent pumping. In particular, two studies, one utilizing mean-field dynamo simulations [@brand92] and the other utilizing turbulent three dimensional magnetoconvection simulations [@ossen02] recognized the possibility that turbulent pumping may contribute to the equatorward propagation of the toroidal field belt. We note that most Babcock-Leighton kinematic flux transport solar dynamo models do not include the process of turbulent pumping of magnetic flux. The few studies that exist on the impact of turbulent pumping in the context of flux transport dynamo models show it to be dynamically important in flux transport dynamics, the maintenance of solar-like parity and solar-cycle memory [@guer08; @kar12; @jiang13]. In their model with turbulent pumping, Guerrero & de Gouveia Dal Pino (2008) used a spatially distributed $\alpha$-coefficient in the near-surface layers to model the Babcock-Leighton poloidal source and a meridional circulation whose equatorward component penetrated up to $0.8R_\odot$, i.e., more than half the depth of the SCZ; therefore, from this modelling it is not possible to segregate the contributions of turbulent pumping and meridional flow (the peak latitudinal component of the former coincides with the equatorward component of the latter) to the toroidal field migration. Here, utilizing a newly developed state-of-the-art flux transport dynamo model where a double-ring algorithm is utilized to model the Babcock-Leighton process, we explore the impact of turbulent pumping in flux transport dynamo models with nonexistent, or shallow meridional circulation. Our results indicate the possibility of an alternative flux transport paradigm for the solar cycle in which turbulent pumping of magnetic flux resolves the problems posed by a shallow (or inconsequential) meridional flow. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --  (a)                                          (b)        (c) -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- Model ===== Our flux transport solar dynamo model solves for the coupled, evolution equation for the axisymmetric toroidal and poloidal components of the solar magnetic fields: $$\label{Eq_2.5DynA} \frac{\partial A}{\partial t} + \frac{1}{s}\left[ \textbf{v}_p \cdot \nabla (sA) \right] = \eta\left( \nabla^2 - \frac{1}{s^2} \right)A + {S_{BL}},$$ $$\begin{aligned} \label{Eq_2.5DynB} \frac{\partial B}{\partial t} + s\left[ \textbf{v}_p \cdot \nabla\left(\frac{B}{s} \right) \right] + (\nabla \cdot \textbf{v}_p)B = \eta\left( \nabla^2 - \frac{1}{s^2} \right)B \nonumber \\ + s\left(\left[ \nabla \times (A\bf \hat{e}_\phi) \right]\cdot \nabla \Omega\right) + \frac{1}{s}\frac{\partial (sB)}{\partial r}\frac{\partial \eta}{\partial r},~~~~~\end{aligned}$$ where, $B$ is the toroidal component of magnetic field and $A$ is the vector potential for the poloidal component of magnetic field. ${\textbf v}_p$ is the meridional flow, $\Omega$ is the differential rotation, $\eta$ is the turbulent magnetic diffusivity and $s = r\sin(\theta)$. For the differential rotation and diffusivity profile, we use an analytic fit to the observed solar differential rotation (the near-surface shear layer is not included) and a two-step turbulent diffusivity profile (which ensures a smooth transition to low levels of diffusivity beneath the base of the convection zone) (For detailed profile, see Hazra & Nandy 2013). We use the same meridional flow profile as defined in Hazra & Nandy (2013). Our flow profile has penetration depth of $0.65R_\odot$ to represent deep meridional flow situation, and $0.90~R_\odot$ to represent shallow meridional flow situation. We set the peak speed of the meridional flow to be 15 ms$^{-1}$ (near mid-latitudes). The second term on the RHS of the toroidal field evolution equation acts as the source term for the toroidal field (rotational shear), while in the poloidal field evolution equation, the source term, ${S_{BL}}$, is due to the Babcock-Leighton mechanism. Here we use a double-ring algorithm for buoyant sunspot eruptions that best captures the Babcock-Leighton mechanism for poloidal field generation [@durney97; @nandy01; @munoz10; @haz13] and which has been tested thoroughly in other contexts. Specifics about our double ring algorithm can be found in Hazra & Nandy (2013) and Hazra (2016; PhD Thesis). --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --  --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- Results ======= To bring out the significance of the recent observations, we first consider a single cell, shallow meridional flow, confined only to the top 10% of the convection zone (Fig. 1, right-hemisphere). In the first scenario we seek to answer the following question: Can solar-like cycles be sustained through magnetic field dynamics completely confined to the top 10% of the Sun? In these simulations initialized with antisymmetric toroidal field condition (with initial B $\sim$ 100 kG), we first allow magnetic flux tubes to buoyantly erupt from 0.90 $R_{\odot}$ (i.e., the depth to which the shallow flow is confined) when they exceed a buoyancy threshold of $10^4$ Gauss (G). In this case we find that the simulated fields fall and remain below this threshold (at all latitudes at 0.90 $R_{\odot}$) with no buoyant eruptions, implying that a Babcock-Leighton type solar dynamo cannot operate in this case. Dikpati [*et al.*]{} (2002) considered the contribution of the near-surface shear layer in their simulations (which we have not) and concluded that this near-surface layer contributes only about 1 kG to the total toroidal field production and hence insufficient to drive a large-scale dynamo. Guerrero & de Gouveia Dal Pino (2008) also utilized a near-surface shear layer with radial pumping and found solar-like solutions only under special circumstances; however, given that for this particular case they utilized a local $\alpha$-effect for the latter simulations (with a spatially distributed $\alpha$-effect in the near-surface layer) it is not evident that these simulations are relatable to the Babcock-Leighton solar dynamo concept. The upper layers of the SCZ is highly turbulent and storage and amplification of strong magnetic flux tubes may not be possible in these layers [@park75; @moren83] and therefore this result is not unexpected. While Brandenburg (2005) has conjectured that the near-surface shear layer may be able to power a large-scale dynamo, this remains to be convincingly demonstrated in the context of a Babcock-Leighton dynamo. In the second scenario with a shallow meridional flow, we allow magnetic flux tubes to buoyantly erupt from 0.71 $R_{\odot}$, i.e. from base of the convection zone. In this case we get periodic solutions but analysis of the butterfly diagrams (taken both at the base of SCZ and near solar surface) shows that the toroidal field belts have almost symmetrical poleward and equatorward branches with no significant equatorward migration (see Fig. 2). Moreover, as already noted by Guerrero, G. & de Gouveia Dal Pino (2008), the solutions with shallow meridional flow always display quadrupolar parity in contradiction with solar cycle observations. Clearly, a shallow flow poses a serious problem for solar cycle models. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --  (a)                                               (b)      (c) ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ --  ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -- We now introduce both radial and latitudinal turbulent pumping in our dynamo model to explore whether a Babcock Leighton flux transport dynamo can operate with meridional flow which is much shallower than previously assumed; we also extend this study to the scenario where meridional flow is altogether absent. The turbulent pumping profile is determined from independent MHD simulations of solar magnetoconvection [@ossen02; @kapla06]. Profiles for radial and latitudinal turbulent pumping ($\gamma_r$ and $\gamma_\theta$) are: $$\begin{aligned} \gamma_r = - \gamma_{0r} \left[ 1 + \rm{erf}\left( \frac{r - 0.715R_\odot}{0.015R_\odot}\right) \right] \left[ 1 - \rm{erf} \left( \frac{r-0.97R_\odot}{0.1R_\odot}\right) \right] \nonumber \\ \times \left[ \rm{exp}\left( \frac{r-0.715R_\odot}{0.25R_\odot}\right) ^2 \rm{cos}\theta +1\right] ~~~~\end{aligned}$$ $$\begin{aligned} \gamma_\theta = \gamma_{0\theta} \left[1+\mathrm{erf}\left(\frac{r-0.8R_\odot}{0.55R_\odot}\right)\right] \left[1-\mathrm{erf}\left(\frac{r-0.98R_\odot}{0.025R_\odot}\right)\right] \times \cos \theta \sin^4 \theta ~~~~\end{aligned}$$ The value of $\gamma_{0r}$ and $\gamma_{0\theta}$ determines the amplitude of $\gamma_r$ and $\gamma_\theta$ respectively. Fig. 3 (top and bottom plot) shows that radial pumping speed (dashed lines) is negative throughout the convection zone corresponding to downward advective transport and vanishes below $0.7R_\odot$. The radial pumping speed is maximum near the poles and decreases towards the equator. Fig. 3 (top and bottom plot) shows that the latitudinal pumping speed (solid lines) is positive (negative) in the convection zone in the northern (southern) hemisphere and vanishes below the overshoot layer. This corresponds to equatorward latitudinal pumping throughout the convection zone. Dynamo simulations with turbulent pumping generate solar-like magnetic cycles (Fig. 4 and Fig. 5). Now the toroidal field belt migrates equatorward, the solution exhibits solar-like parity and the correct phase relationship between the toroidal and poloidal components of the magnetic field (see Fig. 5). Evidently, the coupling between the poloidal source at the near-surface layers with the deeper layers of the convection zone where the toroidal field is stored and amplified, the equatorward migration of the sunspot-forming toroidal field belt and correct solar-like parity is due to the important role played by turbulent pumping. We note if the speed of the latitudinal pumping in on order of 1.0 ms$^{-1}$ the solutions are always of dipolar parity irrespective of whether one initializes the model with dipolar or quadrupolar parity. Interestingly, the latitudinal migration rate of the sunspot belt as observed is of the same order. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --  (a)                                              (b)      (c) ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- The above result begs the question whether flux transport solar dynamo models based on the Babcock-Leighton mechanism that include turbulent pumping can operate without any meridional plasma flow. To test this, we remove meridional circulation completely from our model and perform simulations with turbulent pumping included. We find that this model generates solar-like sunspot cycles with periodic reversals (see Fig. 6) which are qualitatively similar to the earlier solution with both pumping and shallow meridional flow. However, we find that the surface magnetic field dynamics related to polar field reversal is limited to within 60 degrees latitudes in both the hemispheres. At higher latitudes (near the poles) the field is very weak and almost non-varying over solar cycle timescales. This is expected if the surface magnetic field dynamics is governed primarily by diffusion. Based on this result, we argue that this scenario of non-existent meridional circulation is not supported by current observations of surface dynamics which seem to suggest that the fields do migrate all the way to the poles. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --  -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- Two important characteristics associated with the solar magnetic cycle are its amplitude and periodicity. While the periodicity of the cycle predominantly depends on the recycling time between toroidal and poloidal field, its amplitude depends on a variety of factors including dynamo source strengths and relative efficacy of transport timescales with respect to the turbulent diffusion timescale. We explore the dependency of the solar cycle period and amplitude to variations in the transport coefficients to explore the subtleties of the interplay between diverse flux transport processes. Figure 7 shows the dependency of cycle amplitude and periodicity on different velocity components like turbulent pumping and (shallow) meridional flow. A parametric analysis of this dependency yields the following relationships for cycle period ($T$) and cycle amplitude (Amp): $$T \simeq 9.7 ~ \gamma_r^{-0.25} \gamma_\theta^{-0.26} v^{-0.068},$$ $$Amp \simeq 11.76 ~ \gamma_r^{1.07} \gamma_\theta^{-0.27} v^{-0.16},$$ which is gleaned from simulations within the following ranges: $0.25 ~ms^{-1} \leq \gamma_r \leq 1.25 ~ms^{-1}$, $0.25 ~ms^{-1} \leq \gamma_\theta \leq 1.25 ~ms^{-1}$ and $2 ~ms^{-1} \leq v \leq 15 ~ms^{-1}$; $\gamma_r$ and $\gamma_\theta$ are radial and latitudinal turbulent pumping speeds, and $v$ is the shallow meridional flow speed. This analysis shows that cycle period and amplitude are both governed by diverse transport coefficients such as meridional flow speed, and radial and latitudinal components of turbulent pumping. As radial turbulent pumping carries the flux directly to the base of the convection zone where toroidal field is amplified, increase in the radial turbulent pumping speed leads to a decrease in cycle period. Increasing latitudinal pumping also has a similar effect on period which is similar to what is achieved by increasing meridional flow speed, namely a faster transport through the shear layer and thus shorter cycle periods. The cycle amplitude decreases on increasing the latitudinal pumping or meridional flow speed and this is due to the fact that less time is available for toroidal field induction when it is swept at a faster rate through the rotational shear layers. In surface flux transport models, a similar effect is found but due to a different reason – wherein a faster meridional flow reduces the polar field strength because it takes flux of both polarity and deposits this at the poles (in effect carrying less net flux to the poles); in these simulations with a shallow meridional flow and the double-ring algorithm a similar mechanism could also be contributing to an overall reduction of the field strength. What is interesting to note though is the positive dependence of cycle amplitude on the radial pumping speed. We believe that a faster radial pumping moves the poloidal field down to the generating layers of the toroidal field in the deeper parts of the convection zone faster, thus allowing for less turbulent decay in the poloidal field strength; this eventually results in a stronger poloidal field in the SCZ which generates a stronger toroidal component. We note that the derived exponents for the cycle period above differ from that determined by Guerrero & de Gouveia Dal Pino (2008). The cycle period in our simulations is more strongly dependent on the latitudinal speed of turbulent pumping and less so on meridional circulation, whereas in Guerrero & de Gouveia Dal Pino (2008) it is the exact reverse. In our model the meridional flow is very shallow and limited to only the top $10 \%$ of the SCZ, whereas in the model setup of Guerrero & de Gouveia Dal Pino (2008), the meridional flow penetrates down to about 0.8 $R_\odot$; this we believe makes their dynamo cycle periods more sensitive to meridional flow as compared to latitudinal pumping. Generally, we find solar-like solutions in a modest turbulent pumping speed range on the order of 1 ms$^{-1}$. This parameter study shows that our result are robust to reasonable variations in turbulent pumping coefficients and also points to how the latter may determine solar cycle strength and periodicity. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --   ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- As there is some uncertainty regarding the exact details of turbulent pumping profiles, we have tested an alternative turbulent pumping profile based on Warnecke *et al.* (2016). Recent magnetoconvection simulations performed by Warnecke et al. (2016) suggest that radial pumping is downward throughout the convection zone below $45^{\circ}$ and upward above $45^{\circ}$, while latitudinal pumping is poleward at the surface and equatorward at the base of the convection zone. Our generated turbulent pumping profiles in the northern hemisphere (defined within $0 \leq \theta \leq \pi/2$) based on the suggestions of Warnecke *et al.* (2006) are: $$\begin{aligned} \gamma_r = - \gamma_{0r} \left[ 1 + \rm{erf}\left( \frac{r - 0.715R_\odot}{0.05R_\odot}\right) \right] \left[ 1 - \rm{erf} \left( \frac{r-0.98R_\odot}{0.08R_\odot}\right) \right] \nonumber \\ \times \sin(4 \theta), ~~~~\end{aligned}$$ $$\begin{aligned} \gamma_\theta = \left\{\begin{array}{cc} \gamma_{0\theta} \sin \left[\frac{2 \pi (r- R_p)}{R_0-R_p}\right] \times \cos \theta \sin^4 \theta ~~~~ & r \geq R_p\\ 0 & r<R_p \end{array}\right.,\end{aligned}$$ where $R_p= 0.76R_\odot$ i.e. the penetration depth of the latitudinal pumping. The amplitudes of $\gamma_r$ and $\gamma_\theta$ are determined by the value of $\gamma_{0r}$ and $\gamma_{0\theta}$ respectively. Turbulent pumping profiles in the southern hemisphere are generated by replacing colatitude $\theta$ by $(\pi- \theta)$. Fig. 8 (the top and middle panels) show that our generated turbulent pumping profiles capture the basic essence of the suggestions made by Warnecke *et al.* (2016). Our simulations (with shallow meridional flow) and the more complex turbulent pumping profile gleaned from Warnecke *et al.* (2016) reproduce broad features of the solar cycle and are qualitatively similar to those detailed earlier. Discussions =========== In summary, we have demonstrated that flux transport dynamo models of the solar cycle based on the Babcock-Leighton mechanism for poloidal field generation does not require a deep equatorward meridional plasma flow to function effectively. In fact, our results indicate that when turbulent pumping of magnetic flux is taken in to consideration, dynamo models can generate solar-like magnetic cycles even without any meridional circulation although the surface magnetic field dynamics does not reach all the way to the polar regions in this case. Our conclusions are robust across a modest range of plausible parameter space for turbulent pumping coefficients and also indicate some tolerance for diverse pumping profiles. These findings have significant implications for our understanding of the solar cycle. First of all, the serious challenges that were apparently posed by observations of a shallow (and perhaps complex, multi-cellular) meridional flow on the very premise of flux transport dynamo models stands resolved. Turbulent pumping essentially takes over the role of meridional circulation by transporting magnetic fields from the near-surface solar layers to the deep interior, ensuring that efficient recycling of toroidal and poloidal field components across the SCZ is not compromised. While these findings augur well for dynamo models of the solar cycle, they also imply that we need to revisit many aspects of our current understanding if indeed meridional circulation is not as effective as previously thought. For example, our simulations indicate that variations in turbulent pumping speeds can be an effective means for the modulation of solar cycle periodicity and amplitude. It has been argued earlier that the interplay between competing flux transport processes determine the dynamical memory of the solar cycle governing solar cycle predictability [@yeat08]. If turbulent pumping is the dominant flux transport process as seems plausible based on the simulations presented herein, the cycle memory would be short and this is indeed supported by independent studies [@kar12] and solar cycle observations [@munoz13]. It is noteworthy that on the other hand, if meridional circulation were to be the dominant flux transport process, the solar cycle memory would be relatively longer and last over several cycles. This is not borne out by observations. Previous results in the context of the maintenance of solar-like dipolar parity have relied on a strong turbulent diffusion to couple the Northern and Southern hemispheres of the Sun [@chat04], or a dynamo $\alpha$-effect which is co-spatial with the deep equatorward counterflow in the meridional circulation assumed in most flux transport dynamo models [@dikp01]. However, our results indicate that turbulent pumping is equally capable of coupling the Northern and Southern solar hemispheres and aid in the maintenance of solar-like dipolar parity. This is in keeping with earlier, independent simulations based on a somewhat different dynamo model [@guer08]. Most importantly, our results point out an alternative to circumventing the Parker-Yoshimura sign rule constraint [@park55; @yosh75] in Babcock-Leighton type solar dynamos that would otherwise imply poleward propagating sunspot belts in conflict with observations. Brandenburg [*et al.*]{} (1992) and Ossendrijver [*et al.*]{} (2002) had already pointed towards this possibility in the context of mean-field dynamo models. While a deep meridional counterflow is currently thought to circumvent this constraint and force the toroidal field belt equatorward, our results convincingly demonstrate that the latitudinal component of turbulent pumping provides a viable alternative to overcoming the Parker-Yoshimura sign rule in Babcock-Leighton models of the solar cycle (even in the absence of meridional circulation). We note however that our theoretical results should not be taken as support for the existence of a shallow meridional flow, rather we point out that flux transport dynamo models of the solar cycle are equally capable for working with a shallow or non-existent meridional flow, as long as the turbulent pumping of magnetic flux is accounted for; this is particularly viable when turbulent pumping has a dynamically important latitudinal component. Taken together, these insights suggest a plausible new paradigm for dynamo models of the solar cycle, wherein, turbulent pumping of magnetic flux effectively replaces the important roles that are currently thought to be mediated via a deep meridional circulation within the Sun’s interior. Since the dynamical memory and thus predictability of the solar cycle depends on the dominant mode of magnetic flux transport in the Sun’s interior, this would also imply that physics-based prediction models of long-term space weather need to adequately include the physics of turbulent pumping of magnetic fields. We acknowledge the referee of this manuscript for useful suggestions. We thank Jörn Warnecke for helpful discussions related to the adaptation of the turbulent pumping profile from Warnecke [*et al.*]{} 2016. We are grateful to the Ministry of Human Resource Development, Council for Scientific and Industrial Research, University Grants Commission of the Government of India and a NASA Heliophysics Grand Challenge Grant for supporting this research. natexlab\#1[\#1]{} Augustson, K., Brun, A.S., Miesch, M. & Toomre, J. 2015, ApJ, 809, 149 Babcock, H.W. 1961, ApJ, 133, 572 Brandenburg, A., Moss, D. & Tuominen, I. 1992, in The Solar Cycle, ed. K. L., Harvey, ASP Conf. Ser., 27, 542 Brandenburg, A., Jennings, R.L., Nordlund, A., Rieutord, M., Stein, R.M. & Tuominen, I. 1996, JFM, 306, 325 Brandenburg, A. 2005,ApJ, 625, 539 Belusz, B., Dikpati, M. & Forgacs-Dajka 2015, ApJ, 806, 169 Charbonneau, P., Dikpati, M. 2000, ApJ, 543, 1027 Charbonneau, P. 2010, Living Rev. Sol. Phys., 7, 3 Chatterjee, P., Nandy, D., & Choudhuri, A.R. 2004, A&A, 427, 1019 Choudhuri, A. R., Schüssler, M., & Dikpati, M. 1995, A&A, 303, L29 Choudhuri, A.R., Chatterjee, P., & Nandy, D. 2004, ApJL, 615, L57 Choudhuri, A.R. & Karak, B.B. 2012, PhRvL, 109, 171103 Dasi-Espuig, M. [*et al.*]{} 2010, A&A, 518, A7 Dikpati, M., & Charbonneau, P. 1999, ApJ, 518, 508 Dikpati, M., & Gilman, P.A. 2001, ApJ, 559, 428 Dikpati, M., Corbard, T., Thompson, M.J. & Gilman, P.A. 2002, ApJ, 575, L41 Dorch, S.B.F. & Nordlund, A. 2001, A&A, 365, 562 Durney, B.R. 1997, ApJ, 486, 1065 Gilman, P.A. 1983, ApJ, 53, 243 Glatzmaier, G.A. 1985, ApJ, 291, 300 Guerrero, G. & de Gouveia Dal Pino, E.M. 2007, A&A, 464,341 Guerrero, G. & de Gouveia Dal Pino, E.M. 2008, A&A, 485, 267 Hale, G.E. 1908, ApJ, 133, 572 Hale, G.E., Ellerman, F., Nicholson, F.R. & Joy, A.B. 1919, ApJ, 49, 153 Hathaway, D. H., Nandy, D., Wilson, R. M., & Reichmann, E. J. 2003, ApJ, 589, 665. Hathaway, D.H. & Rightmire, L. 2010, Science 327, 1350 Hathaway, D.H. & Rightmire, L. 2011,  729, 80 Hathaway, D.H. , 2012, ApJ, 760, 84 Hazra, S. & Nandy, D. 2013, In ASI Conf. Ser. 10, International Symposium on Solar Terrestrial Physics, ed. N. Gopalswamy, S.S. Hasan, P.B. Rao & P. Subramanian, 115 Hazra, G., Karak, B.B. & Choudhuri, A.R. 2014, ApJ, 782, 93 Hazra, S., Passos, D. & Nandy, D. 2014, ApJ, 789, 5 Hazra, S. 2016, PhD Thesis, IISER Kolkata, arXiv:1604.00563 Jiang, J., Cameron, R. H., Schmitt, D. & Isik, E. 2013, A&A, 553, A128 Käpylä, P.J., Korpi, M.J., Tuominen, I. 2006, Astron Nachr, 327, 884 Karak, B.B., & Nandy, D. 2012, ApJL, 296, 1671 Leighton, R.B. 1969, ApJ, 156, 1 Mitra-Kraev, U., & Thompson, M.J. 2007, Astronomische Nachrichten, 328,1009 Moreno-Insertis, F. 1983, A&A, 122, 241 Moreno-Insertis, F., Schüssler, M., Ferriz-Mas, A., A&A, 264, 686 Muñoz-Jaramillo, A., Nandy, D., & Martens, P. C. H. 2009, ApJ, 698, 461 Muñoz-Jaramillo, A., Nandy, D., Martens, P.C.H. & Yeates, A.R. 2010, ApJL, 720, L20 Muñoz-Jaramillo, A., Dasi-Espuig, M., Balmceda, L.A., Kirova, N.A., & Deluca, E.E. 2013, ApJL, 767, L25 Nandy, D., & Choudhuri, A. R. 2001, ApJ, 551, 576 Nandy, D., & Choudhuri, A. R. 2002, Science, 296, 1671 Nandy, D., Mu[ñ]{}oz-Jaramillo, A., & [Martens]{}, P. C. H. 2011, Nature, 471, 80 Ossendrijver, M., Stix, M., Brandenburg, A., & R[ü]{}diger, G. 2002, A&A, 394, 735 Ossendrijver M 2003, A&A Rev, 11, 287 Parker, E.N. 1955, ApJ, 122, 293 Parker, E.N. 1975, ApJ, 202, 523 Passos, D., Nandy, D., Hazra, S., & Lopes, I. 2014, A&A, 563, A18 Passos, D., Charbonneau, P. & Miesch, M. 2015, ApJL, 800, L18 Pipin, V.V. & Seehafer, N. 2009, ApJ, 493, 819 Racine, E., Charbonneau, P., Bouchat, A. & Smolarkiewicz, P.K. 2011, ApJ, 735, 46 Rogachevskii, I., Kleeorin, N., Käpylä, P.J. & Brandenburg, A. 2011, Phys Rev E, 84, 056314 Simard, C., Charbonneau, P. & Dube, C. 2016, ArXiv 1604.01533v1 Tobias, S.M., Brummell, N.H., Clune, T.L. & Toomre, J. 2001, ApJ, 549, 1183 Warnecke, J., Rheinhardt, M., Käpylä, M.J, Käpylä, P.J. & Brandenburg, A. 2016, ArXiv 1601.03730 Yoshimura, H. 1975, ApJ, 201, 740 Yeates, A.R., Nandy, D. & Mackay, D.H. 2008, ApJ, 673, 544 Zhao, J., [Bogart]{}, R.S., Kosovichev, A.G., Duvall, Jr., T.L. & Hartlep, T. 2013, ApJL, 774, L29

--- abstract: 'We investigated the spin coherence of high-mobility two-dimensional electron gases confined in multilayer GaAs quantum wells. The dynamics of the spin polarization was optically studied using pump-probe techniques: time-resolved Kerr rotation and resonant spin amplification. For double and triple quantum wells doped beyond the metal-to-insulator transition, the spin-orbit interaction was tailored by the sample parameters of structural symmetry (Rashba constant), width and electron density (Dresselhaus linear and cubic constants) which allows us to attain long dephasing times in the nanoseconds range. The determination of the scales: transport scattering time, single-electron scattering time, electron-electron scattering time, and spin polarization decay time further supports the possibility of using n-doped multilayer systems for developing spintronic devices.' author: - 'S. Ullah' - 'G. M. Gusev' - 'A. K. Bakarov' - 'F. G. G. Hernandez' bibliography: - 'JAP\_FGGH\_final.bib' title: | Long-lived nanosecond spin coherence in high-mobility 2DEGs\ confined in double and triple quantum wells --- Introduction ============ Long-lived spin coherence time ($T_{2}^{*}$) for ensembles is a milestone for practical applications of spintronic devices.[@wu] The tunability of $T_{2}^{*}$ have been widely studied in semiconductor quantum wells (QWs) with a large variety of optical techniques developed for the study of spin polarization dynamics and spin relaxation mechanisms.[@5; @6; @7; @8] In n-type samples, for example, it was observed that the doping level has a major role to attain long coherence time or to limit it with $T_{2}^{*}$ changing from tens of picoseconds up to nanoseconds.[@13; @22; @28; @30] The turning point, where $T_{2}^{*}$ decreases with an increase of the electron concentration, was found at the metal-to-insulator transition for bulk[@dzhioev02; @romer] ($2\times10^{16}$ cm$^{-3}$) and GaAs QWs[@sandhu] ($5\times10^{10}$ cm$^{-2}$). Beyond this point, the Dyakonov-Perel (DP) spin relaxation mechanism is dominant and controlled by electron-electron collisions.[@leyland] The DP mechanism defines that the decay time of the spin polarization $t_z$ (along the QW out-of-plane direction) is limited by the spin-orbit interaction which give us a path to control spin coherence. It can be calculated according to: $t_z^{-1}=8D_sm^2\hbar^{-4}[\alpha^{2}+(\beta_1-\beta_3)^2+\beta_3^2)]$, where $D_s$ is the spin diffusion constant, $\alpha$ is the Rashba coefficient due to structural inversion asymmetry, and $\beta_1$ and $\beta_3$ are the linear and cubic Dresselhaus constants due to bulk inversion asymmetry.[@walser; @kainz] Recently, the authors demonstrated that multilayer QWs are exceptional platforms for the investigation of current-induced spin polarization effects.[@31; @3] While such complex systems also offer new possibilities for applications, for example in the production of spin blockers[@souma] and filters,[@23] the study of long-lived spin coherence in double (DQW) and triple quantum wells (TQW) is still required. Here, we report on the coherent spin dynamics in multilayer quantum wells using time-resolved Kerr rotation (TRKR) and resonant spin amplification (RSA). The sample structure allowed us to tailor the spin-orbit constants by the well width, symmetry and subband concentration parameters. Remarkably, it results in coherence times in the nanoseconds range even for DQW and TQW samples with individual subband density beyond the metal-to-insulator transition. Materials and Experiment ======================== We investigated two different samples grown in the \[001\] direction, one double and one triple quantum well, both containing a dense two-dimensional electron gas (2DEG) with equal total density. For both samples, the barriers were made of short-period AlAs/GaAs superlattices (SPSL) in order to shield the doping ionized impurities and efficiently enhance the mobility.[@bakarov] The density of the Si delta-doping was $2.2\times10^{12}$cm$^{-2}$ symmetrically separated from the QW by 7 periods of the SPSL with 4 AlAs monolayers and 8 GaAs monolayers per period. The DQW sample consists of a wide doped GaAs well with w $=45$ nm, total electron density $n_{t}=9.2\times10^{11}$cm$^{-2}$ and mobility $\mu=1.9\times10^{6}$cm$^2$/Vs at low temperature. The electronic system has a DQW configuration with symmetric and antisymmetric wave functions for the two lowest subbands with subband separation $\Delta_{12}=1.4$ meV and approximately equal subband density $n_{s}$.[@wiedmannDQW] Fig. 1(a) shows the calculated DQW band structure and the charge density for both subbands. The second sample is a symmetrically doped GaAs TQW with 2 nm-thick $Al_{0.3}Ga_{0.7}As$ barriers, $n_{t}=9\times10^{11}$cm$^{-2}$ and $\mu=5\times10^{5}$cm$^2$/Vs measured at low temperature. The central well width is 22 nm and both side wells have equal width of 10 nm. The central well has a larger width in order to be populated because the electron density tends to concentrate mostly in the side wells as a result of electron repulsion and confinement. The estimated density in the central well is 35% smaller than in the side wells. The coupling strength between the quantum wells is characterized by the separation energies $\Delta_{ij}$ of the three occupied subbands ($i,j=1,2,3$) given by $\Delta_{12}=1.0$ meV, $\Delta_{23}=2.4$ meV, $\Delta_{13}=3.4$ meV.[@wiedmannTQW] TRKR and RSA were used to probe the coherent spin dynamics in the electron gas. For optical excitation, we used a mode-locked Ti-sapphire laser with pulse duration of 100 fs and repetition rate of $f_{rep}=$ 76 MHz corresponding to a repetition period ($t_{rep}$) of 13.2 ns. The time delay $\Delta t$ between pump and probe pulses was varied by a mechanical delay line. The pump beam was circularly polarized by means of a photo-elastic modulator operated at a frequency of 50 kHz. The rotation of the probe polarization was recorded as function of $\Delta t$ and detected with a balanced bridge using coupled photodiodes. The laser wavelength was tuned looking for the TRKR energy dependence in each sample. The samples were immersed in the variable temperature insert of a split-coil superconductor magnet in the Voigt geometry. Results and Discussion ====================== The time evolution of the spin dynamics for the DQW is displayed in Fig. 1(b) up to 2 T with pump/probe power of 1 mW/300 $\mu$W. The TRKR oscillations are associated with the precession of coherently excited electron spins about an in-plane magnetic field. To obtain the spin coherence time, the evolution of the Kerr rotation angle can be described by an exponentially damped harmonic: $$\theta_{K}(\Delta t) = A \exp(-\Delta t/T_{2}^{*})\cos(\omega_{L}\Delta t + \phi)$$ where $A$ is the initial spin polarization build-up by the pump, $\phi$ is the oscillation phase, and $\omega_{L}=g\mu_{B}$B/$\hbar$ is the Larmor frequency with magnetic field B, electron g-factor (absolute value) $g$, Bohr magneton $\mu_{B}$, and reduced Planck’s constant $\hbar$. The magnetic field dependence of $\omega_{L}$ and T$_{2}^{*}$ are shown in Fig. 1(c) and (d). Solid lines are fits to the data. One can clearly see that the spin precession frequency increases with B following the linear dependence of the Larmor frequency on the applied field. The value of the fitted g-factor is 0.453 which is close to absolute value for bulk GaAs and similar to the value measured for a quase-two-dimensional system in a single barrier heterostructure with two-subbands occupied.[@zhangEPL] {width="1\columnwidth"} According to the Dyakonov-Perel mechanism, the observed exponential decay at B $=$ 0 corresponds to the strong scattering regime. In the opposite case, where the spin precess more than a revolution before being scattered, an oscillatory behavior would be expected.[@leyland; @brand] The measured value for the decay time of the spin polarization along the z-direction (out-of-plane) is 1.1 ns at zero external field. For our symmetric, wide and dense quantum well; we estimate $\alpha\simeq0$, $\beta_1=-\gamma(\pi/w)^2=0.49\times10^{-13}$eVm, and $\beta_3=-\frac{1}{2}\gamma\pi n_s=0.70\times10^{-13}$eVm for the first subband using $\gamma=-$10 eV$\AA^3$.[@walserprb] The charge diffusion constant can be estimated, using the effective mass $m$ and the electron’s charge $e$, from the Fermi velocity $v_F=\hbar\sqrt{2\pi n_s}/m$ and the transport scattering time $\tau=\mu m/e=70$ ps by $D_c=v_F^2\tau/2=$ 3 m$^2$/s. The diffusion constant for spins is approximately two orders of magnitude smaller than for charge.[@walser] Scaling $D_s=100$ to 300 cm$^2$/s, we obtain $t_z\sim[8D_sm^2\hbar^{-4}\beta_3^2]^{-1}=$ 1.1 to 3.3 ns. The data at B $=$ 0 thus agrees with a DP mechanism where the spin dynamics is dominated by the cubic Dresselhaus term. The cancellation of $\alpha\simeq0$ and $\beta_1-\beta_3\simeq0$ due to the sample parameters shows a practical path for long-lived spin coherence in highly doped QWs. Increasing the magnetic field up to 0.5 T, we found a systematic increase of T$_{2}^{*}$. In this situation, the cyclotron motion acts as momentum scattering and leads to a less efficient spin relaxation in agreement with the DP model.[@griesbeck] It is important to note that the in-plane magnetic field was applied using Voigt configuration and the the cyclotron motion is perpendicular to the QW plane. The increase follows a quadratic dependence[@41] with $T_{2}^{*}(B)/T_{2}^{*}(0)=1+(\omega_{c}\tau_p^*)^2$ where $\omega_{c}$ is the cyclotron frequency and $\tau_p^*$ is the single-electron momentum scattering time. We found $\tau_p^*=$ 0.92 ps in agreement with the magnitude of the quantum lifetime measured by transport from the Dingle factor of the magneto-intersubband oscillations on the same sample.[@wiedmann2010] The value for $\tau_p^*$ is also in agreement with the determination of approximately 0.5 ps for QWs of shorter width.[@walserprb] One of the reasons for the large difference between $\tau$ and $\tau_p^*$ is the insensibility of the first to electron-electron scattering. The ratio of $\tau/\tau_p^*\simeq100$ implies that the dominant scattering from impurities is due to remote instead of background impurities.[@macleod] If we consider that $1/\tau_p^*=1/\tau+1/\tau_{ee}$, we get a time scale of $\tau_{ee}=\tau_p^*$ which demonstrates that electron-electron collisions dominate the microscopic scattering mechanisms as expected.[@leyland] Additionally, a further increase of the magnetic field leads to a strong decay due to the spread of the g-factor within the measured ensemble.[@greilichprl2006; @zhukov2007] The size of the inhomogeneity $\Delta g$ can be inferred by fitting the data according to $1/T_{2}^{*}(B)=1/T_{2}^{*}(0)+\Delta g\mu_BB/\sqrt{2}\hbar$ as shown in Fig. 1(d). From the 1/B dependence,[@22; @28] we obtain $\Delta g=0.002$ or 0.44% and $T_{2}^{*}(0)=$ 2 ns. The optical power influence on the spin dynamics for the DQW sample is shown in Fig. 2(a) at 1 T. Only at low pump power, we observed negative delay oscillations of considerably large amplitude. To find electron spin polarization before the pump pulse arrival indicates that the spin polarization persists from the previous pump pulse, which took place $t_{rep}=13.2$ ns before. The excitation power dependence of T$_{2}^{*}$ was plotted in Fig. 2(b) yielding an exponential decay. For single QW structures, the decrease of the coherence time at high pump density was associated with the electrons delocalization caused by their heating due to the interaction with the photogenerated carriers.[@zhukov2007] A similar decrease was also attributed to an increased efficiency of the Bir-Aronov-Pikus mechanism induced by the larger hole photogenerated density in GaAs QWs.[@37] However, it is unlikely to be relevant in our dense 2DEG where the photogenerated hole loses its spin and energy quickly and fast recombines with an electron from the 2DEG. Nevertheless, being a key parameter for spin devices, we note that T$_{2}^{*}$ remains near the nanoseconds range when the power is raised by almost one order of magnitude. At higher excitation power, an additional short-lived component in the signal becomes more significant. In systems where T$_{2}^{*}$ is comparable or longer than the laser repetition period, one can use the RSA technique to extract the spin dephasing time by scanning the magnetic field at a fixed pump-probe delay.[@22] We note that the 2DEG dynamics is associated with the long lasting oscillations, rather than with excitons or photo-excited electrons.[@zhukov2007] {width="1\columnwidth"} {width="1\columnwidth"} Fig. 3(a) displays the RSA signals measured for different $\Delta t$ with pump/probe power of 1 mW/300 $\mu$W. We observed a series of sharp resonance peaks as a function of B corresponding to the electron spin precession frequencies which are commensurable with the pump pulse repetition period obeying the periodic condition: $\Delta B = (h f_{rep})/(g \mu_B)$.[@22] As function of the magnetic field, the RSA peaks amplitude decreases as a result of the g-factor variation within the measured ensemble as noted above. The RSA resonances are modulated by a slow oscillation that depends on $f_{d}=1/\Delta t$ according to the same periodic condition. We will focus on the zero field resonance. T$_{2}^{*}$ can be directly evaluated from the width of the zero-th resonance using a Hanle (Lorentzian) model:[@3; @22] $$\theta_{K}=A/[(\omega_{L}T_{2}^{*})^{2}+1]$$ with half-width $B_{1/2}=\hbar/(g\mu_{B}T_{2}^{*})$. The fitting result is displayed in Fig. 3(b) for negative and positive delays. The extracted values for T$_{2}^{*}$ and for the amplitude are shown in Fig. 3(c) and (d) as function of $\Delta t$. For positive delays, both quantities display an exponential decay (solid line). Increasing the pump-probe delay cause the broadening of RSA peaks according to a shorter spin dephasing time. However, the system coherence is recovered just before pump arrival for the long-lived spin component in the system dynamics.[@yugova2012] The RSA amplitude measured at negative delay was T$_{2}^{*}=$ 4.4 ns. {width="1\columnwidth"} Concerning the subband dependence, the spin relaxation time was calculated to be identical in an electron system with two occupied subbands, although the higher subband may have a much larger inhomogeneous broadening, due to strong intersubband Coulomb scattering.[@weng; @zhangEPL] In our samples, we studied the pump/probe wavelength dependence as reported in TRKR[@zhangEPL] and photoluminescence[@pussep2; @pussep3] studies on similar multilayer systems. Figure 4(a) displays the RSA scans of the DQW sample for different pump-probe wavelengths at fixed delay. Panel 4(b) shows a comparison between the zero-field resonances where the solid line is a Hanle fit to the data as described above. T$_{2}^{*}$ and the amplitude obtained from (b) increase with the pump-probe wavelength as shown in Figures 4(c) and 4(d). Increasing the pump-probe energy about 3 meV ($\simeq2\Delta_{12}$), from 817 nm to 815 nm, leads to a T$_{2}^{*}$ decrease of less than 10% in Figure 4(d). In comparison, Figure 1 shows negative delay oscillations in the same wavelength range.[@SM] This small change could be associated with the relative similitude between the charge density distribution for both subbands. On the other side, fast intersubband scattering may be hiding differences expected in the spin-orbit interaction for the second subband.[@egues] {width="1\columnwidth"} Finally, we focus on the results for the TQW sample. Fig. 5(a) shows the calculated band diagram and charge density for three occupied subbands. The TRKR scans measured as function of the magnetic field yield g $=$ 0.452. Due to the long spin coherence comparable with the laser repetition period, there is almost no decay over the measured time window (2.5 ns). In analogy to the DQW sample, we used the constructive interference of the coherence oscillations from successive pulses to extract the spin coherence time by the RSA technique. Fig. 5(c) and (d) show the magnetic field scans of the KR amplitude performed at different pump/probe separation for 821 and 823 nm, respectively. From the Lorentzian fit of the zero-field peak, the spin dephasing for the TQW sample was obtained revealing the longest T$_{2}^{*}=$ 10.42 ns at negative delay as for the DQW. In this case, the same energy increase ($\sim$3 meV $\simeq\Delta_{13}$), leads to strong T$_{2}^{*}$ decrease of almost 50%/30% at negative/positive delay. We note that, contrary to the DQW case, the third subband for the TQW have opposite charge distribution if compared with the lower subbands. While the third subband has the charge density more localized in the central well, the electrons in the first and second subbands are distributed in the side wells. Conclusions =========== In conclusion, we have studied the spin dynamics of a two-dimensional electron gas in multilayer QWs by TRKR and RSA. The dependence of spin dephasing time on the experimental parameters: magnetic field, pump power, and pump-probe delay was demonstrated. In the DQW sample, T$_{2}^{*}$ extends to 4.4 ns. Additionally, for the TQW sample, T$_{2}^{*}$ exceeding 10 ns was observed. The results found are among the longest T$_{2}^{*}$ reported for samples of similar doping level [@sandhu; @41] and comparable with nominally undoped narrow GaAs QWs [@dzhioev] and low density 2DEGs in CdTe QWs [@zhukov2007]. The measured long spin dephasing time was tailored by the control of the QW width, symmetry and electron density. The spin dynamics is dominated through the cubic Dresselhaus interaction by the DP mechanism. All the relevant time scales were determined indicating the importance of each scattering mechanism in the spin dynamics. We demonstrate that the wave function engineering in multilayer QWs may provide practical paths to control the dynamics in spintronic devices. Acknowledgments =============== F.G.G.H. acknowledges financial support from Grants No. 2009/15007-5, No. 2013/03450-7, and No. 2014/25981-7 of the São Paulo Research Foundation (FAPESP). S.U. acknowledges TWAS/CNPq for financial support.

--- abstract: 'We consider the problem of predicting the future trajectory of scene agents from egocentric views obtained from a moving platform. This problem is important in a variety of domains, particularly for autonomous systems making reactive or strategic decisions in navigation. In an attempt to address this problem, we introduce TITAN (Trajectory Inference using Targeted Action priors Network), a new model that incorporates prior positions, actions, and context to forecast future trajectory of agents and future ego-motion. In the absence of an appropriate dataset for this task, we created the TITAN dataset that consists of 700 labeled video-clips (with odometry) captured from a moving vehicle on highly interactive urban traffic scenes in Tokyo. Our dataset includes 50 labels including vehicle states and actions, pedestrian age groups, and targeted pedestrian action attributes that are organized hierarchically corresponding to atomic, simple/complex-contextual, transportive, and communicative actions. To evaluate our model, we conducted extensive experiments on the TITAN dataset, revealing significant performance improvement against baselines and state-of-the-art algorithms. We also report promising results from our Agent Importance Mechanism (AIM), a module which provides insight into assessment of perceived risk by calculating the relative influence of each agent on the future ego-trajectory. The dataset is available at <https://usa.honda-ri.com/titan>' author: - | Srikanth Malla                  Behzad Dariush                  Chiho Choi\ Honda Research Institute USA\ [{smalla, bdariush, cchoi}@honda-ri.com ]{} bibliography: - 'egbib.bib' title: 'TITAN: Future Forecast using Action Priors' --- Introduction ============ The ability to forecast future trajectory of agents (individuals, vehicles, cyclists, etc.) is paramount in developing navigation strategies in a range of applications including motion planning and decision making for autonomous and cooperative (shared autonomy) systems. ![Our goal is to predict the future trajectory of agents from egocentric views obtained from a moving platform. We hypothesize that prior actions (and implicit intentions) play an important role in future trajectory forecast. To this end, we develop a model that incorporates prior positions, actions, and context to forecast future trajectory of agents and future ego-motion. This figure is a conceptual illustration that typifies navigation of ego-vehicle in an urban scene, and how prior actions/intentions and context play an important role in future trajectory forecast. We seek to also identify agents (depicted by the red bounding box) that influence future ego-motion through an Agent Importance Mechanism (AIM) . []{data-label="fig:block_diagram"}](figures1/frontpage.pdf){width="45.00000%"} We know from observation that the human visual system possesses an uncanny ability to forecast behavior using various cues such as experience, context, relations, and social norms. For example, when immersed in a crowded driving scene, we are able to reasonably estimate the intent, future actions, and future location of the traffic participants in the next few seconds. This is undoubtedly attributed to years of prior experience and observations of interactions among humans and other participants in the scene. To reach such human level ability to forecast behavior is part of the quest for visual intelligence and the holy grail of autonomous navigation, requiring new algorithms, models, and datasets. In the domain of behavior prediction, this paper considers the problem of future trajectory forecast from egocentric views obtained from a mobile platform such as a vehicle in a road scene. This problem is important for autonomous agents to assess risks or to plan ahead when making reactive or strategic decisions in navigation. Several recently reported models that predict trajectories incorporate social norms, semantics, scene context, etc. The majority of these algorithm are developed from a stationary camera view in surveillance applications, or overhead views from a drone. The specific objective of this paper is to develop a model that incorporates prior positions, actions, and context to simultaneously forecast future trajectory of agents and future ego-motion. In a related problem, the ability to predict future actions based on current observations has been previously studied in [@lan2014hierarchical; @soomro2016predicting; @singh2017online; @singh2018predicting; @sun2019relational]. However, to the best of our knowledge, action priors have not been used in forecasting future trajectory, partly due to a lack of an appropriate dataset. A solution to this problem can help address the challenging and intricate scenarios that capture the interplay of observable actions and their role in future trajectory forecast. For example, when the egocentric view of a mobile agent in a road scene captures a delivery truck worker closing the tailgate of the truck, it is highly probable that the worker’s future behavior will be to walk toward the driver side door. Our aim is to develop a model that uses such action priors to forecast trajectory. The algorithmic contributions of this paper are as follows. We introduce TITAN (Trajectory Inference using Targeted Action priors Network), a new model that incorporates prior positions, actions, and context to simultaneously forecast future trajectory of agents and future ego-motion. Our framework introduces a new interaction module to handle dynamic number of objects in the scene. While modeling pair-wise interactive behavior from all agents, the proposed interaction module incorporates actions of individuals in addition to their locations, which helps the system to understand the contextual meaning of motion behavior. In addition, we propose to use multi-task loss with aleatoric homoscedastic uncertainty [@kendall2018multi] to improve the performance of multi-label action recognition. For ego-future, Agent Importance Mechanism (AIM) is presented to identify objects that are more relevant for ego-motion prediction. Apart from algorithmic contributions, we introduce a novel dataset, referred to as TITAN dataset, that consists of 700 video clips captured from a moving vehicle on highly interactive urban traffic scenes in Tokyo. The pedestrians in each clip were labeled with various action attributes that are organized hierarchically corresponding to atomic, simple/complex contextual, transportive, and communicative actions. The action attributes were selected based on commonly observed actions in driving scenes, or those which are important for inferring intent (*e.g.*, waiting to cross). We also labeled other participant categories, including vehicle category (4 wheel, 2 wheel), age-groups, and vehicle state. The dataset contains synchronized ego-motion information from an IMU sensor. To our knowledge, this is the only comprehensive and large scale dataset suitable for studying action priors for forecasting the future trajectory of agents from ego-centric views obtained from a moving platform. Furthermore, we believe our dataset will contribute to advancing research for action recognition in driving scenes. {width="\textwidth"} Related Work ============ Future Trajectory Forecast -------------------------- **Human Trajectory Forecast** Encoding interactions between humans based on their motion history has been widely studied in the literature. Focusing on input-output time sequential processing of data, recurrent neural network (RNN)-based architectures have been applied to the future forecast problem in the last few years  [@social_lstm; @lee2017desire; @social_gan; @xu2018encoding; @zhang2019sr]. More recently, RNNs are used to formulate a connection between agents with their interactions using graph structures [@social_attention; @ma2019trafficpredict]. However, these methods suffer from understanding of environmental context with no or minimal considerations of scene information. To incorporate models of human interaction with the environment, [@xue2018ss] takes local to global scale image features into account. More recently, [@choi2019looking] visually extracts relational behavior of humans interacting with other agents as well as environments. **Vehicle Trajectory Forecast** Approaches for vehicle motion prediction have developed following the success of interaction modeling using RNNs. Similar to human trajectory forecast, [@deo2018multi; @park2018sequence; @ma2019trafficpredict; @li2019interaction] only consider the past motion history. These methods perform poorly in complex road environments without the guidance of structured layouts. Although the subsequent approaches [@rhinehart2018r2p2; @li2019conditional; @choi2019drogon] partially overcome these issues by using 3D LiDAR information as inputs to predict future trajectories, their applicability to current production vehicles is limited due to the higher cost. Recent methods [@bhattacharyya2018long; @hevi_dataset; @malla2019nemo] generate trajectories of agents from an egocentric view. However, they do not consider interactions between road agents in the scene and the potential influence to the ego-future. In this work, we explicitly model pair-wise interactive behavior from all agents to identify objects that are more relevant for the target agent. Action Recognition ------------------ With the success of 2D convolutions in image classification, frame-level action recognition has been presented in [@karpathy2014large]. Subsequently, [@simonyan2014two] separates their framework into two streams: one to encode spatial features from RGB images and the other to encode temporal features from corresponding optical flow. Their work motivated studies that model temporal motion features together with spatial image features from videos. A straightforward extension has been shown in [@tran2015learning; @varol2017long], replacing 2D convolutions by 3D convolutions. To further improve the performance of these models, several research efforts have been provided such as I3D [@i3d] that inflates a 2D convolutional network into 3D to benefit from the use of pre-trained models and 3D ResNet [@r3d] that adds residual connections to build a very deep 3D network. Apart from them, other approaches capture pair-wise relations between actor and contextual features [@acrn_cvpr] or those between pixels in space and in time [@non_local_neuralnets]. More recently, Timeception [@hussein2019timeception] models long range temporal dependencies, particularly focusing on complex actions. Datasets -------- **Future Trajectory** \[sec:dataset\_futuretraj\] Several influential RGB-based datasets for pedestrian trajectory prediction have been reported in the literature. These datasets are typically created from a stationary surveillance camera [@ucy_dataset; @eth_dataset; @actEV_dataset], or from aerial views obtained from a static drone-mounted camera [@stanford_drone_dataset]. In driving scenes, the 3D point cloud-based datasets [@kitti_dataset; @h3d_dataset; @lyft_dataset; @nuscenes_dataset; @waymo_open_dataset; @argoverse_dataset] were originally introduced for detection, tracking, etc., but recently used for vehicle trajectory prediction as well. Also, [@hevi_dataset; @chandra2019traphic] provide RGB images captured from an egocentric view of a moving vehicle and applied to future trajectory forecast problem. The JAAD [@jaad_dataset], CMU-UAH [@minguez2018pedestrian], and PIE [@pie_dataset] datasets are most similar to our TITAN dataset in the sense that they are designed to study the intentions and actions of objects from on-board vehicles. However, their labels are limited to simple actions such as walking, standing, looking, and crossing. These datasets, therefore, do not provide an adequate number of actions to use as priors in order to discover contextual meaning of agents’ motion behavior. To address these limitations, our TITAN dataset provides 50 labels including vehicle states and actions, pedestrian age groups, and targeted pedestrian action attributes that are hierarchically organized as illustrated in the supplementary material. **Action Recognition** \[sec:dataset\_action\] A variety of datasets have been introduced for action recognition with a single action label [@hmdb_dataset; @ucf_dataset; @karpathy2014large; @marszalek2009actions; @kay2017kinetics] and multiple action labels [@charades; @yeung2018every; @caba2015activitynet] in videos. Recently released datasets such as AVA [@ava_dataset], READ [@fontana2018action], and EPIC-KITCHENS [@Damen2018EPICKITCHENS] contain actions with corresponding localization around a person or object. Our TITAN dataset is similar to AVA in the sense that it provides spatio-temporal localization for each agent with multiple action labels. However, the labels of TITAN are organized hierarchically from primitive atomic actions to complicated contextual activities that are typically observed from on-board vehicles in driving scenes. {width="\textwidth"} TITAN Dataset ============= In the absence of an appropriate dataset suitable for our task, we introduce the TITAN dataset for training and evaluation of our models as well as to accelerate research on trajectory forecast. Our dataset is sourced from 10 hours of video recorded at 60 FPS in central Tokyo. All videos are captured using a GoPro Hero 7 Camera with embedded IMU sensor which records synchronized odometry data at 100 HZ for ego-motion estimation. To create the final annotated dataset, we extracted 700 short video clips from the original (raw) recordings. Each clip is between 10-20 seconds in duration, image size width:1920px, height:1200px and annotated at 10 HZ sampling frequency. The characteristics of the selected video clips include scenes that exhibit a variety of participant actions and interactions. The taxonomy and distribution of all labels in the dataset are depicted in Figure \[fig:titan\_dataset\]. The total number of frames annotated is approximately 75,262 with 395,770 persons, 146,840 4-wheeled vehicles and 102,774 2-wheeled vehicles. This includes 8,592 unique persons and 5,504 unique vehicles. For our experiments, we use 400 clips for training, 200 clips for validation and 100 clips for testing. As mentioned in Section \[sec:dataset\_futuretraj\], there are many publicly available datasets related to mobility and driving, many of which include ego-centric views. However, since those datasets do not provide action labels, a meaningful quantitative comparison of the TITAN dataset with respect to existing mobility datasets is not possible. Furthermore, a quantitative comparison with respect to action localization datasets such as AVA is not warranted since AVA does not include ego-centric views captured from a mobile platform. In the TITAN dataset, every participant (individuals, vehicles, cyclists, etc.) in each frame is localized using a bounding box. We annotated 3 labels (person, 4-wheeled vehicle, 2-wheeled vehicle), 3 age groups for person (child, adult, senior), 3 motion-status labels for both 2 and 4-wheeled vehicles, and door/trunk status labels for 4-wheeled vehicles. For action labels, we created 5 mutually exclusive person action sets organized hierarchically (Figure \[fig:titan\_dataset\]). In the first action set in the hierarchy, the annotator is instructed to assign exactly one class label among 9 atomic whole body actions/postures that describe primitive action poses such as sitting, standing, standing, bending, etc. The second action set includes 13 actions that involve single atomic actions with simple scene context such as jaywalking, waiting to cross, etc. The third action set includes 7 complex contextual actions that involve a sequence of atomic actions with higher contextual understanding, such as getting in/out of a 4-wheel vehicle, loading/unloading, etc. The fourth action set includes 4 transportive actions that describe the act of manually transporting an object by carrying, pulling or pushing. Finally, the fifth action set includes 4 communicative actions observed in traffic scenes such as talking on the phone, looking at phone, or talking in groups. In each action sets 2-5, the annotators were instructed to assign ‘None’ if there is no label. This hierarchical strategy was designed to produce unique (unambiguous) action labels while reducing the annotators’ cognitive work-load and thereby improving annotation quality. The tracking ID’s of all localized objects are associated within each video clip. Example scenarios are displayed in Figure \[fig:titan\_scenes\]. {width="\textwidth"} Methodology =========== Figure \[fig:block\_diagram\] shows the block diagram of the proposed TITAN framework.A sequence of image patches $\mathcal{I}^i_{t=1:T_{obs}}$ is obtained from the bounding box[^1] $x^i=\{c_u,c_v,l_u,l_v\}$ of agent $i$ at each past time step from 1 to $T_{obs}$, where $(c_u,c_v)$ and $(l_u,l_v)$ represent the center and the dimension of the bounding box, respectively. The proposed TITAN framework requires three inputs as follows: $\mathcal{I}^i_{t=1:T_{obs}}$ for the action detector, $x^i_t$ for both the interaction encoder and past object location encoder, and $e_t=\{\alpha_t,{\omega}_t\}$ for the ego-motion encoder where $\alpha_t$ and $\omega_t $ correspond to the acceleration and yaw rate of the ego-vehicle at time $t$, respectively. During inference, the multiple modes of future bounding box locations are sampled from a bi-variate Gaussian generated by the noise parameters, and the future ego-motions $\hat{e_t}$ are accordingly predicted, considering the multi-modal nature of the future prediction problem. Henceforth, the notation of the feature embedding function using multi-layer perceptron (MLP) is as follows: $\Phi$ is without any activation, and $\Phi_r$, $\Phi_t$, and $\Phi_s$ are associated with ReLU, tanh, and a sigmoid function, respectively. Action Recognition {#sec:action_rec} ------------------ We use the existing state-of-the-art method as backbone for the action detector. We finetune single-stream I3D [@i3d] and 3D ResNet [@r3d] architecture pre-trained on Kinetics-600 [@carreira2018short]. The original head of the architecture is replaced by a set of new heads (8 action sets of TITAN except age group and type) for multi-label action outputs. The action detector takes $\mathcal{I}^i_{t=1:T_{obs}}$ as input, which is cropped around the agent $i$. Then, each head outputs an action label including a ‘None’ class if no action is shown. From our experiments, we observed that certain action sets converge faster than others. This is due in part because some tasks are relatively easier to learn, given the shared representations. Instead of tuning the weight of each task by hand, we adopt the multi-task loss in [@kendall2018multi] to further boost performance of our action detector. Note that each action set of the TITAN dataset is mutually exclusive, thus we consider the outputs are independent to each other as follows: $$p(y_m,..,y_n|f(\mathcal{I}))=\prod_{i=m}^{n} p(y_i|f(\mathcal{I})), \label{eq:multi_task_out}$$ where $y_i$ is the output label of $i^{th}$ action set and $f$ is the action detection model. Then, multi-task loss is defined as: $$\mathcal{L}_a=\sum_{i=m}^{n}\frac{ce(\widehat{cls}_i, {cls}_i)}{\sigma_i^2}+log\ \sigma_i, \label{eq:multi_task_loss}$$ where $ce$ is the cross entropy loss between predicted actions $\widehat{cls}_i$ and ground truth ${cls}_i$ for each label $i=m:n$. Also, $\sigma_i$ is the task dependent uncertainty (aleatoric homoscedastic). In practice, the supervision is done separately for vehicles and pedestrians as they have different action sets. The efficacy of the multi-task loss is detailed in the supplementary material, and the performance of the action detector with different backbone is compared in Table \[tbl:action\_recog\]. Future Object Localization {#sec:fol_methodology} -------------------------- Unlike existing methods, we model the interactions using the past locations of agents conditioned on their actions, which enables the system to explicitly understand the contextual meaning of motion behavior. At each past time step $t$, the given bounding box $x^i_t=\{c_u,c_v,l_u,l_v\}_t$ is concatenated with the multi-label action vector $a^i_t$. We model the pair-wise interactions between the target agent $i$ and all other agents $j$ through MLP, $v^{ij}_t=\Phi_r(x^{i}_t\boxtimes a^{i}_t\boxtimes x^{j}_t \boxtimes a^{j}_t)$ where $\boxtimes$ is a concatenation operator. The resulting interactions $v_t^{ij}$ are evaluated through the dynamic RNN with GRUs to leave more important information with respect to the target agent, $h^{i(j+1)}_t = GRU(v^{ij}_t, h_t^{ij};W_{\textnormal{INT}})$, where $W_{\textnormal{INT}}$ are the weight parameters. Note that we pass the messages of instant interaction with each agent at time $t$, which enables us to find their potential influence at that moment. Then, we aggregate the hidden states to generate interaction features $\psi_{t}^i=\frac{1}{n}\sum_i h^{ij}_t$ for the target agent $i$, computed from all other agents in the scene at time $t$ as in Figure \[fig:interaction\_encoder\]. ![Interaction encoding for agent $i$ against others at time $t$.[]{data-label="fig:interaction_encoder"}](figures1/interaction_encoder.png){width="47.00000%"} The past ego motion encoder takes $e_t=(\alpha_t,\omega_t$) as input and embeds the motion history of ego-vehicle using the GRU. We use each hidden state output $h_t^e$ to compute future locations of other agents. The past object location encoder uses the GRU to embed the history of past motion into a feature space. The input to this module is a bounding box $x^i$ of the target agent $i$ at each past time step, and we use the embedding $\Phi(x^i_t)$ for the GRU. The output hidden state $h_t^p$ of the encoder is updated by $\hat{h_{t}^p}=\Phi(H_{t}^{xi}\boxtimes h_t^p)$, where $H_{t}^{xi}=\Phi_r(a^i_{t}) \boxtimes \psi_{t}^i \boxtimes \Phi_r(h_t^e)$ is the concatenated information. Then, $\hat{h_{t}^p}$ is used as a hidden state input to the GRU by $h_{t+1}^p=GRU(\hat{h_t^p},\Phi(x_t^i);W_{\textnormal{POL}})$, where $W_{\textnormal{POL}}$ are the weight parameters. We use its final hidden state as an initial hidden state input of the future object location decoder. The future bounding boxes of the target agent $i$ are decoded using the GRU-based future object location decoder from time step $T_{obs}+1$ to $T_{pred}$. At each time step, we output a 10-dimensional vector where the first 5 values are the center $\mu_c=(c_u, c_v)$, variance $\sigma_c =(\sigma_{cu}, \sigma_{cv})$, and its correlation $\rho_c$ and the rest 5 values are the dimension $\mu_l=(l_u, l_v)$, variance $\sigma_l=(\sigma_{lu}, \sigma_{lv})$, and its correlation $\rho_l$. We use two bi-variate Gaussians for bounding box centers and dimensions, so that they can be independently sampled. We use the negative log-likelihood loss function as: $$\begin{split} \mathcal{L}_O = -\frac{1}{T}\sum_{t=T_{obs}+1}^{T_{pred}} &log\ p(c| \mu_{c}^t,\sigma_{c}^t, \rho_{c}) p(l| \mu_{l}^t,\sigma_{l}^t,\rho_{l}). \end{split}$$ Future Ego-motion prediction ---------------------------- We first embed the predicted future bounding box of all agents $\hat{X}=\{\hat{x}^1,...,\hat{x}^N\}$ through MLP at each future time step $T_{obs}+1$ to $T_{pred}$. We further condition it on the previously computed action labels in a feature space through $H_t^{ei} = \Phi(r^i_{T_{obs}}\boxtimes \Phi_r(\hat{x}_t^i))$, where $r^i_{T_{obs}}=\Phi_r(a_{T_{obs}}^i)$. By using the action labels as a prior constraint, we explicitly lead the model to understand about the contextual meaning of locations. The resulting features of each agent $i$ are weighted using the AIM module $\hat{H}_t^{ei} = w_t^i*{H}_t^{ei}$, where the weights $w_t^i = \Phi_t(H_t^{ei})$, similar to self-attention [@selfattention]. Then, we sum all features $H^e_t=\sum_i \hat{H}_t^{ei}$ for each future time step. This procedure is detailed in Figure \[fig:attention\_module\]. Note that our AIM module is simultaneously learned with the future ego-motion prediction, which results in weighting other agents more or less based on their influence/importance to the ego-vehicle. It thus provides insight into assessment of perceived risk while predicting the future motion. We qualitatively evaluate it in Sec. \[experiments\]. ![ Agent Importance Mechanism (AIM) module.[]{data-label="fig:attention_module"}](figures1/attention_module.png){width="47.00000%"} The last hidden state $h_{T}^e$ of the past ego motion encoder is concatenated with $H^e_t$ through $\hat{h}_{T}^e=\Phi(H^{e}_t\boxtimes h_{T}^e)$ and fed into the future ego motion decoder. The intermediate hidden state $h_t^f$ is accordingly updated by $H^e_t$ at each future time step for recurrent update of the GRU. We output the ego-future using each hidden state $h_t^f$ through $\hat{e}_{t}^i=\Phi(h_t^f)$ at each future time $T_{obs}+1$ to $T_{pred}$. For training, we use task dependent uncertainty with L2 loss for regressing both acceleration and angular velocity as shown below: $$\mathcal{L}_E = \frac{{\left\lVert\alpha_t-\hat{\alpha}_t\right\rVert}^2}{\sigma_1^2}+\frac{{\left\lVert\omega_t-\hat{\omega}_t\right\rVert}^2}{\sigma_2^2}+log \sigma_1 \sigma_2.$$ Note that the predicted future ego-motion is deterministic in its process. However, its multi-modality comes from sampling of the predicted future bounding boxes of other agents. In this way, we capture their influence with respect to the ego-vehicle, and AIM outputs the importance weights consistent with the agents’ action and future motion. {width="98.00000%"} \[fig:ours\] {width="\textwidth"} Experiments =========== In all experiments performed in this work, we predict up to 2 seconds into the future while observing 1 second of past observations as proposed in  [@malla2019nemo]. We use average distance error (ADE), final distance error (FDE), and final intersection over union (FIOU) metrics for evaluation of future object localization. We include FIOU in our evaluation since ADE/FDE only capture the localization error of the final bounding box without considering its dimensions. For action recognition, we use per frame mean average precision (mAP). Finally, for ego-motion prediction, we use root mean square error (RMSE) as an evaluation metric. Action Recognition {#action-recognition-1} ------------------ We evaluate two state-of-the-art 3D convolution-based architectures, I3D with InceptionV1 and 3D ResNet with ResNet50 as backbone. Both models are pre-trained on Kinetics-600 and finetuned using TITAN with the multi-task loss in Eqn. \[eq:multi\_task\_loss\]. As detailed in Sec. \[sec:action\_rec\], we modify the original structure using new heads that corresponds to the 8 action sets of the TITAN dataset. Their per frame mAP results are compared in Table \[tbl:action\_recog\] for each action set. We refer to the supplementary material for the detailed comparison on individual action categories. Note that we use the I3D-based action detector for the rest of our experiments. Method I3D [@i3d] --------- --------------- ------------- ---------- Backbone InceptionV1 ResNet50 atomic 0.9219 0.7552 simple 0.5318 0.3173 person complex 0.9881 0.9880 communicative 0.8649 0.8648 transportive 0.9080 0.9081 overall 0.8429 0.7667 motion 0.9918 0.7132 vehicle trunk 1.00 1.00 doors 1.00 1.00 overall 0.9921 0.9044 0.8946 0.8128 : Action recognition results (mAP) on TITAN. \[tbl:action\_recog\] Future Object Localization {#future-object-localization} -------------------------- The results of future object localization performance is shown in Table \[tbl:fol\]. The constant velocity (Const-Vel [@constvel]) baseline is computed using the last two observations for linearly interpolating future positions. Since the bounding box dimensions error is not captured by ADE or FDE, we evaluate on FIOU using two baselines: 1) without scaling the box dimensions, and 2) with scaling linearly the box dimensions. Titan\_vanilla is an encoder and decoder RNN without any priors or interactions. It shows better performance than linear models. Both Social-GAN [@social_gan] and Social-LSTM [@social_lstm] improve the performance in ADE and FDE compared to the simple recurrent model (Titan\_vanilla) or linear approaches. Note that we do not evaluate FIOU for Social-GAN and Social-LSTM since their original method is not designed to predict dimensions. Titan\_AP adds action priors to the past positions and performs better than Titan\_vanilla, which shows that the model better understands contextual meaning of the past motion. However, its performance is worse than Titan\_EP that includes ego-motion as priors. This is because Titan\_AP does not consider the motion behavior of other agents in egocentric view. Titan\_IP includes interaction priors as shown in Figure \[fig:interaction\_encoder\] without concatenating actions. Interestingly, its performance is better than Titan\_AP (action priors) and Titan\_EP (ego priors) as well as Titan\_EP+AP (both ego and action priors). It validates the efficacy of our interaction encoder that aims to pass the interactions over all agents. This is also demonstrated by comparing Titan\_IP with two state-of-the-art methods. With ego priors as default input, interaction priors (Titin\_EP+IP) finally perform better than Titan\_IP. Interactions with action information (Titan\_EP+IP+AP) significantly outperforms all other baselines, suggesting that interactions are important and can be more meaningful with the information of actions[^2]. Method ADE $\downarrow$ FDE $\downarrow$ FIOU $\uparrow$ ------------------------------------- ------------------ ------------------ ----------------- Const-Vel (w/o scaling) [@constvel] 44.39 102.47 0.1567 Const-Vel (w/ scaling) [@constvel] 44.39 102.47 0.1692 Social-LSTM [@social_lstm] 37.01 66.78 - Social-GAN [@social_gan] 35.41 69.41 - Titan\_vanilla 38.56 72.42 0.3233 Titan\_AP 33.54 55.80 0.3670 Titan\_EP 29.42 41.21 0.4010 Titan\_IP 22.53 32.80 0.5589 Titan\_EP+AP 26.03 38.78 0.5360 Titan\_EP+IP 17.79 27.69 0.5650 Titan\_EP+IP+AP (ours) **11.32** **19.53** **0.6559** : Quantitative evaluation for future object localization. ADE are FDE in pixels on the original size 1920x1200. \[tbl:fol\] The qualitative results are shown in Figure \[fig:ours\]. The proposed method predicts natural motion for the target with respect to their detected actions (listed below each example). In Figure \[fig:comp\], we compare ours with the baseline models. The performance improvement against Titan\_EP+IP further validates our use of action priors for future prediction. Additional results can be found in the supplementary material. Method acc RMSE $\downarrow$ yaw rate RMSE $\downarrow$ ----------------------- ----------------------- ---------------------------- Const-Vel [@constvel] 1.745 0.1249 Const-Acc 1.569 0.1549 Titan\_vanilla 1.201 0.1416 Titan\_FP 1.236 0.1438 Titan\_FP+AP 1.182 0.1061 Titan\_AIM\_FP 1.134 0.0921 Titan\_AIM (ours) **1.081** **0.0824** : Comparison of Future ego motion prediction. acceleration error in $m/s^2$ and yaw rate error in $rad/s$. \[tbl:fe\] Future Ego-Motion Prediction ---------------------------- The quantitative results for future ego-motion prediction are shown in Table \[tbl:fe\]. Between Const-Vel [@constvel] and Const-Acc (acceleration), the Const-Vel baseline performs better in predicting angular velocity (yaw-rate) and Const-Acc performs better for predicting acceleration. Titan\_vanilla only takes the past ego-motion as input, performing better than Const-Vel and Const-Acc for acceleration prediction. Although incorporating information of other agents’ future predictions (Titan\_FP) does not improve the performance over Titan\_vanilla, the addition of their action priors (Titan\_FP+AP) shows better performance for both acceleration and yaw rate prediction. By adding just future position in the AIM module (Titan\_AIM\_FP), the system can weight the importance of other agents’ behavior with respect to the ego-future, resulting in decreased error rates. Finally, by incorporating future position and action in the AIM module as a prior yields the best performance, Titan\_AIM. To show the interpretability of which participant is more important for ego-future, we visualize the importance weights in Figure \[fig:attn\]. In particular, the top row illustrates that the importance weight of the pedestrian increases as the future motion direction (in white arrow) is towards the ego-vehicle’s future motion. Although the agent is closer to the ego-vehicle at a later time step, the importance decreases as the future motion changes. This mechanism provides insight into assessment of perceived risk for other agents from the perspective of the ego-vehicle. Conclusion ========== We presented a model that can reason about the future trajectory of scene agents from egocentric views obtained from a mobile platform. Our hypothesis was that action priors provide meaningful interactions and also important cues for making future trajectory predictions. To validate this hypothesis, we developed a model that incorporates prior positions, actions, and context to simultaneously forecast future trajectory of agents and future ego-motion. For evaluation, we created a novel dataset with over 700 video clips containing labels of a diverse set of actions in urban traffic scenes from a moving vehicle. Many of those actions implicitly capture the agent’s intentions. Comparative experiments against baselines and state-of-art prediction algorithms showed significant performance improvement when incorporating action and interaction priors. Importantly, our framework introduces an Agent Importance Mechanism (AIM) module to identify agents that are influential in predicting the future ego-motion, providing insight into assessment of perceived risk in navigation. For future work, we plan to incorporate additional scene context to capture participant interactions with the scene or infrastructure. **Acknowledgement** We thank Akira Kanehara for supporting our data collection and Yuji Yasui, Rei Sakai, and Isht Dwivedi for insightful discussions. {width="99.00000%"} Category Set \# Classes \# Instances Description ---------- -------------------- ------------ -------------- -------------------------------------------------------------- Atomic 9 392511 atomic whole body actions/postures that describe primi- tive action poses (*e.g.*, sitting, standing, walking, etc.) Simple contextual 13 328337 single atomic actions that include scene context (*e.g.*, jaywalking, waiting to cross) Human Complex contextual 7 5084 a sequence of atomic actions with increased complexity Action and/or higher contextual understanding Transportive 4 35160 manually transporting an object by carrying, pulling, or pushing. Communicative 4 57030 communicative actions (e.g. talking on the phone, look- ing at phone, or talking in groups.) Motion status 3 249080 motion status of 2-wheeled and 4-wheeled vehicles (parked / moving / stationary) Vehicle Trunk status 2 146839 trunk for 4-wheeled vehicles State (open / closed) Door status 2 146839 door status for 4-wheeled vehicles (open / closed) Age group 3 395769 subjective categorization of pedestrians into age groups Other (child / adult / senior) Labels Object type 3 645384 participant types categorized into pedestrian / 2-wheeled / 4- wheeled vehicles Details of the TITAN Dataset ============================ Figure \[fig:main\] illustrates the labels of the TITAN dataset, which are typically observed from on-board vehicles in driving scenes. We define 50 labels including vehicle states and actions, pedestrian age groups, and targeted pedestrian action attributes that are hierarchically organized from primitive atomic actions to complicated contextual activities. Table \[tbl:titan\_dataset\] further details the number of labels, instances, and descriptions for each action set in the TITAN dataset. For pedestrians, we categorize human actions into 5 sub-categories based on their complexities and compositions. Moreover, we annotate vehicle states with 3 sub-categories of motion, and trunk / door status. Note that the trunk and door status is only annotated for 4-wheeled vehicles. Vehicles with 3-wheels without trunk but with doors are annotated as 4-wheeled and trunk open. Also, 3-wheeled vehicles with no trunk and doors are annotated as 2-wheeled vehicles. The list of classes for human actions is shown in Table \[tbl:per\_class\_results\]. The annotators were instructed to only localize pedestrians and vehicles with a minimum bounding box size of $70\times10$ $pixels$ and $50\times10$ $pixels$ in the image, respectively. Several example scenarios of TITAN are depicted in Figure \[fig:titan\_dataset\]. In each scenario, four frames are displayed with a bounding box around a road agent. We also provide their actions below each frame. Note that only one agent per frame is selected for the purpose of visualization. The same color code is used for each action label, which can be found in Figure 2 of the main manuscript. Additional Evaluation ===================== In this section, we provide additional evaluation results of the proposed approach. Per-Class Quantitative Results ------------------------------ In Table \[tbl:per\_class\_results\], we present per-class quantitative results of the proposed approach, which are evaluated using the test set of TITAN. Note that the number of instances for some actions (*e.g.*, *kneeling*, *jumping*, etc.) are zero, although they are present in the training and validation set. This is because we randomly split 700 clips of TITAN into training, validation, and test set. We will regularly update TITAN to add more clips with such actions. We observe that the error rate for some classes are either much lower or higher than other classes. For example, scenarios depicting *getting into a 4 wheel vehicle*, *getting out of a 4 wheel vehicle*, and *getting on a 2 wheel vehicle* show very small FDE as compared to others. Also, scenarios depicting *entering a building* has a larger ADE and FDE than other scenarios. The reason for this can be explained by considering interactions of agents. When a person is *getting into a vehicle*, the proposed interaction encoder builds a pair-wise interaction between the person (subject that the action generates) and the vehicle (object that the subject is related to). It further validates the efficacy of our interaction encoding capability. In contrast, no interactive object is given to the agent for *entering a building* class since we assume agents are either pedestrians or vehicles. As mentioned in the main manuscript, we plan to incorporate additional scene context such as topology or semantic information. Efficacy of Multi-Task Loss --------------------------- The comparative results of the I3D action recognition module with and without the multi-task (MT) loss is shown in Table \[tbl:mt\_loss\]. The performance improvement for atomic and simple contextual actions for pedestrians and motion status for vehicles with the MT loss validates its efficacy of modeling aleatoric homoscedastic uncertainty of different tasks. {width="99.00000%"} Action Set Class ADE$\downarrow$ FDE$\downarrow$ FIOU$\uparrow$ \#Instances -------------------- ------------------------------------ ----------------- ----------------- ---------------- ------------- standing 10.56 18.63 0.6128 1476 running 12.39 19.95 0.6179 89 bending 12.76 20.85 0.6560 156 kneeling 0.00 0.00 0.00 0 Atomic Action walking 13.31 23.15 0.5712 6354 sitting 11.10 20.74 0.6282 337 squatting 11.90 18.82 0.5598 4 jumping 0.00 0.00 0.00 0 laying down 0.00 0.00 0.00 0 none of the above 9.69 16.43 0.7408 7237 crossing at pedestrian crossing 13.22 21.59 0.5976 881 jaywalking 13.10 21.91 0.6148 340 waiting to cross street 11.49 21.75 0.5783 65 motorcycling 20.00 31.81 0.5494 4 biking 13.22 21.13 0.6283 287 walking along the side of the road 11.33 24.50 0.5516 2668 Simple-Contextual walking on the road 13.41 22.30 0.5794 2486 cleaning (ground, surface, object) 11.67 22.58 0.6502 19 closing 9.84 20.50 0.4947 14 opening 12.99 29.89 0.1995 13 exiting a building 13.56 28.09 0.5264 61 entering a building 28.06 53.02 0.2259 6 none of the above 9.85 16.76 0.7201 8809 unloading 11.07 18.45 0.7082 37 loading 11.59 18.54 0.6652 40 getting in 4 wheel vehicle 8.39 10.80 0.5682 10 Complex-Contextual getting out of 4 wheel vehicle 9.63 9.58 0.7972 3 getting on 2 wheel vehicle 7.73 11.16 0.7619 10 getting off 2 wheel vehicle 0 0 0 0 none of the above 11.32 19.54 0.6557 15553 looking at phone 12.12 21.48 0.6435 392 Communicative talking on phone 11.69 19.39 0.6056 268 talking in group 11.70 20.82 0.6025 461 none of the above 11.28 19.43 0.6588 14532 pushing 12.57 23.07 0.6148 232 Transportive carrying with both hands 11.39 20.23 0.6477 445 pulling 12.01 21.29 0.5198 76 none of the above 11.29 19.44 0.6574 14900 stopped 8.96 23.08 0.6148 232 Motion-Status moving 9.18 20.23 0.6477 445 parked 9.93 21.29 0.5199 76 none of the above 12.72 19.44 0.6574 14900 \[tbl:per\_class\_results\] {width="99.00000%"} {width="99.00000%"} {width="95.00000%"} Method w/ MT loss$\uparrow$ w/o MT loss$\uparrow$ -- --------------- ---------------------- ----------------------- atomic 0.9219 0.7552 simple 0.5318 0.3173 complex 0.9881 0.9880 communicative 0.8649 0.8647 transportive 0.9080 0.9080 overall 0.8429 0.7667 motion 0.9918 0.7130 trunk 1.00 1.00 doors 1.00 1.00 overall 0.9921 0.9043 0.8946 0.8127 : Action recognition results (mAP) on TITAN.[]{data-label="tbl:mt_loss"} Additional Qualitative Results ------------------------------ Figure \[fig:ours\] and \[fig:comparison\] show the prediction results of the proposed approach for future object localization. Titan\_EP+IP+AP consistently shows better performance against the baseline model and the state-of-the-art methods. We also observed that t In Figure \[fig:attn\], the proposed Agent Importance Module (AIM) is evaluated on additional sequences. The ego-vehicle decelerates due to the crossing agent, and our system considers this agent as having a higher influence (or importance)than other agents. Agents with high importance are depicted with a red over-bar. Particularly in scenario 10, when the person walks along the road in the longitudinal direction, its importance is relatively low. However, the importance immediately increases when the motion changes to the lateral direction. Implementation ============== TITAN framework is trained on a Tesla V100 GPU using PyTorch Framework. We separately trained action recognition, future object localization, and future ego-motion prediction modules. During training, we used ground-truth data as input to each module. However, during testing, the output results of one module are directly used for later tasks. Future Object Localization {#future-object-localization-1} -------------------------- During training, we used a learning rate of 0.0001 with RMSProp optimizer and trained for 80 epochs using a batch size of 16. We used hidden state dimension of 512 for both encoder and decoder. A size of 512 is used for the embedding size of action, interaction, ego-motion and bounding box. The input box dimension is 4, action dimension is 8, and ego-motion dimension is 2. The original image size width is 1920 $pixels$ and height is 1200 $pixels$ and accordingly cropped using the bounding box dimension. It is further resized to $228\times228$ for the I3D-based action detector. The bounding box inputs and outputs are normalized between 0 to 1 using image dimensions. Layer Kernal shape Output shape ---- ------------------------------------------ --------------- ---------------- 0 ego$\_$box$\_$embed.Linear$\_$0 \[4, 512\] \[1, 10, 512\] 1 ego$\_$box$\_$embed.ReLU$\_$1 - \[1, 10, 512\] 2 ego$\_$action$\_$embed.Linear$\_$0 \[8, 512\] \[1, 512\] 3 ego$\_$action$\_$embed.ReLU$\_$1 - \[1, 512\] 4 ego$\_$motion$\_$embed.Linear$\_$0 \[2, 512\] \[1, 10, 512\] 5 ego$\_$motion$\_$embed.ReLU 1 - \[1, 10, 512\] 6 box$\_$encoder.GRUCell$\_$enc - \[1, 512\] 7 motion$\_$encoder.GRUCell$\_$enc - \[1, 512\] 8 int$\_$encoder.embed.Linear$\_$0 \[24, 512\] \[1, 512\] 9 int$\_$encoder.embed.ReLU$\_$1 - \[1, 512\] 10 int$\_$encoder.encode.GRUCell$\_$enc - \[1, 512\] 11 concat$\_$to$\_$hidden.Linear$\_$0 \[2048, 512\] \[1, 512\] 12 concat$\_$to$\_$hidden.ReLU$\_$1 - \[1, 512\] 13 14 15 pred.GRUCell$\_$dec - \[1, 512\] 16 pred.hidden$\_$to$\_$input.Linear$\_$0 \[512, 512\] \[1, 512\] 17 pred.hidden$\_$to$\_$input.ReLU$\_$1 - \[1, 512\] 18 pred.hidden$\_$to$\_$output.Linear$\_$0 \[512, 10\] \[1, 10\] 19 pred.hidden$\_$to$\_$output.Sigmoid$\_$1 - \[1, 10\] 20 : Future Object Localization model summary with an example batch size of 1[]{data-label="tbl:fol_model"} The model summary for Future Object Localization is shown in Table \[tbl:fol\_model\]. We embed the bounding box (through 0 and 1), action (2-3), ego-motion (4-5) at each time step, and pairwise interaction encoding (8-12). We concatenate the embedded features through (11-12), which are given from the hidden states of the bounding box encoder GRU (6), the hidden states of the ego encoder GRU (7), encoded interaction (10) and action embedding (3). We encode all information for 10 observation time steps from (14). We decode the future locations using decoder GRU for 20 future time steps (20). Action Recognition {#action-recognition-2} ------------------ We used Kinetics-600 pre-trained weights for both I3D and 3D-ResNet. For I3D, we use layers until Mixed$\_$5c layer of the original structure. We used learning rate of 0.0001 and a batch size of 8. We trained it for 100 epochs. The input size is $3\times10\times244\times244$, where 10 is the number of time steps, 3 is the number of RGB channels. If the agent is occluded and reappears at any time step, we used the last observed crop of image for that the agent. During training we backpropagate the gradients for pedestrians and vehicles with the loss function as shown below: $$\mathcal{L}_{total}=\mathds{1}_p {\mathcal{L}_a}^{i=1:5}+(1-\mathds{1}_p){\mathcal{L}_a}^{i=6:8}, \label{eq:multi_task_loss_sep}$$ where $\mathds{1}_p$ is an indicator function that equals 1 if the agent is a pedestrian and 0 if the agent is a vehicle. We refer to the main manuscript for $\mathcal{L}_a$. Layer Kernal shape Output shape ----- ---------------------------------------- -------------------- ------------------------ 1 i3d.Conv3d$\_$1a$\_$7x7.conv3d \[3, 64, 7, 7, 7\] \[1, 64, 5, 112, 112\] .. 126 i3d.Mixed$\_$5c.b3b.BatchNorm3d \[128\] \[1, 128, 2, 7, 7\] 127 action.hid$\_$to$\_$pred1.Linear$\_$0 \[100352, 10\] \[1, 10\] 128 action.hid$\_$to$\_$pred1.Softmax$\_$1 - \[1, 10\] 129 action.hid$\_$to$\_$pred2.Linear$\_$0 \[100352, 13\] \[1, 13\] 130 action.hid$\_$to$\_$pred2.Softmax$\_$1 - \[1, 13\] 131 action.hid$\_$to$\_$pred3.Linear$\_$0 \[100352, 7\] \[1, 7\] 132 action.hid$\_$to$\_$pred3.Softmax$\_$1 - \[1, 7\] 133 action.hid$\_$to$\_$pred4.Linear$\_$0 \[100352, 4\] \[1, 4\] 134 action.hid$\_$to$\_$pred4.Softmax$\_$1 - \[1, 4\] 135 action.hid$\_$to$\_$pred5.Linear$\_$0 \[100352, 4\] \[1, 4\] 136 action.hid$\_$to$\_$pred5.Softmax$\_$1 - \[1, 4\] 137 action.hid$\_$to$\_$pred6.Linear$\_$0 \[100352, 4\] \[1, 4\] 138 action.hid$\_$to$\_$pred6.Softmax$\_$1 - \[1, 4\] 139 action.hid$\_$to$\_$pred7.Linear$\_$0 \[100352, 3\] \[1, 3\] 140 action.hid$\_$to$\_$pred7.Softmax$\_$1 - \[1, 3\] 141 action.hid$\_$to$\_$pred8.Linear$\_$0 \[100352, 3\] \[1, 3\] 142 action.hid$\_$to$\_$pred8.Softmax$\_$1 - \[1, 3\] : I3D action recognition model summary with an example batch size of 1[]{data-label="tbl:action_model"} The model summary for action recognition is shown in Table \[tbl:action\_model\]. Note that, from mixed$\_$5c layer \[b0, b1b, b2b, b3b\] are concatenated to give a shape of \[1,1024,2,7,7\] which is flattened to give a tensor of shape \[1,100352\] before feeding it to each MLP head for individual action sets. Future Ego-Motion Prediction ---------------------------- Layer Kernal shape Output shape ---- --------------------------------------------- -------------- ------------------ 0 ego$\_$embed.Linear$\_$0 \[2, 128\] \[1, 10, 128\] 1 ego$\_$embed.ReLU$\_$1 - \[1, 10, 128\] 2 ego$\_$encoder.GRUCell$\_$enc - \[1, 128\] 3 4 pred.box$\_$embed.Linear$\_$0 \[4, 128\] \[1, 1, m, 128\] 5 pred.box$\_$embed.ReLU$\_$1 - \[1, 1, m, 128\] 6 pred.action$\_$embed.Linear$\_$0 \[8, 128\] \[1, 1, m, 128\] 7 pred.action$\_$embed.ReLU$\_$1 - \[1, 1, m, 128\] 8 pred.concat$\_$to$\_$hid2.Linear$\_$0 \[256, 128\] \[1, 1, m, 128\] 9 pred.AIM$\_$layer.Linear$\_$0 \[128, 1\] \[1, 1, m, 1\] 10 pred.AIM$\_$layer.Tanh$\_$1 - \[1, 1, m, 1\] 11 pred.concat.concat$\_$0 - \[1, 256\] 12 pred.concat$\_$to$\_$hid.Linear$\_$0 \[256, 128\] \[1, 128\] 13 pred.GRUCell$\_$dec - \[1, 128\] 14 pred.hid$\_$to$\_$pred$\_$input.Linear$\_$0 \[128, 128\] \[1, 128\] 15 pred.hid$\_$to$\_$pred$\_$input.ReLU$\_$1 - \[1, 128\] 16 pred.Linear$\_$hid$\_$to$\_$pred \[128, 2\] \[1, 2\] 17 : Future ego motion prediction model summary with an example batch size of 1, m is the number of agents at that future time step[]{data-label="tbl:fe_model"} We use batch size of 64, learning rate of 0.0001 and trained for 100 epoch with RMSProp optimizer. We use the hidden state dimension of 128 for both encoder and decoder. We use the embedding size of 128. The prediction is done for 20 time steps in future. The input and output dimensions are 2 at each time step. The model summary of the future ego-motion prediction is shown in Table \[tbl:fe\_model\]. We embed the ego motion at each time step (0-1) and use GRU encoder (2) for 10 observation time steps (3). The encoded information is used for the decoder. The embedded future bounding box (4-5) and embedded current action (6-7) are concatenated (8). The agent importance module (AIM) is used to weight the agents at each time step (9-10). We concatenate (11) the AIM output with the past hidden state and embed it (12). The embedded feature is used as an input hidden state. The current hidden state (13) is passed to the next time-step (14-15) using GRU. The output is decoded (16) from the hidden state at each time step (17). As a result, we get for 20 future predictions. [^1]: We assume that the bounding box detection using past images is provided by the external module since detection is not the scope of this paper. [^2]: Using ground-truth actions as a prior, we observed further improvement in overall ADE by 2 *pixels* and overall FDE by 3.5 *pixels*.

--- author: - | Jon A. Bailey, Sunkyu Lee,\ Lattice Gauge Theory Research Center, CTP, and FPRD,\ Department of Physics and Astronomy,\ Seoul National University, Seoul 08826, South Korea\ E-mail: - | Yong-Chull Jang\ Physics Department, Brookhaven National Laboratory, Upton, NY11973, USA - | Jaehoon Leem\ School of Physics, Korea Institute for Advanced Study (KIAS), Seoul 02455, South Korea - | Sungwoo Park\ Los Alamos National Laboratory, Theoretical Division T-2, Los Alamos, NM87545, USA - SWME Collaboration bibliography: - 'refs.bib' title: '2018 Update on $\epsK$ with lattice QCD inputs' --- Introduction ============ This paper is a brief summary of our previous paper [@Bailey:2018feb]. This paper is also an update of our previous papers [@Jang:2017ieg; @Bailey:2015tba; @Bailey:2015frw]. Input parameters: $\Vcb$ and $\xi_0$ {#sec:Vcb} ==================================== In Table \[tab:Vcb\], we present updated results for both exclusive $\Vcb$ and inclusive $\Vcb$. Recently, HFLAV reported them in Ref. [@Amhis:2016xyh]. The results for exclusive $\Vcb$ are obtained using lattice QCD results for the semileptonic form factors of Refs. [@Bailey2014:PhysRevD.89.114504; @Lattice:2015rga; @Detmold:2015aaa]. Here, we use the combined results (ex-combined) for exclusive $\Vcb$ and the results of the $1S$ scheme for inclusive $\Vcb$ to evaluate $\epsK$. For more details on $\Vcb$ and the related caveats, refer to Ref. [@Bailey:2018feb]. The absorptive part of long distance effects on $\epsK$ is parametrized into $\xi_0$. $$\begin{aligned} \xi_0 &= \frac{\Im A_0}{\Re A_0}, \qquad \xi_2 = \frac{\Im A_2}{\Re A_2}, \qquad \Re \left(\frac{\eps'}{\eps} \right) = \frac{\omega}{\sqrt{2} |\eps_K|} (\xi_2 - \xi_0) \,. \label{eq:e'/e:xi0}\end{aligned}$$ There are two independent methods to determine $\xi_0$ in lattice QCD: one is the indirect method and the other is the direct method. In the indirect method, one can determine $\xi_0$ using Eq.  with lattice QCD input $\xi_2$ and with experimental results for $\eps'/\eps$, $\epsK$, and $\omega$. In the direct method, one can determine $\xi_0$ directly using lattice QCD results for $\Im A_0$ combined with experimental results for $\Re A_0$. In Table \[tab:xi0+d0\](), we summarize results for $\xi_0$ calculated by RBC-UKQCD using the indirect and direct methods. Here, we use the results of the indirect method for $\xi_0$ to evaluate $\epsK$. In Ref. [@Bai:2015nea], RBC-UKQCD also reported the S-wave scattering phase shift for the $I=0$ channel: $\delta_0 = 23.8(49)(12)$, which is different from those of the dispersion relations [@Colangelo:2001df; @GarciaMartin:2011cn] by $\approx 3 \sigma$. In Ref. [@Wang:2018Latt], they have accumulated higher statistics to obtain $\delta_0 = 19.1(25)(12)$, which is about $5\sigma$ different from those of the dispersion analyses. They introduce a $\sigma$ operator and make all possible combinations with the $\sigma$ and $\pi-\pi$ operators. Then, RBC-UKQCD has obtained $\delta_0 = 32.8(12)(30)$ which is consistent with those of the dispersion relations. These results are presented in Table \[tab:xi0+d0\]() and Figure \[tab:xi0+d0\](). Input parameters: Wolfenstein parameters, $\BK$, $\xi_\text{LD}$, and others ============================================================================ In Table \[tab:input-WP-eta\](), we summarize the Wolfenstein parameters on the market. The CKMfitter and UTfit collaboration provide the Wolfenstein parameters determined by the global unitarity triangle (UT) fit. Unfortunately, $\epsK$, $\BK$, and $\Vcb$ are used as inputs to the global UT fit, which leads to unwanted correlation with $\epsK$. We want to avoid this correlation, and so take another input set from the angle-only fit (AOF) suggested in Ref. [@Bevan2013:npps241.89]. The AOF does not use $\epsK$, $\BK$, and $\Vcb$ as input to determine the UT apex $(\bar{\rho}, \bar{\eta})$. Here the $\lambda$ parameter is determined from $\Vus$ which is obtained from the $K_{\ell 2}$ and $K_{\ell 3}$ decays using lattice QCD results for the form factors and decay constants. The $A$ parameter is determined from $\Vcb$. In the FLAG review [@Aoki:2016frl], they present lattice QCD results for $\BK$ with $N_f=2$, $N_f=2+1$, and $N_f= 2+1+1$. Here, we use the results for $\BK$ with $N_f=2+1$, which is obtained by taking a global average over the four data points from BMW 11 [@Durr:2011ap], Laiho 11 [@Laiho:2011np], RBC-UKQCD 14 [@Blum:2014tka], and SWME 15 [@Jang:2015sla]. In Table \[tab:input-BK-other\](), we present the FLAG 17 result for $\BK$ with $N_f = 2+1$, which is used to evaluate $\epsK$. The dispersive long distance (LD) effect is defined as $$\begin{aligned} \xi_\text{LD} &= \frac{m^\prime_\text{LD}}{\sqrt{2} \Delta M_K} \,, \qquad m^\prime_\text{LD} = -\Im \left[ \mathcal{P}\sum_{C} \frac{\mate{\wbar{K}^0}{H_\text{w}}{C} \mate{C}{H_\text{w}}{K^0}} {m_{K^0}-E_{C}} \right] \label{eq:xi-LD}\end{aligned}$$ If the CPT invariance is well respected, the overall contribution of the $\xi_\text{LD}$ to $\epsK$ is about $\pm 2\%$. Lattice QCD tools to calculate $\xi_\text{LD}$ are well established in Refs. [@Christ2012:PhysRevD.88.014508; @Bai:2014cva; @Christ:2015pwa]. In addition, there have been a number of attempts to calculate $\xi_\text{LD}$ on the lattice [@Christ:2015phf; @Bai:2016gzv]. In them, RBC-UKQCD used a pion mass of 329 MeV and a kaon mass of 591 MeV, and so the energy of the 2 pion and 3 pion states are heavier than the kaon mass. Hence, the sign of the denominator in Eq. \[eq:xi-LD\] is opposite to that of the physical contribution. Therefore, this work belongs to the category of exploratory study rather than to that of precision measurement. In Ref. [@Buras2010:PhysLettB.688.309], they use chiral perturbation theory to estimate the size of $\xi_\text{LD}$ and claim that $$\begin{aligned} \xi_\text{LD} &= -0.4(3) \times \frac{\xi_0}{ \sqrt{2} } \label{eq:xiLD:bgi}\end{aligned}$$ where we use the indirect results for $\xi_0$ and its error. Here, we call this method the BGI estimate for $\xi_\text{LD}$. In Refs. [@Christ2012:PhysRevD.88.014508; @Christ:2014qwa], RBC-UKQCD provides another estimate for $\xi_\text{LD}$: $$\begin{aligned} \xi_\text{LD} &= (0 \pm 1.6)\%. \label{eq:xiLD:rbc}\end{aligned}$$ Here, we call this method the RBC-UKQCD estimate for $\xi_\text{LD}$. In Table \[tab:input-WP-eta\](), we present higher order QCD corrections: $\eta_{ij}$ with $i,j = t,c$. In Table \[tab:input-BK-other\](), we present other input parameters needed to evaluate $\epsK$. Since Lattice 2017, three parameters: $m_t(m_t)$, $m_{K^{0}}$, $F_K$ have been updated. The $m_t(m_t)$ parameter is the scale-invariant (SI) top quark mass renormalized in the $\MSb$ scheme. The pole mass of top quarks comes from Ref. [@Patrignani:2016xqp]: $ M_t = 173.5 \pm 1.1 \GeV$. We convert the top quark pole mass into the SI top quark mass using the four-loop perturbation formula. For more details, refer to Ref. [@Bailey:2018feb]. Results for $\epsK$ =================== In Fig. \[fig:epsK:cmp:rbc\], we present results for $|\epsK|$ evaluated directly from the standard model (SM) with lattice QCD inputs given in the previous sections. In Fig. \[fig:epsK:cmp:rbc\](), the blue curve represents the theoretical evaluation of $|\epsK|$ using the FLAG-2017 $\BK$, AOF for Wolfenstein parameters, and exclusive $\Vcb$, and the RBC-UKQCD estimate for $\xi_\text{LD}$. The red curve in Fig. \[fig:epsK:cmp:rbc\] represents the experimental value of $|\epsK|$. In Fig. \[fig:epsK:cmp:rbc\](), the blue curve represents the same as in Fig. \[fig:epsK:cmp:rbc\]() except for using the inclusive $\Vcb$. Our results for $|\epsK|$ are summarized in Table \[tab:epsK\]. Here, the superscript ${}^\text{SM}$ means that it is obtained directly from the standard model, the subscript ${}_\text{excl}$ (${}_\text{incl}$) means that it is obtained using exclusive (inclusive) $\Vcb$, and the superscript ${}^\text{Exp}$ represents the experimental value. Results in Table \[tab:epsK\]() are obtained using the RBC-UKQCD estimate for $\xi_\text{LD}$ and those in Table \[tab:epsK\]() are obtained using the BGI estimate for $\xi_\text{LD}$. In Table \[tab:epsK\](), we find that the theoretical evaluation of $|\epsK|$ with lattice QCD inputs (with exclusive $\Vcb$) $|\epsK|^\text{SM}_\text{excl}$ has $4.2\sigma$ tension with the experimental result $|\epsK|^\text{Exp}$, while there is no tension with inclusive $\Vcb$ (heavy quark expansion with QCD sum rules). In Fig. \[fig:depsK:sum:rbc:his\](), we plot the $\Delta \epsK \equiv |\epsK|^\text{Exp} - |\epsK|^\text{SM}_\text{excl}$ in units of $\sigma$ (the total error) as a function of time starting from 2012. In 2012, $\Delta \epsK$ was $2.5\sigma$, but now it is $4.2\sigma$. In Fig. \[fig:depsK:sum:rbc:his\](), we plot the history of the average $\Delta \epsK$ and the error $\sigma_{\Delta \epsK}$. We find that the average has increased with some fluctuations by 27% during the period of 2012-2018, and its error has decreased monotonically by 25% in the same period. In Table \[tab:err-bud+his-DepsK\](), we present the error budget for $|\epsK|^\text{SM}_\text{excl}$. Here, we find that the largest error comes from $\Vcb$. Hence, it is essential to reduce the error in $\Vcb$ significantly. In Table \[tab:err-bud+his-DepsK\](), we present how the values of $\Delta\epsK$ have changed from 2015 [@Bailey:2015tba] to 2018 [@Bailey:2018feb]. Here, we find that the positive shift of $\Delta \epsK$ is about the same for the inclusive and exclusive $\Vcb$. This reflects the changes in other parameters since 2015. We thank Shoji Hashimoto and Takashi Kaneko for helpful discussion on $\Vcb$. The research of W. Lee is supported by the Creative Research Initiatives Program (No. 2017013332) of the NRF grant funded by the Korean government (MEST).  J.A.B. is supported by the Basic Science Research Program of the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2015024974). W. Lee would like to acknowledge the support from the KISTI supercomputing center through the strategic support program for the supercomputing application research (No. KSC-2016-C3-0072). Computations were carried out on the DAVID GPU clusters at Seoul National University.

--- author: - Atsuhisa Ota - and Masahide Yamaguchi bibliography: - 'bib.bib' title: Secondary isocurvature perturbations from acoustic reheating --- Introduction ============ The conserved quantities on superhorizon scales play an important role in inflationary Universe because they connect the primordial perturbations generated during inflation with those at the late Universe. Even though most of characteristic signals of the early universe are washed away due to the thermalization processes, they keep the statistics of the primordial density fluctuations, which enables us to reveal the details of inflationary models. We usually evaluate such quantities when they exit horizons during inflation and consider them as initial conditions of the hot Big Bang universe. The curvature perturbation on the uniform density slice $\zeta$ is one of typical examples of such conserved quantities [@Malik:2003mv; @Lyth:2003im; @Lyth:2004gb]. Suppose the total energy momentum tensor is conserved and we drop the gradient terms, it is well-known that $\zeta$ is conserved even at nonlinear order when there are no non adiabatic pressure perturbations. We can also define the curvature perturbations $\zeta_{\alpha}$ on $\alpha$-fluid uniform density slice, where $\alpha=\nu,~b,~c$ represents neutrino, baryon, or cold dark matter (CDM) while $\gamma$ the photon fluid. Then, the isocurvature perturbations are introduced as $S_{\alpha \gamma}\equiv 3(\zeta_{\alpha}-\zeta_{\gamma})$. It should be noticed that $S_{\alpha \gamma}$ are also conserved at the leading order of the gradient expansion if the energy momentum tensors of $\alpha$- and $\gamma$-fluids are conserved, respectively. The conservation law of the total energy momentum tensor is universal so that the conservation laws of the curvature perturbations have been also considered to be robust as long as the other conditions are satisfied. In this paper, we revisit the above two assumptions for the conservation laws of $\zeta_{\alpha}$: ignoring the gradient terms and the conservation laws of the energy momentum tensors. First, we point out that, at nonlinear order, we cannot justify to drop the gradient terms even when we consider the long wavelength modes; convolutions in Fourier space can pick up products of short wavelength modes, which might be significant. As a result, the total curvature perturbations might not be conserved at nonlinear order even without non-adiabatic pressure perturbations. We then newly introduce a second order conserved quantity in the presence of gradient terms. Second, we discuss energy transfer among components, that still conserves the total energy momentum but violates each one. This would lead to the evolution of superhorizon isocurvature perturbations. The typical example of the above process is acoustic reheating of the photon-baryon plasma [@Jeong:2014gna; @Nakama:2014vla; @Naruko:2015pva]. The short wavelength temperature fluctuations of the cosmic microwave background (CMB) are significantly damping due to imperfectness of the photon-baryon fluid, which produces the second order entropy production and the second order energy transfer between the photons and the baryons. These processes actually happen inside each diffusion scale; the secondary effects fluctuate on scales larger than those of corse graining. The distant patches are not necessarily reheated homogeneously if there exist three or four-point correlations of primordial density perturbations a priori [@Naruko:2015pva]. They are comparable to the non gradient terms at second order because the convolutions pick heat conduction and shear viscosity on small scales up. We investigate these diffusion effects in detail by employing the nonlinear cosmological perturbation theory, which enables us to follow the evolution of the photon distribution function directly. We organize this paper as follows. First of all, we explain our set up for the second order perturbation theory in section \[nonlinearpert\]. Then, we discuss the non conservation of the curvature perturbations in the presence of gradient terms and introduce a new conserved quantity in section \[EMTevol\]. Section \[evophoton\] is devoted to describe the actual time evolution of the photon baryon plasma due to the weak Compton scattering. We comment on several definitions for the isocurvature perturbations during non-equilibrium periods in section \[sec:entropy\]. In the final section, we summarize our conclusions and describe future prospects related to the present results. Set up for second order perturbation theory {#nonlinearpert} =========================================== We need to perturb both the gravity and the matter sectors up to nonlinear order. Here, let us first define the nonlinear metric perturbations. The metric perturbations {#section:metric} ------------------------ We start with writing the spacetime metric in the following 3+1 form: $$\begin{aligned} ds^2&=-\mathcal N^2 d\eta^2 + \gamma_{ij}(\beta^i d\eta + dx^i)(\beta^j d\eta + dx^j)\notag \\ &=(-\mathcal N^2+\beta_k\beta^k) d\eta^2 +2\beta_i dx^i d\eta+ \gamma_{ij} dx^i dx^j.\label{def:metric}\end{aligned}$$ In other words, each component can be written as $$\begin{aligned} g_{\mu\nu}&=\left(\begin{array}{cc}-\mathcal N^2+\beta_k\beta^k & \beta_j \\\beta_i & \gamma_{ij}\end{array}\right),\label{def:comp}\end{aligned}$$ where $\mathcal N$ and $\beta_i$ are the lapse and the shift, respectively. $\gamma_{ij}$ is the spatial metric. Let us consider nonlinear scalar perturbations introduced as $$\begin{aligned} \mathcal N^2&=a^2 e^{2{A}},\\ \beta_i&=a^2e^{{D}}\partial_i e^{B},\label{def:shift}\\ \gamma_{ij}&=a^2e^{2{D}}\delta_{ij}\label{def:gamma},\end{aligned}$$ where $a$ is the scale factor, and we have fixed only the spacial coordinate by vanishing the anisotropic part of $\gamma_{ij}$. The nonlinear metric perturbations can be expanded as $X \equiv \sum_{n=1} X^{(n)}$ for $X=A,B$ and $D$ with $n$ being the order in primordial perturbations. Note that the conformal Newtonian, the uniform density, the spatially flat and the velocity orthogonal isotropic gauges (comoving gauge) are mutually transformed by changing only the time slice. Here, we ignore the vector and the tensor perturbations for simplicity. This would be justified if the primordial vector perturbations and the primordial tensor ones are subdominant compared to the second order scalar ones. We include the curvature perturbation $D$ in Eq. (\[def:shift\]) to simplify the inverse matrix in the following discussions. The inverse matrixes for the induced metric and the shift vector are written as $$\begin{aligned} \gamma^{ij}&=a^{-2}e^{-2{D}}\delta_{ij},\\ \beta^i&=e^{-{D}}\partial_ie^{B}.\end{aligned}$$ Then, we obtain $$\begin{aligned} \beta^k\beta_k&=a^2\partial e^{{B}}\partial e^{{B}},\\ -\mathcal N^2+\beta_k\beta^k&=-a^2e^{2{A}}+a^2\partial e^{{B}}\partial e^{{B}},\label{def:lapse}\end{aligned}$$ where we write as $\partial X\partial Y\equiv \partial_i X\partial_i Y$ for notational simplicity. Eqs. (\[def:shift\]), (\[def:gamma\]) and (\[def:lapse\]) yield $$\begin{aligned} g_{00}&=-a^2e^{2{A}}+a^2e^{2{B}}(\partial {{B}})^2,\\ g_{0i}&=a^2e^{{D}+{B}}\partial_i {B},\\ g_{ij}&=a^2e^{2{D}}\delta_{ij}.\end{aligned}$$ The inverse matrix of Eq. (\[def:comp\]) is well known: $$\begin{aligned} g^{\mu\nu}&=\left(\begin{array}{cc}-\frac{1}{\mathcal N^2} & \frac{\beta^j}{\mathcal N^2} \\\frac{\beta^i}{\mathcal N^2} & \gamma^{ij}-\frac{\beta^i\beta^j}{\mathcal N^2}\end{array}\right).\end{aligned}$$ Then, each component of the inverse matrix can be obtained as $$\begin{aligned} g^{00}&=-a^{-2}e^{-2{A}},\\ g^{0i}&=a^{-2}e^{-2{A}-{D}+{B}}\partial_i{B},\\ g^{ij}&=a^{-2}e^{-2{D}}\delta^{ij}-a^{-2}e^{-2{A}-2{D}+2{B}}\partial_i{{B}}\partial_j{{B}}.\end{aligned}$$ The determinant of $g_{\mu\nu}$ can be also evaluated as $$\begin{aligned} \sqrt{-g}=\mathcal N\sqrt{\gamma}=a^4 e^{{A}+3{D}}.\label{def:det}\end{aligned}$$ The Christoffel symbols at second order --------------------------------------- Here and hereafter we consider only the perturbations up to second order. Up to second order, each component of the metric tensor can be rewritten as $$\begin{aligned} g_{00}&=-a^2e^{2{A}}+a^2(\partial {{B}})^2,\\ g_{0i}&=a^2e^{{D}+{B}}\partial_i {B},\\ g_{ij}&=a^2e^{2{D}}\delta_{ij},\end{aligned}$$ and the inverse matrix components are $$\begin{aligned} g^{00}&=-a^{-2}e^{-2{A}},\\ g^{0i}&=a^{-2}e^{-2{A}-{D}+{B}}\partial_i{B},\\ g^{ij}&=a^{-2}e^{-2{D}}\delta^{ij}-a^{-2}\partial_i{{B}}\partial_j{{B}}.\end{aligned}$$ Let us evaluate the Christoffel symbol $$\begin{aligned} \Gamma^\mu{}_{\nu\rho}\equiv \frac12g^{\mu\alpha}\left(\partial_\rho g_{\alpha\nu}+\partial_\nu g_{\alpha\rho}-\partial_\alpha g_{\nu\rho} \right).\end{aligned}$$ Each component of the symbols can be calculated as $$\begin{aligned} \Gamma^0{}_{00}=&\mathcal H+{A}'+\mathcal H (\partial {B})^2+ \partial A\partial B,\\ \Gamma^0{}_{0i}=&\partial_i{A}+e^{-2{A}+{D}+B}(\mathcal H +{D}' )\partial_i {B}-\frac12\partial_i(\partial{B})^2,\\ \Gamma^0{}_{ij}=& \frac12 \left[\partial_i {B}\partial_j{D} + \partial_j {B}\partial_i{D}\right] -e^{-2{A}+{D}+{B}}\partial_i\partial_j{B} \notag \\ &-\partial_i{B}\partial_j{B} +e^{-2{A}+2{D}}\delta_{ij}\left[ \mathcal H+{D}'-\partial B\partial D\right],\\ \Gamma^{i}{}_{00}=& e^{-{D}+{B}}(\mathcal H\partial_iB + \partial_iB') +(-A'+D'+B')\partial_i B\notag\\ &+e^{-2{D}+2{A}}\partial_i{A}-\frac12\partial_i (\partial{B})^2,\\ \Gamma^i{}_{0j}=&(\mathcal H+{D}')\delta_{ij}-\partial_i {B}\partial_j {A}-\mathcal H\partial_i {B}\partial_j {B}\notag \\ & -\frac12(\partial_i{D}\partial_j {B}-\partial_j{D}\partial_i {B}) ,\\ \Gamma^i{}_{jk}=&-\partial_i{D}\delta_{jk}+\partial_k{D}\delta_{ij}+\partial_j{D}\delta_{ik}+(\partial_i{B})\partial_j\partial_k{B}\notag \\ & -e^{-2{A}+{D}+{B}}(\mathcal H+{D}')\delta_{jk}\partial_i{B}.\end{aligned}$$ Conserved quantity at second order {#EMTevol} ================================== In this section we show the conservation laws of the curvature perturbations and discuss the gradient corrections by full consideration of second order perturbation theory. Divergence of the energy momentum tensor ---------------------------------------- Let $T^{(\alpha)\mu\nu}$ be energy momentum tensors of $\alpha$-fluid. Assuming the conservation of the energy momentum tensor for each fluid component $$\begin{aligned} \nabla_\mu T^{(\alpha)\mu\nu}=0,\label{cons:colless}\end{aligned}$$ the curvature perturbations on $\alpha$-fluid uniform density slice $$\begin{aligned} \zeta_\alpha\equiv D+\frac13\int^{\rho(\eta,\mathbf x)}_{\rho_{\rm rf}(\eta)} \frac{d \rho_\alpha}{\rho_\alpha+P_\alpha},\label{defzetatotal}\end{aligned}$$ are conserved as long as non-adiabatic pressure perturbations and the gradient terms are negligible [@Lyth:2004gb]. Let us first take a closer look at the above theorem. In this section, we do not specify a fluid component explicitly and drop the symbols from expressions. The time component of the covariant divergence can be given as $$\begin{aligned} \nabla_\mu T^\mu{}_0=&\partial_\mu T^\mu{}_0 +T^\mu{}_0 \partial_\mu \ln\sqrt{-g}-\Gamma^\alpha{}_{\mu 0}T^\mu{}_\alpha.\label{tenkaicovemt}\end{aligned}$$ Note that only a spatial gradient term in $$\begin{aligned} \partial_\mu T^\mu{}_0=\partial_0 T^0{}_0 + \partial_i T^i{}_0,\label{partderivs}\end{aligned}$$ is negligible on superhorizon scales. The other gradient terms arising in products of the linear perturbations cannot be dropped without their concrete evaluations since they may have significant contributions on small scales through convolutions in Fourier space. On the other hand, from Eq. (\[def:det\]), the second term in Eq. (\[tenkaicovemt\]) can be easily evaluated as $$\begin{aligned} T^\mu{}_0 \partial_\mu \ln\sqrt{-g}=(4\mathcal H+{A}'+3{D}')T^0{}_0+T^i{}_0\partial_i({A}+3{D}).\label{volumefac}\end{aligned}$$ The term with the Christoffel symbol in Eq. (\[tenkaicovemt\]) is decomposed into 4 parts: $$\begin{aligned} \Gamma^\alpha{}_{\mu 0}T^\mu{}_\alpha&=\Gamma^0{}_{0 0}T^0{}_0+\Gamma^0{}_{i 0}T^i{}_0+\Gamma^i{}_{0 0}T^0{}_i+\Gamma^i{}_{j 0}T^j{}_i.\end{aligned}$$ Each part can be easily calculated as $$\begin{aligned} \Gamma^0{}_{0 0}T^0{}_0=&(\mathcal H+{A}')T^0{}_0+\mathcal H(\partial{B})^2T^0{}_0+(\partial A\partial B) T^0{}_0, \\ \Gamma^0{}_{i 0}T^i{}_0=&\mathcal H \partial_i{B} T^i{}_{0}+\partial_i{A} T^i{}_{0}, \\ \Gamma^i{}_{0 0}T^0{}_i=& \mathcal H\partial_i BT^0{}_i+ \partial_i B'T^0{}_i + \partial_iAT^0{}_i\\ \Gamma^i{}_{j 0}T^j{}_i=& 3P(\mathcal H +{D}') -P(\partial{B}\partial{A})-P\mathcal H(\partial {B})^2,\label{Gamma0ijsubs}\end{aligned}$$ where we have decomposed $T^i{}_j$ into the trace part (that is, the pressure part) and the traceless part (the anisotropic pressure part), $$\begin{aligned} T^i{}_j = P \delta^i{}_j + \widetilde{T}^i{}_j\end{aligned}$$ with $\widetilde{T}^i{}_i = 0$. Note that the anisotropic pressure is at least first order quantity, which would be included in the cubic order terms above; therefore, only the isotropic pressure arises in Eq. (\[Gamma0ijsubs\]). At linear order, the following relation is useful: $$\begin{aligned} T^i{}_0+T^0{}_i=-\partial_iB(T^0{}_0 - P).\label{christoffelpart}\end{aligned}$$ Then, using Eqs. (\[partderivs\]), (\[volumefac\]) and (\[christoffelpart\]), we finally obtain $$\begin{aligned} \nabla_\mu T^\mu{}_0=&\partial_\mu T^\mu{}_0+3(\mathcal H+{D}')(T^0{}_0-P)-(T^0{}_0-P)\partial{B}\partial({A}+3D)-T^0{}_i\partial_i(A+3D+B'). \label{emtdiv0com}\end{aligned}$$ In most of the previous literatures where perfect fluid approximations are assumed, the gradient terms are automatically dropped. On the other hand, in our case, only the second term in Eq. (\[partderivs\]) is negligible, and products of the linear perturbations cannot be necessarily dropped. Let us introduce the energy density $\rho$ and the momentum transfer $q$ as $$\begin{aligned} \rho&\equiv -T^0{}_0,\\ \partial_i q&\equiv \frac{ T^0{}_i}{\rho+P}.\end{aligned}$$ Then, (\[emtdiv0com\]) can be recast into $$\begin{aligned} -\frac{1}{3(\rho+P)}\nabla_\mu T^\mu{}_0&=\mathcal H + D'+\frac{\rho'}{3(\rho+P)}-\frac13\partial{B}\partial({A}+3D) +\frac13\partial q\partial(A+3D+B'). \label{emtdiv0com3}\end{aligned}$$ Note that we have not taken the specific time slice other than the spacial coordinate; therefore Eq. (\[emtdiv0com\]) is useful for conformal Newtonian ($B=0$), uniform density ($\delta \rho=0$), spatially flat ($D=0$) or velocity orthogonal isotropic gauges ($q=0$), respectively.\ Gradient corrections -------------------- We are now ready to discuss the superhorizon conserved quantities in the presence of gradient terms. From Eqs. (\[cons:colless\]), (\[defzetatotal\]), and (\[emtdiv0com3\]), we immediately obtain $$\begin{aligned} \zeta_\alpha'=&\frac13\partial{B}\partial({A}+3D) -\frac13\partial q_\alpha \partial(A+3D+B'). \label{non:cons:zeta}\end{aligned}$$ Eq. (\[non:cons:zeta\]) apparently shows that $\zeta_{\alpha}$ is not conserved in the presence of second order gradient terms. Note that we cannot simply ignore the RHS even for long wavelength modes as we already mentioned.\ As explained in section \[section:metric\], the spacial coordinate is already fixed; the residual linear gauge freedom is given by a shift of the time coordinate $$\begin{aligned} \eta &\to \eta+\alpha.\label{alphadefgauge}\end{aligned}$$ Here, it should be noticed that the source term is composed of the products of linear perturbations; therefore, we only consider the linear gauge transformation here. In response to the above transformation, the metric perturbations obey the following transformation laws [@Ma:1995ey]: $$\begin{aligned} A&= \tilde A-\alpha'-\mathcal H\alpha,\\ B&= \tilde B+ \alpha,\\ D&= \tilde D-\mathcal H\alpha.\end{aligned}$$ On the other hand, the energy density, the pressure and the momentum transfer transform as $$\begin{aligned} \delta \rho &= \delta\tilde \rho -\alpha\rho^{(0)}{}',\\ \delta P &= \delta\tilde P -\alpha P^{(0)}{}',\\ q &= \tilde q + \alpha.\label{gt:q}\end{aligned}$$ Then, we find $$\begin{aligned} A+3D+B'&=\tilde A+3\tilde D+\tilde B'-4\mathcal H \alpha.\label{gauge:a3dbp}\end{aligned}$$ Eqs. (\[non:cons:zeta\]) and (\[gauge:a3dbp\]) motivate us to move on to the gauge which satisfies the following relation: $$\begin{aligned} A+3D+B'=0.\label{mygauge}\end{aligned}$$ This condition is useful since the fluid components and metric perturbations decouple in the covariant derivative of the energy momentum tensor, and gauge fixing is complete from Eq. (\[gauge:a3dbp\]). In this gauge, we find following quantities are conserved: $$\begin{aligned} \xi_{\alpha} \equiv D + \frac16\partial{B}\partial B+\frac13\int^{\rho(\eta,\mathbf x)}_{\rho_{\rm rf}(\eta)} \frac{d \rho_\alpha}{\rho_\alpha+P_\alpha}.\label{cons:grad:second}\end{aligned}$$ Note that $\xi_{\alpha}\to \zeta_{\alpha}$ if we ignore the gradient term. We define the isocurvature perturbations in terms of $\xi_{\alpha}$ in the similar way: $$\begin{aligned} S_{\alpha \gamma}=3(\xi_{\alpha}-\xi_{\gamma}),\label{def:iso:xi}\end{aligned}$$ which are also conserved if the energy momentum tensors are conserved and non-adiabatic pressure perturbations are absent. Thus the curvature perturbations on the uniform density slice are no more conserved in the presence of gradient terms. Instead, we introduced another conserved quantity $\xi$ at second order. $\xi$ is no more the curvature perturbation on the uniform density slice since we moved to another specific time slicing. In the next section, we consider the time evolution of $\xi$ in the presence of a collision process. Energy transfer and time evolution of the isocurvature perturbations {#evophoton} ==================================================================== The local Minkowski frame for collision processes ------------------------------------------------- Here, we discuss the collision processes for the weak Compton scattering, which are described by the quantum electrodynamics (QED) in the local Minkowski coordinate. To relate the local frame with the global one defined in Eq. (\[def:metric\]), let us consider the following coordinate transformations [@Pitrou:2007jy; @Naruko:2013aaa]: $$\begin{aligned} g_{\mu\nu}=\eta_{\bar\alpha\bar\beta}e^{\bar\alpha}{}_\mu e^{\bar\beta}{}_\nu,\end{aligned}$$ where each vierbein is defined as $$\begin{aligned} e^{\bar 0}{}_0&=ae^A,\\ e^{\bar 0}{}_i&=0,\\ e^{\bar a}{}_0&=ae^B\partial_{\bar a}B,\\ e^{\bar a}{}_i&=ae^{D}\delta_{\bar ai}.\end{aligned}$$ For the inverse matrix, the coordinate transformation becomes $$\begin{aligned} g^{\mu\nu}=e^\mu{}_{\bar\alpha} e^\nu{}_{\bar\beta}\eta^{\bar\alpha\bar\beta},\end{aligned}$$ where we have introduced $$\begin{aligned} e^{0}{}_{\bar 0}&=a^{-1}e^{-A},\\ e^{0}{}_{\bar a}&=0,\\ e^{i}{}_{\bar 0}&=-a^{-1}e^{-A-D+B}\partial_{i}B,\\ e^{i}{}_{\bar a}&=a^{-1}e^{-D}\delta_{i\bar a}.\end{aligned}$$ Next, let us consider the physical momentum $\tilde p_{\bar\alpha}$ of a particle in the local Minkowski frame. The momentum satisfies $$\begin{aligned} \tilde p_{\bar\alpha}\tilde p^{\bar\alpha}=\eta^{\bar\alpha\bar\beta}\tilde p_{\bar\alpha}\tilde p_{\bar\beta}=\eta_{\bar\alpha\bar\beta}\tilde p^{\bar\alpha}\tilde p^{\bar\beta}=-m^2,\end{aligned}$$ where $m$ is the mass of the particle. The evolution of the photon momentum in the expanding universe is written as $$\begin{aligned} \tilde p^{\bar\alpha}\propto \frac{1}{a}.\end{aligned}$$ Then, it would be more convenient to introduce the comoving momentum so as to subtract the background spacetime evolution. For this purpose, we define the comoving momentum of the conformal flat coordinate as $$\begin{aligned} p^{\bar\alpha}\equiv a \tilde p^{\bar\alpha}.\end{aligned}$$ The energy and the spacial direction of the photon are also introduced as $$\begin{aligned} p&\equiv p^{\bar 0},\label{pdef}\\ n^{\bar a}&\equiv \frac{p^{\bar a}}{p}.\end{aligned}$$ Then we can write the conjugate momentum, $P^\mu=e^\mu{}_{\bar\alpha}\tilde p^{\bar\alpha}$, associated with the spacial coordinate by using $p$ and $n^i$ as $$\begin{aligned} P^0&=\frac{\tilde p^{\bar 0}}{ae^A}=\frac{p}{a^2e^A},\label{P0def}\\ P^i&=\frac{p}{a^2e^D}(n^i-e^{B-A}\partial_i B),\\ P_0&=-pe^{A}(1- e^{B-A} n\partial {B}).\end{aligned}$$ 0 Now let us consider the time derivative of $p$ that will arise in the Boltzmann equation. First, we straightforwardly write the time derivative of Eq. (\[P0def\]): $$\begin{aligned} \frac{d}{d\eta}\ln P^0=\frac{d\ln p}{d\eta} -\frac{dA}{d\eta}-2\mathcal H.\label{logP0prime}\end{aligned}$$ Combining Eq. (\[logP0prime\]) with the zeroth component of the geodesic equation $$\begin{aligned} \frac{1}{P^0}\frac{dP^0}{d\eta}=-\Gamma^0{}_{\alpha\beta}\frac{P^\alpha P^\beta}{{(P^0)}^2},\end{aligned}$$ we obtain $$\begin{aligned} \frac{d\ln p}{d\eta}=&-\frac{\partial D}{\partial \eta}-e^{A-D}(n\partial)A+e^{B-D}(n\partial)^2B\notag \\ &+\partial A\partial B+\partial B\partial D-(n\partial B)(n\partial D)+(n\partial B)^2.\label{logpbibun}\end{aligned}$$ Time evolution of the photon energy momentum tensor --------------------------------------------------- In order to elucidate a concrete collision process, we start with constructing the photon energy momentum tensor from the phase space distribution function $f_{\gamma}$: $$\begin{aligned} T^{(\gamma)\mu\nu} &\equiv 2\int \frac{d^4P}{\sqrt{-g}(2\pi)^4}2\pi\delta(P_\alpha P^\alpha)\theta(P^0)2P^\mu P^{\nu} f_\gamma, \label{def:EMT}\end{aligned}$$ where $\theta$ is a step function, $P$’s in this expression are conjugate momenta $P_\mu$, and $\alpha$ implies a fluid component. Then the covariant derivative of Eq. (\[def:EMT\]) is given by $$\begin{aligned} \nabla_\mu T^{(\gamma)\mu}{}_{\nu}=2 \int \frac{d^3 P}{\sqrt{-g}(2\pi)^3P^0} P_{\nu} \frac{df_\gamma}{d\lambda},\label{div:emt:f}\end{aligned}$$ where $\lambda$ is an affine parameter and $P^0=d\eta/d\lambda$. Under the non canonical coordinate transformation $P_i \to p^{\bar a}$ $$\begin{aligned} P_i=g_{ij} e^{j}{}_{\bar a}\frac{p^{\bar a}}{\bar a},\end{aligned}$$ the Jacobian is transformed as $$\begin{aligned} |g_{ij} e^{j}{}_{\bar a}a^{-1}|=e^{3D}.\end{aligned}$$ Then, the three dimensional volume element in momentum space can be expressed as $$\begin{aligned} d^3P\equiv dP_1dP_2dP_3 = e^{3D}p^2 dp d\mathbf n,\end{aligned}$$ in terms of the momentum in the local conformal Minkowski frame. Using the above expression, Eq. (\[div:emt:f\]) yields $$\begin{aligned} \nabla_\mu T^{(\gamma)\mu}{}_{0}=-\frac{2}{a^{4}} \int \frac{p^{2} dp d\mathbf n}{(2\pi)^3} p(1 - n\partial {B}+\cdots ) \frac{df_\gamma}{d\eta},\label{div:emt:f:loc}\end{aligned}$$ where dots imply second order corrections. The integrand of Eq. (\[div:emt:f:loc\]) is directly related to the collision process through the Boltzmann equation: $$\begin{aligned} \frac{df_\gamma}{d\eta}=\mathcal C[f_{\gamma},\cdots],\end{aligned}$$ where the dots imply the distribution functions of the fluids which interact with the photons. When we consider the weak Compton scattering up to second order, a solution to the above Boltzmann equation can written as the superposition of a local blackbody and the spectral $y$ distortion. In this case, the collision term can be decomposed into the following form [@Ota:2016esq] $$\begin{aligned} \mathcal C[f]=\mathcal A \mathcal G(p) +\mathcal B\mathcal Y(p),\label{col:exp}\end{aligned}$$ where we have also introduced $$\begin{aligned} \mathcal G(p)&\equiv \left(-p\frac{\partial }{\partial p}\right)f^{(0)}(p),\\ \mathcal Y(p)&\equiv \left(-p\frac{\partial }{\partial p}\right)^{2}f^{(0)}(p)- 3\mathcal G(p),\end{aligned}$$ with $f^{(0)}(p) \equiv (e^{p/T_{\rm rf}}-1)^{-1}.$ $p$ is the local frame comoving momentum defined in Eq. (\[pdef\]), and $T_{\rm rf}$ is a (constant) comoving temperature of reference blackbody whose number density and energy density are defined as $$\begin{aligned} N_{\gamma \rm rf}&=2\int\frac{p^{2}dp}{2\pi^{2}}f^{(0)},\\ \rho_{\gamma \rm rf}&=2\int\frac{p^{2}dp}{2\pi^{2}}pf^{(0)}.\end{aligned}$$ We can show that the isotropic component of $\mathcal A$ is zero from the fact that the weak Compton scattering does not change the number of photons. Here we introduce the following number density flux $$\begin{aligned} N^\mu_\gamma\equiv 2\int \frac{d^4P}{\sqrt{-g}(2\pi)^4}2\pi\delta(P_\alpha P^\alpha)\theta(P^0)2P^\mu f_\gamma. \label{def:Nflux}\end{aligned}$$ The covariant derivative of the number flux can be calculated as $$\begin{aligned} \nabla_\mu N_\gamma^\mu=2\int \frac{d^3P}{\sqrt{-g}(2\pi)^3P^0}\frac{d f_\gamma}{d\lambda}.\label{tochu:covN}\end{aligned}$$ Then, substituting Eqs. (\[col:exp\]) into (\[tochu:covN\]), we obtain $$\begin{aligned} \nabla_\mu N_\gamma^\mu=3N_{\gamma \rm rf}\frac{1}{e^{A}a^4}\int \frac{d\mathbf n}{4\pi}\mathcal A=0,\end{aligned}$$ where we have used $$\begin{aligned} 2\int\frac{p^{2}dp }{2\pi^{2}}\mathcal G &= 3N_{\gamma \rm rf}, \\ 2\int\frac{p^{2}dp }{2\pi^{2}}\mathcal Y &= 0.\end{aligned}$$ On the other hand, the dipole component of $\mathcal A$ is not zero. In our notation, the dipole component of $\mathcal A$ and the monopole component of $\mathcal B$ are written as [@Pitrou:2009bc; @Naruko:2013aaa; @Chluba:2012gq; @Ota:2016esq] $$\begin{aligned} \int \frac{d\mathbf n}{4\pi}\mathbf n\mathcal A &=\frac 13n_{\rm e} \sigma_{\rm T}a\hat \partial (v+3i\Theta_{1})+\cdots \label{dip:A} \\ \int \frac{d \mathbf n}{4\pi}\mathcal B&=\frac13n_{\rm e} \sigma_{\rm T}a\hat \partial v\hat\partial (v+3i\Theta_{1})\label{Monopo:B},\end{aligned}$$ where the dots represent the second order corrections, and $\hat \partial$ corresponds to $i\mathbf k/|\mathbf k|$ in Fourier space [^1]. $v=|\mathbf v|$ is the magnitude of the velocity of the baryon fluid, and $\Theta_{1}$ is the dipole component of the photon temperature perturbations. $n_{\rm e}$ is the electron density, $\sigma_{\rm T}$ is the Thomson scattering cross section, and $a$ is a scale factor. Using Eqs. (\[div:emt:f:loc\]), (\[col:exp\]), (\[dip:A\]) and (\[Monopo:B\]), we find $$\begin{aligned} \nabla_\mu T^{(\gamma )\mu}{}_{0}=- \frac{4}{3a^{4}}\rho_{\gamma,{\rm rf}}n_{\rm e} \sigma_{\rm T}a(\hat\partial v-\partial B)\hat\partial (v+3i\Theta_{1}),\label{res:cons:emt}\end{aligned}$$ where we have used $$\begin{aligned} 2\int\frac{p^{2}dp p}{2\pi^{2}}\mathcal G&=4\rho_{\gamma \rm rf},\\ 2\int\frac{p^{2}dp p}{2\pi^{2}}\mathcal Y&=4\rho_{\gamma \rm rf}.\end{aligned}$$ We are now ready to discuss the superhorizon evolution of the isocurvature perturbations in the presence of heat conduction between electrons and photons. From Eqs. (\[emtdiv0com3\]), (\[cons:grad:second\]), and (\[res:cons:emt\]), we find $$\begin{aligned} \xi_\gamma'=& \frac{1}{3} n_{\rm e} \sigma_{\rm T}a(\hat\partial v-\partial B)\hat\partial (v+3i\Theta_{1}), \label{emtdiv0com4}\\ \xi_b'=& -\frac{1}{3R} n_{\rm e} \sigma_{\rm T}a(\hat\partial v-\partial B)\hat\partial (v+3i\Theta_{1}), \label{emtdiv0com5}\\ \xi_c'=&0, \label{emtdiv0com6}\end{aligned}$$ where $R=3\rho_{b}/4\rho_{\gamma}=3a\rho_{b,{\rm rf}}/4\rho_{\gamma,{\rm rf}}$, and we used Eq. (\[emtdiv0com3\]) for the baryon fluid with $$\begin{aligned} \nabla_\mu T^{(\gamma )\mu}{}_{0}+ \nabla_\mu T^{(b)\mu}{}_{0}=0.\end{aligned}$$ Then time derivatives of the isocurvature perturbations defined with Eq. (\[def:iso:xi\]) become $$\begin{aligned} S_{b\gamma }'&=-\frac{(1+R)}{R} n_{\rm e} \sigma_{\rm T}a(\hat\partial v-\partial B)\hat\partial (v+3i\Theta_{1}),\label{Sgabbabprime}\\ S_{c\gamma }'&=- n_{\rm e} \sigma_{\rm T}a(\hat\partial v-\partial B)\hat\partial (v+3i\Theta_{1}).\label{Sgabbabprimec}\end{aligned}$$ These expressions imply that the heat conduction from electron fluid is responsible for the change of the total photon energy while the friction heat from the intrinsic photon shear viscosity $\Theta_2$ is not. This is because the friction heat from the photon anisotropic stress does not increase the net energy in a photon system as long as we deal with background and perturbations as a whole system. Some confusion may occur if one separates the background and perturbations as done in the previous literatures, in which energy transfers from perturbations to the background are discussed. In response to Eq. (\[res:cons:emt\]), the energy momentum conservation for baryons should be also broken while those of the total fluids and the other dark sectors remain conserved. Note that these expressions are independent of the gauge choice (\[mygauge\]) since Eq. (\[gt:q\]) for the baryons and the photons are written as $$\begin{aligned} v&\to \tilde v=v+k\alpha,\\ \Theta_{1}&\to \tilde \Theta_{1}=\Theta_{1}+\frac{ik}{3}\alpha.\end{aligned}$$\ Role of the primordial non Gaussianity -------------------------------------- Eqs. (\[Sgabbabprime\]) and (\[Sgabbabprimec\]) imply that the observed isocurvature perturbations are superposition of the primordial isocurvature and the secondary isocurvature. Suppose we only have the adiabatic perturbations at the beginning, the Fourier space isocurvature perturbations are simply given as $$\begin{aligned} S_{\alpha \gamma,\mathbf k} &= \int \frac{d^{3}k_{1}d^{3}{k_{2}}}{(2\pi)^{6}}(2\pi)^{3}\delta^{(3)}(\mathbf k_{1}+\mathbf k_{2}-\mathbf k) \mathcal S_{\alpha}(\mathbf k_{1},\mathbf k_{2})\zeta_{\mathbf k_{1}}\zeta_{\mathbf k_{2}},\label{So:def}\end{aligned}$$ Here, the transfer functions in Fourier space are introduced as $$\begin{aligned} \mathcal S_{\alpha }(\mathbf k_{1},\mathbf k_{2})= \hat k_{1}\cdot \hat k_{2} \int d\eta w_{\alpha} n_{\rm e}\sigma_{\rm T}a[v(k_{1})-k_{1} B(k_{1})][v(k_{2})+3i\Theta_{1}(k_{2})],\end{aligned}$$ where $w_{b}=(1+R)/R$, $w_{c}=1$. On the other hand, the statistics of the adiabatic perturbations in the Fourier spaces are written as $$\begin{aligned} \langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\rangle &=(2\pi)^3\delta^{(3)}\left[\sum_{i=1}^2 \mathbf k_i\right]P_\zeta(k_1),\label{power:zeta}\\ \langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\rangle &=(2\pi)^3\delta^{(3)}\left[\sum_{i=1}^3 \mathbf k_i\right]B_\zeta(\mathbf k_1,\mathbf k_2,\mathbf k_3),\label{bis:zeta}\\ \langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4}\rangle&=(2\pi)^3\delta^{(3)}\left[\sum_{i=1}^4 \mathbf k_i\right]T_\zeta(\mathbf k_1,\mathbf k_2,\mathbf k_3,\mathbf k_4).\label{tri:zeta}\end{aligned}$$ Then, the cross correlations with the adiabatic perturbations and the auto correlations become $$\begin{aligned} \langle S_{\alpha\gamma ,\mathbf k}\zeta_{\mathbf k'} \rangle &=(2\pi)^{3}\delta(\mathbf k+\mathbf k') P_{\alpha \zeta}(\mathbf k),\\ \langle S_{\alpha \gamma ,\mathbf k}S_{\beta \gamma\mathbf k'} \rangle &=(2\pi)^{3}\delta(\mathbf k+\mathbf k') P_{\alpha \beta}(\mathbf k),\end{aligned}$$ where the powerspectra are calculated as $$\begin{aligned} P_{\alpha \zeta}&=\int \frac{d^{3}k_{1}}{(2\pi)^{3}} \mathcal S_{\alpha}(\mathbf k_{1},\mathbf k- \mathbf k_{1}) B_{\zeta}(\mathbf k_{1},\mathbf k-\mathbf k_{1},\mathbf k),\\ P_{\alpha \beta}&=\prod_{i=\alpha,\beta}\left[\int \frac{d^{3}k^{(i)}_{1}}{(2\pi)^{3}}\mathcal S_{i}(\mathbf k^{(i)}_{1},\mathbf k- \mathbf k^{(i)}_{1}) \right] T_{\zeta}(\mathbf k^{(\alpha)}_{1},\mathbf k-\mathbf k^{(\alpha)}_{1},\mathbf k^{(\beta)}_{1},\mathbf k-\mathbf k^{(\beta)}_{1}).\end{aligned}$$ The scale dependences of the secondary powerspectra depend on the shape of the primordial non Gaussianity. As an example, consider the local forms of bispectra and trispectra: $$\begin{aligned} &B_\zeta(\mathbf k_1,\mathbf k_2,\mathbf k_3)=\frac65f^{\rm loc.}_{\rm NL}\left[P_{\zeta}(k_1)P_{\zeta}(k_2) + (\text{2 perms.}) \right],\label{def:fnl}\\ &T_\zeta(\mathbf k_1,\mathbf k_2,\mathbf k_3,\mathbf k_4)=\tau^{\rm loc.}_{\rm NL}\left[P_{\zeta}(k_1)P_{\zeta}(k_2)P_{\zeta}(|\mathbf k_1+\mathbf k_3|) + (\text{11 perms.}) \right]\label{def:tnl},\end{aligned}$$ where we have omitted terms proportional to $g^{\rm loc.}_{\rm NL}$ for simplicity. Then the dominant contributions become $$\begin{aligned} P_{\alpha \zeta}&\approx \frac{12}{5}f^{\rm loc.}_{\rm NL}P_{\zeta}(k)\times \int \frac{d^{3}k_{1}}{(2\pi)^{3}} \mathcal S_{\alpha}(\mathbf k_{1},-\mathbf k_{1})P_{\zeta}(k_{1}),\\ P_{\alpha \beta}&\approx 4\tau^{\rm loc.}_{\rm NL}P_{\zeta}(k) \times \prod_{i=\alpha,\beta}\left[\int \frac{d^{3}k^{(i)}_{1}}{(2\pi)^{3}}S_{i}(\mathbf k^{(i)}_{1},-\mathbf k^{(i)}_{1})P_{\zeta}(k^{(i)}_{1}) \right].\end{aligned}$$ 0 In Eq. (\[So:def\]), the relative velocity $v+3i\Theta_{1}$ is strongly suppressed during tight coupling approximation, but we have a significant numerical factor coming from $n_{\rm e} \sigma_{\rm T}a$. As a result, the transfer function (\[So:def\]) becomes order of unity if there is no special cancellation. Therefore, the cross- and the auto correlation functions of the secondary isocurvature perturbations are expected to be $P_{\alpha \zeta}\approx f^{\rm loc.}_{\rm NL}\times 10^{-18}$ and $P_{\alpha \beta}\approx \tau^{\rm loc.}_{\rm NL}\times 10^{-27}$, respectively. Thus, the secondary effect on the isocurvature perturbations are negligible if we do not have significant local type primordial non Gaussianity. Thus, the powerspectra of the secondary isocurvature perturbations are the same form with the linear isocurvature powerspectrum. The disconnected part of the trispectrum leads to the following contribution for $\mathbf k\neq 0$ and $\mathbf k'\neq 0$: $$\begin{aligned} P^{(d)}_{\alpha \beta}&\approx \int \frac{d^{3}k_{1}}{(2\pi)^{3}} \prod_{i=\alpha,\beta}\ \mathcal S_{i}(\eta^{i},\mathbf k_{1},-\mathbf k_{1}) P_{\zeta}(k_{1})P_{\zeta}(k_{1}).\end{aligned}$$ Then we obtain $P^{(d)}_{\alpha \beta}\approx{\rm const.}$ for the disconnected trispectrum. This suggests the spectral index is 4, and the powerspectrum is mainly enhanced on scales where the physical process occurs. In other words, the Gaussian fluctuations cannot produce the superhorizon isocurvature modes. 0 Generation of Entropy perturbations {#sec:entropy} =================================== Besides the conserved quantity (\[cons:grad:second\]), one may wonder if we could introduce the similar quantities by using the entropy flux. In this section, we introduce the secondary entropy perturbations, which are not identified with the isocurvature perturbations if we consider non equilibrium universe during recombination. Entropy flux non conservation ----------------------------- Suppose the universe is out of equilibrium states, the standard thermodynamic relation among the entropy density, the energy density and pressure is not applicable. Instead, we introduce the (Shannon) entropy flux, which is defined in terms of a logarithm of the number of states [@Khatri:2012rt], $$\begin{aligned} S^\mu_\gamma&\equiv 2\int \frac{d^4P}{\sqrt{-g}(2\pi)^4}2\pi\delta(P_\alpha P^\alpha)\theta(P^0)2 P^\mu \mathcal F \label{def:shannon},\\ \mathcal F&\equiv \left[(f_\gamma+1)\ln(f_\gamma+1)-f_\gamma\ln f_\gamma \right].\label{def:calF}\end{aligned}$$ Note that this definition reproduces the thermodynamic entropy density for the Planck distribution. The covariant divergence of this entropy flux can be calculated as $$\begin{aligned} \nabla_\mu S^\mu_\gamma =2\int \frac{d^3P}{\sqrt{-g}(2\pi)^3P^0}\frac{d \mathcal F}{d\lambda}.\label{tochu:divS}\end{aligned}$$ A solution to the Boltzmann equation with the weak Compton collision process can be written as a superposition of the local blackbody and the spectral $y$ distortion up to the second order in the primordial fluctuations [@Pitrou:2009bc; @Naruko:2013aaa]. Such an ansatz can be expanded as follows: $$\begin{aligned} f_{\gamma}=&f^{(0)}(p)+\left[\Theta +\frac32\Theta^2\right] \mathcal G(p) + \left[\frac12\Theta^2+y\right]\mathcal Y(p),\label{def:non-thermal_anz}\end{aligned}$$ where $\Theta=\Theta^{(1)}+\Theta^{(2)}$ and $y=y^{(2)}$ are the temperature perturbation and spectral $y$ distortion, respectively. Then, Eq. (\[tochu:divS\]) vanishes at zeroth and first orders of the perturbations, but there exist non-zero contributions at second order, which is manifest from the following expression, $$\begin{aligned} \frac{1}{P^0}\frac{d \mathcal F}{d\lambda} &= \frac{d \mathcal F}{d\eta}=\frac{p}{T_{\rm rf}}\left[ \left(1-\Theta\right)\mathcal A\mathcal G+\mathcal B\mathcal Y\right].\label{calc:F2}\end{aligned}$$ Here we have replaced the Liouville term with the collision terms by using the Boltzmann equation. Using the Boltzmann equation for the $y$ distortion [@Pitrou:2009bc; @Naruko:2013aaa; @Chluba:2012gq; @Ota:2016esq], $$\begin{aligned} y'&=\mathcal B-\Theta\mathcal A, \label{y:eq}\end{aligned}$$ with Eqs. (\[tochu:divS\]), and (\[calc:F2\]), we find $$\begin{aligned} \nabla_\mu S^\mu_\gamma= \frac{4\pi^2}{15 a}\left(\frac{T_{\rm rf}}{a}\right)^3 y'_0.\label{eq:14}\end{aligned}$$ Thus, entropy increases with the generation of the spectral $y$ distortion. The physical entropy density can be defined as $S_\gamma\equiv - n_{\mu} S^{\mu}_\gamma$ with $ n_{\mu}\equiv {\nabla_\mu \eta}(-\nabla_\nu\eta \nabla^\nu\eta)^{-\frac12}$ being the normalized 1-form orthogonal to a constant $\eta$ hypersurface. One may wonder if Eq. (\[eq:14\]) can also be derived from the standard thermodynamic relation, $$\begin{aligned} \frac{dS_\gamma}{dt}=\frac{1}{T}\frac{dQ}{dt},\label{defthent}\end{aligned}$$ where $Q$ is thermodynamical heat. What we found is not a reinterpretation of this relation because we identify “heat” for the photon baryon fluid in the presence of non-equilibrium effect; thermodynamic arguments are not applicable to. Thus, the generation of $y$ distortion is not directly identified with the entropy perturbation production without a kinetic description based on the Boltzmann equation.\ Entropy perturbations at second order ------------------------------------- We are now ready to introduce a quantity $$\begin{aligned} \zeta^{(S)}_\gamma\equiv D+\frac{A}{3}+\frac13\ln \left( \frac{S^0}{S^0_{\rm rf}}\right),\label{zeta_S}\end{aligned}$$ where $S^{0}_{\rm rf}= 4\pi^2T_{\rm rf}^3/(45a^4)$. This quantity is conserved as long as entropy flux conserves at leading order of the gradient expansion. Let us check this statement by considering the covariant derivative of the entropy flux: $$\begin{aligned} \nabla_\mu S^\mu_\gamma=\partial_\mu S^{\mu}+S^\mu\partial_\mu\ln \sqrt{-g}.\end{aligned}$$ Dropping a gradient term $\partial_i S^i$, we find $$\begin{aligned} \zeta_\gamma ^{(S)}{}'= -\frac{S^i}{3S^0}\partial_i(A+3D)+y'_0,\end{aligned}$$ where we have used Eq. (\[eq:14\]). The first term represents a volume effect, which is manifest only when we take into account the next leading order of the gradient expansion. $\zeta^{(S)}_{\gamma}$ is conserved even at second order when the scattering is negligible, but only if we move on to $A+3D=0$ gauge, where the volume element does not fluctuate. However, note that gauge is not completely fixed on this slice. The second term arises as a result of the entropy production, which, in this paper, we should keep since the imperfectness of a fluid on subhorizon scales could be non negligible due to convolutions.\ The entropy density is not necessarily proportional to the number density if both of them are evaluated for a non-equilibrium state. In our case, its discrepancy is expressed in terms of $y$ distortion, which characterizes the deviation from the thermodynamic system. The curvature perturbations on the uniform number density slice can be also defined through the same procedures with the entropy: $$\begin{aligned} \zeta^{(N)}_\gamma\equiv D+\frac{A}{3}+\frac13\ln \left( \frac{N^0}{N^0_{\rm rf}}\right),\end{aligned}$$ where $N^{0}_{\rm rf}= 2\zeta(3) T_{\rm rf}^3/(\pi^2 a^4)$. Using the number flux conservation laws and dropping $\partial_i N^i$, we find $$\begin{aligned} \zeta_\gamma ^{(N)}{}'=-\frac{N^i}{3N^0}\partial_i(A+3D).\end{aligned}$$ Thus $\zeta_\gamma ^{(N)}$ is also a conserved quantity if we have the number conservation law and take the leading order of the gradient expansion. Note that $\zeta^{(N)}_\gamma$ is also conserved in $A+3D=0$ gauge even at second order without truncating the higher order gradient corrections.\ Now let us consider the following isocurvature perturbations: $$\begin{aligned} S^{(NS)}_{\alpha \gamma} \equiv \zeta^{(N)}_{\alpha} - \zeta^{(S)}_{\gamma}.\label{def:iso:NS}\end{aligned}$$ This is a covariant extension of $\delta \left(N_{\alpha}/S_{\gamma}\right)$ at nonlinear order. It should be noticed that the following relation $$\begin{aligned} \frac{N^i}{N^0}=\frac{S^i}{S^0},\end{aligned}$$ applies at linear order even for the present case since the spectral distortion is a second order effect. Then we obtain $$\begin{aligned} S^{(NS)'}_{\alpha \gamma} = -y_{0}'.\end{aligned}$$ Thus, the entropy perturbations are also conserved quantity in the presence of gradient terms if the photon entropy flux and $\alpha$-fluid number density flux are conserved. We may also consider isocurvature perturbations defined as $$\begin{aligned} S^{(N)}_{\alpha \gamma}\equiv 3(\zeta^{(N)}_\alpha-\zeta^{(N)}_\gamma),\label{def:photoniso:nn}\end{aligned}$$ which are conserved if each number density flux are conserved. The above discordance between Eqs. (\[def:iso:NS\]) and (\[def:photoniso:nn\]) motivates us to newly define the *photon isocurvature perturbation* as a fluctuation of a fraction between the photon number density and the photon entropy density $$\begin{aligned} S^{(NS)}_{\gamma\gamma}=-3y_0.\end{aligned}$$ This is nothing but the spectral $y$ distortion. For the chemical equilibrium period in the early universe where $y$ distortion is erased, it is obvious that $S^{(N-S)}_{\alpha\gamma}=S^{(N)}_{\alpha\gamma}=S_{\alpha\gamma}$ due to thermodynamic relations.\ Thus, Eqs. (\[def:iso:NS\]) and (\[def:photoniso:nn\]) can be also defined as superhorizon conserved quantities without scattering processes. However, in contrast to the conservation laws of energy momentum tensor, the conservation laws for the number flux and the entropy flux are not necessarily established in the whole cosmic history. Therefore, Eq. (\[def:iso:xi\]) is much more important than the others. Conclusions {#conclusion} =========== In this paper, we revisited the two assumptions for the conservation laws of the superhorizon isocurvature perturbations: the negligibility of the gradient terms and the energy conservation laws for the component fluids. We pointed out that the second order gradient terms are not necessarily dropped even if we consider the long wavelength modes. Then, we have introduced new second order quantities, which are conserved even in the presence of gradient terms if there are no non-adiabatic pressure perturbations. It should be noticed that they coincide with the curvature perturbations on the uniform density slice only when we can ignore the gradient terms. The total energy momentum tensor is always conserved, but that for each component fluid is not necessarily conserved. As such an example, we discuss the weak Compton scattering that transfers the energy between the photons and baryons. We found that the secondary isocurvature perturbations are generated due to this energy transfer. The powerspectra of secondary isocurvature perturbations become scale invariant if we consider the local form of the primordial tri- and bispectrum. On the other hand, the disconnected part of the trispectrum only produces the isocurvature perturbations on scales where the actual physical process occurs. We also commented on the entropy perturbations, which are usually equivalent to the isocurvature perturbations in thermal equilibrium states. However, in our case, we cannot identify these two quantities when the universe is dominated by the weak Compton scattering and is not in thermal equilibrium. We found that the entropy perturbations can be understood in terms of the spectral $y$ distortion, which is a non thermal deviation from the blackbody spectrum produced in the weak Compton scattering dominated universe. The new quantity $\xi$ we have introduced in this paper is still gauge dependent. However, it should be noticed that we can always define the gauge invariant quantities recursively even at nonlinear order as pointed out in Ref. [@Nakamura:2014kza]. Using this formalism, the gauge invariant expressions for $\xi$ would be investigated in future works. Though we only consider the weak Compton scattering, it would also be interesting if we consider the similar heat conduction from the other species such as neutrinos in the earlier epoch. This would lead to a new constraint on curvature perturbations with extremely short wavelength though it requires explicit evaluation for each scattering process, which is left for our future works. So far, we have discussed the late epoch when the universe is in neither kinetic nor chemical equilibrium. In the early epoch, the full considerations of the Compton collision terms are necessary. When there exist relativistic electrons that can sufficiently transfer the photon energy, local kinetic equilibrium is expected. In this case, the $y$ distortion may be transformed into the $\mu$ distortion, which is defined as chemical potential of a Bose distribution function. In the earlier epoch, the number changing process such as the double Compton effects, Bremsstrahlung or pair annihilation are also non-negligible. They adjust the number density and erase the spectral distortions so as to realize chemical equilibrium. Referring to Eqs. (\[tochu:covN\]) and (\[div:emt:f\]), such violation of photon number density conservation would break photon energy conservation as well. Then, secondary isocurvature perturbations might be additionally generated on superhorizon scales, but further study is necessary to make a clearer statement. We would like to thank Misao Sasaki and Atsushi Naruko for useful discussion on conservation of isocurvature perturbations on superhorizon scales. The authors are grateful to Kouji Nakamura and Karim Malik for helpful discussions. We also would like to thank Rampei Kimura for careful reading of our manuscript. This work was supported in part by JSPS Grant-in-Aid for PD Fellows (A.O.), JSPS Grant-in-Aid for Scientific Research Nos. 25287054 (M.Y.) and 26610062 (M.Y.), MEXT KAKENHI for Scientific Research on Innovative Areas “Cosmic Acceleration” No. 15H05888 (M.Y.). [^1]: In Ref. [@Ota:2016esq], the angular dependence was not properly treated, and $\hat \partial$ was dropped.

--- abstract: 'Numerical simulation of Fresnel diffraction with fast Fourier transform (FFT) is widely used in optics, especially computer holography. Fresnel diffraction with FFT cannot set different sampling rates between source and destination planes, while shifted-Fresnel diffraction can set different rates. However, an aliasing error may be incurred in shifted-Fresnel diffraction in a short propagation distance, and the aliasing conditions have not been investigated. In this paper, we investigate the aliasing conditions of shifted-Fresnel diffraction and improve its properties based on the conditions.' address: ' $^1$Graduate School of Engineering, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan ' author: - 'Tomoyoshi Shimobaba,$^{1}$ Takashi Kakue,$^1$ Naohisa Okada,$^1$ Minoru Oikawa,$^1$ Yumi Yamaguchi,$^1$ and Tomoyoshi Ito$^{1}$' title: 'Aliasing-reduced Fresnel diffraction with scale and shift operations' --- [*Keywords*]{}: Diffraction, Hologram, Holography, Computer-generated hologram, Digital holography Introduction ============ Numerical implementation of Fresnel diffraction [@goodman] with fast Fourier transform (FFT) is widely used in optics, especially computer-generated holograms (CGH) and digital holography [@poon; @dh]. CGH is a technique for generating a hologram pattern on a computer by simulating diffracted light from objects, and its applications include three-dimensional display and the design of optical elements. Digital holography reconstructs the light of an object including the amplitude and phase on a computer from a hologram captured by imaging devices such CCD cameras. In these fields, Fresnel diffraction and the angular spectrum method are used as key techniques. These diffraction calculation can be accelerated by FFT-implemented convolution; however it imposes the limitation that the sampling rate on a source plane is the same as that on a destination plane due to the property of FFT. If we can change the sampling rate, we will be able to extend the applications of diffraction calculations. In the angular spectrum method, some studies address different sampling rates on source and destination planes. Reference [@sasm1] proposed the scaled angular spectrum method using scaled-FFT, which is one version of FFT introducing a scaling parameter. This angular spectrum method requires four FFTs. Another one was proposed by us [@sasm2]. Our scaled angular spectrum method was implemented by non-uniform FFT. This angular spectrum method requires two FFTs and one interpolation. Fresnel diffraction to address the scale operation has already been realized by some studies. Double-step Fresnel diffraction executes the scale operation by double Fresnel diffraction calculations [@yamaguchi]. Shifted-Fresnel diffraction addresses the shift and scale operations by scaled-FFT [@shift1]. Reference [@shift2] proposed the same method as the shifted-Fresnel diffraction by Bluestein transform. Scaled-FFT and Bluestein transform are essentially the same. In holography, the scale operation of shifted-Fresnel diffraction [@shift1] is used in several applications: for example, wavefront-recording method [@wrp1; @wrp2; @wrp3] for accelerating CGH generation, digital holographic microscopy observable at any magnification [@dhm], and zoomable holographic projection without a zoom lens [@proj]. Shifted-Fresnel diffraction yields new applications; however, the calculations may incur a serious aliasing error in a short propagation distance and the aliasing error has not been investigated. In this paper, we investigate the aliasing conditions of shifted-Fresnel diffraction and improve the problems incurred based on the conditions. The improved diffraction is referred as to aliasing-reduced shifted and scaled (ARSS)-Fresnel diffraction. In Section 2, we explain ARSS-Fresnel diffraction. In Section 3, we show a comparison of diffracted results between ARSS-Fresnel diffraction and shifted-Fresnel diffraction. Section 4 concludes this work. Aliasing-reduced Fresnel diffraction with the scale and shift operations ======================================================================== Fresnel diffraction is expressed by, $$\begin{aligned} u_2(x_2) = \frac{\exp(i k z)}{i \lambda z} \int u_1(x_1) \exp( \frac{i \pi}{\lambda z} (x_2-x_1)^2 ) dx_1 \label{eqn:fre}\end{aligned}$$ where $k$ is the wave number, $\lambda$ is the wavelength of light, $u_1(x_1)$ is the source plane, $u_2(x_2)$ is the destination plane, and $z$ is the propagation distance between the source and destination planes. We derive ARSS-Fresnel diffraction by the following relation: $$(x_2-s x_1+o)^2=s(x_2-x_1)^2+(s^2-s)x_1^2+(1-s)x_2^2+2 o x_2 -2 s o x_1 + o^2 \label{eqn:relation}$$ where $s$ is the scaling parameter, $x_1=p m_1$ and $x_2=p m_2$. $m_1$ and $m_2$ are integers with ranges of $m_1, m_2 \in [-N/2, N/2-1]$ where $N$ is the number of sampling points. $p$ is the sampling rate on the destination plane and $o$ is offset from the origin. The sampling rate on the source plane is determined by $s p$. We can obtain the following equation by substituting the relation into Eq.(\[eqn:fre\]): $$\begin{aligned} u_2(x_2) &=& C_z \int u_1(x_1) \exp(\frac{i \pi}{\lambda z}((s^2-s)x_1^2-2sox_1)) \exp(\frac{i \pi s(x_2-x_1)^2}{\lambda z}) dx_1 \nonumber \\ \label{eqn:prev_arss}\end{aligned}$$ We can obtain ARSS-Fresnel diffraction by using the convolution theorem and introducing the band-limiting function to reduce aliasing: $$\begin{aligned} u_2(x_2) &=& C_z {\mathcal{F}^{-1} \biggl[ {\mathcal{F} \biggl[u_1(x_1) \exp(i \phi_u) \biggl]} {\mathcal{F} \biggl[\exp(i \phi_h) {\rm Rect}(\frac{x_h}{2 x_{max}}) \biggl]} \biggl]} \label{eqn:arss}\end{aligned}$$ where ${\rm Rect}(\cdot)$ is the rectangular function, $x_{h}=p m_{h}$ ( $m_h \in [-N/2, N/2-1]$), and $\exp(i \phi_u)$, $\exp(i \phi_h)$ and $C_z$ are defined by, $$\begin{aligned} \exp(i \phi_u) = \exp(i \pi \frac{(s^2-s)x_1^2-2sox_1}{\lambda z}) \\ \exp(i \phi_h) = \exp(i \pi \frac{sx_h^2}{\lambda z}) \\ C_z=\frac{\exp(i \phi_c)}{ i \lambda z }=\frac{\exp(i kz + \frac{i \pi}{\lambda z}((1-s)x_2^2+2ox_2+o^2))}{i \lambda z} \label{eqn:cz}\end{aligned}$$ Band-limitation function will be acceptable other than the rectangular function, for example, the circular function, window functions (e.g. Hanning and Hamming window functions) and so forth. It is straightforward way to extend to two-dimensional ARSS-Fresnel diffraction because Fresnel diffraction allows separation of variables. Figure \[fig:sfre-ex\] shows the light intensity distributions calculated by shifted-Fresnel diffraction [@shift1; @shift2]. The calculation parameters are $N=1,024$, $\lambda=633$ nm and $z=0.1$ m. The sampling rates on the source and destination planes indicate $p_1$ and $p_2$, respectively. The scaling parameter $s (=p_1/p_2)$ of Figs \[fig:sfre-ex\] (a), (b) and (c) are $s=1$($p_1$ and $p_2$ are 6 $\mu$ m), $s=6/4$ and $s=6/10$, respectively. ![Light intensity distributions calculated by shifted-Fresnel diffraction [@shift1; @shift2]. (a) $s=1$ (b) $s=6/4$ (c) $s=6/10$.[]{data-label="fig:sfre-ex"}](fig-shift-fre-example){width="12cm"} As shown in the figure, shifted-Fresnel diffraction can magnify diffracted results by the scaling parameter; however, no countermeasure for aliasing has been devised. For instance, when changing $z$ from 0.1 m to 0.05 m and 0.03 m, the results are shown in Fig.\[fig:sfre-aliasing\]. The upper images show the real parts and the bottom images show the light intensity of shifted-Fresnel diffraction. In the case of $z=0.1$m, the results have good quality without aliasing. However, in the cases of $z=0.05$m and $z=0.03$m, the results are extremely contaminated by aliasing. ![Real part and intensity of shifted-Fresnel diffraction at the propagation distance of 0.1 m, 0.05 m and 0.03 m.[]{data-label="fig:sfre-aliasing"}](fig-shift-fre-aliasing){width="14cm"} Aliasing conditions ------------------- Although ARSS-Fresnel diffraction is derived in a different way to shifted-Fresnel diffraction, ARSS-Fresnel diffraction of Eq.(\[eqn:arss\]) is essentially the same as shifted-Fresnel diffraction, except for the introduction of the band-limitation function. In this subsection, we clarify the aliasing conditions of ARSS-Fresnel diffraction. Equation (\[eqn:arss\]) involves three chirp functions, namely $\exp(i \phi_c)$, $\exp(i \phi_u)$ and $\exp(i \phi_h)$. The cause of aliasing is the chirp functions. Figure \[fig:zone\] shows the real part of the chirp functions, under the calculation conditions of $N=1,024$, $o=0$, $\lambda=633$ nm, $p=10 \mu$m and $s=6/10$. In the case of $z=0.1$ m, the aliasing is not incurred in the chirp functions; however, in the cases of $z=0.03$ m, $\exp(i \phi_c)$ and $\exp(i \phi_h)$ are contaminated by aliasing. ![Real part of the chirp functions at the propagation distance of $z=0.1$ m and $z=0.03$ m. []{data-label="fig:zone"}](fig-zoneplates){width="14cm"} Therefore, we need to band-limit these functions. The aliasing-free areas of the chirp functions can be calculated by [@goodman], $$\frac{1}{p} \geq 2{\left |f_c \right |} = 2 {\left |\frac{1}{2 \pi} { \frac{\partial \phi_c}{\partial x_2} } \right |}={\left | \frac{2(1-s)x_2+2so}{\lambda z} \right |} \label{eqn:freq_cz}$$ $$\frac{1}{p} \geq 2|f_u| = 2 {\left | \frac{1}{2 \pi} { \frac{\partial \phi_u}{\partial x_1} } \right |} = {\left | \frac{2(s^2-s)x_1-2so}{\lambda z} \right |} \label{eqn:freq_u1}$$ $$\frac{1}{p} \geq 2|f_h| = 2 {\left | \frac{1}{2 \pi} { \frac{\partial \phi_h}{\partial x_h} } \right |} = {\left | \frac{2 s x_h}{\lambda z} \right |} \label{eqn:freq_h}$$ From Eqs.(\[eqn:freq\_cz\])-(\[eqn:freq\_h\]), the aliasing-free areas in pixel units are as follows: $${\left | m_2 \right |} \leq {\left | \frac{\lambda {\left |z \right |} -{\left |2so \right |} }{2(1-s) p^2} \right |} \label{eqn:r_cz}$$ $${\left | m_1 \right |} \leq {\left | \frac{\lambda {\left |z \right |} -{\left |2so \right |} }{2(s^2-s) p^2} \right |} \label{eqn:r_u1}$$ $${\left | m_h \right |} \leq {\left | \frac{\lambda z }{2s p^2} \right |} \label{eqn:r_h}$$ According to Eq.(\[eqn:r\_h\]), the band-limited chirp function of $\exp(i \phi_h)$ at $z=0.03$m is shown in Fig.\[fig:zone\]. Changing $z$ between 0.01 m and 0.25 m, Fig.\[fig:plot1\] plots the aliasing-free areas in pixel units of the chirp functions under the calculation conditions of $N=1,024$, $o=0$, $\lambda=633$ nm, $p=10 \mu$m and $s=6/10$. In the figure, the green, red, and light blue plots indicate the aliasing-free areas of $\exp(i \phi_h)$, $\exp(i \phi_u)$ and $\exp(i \phi_c)$, respectively. The blue line indicates the half area ($N/2$) of the source and destination planes. The aliasing is incurred when each plot falls below the blue line. In Fig.\[fig:plot1\](a), the effective distance without aliasing is over about 0.07 m because the chirp functions $\exp(i \phi_c)$ and $\exp(i \phi_u)$ do not incur aliasing over about 0.07 m, and we need to band-limit $\exp(i \phi_h)$ in about 300 pixels ($x_{max} \approx 300 p$ in Eq.(\[eqn:arss\])). In Fig.\[fig:plot1\](b), the chirp functions $\exp(i \phi_u)$ and $\exp(i \phi_h)$ for $p=20 \mu$m and $s=6/20$ do not incur aliasing over about 0.18 m, while $\exp(i \phi_c)$ still incurs aliasing at about 0.18 m. If we band-limit $\exp(i \phi_c)$ in the aliasing-free area of about 200 pixels, no aliasing is incurred; however, the complex amplitude resulting in Eq.(\[eqn:arss\]) outside the aliasing-free area was eliminated because zero area of $\exp(i \phi_c)$ outside the aliasing-free area is multiplied with the result of inverse FFT of Eq.(\[eqn:arss\]). Therefore, for complex amplitude, we need to satisfy the following relation to avoid aliasing: $${\left |m_1 \right |}, {\left |m_2 \right |} \geq N/2 \label{eqn:aliasing_complex}$$ For light intensity, we need to satisfy the following relation to avoid aliasing: $${\left |m_1 \right |} \geq N/2 \label{eqn:aliasing_intensity}$$ because the $\exp(i \pi \phi_c)$ can be ignored in the light intensity. The aliasing condition of the light intensity ${\left |u_2(x_2) \right |}^2$ is mitigated compared to that of complex amplitude. ![Aliasing-free area in pixel units of the chirp functions under the calculation condition of $\lambda=633$ nm. (a) $s=6/10$ ($p=10 \mu$m) (b) $s=6/20$ ($p=20 \mu$m).[]{data-label="fig:plot1"}](fig-range1){width="17cm"} In Fig.\[fig:plot2\](a), the effective distance without aliasing in the scaling parameter of $s=6/4$ ($p=4 \mu$m) is over about 0.02 m because the chirp functions $\exp(i \phi_c)$ and $\exp(i \phi_u)$ do not incur aliasing over about 0.02 m, and we need to band-limit $\exp(i \phi_h)$ in the aliasing-free area of about 250 pixels ($x_{max} \approx 250 p$ in Eq.(\[eqn:arss\])). In Fig.\[fig:plot2\](b), the chirp functions $\exp(i \pi \phi_c)$ and $\exp(i \pi \phi_h)$ in the scaling parameter of $s=6/2$ ($p=2 \mu$m) do not incur aliasing over 0.02 m, while $\exp(i \phi_u)$ incurs aliasing at the distance. If we band-limit $\exp(i \phi_u)$ in the aliasing-free area of about 250 pixels, the aliasing is not incurred; however, the complex amplitude and intensity of Eq.(\[eqn:arss\]) was eliminated outside the 250 pixels. ![Aliasing-free area in pixel units of the chirp functions under the calculation condition of $\lambda=633$ nm. (a) $s=6/4$ ($p=4 \mu$m) (b) $s=6/2$ ($p=2 \mu$m).[]{data-label="fig:plot2"}](fig-range2){width="17cm"} Results ======= Figure \[fig:result1\] shows the intensity distributions calculated by ARSS-Fresnel diffraction and shifted-Fresnel diffraction [@shift1; @shift2], under the calculation conditions of $N=1,024$, $\lambda=633$ nm, $p=10 \mu$m, and $s=6/10$. We change the propagation distance from $z=0.1$ m to 0.07 m and 0.04 m. As we can see, the results of ARSS-Fresnel diffraction do not incur aliasing, while those of shifted-Fresnel diffraction incur aliasing at $z=0.07$ m and $0.04$ m. ![Intensity distributions calculated by ARSS-Fresnel diffraction and shifted-Fresnel diffraction. The scaling parameter is $s=6/10$.[]{data-label="fig:result1"}](fig-result1){width="13cm"} Figure \[fig:result2\] shows the intensity distributions calculated by ARSS-Fresnel diffraction and shifted-Fresnel diffraction, under the calculation conditions of $\lambda=633$ nm, $p=4 \mu$m, and $s=6/4$. As we can see, the results of ARSS-Fresnel diffraction do not incur aliasing, while those of shifted-Fresnel diffraction incur aliasing at $0.04$ m. From Figs.\[fig:result1\] and \[fig:result2\], ARSS-Fresnel diffraction can be applied to a wider propagation distance than shifted-Fresnel diffraction. ![Intensity distributions calculated by ARSS-Fresnel diffraction and shifted-Fresnel diffraction. The scaling parameter is $s=6/4$.[]{data-label="fig:result2"}](fig-result2){width="13cm"} Conclusion ========== We clarified the aliasing conditions of shifted-Fresnel diffraction and improved the diffraction, which was named ARSS-Fresnel diffraction. ARSS-Fresnel diffraction is useful for CGH calculation and digital holography because of the scale and shift property. Acknowledgement {#acknowledgement .unnumbered} =============== This work is supported by Japan Society for the Promotion of Science (JSPS) KAKENHI (Grant-in-Aid for Scientific Research (C) 25330125) 2013, and KAKENHI (Grant-in-Aid for Scientific Research (A) 25240015) 2013. References {#references .unnumbered} ========== [99]{} J.W.Goodman,“Introduction to Fourier Optics (3rd),” Roberts & Co (2004). T.C. Poon (ed.), “Digital Holography and Three-Dimensional Display,” Springer (2006). U. Schnars and W. Jueptner, “Digital Holography - Digital Hologram Recording, Numerical Reconstruction, and Related Techniques,” Springer (2005). S. Odate, C. Koike, H. Toba, T. Koike, A. Sugaya, K. Sugisaki, K. Otaki, and K. Uchikawa, “Angular spectrum calculations for arbitrary focal length with a scaled convolution,” Opt. Express 19, 14268-14276 (2011) T. Shimobaba, K. Matsushima, T. Kakue, N. Masuda and T. Ito, “Scaled angular spectrum method,” Optics Letters, 37, 4128-4130 (2012.09) F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett. [**29**]{}, 1668–1670 (2004). R. P. Muffoletto, J. M. Tyler, and J. E. Tohline, “Shifted Fresnel diffraction for computational holography,” Opt. Express 15, 5631-5640 (2007) J. F. Restrepo and J. G. Sucerquia, “Magnified reconstruction of digitally recorded holograms by Fresnel-Bluestein transform,” Appl. Opt. 49, 6430-6435 (2010) T. Shimobaba, N. Masuda and T. Ito, “Simple and fast calclulation algorithm for computer-generated hologram with wavefront recording plane,” Optics Letters, 34, 20, 3133-3135 (2009) T. Shimobaba, H. Nakayama, N. Masuda and T. Ito, “Rapid calculation algorithm of Fresnel computer-generated-hologram using look-up table and wavefront-recording plane methods for three-dimensional display,” Opt. Express [**18**]{}, 19504-19509 (2010) . J. Weng, T. Shimobaba, N. Okada, H. Nakayama, M. Oikawa, N. Masuda and T. Ito, “Generation of real-time large computer generated hologram using wavefront recording method,” Optics Express , 20, 4018-4023 (2012.02) T. Shimobaba, N. Masuda, Y. Ichihashi and T. Ito, “Real-time digital holographic microscopy observable in multi-view and multi-resolution,” Journal of Optics, 12, 065402 (4pp) (2010.6) T. Shimobaba, T. Kakue, N. Masuda and T. Ito, “Numerical investigation of zoomable holographic projection without a zoom lens,” Journal of the Society for Information Display [**20**]{}, 533–538 (2012).

--- abstract: 'We use the ratio $L_{\rm FIR}/L_{\rm B}$ and the IRAS color index S$_{25}$/S$_{12}$ (both widely used as indices of relative star formation rates in galaxies) to analyse subsets (containing no known AGNs or merging/interacting galaxies) of: (a) the IRAS Bright Galaxy Sample, (b) galaxies from the optically complete RSA sample which have IRAS detections in all four bands, and (c) a volume-limited IR-unselected sample. We confirm that IR-bright barred (SB) galaxies do, on average, have very significantly higher values of the FIR-optical and S$_{25}$/S$_{12}$ ratios (and presumably, higher relative star formation rates, SFR) than that do unbarred ones; the effect is most obvious in the IR colors. We also confirm that these differences are confined to early-type (S0/a - Sbc) spirals and are not evident among late-type systems (Sc - Sdm). [*Unlike others, we see no enhancement of the SFR in weakly-barred (SAB) galaxies.*]{} We further confirm that the effect of bars on the SFR is associated with the relative IR luminosity and show that it is detectable only in galaxies with $L_{\rm FIR}/L_{\rm B}$ ${\vcenter{\hbox{$>$}\offinterlineskip\hbox{$\sim$}}}$ 1/3, suggesting that as soon as they have any effect, bars translate their host galaxies into this relatively IR-luminous group. Conversely, for galaxies with $L_{\rm FIR}/L_{\rm B}$ below$\sim$ 0.1 this luminosity ratio is [*lower*]{} among barred than unbarred systems, again confirming and quantifying an earlier result. Although there is no simple physical relation between HI content and star formation, a strong correlation of HI content with the presence of bars has been found for early-type spirals with $L_{\rm FIR}/L_{\rm B}$ ${\vcenter{\hbox{$>$}\offinterlineskip\hbox{$\sim$}}}$ 1/3. This suggests that the availability of fuel is the factor determining just which galaxies undergo bar-induced starbursts.' author: - 'J.H.Huang' - 'Q.S. Gu' - 'H.J. Su' - 'T.G. Hawarden' - 'X.H. Liao' - 'G.X. Wu' date: 'Received;accepted' title: 'The Bar-enhanced Star-formation Activities in Spiral Galaxies' --- Introduction ============ For well over a decade, models of the dynamics of the interstellar medium (ISM) in spiral galaxies (e.g. Roberts, Huntley & van Albada, ( 1979 ); Schwarz ( 1984 ); Combes & Gerin ( 1985 ); Byrd et al ( 1986 ); Noguchi ( 1986, 1988 ), [*inter alia*]{}, see also the review by Athanassoula 1992 ) have suggested that the presence of a central bar generates an inflow of gaseous interstellar medium ( ISM ) which accumulates at the inner Lindblad resonance ( ILR ), if it exists, or else near the nucleus. Such inflows are obviously potential raw material for a burst of star formation in the center of the galaxy, and it has been known for three decades that bars are indeed strongly associated with the presence of “hot-spots" and other peculiar central structures in spirals (e.g. Sérsic& Pastoriza ( 1965 and 1967 ). Nevertheless, there is still an active debate on the nature and degree of the dependence of star formation activity on barred morphology. Observations at radio wavelengths by Hummel ( 1981 ) suggested that the radio luminosity of central sources in barred spirals (SB+SAB types) are on average a factor 2 more powerful than those in ordinary spirals (SA type). The extension of this work by Puxley et al. ( 1988 ) showed a strong correlation between the presence of a compact radio nucleus and barred morphology (SB+SAB). In the mid and far-IR the IRAS catalogue has permitted a wide range of studies. de Jong et al. ( 1984 ) found that IRAS had a higher detection rate for barred than for unbarred spirals and that the barred systems tended to have hotter FIR colors. Hawarden et al. ( 1986a,b ) found a strong dependence of IRAS [*mid*]{}-IR colors on barredness. Devereux ( 1987 ) also found this effect, and noted that the presence of bars was strongly correlated with the concentration near the nucleus (${\vcenter{\hbox{$<$}\offinterlineskip\hbox{$\sim$}}}$ $\rm 5^{\prime\prime}$) of a large fraction of the total emission at 10$\mu$m wavelength. A similar conclusion to that of Hawarden et al. was reached by Dressel ( 1988 ): the presence of a bar appeared clearly to affect the SFR in S0, Sa and Sb galaxies but (in agreement with Devereux) not in late types. Enhancement of star formation rates inferred from optical (H$\alpha$) data was also seen in barred galaxies by Arsenault ( 1989 ). On the other hand, 10$\mu$m observations of a nearby, relatively faint galaxy sample ( Devereux et al. 1987 ) show no evidence for enhanced near-nuclear emission in barred systems. Eskridge & Pogge ( 1991 ) argued that the presence or absence of bars does not affect the SFR in S0 galaxies, while Isobe & Feigelson ( 1992 ) found that barred galaxies in a volume-limited (low-luminosity) sample had [*lower*]{} overall FIR luminosities than unbarred galaxies. A search in the near-IR for bars in an IR-bright sample of 16 non-Seyfert, non-interacting galaxies by Pompea & Rieke ( 1990 ) appeared to show that IR bars are not a necessary prerequisite for strong infrared activity in such isolated, non-AGN galaxies. Perhaps these apparently discrepant studies can be understood if we recall that some groups worked on optically-selected samples of normal spiral galaxies while others used samples extracted from the IRAS catalogue, some deliberately selected to be IR-Bright. Statistically, the sources in the IRAS catalogue, especially IR-Bright galaxies, differ markedly from normal spirals ( Mazzarella et al 1991 ), so differences in current star formation levels and their governing factors are likewise to be expected. Such differences were foreshadowed by Devereux et al. ( 1987 ) who found in studying the IR characteristics of normal nearby galaxies that dependence of SFR on bar morphology is related to IR luminosities, being absent in low-luminosity systems. To help understand such effects we have constructed a new IR-luminous sample, the details of which are presented in Section 2. The results of our analysis of this sample and similar analyses of two other samples taken, or updated, from previous studies are presented in Section 3, with a comparison of their properties. The combined analyses of these samples suggests a simple unified picture of the influence of barred morphology on star formation in, and on the IR properties of, spiral galaxies. A New Sample, and Statistical methods ===================================== The IR-Bright Galaxy Sample --------------------------- To explore the effects of sample properties on the apparent influence of barred morphology on the far-infrared properties for spiral galaxies, we have constructed a sample which is deliberately intended to be IR-bright. It is derived from IRAS Bright Galaxies Sample ( Soifer et al. 1989 ), the main selection criterion for which is a 60 $\mu m$ flux density $\geq$ 5.4 Jy, and which includes a variety of classes of galaxies, including AGN, mergers and strongly interacting galaxies. All of these tend to contain warm dust, whether or not from star formation ( Miley et al 1985; Lonsdale et al 1984; Sanders et al 1988; Mazzarella et al 1991; Surac et al 1993; and references therein ) and independent of the presence of barred morphology. In order to isolate the effects of just this morphology on star formation, we define a subsample by excluding all active nuclei (Seyferts and LINERs) listed by Véron-Cetty &Véron ( 1993 ), and all mergers and strongly interacting galaxies listed by Lonsdale et al. ( 1984 ). We also require that each object be assigned a definite Hubble type S in the Reference Catalog, which will also exclude some morphologically peculiar galaxies. We refer to the resulting subsample as the IR-Bright Galaxy Sample (the IRBG sample). Fig 1 shows a cumulative distribution of log $L_{\rm FIR}/L_{\rm B}$ for this sample. It is immediately apparent that the SB systems differ from the SAs. However, rather unexpectedly, the SAB galaxies are indistinguishable from the unbarred systems; in the next section we provide statistical confirmation, while later we return to this issue with a test and sample in which barred galaxies show an even stronger difference from the SA systems; once more no difference between SAB and SA systems is discernable. In sharp contrast, Fig 1 of Hawarden et al. ( 1986b ) shows a clear difference between their SAB and SA samples. The former contains  12 galaxies with IRAS 25 $\mu m$ fluxes 2.2 or more times stronger than the 12 $\mu m$ flux, and above and below this threshold. From a 2$\times$2 $\chi^{\rm 2}$ test with Yates’ corrections the probability that the two groups are the same P$_{\rm null}$ is $<$ 0.005. We note that the sample of Pompea & Rieke ( 1990 ) (discussed below) also suggests that SAB systems differ from SAs in having higher SFRs. These contradictions will be explored elsewhere; in this analysis we simply omit all SAB galaxies, except in the discussion of the work by Pompea & Rieke. Details of the 123 objects remaining in the present sample (the BG sample), ranging from S0/a to Sdm in type, are listed in Table 1 as follows: column (1), the NGC, UGC, and IC number; column (2), the morphological type from RC3 ( de Vaucouleurs et al 1991 , hereafter RC3 ); column (3), the recessional velocities corrected to the Galactic Center from RC3; column (4), distance in Mpc. Following Soifer et al ( 1989 ), where a “primary distance/Fisher-Tully distance" was available this is given in preference to the radial velocity distance. Primary distances are taken from Sandage & Tammann ( 1981 ), while Fisher-Tully distances are taken from Aaronson et al ( 1982 ) or Aaronson and Mould ( 1983 ), and adjusted to a Virgo distance of 17.6 Mpc ( corresponding to a Hubble constant of 75 $$km$~$s$^{-1}~$Mpc$^{-1}$ ); column (5), the total blue magnitude from RC3; column (6), logarithm (base 10) of the FIR luminosity in $$L$_{\odot}$ from 40 to 120$\mu m$, which is given by ( Lonsdale et al 1985 )\ $ L_{\rm FIR}~=~3.75~10^{5}~D^{2}~(2.58~S_{60}+S_{100}) $\ where $S_{60}$ and$S_{100}$ are the flux densities at 60$\mu m$ and 100$\mu m$ in Jy, from Soifer et al ( 1989 ), D is the distance in Mpc; column (7), logarithm (base 10) of the blue luminosity in $$L$_{\odot}$, computed from $B_{\rm T}^{\rm o}$ using the formula ( Pogge & Eskridge 1993 )\ $ log~L_{\rm B}=12.208-0.4B_{\rm T}^{\rm o}+log(1+z)+2logD $\ where $z$ is the redshift of the galaxy. ---------------------------------------------- ------------- --------------- ------------ ------------ ------------ ------------ ------------ ------------ ------------ name option early late early late early late early late barred -0.0455(33\*) -0.384(23) -0.870(35) -0.888(19) -1.356(8) -1.062(13) -0.569(68) -0.724(46) unbarred -0.298(36) -0.420(33) -0.830(38) -0.860(28) -1.105(10) -1.037(15) -0.622(65) -0.686(61) $KS^{\dag}$ 0.03 0.71 0.76 0.37 0.33 0.94 0.20 0.98 $t^{\ddag}$ 0.005 0.65 0.72 0.82 0.42 0.85 0.62 0.65 barred 0.0362(29) -0.251(16) -0.080(7) -0.231(2) ¶ ¶ 0.015(34) -0.251(16) [$\overline{log L_{\rm FIR}/L_{\rm B}}$]{} unbarred -0.153(27) -0.196(18) -0.312(9) -0.360(6) -0.179(31) -0.197(20) [[($L_{\rm FIR}/L_{\rm B} > 1/3)^{\S}$]{}]{} $KS^{\dag}$ 0.05 0.70 0.23 0.68 0.03 0.70 $t^{\ddag}$ 0.02 0.45 0.10 0.27 0.007 0.45 barred -0.638(4) -0.689(7) -1.067(28) -0.965(17) -1.716(6) -1.062(13) -1.153(34) -0.977(30) [$\overline{log L_{\rm FIR}/L_{\rm B}}$]{} unbarred -0.734(9) -0.688(15) -0.990(29) -0.990(22) -1.196(9) -1.108(14) -1.025(34) -0.924(41) [[($L_{\rm FIR}/L_{\rm B} < 1/3)^{\S}$]{}]{} $KS^{\dag}$ 0.92 0.99 0.70 0.46 0.08 0.80 0.47 0.85 $t^{\ddag}$ 0.36 0.99 0.40 0.84 0.03 0.67 0.188 0.457 ---------------------------------------------- ------------- --------------- ------------ ------------ ------------ ------------ ------------ ------------ ------------ [l|l||c|c|c|c]{} &barred&&&&\ &unbarred&1.749(73)&1.225(65)&&1.564(107)\ &$KS^{\dag}$&0.00348&0.030&&0.009\ &$t^{\ddag}$&0.0015&0.020&&0.003\ &barred&1.570(11)&1.351(39)&¶&1.273(41)\ [$\overline{S_{\rm 25\mu m}/S_{\rm 12\mu m}}$]{}&unbarred&1.237(24)&1.160(50)&&1.190(53)\ [[($L_{\rm FIR}/L_{\rm B} < 1/3)^{\S}$]{}]{}&$KS^{\dag}$&0.04&0.39&&0.63\ &$t^{\ddag}$&0.056&0.17&&0.47\ The numbers in parentheses are number of sources for each subsample\ The probability that the barred and unbarred systems are from the same parent population,\ derived from the KS test.\ The probability that the barred and unbarred systemsare from the same parent population,\ derived from the t test.\ §The statistics are performed for sources with $L_{\rm FIR}/L_{\rm B} > $ or $<$ 1/3.\ ¶The sample is too small, or no data available for statistics. Statistics and Statistical Methods ---------------------------------- Our sample is not large enough to analyse for the effects of barredness in each morphological subtype. Instead, following Combes & Elmegreen ( 1993 ), we have separated our sample into just two groups: early types (S0/a through Sbc) and late types (Sc through Sdm). We treat the S0/a galaxies as spirals because the distribution of their relative FIR emission differs from that of lenticulars ( Eskridge & Pogge 1991 ), and the distribution of their HI content closely resembles that of the Sa systems ( Wardle & Knapp 1986 ). The far-IR luminosity $L_{\rm FIR}$ measures not only the star-forming rate, SFR, but also the [*size*]{} of a galaxy. We must therefore normalize the total luminosities, e.g. to the actual projected area, or to the optical luminosity of the galaxy. Mazzarella et al. ( 1991 ) showed that both these normalizations give similar results. Since the quantity $L_{\rm FIR}/L_{\rm B}$ is now widely used as an indicator of relative star formation rate in analysing the infrared properties of galaxies ( e.g. Keel 1993; Combes et al 1994; Helou & Bicay 1993 ), we will use this quantity to discuss the bar-enhancement of star formation in this paper. Hawarden et al. ( 1986a ) discuss the use of the mid-IR [*color*]{} ${\rm S_{25}/S_{12}}$, as a sensitive indicator of elevated star formation activity. Hawarden et al ( 1986b ) employed this parameter in their analysis; we will also do so, in parallel with $L_{\rm FIR}/L_{\rm B}$. We have adopted two main tests for the samples examined here. To verify the similarity or difference of samples we compare cumulative distributions of the property being examined (generally $L_{\rm FIR}/L_{\rm B}$ or ${\rm S_{25}/S_{12}}$) by means of the two-sample Kolmogorov-Smirnov test (KS). We estimate the significance of differences in the mean properties of samples by the student’s t test. Occasionally, where obvious (or predefined) dividing lines exist in two properties, the difference between two samples is illustrated by a simple 2 x 2 $\chi^2$ test with Yates’ corrections for small numbers. The basic statistical results presented in this paper are summarized in Table 2. The following section discusses the individual results in more detail. Comparison of Samples ===================== Analysis of the BG sample ------------------------- Fig 2 shows cumulative distributions of $\log L_{\rm FIR}/L_{\rm B}$ for barred and unbarred galaxies in the BG sample, the early types in Fig 2a and the late types in Fig. 2b. In the latter figure the late-type SBs have mean $\log$ $L_{\rm FIR}/L_{\rm B}$ = -0.38, which is not significantly different from that of the SAs ($p_{\rm t}\sim$ 0.65). Similarly, the KS test indicates a probability P$_{\rm null}$ = 0.71 that the 23 barred and 33 unbarred late-type galaxies are drawn from the same parent population with repect to the optical-FIR luminosity ratio. However among the early-type galaxies the mean of $L_{\rm FIR}/L_{\rm B}$ for SBs is about 1.8 times higher than for SAs, and the difference is significant: $p_{\rm t} \sim$ 0.005. The two sample KS test now gives P$_{\rm null}$ $\sim$ 0.03, suggesting a real bar-enhancement of the SFR for the early-type systems. We also note in Fig 2a that this difference only becomes apparent for $L_{\rm FIR}/L_{\rm B}$ ${\vcenter{\hbox{$>$}\offinterlineskip\hbox{$\sim$}}}$ 1/3 Elmegreen & Elmegreen ( 1985, 1989 ) and Combes & Elmegreen ( 1993 ) have discussed the different properties of bars in early- and late-type spirals. Those in early-type systems are relatively long and strong, with quite flat intensity distributions, while those in late-type galaxies are rather short and weak and their intensities vary exponentially along the bar. The weaker bars in the later types are likely to be less effective at driving the inward flow of gas, and consequently to have lower levels of induced star formation. Furthermore, in late-type galaxies the nucleus contributes on average less than 10% to the FIR luminosity ( Devereux et al. 1987 ), so a change in that fraction, even by quite a large factor, will have a small effect on the overall properties of the galaxy. Conversely, the strong bars common in early-type galaxies may be expected to be much more efficient movers of gas, and, being longer, to have a larger collection range from which to supply the inflow; such features may reasonably be expected to generate powerful enhancements in SFRs. Moreover, the nuclear contribution to the FIR luminosity in early types is about 30% ( Devereux et al. 1987 ) so such enhancements will have greater impact on the overall properties of the system. The apparent confinement of the SRF-enhancing effects of bars to early type systems is therefore easily understood, at least qualitatively. Comparison with the results obtained by Hawarden et al. ( 1986 ) ---------------------------------------------------------------- Hawarden et al. ( 1986b ) and Puxley et al. ( 1988 ) examined samples comprising galaxies from the optically-complete magnitude-limited Revised Shapley Ames catalogue (RSA: Sandage & Tammann 1981 ) with detections in all four bands in the IRAS Point Source Catalog ( 1985 ). They found that galaxies with IRAS flux ratio $S_{\rm 25\mu m}/S_{\rm 12\mu m}~>~2.2$ are almost of types SB or SAB. We have produced an updated sample by selecting from the RSA those galaxies with morphological types between S0/a and Sdm in the RC3 ( de Vaucouleurs et al. 1993 ) which have detections in all four IRAS bands in the IRAS PSC (Version 2). After excluding all known AGN (Seyfert 1 and 2 and, unlike Hawarden et al., also all LINERs) in the catalogue by Véron-Cetty & Véron ( 1993 ) and, for uniformity (Section 2) all SAB systems, as well as the morphologically peculiar, post-merger system NGC 2146 (RC3: of SBabP) we are left with a list of 120 objects (hereafter the HP sample). Fig 3a and Fig 3b illustrate the distributions of $\log L_{\rm FIR}/L_{\rm B}$ for early- and late-type galaxies, respectively, in this sample. The lack of any major difference between SB and SA systems, whether early or late type, is obvious and confirmed by the KS test : the probability that the barred and unbarred sets are from the same parent population is 0.76 (early-type) and 0.37 (late-type), respectively. Consistent with this result, the mean $\log$ $L_{\rm FIR}/L_{\rm B}$ for early- and late-type barred/unbarred galaxies (-0.870/-0.830 and -0.888/-0.860 respectively) are not significantly different either ($p_{\rm t} \sim $ 0.72, 0.82). All the above statistical results indicate that the presence of bars, whether for early- or late-type galaxies, does not measurably enhance the SFRs in this hybrid optical/IR selected sample. However, the mean $L_{\rm FIR}/L_{\rm B}$ for the galaxies in Fig. 3 is well below that of the BG sample. We remarked in Section 3.1 that the apparent enhancement of $L_{\rm FIR}/L_{\rm B}$ in SBs relative to SAs in the BG sample is only apparent when $L_{\rm FIR}/L_{\rm B} >$ 1/3. If we divide the distribution in Fig 3a into two parts at this value of $L_{\rm FIR}/L_{\rm B}$, we find that for systems with $L_{\rm FIR}/L_{\rm B}~<~1/3$ , the mean value of the ratio for 28 barred galaxies is indistinguishable from that for the 29 unbarred systems (0.086/0.102) ($p_{\rm t} \sim$ 0.40), but among galaxies with $L_{\rm FIR}/L_{\rm B}~>~1/3$ the mean for the seven SB objects is 1.7 times higher than that for the 10 SAs, a marginally significant difference ($p_{\rm t} \sim$ 0.1). The initial apparently null result does not directly contradict the results of Hawarden et al. ( 1986b ), as their study concentrated on IR rather than IR/Optical colors and luminosities. We therefore show in Fig 4 the distribution of the ratio $S_{\rm 25}/S_{\rm 12}$ – used by Hawarden et al. ( 1986b ) – for the HP sample. Fig 4a illustrates the distribution for the whole HP sample. Barred and unbarred galaxies are now distributed very differently: a 2$\times$2 $\chi^{\rm 2}$ test with division of the samples about $S_{\rm 25}/S_{\rm 12}$ = 2.2 indicates that the probability that the barred and unbarred galaxies are drawn from the same population is $<<$ 0.001. The difference is evidently real, in agreement with the results of Hawarden et al ( 1986b ); the mean $S_{\rm 25}/S_{\rm 12}$ for barred galaxies is significantly higher than that for unbarred sources(1.583/1.225, $p_{\rm t} \sim $ 0.020), in agreement with the KS test, which gives P$_{\rm null}$ $\sim$ 0.03. Fig 4b shows the distribution of ${\rm S_{25}/S_{12}}$ for sources in the HP sample with $L_{\rm FIR}/L_{\rm B}~<~1/3$. Now the mean $S_{\rm 25}/S_{\rm 12}$  for SBs is not significant different from that for SA systems (1.351/1.160, $p_{\rm t}\sim $ 0.17, or P$_{\rm null}$ $\sim $ 0.39 from the KS test). Again, this time from analysis of the ${\rm S_{25}/S_{12}}$ colors, the effects of barred morphology are seen only when $L_{\rm FIR}/L_{\rm B}$ $>$ 1/3. Comparison to results obtained by Isobe & Feigelson ( 1992 ) ------------------------------------------------------------ Isobe & Feigelson ( 1992 ) have recently carried out survival analysis on a volume-limited sample ( v $\leq$ 1400 km $$s$^{-1}$  ) selected from the Zwicky Catalog, and concluded that barred galaxies are systematically fainter in their FIR emission than unbarred galaxies, which is just the opposite to what Hawarden et al ( 1986b ) and we obatined. To investigate the difference between their results and those of Hawarden et al ( 1986b ), Isobe & Feigelson examined their data sets omitting the nondetections and survival analysis method, and found that the type SB galaxies are not very different from the SA galaxies ( P $\sim$  0.12 ). They concluded then that the difference may be due to their use of different samples, but not due to methodological bias. To make a comparison between Isobe & Feigelson’s results and ours, we have adopted their approach to omit survival analysis method and sources with no detections at 12$\mu m$ and 25$\mu m$ from their sample, the SAB type galaxies are also omitted as before. The distributions of log $L_{\rm FIR}/L_{\rm B}$ for the resulting sample ( IF sample hereafter ) are indicated in Fig 5a and Fig 5b, for early- and late-type galaxies, respectively. It is obvious from Fig 5b that the mean $L_{\rm FIR}/L_{\rm B}$ for late-type barred sources ( $$N$_{\rm B}$ = 13 ) is not different from that of unbarred ( $$N$_{\rm N}$ = 15 ) galaxies ( 0.0867/0.0918, $p_{\rm t} \sim$ 0.85 ) . While the early-type subsample contains only 18 galaxies (Fig 5a) it immediately appears that, as found by IF, barred galaxies are less luminous than unbarred galaxies ,by a factor of about 1.8. Summary of comparison --------------------- Generally, the mean $L_{\rm FIR}/L_{\rm B}$ for the early-type galaxies in the IF sample is about 11. times lower than that of the early-type BG sample ( 0.0613/0.702, $p_{\rm t} \sim 5.8\times10^{-7}$ ), and 2.3 times lower than that of the early-type HP sample ( 0.0613/0.141, $p_{\rm t} \sim$  0.022 ). Statistically speaking, they are all significantly different. In terms of $L_{\rm FIR}/L_{\rm B}$, the above various samples cover over varying intensities of relative star formation rate. Especially, early-type IF sample covers over rather weak region of $L_{\rm FIR}/L_{\rm B}$, while BG sources cover over a relative strong region of $L_{\rm FIR}/L_{\rm B}$. From the analyses in Sec.3.1, 3.2, and 3.3 we have seen that the bar-enhancement becomes striking over the region of strong $L_{\rm FIR}/L_{\rm B}$. And the bar-reduced SFR applies to a region of rather weak $L_{\rm FIR}/L_{\rm B}$. Different conclusions come from samples with different $L_{\rm FIR}/L_{\rm B}$ coverage. To illustrate this further, and to explore transition thresholds, we have therefore combined all the samples defined above to give a sample with a much wider range of $L_{\rm FIR}/L_{B}$ than any of the component subsamples. Despite being statistically more complicated, its large dynamic range will assist the investigation of transitions between IR property regimes. Fig 6a shows the $L_{\rm FIR}/L_{B}$ distribution of early spirals for the “combined" sample. As expected, it illustrates clearly the dependences on barred morphology in the different IR regimes. For low SFRs, among galaxies with $L_{\rm FIR}/L_{B}~<~$ 1/10, this ratio is 1.5 times [*lower*]{} among barred than among unbarred galaxies (at significance $p_{\rm t} \sim$ 0.07), consistent with the results by Isobe & Feigelson ( 1992 ). Conversely, among high-SFR galaxies with $L_{\rm FIR}/L_{\rm B}~>~$ 1/3, the mean value of this ratio is 1.6 times [*higher*]{} for barred than for unbarred systems ($p_{\rm t} \sim$  0.007), which is in agreement with the results of Hawarden et al ( 1986b ), Devereux ( 1987 ), Puxley et al. ( 1988 ), and Dressel ( 1988 ). For the region in between, 1/10 $<~L_{\rm FIR}/L_{\rm B}~<~$ 1/3, the distributions of barred and unbarred systems are similar. Also as might be expected from Devereux’ results ( 1987 ), the late type systems from the combined sample, Fig 6b, show no significant differences between barred and unbarred systems ($p_{\rm t} \sim$  0.65). Fig 7a illustrates the distribution of the $S_{\rm 25}/S_{\rm 12}$ color for the entire “combined" sample. Barred and unbarred distributions are obviously very dissimilar (P $\sim$  0.011 from KS test), the barred systems having a markedly wider distribution which is clearly centered at high values of the color ratio: the mean for barred galaxies (1.994) is very significantly higher than for unbarred (1.564), with $p_{\rm t} \sim$  0.003. On the other hand, Fig 7b shows $S_{\rm 25}/S_{\rm 12}$  distribution of the sources for the “combined” sample with $L_{\rm FIR}/L_{\rm B} <$ 1/3. Once again the distributions of barred and unbarred galaxies are nearly the same. Statistically, both the KS test and the t-test confirm this ( $\rm P \sim$ 0.34, $p_{\rm t} \sim$ 0.31 ). These statistical results strongly suggest that the effect of bars on SFR becomes prominent for a sample with $L_{\rm FIR}/L_{\rm B} >$ 1/3. On the results obtained by Pompea & Rieke ( 1990 ) -------------------------------------------------- Pompea & Rieke have observed 15 non-Seyfert, non-interacting galaxies in NIR bands ( 1990 ), and found that their observations did not support the suggestion by Hawarden et al ( 1986b ) of bars are a ubiquitous feature of galaxies with 25$\mu m$ excess. In fact, the 15 sources observed by Pompea & Rieke ( 1990 ) are IR active, their $L_{\rm FIR}/L_{\rm B}$ are all larger than 1/3, see Table 3, 11 of them are in IRBGS described in Sec 2. . Three of the remaining 4 galaxies, NGC 2146, NGC 5665, and NGC 6574, have flux densities at 60$\mu m$ being larger than 5.4 Jy, the basic selection criterion of IRBGS. The last one, NGC 2784, has been assigned as lenticular galaxies in RC3. Basically, the Pompea & Rieke’s sources should belong to IRBGS. The suggestion by Hawarden et al ( 1986b ) mentioned above is mainly based on the fact that galaxies with IRAS flux ratio $S_{\rm 25}/S_{\rm 12} >$ 2.2 are exclusively barred ( with two exceptions ) as shown in their Fig 1. The number ratio of unbarred to barred galaxies with $S_{\rm 25}/S_{\rm 12} >$ 2.2 is 2/23, about 1/11. And they speculated that these two sources may in fact be barred. [*This is, however, not the case for IRBGS*]{}, as shown in Fig 8. The number ratio of unbarred to barred sources with $S_{\rm 25}/S_{\rm 12} >$ 2.2 is 12/26, about 1/2, suggesting that unbarred sources of $S_{\rm 25}/S_{\rm 12} >$ 2.2 are not rare in IRBGS. According to Pompea & Rieke ( 1990 ), 10 of their 15 sources have flux ratio $S_{\rm 25}/S_{\rm 12} >$ 2.2, but only 3 of these galaxies ( NGC 3504, NGC 4536, and NGC 5713 ) are barred, ( not the Type listed in Table 3 which are from RC3 ). We have noticed, however, among the remaining 7 unbarred sources, one galaxies ( NGC 2146 ) is a peculiar galaxies as we pointed out above, other two galaxies ( NGC 253, and NGC 2782 ) are AGN listed in Catalogue by Véron-Cetty & Véron ( 1993 ). After removing these three galaxies,the observed number ratio of unbarred to barred sources with $S_{\rm 25}/S_{\rm 12} >$ 2.2 is 4/3. Considering the small size of Pompea & Rieke’s sample of 15, especially the total number of 10 sources with $S_{\rm 25}/S_{\rm 12} >$ 2.2 as compared to the number of 12 unbarred galaxies with $S_{\rm 25}/S_{\rm 12} >$ 2.2 in IRBGS, we might not be able to say something about the difference between the observed and statistical number of ratio of unbarred and barred galaxies ( 4/3 to 1/2 ). But the problem here is certainly not so severe as originally thought. [llrcc]{} Name & Type & S$_{60\mu m}$ & $log~[L_{\rm FIR}/L_{\rm B}]$ & EX$\ddag$\ NGC 253 & .SXS5.. & 980.080 & -0.222 & yes\ NGC 922 & .SBS6P. &   5.907 & -0.309 &\ NGC 2782 & .SXT1P. &   9.632 & -0.245 & yes\ NGC 2990 & .S..5\*. &   5.504 & -0.153 &\ NGC 3310 & .SXR4P. &  35.972 & -0.115 & yes\ NGC 3504 & RSXS2.. &  22.816 & -0.064 & yes\ NGC 4433 & .SXS2.. &  14.207 &  0.202 & yes\ NGC 4536 & .SXT4.. &  32.853 & -0.291 & yes\ NGC 5653 & PSAT3.. &  11.193 &  0.143 &\ NGC 5713 & .SXT4P. &  23.247 & -0.058 & yes\ NGC 5861$\dag$ & .SXT5.. &  11.896 & -0.336 &\ NGC 5936 & .SBT3.. &   9.270 &  0.137 & yes\ NGC 2146 & .SBS2P. & 153.624 &  0.370 & yes\ NGC 5665 & .SXT5P\$ &    6.552 & -0.216 &\ NGC 6574 & .SXT4\*. &  14.800 & -0.089 &\ NGC 2764 & .L...\*. &   3.980 & -0.041 & yes\ $\ddag$ refered to ${\rm S_{25}/S_{12}}$ excess.\ $\dag$ not observed by Pompea & Rieke. Discussion ========== Gas content among Samples ------------------------- The gaseous ISM is the raw material from which young stars form, and it is reasonable to expect that its availability will affect the vigour of that process. Conversely, therefore, we may expect that samples of galaxies with different SFRs will exhibit correlated differences in gas content. We here explore these possibilities among the early-type galaxies where the effects of barred morphology are most apparent. Although star formation occurs in molecular, rather than atomic gas, and the resulting IR emission is mostly from dust associated with the molecular material, the current data sets on molecular gas in early-type spirals are still too small to provide good coverage of our sample lists and we must await additional observations. However, plenty of HI data are conveniently available ,e.g. from RC3. We now employ these data in a similar manner to some other groups ( see, e.g. Eskridge & Pogge 1991, and references therein ). Fig 9 demonstrates a clear correlation between FIR emission and HI content in the early-type galaxies of all samples, with correlation coefficient R = 0.66 for the BG sample, and R=0.60 for the HP sample (including the early-type galaxies from the IF sample, because there are insufficient HI data to perform a meaningful analyses for the IF sample alone), showing in Fig 9a and 9b respectively. What the statistical correlation in Fig 9 indicated is the following: galaxies with large FIR emission also tend to have large HI component, or [*vice versa*]{}. No trace of association with barredness is apparent. However a much more suggestive result emerges if we subdivide the “combined” sample, as before, at the value of relative IR luminosity where we have found that the IR properties begin to depend on barred morphology, $L_{\rm FIR}/L_{\rm B}$ $\sim$ 1/3, and examine the distributions of HI content. The results are shown in Fig 10. Once again the relatively IR-brighter systems show the morphological dependence, see Fig 10a, while the IR-faint objects do not, see Fig 10b. The barred galaxies above $L_{\rm FIR}/L_{\rm B}$ $>$ 1/3 have mean HI content 1.9 times higher than that of the unbarred objects, a significant discrepancy at $p_{\rm t} \sim $ 0.07. We have shown in Sec 3.4 that for galaxies with $L_{\rm FIR}/L_{\rm B} >$  1/3 in “ combined” sample, the bars strongly enhance the star formation rate, while not for sources with $L_{\rm FIR}/L_{\rm B} <$ 1/3 ( not including early-type IF sample). Thus it follows that for galaxies of $L_{\rm FIR}/L_{\rm B} >$ 1/3, the bar-enhanced SFR is associated with the bar-related HI content excess, which we consider, in circumstances, to be highly suggestive of a common mechanism. The results of this investigation will be discussed elsewhere ( Gu et al 1996 ). We have no intension of directly connecting the HI content with the effect of bars on star formation, shown in Fig 10a, but we would emphasize that the physical nature of enhancements in SFR remains much less clear ( see, e.g. Keel 1993 ). Several recent studies of disk galaxies indicate that substantial amount of FIR flux come from regions which are spatially distinct from either resolved regions of massive star formation or strong CO sources ( Jackson et al 1991; Smith et al 1991 ). One of the possibilities is that the bulk of the FIR comes from something akin to Galactic cirrus. To further clarify the effect of bar on SFR would need more HI and CO data, especially those observations with spatial distribution, which are under planning. Morphological classification ---------------------------- The uncertainties in morphological classification of galaxies will definitely affect our analyses. The way of dividing a sample into early- and late-type systems would partly reduce this kind of influence. In the above separation, we have just followed Combes & Elmegreen ( 1993 ) and Devereux et al ( 1987 ) to take all galaxies before Sbc type as early-type spirals. And we have found that the statictical difference obtained in Sec. 3 between early-type barred and unbarred galaxies would disappear, like those results obtained for late-type systems, if we put galaxies of Sc type into early-type spirals, indicating that the dividing line between early- and late-type spirals set by Combes & Elmegreen ( 1993 ) is basically reasonable. There is another uncertainty in morphological classification in RC3, i.e. a number of sources have been assigned to S type rather than SA or SB type. In our analyses in Sec 3 or 4, they have been taken as SA type galaxies. If we put all of them into SB types instead, the statistical difference between early-type barred and unbarred systems will be strengthened, and the conclusions drawn for late-type galaxies remain as they were statistically. It follows that the uncertainties of not assigning definite Hubble type may not introduce significant modification to our results. Conclusions =========== Our analyses have led us to the following conclusions: 1. Stellar bars in spiral galaxies do indeed affect star formation rates, but only in types S0/a - Sbc, not in later classes. 2. The influence of bars on star formation rate is perceptible only in galaxy samples which locate at the relatively high end of the $L_{\rm FIR}/L_{\rm B}$ range, probably because most galaxies so affected have thereby been moved into those samples. The enhancement effects of barred morphology on SFRs become apparent for $L_{\rm FIR}/L_{\rm B}$ $>$ 1/3, but can be discerned in less IR-luminous systems in the mid-IR color introduced by Hawarden et al. ( 1986a ), which is a more sensitive indicator of the SFR than a simple comparison of luminosities or of FIR excesses. Our analyses indicate that sources with $L_{\rm FIR}/L_{\rm B} >$ 1/3 play an important role in the statistics of IRAS color ($S_{\rm 25}/S_{\rm 12}$) approach. 3. At the other end of the $L_{\rm FIR}/L_{\rm B}$ range bars act to reduce the IR luminosity of a galaxy, though probably not its SFR. 4. There is therefore a huge and highly significant difference between the effects of barredness in the most IR-luminous sample (the BG sample ) and the least IR-luminous (the IF sample) in our study. Most of the different conclusions about the influence of this morphology on SFRs arises from studies of samples falling at different locations within the $L_{\rm FIR}/L_{\rm B}$ range. 5. The fact that distributions of HI content behave similarly to those of IR properties suggests that the availability of fuel is a governing factor in the effects of bars on star formation rates. [*Acknowledgements*]{} We would like to thank Dr. Lequeux for his critical comments and instructive suggestions, that significantly strengthened the analyses in this paper. The authors would thank Dr. Zhong Wang for his valuable discussion. This work has been supported by grants from National Science and Technology Commission and National Natural Science Foundation. Aaronson, M., Huchra, J., Mould, J., et al, 1982, ApJS 50, 241 Aaronson, M., & Mould, J.1983, ApJ 265, 1 Athanassoula, E. 1992. MNRAS, 259, 345 Byrd, G.G., Valtonen, M.J., Sundelius, B., and Valtoja, L. 1986, A&A 166,75 Combes, F., & Elmegreen, B.G. 1993, A&A 271, 391 Combes, F., Prugeniel, P, Rampazzo, R., & Sulentic, J.W. 1994, A&A 281, 725 de Vaucouleurs, G., de Vaucouleurs, A., Corwin Jr., H.G., Buta, R.J., Paturel, G., & Fougue, P. 1991, Third Reference Catalogue of Bright Galaxies, Springer Verlag, New York Devereux, N.A., Becklin, E.E., & Scoville, N. 1987, ApJ 312, 529 Devereux, N.A. 1987, ApJ 323, 91 Devereux, N.A. & Young, J.T. 1991, ApJ 371, 515 Dressel, L.L. 1988, ApJ 329, L69 Elmegreen, B.G., & Elmegreen, D.M. 1985, ApJ 238, 438 Elmegreen, B.G., & Elmegreen, D.M. 1989, ApJ 342, 677 Eskridge, P.B., & Pogge, R.W. 1991, AJ 101, 2056 Gu, Q.S., Huang, J.H., Liao, X.H., & Su, H.J. 1996, in preparation Hawarden, T.G., Fairclough, J.H., Joseph, R.D., Leggett, S.K., & Mountain, C.M. 1986a, in [*New Light on Dark Matter*]{}, ed. F.P.Israel, Reidel, Dordrecht, 455 Hawarden, T.G., Mountain, C.M., Leggett, S.K.& Puxley, P.J., 1986b, MNRAS 221, 41p Helou, G. & Bicay, M.D. 1993, ApJ 415, 93 Hummel, E. 1981, A&A 93, 93 IRAS Point Source Catalog, Version 2, 1988, Joint IRAS Science Working Group, (Washington,D.C.:GPO) Isobe,T., & Feigelson, E.D. 1992, ApJS 79, 197 Jackson,J.M., Eckart, A., Cameron, M., Wild, W., Ho, P.T.P., Pogge,R.W., and Harris, A.I. 1991, ApJ 373 Keel, W.C. 1993, AJ 106, 1771 Lonsdale, C.J., Persson, S.E., & Matthews, K. 1984, ApJ 287, 95 Lonsdale, C.J., Helou, G., Good, J.C., & Rice, W.L. 1985, Catalogued Galaxies and Quasars Observed in the IRAS Survey, Washington D.C. Mazzarella, J.M., Bothun, G.D., & Boroson, T.A. 1991, AJ 101, 2034 Miley, G.K., Neugebauer, G., & Soifer, B.T. 1985, ApJ 293, L11 Noguchi, M. 1988a, A&A 201, 37 Noguchi, M. 1988b, A&A 203, 259 Pogge, R.W., & Eskridge, P.B. 1993, AJ 106, 1405 Pompea, S.M., & Rieke, G.H. 1990, ApJ 356, 416 Puxley, P.J., Hawarden, T.G., and Mountain, C.M. 1988, MNRAS 231, 465 Roberts, W.W., Huntley, J.M. & van Albada, G.D. 1979, ApJ 233, 67 Sandage, A, & Tammann, G.A. 1981, A Revised Shapley-Ames Catalog of Bright Galaxies, Carnegie Institute, Washington, DC Sanders, D.B., Soifer, B.T., Elias, J.H., Madore, B.F., Matthews , K., Neugebauer G., & Scoville, N.Z. 1988, ApJ 325, 74 Sérsic, J.L & Pastoriza, M. 1965, PASP, 77, 287 Sérsic, J.L & Pastoriza, M. 1967, PASP, 79, 152 Soifer, B.T., Boehmer, L., Neugebauer, G., & Sanders, D.B. 1989, AJ 98, 766 Smith, B.J., Lester, D.F., Harvey, P.M., and Pogge, R.W. 1991, ApJ 373 Schwarz, M.P. 1984, MNRAS, 209, 93 Thronson, H.A., & Telesco, C.M. 1986, ApJ 311, 98 Véron-Cetty, M.P., & Véron, P. 1993, ESO Scientific ReportNo.13 Wardle, M., & Knapp, G.R. 1986, AJ 91, 23

--- abstract: 'The correlation-driven Mott transition is commonly characterized by a drop in resistivity across the insulator-metal phase boundary; yet, the complex permittivity provides a deeper insight into the microscopic nature. We investigate the frequency- and temperature-dependent dielectric response of the Mott insulator $\kappa$-(BEDT-TTF)$_{2}$Cu$_2$(CN)$_3$ when tuning from a quantum spin liquid into the Fermi-liquid state by applying external pressure and chemical substitution of the donor molecules. At low temperatures the coexistence region at the first-order transition leads to a strong enhancement of the quasi-static dielectric constant $\epsilon_1$ when the effective correlations are tuned through the critical value. Several dynamical regimes are identified around the Mott point and vividly mapped through pronounced permittivity crossovers. All experimental trends are captured by dynamical mean-field theory of the single-band Hubbard model supplemented by percolation theory.' author: - 'A. Pustogow' - 'R. Rösslhuber' - 'Y. Tan' - 'E. Uykur' - 'M. Wenzel' - 'A. Böhme' - 'A. Löhle' - 'R. Hübner' - 'Y. Saito' - 'A. Kawamoto' - 'J. A. Schlueter' - 'V. Dobrosavljević' - 'M. Dressel' date: - - - title: 'Low-Temperature Dielectric Anomalies at the Mott Insulator-Metal Transition' --- [^1] [^2] [^3] The insulator-metal transition (IMT) remains the main unresolved basic science problem of condensed-matter physics. Especially intriguing are those IMTs not associated with static symmetry changes, where conventional paradigms for phase transitions provide little guidance. Early examples of such behavior are found in certain disorder-driven IMTs [@Shklovskii1984]. In recent years, IMTs with no symmetry breaking were also identified around the Mott transition [@Imada1998], which bears close connection to exotic states of strongly-correlated electron matter such as superconductivity in the cuprates. From a theoretical point of view, the single-band Hubbard model is at present well understood [@Georges1996; @Vollhardt2012], and is found to be in excellent agreement with experiments [@Limelette2003; @*Limelette2003a; @Hansmann2013; @Kagawa2005; @*Kagawa2004; @*Furukawa2015]. While commonly concealed by antiferromagnetism, recent development in the field of organic quantum spin liquids (QSL) enabled us to study the low-temperature Mott IMT in absence of magnetic order [@Kurosaki2005; @*Shimizu2003; @Shimizu2016; @Itou2017; @Li2019; @Furukawa2018; @Pustogow2018], revealing finite-frequency precursors of the metal already on the insulating side [@Pustogow2018]. The Mott insulator and the correlated metal converge at the critical endpoint $T_{\rm crit}$ (Fig. \[fig:phase-diagram\]). The former is bounded by a quantum-critical region along the quantum Widom line (QWL) [@Terletska2011; @*Vucicevic2013; @Furukawa2015; @Pustogow2018; @Dobrosavljevic1997]. On the metallic side, resistivity maxima at the “Brinkman-Rice” temperature $T_{\rm BR}$ signal the thermal destruction of resilient quasiparticles [@Radonjic2012; @*Deng2013] and the crossover to semiconducting transport. Below $T_{\rm crit}$, the IMT is of first order and comprises an insulator-metal coexistence regime [@Georges1996; @Terletska2011; @Vucicevic2013]. It is currently debated whether electrodynamics is dominated by closing of the Mott gap or by spatial inhomogeneity, fueled by recent low-temperature transport studies [@Furukawa2018]. ![(a) Tuning the bandwidth $W$, for instance by chemical or physical pressure, transforms a Mott insulator to a correlated metal. Dynamical mean-field theory predicts a first-order transition with phase coexistence up to the critical endpoint [@Georges1996], and a quantum-critical regime associated with the quantum Widom line (QWL) above $T_{\rm crit}$ [@Vucicevic2013]. The metallic state is confined by the Brinkman-Rice temperature $T_{\rm BR}$, the coherent Fermi-liquid regime by $T_{\rm FL}$. When interactions $U$ are comparable to $W$, and $ T \gg T_{\rm crit}$, semiconducting behavior prevails; neither a gap nor a quasiparticle peak are stabilized. (b-d) Resistivity signatures of the crossovers.[]{data-label="fig:phase-diagram"}](phase-diagram38.pdf){width="0.8\columnwidth"} Phase coexistence around the first-order line ($T<T_{\rm crit}$) emerges from bistability of the insulating and metallic phases between closing of the Mott gap at $U_{c1}$ and demise of the metal at $U_{c2}$ [@Georges1996]. One generally expects hysteretic behaviors when tuning across a first-order transition. Seminal transport and susceptibility experiments indeed found a pronounced hysteresis in Mott insulators with a magnetically-ordered ground state [@Lefebvre2000; @Limelette2003; @*Limelette2003a]. Unfortunately, analogue measurements with continuous pressure tuning are not feasible on QSL compounds, such as , due to the low temperatures ($T<20$ K) and high pressures ($p>1$ kbar) required to cross the first-order IMT. A more direct insight into the coexistence region was provided by spatially resolved optical spectroscopy [@Sasaki2004]. The most compelling results came from near-field optical experiments on vanadium oxides by Basov and collaborators [@Qazilbash2007; @*Huffman2018; @McLeod2016] where a spatial separation of metallic and insulating regions upon heating could be visualized, in accord with x-ray studies [@Lupi2010; @Hansmann2013]; the range with hysteresis in $\rho(T)$ coincides with the observed phase coexistence. Although recent developments in cryogenic near-field instrumentation are rather promising [@McLeod2016; @Post2018; @*Pustogow2018s], they fall short of covering the regime $T<T_{\rm crit} \approx 15$ K required here and do not allow for pressure tuning. For this reason, we suggest dielectric spectroscopy as novel bulk-sensitive method in order to reveal the coexistence regime, distinguish the individual phases and obtain a deeper understanding of the dynamics around the IMT. The complex conductivity $\sigma_1+{\rm i}\sigma_2$ not only reveals the closing of the Mott gap but yields insight into the growth of metallic regions and the formation of quasiparticles as correlation effects decrease. In this Letter we tackle the fundamental question whether the electrodynamic response around the Mott IMT is overwhelmed by the gradual decrease of the Mott-Hubbard gap within a homogeneous insulating phase, or whether the effects of phase coexistence dominate. Furthermore, is it possible to distinguish on the metallic side between the coherent (quasiparticle) low-$T$ regime and incoherent transport at high-$T$? To answer these questions, we present temperature- and frequency-dependent dielectric measurements on a genuine Mott compound that is bandwidth-tuned across its first-order IMT. In addition to hydrostatic pressure we developed a novel approach of chemically substituting the organic donor molecules. The experimental findings are complemented by dynamical mean-field theory (DMFT) calculations, incorporating spatial inhomogeneities in a hybrid approach. We conclude that electronic phase segregation plays a crucial role, leading to percolative phenomena due to the separation of insulating and metallic regions, also allowing clear and precise mapping of different dynamical regimes around the IMT. ![The dielectric conductivity and permittivity of  were measured as a function of temperature and frequency for various applied (a-e) pressures and (f-j) chemical substitutions \[introduction of Se-containing BEDT-STF molecules illustrated in (h) and (k)\] that drive the system across the Mott transition. (a,f) Starting from the insulator, $\sigma_1(T)$ grows with increasing $p$ or $x$; a metallic phase is stabilized below $T_{\rm BR}$, in accord with dc transport [@Kurosaki2005; @Furukawa2015; @Furukawa2018]. (b,c,g,h) In the Mott-insulating state $\epsilon_1(T)$ exhibits relaxor-ferroelectric behavior similar to the parent compound [@Abdel-Jawad2010; @Pinteric2014]. Extrinsic high-temperature contributions are subtracted. (d,i) The strong enhancement of $\epsilon_1(T)$ at the transition is a hallmark of a percolative IMT, as sketched in (l-o). (e,j) When screening becomes dominant in the metal, $\epsilon_1$ turns negative; $\sigma_1$ exhibits Fermi-liquid behavior below $T_{\rm FL}$. []{data-label="fig:sigma-eps_T"}](Fig_2_44.pdf){width="1\columnwidth"} We have chosen  single crystals for our investigations because this paradigmatic QSL candidate is well characterized by electric, optical and magnetic measurements [@Kurosaki2005; @*Shimizu2003; @Kezsmarki2006; @Kanoda2011; @*Zhou2017; @Furukawa2018; @Pustogow2018; @Culo2019]. Although the dimerized charge-transfer salt possesses a half-filled conduction band, strong electronic interaction $U \approx 250$ meV stabilizes a Mott-insulating state below the QWL ($T_{\rm QWL}\approx 185$ K at ambient pressure [@Pustogow2018; @Furukawa2015]). The effective correlation strength $U/W$ can be reduced by increasing the bandwidth $W$; for pressure $p > 1.4$ kbar the metallic state is reached at low temperatures [^4], with $T_{\rm crit} \approx 15$-20 K. In addition, we exploited a novel route of partially replacing the S atoms of the donor molecules by Se, where more extended orbitals lead to larger bandwidth \[see sketches in Fig. \[fig:sigma-eps\_T\](h,k)\]. The substitutional series  ($0\leq x \leq 1$, abbreviated ) spans the interval ranging from a Mott insulator to a Fermi-liquid metal [^5]. Details on the sample characterization and experimental methods are given in Refs. . Here we focus on the out-of-plane dielectric response measured from $f=7$ kHz to 5 MHz down to $T=5$ K. Both physical pressure and STF-substitution allow us to monitor the permittivity while shifting the system across the first-order IMT. Fig. \[fig:sigma-eps\_T\] displays the temperature-dependent conductivity and permittivity data of  when $p$ rises (a-e) and $x$ increases in  (f-j). The insulating state ($p<1.4$ kbar, $x<0.1$), characterized by ${\rm d}\sigma_1/{\rm d}T>0$, generally features small positive $\epsilon_1\approx 10$. The relaxor-like response previously observed in the parent compound below 50 K has been subject of debate [@Abdel-Jawad2010; @Pinteric2014]. The metallic state ($p>3$ kbar, $x>0.2$) is defined by ${\rm d}\sigma_1/{\rm d}T<0$ and, concomitantly, $\epsilon_1<0$ that becomes very large at low $T$ as itinerant electrons increasingly screen [^6]. This onset of metallic transport identifies $T_{\rm BR}$ [@Radonjic2012]; while thermal fluctuations prevail at higher $T$, the quasiparticle bandwidth is the dominant energy scale for $T<T_{\rm BR}$. Below $T_{\rm FL}$ the resistivity $\rho(T)\propto T^2$ indicates the Fermi-liquid state. Right at the first-order IMT, however, the dielectric behavior appears rather surprising. When approaching the low-temperature phase boundary, $\epsilon_1$ rapidly increases by several orders of magnitude. This colossal permittivity enhancement is more pronounced in the quasi-static limit, $\epsilon_1\approx 10^5$ at $f=7.5$ kHz, and the peak value approximately follows a $f^{-1.5}$ dependence. The overall range in $T$ and $p$/$x$ of the divergency is robust and does not depend on the probing frequency; detailed analysis on the dynamic properties is given in [@Rosslhuber2019; @Saito2019]. In Fig. \[fig:sigma-eps\_p\_x\_DMFT\](a,b) the pressure evolution of $\sigma_1$ and $\epsilon_1$ is plotted for fixed $T$. At $T=10$ K, $\sigma_1(p)$ rises by six orders of magnitude in the narrow range of 1 kbar and $\Delta x = 0.1$. This behavior flattens to a gradual transition above 20 K, associated with the quantum-critical crossover at the QWL. The inflection point shifts to higher $p$, in accord with the positive slope of the phase boundary [@Pustogow2018] associated with the rising onset of metallicity at $T_{\rm BR}$. The  series exhibits similar behavior \[Fig. \[fig:sigma-eps\_p\_x\_DMFT\](c,d)\]: around the critical concentration of $x\approx 0.12$ a drastic increase in $\sigma_1$ is observed at low $T$ that smears out as $T$ rises. The maximum in $\epsilon_1(x)$ is reached for $x=0.16$ but broadens rapidly upon heating. ![(a) The Mott IMT of $\kappa$-(BEDT-TTF)$_2$Cu$_2$(CN)$_3$ appears as a rapid increase of $\sigma_1(p)$ that smoothens at higher $T$; above $T_{\rm crit}$ a gradual crossover remains. (b) $\epsilon_1(p)$ exhibits a sharp peak below $T_{\rm crit}$. The results at $f=380$ kHz are plotted on a logarithmic scale. (c,d) Similar behavior is observed for chemical BEDT-STF substitution. (e,f) Fixed-temperature line cuts of our hybrid DMFT simulations (see text) as a function of correlation strength $U/W$ and $T/W$ [@Vucicevic2013; @Pustogow2018] resemble the experimental situation in minute detail, including the shift of the IMT with $T$. The lack of saturation of $\sigma_1(T\rightarrow 0)$ seen in DMFT modeling reflects the neglect of elastic (impurity) scattering in the metal (outside the coexistence region).[]{data-label="fig:sigma-eps_p_x_DMFT"}](Fig_3_41.pdf){width="1\columnwidth"} {width="90.00000%"} The Mott IMT is based on the idea that a reduction of electronic correlations, i.e. rise of $W/U$, gradually closes the Mott-Hubbard gap: a coherent charge response develops, causing a finite metallic conductivity. Pressure-dependent optical studies on several organic Mott insulators actually observe this behavior [@Faltermeier2007; @*Merino2008; @*Dumm2009; @Li2019]. It was pointed out [@Aebischer2001] that even in the case of certain second-order phase transitions, a continuously vanishing charge gap might induce an enhancement in $\epsilon_1$, perhaps leading to a divergence at low $T$. Probing the optical response at THz frequencies is actually a convenient method to monitor the gap contribution to the permittivity. From $p$ and $T$ sweeps across the Mott IMT of very different materials an increase by a factor of 10 is consistently reported [@Qazilbash2007; @Li2019]; in the case of  we find it even smaller [@Saito2019]. Hence, the dielectric catastrophe of $\epsilon_1 \approx 10^5$ observed in our pressure and substitution-dependent dielectric experiments evidences an additional effect. Treating the fully-frustrated model at half filling, even at $T=0$ DMFT finds metallic and insulating solutions coexisting over an appreciable range of $U/W$ [@Georges1996]. This may result in a spatial segregation of these distinct electronic phases. Such a picture resembles composite materials, such as microemulsions, composites or percolating metal films [@vanDijk1986; @Clarkson1988a; @*Clarkson1988b; @Pecharroman2000; @Nan2010; @Hovel2010]. Classical percolation is not a thermodynamic phase transition, but a statistical problem that has been studied theoretically for decades by analytical and numerical methods in all details; one of the key predictions is the divergency of $\epsilon_1$ when approaching the transition from either side, with characteristic scaling [@Dubrov1976; @Efros1976; @Bergman1977; @*Bergman1978]. Over a large parameter range the dielectric properties are well described by the Maxwell-Garnett or Bruggeman effective medium approaches [@Choy2015]. To further substantiate this physical picture in quantitative details, we carried out theoretical modeling of the systems under study. We calculated $\epsilon_1+{\rm i}\epsilon_2$ using DMFT for a single-band Hubbard model, and obtained the respective responses for both the insulating and the metallic phase around the Mott point [@Vucicevic2013]. The DMFT phase diagram (Fig. \[fig:phase-diagram\]) also features an intermediate coexistence region below $ k_{B} T_{\rm crit} \approx 0.02 W$. In accord with the analysis of our experimental results, we assumed a smoothly varying metallic fraction $x$ within the phase coexistence region. To obtain the total dielectric response, we solved an appropriate electrical-network model representing such spatial inhomogeneity utilizing the standard effective-medium approximation (for details, see Ref. ). Our hybrid DMFT simulation yields excellent agreement with experiment – both pressure-tuning and chemical substitution – as illustrated in Fig. \[fig:sigma-eps\_p\_x\_DMFT\](e,f) and in the false-color plots in Fig. \[fig:contour-plots\]. We find that the colossal peak in $\epsilon_1$ is confined to the spatially inhomogeneous coexistence regime, exactly as observed in experiments. As correlation effects diminish, the dynamical conductivity (upper panels) increases from the Mott insulator to the Fermi liquid. The step in $\sigma_1$ and drop in $\epsilon_1$ appear abruptly in the model but more smoothly in experiment, most likely due to inhomogeneities which broaden the coexistence regime by providing nucleation seeds for the incipient phase. Note, the percolative transition region narrows for $T\rightarrow T_{\rm crit}$ and vanishes above that; the metallic fraction of the simulation is indicated in Fig. \[fig:contour-plots\](f). The inset of panel (f) clearly demonstrates that the colossal permittivity enhancement appears exclusively for a percolating mixture of metallic and insulating regions, and not for the pure phases. These results render the gap closing irrelevant for the electrodynamics at the low-temperature Mott IMT. While the transition is of first-order at half filling, doping [@Hebert2015] or disorder [@Gati2018; @Urai2019] effectively move $T_{\rm crit}\rightarrow 0$, eventually turning it into a true quantum-critical point. We further point out that the discussed mechanism of a percolative enhancement of $\epsilon_1$ may also apply to related organic compounds subject to first-order transitions. Similar dielectric anomalies in  and  were previously assigned to ferroelectricity [@Gati2018a] and multiferroicity [@Lunkenheimer2012], respectively. The former exhibits a peak in $\epsilon_1$ right at its charge-order IMT, where phase coexistence is evident [@Gati2018a; @Hassan2019]. The latter is located extremely close to the Mott IMT [@Limelette2003a], so the coexistence region is likely entered already at ambient pressure [^7]. While previous transport and optical studies [@Furukawa2015; @Pustogow2018] already provided hints favoring the DMFT scenario, they do not map out the predicted dynamical regimes, especially regarding a well-defined coexistence region at $T < T_{\rm crit}$. Our new dielectric data, however, reveal all phases in vivid detail and in remarkable agreement with the respective crossover lines obtained from dc transport. Indeed, we recognize the gapped Mott insulator by essentially constant $\epsilon_1$ (light red) bounded precisely by the QWL [@Vucicevic2013], while also below $T_{\rm BR}$ [@Radonjic2012; @*Deng2013] the response clearly follows the dielectric behavior expected for a metal ($\epsilon_1 < 0$, blue). Most remarkably, these two crossover lines converge towards $T_{\rm crit}$, which marks the onset of the coexistence region, just as anticipated from Fig. \[fig:phase-diagram\]. The emergence of phase segregation is evidenced by the huge peak of $\epsilon_1$ in excellent agreement with our current DMFT-based modeling. The sharply defined boundaries of this dielectric anomaly imply that the corresponding inhomogeneities are [*the consequence and not the cause*]{} of phase separation, the latter resulting from strong correlation effects inherent to Mottness. Our findings leave little room for doubt that the DMFT scenario offers a rather accurate picture of the Mott IMT, in contrast to other theoretical viewpoints [@Senthil2008] which focus on the spin degrees of freedom in the QSL. This also confirms recent experimental and theoretical results [@Lee2016; @Pustogow2018spinons] suggesting that such gapless spin excitations, while dominant deep within the low-temperature Mott-insulating phase, are quickly damped away by the onset of charge fluctuations close to the IMT. We also demonstrated that the novel chemical method of partially substituting the organic donor molecules of the fully-frustrated Mott insulator  by BEDT-STF yields similar bandwidth-tuning like physical pressure. By comparing the boundaries of the Mott state and the correlated metal (QWL, $T_{\rm BR}$, $T_{\rm FL}$) we find that 1 kbar is equivalent to $\Delta x \approx 0.06$. The pronounced divergency in $\epsilon_1$ evidences a spatial coexistence of metallic and insulating electronic phases around the first-order IMT that can be circumstantially described by percolation theory. Our results yield that the Mott gap has a minor effect on the dielectric properties while the effects of phase coexistence dominate. We appreciate discussions with S. Brown, B. Gompf and I. Voloshenko. We acknowledge support by the DFG via DR228/52-1. A.P. acknowledges support by the Alexander von Humboldt Foundation through the Feodor Lynen Fellowship. Work in Florida was supported by the NSF Grant No. 1822258, and the National High Magnetic Field Laboratory through the NSF Cooperative Agreement No. 1157490 and the State of Florida. E.U. receives support of the European Social Fund and the Ministry of Science Research and the Arts of Baden-Württemberg. J.A.S. acknowledges support from the Independent Research/Development program while serving at the National Science Foundation. A.P., R.R. and Y.T. contributed equally to this work. [67]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop [**]{} (, , ) [****, ()](https://link.aps.org/doi/10.1103/RevModPhys.70.1039) [****, ()](https://link.aps.org/doi/10.1103/RevModPhys.68.13) [****,  ()](\doibase 10.1002/andp.201100250) [****,  ()](http://science.sciencemag.org/content/302/5642/89.abstract) [****,  ()](https://link.aps.org/doi/10.1103/PhysRevLett.91.016401) [****,  ()](\doibase 10.1002/pssb.201248476) @noop [****,  ()]{} [****,  ()](\doibase 10.1103/PhysRevB.69.064511) @noop [****,  ()]{} [****,  ()](https://link.aps.org/doi/10.1103/PhysRevLett.95.177001) [****,  ()](https://link.aps.org/doi/10.1103/PhysRevLett.91.107001) [****,  ()](https://link.aps.org/doi/10.1103/PhysRevLett.117.107203) [****,  ()](http://advances.sciencemag.org/content/3/8/e1601594.abstract) [****,  ()](\doibase 10.1103/PhysRevB.99.115137) [****,  ()](\doibase 10.1038/s41467-017-02679-7) [****,  ()](\doibase 10.1038/s41563-018-0140-3) [****,  ()](https://link.aps.org/doi/10.1103/PhysRevLett.107.026401) [****, ()](https://link.aps.org/doi/10.1103/PhysRevB.88.075143) [****,  ()](\doibase 10.1103/PhysRevLett.79.455) [****,  ()](https://link.aps.org/doi/10.1103/PhysRevB.85.085133) [****,  ()](https://link.aps.org/doi/10.1103/PhysRevLett.110.086401) @noop [****,  ()]{} [****,  ()](\doibase 10.1103/PhysRevLett.92.227001) [****,  ()](\doibase 10.1126/science.1150124) [****,  ()](\doibase 10.1103/PhysRevB.97.085146) [****, ()](\doibase 10.1038/nphys3882) [****,  ()](\doibase 10.1038/ncomms1109) [****,  ()](\doibase 10.1038/s41567-018-0201-1) [****,  ()](\doibase 10.1126/sciadv.aau9123) [****,  ()](\doibase 10.1103/PhysRevB.82.125119) [****,  ()](https://link.aps.org/doi/10.1103/PhysRevB.90.195139) [****, ()](https://link.aps.org/doi/10.1103/PhysRevB.74.201101) [****,  ()](\doibase 10.1146/annurev-conmatphys-062910-140521) [****, ()](https://link.aps.org/doi/10.1103/RevModPhys.89.025003) [****,  ()](\doibase 10.1103/PhysRevB.99.045114) @noop [****,  ()](\doibase 10.1103/PhysRevB.98.205141) @noop [ ]{} [****, ()](https://link.aps.org/doi/10.1103/PhysRevB.76.165113) [****,  ()](https://link.aps.org/doi/10.1103/PhysRevLett.100.086404) [****, ()](https://link.aps.org/doi/10.1103/PhysRevB.79.195106) [****, ()](\doibase 10.1103/PhysRevLett.86.468) [****, ()](\doibase 10.1063/1.451588) [****, ()](\doibase 10.1103/PhysRevA.37.2070) [****,  ()](\doibase 10.1103/PhysRevA.37.2079) [****,  ()](\doibase 10.1002/(SICI)1521-4095(200002)12:4<294::AID-ADMA294>3.0.CO;2-D) [****, ()](\doibase 10.1146/annurev-matsci-070909-104529) [****, ()](https://link.aps.org/doi/10.1103/PhysRevB.81.035402) @noop [****,  ()]{} @noop [****,  ()]{} [****,  ()](\doibase 10.1103/PhysRevLett.39.1222) [****,  ()](\doibase https://doi.org/10.1016/0370-1573(78)90009-1) @noop [**]{},  ed. (, , ) [****,  ()](\doibase 10.1103/PhysRevB.92.195112) [****,  ()](\doibase 10.1103/PhysRevB.97.075115) [****,  ()](\doibase 10.1103/PhysRevB.99.245139) [****,  ()](\doibase 10.1103/PhysRevLett.120.247601) [****,  ()](http://dx.doi.org/10.1038/nmat3400 http://10.0.4.14/nmat3400 https://www.nature.com/articles/nmat3400{#}supplementary-information) @noop [ ]{} [****,  ()](https://link.aps.org/doi/10.1103/PhysRevB.78.045109) [****,  ()](https://link.aps.org/doi/10.1103/PhysRevLett.117.136601) [****,  ()](\doibase 10.1103/PhysRevLett.121.056402) [^1]: authors contributed equally [^2]: authors contributed equally [^3]: authors contributed equally [^4]: The superconducting state at $T\approx 4$ K [@Kurosaki2005; @Furukawa2018] is below the temperature accessible to us here. [^5]: BEDT-TTF stands for bis-ethylene-dithio-tetrathiafulvalene. Substituting two of the inner sulfur atoms by selenium leads to bis-ethylenedithio-diselenium-dithiafulvalene, abbreviated BEDT-STF [@Saito2019]. [^6]: Comparison of the results in Fig. \[fig:sigma-eps\_T\](j) with optical data measured on the same substitution yields fair agreement of the metallic values $\epsilon_1 < 0$ [@Saito2019]. Technical details of the dielectric experiments can be found in Ref. [@Rosslhuber2019]. [^7]: While $\epsilon_1$ initially increases upon cooling in , it peaks at the antiferromagnetic transition and reduces at lower $T$. This could be a consequence of the metallic fraction first increasing as the insulator-metal phase boundary approaches the ambient-pressure position, but then reducing below $T_{\rm N}$ because of the negative slope of the boundary between antiferromagnet and metal which moves the IMT further away from $p=0$.

--- abstract: 'We study the effects of an electron-electron effective interaction on the formation of entangled states in a two-qubit system, driven by the coupling of electronic states with vibrational modes. The system is composed by four quantum dots separated in pairs, each pair with one excess electron, which is able to tunnel between the dots. Also, the dots from each pair are coupled with different vibrational modes. The combined action of both, this effective interaction and the electronic tunneling explains not only features on the spectrum and the eigenstates of the Hamiltonian, but also the formation of electronic Bell states by exploiting the quantum dynamics of the system.' author: - 'F. M. Souza' - 'P. A. Oliveira' - 'L. Sanz' title: 'Quantum entanglement driven by electron-nanomechanical coupling' --- Introduction {#sec:intro} ============ For many years, the single-molecule electronics has being an outstanding issue due to its potentiality for future implementations of a cheaper and faster single-electron transistor [@xiang2016; @Ratner05]. There are already many successful examples of single-molecule devices that operate in the Coulomb-blockade regime, presenting transistor-like behavior [@park2002]. To increase further the functionality of a single-electron transistor, it is possible to couple nanomechanical and electronic degrees of freedom [@park2000; @steele20092; @lassagne2009]. It is well known that this kind of interaction plays a signifcant role, bringing a wealth of interesting effects, such as quantum-shuttles in quantum dots (QDs) systems [@gorelik1998; @armour2002; @donarini2005], local cooling [@kepesidis2016], phonon-assisted transport in molecular quantum dot junctions [@walter2013; @Sowa17], and Franck-Condon blockade [@leturcq2009]. Concerning this single electron systems, there is an emerging interest in the interplay between the electronic degrees of freedom, confined inside nanoparticles, and discrete vibrational modes. In particular, this problem has being analyzed considering two different contexts: carbon nanotubes quantum dots and vibrational modes of a cavity coupled with quantum dots. Carbon nanotubes (CNT) become one of the most successful new materials in front of their wide set of direct applications [@dresselhaus2001]. These include the implementation of ultrahigh tunable frequency resonators [@sazonova2004; @deng2016], nanoradios[@jensen2007], and ultrasensitive mass sensors [@chiu2008; @jensen2008]. Moreover, when operated as mechanical ressonators, nanotubes show high quality factors [@lassagne2009; @laird2012; @moser2016] being possible, for instance, to excite, detect and control specifical vibrational modes of a CNT with a current being injected from a scanning tunneling microscopy (STM) tip into a CNT [@leroy2004]. Additionally, it have been reported strong coupling regimes between single-electron tunneling and nanomechanical motion on a suspended nanotube, tuned via electrical gates [@benyamini2014]. Regarding applications in micro and nanoelectronics, carbon nanotubes presented balistic conduction [@laird2015] and Coulomb blockade effect in single and double nanotube based quantum dot devices [@steele2009]. Particularly, it was proposed a mechanically induced “two-qubit" quantum gate and the generation of entanglement between electronic spin states in CNT [@wang2015]. Also, phonons on CNTs showed its potentiality as “flying” qubits for electron spin communications over long distances [@deng2016]. Alternatively, regarding the second experimental context, the coupling between electronic degrees of freedom and vibrational modes can also be accomplished in piezoelectric phononic cavities, based on Bragg mirrors that confine a vibrational mode [@Chen15]. This coupling was experimentally demonstrated in transport measurements as a phonon assisted tunneling in a double quantum dot structure embedded in the cavity. The electron-vibrational mode coupling factor in this quantum dots system is ten times bigger if compared with couplings found on cavity quantum electrodynamics (CQED) and can be enhanced, on demand, by a factor of 20-500 of its regular value [@Chen15]. In front of the advantages cited above, both systems have a great potentiality as a solid state based quantum information processing. In the present work, we explore the properties of a Hamiltonian which describes two electrons on four quantum dots, considering a general model of coupling between electrons and vibrational modes, which applies to both experimental setups. Our proposal is based on charge-qubits, instead of spin, where the states of two qubits are defined depending of the occupation of the quantum dots. We are interested on the interplay between the electron-vibrational mode coupling and the tunneling of electrons between adjacent quantum dots. In Sec. \[sec:model\], by using the unitary transformation of Lang-Firsov, we demonstrate that both couplings are responsible for the apparition of an effective electron-electron interaction. Section \[sec:eigenproblem\] is devoted to the exploration of the signatures of this effective interaction and correlated phenomena on the spectrum and eigenstates of the model. Finally, in Sec. \[sec:dynamics\] , using our previous experience on quantum dynamics on coupled quantum dots [@Oliveira15; @souza2017], we study the formation of Bell states under specific conditions. Section \[sec:summary\] contains our final remarks. Model {#sec:model} ===== Consider a multipartite system with two main parts: the electronic subspace $\mathcal{D}$, consisting on four quantum dots, and the bosonic subspace $\mathcal{V}$, with two devices containing vibrational modes. Vibrational mode $1$ couples with electronic levels of the dots $1$ and $3$, while vibrational mode $2$ couples with dots $2$ and $4$. Tunneling is allowed between dots $1$ ($3$) and $2$ ($4$) so the pair $1-2$ ($3-4$) can be described as a qubit. A possible experimental setup is illustrated in Fig. \[fig:system\], where the bosonic devices are carbon nanotubes. Electronic levels on the quantum dots can be populated by using sources and drains, with additional gates to control the process of tunneling between the dots [@Shinkai07]. The electron-vibrational interaction does not change the electronic population, although creates or destroys vibrational excitations in both devices. Although electrons in different qubits could interact, for instance, by Coulomb interaction [@Hayashi03], we assume this interaction is weak enough to be ignored. ![A possible experimental setup of the system of interest: quantum dots $1$ and $3$ ($2$ and $4$) are coupled with the vibrational (bosonic) mode $1$ ($2$) on carbon nanotubes. Also, the dots 1 (3) and 2 (4) are coupled to each other by tunneling, encoding a qubit.[]{data-label="fig:system"}](figure1){width="0.5\linewidth"} The Hamiltonian that describes the two charged qubits and vibrational modes is given by $$\label{eq:Hgeneral} H=H_{\mathcal{D}} \otimes I_{\mathcal{V}}+I_{\mathcal{D}} \otimes H_{\mathcal{V}}+V_{\mathcal{DV}}.$$ Here $H_{\mathcal{D}}$ and $H_{\mathcal{V}}$ are the free Hamiltonians of the quantum dots and vibrational modes subspaces, respectively, $V_{\mathcal{DV}}$ is the dots-vibrational modes coupling and $I_{\mathcal{D(V)}}$ is the identity matrix for quantum dots ($\mathcal{D}$) or the vibrational modes ($\mathcal{V}$) subspaces. While each pair of dots is a $2$-dimensional subspace, the vibrational mode spans on an infinite dimension subspace $N_v$, where $N_{v}\rightarrow \infty$ for $v$-th subspace ($v=1,2$). The elements of the computational basis have the general form ${\mathinner{|{n_1 n_2 n_3 n_4}\rangle}}_{\mathcal{D}}\otimes{\mathinner{|{N_1 N_2}\rangle}}_{\mathcal{V}}={\mathinner{|{n_1 n_2 n_3 n_4; N_1 N_2}\rangle}}$ with the first four indexes indicating the occupation of the specific dot ($0$-empty and $1$-occupied) and the last two being the population of the vibrational modes. One example is the state ${\mathinner{|{1010,00}\rangle}}$ which represents the situation where dots 1 and 3 contains a single electron each and there is no excitations in the vibrational modes. We proceed to write each term of the Hamiltonian (\[eq:Hgeneral\]). The free Hamiltonian for the four quantum dots subsystem is written as ($\hbar=1$): $$\begin{aligned} \label{eq:Hmol} H_{\mathcal{D}}&=&\left[\sum_{i=1,2}\varepsilon_i N^{\mathcal{D}}_i+\Delta_{12}\left(S_1^\dagger S_2+S_2^\dagger S_1\right)\right]\otimes I^{\otimes 2}\\ &&+ I^{\otimes 2}\otimes\left[\sum_{j=3,4}\varepsilon_j N^{\mathcal{D}}_j+\Delta_{34}\left(S_3^\dagger S_4+S_4^\dagger S_3\right)\right],\nonumber\end{aligned}$$ where $S_{i(j)}^\dagger$ $\left[S_{i(j)}\right]$ are the creation (annihilation) operators for the $i(j)$-th quantum dot and $N^{\mathcal{D}}_{i(j)}=S_{i(j)}^\dagger S_{i(j)}$ ($i=1,2$ while $j=3,4$). The parameters $\varepsilon_{i(j)}$ are the electronic levels for each dot while $\Delta_{12(34)}$ describes the tunneling coupling. If we consider a single vibrational mode per bosonic subsystem, the free Hamiltonian $ H_{\mathcal{V}}$ becomes $$H_{\mathcal{V}} = \omega_1 B_1^\dagger B_1 \otimes I^{\otimes N_2} + I^{\otimes N_1} \otimes \omega_{2} B^\dagger_2 B_2,$$ where $\omega_{1(2)}$ is the energy of the corresponding vibrational mode. Here $B^{\dagger}_{v}$ ($B_v$) creates (annihilates) an excitation in a $v$-th vibrational mode subspace. Now we focus on the term $V_{\mathcal{DV}}$, which provides the electron-vibrational mode coupling. This coupling happens when a single electron on dot $i$ interacts with the vibrational subsystem thus creating or annihilating one excitation in the corresponding vibrational mode. We consider that the electron-vibrational mode coupling is the same for both bosonic modes, with a strength given by $g_v$. The term $V_{\mathcal{DP}}$ is written as $$\begin{aligned} \label{eq:Vmp} &&V_{\mathcal{DV}}=g_1\left(N^{\mathcal{D}}_1\otimes I^{\otimes 2}+I^{\otimes 2}\otimes N^{\mathcal{D}}_3\right) \otimes \left({B}_{1}^\dagger+{B}_{1}\right)\otimes I^{\otimes N_2}\nonumber\\ &&\;\;+g_2\left(N^{\mathcal{D}}_2\otimes I^{\otimes 2}+I^{\otimes 2}\otimes N^{\mathcal{D}}_4\right) \otimes I^{\otimes N_1}\otimes \left({B}_{2}^\dagger+{B}_{2}\right).\end{aligned}$$ As pointed out by Sowa *et. al* [@Sowa17], there can be a phase difference in the coupling parameters $g_v$ given by where $\phi_v=\mathbf{k}_v \cdot \mathbf{d}_v$, with $\mathbf{k}_v$ being the wavevector of the $v$ vibrational mode and $\mathbf{d}_v$ the distance between dots coupled with this specific mode. Here, we assume that the distance between dots inside a molecule is smaller than the vibrational wavelength so the phase difference can be ignored and $g_1=g_2=g$. In order to analyze the action of electron-vibrational mode and tunneling couplings, we apply the Lang-Firsov [@Mahanbook] unitary transformation over the Hamiltonian in Eq.(\[eq:Hgeneral\]) calculating $\bar{H} = e^{S} H e^{-S}$, where $$\begin{aligned} &&S=\alpha_1 \left(N^{\mathcal{D}}_1\otimes I^{\otimes 2}+I^{\otimes 2}\otimes N^{\mathcal{D}}_3\right) \otimes \left(B_1^\dagger - B_1\right)\otimes I^{\otimes N_2} \nonumber \\ &&+\alpha_2 \left(N^{\mathcal{D}}_2\otimes I^{\otimes 2}+I^{\otimes 2}\otimes N^{\mathcal{D}}_4\right) \otimes I^{\otimes N_1}\otimes \left(B_2^\dagger - B_2\right),\end{aligned}$$ with $\alpha_v=\frac{g}{\omega_v}$. This calculation results on a new form for the Hamiltonian written as $$\label{eq:Htrans} \bar{H}=\left(\bar{H}_{\mathcal{D}}+V^{C}_{\mathrm{eff}}\right)\otimes I_{\mathcal{V}}+I_{\mathcal{D}}\otimes H_{\mathcal{V}}+\Delta^T_{\mathcal{DV}},$$ where $$\label{eq:Hmtrans} \bar{H}_{\mathcal{D}}=\sum_{i=1,2}\widetilde{\varepsilon}_i N^{\mathcal{D}}_i\otimes I^{\otimes 2}+I^{\otimes 2}\otimes \sum_{j=3,4}\widetilde{\varepsilon}_j N^{\mathcal{D}}_j,$$ is the transformed Hamiltonian for the dots with $\widetilde{\varepsilon}_{i(j)}$ being an energy level shifted due to the action of the electron-vibrational mode coupling. Specifically, $\widetilde{\varepsilon}_{1(3)}=\varepsilon_{1(3)}-\alpha_1^2\omega_1$ while $\widetilde{\varepsilon}_{2(4)}=\varepsilon_{2(4)}-\alpha_2^2\omega_2$. Apart from this dressed uncoupled electronic Hamiltonian, we want to highlight two new terms on Eq.(\[eq:Htrans\]). The first one can be seen as an effective electron-electron interaction $$\label{eq:Heeeff} V^{C}_{\mathrm{eff}}=-2\alpha_1^2\omega_1N^{\mathcal{D}}_1\otimes N^{\mathcal{D}}_3-2\alpha_2^2\omega_2N^{\mathcal{D}}_2\otimes N^{\mathcal{D}}_4,$$ which cooperates with tunneling in order to generate maximally entangled states. The last term $$\begin{aligned} \label{eq:Hvmvmeff} &&\Delta^T_{\mathcal{DV}}=\left[\left(\Delta_{12} S_1^\dagger S_2\right)\otimes I^{\otimes 2} + I^{\otimes 2}\otimes\left(\Delta_{34}S_3^\dagger S_4\right)\right]\otimes X_{12}\nonumber\\ &&\;\;+\left[\left(\Delta_{12} S_1 S_2^\dagger\right)\otimes I^{\otimes 2} + I^{\otimes 2}\otimes \left(\Delta_{43} S_3 S_4^\dagger\right)\right]\otimes X^{\dagger}_{12},\end{aligned}$$ describes an effective interaction between the two vibrational modes considered on our problem. Here $X_{12}=e^{-\alpha_1\left(B_1 - B^\dagger_1\right)}\otimes e^{-\alpha_2\left(B^\dagger_2- B_2\right)}=D_1(\alpha_1)\otimes D_2(\alpha_2)$, being a tensorial product of displacement operators for the quantum harmonic oscillator [@Scullybook]. The new transformed Hamiltonian, Eq. (\[eq:Htrans\]) and its terms Eqs. (\[eq:Hmtrans\])-(\[eq:Hvmvmeff\]), highlights important effects of the couplings considered on this particular physical system. The first is a shift on the value of the electronic levels which depends on both, the coupling parameter $g$ and $\omega_v$. The second is the effective electron-electron interaction which couples the electrons from different qubits, which is mediated by the electron-vibrational mode coupling. As we discuss below, the effective electron-electron interaction, together with electronic tunneling, is behind the apparition of entangled eigenstates of the full Hamiltonian and the subsequent possibility of generation, by quantum dynamics, of Bell states. Spectral analysis {#sec:eigenproblem} ================= We proceed to explore the characteristics of energy spectrum and eigenstates of the Hamiltonian in Eq. (\[eq:Hgeneral\]). Along with the study of energy spectrum, we are interested on the entanglement properties of the eigenstates. It is well known that Coulomb interaction is behind the formation of entangled states in coupled quantum dots molecule [@Fujisawa11; @Oliveira15], once this interaction couples two single electrons, making viable the encoding of two qubits. In the present problem, we expected the apparition of signatures of the effective electron-electron interaction on the entanglement degree of the eigenstates. To perform our numerical analysis, both basis associated with the vibrational modes are truncated at $N_1=N_2=13$, although both basis for vibrational modes have infinite dimension. This number of computational states is enough to guarantee the accuracy of the calculation of lower energies and eigenstates. To analyze the formation of entangled states, we first build up a density matrix for each eigenstate in the complete basis $\hat{\rho}_l={\mathinner{|{\psi_l}\rangle}}{\mathinner{\langle{\psi_l}|}}$, where ${\mathinner{|{\psi_l}\rangle}}$ is the $l$th eigenstate of Hamiltonian (\[eq:Hgeneral\]). The second step is the calculation of the reduced $4\times 4$ density matrix for the two qubits, by tracing out the degrees of freedom of the vibrational modes so $\hat{\rho}_{\mathcal{D},l}=\mathrm{Tr}_{\mathcal{V}}[\hat{\rho}_l]$. Then, the concurrence, defined by Wootters [@Wootters98], is used as a measurement of entanglement. ![The first $24$ eigenvalues of Hamiltonian (\[eq:Hgeneral\]) varying the detuning $\delta=\varepsilon_1-\varepsilon_2$ considering $\varepsilon_3=\varepsilon_4=0$ and $g=0.5\omega$ and $\Delta_{12}=\Delta_{34}=5\times 10^{-2}\omega$. The energy increases from panel (c) to panel (a) showing the first four states (black solid lines and squares), the next eight states (red dashed lines and circles), and the following twelve states (blue dotted lines and triangles).[]{data-label="fig:eigenproblem"}](figure2){width="0.7\linewidth"} An auxiliary Hermitian operator [@Hill97] $R_l$ is defined as $R_l=\sqrt{\sqrt{\hat{\rho}_{\mathcal{D},l}}\;\widetilde{\hat{\rho}_{\mathcal{D},l}}\sqrt{\hat{\rho}_{\mathcal{D},l}}}$, where $\widetilde{\hat{\rho}_{\mathcal{D},l}}=(\sigma_y \otimes \sigma_y)\hat{\rho}^\star_{\mathcal{D},l}(\sigma_y \otimes \sigma_y)$, is the spin-flipped matrix with $\hat{\rho}^\star_{\mathcal{D},l}$ being the complex conjugate of $\hat{\rho}_{\mathcal{D},l}$. The concurrence is obtained once $C=\mathrm{max}(0,\lambda_1-\lambda_2-\lambda_3-\lambda_4)$ where $\lambda_k$ ($k=1...4$) are the eigenvalues of the operator $R_l$ in decreasing order. Figures \[fig:eigenproblem\](a) to (c) show the behavior of the first $24$ eigenvalues of the Hamiltonian as a function of detuning $\delta=\varepsilon_1-\varepsilon_2$, considering $\varepsilon_3=\varepsilon_4=0$, a resonance condition for the dots 3 and 4. We consider that both vibrational modes have the same frequency value so $\omega_1=\omega_2=\omega$. The coupling parameters are defined in terms of the frequency $\omega$ being $g=\omega/2$ the electron-vibrational mode coupling and $\Delta_{12}=\Delta_{34}=\Delta/2=\omega/20$ the tunneling rates. The energy spectrum shows the emergence of branches, spanned on an energy interval $\Delta E=\omega$, each with an increasing number of inner states as energy increases: while the first branch (solid black lines) has the four states shown in Fig. \[fig:eigenproblem\] (c) including the ground level, the second branch (dashed red lines) has eight states, as seen in Fig. \[fig:eigenproblem\] (b). The subsequent branch (dotted blue lines) contains twelve eigenstates, shown in Fig. \[fig:eigenproblem\] (a). The branches share some common features. The first is the appearance of anticrossings at $\delta=\pm 0.5$, due to first order transitions that switch only one electron per time, e.g., ${\mathinner{|{1001}\rangle}}_{\mathcal{D}}$ $\leftrightarrow$ ${\mathinner{|{1010}\rangle}}_{\mathcal{D}}$. The second is a little anticrossing arising at $\delta=0$, related to higher order transition processes. For instance, two electrons can start at dots 1 and 4 (state ${\mathinner{|{1001}\rangle}}_{\mathcal{D}}$) ending at dots 2 and 3 (state ${\mathinner{|{0110}\rangle}}_{\mathcal{D}}$)[^1]. For each branch, the inner states have interesting properties concerning entanglement as can be seen from Fig. \[fig:eigenconcurrence\] (a) to (c), where we show the behavior of concurrence as a function of detuning $\delta$. Comparing our findings with results on a previous work [@Oliveira15], some similarities let us to conclude that the effective electron-electron coupling is behind the apparition of dressed Bell states as eigenstates. A signature of this fact is the increasing value of maximally entangled states at $\delta=0$, as shown by the scattered plots: there is one on the black branch \[filled squares on Fig. \[fig:eigenconcurrence\] (a) and Fig. \[fig:eigenproblem\] (c)\], two inside the red branch \[filled and open circles on Fig. \[fig:eigenconcurrence\] (b) and Fig. \[fig:eigenproblem\] (b)\] and three on the blue branch \[filled and open triangles point up and down on Fig. \[fig:eigenconcurrence\] (c) and Fig. \[fig:eigenproblem\] (a)\]. Additionally, some satellite peaks at $\delta=\pm 0.5$ are observed, having a lower value of concurrence. While those secondary peaks are related to first order processes, the sharp peak at zero energy arises from second and higher order transitions. Those features of the eigenstates can be explored theoretically through a straightforward calculation (see Appendix \[ap:dressedbell\] for details) which consists on perform a basis transformation going from the electronic computational four-dimensional basis given by $\left\{{\mathinner{|{1010}\rangle}},{\mathinner{|{0101}\rangle}},{\mathinner{|{1001}\rangle}},{\mathinner{|{0110}\rangle}}\right\}_{\mathcal{D}}$ to an electronic Bell basis ordered as $\{{\mathinner{|{\Psi_{-}}\rangle}},{\mathinner{|{\Phi_{-}}\rangle}},{\mathinner{|{\Psi_{+}}\rangle}},{\mathinner{|{\Phi_{+}}\rangle}}\}_{\mathcal{D}}$, where ${\mathinner{|{\Psi_{\pm}}\rangle}}_{\mathcal{D}}=\frac{1}{\sqrt{2}}\left({\mathinner{|{1001}\rangle}}\pm{\mathinner{|{0110}\rangle}}\right)$ and ${\mathinner{|{\Phi_{\pm}}\rangle}}_{\mathcal{D}}=\frac{1}{\sqrt{2}}\left({\mathinner{|{1010}\rangle}}\pm{\mathinner{|{0101}\rangle}}\right)$. For the specific case of equal tunneling rates and $\delta=0$, the calculation shows that the terms of the Hamiltonian regarding ${\mathinner{|{\Psi_{-}}\rangle}}_{\mathcal{D}}$ can be written as a tensorial product given by $$\begin{aligned} \label{eq:HtermsPsi-} H_{\mathrm{with }{\mathinner{|{\Psi_-}\rangle}}}&=&({\mathinner{|{\Psi_-}\rangle}}{\mathinner{\langle{\Psi_-}|}})_{\mathcal{D}}\otimes\big\{\big[\mathbf{{\mathinner{|{00}\rangle}}_{\mathcal{V}}}\big(E_{00}{\mathinner{\langle{00}|}}_{\mathcal{V}}+g{\mathinner{\langle{10}|}}_{\mathcal{V}} +g{\mathinner{\langle{01}|}}_{\mathcal{V}}\big)\big] +\big[\mathbf{{\mathinner{|{01}\rangle}}_{\mathcal{V}}}\big(E_{01}{\mathinner{\langle{01}|}}_{\mathcal{V}}+g{\mathinner{\langle{11}|}}_{\mathcal{V}}+\sqrt{2}g{\mathinner{\langle{02}|}}_{\mathcal{V}}\big)\nonumber\\ &&+\mathbf{{\mathinner{|{10}\rangle}}_{\mathcal{V}}}\big(E_{10}{\mathinner{\langle{10}|}}_{\mathcal{V}}+g{\mathinner{\langle{11}|}}_{\mathcal{V}}+\sqrt{2}g{\mathinner{\langle{20}|}}_{\mathcal{V}}\big)\big] +\big[\mathbf{{\mathinner{|{11}\rangle}}_{\mathcal{V}}}\big(E_{11}{\mathinner{\langle{11}|}}_{\mathcal{V}}+\sqrt{2}g{\mathinner{\langle{21}|}}_{\mathcal{V}}+\sqrt{2}g{\mathinner{\langle{12}|}}_{\mathcal{V}}\big)\nonumber\\ &&+\mathbf{{\mathinner{|{02}\rangle}}_{\mathcal{V}}}\big(E_{02}{\mathinner{\langle{02}|}}_{\mathcal{V}}+...\big)+\mathbf{{\mathinner{|{20}\rangle}}_{\mathcal{V}}}\big(E_{20}{\mathinner{\langle{20}|}}_{\mathcal{V}}+...\big)\big]+...+\mathrm{h.c.}\big\}.\end{aligned}$$ Other terms on Hamiltonian cannot be written as a tensorial product of the form ${\mathinner{|{\psi}\rangle}}{\mathinner{\langle{\psi}|}}_{\mathcal{D}}\otimes\sum \alpha{\mathinner{|{N'_1N'_2}\rangle}}_{\mathcal{V}}{\mathinner{\langle{N_1N_2}|}}$: terms with ${\mathinner{|{\Psi_+}\rangle}}$ are coupled with ${\mathinner{|{\Phi_+}\rangle}}$ by electron-vibrational mode interaction, while elements ${\mathinner{|{\Phi_{+}}\rangle}}$ and ${\mathinner{|{\Phi_{-}}\rangle}}$ are also coupled to each other by tunneling (see Appendix \[ap:dressedbell\]). In the Eq. \[eq:HtermsPsi-\], we use bold type and the square brackets, $[\;]$, to emphasize the new dressed basis $\left\{{\mathinner{|{\psi_{\mathrm{Bell}},N_1N_2}\rangle}}\right\}$. The number of eigenstates per branch and the number of maximally entangled molecular states at $\delta=0$ are linked with the dimension of original subspaces with the same value of the sum $N_1+N_2$, as can be seen from Eq.(\[eq:HmatrixBell\]). Although these subspaces are coupled with each other, each branch can be seen as dressed Bell states, with an energy increasing as $N=N_1+N_2$ grows. ![Behavior of the concurrence, $C$, for the electronic part of the first $24$ eigenstates. We use the same physical parameters and the same color, lines and symbol conventions that in Fig. \[fig:eigenproblem\]. Note that the condition $\delta=0$ is related with the apparition of electronic maximally entangled states, with $C=1$.[]{data-label="fig:eigenconcurrence"}](figure3){width="0.8\linewidth"} Dynamical generation of electronic Bell states {#sec:dynamics} ============================================== After studying the properties of the eigenstates of the model, it is worthwhile to explore the preparation of electronic entangled states by quantum dynamics. We again use a numerical approach, considering the general Hamiltonian (\[eq:Hgeneral\]), to simulate the quantum dynamics through the evolution of the density matrix $\rho(t)={\mathinner{|{\psi(t)}\rangle}}{\mathinner{\langle{\psi(t)}|}}$. Tracing out the vibrational degrees of freedom, it results on the electronic reduced density matrix $$\rho_{\mathcal{D}}(t)=\mathrm{Tr}_{\mathcal{V}}[\rho(t)],$$ used to explore the behavior of the electronic part of the system through the calculation of physical properties as populations, fidelity, and concurrence. Initially, we perform a test to check the precision of our calculation together with our findings about the special character of the electronic Bell state ${\mathinner{|{\Psi_-}\rangle}}_{\mathcal{D}}$ as discussed on Sec. \[sec:eigenproblem\]. We consider $\Delta_{12}=\Delta_{34}$ (equal tunneling couplings), detuning $\delta=0$ and the initial state given by $$\rho(0)= \left({\mathinner{|{\Psi_-}\rangle}}{\mathinner{\langle{\Psi_-}|}}\right)_{\mathcal{D}}\otimes{\mathinner{|{00}\rangle}}_{\mathcal{V}}{\mathinner{\langle{00}|}}.$$ After the calculation of $\rho(t)$ and the $ \rho_{\mathcal{D}}(t)$, we obtain the fidelity of the evolved electronic state with this initial state being $\mathcal{F}(t)=\mathrm{Tr}_{\mathcal{D}}[\rho_{\mathcal{D}}(t)\rho_{\mathcal{D}}(0)]$. Our results shown that the fidelity remains constant with value $\mathcal{F}(t)=1$, showing that, from the electronic point of view, any initial state with $\left({\mathinner{|{\Psi_-}\rangle}}{\mathinner{\langle{\Psi_-}|}}\right)_{\mathcal{D}}$ as the electronic part, acts as a stationary state. In order to explore the generation of electronic entangled states, we now assume the system is being prepared as $$\rho_0={\mathinner{|{1001}\rangle}}_{\mathcal{D}}{\mathinner{\langle{1001}|}}\otimes{\mathinner{|{00}\rangle}}_{\mathcal{V}}{\mathinner{\langle{00}|}},$$ which is an experimentally feasible initial state, once experimental setups include a set of sources and drains attached to the quantum dots, that can inject electrons. We consider the condition of $\delta=0$ and the same choices of physical parameters used in Figs. \[fig:eigenproblem\]-\[fig:eigenconcurrence\]. In Fig. \[fig:popconcfid\] (a), we show the evolution of the populations of electronic states $\left\{{\mathinner{|{1001}\rangle}},{\mathinner{|{0110}\rangle}},{\mathinner{|{0101}\rangle}},{\mathinner{|{1010}\rangle}}\right\}$. At initial times, the values is given by $P_{1001}=1$ (black line), $P_{0110}=0$ (red line), and $P_{0101}=P_{1010}=0$ (green line) in agreement with the initial condition. As time evolves, $P_{1001}$ and $P_{0110}$ oscillates out of phase between zero (non-occupied) and $1$ (occupied), while $P_{1010}$ and $P_{0101}$ remain oscillating close to zero. This is a promising sign toward the formation of a state similar with ${\mathinner{|{\Psi_{\pm}}\rangle}}$ states on Bell basis, once the population of states ${\mathinner{|{1001}\rangle}}$ and ${\mathinner{|{0110}\rangle}}$ is $0.5$ at $\omega t = 300$. We also plot the concurrence, solid blue line in Fig. \[fig:popconcfid\] (a), which reaches the value $C\approx 1$ when $P_{1001}=P_{0110}$, indicating the actual formation of a maximally entangled electronic state. A similar effect was originally reported in Ref.  in the context of two-qubits coupled via Coulomb interaction. In the present case, this evolution shows that the effective electron-electron interaction mediated by the vibrational modes turns out to be responsible for the coupling between the qubits. In order to give a closer look at the entangled states created by quantum dynamics, we compute the fidelity compared with a pre-defined target Bell state, $\mathcal{F}(t)=\mathrm{Tr}_{\mathcal{D}}[\rho_{\mathcal{D}}(t)\rho_{\mathcal{D}}^{\mathrm{tar}}]$. For this calculation, we use as target state the electronic density matrix $\rho_{\mathcal{D}}^{\mathrm{tar}}={\mathinner{|{\Psi(\varphi)}\rangle}}_{\mathcal{D}}{\mathinner{\langle{\Psi(\varphi)}|}}$, where $${\mathinner{|{\Psi(\varphi)}\rangle}}_{\mathcal{D}} = \frac{1}{\sqrt{2}} [|1001\rangle_{\mathcal{D}}+e^{i\varphi} |0110\rangle_{\mathcal{D}}], \label{eq:psi_fase}$$ where $\varphi$ is a relative phase between the state. Note that if $\varphi=0$ ($\varphi=\pi$) then ${\mathinner{|{\Psi(0)}\rangle}}\equiv{\mathinner{|{\Psi_+}\rangle}}$ (${\mathinner{|{\Psi(\pi)}\rangle}}\equiv{\mathinner{|{\Psi_-}\rangle}}$). In Fig. \[fig:popconcfid\] (b) we show how $\mathcal{F}$ evolves with time for different values of $\varphi$. For $\varphi=0$ the fidelity remains stationary, with a value of $0.5$, while for $\varphi=\pi$, the fidelity shows small oscillations around those value. By setting the $\varphi=\pm \pi/2$, we fulfill our goal of find the correct relative phase of the dynamically created entangled state, once the fidelity for both cases oscillates out of phase between $0$ and $1$. The concurrence maximum values being $1$ let us to conclude that the electronic entangled state alternates between ${\mathinner{|{\Psi(\pi/2)}\rangle}}_{\mathcal{D}}$, at $\omega t=300$ and ${\mathinner{|{\Psi(-\pi/2)}\rangle}}_{\mathcal{D}}$, at $\omega t=600$. This means that the dynamics shows not only the ability of create Bell states related with ${\mathinner{|{\Psi_{\pm}}\rangle}}_{\mathcal{D}}$ but also an additional ingredient, which is the imprint of a relative phase $\varphi$. Finally in Fig. \[fig:concvsg\] we explore the effect of the electron-vibrational mode coupling parameter, $g$, on the formation of the electronic entangled states, going into a weak coupling regime. We first notice that, by decreasing the value of $g$, we still obtain values of $C\approx 1$ at some evolved time. Second feature is that it becomes clear that the period of oscillations of the formation of the electronic Bell states is governed by this coupling parameter. This shows that the formation of electronic maximally entangled states is a quite robust effect, which can become accessible in a wide range of experimental devices, even for weak values of electron-vibrational mode coupling. ![Dynamics of populations, concurrence, and fidelity of electrons in quantum dots considering $\delta=\varepsilon_1-\varepsilon_2=0$, $\varepsilon_3=\varepsilon_4=0$, $g=0.5\omega$ and $\Delta_{12}=\Delta_{34}=5\times 10^{-2}\omega$. (a) Populations $P_{1001}$ (black line), $P_{0110}$ (red line) and $P_{1010}=P_{0101}$ (green line), and concurrence (blue line) as functions of $\omega t$. (b) Dynamics of the fidelity between the evolved state and the target state given by Eq. (\[eq:psi\_fase\]), for different values of relative phase $\varphi$: $\varphi=-\pi/2$ (blue line), $\varphi=0$ (green line), $\varphi=+\pi/2$ (black line), and $\varphi=\pi$ (red line).[]{data-label="fig:popconcfid"}](figure4){width="1\linewidth"} ![Concurrence as a function of $\omega t$ for the same values of $\delta$, $\varepsilon_3$, $\varepsilon_4$, $\Delta_{12}$ $\Delta_{34}$ used in Fig. \[fig:popconcfid\], considering several values of electron-vibrational mode coupling $g$: $g=0.5\omega$ (black line), $g=0.1\omega$ (red line), and $g=0.05\omega$ (blue line).[]{data-label="fig:concvsg"}](figure5){width="1\linewidth"} Conclusion {#sec:summary} ========== In this work, we analyze a general model which describes a system where two electrons inside a set of four quantum dots interact with vibrational modes, shown in Fig. \[fig:system\]. The goal is to explore the interplay between tunneling, detuning of the electronic levels inside the quantum dots, and the electron-vibrational mode coupling. The model can describe several experimental scenarios, including electrons inside carbon nanotubes quantum dots or the coupling between electrons and an acoustic cavity. Our findings include the study of the characteristics of the spectrum, the eigenstates, and the quantum dynamics of the system, focusing on the search of electronic maximally entangled states. We find that the electron-vibrational mode coupling is responsible for the apparition of dressed electronic Bell states. Concerning the dynamics, the generation of those entangled states is possible for a wide range of physical parameters. Acknowledgments =============== This work was supported by CNPq (grant 307464/2015-6), and the Brazilian National Institute of Science and Technology of Quantum Information (INCT-IQ). The matrix representation of hamiltonian on the dressed Bell basis {#ap:dressedbell} ================================================================== The entanglement behavior of the system of interest can be explored by writing down its Hamiltonian in terms of the electronic Bell states. Let us calculate the representation of the original Hamiltonian (\[eq:Hgeneral\]) as a matrix written in the dressed basis ${\mathinner{|{\psi_{\mathrm{Bell}},N_1N_2}\rangle}}$, where the electronic part is ordered following $${\mathinner{|{\psi_{\mathrm{Bell}}}\rangle}}_{\mathcal{D}}=\{{\mathinner{|{\Psi_{-}}\rangle}},{\mathinner{|{\Phi_{-}}\rangle}},{\mathinner{|{\Psi_{+}}\rangle}},{\mathinner{|{\Phi_{+}}\rangle}}\}_{\mathcal{D}},$$ where ${\mathinner{|{\Psi_{\pm}}\rangle}}_{\mathcal{D}}=\frac{1}{\sqrt{2}}\left({\mathinner{|{1001}\rangle}}\pm{\mathinner{|{0110}\rangle}}\right)$ and ${\mathinner{|{\Phi_{\pm}}\rangle}}_{\mathcal{D}}=\frac{1}{\sqrt{2}}\left({\mathinner{|{1010}\rangle}}\pm{\mathinner{|{0101}\rangle}}\right)$. We choose to keep together the states with the same number of total excitations $N=N_1+N_2$. This choice remarks the fact that the basis for the whole system has an internal structure of coupled subspaces. For each value of $N$, there is an associated family of subspaces $\mathcal{S}_{\mathrm{B},(N_1N_2)}$: $N=0$ has only the subspace $\mathcal{S}_{\mathrm{B},(00)}$ with four inner states, $N=1$ has two subspaces being $\mathcal{S}_{\mathrm{B},(10)}$ and $\mathcal{S}_{\mathrm{B},(01)}$ (eight inner states), $N=3$ has twelve inner states associated with $\mathcal{S}_{\mathrm{B},(11)}$, $\mathcal{S}_{\mathrm{B},(20)}$, and $\mathcal{S}_{\mathrm{B},(02)}$, and so on. Let us write the matrix representation of the Hamiltonian for this first six $4$D subspaces $\mathcal{S}_{\mathrm{B},(N_1N_2)}$, ordered as $\{\mathcal{S}_{B,(00)},\mathcal{S}_{B,(01)},\mathcal{S}_{B,(10)},\mathcal{S}_{B,(11)},\mathcal{S}_{B,(02)},\mathcal{S}_{B,(20)}\}$: $$\label{eq:HmatrixBell} H=\left( \begin{array}{c|cc|ccc} B_{00} & G_{2} & G_{1} & 0 & 0 & 0 \\ \hline G_{2} & B_{01} & 0 & G_{1} & \sqrt{2}G_{2} & 0 \\ G_{1} & 0 & B_{10} & G_{2} & 0 & \sqrt{2}G_{1} \\ \hline 0 & G_{1} & G_{2} & B_{11} & 0 & 0 \\ 0 & \sqrt{2}G_{2} & 0 & 0 & B_{02} & 0 \\ 0 & 0 & \sqrt{2}G_{1} & 0 & 0 & B_{02} \\ \end{array} \right).$$ using the order ${\mathinner{|{\Psi_{-},N_1N_2}\rangle}}$, ${\mathinner{|{\Phi_{-},N_1N_2}\rangle}}$, ${\mathinner{|{\Psi_{+},N_1N_2}\rangle}}$, and ${\mathinner{|{\Phi_{+},N_1N_2}\rangle}}$, the $4$D matrices $B_{N_1N_2}$ and $ G_{v}$ are defined as $$\begin{aligned} \label{eq:4DBellmatrix} B_{N_1N_2}&=&\left( \begin{array}{cccc} E_{N_1N_2} & \Delta_- & \delta_-/2 & 0 \\ \Delta_- & E_{N_1N_2} & 0 & \delta_+/2 \\ \delta_-/2 & 0 & E_{N_1N_2} & \Delta_+ \\ 0 & \delta_+/2 & \Delta_+ & E_{N_1N_2} \\ \end{array} \right),\end{aligned}$$ and $$\label{eq:4DGmatrix} G_{v}=\left( \begin{array}{cccc} g_{v} & 0 & 0 & 0 \\ 0 & g_{v}/2 & 0 & (-1)^{(v-1)}g_{v}/2 \\ 0 & 0 & g_{v} & 0 \\ 0 & (-1)^{(v-1)}g_{v}/2 & 0 & g_{v}/2 \\ \end{array} \right),$$ where $E_{N_1N_2}=\sum_{i=1,2}\sum_{j=3,4}\sum_{v=1,2}\left(\varepsilon_i+\varepsilon_j+\omega_v\right)$ are the energy of the state ${\mathinner{|{\psi_{\mathrm{Bell}},N_1N_2}\rangle}}$, the tunneling couplings are defined as $\Delta_{\pm}=\Delta_{34}\pm\Delta_{12}$ and $\delta_{\pm}=\delta_{12}\pm\delta_{34}$, with $\delta_{lm}=\varepsilon_l-\varepsilon_m$ ($l=1,3$ and $m=2,4$). The first matrix resembles the rotated matrix on Bell basis, whose properties discussed in details on Ref. , and the matrices $G_v$ depends on $g_v$ and carry on the effect of electron-vibrational mode coupling, where the factor $\sqrt{N_v}$ appears on the specific elements of the matrix (\[eq:HmatrixBell\]) which depends on the values of $N_v$ of the coupled subspaces. If we consider a full resonance condition between electronic levels $\delta_{\pm}=0$, the matrix $B_{N_1N_2}$ becomes $$\begin{aligned} \label{eq:4DBellmatrixv1} B_{N_1N_2}&=&\left( \begin{array}{cccc} E_{N_1N_2} & 0 & 0 & 0 \\ 0 & E_{N_1N_2} & 0 & 0 \\ 0 & 0 & E_{N_1N_2} & \Delta_+ \\ 0 & 0 & \Delta_+ & E_{N_1N_2} \\ \end{array} \right).\end{aligned}$$ At first sight, it seems that the states with electronic part being ${\mathinner{|{\Psi{-}}\rangle}}_\mathcal{D}$ and ${\mathinner{|{\Phi{-}}\rangle}}_\mathcal{D}$ are decoupled, at the same time that ${\mathinner{|{\Psi{+}}\rangle}}_\mathcal{D}$ and ${\mathinner{|{\Phi{+}}\rangle}}_\mathcal{D}$ are not, in the same way that in Ref. . Nevertheless, if elements for the first two lines on matrix (\[eq:HmatrixBell\]), associated with ${\mathinner{|{\Psi_-,00}\rangle}}$ and ${\mathinner{|{\Psi_-,00}\rangle}}$ respectively, are written using the notation ${\mathinner{|{\;}\rangle}}{\mathinner{\langle{\;}|}}$ we obtain: $$\begin{aligned} \label{eq:HtermsPsi-Phi-} H&=&{\mathinner{|{\Psi_-,00}\rangle}}\big(E_{00}{\mathinner{\langle{\Psi_-,00}|}}+g_2{\mathinner{\langle{\Psi_-,01}|}}+g_1{\mathinner{\langle{\Psi_-10}|}}\big)\nonumber\\ &+&{\mathinner{|{\Phi_-,00}\rangle}}\left(E_{00}{\mathinner{\langle{\Phi_-,00}|}} +\frac{g_2}{2}{\mathinner{\langle{\Phi_-,01}|}}-\frac{g_2}{2}{\mathinner{\langle{\Phi_+,01}|}}\right.\nonumber\\ &&\left.+\frac{g_1}{2}{\mathinner{\langle{\Phi_-,10}|}}+\frac{g_1}{2}{\mathinner{\langle{\Phi_+,10}|}}\right)+...\end{aligned}$$ The term with ${\mathinner{|{\Psi_-}\rangle}}_{\mathcal{D}}$ can be written as $\big({\mathinner{|{\Psi_-}\rangle}}{\mathinner{\langle{\Psi_-}|}}\big)_{\mathcal{D}}\otimes\big(E_{00}{\mathinner{|{00}\rangle}}{\mathinner{\langle{00}|}}+g_2{\mathinner{|{00}\rangle}}{\mathinner{\langle{01}|}}+g_1{\mathinner{|{00}\rangle}}{\mathinner{\langle{10}|}}\big)$, while the others do not permit the same. Continuing with the calculation, we realize that only the terms on Hamiltonian associated with ${\mathinner{|{\Psi_-}\rangle}}_{\mathcal{D}}$ are decoupled, at least from the electronic point of view, from the rest of the Bell basis. In this way, there is a Bell state, dressed by vibrational modes, becoming an eigenstate of the Hamiltonian (\[eq:Hgeneral\]) for the specific condition of equal tunneling couplings, $\Delta_{34}=\Delta_{12}$ and full resonance between the electronic levels. [37]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop [****,  ()]{} [****,  ()](\doibase 10.1002/qua.20484) @noop [****,  ()]{} @noop [****, ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [**]{} (, , ) @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} [****, ()](\doibase 10.1038/nnano.2009.71) @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [**]{},  ed. (, , ) @noop [**]{} (, ) @noop [****,  ()]{} @noop [****,  ()]{} [****,  ()](\doibase 10.1103/PhysRevLett.78.5022) [^1]: Although not shown here, the second-order time-dependent perturbation theory can be used to show this transition

--- abstract: 'Work in time-domain astronomy necessitates robust, automated data processing pipelines that operate in real time. We present the BANZAI pipeline which processes the thousands of science images produced across the Las Cumbres Observatory Global Telescope (LCOGT) network of robotic telescopes each night. BANZAI is designed to perform near real-time preview and end-of-night final processing for four types of optical CCD imagers on the three LCOGT telescope classes. It performs instrumental signature removal (bad pixel masking, bias and dark removal, flat-field correction), astrometric fitting and source catalog extraction. We discuss the design considerations for BANZAI, including testing, performance, and extensibility. BANZAI is integrated into the observatory infrastructure and fulfills two critical functions: (1) real-time data processing that delivers data to users quickly and (2) derive metrics from those data products to monitor the health of the telescope network. In the era of time-domain astronomy, to get from these observations to scientific results, we must be able to automatically reduce data with minimal human interaction, but still have insight into the data stream for quality control.' author: - Curtis McCully - 'Nikolaus H. Volgenau' - 'Daniel-Rolf Harbeck' - 'Tim A. Lister' - 'Eric S. Saunders' - 'Monica L. Turner' - 'Robert J. Siverd' - Mark Bowman bibliography: - 'main.bib' title: 'Real-time processing of the imaging data from the network of Las Cumbres Observatory Telescopes using BANZAI' --- Introduction {#sect:intro} ============ In the era of time-domain astronomy, we need to rethink the way that we handle observational data. The traditional procedure of taking data manually, visually inspecting every image, and spending significant a amount of time and effort to produce final reduced products does not scale to upcoming surveys such as the Zwicky Transient Facility[@ZTF] (ZTF) and the Large Synoptic Survey Telescope[@LSST] (LSST), which will produce hundreds of thousands to millions of alerts per night. The volume of data is not the only consideration. We are starting to find new classes of astrophysical sources that vary on a vast range of time scales: from seconds for fast radio bursts [@Katz2016], to minutes for gamma-ray burst afterglows, to hours for kilonovae [@LIGO2017], to days for supernovae. Thus, work in time-domain astronomy requires fast-turnaround data reduction without human intervention. Las Cumbres Observatory Global Telescope[@Brown2013] (LCOGT) is pioneering the future of time domain follow-up. We currently operate a network of 21 small-aperture telescopes around the world which produce nearly 50,000 images ($\sim 1.5$ TB uncompressed raw data) per month. Rather than acting individually, our telescopes operate in unison. Specifically, users submit requests to a single access point, which are then dynamically scheduled across the network resources[@Saunders2014]. From a user’s perspective, LCOGT is a single facility, so it is necessary to produce homogeneous data for heterogeneous instruments by removing the instrumental signatures across the whole network of telescopes. To meet these challenges we have produced the “Beautiful Algorithms to Normalize Zillions of Astronomical Images” (BANZAI) pipeline[@banzai]. BANZAI reduces all of the images taken with LCOGT in real time. Below, we present the design choices we made for the BANZAI pipeline and how the pipeline is integrated into the overall dataflow of the observatory. Finally, we discuss the lessons LCOGT has learned which illustrate the challenges of running a high volume data processing service. Data flow from telescope to user ================================ LCOGT provides three tiers of data to users: raw, preview, and processed. The “raw” files contain unprocessed, 16-bit image data exactly as it was downloaded from the CCD. The “preview” and “processed” files have been calibrated to remove the instrument signature and also include source catalogs as additional extensions. Although both preview and processed files follow the same procedure, they may differ in which calibration files were used. “Preview” files are produced on-the-fly using recent calibration data, often from the previous day. “Processed” files are produced at the end of each night, using master calibration files from the same day as the observations (if available). All of our files are provided in Rice-compressed, “fpacked”[@fpack1; @fpack2] format. The interaction between users and the LCOGT telescopes is significantly different than with traditional facilities. For classical observing, an astronomer would travel to the telescope, observe through the night, and take their data home on an external hard drive (or even DAT tapes) to reduce. Processing data from a single observational run often required significant time and effort. Queue observing solves the issue of needing to travel to the telescope which makes rapid follow-up possible. However, many queue scheduled observatories still require manual data reduction. Both rapid response and automatic, robust data reduction are necessary to follow up alerts from upcoming surveys like ZTF and LSST. ![ \[fig:dataflow\] Schematic of the flow of data from the end of exposure to being available to users. Once the shutter closes, the FITS file is placed on the queue to be shipped back to the LCOGT headquarters. After it arrives, BANZAI pulls it off the FITS Exchange and processes the data. After the data is processed it gets ingested into the archive (hosted in the cloud) where it can be downloaded by users.](LCO-Dataflow.pdf){width="\textwidth"} Instead, LCOGT dynamically responds to changes in the system. When a user submits a request, a dynamic scheduler then optimizes when the observation should be attempted and at which site[@Saunders2014]. Figure \[fig:dataflow\] illustrates the dataflow after an observation is taken. Once the shutter closes, the telescope writes a FITS file that is “fpacked”[@fpack1; @fpack2] to on-site storage. The name and location of this file are added to a transfer queue (we use RabbitMQ[@rabbitmq], but we are not limited to this choice of implementation). The “Shipper” then transfers the file back to the LCOGT headquarters, and puts it on another queue, the “FITS Exchange” (again implemented using RabbitMQ[@rabbitmq]). The raw frames are then pushed to the science archive, which is hosted in the cloud using Amazon Web Services[@aws], where they can be downloaded by users. BANZAI also listens to the FITS Exchange, reducing images as they are added running in a “preview” mode. These processed images are then placed back on the FITS Exchange to be ingested into the science archive. BANZAI does standard image processing on every image taken with LCOGT: we mask bad pixels, subtract the overscan, subtract a master bias frame and a master dark, divide out a master flat, extract photometry for the sources in the field, and solve for the astrometry. We use SEP[@sep] to do source extraction and Astrometry.net[@Lang10] to solve for the astrometry. Raw files compared to a stack of reduced images from BANZAI are shown in Figure \[fig:example\_reduction\]. The median time from shutter close to the “preview“ data being available for download by users is 10 minutes (the full distribution is shown in Figure \[fig:archive\_lag\]). The tail to longer delays is often due to network issues. The median time for BANZAI to process an image is 2 minutes; Figure \[fig:processing\_lag\] shows the distribution of processing times for images. The long tail of processing times is due to dense fields, e.g. microlensing fields near the galactic center, that take longer to extract photometry and solve the astrometry than sparse fields. At the end of the local night for each site, we reprocess the images using master calibration images produced from calibrations taken during evening and morning twilight of that night. These constitute the science-quality reductions for users. By having both a preview and an end-of-night mode we satisfy users that need data quickly for rapidly evolving transients, but also use the best calibration data for the final reduction. ![ \[fig:example\_reduction\] An example of reduced frames from BANZAI. The left shows some example raw frames and the right panel shows a stack of BANZAI processed images.](banzai_example_reduction.pdf){width="\textwidth"} ![ \[fig:archive\_lag\] Distribution of time between the end of the observation and availability on the archive. The median time between shutter close and being available to users is 10 minutes. Network issues and minor isolated pipeline failures drive the tail of the distribution to longer delays.](archive_lag.pdf){width="\textwidth"} ![ \[fig:processing\_lag\] Distribution of time it takes BANZAI to process an image. The median reduction time is 35 s. Dense fields like those taken near the galactic center for microlensing take longer to extract photometry and solve for the astrometry, driving the tail of the distribution to longer processing times.](processing_lag.pdf){width="\textwidth"} BANZAI architecture =================== There are several architectural decisions that we made when developing BANZAI to minimize the barrier to entry for the code, to encourage transparency, and to allow for outside contributions from the broader astronomical community. The first choice is that we implemented BANZAI primarily in Python. Python has become one of the standard languages for astronomy, making it possible for a broad set of people to contribute to the development of the code base. The sections of the code that require high performance are written in Cython, with a few sorting algorithms implemented in C. Code Availability ----------------- LCOGT is committed to being an open-source organization so the BANZAI pipeline is publicly available on GitHub. Being open-source encourages best practices like transparency, code readability, and maintainability. The open source community has also built a variety of useful tools like Coveralls to measure test coverage and ReadTheDocs[@readthedocs] that enables clear and easily accessible documentation. Because BANZAI is an open-source project, people outside of LCOGT can improve the quality of data that it produces because the code is not a proprietary black box. This improves the reproducibility of scientific results and allows for customizations for specific projects. By hosting BANZAI on GitHub, every commit is recorded, making every individual person’s contributions apparent. Object model ------------ BANZAI is built as individual stages which are chained together to process images. The stage object is implemented as an abstract class which is used as the template for the Template Method pattern[@design_patterns]; the superclass contains all of the logic to pass images to the stage and for the top-level script to run the stage. The only piece that an individual stage needs to implement is a single method that takes a list of images and returns a list of processed images. The Template Method pattern employed here keeps the infrastructure for the pipeline simple and self-contained; the individual stages do not need to reimplement wrapper code, a common problem with pipelines. All of the images are kept in memory rather than being written to disk for each stage which keeps the file size small and the performance high. Master calibrations ------------------- Each night, every telescope in the LCOGT network takes calibration frames, e.g. biases, darks, and sky flats. BANZAI includes two utility template stage classes for these observations: a calibration stacker and a stage to correct science frames using a calibration stack. All of the file structure and retrieval logic for the most recent master calibration frames are built into the template classes. The specific stages, e.g., subtracting a dark frame, only have to implement the specialized logic to subtract the scaled dark from an image. Again the generic infrastructure is in a single location, making extension simple and limiting bugs from propagating via copy and paste. The master calibration metadata, such as the file path, are stored in a database. We use MySQL for our production instance but BANZAI itself is agnostic to the choice of database technology. We accomplish this by writing all of the queries in BANZAI using the Object Relational Model in SQLAlchemy[@sqlalchemy]. SQLAlchemy provides a unified frontend for most common database backends. This enables faster deployment and simpler testing; e.g. we use SQLite for testing and MySQL for production. Tests on the master calibration frames have been crucial for high-quality results. If a bad calibration frame slips into the reductions (e.g. a camera warms unexpectedly), images from multiple nights can be affected. To safeguard against this, we do two sets of comparisons on every calibration frame. The first is to compare the new calibration frames to previous “good” master calibration frames. A new “good” frame is created using the frames that pass this check. This allows the master calibration frames to evolve slowly over time, but rejects outliers. Prompt notification (early enough to allow intervention when needed) is an added benefit of real-time alerting. The second set of comparisons are done between all of the frames taken in the evening and morning twilight of an observing night. We employ a pixel-by-pixel comparison and reject any frames that fall outside the expected distribution (which is estimated using the median absolute deviation). Deployment ---------- The BANZAI code is deployed inside a Docker[@docker] container which allows rapid deployment on any machine as all of the dependencies are encapsulated in the container. If a user wants or needs to install the code locally instead of using Docker, the *Dockerfile* acts as a recipe to install the necessary dependencies. Using Docker also allows us to run multiple instances of the pipeline in parallel, e.g. a preview mode that produces fast-turnaround reductions and an end-of-night mode that applies master calibration frames taken in the morning after a given observation. The lifecycles of the Docker containers are managed by Rancher[@rancher], a container orchestration platform. Rancher schedules CPU resources, creates and destroys containers, and maps network ports between containers. The frontend streamlines deploying upgrades: simply changing a version in a web form and clicking “upgrade” is all that is required to deploy a new version of the pipeline. Rancher also provides access to a terminal inside containers which can be used to debug issues. ![\[fig:dq\_report\] An example summary report on the state of data processing for the LCOGT network. Each column corresponds to an individual instrument. The top rows show the total number of images taken and processed from last night. The middle section shows the results from a variety of data quality tests. The bottom colored section shows the ages of flat-fields for common filters. Green denotes that the flat-fields are recent, less than a week old. Red shows filters for which a flat has not been taken for more than two weeks and require human intervention.](banzai_dq.pdf){width="\textwidth"} Testing ------- Unit tests for BANZAI are automatically run using continuous integration (CI) services: we use a public Travis-CI site[@travis] and an internal Jenkins[@jenkins] server. The Travis-CI page allows outside users to see the results of the unit tests, while the internal Jenkins server allows us to test deployment and to do integration and end-to-end tests that require large test datasets. Failure Detection and Performance Metrics ========================================= One of the key challenges to run a data reduction service at this scale is detecting failures automatically. LCOGT produces thousands of images per night, more than is reasonable for any human being to check manually. Instead we have to rely on automated checks to detect and, if necessary, reject failed frames. The results of these checks and performance metrics are stored in ElasticSearch[@elasticsearch], a NoSQL database. The NoSQL format for ElasticSearch makes it simple to add metrics or change the format of stored results. ElasticSearch also provides a web API to query the results of our tests. Using the APIs, we have developed several monitoring tools. One is a summary report that is emailed to several members of the LCOGT operations team. An example table is shown in Figure \[fig:dq\_report\]. This report is designed to quickly visualize the age of the master calibration frames (less than a week old is shown in green) and to summarize the occurrence of known failure modes across the LCOGT network. We also use the metrics to generate real-time alerts. These alerts can be sent via email or to the operations team in Slack[@slack], a real-time communication platform. As these alerts are generated less than 10 minutes after shutter close, issues can be addressed before the whole observing night is affected and errors are more quickly detected rather than needing to wait for errors to be reported by users. ![\[fig:grafana\] An example dashboard that can be used to monitor the data quality being produced by the LCOGT network. The top panel shows the FWHM for each of our telescopes. Individual sites or cameras (shown with different colored points) can be selected using the dropdown menus at the top. The middle panel shows the offset between the requested center of the frame and the actual pointing. The bottom panel shows the number of frames per telescope that were not able to solve for the astrometry. The astrometry failure rate for BANZAI is 12.4% for the last six months. About half of those failures are due to focus issues on the 0.4-m telescopes. The rest are typically due to the sky transparency being too low.](grafana.png){width="\textwidth"} We also use metrics calculated by BANZAI to preform long-term monitoring of the network (e.g. the optical performance of LCOGT[@Harbeck2018]). As with the summary reports, we can use the ElasticSearch APIs to visualize data-quality metrics. We use Grafana[@grafana] to plot time series of metrics based on image quality, telescope pointing, etc. Figure \[fig:grafana\] shows an example dashboard with the Full Width Half Maximum (FWHM), the pointing quality, and the number of astrometric solution failures from the last week. A few example data-quality tests are described below. There are two main failure modes of the cameras at LCOGT. The first is when the camera produces an image that is only electric noise around the value of 1000 (this value is an artifact of the firmware). Frames are automatically rejected if a significant fraction of the image exactly equals 1000. When this happens, an alert is posted to Slack so that the operations team can restart the camera. The second failure mode comes from electrical interference. To detect this, we employ a 2D Fast Fourier Transform (FFT) and check for excess power due to pattern artifacts in the frame. This is recorded in ElasticSearch but does not currently generate an alert. Tuning the threshold to detect this pattern noise is still a work in progress to avoid false positives. This was producing an overwhelming number of alerts, making them counterproductive. *Limiting alert fatigue is essential for monitoring a dataflow of this size.* Detecting and recovering from failures is an area of active development. Conclusion ========== Data reduction at the scale of the LCOGT network is not without challenges: reducing thousands of images per night requires a high-performance pipeline, which was the motivation for BANZAI. Our median time to process an image is 35 seconds (see Figure \[fig:processing\_lag\]), and the median time between the shutter closing and the processed image being available to the user on the science archive is less than 10 minutes as shown in Figure \[fig:archive\_lag\]. As LCOGT is producing data 24 hours a day, 7 days a week, robust deployment is essential: BANZAI is deployed in a Docker container that is managed by Rancher, making it easy to upgrade and redeploy the pipeline as necessary. As new instruments come online, BANZAI needs to be able to adapt, so the code is designed with the idea of extensibility in mind. Monitoring a dataflow of this size without human intervention is another significant challenge. To solve this, we have tightly integrated the BANZAI pipeline into telescope operations via queues and APIs. We also send status and metrics to an ElasticSearch database that we use to produce summary plots and metrics to monitor the health of the telescope network. As future surveys like ZTF and LSST begin, going from follow-up observations to scientific results will require that facilities be able to reduce data with minimal human interaction, but still have insight into the datastream. The lessons we have learned at LCOGT will help to make that possible. We thank Martin Norbury, Todd Boroson, and Stefano Valenti for discussion about the design of BANZAI. CM was supported by supported by NSF grant AST-1313484.

--- abstract: 'Dans ce papier, on résout d’abord le $\partial\bar\partial$ pour les courants prolongeables définis dans $\C^n$ privé d’une boule $B$ de $\C^n$, ensuite dans une variété analytique complexe $X$, on le résout pour un domaine $D=X\setminus\bar{\Omega}$, où $\Omega$ est un domaine borné de $X$ défini par $\{z\in X\,\ /\,\ \varphi(z)<0\}$, (avec $\varphi$ une fonction d’exhaustion strictement plurisousharmonique). [A]{}[BSTRACT.]{} In this present paper, we first solve the $\partial\bar\partial$ for extendable currents defined in $\C^n\setminus B$, where $B$ is a ball of $\C^n$, then in a analytic complex manifold $X$, and in a domain $D=X\setminus\bar{\Omega}$ where $\Omega$ is a bounded domain of $X$ defined by $\{z\in X\,\ /\,\ \varphi(z)<0\}$, ($\varphi$ is an exhaustion strictly plurisubharmonic function). 1.3mm *Classification mathématique 2010 :* 32F32.' address: | Département de Mathématiques\ UFR des Sciences et Thechnologies\ Université Assane Seck de Ziguinchor, BP: 523 (Sénégal) author: - | Eramane Bodian & Ibrahima Hamidine\ & Salomon Sambou title: 'Résolution du $\partial\bar\partial$ pour les courants prolongeables définis dans un anneau' --- ¶ \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] Introduction {#introduction .unnumbered} ============ Soit $B\subset\C^n$ la boule unité, on se pose la question suivante : si $T$ est un courant prolongeable $d$-ferm[é]{} sur $\C^n\setminus \bar B$, existe-t-il un courant prolongeable sur $\C^n\setminus \bar B$ tel que $\partial \bar{\partial} u = T$? Tenant compte de consid[é]{}rations classiques, nous devons pour r[é]{}pondre [à]{} cette question, avoir [à]{} r[é]{}soudre l’[é]{}quation $$\label{Equa1} d u =T,$$ o[ù]{} $T$ est un courant prolongeable, la solution obtenue se d[é]{}compose sans perte de généralités en une partie $\partial$-ferm[é]{}e et l’autre $\bar{\partial}$- ferm[é]{}e. $\C^n\setminus \bar B$ a les conditions géométriques nécessaires à la résolution du $\partial$ et $\bar \partial$ pour les courants prolongeables (voir [@Samb]). Partant de r[é]{}sultats connus de cohomologie de de Rham et de l’analogue convexe (voir [@SBD]), alors l’[é]{}quation admet une solution. La r[é]{}solution du $\partial \bar{\partial}$ devient alors une cons[é]{}quence des r[é]{}sultats de r[é]{}solution du $\bar{\partial}$ pour les courants prolongeables obtenus dans [@Samb]. Dans le cas d’une variété, on introduit la notion d’extension contractile et on résout le $\partial\bar{\partial}$ dans ce cadre. Pr[é]{}liminaires et notations ============================== Soit $B$ la boule de $\C^n$. $~~ $\ Un courant $T$ défini sur $\C^n\setminus \bar B$ est dit prolongeable s’il existe un courant $\check{T}$ défini sur $\C^n$ tel que $\check{T}_{|(\C^n\setminus \bar B)}=T$. D’après Martineau [@Mart], puisque $\stackrel{\circ}{(\overline{\C^n\setminus \bar B})}=\C^n\setminus \bar B$, les courants prolongeables de degré $p$ sur $\C^n\setminus \bar B$ sont égaux au dual topologique des $(2n - p)$-formes diff[é]{}rentielles de classe $\CC^{\infty}$ sur $\C^n$ à support compact sur $\C^n\setminus B$. On note $\check{\mathcal{D}}'^p(\C^n\setminus \bar B)$ l’espace des $p$-courants définis sur $\C^n\setminus \bar B$ et prolongeables à $\C^n$, $\A_c^p (\C^n\setminus \bar B)$ les $p$-formes diff[é]{}rentielles de classe $\CC^{\infty}$ sur $\C^n$ à support compact dans $\C^n\setminus \bar{B}$. Sur $\C^n$, on note $\check{\mathcal{D}}'^{p, q} (\C^n\setminus \bar{B})$ l’espace des $(p, q)$-courants prolongeables d[é]{}finis sur $\C^n\setminus \bar{B}$ et $\A_c^{p, q} (\C^n\setminus \bar{B})$ l’espace des $(p, q)$-formes diff[é]{}rentielles [à]{} support compact dans $\C^n\setminus \bar{B}$. On note $\check{{\rm H}}^p(\C^n\setminus \bar{B})$ le $p^{{\rm ieme}}$ groupe de cohomologie de de Rham des courants prolongeables d[é]{}finis sur $\C^n\setminus \bar{B}$, $\check{{\rm H}}^{p, q} (\C^n\setminus \bar{B})$ le $(p,q)^{{\rm ieme}}$ groupe de cohomologie de Dolbeault des courants prolongeables d[é]{}finis sur $\C^n\setminus \bar{B}$. Si $F \subset \C^n$, alors ${\rm H}_{\infty}^p (F)$ d[é]{}signe le $p^{{\rm ieme}}$ groupe de cohomologie de de Rham des $p$-formes diff[é]{}rentielles de classe $\CC^{\infty}$ d[é]{}finis sur $\C^n$, ${\rm H}_{\infty, c}^p (\C^n)$ est le groupe de cohomologie de de Rham des $p$-formes diff[é]{}rentiables de classe $\CC^{\infty}$ sur $\C^n$ [à]{} support compact et enfin $\A^p (F)$ l’espace des $p$-formes diff[é]{}rentielles de classe $\CC^{\infty}$ sur $F$. On note aussi, pour tout domaine $D$ de $\C^n$, $\flat D$ le bord de $D$. Résolution de l’équation $du=T$ =============================== Tout le long de cette section, nous considérons $$S=\{z\in\C^n\, ; |z|=1\}$$ et $$B=\{ z\in \C^n\, ; |z|<1\},$$ la sphère et la boule unité respectivement dans $\C^n$. Considérons la suite courte suivante, pour $0\leq p\leq 2n$ $$0\to \A^p(\C^n)\to \A^p(\C^n\setminus \bar B)\oplus\A^p(\bar B)\to \A^p(S)\to 0.$$ Sur le plan de cohomologie de de Rham, on a la suite longue de cohomologie suivante : $$\label{suitelong1} \begin{split} & 0 \to {\rm H}^0(\C^n)\to {\rm H}^0(\C^n\setminus \bar B)\oplus {\rm H}^0(\bar B)\to{\rm H}^0(S)\to\\ & {\rm H}^1(\C^n)\to {\rm H}^1(\C^n\setminus \bar B)\oplus {\rm H}^1(\bar B)\to{\rm H}^1(S)\to \ldots\to \\ & {\rm H}^{2n-1}(\C^n)\to {\rm H}^{2n-1}(\C^n\setminus \bar B)\oplus {\rm H}^{2n-1}(\bar B)\to {\rm H}^{2n-1}(S)\\ &\to {\rm H}^{2n}(\C^n)\to {\rm H}^{2n}(\C^n\setminus \bar B)\oplus {\rm H}^{2n}(\bar B)\to 0 \end{split}$$ On sait que $${\rm H}^p(\C^n\setminus \bar B)={\rm H}^p(\C^n\setminus B).$$ \[remark1\] $$\left\{ \begin{array}{ll} {\rm H}^p(S)=0, & \hbox{ si }\, 1< p < 2n-1\\ {\rm H}^0(S)={\rm H}^{2n-1}(S)=\R, \\ {\rm H}^p(\R^{2n})=0, & \hbox{ si }\, p\geq 1\\ {\rm H}^0(\R^{2n})=\R, \\ {\rm H}^p(B)=0, & \hbox{ si }\, p\geq 1\\ {\rm H}^0(B)=\R. \end{array} \right.$$ \[thm1\]$~~ $\ $$\check{{\rm H}}^j(\C^n\setminus\bar B)=0 \mbox{ pour } 2\leq j\leq 2n-2.$$ Pour démontrer le théorème \[thm1\], on a besoin du lemme suivant : \[lem1\]$~~ $\ $\A_c^p(\C^n\setminus B)\cap \ker d=d(\A_c^{p-1}(\C^n\setminus B)\big)$ pour $2\leq p\leq 2n-1$. $~~ $\ On utilse les résultats suivants: $${\rm H}_c^p(\C^n)=0, \mbox{ si } p \leq 2n-1$$ $${\rm H}_c^{2n}(\C^n)=\R, \mbox{ pour } p=2n.$$ Si $f\in\A_c^p(\C^n\setminus B)\cap \ker d$, alors $f\in\A_c^p(\C^n)\cap \ker d$ si $1\leq p\leq 2n-1$. ${\rm H}_c^{p}(\C^n)=0$, alors il existe $u\in \A_c^{p-1}(\C^n)$ telle que $du=f$. Si $p=1$, $u$ est une $0$-forme différentielle à support compact. Alors $du_{|_B}=0$. Ainsi $u=cst$ sur $B$. Par suite $du_{|_{\C^n\setminus\bar{B}}}\not\in \A_c^p(\C^n\setminus B)$ sauf pour $u$ identiquement nulle. Si $p\geq 2$, $du_{|_B}=0$. Puisque $${\rm H}^{p-1}(B)={\rm H}^{p-1}(\bar{B})=0, \mbox{ pour } 1\leq p-1;$$ i.e; $p\geq 2$, il existe $v\in \A^{p-2}(\bar{B})$ tel que $dv=u$ sur $B$. Posons $\tilde{v}$ une extension à support compact dans $\C^n$ de $v$, on a $$\tilde{u}=u-d\tilde{v}$$ qui est un élément de $\A_c^{p-1}(\C^n\setminus B)$ tel que $d\tilde{u}=f$. $~~ $\ *Étape 1:* Soit $T\in \check{\mathcal{D}'}^p(\C^n\setminus \bar{B})\cap\ker d$, $2\leq p\leq 2n-2.$ L’espace $d\A_c^{2n-p}(\C^n\setminus \bar{B})$ est fermé dans $\A_c^{2n-p+1}(\C^n\setminus B)$ pour $2\leq 2n-p+1\leq 2n-1$, (voir par exemple [@Samb], remarque 2). Pour $K$ un compact de $\C^n\setminus B$, notons $\A_{c,K}^{2n-p+1}(\C^n\setminus B)$ le sous-espace des formes différentielles appartenant à $\A_c^{2n-p+1}(\C^n\setminus B)]$ et qui ont leur support dans $K$. L’espace $\A_{c,K}^{2n-p+1}(\C^n\setminus B)\cap d\A_c^{2n-p}(\C^n\setminus B)$ est fermé dans $\A_{c,K}^{2n-p+1}(\C^n\setminus B)$ qui est un espace de Fréchet, par conséquent, c’est un espace de Fréchet. $$\A_{c,K}^{2n-p+1}(\C^n\setminus B)\cap d\A_c^{2n-p}(\C^n\setminus B)=\bigcup_{\nu\in\N}\big(\A_{c,K}^{2n-p+1}(\C^n\setminus B)\cap d\A_{K_\nu}^{2n-p}(\C^n\setminus B)\big);$$ avec $K_\nu=\{z\in\C^n \; |z|\leq R_\nu,\,\ R_\nu\in\R_+^*\}\setminus B $ et $R_\nu > 1$ une suite exhaustive de compacts dans $\C^n\setminus B$. Il existe $\nu_0$ tel que $\A_c^{2n-p+1}(\C^n\setminus B)\cap d\A_{K_{\nu_0}}^{2n-p}(\C^n\setminus B)$ soit de deuxième catégorie de Baire. L’opérateur $d$ est alors un opérateur fermé de domaine de définition $$\{ \varphi\in\A_{K_{\nu_0}}^{2n-p}(\C^n\setminus B)| d\varphi\in \A_c^{2n-p+1}(\C^n\setminus B)\}$$ entre les espaces de Fréchet $\A_{K_{\nu_0}}^{2n-p}(\C^n\setminus B)$ et $\A_c^{2n-p+1}(\C^n\setminus B)\cap d\A_c^{2n-p} (\C^n\setminus B)$ dont l’image est de seconde catégorie de Baire. Le théorème de l’application ouverte implique que cet opérateur est surjectif et ouvert (voir par exemple [@Samb], Lemme 3.1). Donc $$d\A_{c,K_{\nu_0}}^{2n-p}(\C^n\setminus B)\cap \A_{c,K}^{2n-p+1}(\C^n\setminus B)=\A_{c,K}^{2n-p+1}(\C^n\setminus B)\cap d\A_{c}^{2n-p}(\C^n\setminus B).$$ Posons $\tilde{K}=K_{\nu_0}$. L’application $$\begin{split} {\rm L}_T^K~: \A_{c,K}^{2n-p+1}(\C^n\setminus B)\cap d\A_{c,\tilde{K}}^{2n-p}(\C^n\setminus B)&\to \C \\ d\varphi & \mapsto \langle{\rm T},\varphi\rangle \end{split}$$ est bien définie. En effet, si $d\varphi=d\varphi'$, on a $d(\varphi-\varphi')=0$, $\varphi-\varphi'$ est une $(2n-p)$-forme différentielle, $d$-fermée à support dans $\tilde{K}$, en particulier dans $\C^n\setminus \bar B$. Par conséquent, il existe $\theta\in\A_{c}^{2n-p-1}(\C^n\setminus B)$ tel que $\varphi-\varphi'=d\theta$. Par densité de $\A_{c}^{2n-p-1}(\C^n\setminus\bar B)$ dans $\A_{c}^{2n-p-1}(\C^n\setminus B)$, il existe une suite $(\theta_j)_{j\in\N}$ d’éléments de $\A_{c}^{2n-p-1}(\C^n\setminus \bar B)$ qui converge uniforment vers $\theta$ dans $\A_{c}^{2n-p-1}(\C^n\setminus B)$ et par conséquent $$\langle T,\varphi\rangle=\langle T,\varphi'\rangle+\langle T,d\theta\rangle=\langle T,\varphi'\rangle$$ car $T$ étant $d$-fermé, $$\langle T,d\theta\rangle=\lim_{j\to+\infty}\langle T,d\theta_j\rangle=0.$$ Donc $${\rm L}_T^K(d\varphi)={\rm L}_T^K(d\varphi').$$ L’application ${\rm L}_T^K$ est linéaire et aussi continue comme composée de deux applications continues (et de la dualité entre $\check{\mathcal{D}}_D^p(\C^n)$ et $\A_c^{2n-p}(\C^n\setminus B)$) : $$T~: \A_{c,\tilde{K}}^{2n-p}(\C^n\setminus B)\to \C$$ et $$\delta~: \A_{c,K}^{2n-p+1}(\C^n\setminus B)\cap d\big[\A_{c,\tilde{K}}^{2n-p}(\C^n\setminus B)\big]\to \A_{c,\tilde{K}}^{2n-p}(\C^n\setminus B)$$ qui vérifie $d\circ \delta={\rm Id}$ et qui est obtenue par application du théorème de l’application ouverte appliqué à $$\begin{split} d~: \{ \varphi\in \A_{c,\tilde{K}}^{2n-p}(\C^n\setminus B)/\,\ d\varphi\in \A_{c,K}^{2n-p+1}(\C^n\setminus B)\}\subset \A_{c,\tilde{K}}^{2n-p}(\C^n\setminus B)\\ \qquad \qquad \to \A_{c,K}^{2n-p+1}(\C^n\setminus B)\cap d\big[\A_{c,\tilde{K}}^{2n-p}(\C^n\setminus B)\big]. \end{split}$$ D’après le théorème de Hahn-Banach, on peut [é]{}tendre ${\rm L}_T^K$ en un op[é]{}rateur lin[é]{}aire et continu : $$\tilde{\rm L}_T^K~: \A_c^{2n-p+1} (\C^n\setminus B) \to\C$$ qui est linéaire et continu. Donc $\tilde{{\rm L}}_T^K$ est un courant prolongeable défini dans $\C^n\setminus \bar B$ et $d \tilde{\rm L}_T^K = (- 1)^{2n-p} T$ sur $\stackrel{\circ}{K}$ car ${\rm supp}\varphi\subset K$, $$d\varphi \in\A_{c,K}^{2n-p+1}(\C^n\setminus B)$$ et $$\langle\tilde{{\rm L}}_T^K,d\varphi\rangle=(-1)^{2n-p}\langle T,\varphi\rangle.$$ On pose $S^{(K)}=(-1)^{2n-p}\tilde{{\rm L}}_T^K.$ D’o[ù]{} $S^{(K)} = (- 1)^{2n-p} \tilde{{\rm L}}_T^K $ est un courant prolongeable solution de $d u = T$ sur $K$. *Étape 2:* Soit maintenant $K_1$, $K_2$ et $K_3$ trois compacts d’intérieur non vide de $\C^n\setminus B$ tels que $\stackrel{\circ}{K_1}\subset\subset\stackrel{\circ}{K_2}\subset\subset\stackrel{\circ}{K_3}$ et $\stackrel{\circ}{K_i}\cup \bar B=\{z\in\C^n\,\; \mid z \mid<\eta_i\}$, $i=1,2,3$. Soit $T$ un courant prolongeable sur $\C^n\setminus\bar{B}$ tel qu’il existe $S_2$ et $S_3$ deux $p-1$ courants définis sur $\stackrel{\circ}{K_2}$ et $\stackrel{\circ}{K_3}$ et prolongeables à $\C^n$ tels que, pour tout indice $i=2,3$, $d S_i=T$ sur $\stackrel{\circ}{K_i}$ et soit $\epsilon > 0$, alors il existe un courant prolongeable $\tilde{S_3}$ défini sur $\stackrel{\circ}{K_3}$ tel que : $d\tilde{S_3}=T$ sur $\stackrel{\circ}{K_3}$ et $\tilde{S_3}_{|\stackrel{\circ}{K_1}}=(S_2)_{|\stackrel{\circ}{K_1}}$ si $2\leq p\leq 2n-1.$ En effet, comme $dS_2=T$ sur $\stackrel{\circ}{K_2}$ et $dS_3=T$ sur $\stackrel{\circ}{K_3}$, $d(S_2-S_3)=0$ sur $\stackrel{\circ}{K_2}$. Puisque sur $\stackrel{\circ}{K_2}$, on peut résoudre le $d$ pour les formes différentielles à support compact dans $\stackrel{\circ}{K_2}\cup\flat B$ de degré $p$ avec $2\leq 2n-p+1\leq 2n-1$ et $d\big[\A_c^{2n-p-1}(\stackrel{\circ}{K_2}\cup\flat B)\big]$ est fermé dans $\A_c^{2n-p-1}(\stackrel{\circ}{K_2}\cup\flat B)$, on a d’après l’étape 1 et pour $K$ un compact tel que $\stackrel{\circ}{K_1}\subset\subset K\subset\subset \stackrel{\circ}{K_2}$ un courant $S^{(K)}$ sur $\stackrel{\circ}{K}$ prolongeable à $\stackrel{\circ}{K_2}\cup\bar B$ tel que $S_2-S_3=dS^{(K)}$ sur $\stackrel{\circ}{K}$. Soient $\chi$ une fonction dans $\CC^\infty(\C^n)$ à support compact dans $\stackrel{\circ}{K}\cup \bar{B}$ qui vaut $\bf 1$ dans $K_1$ et $\tilde{S}^{(K)}$ une extension de $S^{(K)}$ à $\C^n$ $$S_3+d(\chi\tilde{S}^{(K)})=S_2-d\big((1-\chi)\tilde{S}^{(K)}\big) \mbox{ sur } \stackrel{\circ}{K_1}.$$ On pose $$\tilde{S}_3=S_3+d(\chi\tilde{S}^{(K)}).$$ *Étape 3:* Considérons une suite exhaustive $(K_j)_{j\in\N}$ de compacts de $\C^n\setminus B$. Supposons que $\stackrel{\circ}{K_j}\cup\bar B=\{z\in\C^n\,\: \mid z \mid<\eta_j\}$ où $(\eta_j)_{j\in\N}$ sont des réels tels que $\eta_j <\eta_{j+1}$. Pour $2\leq p\leq 2n-1$, on associe à $(K_j)_{j\in \N}$ grâce aux étapes 1 et 2 une suite de courants $(S_j)_{j\in\N}$ définis dans $K_j$ et prolongeables à $\C^n$ telle que $d S_j=T$ sur $\stackrel{\circ}{K_j}$ et si $j$, $j+1$, $j+2$ sont trois indices consécutifs, $S_{j+2}=S_{j+1}$ sur $\stackrel{\circ}{K_j}$. La suite $(S_j)_{j\in N}$ converge vers un courant $S$ défini sur $\C^n\setminus \bar{B}$ et prolongeable. De plus, $S$ est solution de l’équation $du=T$ dans $\C^n\setminus\bar{B}$. Résolution du $\partial\bar\partial$ pour les courants prolongeables ==================================================================== Tenant compte du th[é]{}or[è]{}me \[thm1\] et des r[é]{}sultats de r[é]{}solution du $\bar{\partial}$ pour les courants prolongeables obtenus par S. Sambou dans [@Samb], on a le th[é]{}or[è]{}me suivant : \[thm3\]$~~ $\ Soit $T$ un $(p,q)$-courant prolongeable d[é]{}fini sur $\C^n\setminus \bar{B}$. Supposons que $d T = 0$; $1\leqslant p \leqslant n$ et $1 \leqslant q \leqslant n$, alors il existe un $(p-1,q-1)$-courant $S$ d[é]{}fini sur $\C^n\setminus \bar{B}$, prolongeable tel que $\partial \bar{\partial} S = T$, pour $2 \leqslant p + q \leqslant 2n-1$. Soit $T$ un $(p, q)$-courant, $1 \leqslant p \leqslant n$ et $1 \leqslant q \leqslant n$, $d$-ferm[é]{} d[é]{}fini sur $\C^n\setminus \bar{B}$ et prolongeable avec $2 \leqslant p + q \leqslant 2n-1$. Puisque le th[é]{}or[è]{}me \[thm1\] nous assure que $\check{{\rm H}}^{p + q} (\C^n\setminus \bar{B})=0$, il existe un courant prolongeable $\mu$ d[é]{}fini sur $\C^n\setminus \bar{B}$ tel que $d \mu = T$. $\mu$ est un $(p + q - 1)$-courant, il se d[é]{}compose en un $(p - 1, q)$-courant $\mu_1$ et en un $(p, q - 1)$-courant $\mu_2$. On a $$d \mu = d (\mu_1 + \mu_2) = d \mu_1 + d \mu_2 = T.$$ Comme $d=\partial+\bar\partial$, on a, pour des raisons de bidegré, $\partial\mu_2=0$ et $\bar\partial\mu_1=0$. On obtient $\mu_1=\bar\partial u_1$ et $\mu_2=\partial u_2$ avec $u_1$ et $u_2$ des courants prolongeables définis sur $\C^n\setminus \bar{B}$, (voir [@Samb], section 3). On a : $$\begin{aligned} T &= &\partial \mu_1 + \bar{\partial} \mu_2\\ &=&\partial \bar{\partial} u_1 + \bar{\partial} \partial u_2\\ &=&\partial \bar{\partial} (u_1 - u_2)\end{aligned}$$ Posons $S = u_1 - u_2$ , $S$ est un $(p - 1, q - 1)$-courant prolongeable d[é]{}fini sur $\C^n\setminus \bar{B}$ tel que $\partial \bar{\partial} S = T$. Résolution du $\partial\bar\partial$ sur une variété analytique complexe ======================================================================== On va maintenant considérer $X$ comme une variété différentiable de dimension $n$. Soit $X$ une variété différentiable de dimension $n$ et $\omega\subset X$ un domaine contractile. On dit que $X$ est une extension contractile de $\Omega$, s’il existe une suite $\left(\Omega_{n} \right)_{n} $ exhaustive de domaines contractiles telle que $$\forall n, \Omega\subset\subset \Omega_{n}\subset\subset X.$$ Quand $X=\mathbb{C}^{n}$, alors $\mathbb{C}^{n}$ est une extension contractile de la boule unité $B$.\ \ On a pour les extensions contractiles, le théorème suivant: \[thm4\]$~~ $\ Soit $X$ une variété analytique complexe de dimension $n$ et soit $D\subset\subset X$ un domaine contractile fortement pseudoconvexe. Supposons que $X$ est une extension $(n-1)$-convexe avec $ H^{j}(\flat D)=0 \quad 2\leqslant j\leqslant 2n-2$ de $D$ et une extension contractile de $D$. Posons $\Omega=X\setminus\bar{D}$. Si $\stackrel{\circ}{\bar{\Omega}}=\Omega$, alors pour tout $(p,q)$ courant $T$ défini sur $\Omega$, $d$-fermé et prolongeable, il existe un $(p-1,q-1)$ courant $S$ défini sur $\Omega$ et prolongeable tel que $\partial\bar{\partial}S=T$ pour $1\leqslant p\leqslant n-1$ et $1\leqslant q\leqslant n-1$. Pour démontrer le théorème \[thm4\], nous avons besoin du lemme suivant : $$\A_c^r(\bar\Omega)\cap\ker d=d\big(\A_c^{r-1}(\bar\Omega)\big) \mbox{ pour } 1\leq r \leq 2n-1.$$ $~~ $\ On a $X=\cup D_{\nu} \quad , D\subset\subset D_{\nu}\subset\subset X$ et $D_{\nu}$ est contractile.\ Si $f\in\A_c^r(\bar{\Omega)}\cap\ker d \quad \exists \nu_{0}\in \mathbb{N}$ tel que $f\in \A_c^r(D_{\nu_{0}})\cap \ker d$. Or $ H^{j}(D_{\nu_{0}})=0 ,\quad \textrm{pour} j\geq 1 $. Par dualité de Poincaré $H^{j}_{c}(D_{\nu_{0}})=0 \quad \textrm{pour} j<2n$. Il existe $g\in \A_c^{r-1}(D_{\nu_{0}})$, donc $g\in \A_c^{r-1}(X)$ telle que $dg=f$.\ $dg_{\mid D}=0$, si $r=1$ alors $g$ est une constante sur $D$. Si $r>1$, il existe $u$ une $(r-2)$ forme différentielle sur $\bar{D}$ telle que $ g_{\mid D}=du $. Soit $\tilde{u}$ une extension de $u$ à support compact dans $X$, $h=g-d\tilde{u}$ convient. Nous pouvons établir donc la preuve du théorème \[thm4\] $~~ $\ Soit une suite exhaustive de compacts $K_{\nu}$ de $\Omega$. $$K_{\nu}=\bar{D_{\nu}}\setminus \bar{D}$$ et quelque soit $\nu$, $K_\nu$ est un compact d’intérieur non vide. Pour $K$ un compact de $\Omega$, l’opérateur ${\rm L}_T^K$ est bien défini, linéaire et continu, cf. étape 1 de la démonstrations du théorème \[thm1\]. *Étape 1:*\ D’après le théorème de Hahn-Banach, on peut [é]{}tendre ${\rm L}_T^K$ en un op[é]{}rateur lin[é]{}aire et continu : $$\tilde{\rm L}_T^K~: \A_c^{2n-p+1} (\Omega) \to\C$$ qui est linéaire et continu. Donc $\tilde{{\rm L}}_T^K$ est un courant prolongeable défini dans $ \bar\Omega$ et $ d\tilde{\rm L}_T^K = (- 1)^{2n-p} T$ sur $\stackrel{\circ}{K}$ car ${\rm supp}\varphi\subset K$, $$d\varphi \in\A_{c,K}^{2n-p+1}(\Omega)$$ et $$\langle\tilde{{\rm L}}_T^K,d\varphi\rangle=(-1)^{2n-p}\langle T,\varphi\rangle.$$ On pose $S^{(K)}=(-1)^{2n-p}\tilde{{\rm L}}_T^K.$ D’o[ù]{} $S^{(K)} = (- 1)^{2n-p} \tilde{{\rm L}}_T^K $ est un courant prolongeable solution de $d u = T$ sur $K$. *Étape 2:*\ Soit maintenant $K_1$, $K_2$ et $K_3$ trois compacts d’intérieur non vide de $\Omega$ tels que $\stackrel{\circ}{K_1}\subset\subset\stackrel{\circ}{K_2}\subset\subset\stackrel{\circ}{K_3}$ et $\stackrel{\circ}{K_i}\cup \bar D=\{z\in X\,\ ; |z|<\eta_i\}$, $i=1,2,3$. Soit $T$ un courant prolongeable sur $\bar{\Omega}$ tel qu’il existe $S_2$ et $S_3$ deux $(p-1)$ courants définis sur $\stackrel{\circ}{K_2}$ et $\stackrel{\circ}{K_3}$ et prolongeables à $X$ tels que, pour tout indice $i=2,3$, $d S_i=T$ sur $\stackrel{\circ}{K_i}$, alors il existe un courant prolongeable $\tilde{S_3}$ défini sur $\stackrel{\circ}{K_3}$ tel que : $d\tilde{S_3}=T$ sur $\stackrel{\circ}{K_3}$ et $\tilde{S_3}_{|\stackrel{\circ}{K_1}}=(S_2)_{|\stackrel{\circ}{K_1}}$ si $2\leq p.$ En effet, comme $d S_2=T$ sur $\stackrel{\circ}{K_2}$ et $d S_3=T$ sur $\stackrel{\circ}{K_3}$, $d(S_2-S_3)=0$ sur $\stackrel{\circ}{K_2}$. Puisque sur $\stackrel{\circ}{K_2}$, on peut résoudre le $d$ pour les formes différentielles à support compact dans $\stackrel{\circ}{K_2}\cup\,\ \flat D$ de degré $p$ avec $2\leq 2n-p+1\leq 2n-1$ et $d\big[\A_c^{2n-p-1}(\stackrel{\circ}{K_2}\cup\,\ \flat D)\big]$ est fermé dans $\A_c^{2n-p-1}(\stackrel{\circ}{K_2}\cup\,\ \flat D)$, on a d’après l’étape 1 et pour $K$ un compact tel que $\stackrel{\circ}{K_1}\subset\subset K\subset\subset \stackrel{\circ}{K_2}$ un courant $S^{(K)}$ sur $\stackrel{\circ}{K}$ prolongeable à $\stackrel{\circ}{K_2}\cup\bar D$ tel que $S_2-S_3=d S^{(K)}$ sur $\stackrel{\circ}{K}$. Soient $\chi$ une fonction de classe $\CC^\infty$ à support compact dans $\stackrel{\circ}{K}\cup \bar{D}$ qui vaut $1$ dans $K_1$ et $\tilde{S}^{(K)}$ une extension de $S^{(K)}$ à $X$ $$S_3+d(\chi\tilde{S}^{(K)}=S_2-d\big((1-\chi)\tilde{S}^{(K)}\big) \mbox{ sur } \stackrel{\circ}{K_1}.$$ On pose $$\tilde{S}_3=S_3+d(\chi\tilde{S}^{(K)}).$$ *Étape 3:*\ Considérons une suite exhaustive $(K_j)_{j\in\N}$ de compacts de $\Omega$. Supposons que $\stackrel{\circ}{K_j}\cup\bar D=\{z\in X\,\: \mid z \mid<\eta_j\}$ où $(\eta_j)_{j\in\N}$ sont des réels tels que $\eta_j <\eta_{j+1}$. Pour $2\leq p$, on associe à $(K_j)_{j\in \N}$ grâce aux étapes 1 et 2 une suite de courants $(S_j)_{j\in\N}$ définis dans $K_j$ et prolongeables à $X$ telle que $d S_j=T$ sur $\stackrel{\circ}{K_j}$ et si $j$, $j+1$, $j+2$ sont trois indices consécutifs, $S_{j+2}=S_{j+1}$ sur $\stackrel{\circ}{K_j}$. La suite $(S_j)_{j\in N}$ converge vers un courant $S$ défini sur $\Omega $ et prolongeable à $X$. De plus, $S$ est solution de l’équation $d u=T$ dans $\Omega$. *Étape 4 :*\ Soit $T$ un $(p, q)$-courant, $1 \leqslant p \leqslant n$ et $1 \leqslant q \leqslant n$, $d$-ferm[é]{} d[é]{}fini sur $\Omega$ et prolongeable avec $2 \leqslant p + q \leqslant 2n-2$. Puisque le th[é]{}or[è]{}me \[thm1\] nous assure que $\check{{\rm H}}^{p + q} (\Omega)=0$, il existe un courant prolongeable $\mu$ d[é]{}fini sur $\Omega$ tel que $d \mu = T$. $\mu$ est un $(p + q - 1)$-courant, il se d[é]{}compose en un $(p - 1, q)$-courant $\mu_1$ et en un $(p, q - 1)$-courant $\mu_2$. On a $$d \mu = d (\mu_1 + \mu_2) = d \mu_1 + d \mu_2 = T.$$ Comme $d=\partial+\bar\partial$, on a, pour des raisons de bidegré, $\partial\mu_2=0$ et $\bar\partial\mu_1=0$. On obtient $\mu_1=\bar\partial u_1$ et $\mu_2=\partial u_2$ avec $u_1$ et $u_2$ des courants prolongeables définis sur $\Omega$, (voir [@Samb], section 3). On a : $$\begin{aligned} T &= &\partial \mu_1 + \bar{\partial} \mu_2\\ &=&\partial \bar{\partial} u_1 + \bar{\partial} \partial u_2\\ &=&\partial \bar{\partial} (u_1 - u_2)\end{aligned}$$ Posons $S = u_1 - u_2$ , $S$ est un $(p - 1, q - 1)$-courant prolongeable d[é]{}fini sur $\Omega$ tel que $\partial \bar{\partial} S = T$. [33333]{} C. Godbillon. Topologie algébrique, Hermann Paris, 1971. A. Martineau. Distribution et valeurs au bord des fonctions holomorphes. Strasbourg RCP 25 (1966). S. Sambou, E. Bodian, D. Diallo. Résolution du $\partial\bar\partial$ pour les courants prolongeables définis sur la boule euclidienne de $\C^n$. A paraître dans Mathmatical report of royal academy society of Canada. S. Sambou. Résolution du $\bar\partial$ pour les courants prolongeables définis dans un anneau. Annales de la faculté des sciences de Toulouse $6^e$ série, tome 11, 1 (2002), p. 105-129. S. Sambou. Résolution du $\bar\partial$ pour les courants prolongeables. Math. Nachrichten [**235**]{} (2002), pg. 179-190.

--- abstract: 'Consider an arbitrary automorphism of an Enriques surface with its lift to the covering $K3$ surface. We prove a bound of the order of the lift acting on the anti-invariant cohomology sublattice of the Enriques involution. We use it to obtain some mod 2 constraint on the original automorphism. As an application, we give a necessary condition for Salem numbers to be dynamical degrees on Enriques surfaces and obtain a new lower bound on the minimal value. In the Appendix, we give a complete list of Salem numbers that potentially may be the minimal dynamical degree on Enriques surfaces and for which the existence of geometric automorphisms is unknown.' address: - 'Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan' - 'Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, 2641 Yamazaki, Noda, Chiba 278-8510, Japan' - 'Institute of Mathematics, Jagiellonian University, ul. [Ł]{}ojasiewicza 6, 30-348 Kraków, Poland' author: - Yuya Matsumoto - Hisanori Ohashi - 'S[Ł]{}awomir Rams' title: On automorphisms of Enriques surfaces and their entropy --- [^1] introduction ============ It is known that the only compact Kähler surfaces that admit automorphisms of positive topological entropy are rational, Enriques, $K3$ surfaces and complex tori (see e.g. [@cantat $\S$ 2.5]). Salem numbers that can be realized as the dynamical degrees of automorphisms of 2-dimensional tori are fully characterized in [@reschke12 Thm 1.1] in terms of values of the minimal polynomials. The same question is solved for rational surfaces in terms of Weyl groups in [@Uehara]. These are exactly the description of the dynamical spectrum $$\Lambda(\mathcal{C})=\{\lambda(f)\in\mathbb{C}\mid \lambda(f)\text{is the dynamical degree of $f\in \mathrm{Aut}(S)$ for some $S\in \mathcal{C}$}\}$$ where the class of surfaces $\mathcal{C}$ is taken to be $2$-tori or rational surfaces. For $K3$ surfaces, the recent preprint [@BF-T] describes the case of degree 22 Salem numbers. Other degrees on $K3$ surfaces and also on Enriques surfaces the description of $\Lambda(\mathcal{C})$ remains open. The purpose of this note is to give a new property which is satisfied by all automorphisms of Enriques surfaces. As a consequence, we obtain a new constraint on the Salem numbers that appear as the dynamical degrees of automorphisms of Enriques surfaces, namely a property of $\Lambda (\text{Enriques})$. It should be noted that despite its ergodic interpretation [@cantat $\S$.2.2.2], the problem we consider is purely algebraic, in the sense that the dynamical degree of an automorphism of an Enriques surface $S$ can be detected as the spectral radius of the map it induces on $\mathrm{Num}(S)$ (see $\S$.\[sect-dynamical-degrees\]). To state the theorem, let $S$ be an Enriques surface and let $\tilde{S}$ be its $K3$-cover. We denote by $\varepsilon$ the covering involution of the double étale cover $\pi: \tilde{S} \rightarrow S$ and put $N$ to denote the orthogonal complement of the $\varepsilon$-invariant sublattice $H^2(\tilde{S},\Z)^{\varepsilon}$ in the lattice $H^2(\tilde{S},\Z)$: $$\label{eq-latN} N = (H^2(\tilde{S},\Z)^{\varepsilon})^{\perp} \, .$$ Recall that for an arbitrary automorphism $f\in \mathrm{Aut}(S)$, there exists a lift $\tilde{f} \in \mathrm{Aut}(\tilde{S})$. Obviously the lift in question is not unique (given $\tilde{f}$, the automorphism $\tilde{f} \circ \varepsilon$ is also a lift of $f$), but the constraints we prove are valid for any choice of $\tilde{f}$. As is well-known (see e.g. [**[@namikawa]**]{}), the lattice $N$ is stable under the cohomological action $\tilde{f}^*$, hence the restriction $${f_N}:= \tilde f^* \rvert_N$$ is an automorphism (isometry) of $N$. It is easy to see that the order $\ord({f_N})$ is finite, Lemma \[Nfin\]. Here we show a more precise constraint on the order of the map ${f_N}$: \[th1\] Let $S$ be an Enriques surface and let $f\in \mathrm{Aut}(S)$. Then, the order of ${f_N}$ is an integer which divides at least one of the integers $$120, 90, 84, 72, 56, 48.$$ Equivalently and explicitly, these are one of the 31 integers $$\label{eq-T1} 120,90,84,72,60,56, 48,45,42,40,36,30,28, 24,21,20,18, 16, 15,14,12,10,\dots,1.$$ We use the above theorem to derive the following mod 2 constraint for a Salem number to be the dynamical degree of automorphisms of Enriques surfaces (which refines [@third Lemma 4.1]): \[mod2\] Let $f$ be an automorphism of an Enriques surface $S$ and let $s_{\lambda}$ be the minimal polynomial of its dynamical degree $\lambda(f)$. Then the modulo $2$ reduction of $s_{\lambda}$ is a product of (some of) the following polynomials $$\begin{aligned} F_{ 1}(x) &= x + 1, \quad F_{ 3}(x) = x^2 + x + 1, \quad F_{ 5}(x) = x^4 + x^3 + x^2 + x + 1, \\ F_{ 7}(x) &= x^6 + x^5 + x^4 + x^3 + x^2 + x + 1, \\ F_{ 9}(x) &= x^6 + x^3 + 1, \\ F_{15}(x) &= x^8 + x^7 + x^5 + x^4 + x^3 + x + 1, \\\end{aligned}$$ Here each $F_m(x) \in \F_2[x]$ is the modulo $2$ reduction of the $m$-th cyclotomic polynomial $\Phi_m(x)\in \mathbb{Z}[x]$. Among these six polynomials, $F_7(x)$ and $F_{15}(x)$ are products of two distinct irreducible factors, each of which is not self-reciprocal, whereas the other four are irreducible. We do not know whether all the six factors above do appear among factorizations of the minimal polynomial of the map induced by an automorphism of Enriques surfaces, but we are able to give examples where $F_1(x), F_3(x), F_5(x)$ do come up (see Example \[example-existence-3factors\].a). On the other hand, we can check that they all appear from some [*[lattice isometry]{}*]{} of $U\oplus E_8$ by using lattice theory and ATLAS table, for example. A closely related problem to the description of the dynamical spectrum is to find the minimal nontrivial dynamical degree $1\neq \lambda\in \Lambda(\mathcal{C})$. This question was answered for complex tori (see [@mcmullen11 Thm 1.3]), rational and $K3$ surfaces by McMullen (see [@mcmullen07; @mcmullen11; @mcmullen16]), whereas the smallest dynamical degree attained by automorphisms of Enriques surfaces is yet to be found ([@third Question 4.5.(3)]). By [@third Remark 4.4], none of the smallest five Salem numbers (including the ones of degree $>10$) can be realized on Enriques surfaces. Although explicit descriptions of automorphism groups of several special families (see [@BP; @MO15]) of Enriques surfaces are known, our present knowledge seems not to be enough to determine the minimal dynamical degree of automorphisms of surfaces in this class. Presently the smallest known dynamical degree of an automorphism of an Enriques surface is the one constructed by Dolgachev [@dolgachev16 Table 2], who found an automorphism of an Enriques surface (of Hesse type) of dynamical degree $\lambda_D=2.08101\ldots$ (see Example \[Dolgachevexample\] for more details). As to this respect, in Example \[MO15family\], we give an additional study on the family of [@MO15] to show that all nontrivial dynamical degrees of automorphisms of Enriques surfaces in [@MO15] are at least $\lambda_D$. The result thus fails to give a new lower bound, but gives a good account for what is going on. To fill the gap, the constraint given by Thm \[mod2\] works to give the following slightly better theoretical lower bound: \[th2\] The dynamical degree of an automorphism of an Enriques surface is greater than or equal to the Salem number $\lambda=1.35098\cdots$ given by the polynomial $$x^{10}-x^9-x^6+x^5-x^4-x+1.$$ We use Theorem \[mod2\] together with [@Gross--McMullen], combined with Dolgachev’s example [@dolgachev16 Table 2] to show that the smallest (non-trivial) dynamical degree of an automorphism of an Enriques surface must be one of the 39 Salem numbers which we list in $\S.$\[sect-appendix\] Appendix. Our approach is inspired by Oguiso’s proof of [@third Thm 1.2]. We consider mod 2 reduction of the cohomological action, and apply results from [@Gross--McMullen] to obtain a detailed picture. It should be noted that our approach does rule out numerous Salem numbers (see Example \[rem-possibilities\]), but Thm \[mod2\] cannot lead to a necessary and sufficient condition for a Salem number to be the dynamical degree of an automorphism of an Enriques surface. One possible way of determining the exact minimal value of non-trivial dynamical degrees of automorphisms of Enriques surfaces will be that the remaining $39$ cases can be treated efficiently by some refinement of McMullen’s method [@mcmullen16], but this task exceeds the scope of this paper. [**Convention:**]{} In this note we work over the field of complex numbers $\C$. Proof of Theorem \[th1\] {#sect-th1} ======================== We maintain the notation of the introduction: $S$ is assumed to be an Enriques surface and $f\in \mathrm{Aut}(S)$ is an automorphism. As is well-known, the canonical cover $\tilde{S}=\mathrm{Spec}(\mathcal{O}\oplus \mathcal{O}(K_S))$ is a $K3$ surface and the morphism $\pi: \tilde{S} \rightarrow S$ is a double étale cover. We denote by $\varepsilon$ the covering involution of $\pi$. Since the automorphism $f$ preserves the canonical class $K_S\in \mathrm{Pic}(S)$, $f$ lifts to an automorphism $\tilde{f}$ of $\tilde{S}$. Recall that for an Enriques surface the lattice $\mathrm{Num}(S)$ is the free part of the cohomology group $H^2(S,\Z)$. Let $$M:=H^2(\tilde{S},\Z)^{\varepsilon}$$ be the $\varepsilon$-invariant sublattice of the cohomology lattice $H^2(\tilde{S},\Z)$ and let $N:= M^\perp$ be its orthogonal complement. The direct orthogonal sum $M \oplus N$ is a finite index sublattice of the lattice $H^2(\tilde{S},\Z)$. Moreover, we know by [@namikawa Proposition (2.3)] that $M$ coincides with the pullback of $H^2(S,\Z)$ by $\pi$, hence we have the isomorphisms $$\label{eq-enriques-isomorphisms} M \simeq \mathrm{Num}(S)(2) \simeq U(2)\oplus E_8(2) \text{ and } N\simeq U\oplus U(2)\oplus E_8(2),$$ where $U$ denotes the unimodular hyperbolic plane and $E_8$ is the unique even unimodular negative-definite lattice of rank 8. Moreover, for a lattice $L$ and $n\in \Q$, $L(n)$ denotes the lattice whose underlying abelian group is the same as $L$ and the bilinear form is multiplied by $n$. Basic facts concerning integral symmetric bilinear forms can be found in [@nikulin-sym]. Since the lift $\tilde{f}$ commutes with the involution $\varepsilon$, the map it induces on the cohomology lattice preserves sublattices $M$ and $N$. We put $${f_M}:= \tilde f^* \rvert_M \mbox{ and } {f_N}:= \tilde f^* \rvert_N \, .$$ Obviously, ${f_N}$ induces an isometry of the quadratic space $N \otimes \R$ of signature $(2,10)$ that preserves the original lattice $N =: N_{\Z} \subset N \otimes \R$ and the Hodge structure of $N$. Since the latter is exactly given by an oriented positive 2-plane in $N \otimes \R$, we have $${f_N}\in O(N_{\Z})\cap (O(2)\times O(10)).$$ Since the right-hand side is a discrete subgroup in a compact group, we obtain the following well-known fact. \[Nfin\] The map ${f_N}$ is of finite order. By Lemma \[Nfin\], the characteristic polynomial of ${f_N}$ is a product $$\label{eq-charpolfac} {p}_N(x)=\det (xI-{f_N}) = \prod_{i=1}^k \Phi_{n_i}(x)$$ of cyclotomic polynomials $\Phi_{n_i}(x)$ for a collection of positive integers $\{n_i :i=1, \ldots, k\}$ with $\sum_{i=1}^k {\varphi}(n_i) = 12$, where ${\varphi}(\cdot)$ stands for the Euler totient function. Obviously, the order of ${f_N}$ is just the least common multiple $$\label{eq-ord-tdfn} \ord({f_N}) = \operatorname{lcm}\{n_i, i = 1, \ldots,k\}.$$ The integers $m$ for which ${\varphi}(m)\leq 12$ are as follows. \[table-pphi\] ${\varphi}(m)$ $m$ ------------------------------- --------------------- $12$ $13,21,26,28,36,42$ $10$ $11,22$ $ 8$ $15,16,20,24,30$ $ 6$ $7,9,14,18$ $ 4$ $5,8,10,12$ $ 2$ $3,4,6$ $ 1$ $1,2$ The proof of Thm \[th1\] will be based on the following lemmas. \[lem1-new\] Let ${p}_N(x)$ be the characteristic polynomial of the map ${f_N}$. [(a)]{} The reduction $({p}_N(x) \bmod 2)$ is divisible by $(x^2+1)=(x+1)^2$. [(b)]{} ${p}_N(1) {p}_N(-1)$ is either zero or a square in $\mathbb{Q}^{*}$. \(a) Let us consider the action of ${f_N}$ on the reduction $N\otimes \mathbb{F}_2 \cong (1/2)N/N$. Obviously, the reduction contains the 10-dimensional ${f_N}$-invariant subspace $$\label{eq-10dim} N^*/N \subset(1/2)N/N,$$ where $N^*/N$ is the discriminant group of $N$. Thus the reduction $({p}_N(x) \bmod 2)$ is divisible by a degree two polynomial. In fact, there exists an $11$-dimensional canonical subspace $N^{*,+}$ of the reduction $(1/2)N/N$ containing $N^*/N$. We discuss as follows. The residue group $(1/2)N/N^*$ consists of four residue classes modulo $N^*$ and $N^*$ has the property that for all $y\in N^*$, $(y,y)\in \Z$ (namely $\delta (N)=0$ in Nikulin’s notation.) Hence, the induced quadratic form $(1/2)N/N^*\rightarrow \Q/\Z$ is well-defined. Among the four residue classes, there exists a unique nonzero element whose form value is $1/2$, which corresponds to $N^{*,+}$. (The idea of this proof parallels [@allcock]). Thus we obtain (a). \(b) Claim follows from [@BF Proposition 5.1]. \[lem-yuya\] The order $\mathrm{ord}(f_N)$ cannot be one of the integers $11, 22, 35, 70$. We put $p_M(x) = \sum_{i=0}^{10} a_i x^i$ (resp. $p_{N^*/N}(x)$) to denote the characteristic polynomial of ${f_M}$ on $M$ (resp. of the map induced by ${f_N}$ on the discriminant group $N^*/N$). Since $M \oplus N$ is a finite index sublattice of the unimodular lattice $H^2(\tilde{S}, \Z)$ we have a canonical isomorphism of the discriminant groups $$M^*/M \cong N^*/N \, ,$$ which is $({f_M},{f_N})$-equivariant. Moreover, from $M^*/M = \frac{1}{2}M/M$ and the inclusion we infer $$\label{eq-pnstarn} p_{N^*/N}(x) = (p_M(x) \bmod 2) \quad \mbox{and} \quad p_{N^*/N}(x) {\mid}(p_N(x) \bmod 2) .$$ Assume $\mathrm{ord}(f_N) \in \{35, 70\}$. Then, by and the table on p.  we have $$\label{eq-pnx} p_N(x) = \Phi_{5 l_1}(x) \Phi_{7 l_2}(x) \Phi_{l_3}(x) \Phi_{l_4}(x) \quad \mbox{ with } l_1, l_2, l_3, l_4 \in \{1,2\} \, .$$ Thus implies that $$(p_M(x) \bmod 2) = p_{N^*/N}(x) = F_5(x) F_7(x) = x^{10} + x^8 + x^6 + x^5 + x^4 + x^2 + 1 ,$$ so the coefficient $a_5$ and the sums $a_0 + a_2 + a_4$, $a_6 + a_8 + a_{10}$ are odd integers. In particular, the polynomial $p_M(x)$ is self-reciprocal (see e.g. [@cantat]), so the product $p_M(1)p_M(-1)$ can be expressed as $$\begin{aligned} \bigl(\sum_{i:even} a_i\bigr)^2 - \bigl(\sum_{i:odd}a_i\bigr)^2 &=& (2(a_0 + a_2 + a_4))^2 - (2(a_1 + a_3) + a_5)^2 \\ &=& (2 \bmod 4)^2 - (1 \bmod 2)^2 \end{aligned}$$ Thus $p_M(1)p_M(-1) \equiv 3 \bmod 8$, which contradicts [@BF Proposition 5.1]. If $\mathrm{ord}(f_N) \in \{11, 22\}$, then the factorization of $p_N(x)$ is as follows $$\label{eq-pnx11} p_N(x) = \Phi_{11 l_1}(x) \Phi_{l_2}(x) \Phi_{l_3}(x) \quad \mbox{ with } l_1, l_2, l_3 \in \{1,2\} \, .$$ Thus we have $(p_M(x) \bmod 2) = F_{11}(x) $ and, as in the previous case we obtain $p_M(1)p_M(-1) \equiv 3 \bmod 8$. The proof of Lemma \[lem-yuya\] shows that the characteristic polynomial $p_M(x) = \sum_{i=0}^{10} a_i x^i$ of ${f_M}$ cannot be a polynomial such that the coefficient $a_5$ and the sums $a_0 + a_2 + a_4 = a_6 + a_8 + a_{10}$ are odd integers, i.e. the modulo 2 reduction $(p_M(x) \bmod 2)$ cannot be one of the polynomials $$F_5(x) F_7(x), F_{11}(x), F_3(x)^5, F_3(x)^2 F_9(x), F_3(x) F_5(x)^2, F_3(x) F_{15}(x) \, .$$ The next lemma rules out the first two lines of the table on p. . It is also of use in the next section. \[lem12\] If $\Phi_{m}(x)$ comes up in the factorization , then $${\varphi}(m) < 10 \, .$$ If ${\varphi}(m)=12=\operatorname{rank}N$, then the characteristic polynomial ${p}_N(x)$ equals the cyclotomic polynomial $\Phi_m(x)$. The decomposition of $\Phi_m(x) \bmod 2$ into irreducible factors is given in the table below: $m$ irreducible decomposition of $\Phi_m \bmod 2$ ------- ----------------------------------------------- 42,21 $(x^6+x^4+x^2+x+1)(x^6+x^5+x^4+x^2+1)$ 36 $(x^6 + x^3 + 1)^2$ 28 $(x^3 + x + 1)^2 (x^3 + x^2 + 1)^2$ 26,13 $x^{12} + x^{11} + \cdots + x + 1$ Thus ${\varphi}(m) < 12$ by Lemma \[lem1-new\].a. Suppose that ${\varphi}(m)=10$, namely $m=11,22$. Then $\mathrm{ord}(f_N)$ is a multiple of 11, and taking some power we get a contradiction to Lemma \[lem-yuya\]. After these preparations we can give the proof of Thm \[th1\]. Let $p(x) = \prod_{i=1}^k \Phi_{n_i}(x)$ be a product of cyclotomic polynomials for some $n_1$, $\ldots$ $n_k \in \mathbb{N}$ such that $\sum_{i=1}^k {\varphi}(n_i) = 12$ and ${\varphi}(n_i) < 10$ for $i=1, \ldots, k$. If $p(x)$ is the characteristic polynomial of the map ${f_N}$ induced by an automorphism of an Enriques surface, then it satisfies the conditions (a),(b) of Lemma \[lem1-new\] and the order of ${f_N}$ is given by . An enumeration of all cases (by hand or by a help of a computer) and Lemma \[lem-yuya\] show that $n = \operatorname{lcm}\{n_i, i = 1, \ldots,k\}$ is one of the integers that appear in Theorem \[th1\]. \[rem-finite-ohashi\] To illustrate Theorem \[th1\], we shall give a classification of the case when the order of $f\in \mathrm{Aut}(S)$ is finite. Such automorphisms were classified in [@MO1; @ohashi-birep]. The following table gives the complete classification of the pair $(\mathrm{ord} (f),\mathrm{ord} (f_N))$. $\mathrm{ord} (f)$ 1 2 3 4 5 6 8 ---------------------- ----- ----- ----- ------- ------ ----- ----- $\mathrm{ord} (f_N)$ 1,2 1,2 3,6 1,2,4 5,10 3,6 4,8 We list here nontrivial examples exhibiting the pair $(\mathrm{ord} (f),\mathrm{ord} (f_N))$ and leave the proofs to the reader. The pair $(2,1)$ is supplied by No. 18 in [@IO]. When $\mathrm{ord} (f)=3$, $5$ or $6$, $f$ is semi-symplectic by [@MO1 Proposition 4.5] (i.e. $f$ acts trivially on the space $H^0(S, \mathcal{O}(2K_S))$) and we can use the symplectic lift to compute eigenvalues. Examples 1.1 and 1.2 of [@ohashi-birep] give the pairs $(4,4)$ and $(8,8)$. Finally, Example 1.3 of [@ohashi-birep] provides the remaining possibilities for $\mathrm{ord} (f)=4$ or $8$. As Example \[example-existence-3factors\] shows, the order $\mathrm{ord} (f_N)$ is no longer bounded by $10$ when the order of $f \in \mathrm{Aut}(S)$ is infinite (see ). However, we have very few examples: the question of determining the exact list of possible $\ord({f_N})$ remains open. Dynamical degrees {#sect-dynamical-degrees} ================= We maintain the notation of the previous section. For the convenience of the reader we recall the definition of the dynamical degree. Let $f \in \mathrm{Aut}(S)$. The [**dynamical degree $\lambda(f)$ of**]{} $f$ is defined as the spectral radius of the map $f^*:\mathrm{Num}(S) \rightarrow \mathrm{Num}(S)$. One can show that the map $f^*$ has either none or exactly two eigenvalues away from the unit circle in $\C$. If such two eigenvalues come up, they are real and reciprocal. Thus either $\lambda(f) = 1$ or it is the largest real eigenvalue of the map $f^*$ (for a precise discussion of the above notion and its properties see [@cantat $\S$.2.2.2], [@mcmullen11], [@mcmullen16] and references therein). After these preparations we are in position to give the proof of Thm \[mod2\] (c.f. [@third proof of Lemma 4.1]). We put $p_M$ (resp. $p_N$, resp. $p_f$) to denote the characteristic polynomial of ${f_M}$ on $M$ (resp. ${f_N}$ on $N$, resp. $f^*$ on $\mathrm{Num}(S)$). We assume that $\lambda(f) \neq 1$ and denote the minimal polynomial of $\lambda(f)$ by $s_\lambda$. By the first isomorphism in , the action of $f^*$ on the discriminant group of the lattice $\mathrm{Num}(S)(2)$ coincides with the action of ${f_M}$ on the discriminant group $M^*/M$. From $M^*/M = \frac{1}{2}M/M$, we obtain the equality of modulo 2 reductions: $$p_M \equiv p_f \bmod 2 .$$ Moreover yields: $$(p_M \bmod 2) {\mid}(p_N \bmod 2) .$$ Let $h$ be an irreducible factor of the reduction $(s_\lambda \bmod 2)$. We have just shown $h$ appears also in the factorization of $(p_N \bmod 2)$. Then, from and Lemma \[lem12\], $h$ divides $(\Phi_m \bmod 2)$ for certain $m$ such that ${\varphi}(m) \leq 8$. Since $(\Phi_{2^e m} \bmod 2)$ is a power of the reduction $(\Phi_m \bmod 2)$, we may assume $m$ to be odd. Hence, by the table on p. , we have $m = 1,3,5,7,9,15$. Thus either $h = F_m$, with $m=1,3,5,9$ or $h$ appears in the factorization of $F_m$, where $m=7,15$. But, for $m=7,15$ we have $F_m = F_{m,1} \cdot F_{m,2}$ where $$\begin{aligned} F_{7,1} := (x^3 + x + 1), & \quad F_{7,2} := (x^3 + x^2 + 1), \\ F_{15,1} := (x^4 + x + 1), & \quad F_{15,2} := (x^4 + x^3 + 1).\end{aligned}$$ Being a Salem polynomial, $s_\lambda$ is self-reciprocal, and so is its modulo $2$ reduction. Since the polynomials $F_{7,1}$ and $F_{7,2}$ are not self-reciprocal, their multiplicities in $(s_\lambda \bmod 2)$ should coincide with those of their reciprocal counterparts. The same holds for $F_{15,1}$ and $F_{15,2}$. This completes the proof. It is natural to ask which of the six factors given in Thm \[mod2\] do appear in modulo 2 reductions of minimal polynomials of dynamical degrees of automorphisms of Enriques surfaces. We do not know whether the polynomials $F_{7}(x)$, $F_{9}(x)$, $F_{15}(x)$ are realized by automorphisms of Enriques surfaces. To answer the question whether $F_m$ where $m=1,3,5$ come up in $(s_{\lambda} \bmod 2)$, we analyze some automorphisms constructed in [@dolgachev16]. \[example-existence-3factors\] (a) By [@dolgachev16 Sect. 4.5, Table 2] there exists an Enriques surface $S$ of Hesse type and $f \in \mathrm{Aut}(S)$ such that the characteristic polynomial $p_{f^*}(x)$ of $f^* \in \mathrm{Aut}(\mathrm{Num}(S))$ equals[^2] $$\label{eq-f5f3} p_{f^*}(x) = x^{10} - 6x^9 - 7x^8 - 9x^7 - 6x^6 - 10x^5 - 6x^4 - 9x^3 - 7x^2 - 6x +1 .$$ One can check $p_{f^*}(x) \in \Z[x]$ is irreducible and we have $$(p_{f^*}(x) \bmod 2) = F_5(x) \cdot F_3(x) \cdot F_1^4(x).$$ In particular, combined with the proof of Thm \[mod2\] yields that $$\label{eq-order15} 15 {\mid}\ord({f_N}).$$ \(b) According to [@dolgachev16 Sect. 3.2, Case $m = 4$] there exists an Enriques surface $S$ and $f \in \mathrm{Aut}(S)$ such that $p_{f^*}(x)$ is divisible by the following polynomial $$\label{eq-dolgtocheck} x^8 - 165x^7 + 223x^6 - 59x^5 - 133x^4 - 59x^3 + 223x^2 - 165x + 1 \, .$$ Since the modulo 2 reduction of the above polynomial factors as the product $(F_3(x) \cdot F_9(x))$, this would imply that $F_9(x)$ can also appear in the reduction of the minimal polynomial $s_{\lambda}$. Unfortunately, there is a misprint in [@dolgachev16 Sect. 3.2, Case $m = 4$]: the term $x^4$ comes with the coefficient $(-144)$ (instead of $(-133)$ as in ). Thus the question whether the factor $F_9(x)$ is possible or not remains open. The example below shows that even the direct potential refinement of Thm \[mod2\] (i.e. ruling out all/some of the factors $F_7$, $F_9$, $F_{15}$ in the modulo 2 reduction of the characteristic polynomial) cannot lead to a necessary and sufficient condition for a Salem number to be the dynamical degree of an automorphism of an Enriques surface. \[example-strange\] Consider the Salem number $\lambda = 1.64558...$ given by the polynomial $$\label{eq-mini-salem} s_{\lambda} := x^{10}-x^9-x^8-x^2-x+1 \, .$$ As one can easily check, we have $$(s_{\lambda} \bmod 2) = F_3(x) \cdot F_1^8(x).$$ Thus Thm \[mod2\] does not rule out the above number as the dynamical degree of an automorphism of an Enriques surface. But we can apply [@Gross--McMullen Theorem 6.1]. Suppose that the Salem number given by is the dynamical degree of an automorphism of an Enriques surface $S$. The lattice $\mbox{Num}(S)$ is of rank 10, so $s_{\lambda}$ is the full characteristic polynomial of the induced automorphism on $\mbox{Num}(S)$. Also $\mathrm{Num}(S)$ is even unimodular, so by [@Gross--McMullen Theorem 6.1] both $|s_{\lambda}(\pm 1)|$ must be squares, which is not the case. This contradiction shows that cannot be the minimal polynomial of the dynamical degree of an automorphism of an Enriques surface. Presently, the smallest known non-trivial dynamical degree is the one constructed by Dolgachev: \[Dolgachevexample\] By [@dolgachev16 Sect. 4.5, Table 2] there exists an Enriques surface $S$ of Hesse type and $f \in \mathrm{Aut}(S)$ such that the characteristic polynomial $p_{f^*}(x)$ of $f^* \in \mathrm{Aut}(\mathrm{Num}(S))$ has the Salem polynomial $$\label{eq-mp-lambdaD} x^4-x^3-2x^2-x+1$$ as a factor. The largest real root of the above polynomial is $$\lambda_D := 2.08101... \, .$$ This gives the smallest known value of non-trivial dynamical degree of an automorphism of an Enriques surface. \[MO15family\] As an another example, let us consider the Enriques surface $S$ in the family of [@MO15]. In that paper, it is proved that $\mathrm{Aut}(S)\simeq C_2^{*4}\rtimes \mathfrak{S}_4$. We here prove that every element in this group has dynamical degree at least $\lambda_D$. In fact, [@MO15] exhibits 10 smooth rational curves $E_i,\ E_{ij}\ (i,j=1,\dots,4,\ i\neq j)$ on $S$ which form a rational basis of $\mathrm{Num}(S)$. We denote by $L$ the sublattice generated by them. The symmetric group $\mathfrak{S}_4$ acts on the indices of generators, while the generators $s_i$ of the four cyclic groups $C_2$ act by reflections in some divisors $G_i\in L$ of self-intersection $-2$. In particular, $\mathrm{Aut}(S)$ preserves $L$. We use the relations $(G_i, E_{kl})=0$ (for all $i,k,l$) and $(G_i, l)\in 2\mathbb{Z}$ for all $l\in L$. Now from $(G_i, E_{kl})=0$, the six-dimensional subspace $V$ generated by $E_{ij}$ is stable under $\mathrm{Aut}(S)$. Since it is negative-definite, we see that any Salem number on $S$ has degree at most four, the dimension of the complement to $V$. Thus for any element in $\mathrm{Aut}(S)$, the characteristic polynomial $F$ decomposes into degree four (on $L_{\mathbb{C}}/V_{\mathbb{C}}$) and degree six (on $V_{\mathbb{C}}$). Moreover, since $G_i$ intersects evenly with all $l\in L$, it acts trivially on $L/2L$. Hence the degree four part of $(F \bmod 2)$ decomposes either into linear factors or has only one non-linear factor $x^2+x+1$, which arises when the residue class has order 3 in $\mathfrak{S}_4$. Looking through the list in the Appendix, we get the assertion. (This is something unfortunate, although.) We use Theorem \[mod2\] on low-degree Salem numbers that do not exceed $\lambda_D$. \[rem-possibilities\] One can check that there are exactly 133 Salem numbers of degree $\leq 10$ up to Dolgachev’s record $\lambda_D$. (The list of small Salem numbers can be found in [@mossinghoff].) Oguiso’s criterion [@third Lemma 4.1] shows that 28 among them cannot be dynamical degrees of automorphisms of Enriques surfaces. To go further, combining Theorem \[mod2\] and [@Gross--McMullen] we can prove that 65 more cannot be realized by automorphisms of Enriques surfaces. For the convenience of readers we list the 133 Salem numbers in question, their minimal polynomials and their modulo 2 reductions in $\S.$\[sect-appendix\] Appendix. Those numbers excluded by modulo 2 reductions are marked “impossible”. Non-existence that results from [@Gross--McMullen Theorem 6.1] (see Example \[example-strange\]) is marked “impossible (\*)”. (In some cases both apply.) In conclusion, [*there remain $39$ Salem numbers as candidates for the minimal dynamical degree of automorphisms of Enriques surfaces.*]{} On the arXiv[^3] we attach two lists, in plain text format, of the coefficients of the minimal polynomials of (1) these $39$ candidates, and (2) all $133$ Salem numbers of degree $\leq 10$ up to Dolgachev’s record $\lambda_D$. Finally we can give the proof of Corollary \[th2\]. There are only finitely many Salem numbers less than $\lambda = 1.35098...$ and of degree $\leq 10$ (there are none of degree $\leq 6$, one of degree $8$, and six of degree $10$). By a straightforward calculation each of those numbers violates the condition of Thm \[mod2\]. For explicit factorization see the first seven entries of the table in $\S.$\[sect-appendix\] Appendix. [**Acknowledgement:**]{} This project was completed during the workshop “New Trends in Arithmetic and Geometry of Algebraic Surfaces” in BIRS (Banff, Canada). The authors would like to thank the organizers and the staff of BIRS for very good working conditions. We thank the referee for inspiring comments and calling our attention to the paper [@BF]. [99]{} , [The period lattice for Enriques surfaces]{}, [Math. Ann.]{}, **317** (2000), 483–488. , [Automorphisms of Enriques surfaces]{}, [Invent. Math.]{}, **73** (1983), 383–411. , [J. Eur. Math. Soc.]{} **17** (2015), no. 7, 1629–1656. , [Automorphisms of even unimodular lattices and equivariant Witt groups]{}, [arXiv: 1708.05540]{} , [Dynamics of automorphisms of compact complex surfaces]{}, [Frontiers in Complex Dynamics: In celebration of John Milnor’s 80th birthday, 463–514]{}, [Princeton Mathematical Series, Princeton University Press]{} , [Salem numbers and Enriques surfaces]{}, [Experimental Mathematics, DOI: 10.1080/10586458.2016.1261743]{} , [Automorphisms of even unimodular lattices and unramified Salem numbers]{}, [J. Algebra]{}, **257** (2002), no. 2, 265–290. , [Classification of involutions on Enriques surfaces]{}, [Michigan Math. J.]{}, [**63**]{} (2014), 159–188. , [Dynamics on blowups of the projective plane.]{}, [Publ. Math. Inst. Hautes Études Sci.]{}, **105** (2007), 49–89. , [$K3$ surfaces, entropy and glue.]{}, [J. Reine Angew. Math.]{}, **658** (2011), 1–25. , [Automorphisms of projective $K3$ surfaces with minimum entropy.]{}, [Invent. Math.]{}, **203** (2016), no. 1, 179–215. , [Small Salem numbers]{}, <http://www.math.ucla.edu/~mjm/lc/lists.html> , [The automorphism groups of Enriques surfaces covered by symmetric quartic surfaces.]{}, [Recent advances in algebraic geometry, 307–320,]{} [London Math. Soc. Lecture Note Ser., 417, Cambridge Univ. Press, Cambridge, 2015.]{} , [Finite groups of automorphisms of Enriques surfaces and the Mathieu group $M_{12}$]{}, [preprint, arXiv: 1410.7535 (math.AG)]{} , [Periods of Enriques surfaces]{}, [Math. Ann.]{} **270** (1985), no. 2, 201–222. , [Integral symmetric bilinear forms and some of their applications (English translation)]{}, [Math. USSR Izv.]{}, **14** (1980), 103–167. , [The third smallest Salem number in automorphisms of $K3$ surfaces]{}, [Algebraic geometry in East Asia-Seoul 2008]{}, 331–360. , [Bi-canonical representations of finite automorphisms acting on Enriques surfaces]{}, [preprint, arXiv: 1504.00728 (math.AG)]{} , [Salem numbers and automorphisms of complex surfaces]{}, [Math. Res. Lett.]{} **19** (2012), 475–482. , [Rational surface automorphisms with positive entropy]{}, [Ann. Inst. Fourier (Grenoble)]{}, **66** (2016), 377–432. Appendix: List of low-degree Salem numbers up to $\lambda_D$ {#sect-appendix} ============================================================ Below we list all Salem numbers of degree $\leq 10$ up to the dynamical degree $\lambda_D$ of Dolgachev’s example and show the minimal polynomial $s_\lambda$ and its mod 2 factorization. Some of them are impossible by Thm \[mod2\] and marked “impossible”. Others marked “impossible (\*)” are those excluded by [@Gross--McMullen Theorem 6.1]. As noted in Remark \[rem-possibilities\], the list of the coefficients of the minimal polynomials are available in plain text format on the arXiv. ---- ----- ------------ ------------------------------------------------------------------ ------------------- \# deg value minimal polynomial $s_\lambda$ factorization of $s_\lambda \bmod 2$ conclusion 10 1.17628... $x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1$ \* $(x^5 + x^3 + x^2 + x + 1) (x^5 + x^4 + x^3 + x^2 + 1)$ impossible 10 1.21639... $x^{10}-x^6-x^5-x^4+1$ \* $(x^5 + x^4 + x^2 + x + 1) (x^5 + x^4 + x^3 + x + 1)$ impossible 10 1.23039... $x^{10}-x^7-x^5-x^3+1$ \* $x^{10} + x^7 + x^5 + x^3 + 1$ impossible 10 1.26123... $x^{10}-x^8-x^5-x^2+1$ \* $(x^2 + x + 1) (x^8 + x^7 + x^6 + x^4 + x^2 + x + 1)$ impossible 8 1.28063... $x^8-x^5-x^4-x^3+1$ \* $x^8 + x^5 + x^4 + x^3 + 1$ impossible 10 1.29348... $x^{10}-x^8-x^7+x^5-x^3-x^2+1$ \* $(x^5 + x^2 + 1) (x^5 + x^3 + 1)$ impossible 10 1.33731... $x^{10}-x^9-x^5-x+1$ \* $x^{10} + x^9 + x^5 + x + 1$ impossible 1 10 1.35098... $x^{10}-x^9-x^6+x^5-x^4-x+1$ \* $(x^2 + x + 1)^2 (x^3 + x + 1) (x^3 + x^2 + 1)$ 8 1.35999... $x^8-x^7+x^6-2x^5+x^4-2x^3+x^2-x+1$ \* $x^8 + x^7 + x^6 + x^4 + x^2 + x + 1$ impossible 10 1.38363... $x^{10}-x^9-x^7+x^6-x^5+x^4-x^3-x+1$ \* $(x^5 + x^3 + x^2 + x + 1) (x^5 + x^4 + x^3 + x^2 + 1)$ impossible 2 6 1.40126... $x^6-x^4-x^3-x^2+1$ \* $(x^2 + x + 1) (x^4 + x^3 + x^2 + x + 1)$ 3 8 1.42500... $x^8-x^7-x^5+x^4-x^3-x+1$ \* $(x^4 + x + 1) (x^4 + x^3 + 1)$ 10 1.43100... $x^{10}-x^9-x^8+x^7-x^5+x^3-x^2-x+1$ \* $(x^2 + x + 1) (x^8 + x^5 + x^4 + x^3 + 1)$ impossible 10 1.44842... $x^{10}-2x^9+2x^8-2x^7+x^6-x^5+x^4-2x^3+2x^2-2x+1$ \* $(x^5 + x^4 + x^2 + x + 1) (x^5 + x^4 + x^3 + x + 1)$ impossible 4 8 1.45798... $x^8-x^6-x^5-x^3-x^2+1$ \* $(x + 1)^2 (x^6 + x^3 + 1)$ 10 1.47235... $x^{10}-x^9-x^6-x^4-x+1$ \* $(x + 1)^6 (x^4 + x^3 + x^2 + x + 1)$ impossible (\*) 10 1.47960... $x^{10}-2x^8-2x^7+x^6+3x^5+x^4-2x^3-2x^2+1$ \* $(x^5 + x^4 + x^2 + x + 1) (x^5 + x^4 + x^3 + x + 1)$ impossible 5 6 1.50613... $x^6-x^5-x^3-x+1$ \* $(x^2 + x + 1)^3$ 10 1.51386... $x^{10}-x^7-2x^6-x^5-2x^4-x^3+1$ \* $x^{10} + x^7 + x^5 + x^3 + 1$ impossible 8 1.52306... $x^8-x^7-x^6+x^4-x^2-x+1$ \* $x^8 + x^7 + x^6 + x^4 + x^2 + x + 1$ impossible 6 10 1.53292... $x^{10}-x^9-x^8+x^5-x^2-x+1$ \* $(x^4 + x^3 + x^2 + x + 1) (x^6 + x^3 + 1)$ 8 1.54719... $x^8-2x^7+2x^6-3x^5+3x^4-3x^3+2x^2-2x+1$ \* $x^8 + x^5 + x^4 + x^3 + 1$ impossible 7 6 1.55603... $x^6-x^5-x^4+x^3-x^2-x+1$ \* $(x^3 + x + 1) (x^3 + x^2 + 1)$ 8 6 1.58234... $x^6-x^4-2x^3-x^2+1$ \* $(x + 1)^6$ 10 1.59070... $x^{10}-2x^9+x^8-2x^6+3x^5-2x^4+x^2-2x+1$ \* $(x^2 + x + 1) (x^8 + x^7 + x^6 + x^4 + x^2 + x + 1)$ impossible 10 1.59700... $x^{10}-x^8-x^7-x^6-x^5-x^4-x^3-x^2+1$ \* $(x^2 + x + 1)^2 (x^6 + x^3 + 1)$ impossible (\*) 10 1.59866... $x^{10}-2x^9+x^8-x^7+2x^6-3x^5+2x^4-x^3+x^2-2x+1$ \* $(x^5 + x^2 + 1) (x^5 + x^3 + 1)$ impossible 9 8 1.60544... $x^8-2x^7+x^6-x^4+x^2-2x+1$ \* $(x^4 + x^3 + x^2 + x + 1)^2$ 10 1.62501... $x^{10}-x^9-x^8-x^7+x^6+x^5+x^4-x^3-x^2-x+1$ \* $x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$ impossible (\*) 10 1.62754... $x^{10}-2x^9+2x^7-x^6-x^5-x^4+2x^3-2x+1$ \* $(x^5 + x^4 + x^2 + x + 1) (x^5 + x^4 + x^3 + x + 1)$ impossible 10 6 1.63557... $x^6-2x^5+2x^4-3x^3+2x^2-2x+1$ \* $x^6 + x^3 + 1$ 8 1.64003... $x^8-2x^6-x^5+x^4-x^3-2x^2+1$ \* $x^8 + x^5 + x^4 + x^3 + 1$ impossible 10 1.64558... $x^{10}-x^9-x^8-x^2-x+1$ \* $(x + 1)^8 (x^2 + x + 1)$ impossible (\*) 10 1.65740... $x^{10}+x^9-2x^7-4x^6-5x^5-4x^4-2x^3+x+1$ \* $x^{10} + x^9 + x^5 + x + 1$ impossible 11 8 1.66104... $x^8-2x^7+x^6-x^5+x^4-x^3+x^2-2x+1$ \* $(x^2 + x + 1) (x^3 + x + 1) (x^3 + x^2 + 1)$ 10 1.66929... $x^{10}-x^9-x^7-2x^5-x^3-x+1$ \* $(x + 1)^2 (x^8 + x^7 + x^6 + x^4 + x^2 + x + 1)$ impossible 10 1.67310... $x^{10}-2x^9+x^8-x^7+x^5-x^3+x^2-2x+1$ \* $(x^5 + x^2 + 1) (x^5 + x^3 + 1)$ impossible 12 8 1.68491... $x^8-x^7-x^6-x^2-x+1$ \* $(x + 1)^2 (x^2 + x + 1)^3$ 10 1.69017... $x^{10}-x^9-2x^8+x^7+x^6-x^5+x^4+x^3-2x^2-x+1$ \* $(x^5 + x^3 + x^2 + x + 1) (x^5 + x^4 + x^3 + x^2 + 1)$ impossible 13 8 1.69350... $x^8-x^7-x^5-x^4-x^3-x+1$ \* $(x^4 + x + 1) (x^4 + x^3 + 1)$ 10 1.71336... $x^{10}-2x^9+x^8-2x^6+2x^5-2x^4+x^2-2x+1$ \* $(x + 1)^{10}$ impossible (\*) 14 4 1.72208... $x^4-x^3-x^2-x+1$ \* $x^4 + x^3 + x^2 + x + 1$ 10 1.73694... $x^{10}-x^9-x^7-x^6-x^5-x^4-x^3-x+1$ \* $(x^5 + x^3 + x^2 + x + 1) (x^5 + x^4 + x^3 + x^2 + 1)$ impossible 10 1.74492... $x^{10}-2x^9+2x^8-3x^7+2x^6-3x^5+2x^4-3x^3+2x^2-2x+1$ \* $x^{10} + x^7 + x^5 + x^3 + 1$ impossible 10 1.74601... $x^{10}-x^9-x^8-x^7+2x^5-x^3-x^2-x+1$ \* $(x + 1)^4 (x^3 + x + 1) (x^3 + x^2 + 1)$ impossible (\*) 10 1.75173... $x^{10}-2x^9+x^8-x^7+x^6-2x^5+x^4-x^3+x^2-2x+1$ \* $(x + 1)^2 (x^8 + x^5 + x^4 + x^3 + 1)$ impossible 10 1.75309... $x^{10}-x^8-x^7-2x^6-3x^5-2x^4-x^3-x^2+1$ \* $(x^5 + x^2 + 1) (x^5 + x^3 + 1)$ impossible 10 1.76015... $x^{10}-2x^9+x^8-2x^7+2x^6-x^5+2x^4-2x^3+x^2-2x+1$ \* $(x^2 + x + 1) (x^8 + x^7 + x^6 + x^4 + x^2 + x + 1)$ impossible 10 1.76400... $x^{10}-2x^9+x^7-x^5+x^3-2x+1$ \* $x^{10} + x^7 + x^5 + x^3 + 1$ impossible 10 1.76690... $x^{10}-2x^8-2x^7+x^5-2x^3-2x^2+1$ \* $(x^2 + x + 1) (x^4 + x + 1) (x^4 + x^3 + 1)$ impossible (\*) 10 1.77056... $x^{10}-3x^9+4x^8-5x^7+5x^6-5x^5+5x^4-5x^3+4x^2-3x+1$ \* $(x^5 + x^3 + x^2 + x + 1) (x^5 + x^4 + x^3 + x^2 + 1)$ impossible 15 6 1.78164... $x^6-x^5-x^4-x^2-x+1$ \* $(x + 1)^4 (x^2 + x + 1)$ 10 1.78840... $x^{10}-x^9-2x^7-x^5-2x^3-x+1$ \* $x^{10} + x^9 + x^5 + x + 1$ impossible 8 1.79607... $x^8-x^7-x^6-x^4-x^2-x+1$ \* $x^8 + x^7 + x^6 + x^4 + x^2 + x + 1$ impossible 10 1.79978... $x^{10}-3x^8-3x^7+2x^6+5x^5+2x^4-3x^3-3x^2+1$ \* $(x^5 + x^2 + 1) (x^5 + x^3 + 1)$ impossible 16 8 1.80017... $x^8-3x^7+4x^6-5x^5+5x^4-5x^3+4x^2-3x+1$ \* $(x^4 + x + 1) (x^4 + x^3 + 1)$ 17 10 1.80501... $x^{10}-2x^9+x^7-x^6+x^5-x^4+x^3-2x+1$ \* $(x^2 + x + 1)^3 (x^4 + x^3 + x^2 + x + 1)$ 18 8 1.80978... $x^8-x^7-2x^5-2x^3-x+1$ \* $(x + 1)^2 (x^3 + x + 1) (x^3 + x^2 + 1)$ 8 1.81161... $x^8-2x^7+x^5-x^4+x^3-2x+1$ \* $x^8 + x^5 + x^4 + x^3 + 1$ impossible 10 1.82383... $x^{10}-x^8-2x^7-2x^6-2x^5-2x^4-2x^3-x^2+1$ \* $(x + 1)^{10}$ impossible (\*) 19 10 1.82514... $x^{10}-x^9-2x^8+x^6+x^5+x^4-2x^2-x+1$ \* $(x^2 + x + 1)^2 (x^3 + x + 1) (x^3 + x^2 + 1)$ 20 6 1.83107... $x^6-2x^5+x^3-2x+1$ \* $x^6 + x^3 + 1$ 21 8 1.83488... $x^8-x^6-2x^5-3x^4-2x^3-x^2+1$ \* $(x^4 + x^3 + x^2 + x + 1)^2$ 10 1.84835... $x^{10}-x^9-x^8-x^6-x^5-x^4-x^2-x+1$ \* $(x^2 + x + 1)^5$ impossible (\*) 8 1.84959... $x^8+x^7-x^6-4x^5-5x^4-4x^3-x^2+x+1$ \* $x^8 + x^7 + x^6 + x^4 + x^2 + x + 1$ impossible 10 1.85312... $x^{10}-2x^9+x^8-2x^7+2x^6-2x^5+2x^4-2x^3+x^2-2x+1$ \* $(x + 1)^{10}$ impossible (\*) 10 1.85712... $x^{10}-2x^9+x^8-x^7-x^5-x^3+x^2-2x+1$ \* $(x^5 + x^2 + 1) (x^5 + x^3 + 1)$ impossible 10 1.86264... $x^{10}-3x^9+3x^8-x^7-3x^6+5x^5-3x^4-x^3+3x^2-3x+1$ \* $x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$ impossible (\*) 22 8 1.86406... $x^8-x^7-2x^6+2x^4-2x^2-x+1$ \* $(x + 1)^2 (x^3 + x + 1) (x^3 + x^2 + 1)$ 10 1.86876... $x^{10}-x^9-2x^7-3x^5-2x^3-x+1$ \* $x^{10} + x^9 + x^5 + x + 1$ impossible 10 1.87573... $x^{10}-2x^9-x^8+3x^7-3x^5+3x^3-x^2-2x+1$ \* $(x^5 + x^2 + 1) (x^5 + x^3 + 1)$ impossible 23 4 1.88320... $x^4-2x^3+x^2-2x+1$ \* $(x^2 + x + 1)^2$ 10 1.88996... $x^{10}-x^9-x^8-x^6-2x^5-x^4-x^2-x+1$ \* $(x + 1)^2 (x^4 + x + 1) (x^4 + x^3 + 1)$ impossible (\*) 10 1.89360... $x^{10}-x^9-2x^8+x^6+x^4-2x^2-x+1$ \* $(x + 1)^6 (x^4 + x^3 + x^2 + x + 1)$ impossible (\*) 10 1.89663... $x^{10}-2x^9+x^7-x^6-x^4+x^3-2x+1$ \* $(x + 1)^4 (x^6 + x^3 + 1)$ impossible (\*) 10 1.89910... $x^{10}-2x^9+x^6-x^5+x^4-2x+1$ \* $(x^5 + x^4 + x^2 + x + 1) (x^5 + x^4 + x^3 + x + 1)$ impossible 10 1.90562... $x^{10}-x^8-2x^7-3x^6-3x^5-3x^4-2x^3-x^2+1$ \* $(x^3 + x + 1) (x^3 + x^2 + 1) (x^4 + x^3 + x^2 + x + 1)$ impossible (\*) 10 1.90830... $x^{10}-x^9-x^8-x^7-x^5-x^3-x^2-x+1$ \* $(x^2 + x + 1) (x^8 + x^5 + x^4 + x^3 + 1)$ impossible 10 1.91112... $x^{10}-2x^8-2x^7-x^6-x^5-x^4-2x^3-2x^2+1$ \* $(x^5 + x^4 + x^2 + x + 1) (x^5 + x^4 + x^3 + x + 1)$ impossible 10 1.91445... $x^{10}-x^9-x^7-3x^6-x^5-3x^4-x^3-x+1$ \* $(x^5 + x^3 + x^2 + x + 1) (x^5 + x^4 + x^3 + x^2 + 1)$ impossible 24 8 1.91649... $x^8-x^7-x^6-x^5-x^3-x^2-x+1$ \* $(x + 1)^4 (x^4 + x^3 + x^2 + x + 1)$ 8 1.92062... $x^8-3x^7+3x^6-2x^5+x^4-2x^3+3x^2-3x+1$ \* $x^8 + x^7 + x^6 + x^4 + x^2 + x + 1$ impossible 10 1.92606... $x^{10}-x^9-x^8-x^7-x^6+x^5-x^4-x^3-x^2-x+1$ \* $x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$ impossible (\*) 25 8 1.92678... $x^8-2x^6-2x^5-x^4-2x^3-2x^2+1$ \* $(x^2 + x + 1)^4$ 10 1.92990... $x^{10}-x^9-x^8-2x^7+x^6+x^4-2x^3-x^2-x+1$ \* $(x + 1)^2 (x^4 + x + 1) (x^4 + x^3 + 1)$ impossible (\*) 10 1.93231... $x^{10}-2x^9+2x^8-4x^7+3x^6-5x^5+3x^4-4x^3+2x^2-2x+1$ \* $(x^5 + x^4 + x^2 + x + 1) (x^5 + x^4 + x^3 + x + 1)$ impossible 10 1.93295... $x^{10}-3x^9+3x^8-2x^7+x^5-2x^3+3x^2-3x+1$ \* $(x^4 + x^3 + x^2 + x + 1) (x^6 + x^3 + 1)$ impossible (\*) 10 1.93637... $x^{10}-2x^9+x^5-2x+1$ \* $(x^2 + x + 1) (x^4 + x + 1) (x^4 + x^3 + 1)$ impossible (\*) 10 1.94005... $x^{10}-2x^9+x^8-x^7-x^6-x^4-x^3+x^2-2x+1$ \* $(x + 1)^2 (x^8 + x^5 + x^4 + x^3 + 1)$ impossible 26 6 1.94685... $x^6-x^5-x^4-x^3-x^2-x+1$ \* $(x^3 + x + 1) (x^3 + x^2 + 1)$ 10 1.94998... $x^{10}-x^9-2x^8-x^7+x^6+3x^5+x^4-x^3-2x^2-x+1$ \* $(x^5 + x^3 + x^2 + x + 1) (x^5 + x^4 + x^3 + x^2 + 1)$ impossible 8 1.95530... $x^8-2x^7-x^5+3x^4-x^3-2x+1$ \* $x^8 + x^5 + x^4 + x^3 + 1$ impossible 27 6 1.96355... $x^6-2x^5-x^4+3x^3-x^2-2x+1$ \* $(x^2 + x + 1) (x^4 + x^3 + x^2 + x + 1)$ 10 1.97209... $x^{10}-2x^9-x^6+3x^5-x^4-2x+1$ \* $(x^5 + x^4 + x^2 + x + 1) (x^5 + x^4 + x^3 + x + 1)$ impossible 28 6 1.97481... $x^6-2x^5+x^4-2x^3+x^2-2x+1$ \* $(x + 1)^6$ 29 6 1.98779... $x^6-2x^4-3x^3-2x^2+1$ \* $x^6 + x^3 + 1$ 30 8 1.99400... $x^8-2x^7+x^6-2x^5+x^4-2x^3+x^2-2x+1$ \* $(x^4 + x^3 + x^2 + x + 1)^2$ 10 1.99703... $x^{10}-x^9-x^8-x^7-x^6-x^5-x^4-x^3-x^2-x+1$ \* $x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$ impossible (\*) 31 10 1.99852... $x^{10}-2x^9+x^8-2x^7+x^6-2x^5+x^4-2x^3+x^2-2x+1$ \* $(x + 1)^2 (x^2 + x + 1)^4$ 10 2.00145... $x^{10}-x^9-2x^8-x^7+x^6+2x^5+x^4-x^3-2x^2-x+1$ \* $(x + 1)^4 (x^2 + x + 1)^3$ impossible (\*) 10 2.00289... $x^{10}-3x^9+3x^8-3x^7+3x^6-3x^5+3x^4-3x^3+3x^2-3x+1$ \* $x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$ impossible (\*) 10 2.00573... $x^{10}-2x^9-2x+1$ \* $(x + 1)^2 (x^4 + x^3 + x^2 + x + 1)^2$ impossible (\*) 10 2.00624... $x^{10}-x^8-3x^7-3x^6-4x^5-3x^4-3x^3-x^2+1$ \* $(x + 1)^2 (x^8 + x^5 + x^4 + x^3 + 1)$ impossible 10 2.00947... $x^{10}-2x^9+x^8-x^7-2x^6+x^5-2x^4-x^3+x^2-2x+1$ \* $(x^5 + x^2 + 1) (x^5 + x^3 + 1)$ impossible 32 8 2.01128... $x^8-3x^7+3x^6-3x^5+3x^4-3x^3+3x^2-3x+1$ \* $(x^2 + x + 1) (x^6 + x^3 + 1)$ 10 2.01335... $x^{10}-x^9-2x^7-2x^6-3x^5-2x^4-2x^3-x+1$ \* $x^{10} + x^9 + x^5 + x + 1$ impossible 10 2.01488... $x^{10}-2x^8-2x^7-2x^6-3x^5-2x^4-2x^3-2x^2+1$ \* $(x^2 + x + 1) (x^4 + x + 1) (x^4 + x^3 + 1)$ impossible (\*) 10 2.01600... $x^{10}-3x^9+2x^8+x^7-3x^6+3x^5-3x^4+x^3+2x^2-3x+1$ \* $(x^5 + x^3 + x^2 + x + 1) (x^5 + x^4 + x^3 + x^2 + 1)$ impossible 10 2.01671... $x^{10}-2x^9-x^8+2x^7+x^6-3x^5+x^4+2x^3-x^2-2x+1$ \* $(x^3 + x + 1) (x^3 + x^2 + 1) (x^4 + x^3 + x^2 + x + 1)$ impossible (\*) 33 8 2.02202... $x^8-2x^7-2x+1$ \* $(x + 1)^8$ 10 2.02344... $x^{10}-2x^9-x^7+2x^6-x^5+2x^4-x^3-2x+1$ \* $x^{10} + x^7 + x^5 + x^3 + 1$ impossible 10 2.02739... $x^{10}-x^9-x^8-x^7-x^6-2x^5-x^4-x^3-x^2-x+1$ \* $(x + 1)^2 (x^2 + x + 1)^2 (x^4 + x^3 + x^2 + x + 1)$ impossible (\*) 34 8 2.03064... $x^8-x^7-3x^5-x^4-3x^3-x+1$ \* $(x^4 + x + 1) (x^4 + x^3 + 1)$ 10 2.03298... $x^{10}-2x^9+x^6-3x^5+x^4-2x+1$ \* $(x^5 + x^4 + x^2 + x + 1) (x^5 + x^4 + x^3 + x + 1)$ impossible 10 2.03579... $x^{10}-2x^9-x^6+2x^5-x^4-2x+1$ \* $(x + 1)^6 (x^2 + x + 1)^2$ impossible (\*) 10 2.03890... $x^{10}-x^9-2x^8-2x^7+2x^6+3x^5+2x^4-2x^3-2x^2-x+1$ \* $x^{10} + x^9 + x^5 + x + 1$ impossible 35 6 2.04249... $x^6-3x^5+3x^4-3x^3+3x^2-3x+1$ \* $(x^3 + x + 1) (x^3 + x^2 + 1)$ 10 2.04414... $x^{10}-x^9-2x^8-x^7+x^6+x^5+x^4-x^3-2x^2-x+1$ \* $(x^5 + x^3 + x^2 + x + 1) (x^5 + x^4 + x^3 + x^2 + 1)$ impossible 10 2.04776... $x^{10}-3x^9+3x^8-3x^7+2x^6-x^5+2x^4-3x^3+3x^2-3x+1$ \* $(x^2 + x + 1) (x^8 + x^5 + x^4 + x^3 + 1)$ impossible 10 2.04817... $x^{10}-x^9-2x^8-x^5-2x^2-x+1$ \* $x^{10} + x^9 + x^5 + x + 1$ impossible 36 8 2.04952... $x^8-x^7-x^6-x^5-2x^4-x^3-x^2-x+1$ \* $(x + 1)^4 (x^4 + x^3 + x^2 + x + 1)$ 10 2.05286... $x^{10}-x^9-x^8-2x^7-x^5-2x^3-x^2-x+1$ \* $(x^4 + x^3 + x^2 + x + 1) (x^6 + x^3 + 1)$ impossible (\*) 10 2.05353... $x^{10}-2x^9-x^8+2x^7-x^5+2x^3-x^2-2x+1$ \* $(x^2 + x + 1) (x^8 + x^7 + x^6 + x^4 + x^2 + x + 1)$ impossible 10 2.05523... $x^{10}-x^9-x^8-x^7-x^6-3x^5-x^4-x^3-x^2-x+1$ \* $x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$ impossible (\*) 37 10 2.05631... $x^{10}-2x^9-x^7+x^6+x^5+x^4-x^3-2x+1$ \* $(x^2 + x + 1)^3 (x^4 + x^3 + x^2 + x + 1)$ 10 2.05819... $x^{10}-x^9-3x^7-x^6-3x^5-x^4-3x^3-x+1$ \* $(x^5 + x^3 + x^2 + x + 1) (x^5 + x^4 + x^3 + x^2 + 1)$ impossible 38 8 2.06017... $x^8-3x^7+2x^6+x^5-3x^4+x^3+2x^2-3x+1$ \* $(x^4 + x + 1) (x^4 + x^3 + 1)$ 10 2.06226... $x^{10}-2x^8-3x^7-2x^6-x^5-2x^4-3x^3-2x^2+1$ \* $x^{10} + x^7 + x^5 + x^3 + 1$ impossible 10 2.06420... $x^{10}-2x^9+x^8-2x^7-x^5-2x^3+x^2-2x+1$ \* $(x^2 + x + 1) (x^8 + x^7 + x^6 + x^4 + x^2 + x + 1)$ impossible 10 2.06557... $x^{10}-x^8-3x^7-4x^6-5x^5-4x^4-3x^3-x^2+1$ \* $(x^5 + x^2 + 1) (x^5 + x^3 + 1)$ impossible 8 2.06972... $x^8-2x^7+x^5-3x^4+x^3-2x+1$ \* $x^8 + x^5 + x^4 + x^3 + 1$ impossible 10 2.07416... $x^{10}-3x^9+4x^8-6x^7+6x^6-7x^5+6x^4-6x^3+4x^2-3x+1$ \* $x^{10} + x^9 + x^5 + x + 1$ impossible 39 4 2.08101... $x^4-x^3-2x^2-x+1$ \* $(x + 1)^2 (x^2 + x + 1)$ $\exists$ example ---- ----- ------------ ------------------------------------------------------------------ ------------------- [^1]: Y. M. is supported by JSPS KAKENHI Grant Numbers 15H05738 and 16K17560. H. O. is supported by JSPS KAKENHI 15K17521. S. R. is partially supported by National Science Centre, Poland, grant 2014/15/B/ST1/02197. [^2]: There is a misprint in [@dolgachev16 Sect. 4.5, Table 2]: the terms $x^4$, $x^6$ appear with coefficient $6$ in $p_{f^*}(x)$. In we give the correct formula, but the misprint is irrelevant for us because we consider modulo 2 reduction. [^3]: `https://arxiv.org/abs/1707.02563`

--- abstract: 'Using systematic calculations in spinor language, we obtain simple descriptions of the second order symmetry operators for the conformal wave equation, the Dirac-Weyl equation and the Maxwell equation on a curved four dimensional Lorentzian manifold. The conditions for existence of symmetry operators for the different equations are seen to be related. Computer algebra tools have been developed and used to systematically reduce the equations to a form which allows geometrical interpretation.' address: | $^1$ Albert Einstein Institute, Am Mühlenberg 1, D-14476 Potsdam, Germany\ $^2$ The University of Edinburgh, James Clerk Maxwell Building, Mayfield Road, Edinburgh, EH9 3JZ, UK author: - 'Lars Andersson$^1$, Thomas Bäckdahl$^2$, and Pieter Blue$^2$' title: Second order symmetry operators --- Introduction ============ The discovery by Carter [@carter:1968PhRv..174.1559C] of a fourth constant of the motion for the geodesic equations in the Kerr black hole spacetime, allowing the geodesic equations to be integrated, together with the subsequent discovery by Teukolsky, Chandrasekhar and others of the separability of the spin-$s$ equations for all half-integer spins up to $s=2$ (which corresponds to the case of linearized Einstein equations) in the Kerr geometry, provides an essential tool for the analysis of fields in the Kerr geometry. The geometric fact behind the existence of Carter’s constant is, as shown by Walker and Penrose [@PenroseWalker:1970], the existence of a Killing tensor. A Killing tensor is a symmetric tensor $K_{ab} = K_{(ab)}$, satisfying the equation $\nabla_{(a} K_{bc)} = 0$. This condition implies that the quantity $K = K_{ab} \dot \gamma^a \dot \gamma^b$ is constant along affinely parametrized geodesics. In particular, viewed as a function on phase space, $K$ Poisson commutes with the Hamiltonian generating the geodesic flow, $H = \dot \gamma^a \dot \gamma_a$. Carter further showed that in a Ricci flat spacetime with a Killing tensor $K_{ab}$, the operator ${\mathcal K} = \nabla_a K^{ab} \nabla_b$, which may be viewed as the “quantization” of $K$, commutes with the d’Alembertian $\mathcal H = \nabla^a \nabla_a$, which in turn is the “quantization” of $H$, cf. [@carter:1977PhRvD..16.3395C]. In particular, the operator $\mathcal K$ is a symmetry operator for the wave equation $\mathcal H \phi = 0$, in the sense that it maps solutions to solutions. The properties of separability, and existence of symmetry operators, for partial differential equations are closely related [@miller:MR0460751]. In fact, specializing to the Kerr geometry, the symmetry operator found by Carter may be viewed as the spin-0 case of the symmetry operators for the higher spin fields as manifested in the Teukolsky system, see eg. [@kalnins:miller:williams:1989JMP....30.2360K; @kalnins:miller:williams:1989JMP....30.2925K]. In this paper we give necessary and sufficient conditions for the existence of second order symmetry operators, for massless test fields of spin 0, 1/2, 1, on a globally hyperbolic Lorentzian spacetime of dimension 4. (As explained in Section \[sec:indepspinors\], the global hyperbolicity condition can be relaxed.) In each case, the conditions are the existence of a conformal Killing tensor or Killing spinor, and certain auxiliary conditions relating the Weyl curvature and the Killing tensor or spinor. We are particularly interested in symmetry operators for the spin-1 or Maxwell equation. In this case, we give a single auxiliary condition, which is substantially more transparent than the collection previously given in [@KalMcLWil92a]. For the massless spin-1/2 or Dirac-Weyl equation, our result on second order symmetry operators represents a simplification of the conditions given by McLenaghan, Smith and Walker [@mclenaghan:smith:walker:2000RSPSA.456.2629G] for the existence of symmetry operators of order two. The conditions we find for spins 1/2 and 1 are closely related to the condition found recently for the spin-0 case for the conformal wave equation by Michel, Radoux and [Š]{}ilhan [@michel:radoux:silhan:2013arXiv1308.1046M], cf. Theorem \[thm:intro:spin0\] below. A major motivation for the work in this paper is provided by the application by two of the authors [@andersson:blue:2009arXiv0908.2265A] of the Carter symmetry operator for the wave equation in the Kerr spacetime, to prove an integrated energy estimate and boundedness for solutions of the wave equation. The method used is a generalization of the vector fields method [@klainerman:MR784477] to allow not only Killing vector symmetries but symmetry operators of higher order. In order to apply such methods to fields with non-zero spin, such as the Maxwell field, it is desirable to have a clear understanding of the conditions for the existence of symmetry operators and their structure. This serves as one of the main motivations for the results presented in this paper, which give simple necessary and sufficient conditions for the existence of symmetry operators for the Maxwell equations in a 4-dimensional Lorentzian spacetime. The energies constructed from higher order symmetry operators correspond to conserved currents which are not generated by contracting the stress energy tensor with a conformal Killing vector. Such conserved currents are known to exist eg. for the Maxwell equation, as well as fields with higher spin on Minkowski space, see [@anco:pohjanpelto:2003RSPSA.459.1215A] and references therein. In a subsequent paper [@andersson:backdahl:blue:currents] we shall present a detailed study of conserved currents up to second order for the Maxwell field. We will assume that all objects are smooth, we work in signature $(+,-,-,-)$, and we use the 2-spinor formalism, following the conventions and notation of [@PenRin84; @PenRin86]. For a translation to the Dirac 4-spinor notation, we refer to [@PenRin84 Page 221]. Recall that $\Lambda/24$ is the scalar curvature, $\Phi_{ABA'B'}$ the Ricci spinor, and $\Psi_{ABCD}$ the Weyl spinor. Even though several results are independent of the existence of a spin structure, we will for simplicity assume that the spacetime is spin. The 2-spinor formalism allows one to efficiently decompose spinor expressions into irreducible parts. All irreducible parts of a spinor are totally symmetric spinors formed by taking traces of the spinor and symmetrizing all free indices. Making use of these facts, any spinor expression can be decomposed in terms of symmetric spinors and spin metrics. This procedure is described in detail in Section 3.3 in [@PenRin84] and in particular by Proposition 3.3.54. This decomposition has been implemented in the package *SymManipulator* [@Bae11a] by the second author. *SymManipulator* is part of the *xAct* tensor algebra package [@xAct] for *Mathematica*. The package *SymManipulator* includes many canonicalization and simplification steps to make the resulting expressions compact enough and the calculations rapid enough so that fairly large problems can be handled. A Mathematica 9 notebook file containing the main calculations for this paper is available as supplementary data at <http://hdl.handle.net/10283/541>. We shall in this paper consider only massless spin-$s$ test fields. For the spin-$0$ case the field equation is the conformal wave equation $$\label{eq:confwave1} (\nabla^a \nabla_a + 4 \Lambda) \phi = 0,$$ for a scalar field $\phi$, while for non-zero spin the field is a symmetric spinor $\phi_{A \cdots F}$ of valence $(2s,0)$ satisfying the equation $$\label{eq:massless} \nabla^A{}_{A'} \phi_{A \cdots F} = 0 .$$ In this paper we shall restrict our considerations to spins $0,1/2,1$. For $s \geq 3/2$, equation implies algebraic consistency conditions, which strongly restrict the space of solutions in the presence of non-vanishing Weyl curvature. Note however that there are consistent equations for fields of higher spin, see [@PenRin84 §5.8] for discussion. Recall that a Killing spinor of valence $(k,l)$ is a symmetric spinor $L_{A_1\cdots A_k}{}^{A'_1 \cdots A'_l}$, $$\label{eq:KSgeneral} \nabla_{(A_1}{}^{(A'_1} L_{A_2\cdots A_{k+1})}{}^{A'_2 \cdots A'_{l+1})} = 0.$$ A valence $(1,1)$ Killing spinor is simply a conformal Killing vector, while a valence $(2,0)$ Killing spinor is equivalent to a conformal Killing-Yano 2-form. On the other hand, a Killing spinor of valence $(2,2)$ is simply a traceless symmetric conformal Killing tensor. It is important to note that , and are conformally invariant if $\phi$ and $\phi_{A \cdots F}$ are given conformal weight $-1$, and $L^{A_1\cdots A_kA'_1 \cdots A'_l}$ is given conformal weight $0$. See [@PenRin84 sections 5.7 and 6.7] for details. Recall that a symmetry operator for a system ${\mathcal H}\varphi=0$, is a linear partial differential operator ${\mathcal K}$ such that ${\mathcal H}{\mathcal K}\varphi=0$ for all $\varphi$ such that ${\mathcal H}\varphi=0$. We say that two operators ${\mathcal K}_1$ and ${\mathcal K}_2$ are equivalent if ${\mathcal K}_1-{\mathcal K}_2={\mathcal F}{\mathcal H}$ for some differential operator ${\mathcal F}$. We are interested only in *non-trivial* symmetry operators, i.e. operators which are not equivalent to the trivial operator $0$. For simplicity, we will only consider equivalence classes of symmetry operators. To state our main results, we need two auxiliary conditions. \[def:auxcond\] Let $L_{AB}{}^{A'B'}$ be a Killing spinor of valence $(2,2)$. 1. \[point:A0\] $L_{AB}{}^{A'B'}$ satisfies auxiliary condition \[point:A0\] if there is a function $Q$ such that $$\begin{aligned} \nabla_{AA'}Q={}&\tfrac{1}{3} \Psi_{ABCD} \nabla^{(B|B'|}L^{CD)}{}_{A'B'} + \tfrac{1}{3} \bar\Psi_{A'B'C'D'} \nabla^{B(B'}L_{AB}{}^{C'D')} \nonumber\\& + L^{BC}{}_{A'}{}^{B'} \nabla_{(A}{}^{C'}\Phi_{BC)B'C'} + L_{A}{}^{BB'C'} \nabla^{C}{}_{(A'}\Phi_{|BC|B'C')} . \label{eq:A0}\end{aligned}$$ 2. \[point:A1\] $L_{AB}{}^{A'B'}$ satisfies auxiliary condition \[point:A1\] if there is a vector field $P_A{}^{A'}$ such that $$\begin{aligned} \nabla_{(A}{}^{(A'}P_{B)}{}^{B')}= {}&L{}^{CDA'B'} \Psi_{ABCD} - L{}_{AB}{}^{C'D'} \bar\Psi^{A'B'}{}_{C'D'} . \label{eq:A1}\end{aligned}$$ Under conformal transformations such that $L^{ABA'B'}$, $P^{AA'}$ and $Q$ are given conformal weight $0$, the equations and are conformally invariant. We start by recalling the result of Michel et al. for the spin-$0$ conformal wave equation, which we state here in the case of a Lorentzian spacetime of dimension 4. \[thm:intro:spin0\] Consider the conformal wave equation $$\label{eq:confwave} (\nabla^a \nabla_a + 4 \Lambda ) \phi = 0$$ in a 4-dimensional Lorentzian spacetime. There is a non-trivial second order symmetry operator for if and only if there is a non-zero Killing spinor of valence $(2,2)$ satisfying condition \[point:A0\] of Definition \[def:auxcond\]. Previous work on the conformal wave equation was done by [@kamran:mclenaghan:1985LMaPh...9...65K], see also Kress [@kress:thesis], see also [@kalnins:miller:1983JMP....24.1047K]. Symmetry operators of general order for the Laplace-Beltrami operator in the conformally flat case have been analyzed by Eastwood [@eastwood:MR2180410]. Next we consider fields with spins 1/2 and 1. The massless spin-1/2 equations are \[eq:spin1/2eqs\] $$\begin{aligned} \nabla^A{}_{A'} \phi_A &= 0, \label{eq:spin1/2left} \\ \intertext{and its complex conjugate form} \nabla_A{}^{A'} \chi_{A'} &= 0, \label{eq:spin1/2right}\end{aligned}$$ which we shall refer to as the left and right Dirac-Weyl equations [^1]. Analogously with the terminology used by Kalnins et al. [@KalMcLWil92a] for the spin-1 case, we call a symmetry operator $\phi_A \mapsto \lambda_A$, which takes a solution of the left equation to a solution of the left equation a symmetry operator of the *first kind*, while an operator $\phi_A \mapsto \chi_{A'}$ which takes a solution of the left equation to a solution of the right equation a symmetry operator of the *second kind*. If one considers symmetry operators in the Dirac 4-spinor notation, a 4-spinor would correspond to a pair of 2-spinors $(\phi_A, \varphi_{A'})$. Therefore a symmetry operator $(\phi_A, \varphi_{A'})\mapsto (\lambda_A, \chi_{A'})$ for a 4-spinor is formed by a combination of symmetry operators of first $\phi_A\mapsto \lambda_{A}$, and second $\phi_A\mapsto \chi_{A'}$ kind, together with complex conjugate versions of first $\varphi_{A'}\mapsto \chi_{A'}$, and second $\varphi_{A'}\mapsto \lambda_A$ kind symmetry operators. \[thm:intro:spin1/2\] Consider the Dirac-Weyl equations in a Lorentzian spacetime of dimension 4. 1. There is a non-trivial second order symmetry operator of the first kind for the Dirac-Weyl equation if and only if there is a non-zero Killing spinor of valence $(2,2)$ satisfying auxiliary conditions \[point:A0\] and \[point:A1\] of definition \[def:auxcond\]. 2. There is a non-trivial second order symmetry operator of the second kind for the Dirac-Weyl equation if and only if there is a non-zero Killing spinor $L_{ABC}{}^{A'}$ of valence $(3,1)$, such that the auxiliary condition $$\begin{aligned} 0={}&\tfrac{3}{4} \Psi_{ABCD} \nabla^{FA'}L^{CD}{}_{FA'} + \tfrac{5}{6} \Psi_{B}{}^{CDF} \nabla_{(A}{}^{A'}L_{CDF)A'}\nonumber\\& + \tfrac{5}{6} \Psi_{A}{}^{CDF} \nabla_{(B}{}^{A'}L_{CDF)A'} - \tfrac{3}{5} L_{B}{}^{CDA'} \nabla_{(A}{}^{B'}\Phi_{CD)A'B'}\nonumber\\& - \tfrac{3}{5} L_{A}{}^{CDA'} \nabla_{(B}{}^{B'}\Phi_{CD)A'B'} + \tfrac{4}{3} L^{CDFA'} \nabla_{(A|A'|}\Psi_{BCDF)}. \label{eq:A1/2*} \end{aligned}$$ is satisfied. <!-- --> 1. [Under conformal transformations such that $\widehat L^{ABCA'}=L^{ABCA'}$, the equation is conformally invariant. ]{} 2. [We remark that the auxiliary condition \[point:A0\], appears both in Theorem \[thm:intro:spin1/2\], and for the conformal wave equation in Theorem \[thm:intro:spin0\].]{} In previous work, Benn and Kress [@benn:kress:2004CQGra..21..427B] showed that a first order symmetry operator of the second kind for the Dirac equation exists exactly when there is a valence $(2,0)$ Killing spinor. See also Carter and McLenaghan [@carter:mclenaghan:1979PhRvD..19.1093C], and Durand, Lina, and Vinet [@durand:lina:vinet:1988PhRvD..38.3837D] for earlier work. The conditions for the existence of a second order symmetry operator for the Dirac-Weyl equations in a general spacetime were considered in [@mclenaghan:smith:walker:2000RSPSA.456.2629G], see also [@fels:kamran:1990RSPSA.428..229F]. The conditions derived here represent a simplification of the conditions found in [@mclenaghan:smith:walker:2000RSPSA.456.2629G]. Further, we mention that symmetry operators of general order for the Dirac operator on Minkowski space have been analyzed by Michel [@michel:2014arXiv1208.4052]. For the spin-1 case, we similarly have the left and right Maxwell equations \[eq:spin1eqs\] $$\begin{aligned} \nabla^B{}_{A'} \phi_{AB} &= 0, \label{eq:spin1left} \\ \nabla_A{}^{B'} \chi_{A'B'} &= 0 \label{eq:spin1right}\end{aligned}$$ The left-handed and right-handed spinors $\phi_{AB}$, $\chi_{A'B'}$ represent an anti-self-dual and a self-dual 2-form, respectively. Each equation in is thus equivalent to a real Maxwell equation, cf. [@PenRin84 §3.4]. Analogously to the spin-1/2 case, we consider second order symmetry operators of the first and second kind. \[thm:intro:spin1\] Consider the Maxwell equations in a Lorentzian spacetime of dimension 4. 1. \[point:first:maxwell\] There is a non-trivial second order symmetry operator of the first kind for the Maxwell equation if and only if there is a non-zero Killing spinor of valence $(2,2)$ such that the auxiliary condition \[point:A1\] of definition \[def:auxcond\] is satisfied. 2. \[point:second:maxwell\] There is a non-trivial second order symmetry operator of the second kind for the Maxwell equation if and only if there is a non-zero Killing spinor $L_{ABCD}$ of valence $(4,0)$. Note that no auxiliary condition is needed in point *(\[point:second:maxwell\])* of Theorem \[thm:intro:spin1\]. The conditions for the existence of second order symmetry operators for the Maxwell equations have been given in previous work by Kalnins, McLenaghan and Williams [@KalMcLWil92a], see also [@kalnins:mclenaghan:williams:1992grra.conf..129K], following earlier work by Kalnins, Miller and Williams [@kalnins:miller:williams:1989JMP....30.2360K], see also [@kress:thesis]. In [@KalMcLWil92a], the conditions for a second order symmetry operator of the second kind were analyzed completely, and agree with the condition given in point *(\[point:second:maxwell\])* of Theorem \[thm:intro:spin1\]. However, the conditions for a second order symmetry operator of the first kind stated there consist of a set of five equations, of a not particularly transparent nature. The result given here in point *(\[point:first:maxwell\])* of Theorem \[thm:intro:spin1\] provides a substantial simplification and clarification of this previous result. The necessary and sufficient conditions given in theorems \[thm:intro:spin0\], \[thm:intro:spin1/2\], \[thm:intro:spin1\] involve the existence of a Killing spinor and auxiliary conditions. The following result gives examples of Killing spinors for which the auxiliary conditions \[point:A0\], \[point:A1\] and equation are satisfied. \[prop:factorize\] Let $\xi^{AA'}$ and $\zeta^{AA'}$ be (not necessarily distinct) conformal Killing vectors and let $\kappa_{AB}$ be a Killing spinor of valence $(2,0)$. 1. The symmetric spinor $\xi_{(A}{}^{(A'} \zeta_{B)}{}^{B')}$ is a Killing spinor of valence $(2,2)$, which admits solutions to the auxiliary conditions \[point:A0\] and \[point:A1\]. 2. The symmetric spinor $\kappa_{AB}\bar{\kappa}_{A'B'}$ is also a Killing spinor of valence $(2,2)$, which admits solutions to the auxiliary conditions \[point:A0\] and \[point:A1\]. 3. The spinor $\kappa_{(AB}\xi_{C)}{}^{C'}$ is a Killing spinor of valence $(3,1)$, which satisfies auxiliary equation . 4. The spinor $\kappa_{(AB}\kappa_{CD)}$ is a Killing spinor of valence $(4,0)$. \[point:factoredvalence4KS\] The point *(\[point:factoredvalence4KS\])* is immediately clear. The other parts will be proven in Section \[sec:factorizations\]. We now consider the following condition $$\begin{aligned} 0={}&\Psi_{(ABC}{}^{F}\Phi_{D)FA'B'}. \label{eq:intro:AlignmentPsiPhi}\end{aligned}$$ relating the Ricci curvature $\Phi_{ABA'B'}$ and the Weyl curvature $\Psi_{ABCD}$. A spacetime where holds will be said to satisfy the aligned matter condition. In particular this holds in Vacuum and in the Kerr-Newman class of spacetimes. Under the aligned matter condition we can show that the converse of Proposition \[prop:factorize\] part *(\[point:factoredvalence4KS\])* is true. The following theorem will be proved in Section \[sec:factorValence4\]. \[thm:Valence4Factorization\] If the aligned matter condition is satisfied, $\Psi_{ABCD}\neq 0$ and $L_{ABCD}$ is a valence $(4,0)$ Killing spinor, then there is a valence $(2,0)$ Killing spinor $\kappa_{AB}$ such that $$\begin{aligned} L_{ABCD}={}&\kappa_{(AB}\kappa_{CD)}.\end{aligned}$$ If $\Psi_{ABCD}=0$, the valence $(4,0)$ Killing spinor will still factor but in terms of valence $(1,0)$ Killing spinors, which then can be combined into valence $(2,0)$ Killing spinors. However, the two factors might be distinct. A calculation shows that if holds, $\kappa_{AB}$ is a valence $(2,0)$ Killing spinor, then $\xi^{AA'} = \nabla^{BA'} \kappa^{A}{}_{B}$ is a Killing vector field. Taking this fact into account, we have the following corollary to the results stated above. It tells that generically one can generate a wide variety of symmetry operators from just a single valence $(2,0)$ Killing spinor. \[cor:sufficient\] Consider the massless test fields of spins 0, 1/2 and 1 in a Lorentzian spacetime of dimension 4. Assume that there is Killing spinor $\kappa_{AB}$ (not identically zero) of valence $(2,0)$. Then there are non-trivial second order symmetry operators for the massless spin-$s$ field equations for spins 0 and 1, as well as a non-trivial second order symmetry operator of the first kind for the massless spin-$1/2$ field. If, in addition, the aligned matter condition holds, and $\xi_{AA'} = \nabla^{B}{}_{A'} \kappa_{AB}$ is not identically zero, then there is also a non-trivial second order symmetry operators of the second kind for the massless spin-$1/2$ field. We end this introduction by giving a simple form for symmetry operators for the Maxwell equation, generated from a Killing spinor of valence $(2,0)$. \[Thm:SymopMaxwellSimple\] Let $\kappa_{AB}$ be a Killing spinor of valence $(2,0)$ and let $$\begin{aligned} \Theta_{AB}\equiv{}&-2 \kappa_{(A}{}^{C}\phi_{B)C}.\end{aligned}$$ Define the potentials \[eq:intro:ABdef\] $$\begin{aligned} A_{AA'}={}&\bar{\kappa}_{A'}{}^{B'} \nabla_{BB'}\Theta_{A}{}^{B} - \tfrac{1}{3} \Theta_{A}{}^{B} \nabla_{BB'}\bar{\kappa}_{A'}{}^{B'},\\ B_{AA'}={}&\kappa_{A}{}^{B} \nabla_{CA'}\Theta_{B}{}^{C} + \tfrac{1}{3} \Theta_{A}{}^{B} \nabla_{CA'}\kappa_{B}{}^{C} . \end{aligned}$$ Assume that $\phi_{AB}$ is a solution to the Maxwell equation in a Lorentzian spacetime of dimension 4. Let $A_{AA'}, B_{AA'}$ be given by . Then $$\begin{aligned} \chi_{AB}={}&\nabla_{(B}{}^{A'}A_{A)A'},\\ \omega_{A'B'}={}&\nabla^{B}{}_{(A'}B_{|B|B')}\end{aligned}$$ are solutions to the left and right Maxwell equations, respectively. The proof can be found in sections \[sec:symopfirstmaxwellfact\] and \[sec:symopsecondmaxwellfact\]. The general form of the symmetry operators for spins 0, 1/2 and 1 is discussed in detail below. The symmetry operators of the Maxwell equation can in general be written in potential form. See Theorem \[Thm::SymOpFirstKind\] and Theorem \[Thm::SymOpSecondKind\]. The method used in this paper can also be used to show that the symmetry operators $R$-commute with the Dirac and Maxwell equations. Recall that an operator $S$ is said to $R$-commute with a linear PDE $L\phi=0$ if there is an operator $R$ such that $LS=RL$. Even providing a formula for the relevant $R$ operators would require additional notation, so we have omitted this result from this paper. Overview of this paper {#overview-of-this-paper .unnumbered} ---------------------- In Section \[sec:prel\] we define the fundamental operators $\sDiv, \sCurl, \sCurlDagger, \sTwist$ obtained by projecting the covariant derivative of a symmetric spinor on its irreducible parts. These operators are analogues of the Stein-Weiss operators discussed in Riemannian geometry and play a central role in our analysis. We give the commutation properties of these operators, derive the integrability conditions for Killing spinors, and end the section by discussing some aspects of the methods used in the analysis. Section \[sec:spin0\] gives the analysis of symmetry operators for the conformal wave equation. The results here are given for completeness, and agree with those in [@michel:radoux:silhan:2013arXiv1308.1046M] for the case of a Lorentzian spacetime of dimesion 4. The symmetry operators for the Dirac-Weyl equation are discussed in Section \[sec:spin1/2\] and our results for the Maxwell case are given in Section \[sec:spin1\]. Special conditions under which the auxiliary conditions can be solved is discussed in Section \[sec:factorizations\]. Finally, Section \[sec:symopfactored\] contains simplified expressions for the symmetry operators for some of the cases discussed in Section \[sec:factorizations\]. Preliminaries {#sec:prel} ============= Fundamental operators --------------------- Let $S_{k,l}$ denote the vector bundle of symmetric spinors with $k$ unprimed indices and $l$ primed indices. We will call these spinors symmetric valence $(k,l)$ spinors. Furthermore, let $\mathcal{S}_{k,l}$ denote the space of smooth ($C^\infty$) sections of $S_{k,l}$. For any $\varphi_{A_1\dots A_k}{}^{A_{1}'\dots A_{l}'}\in \mathcal{S}_{k,l}$, we define the operators $\sDiv_{k,l}:\mathcal{S}_{k,l}\rightarrow \mathcal{S}_{k-1,l-1}$, $\sCurl_{k,l}:\mathcal{S}_{k,l}\rightarrow \mathcal{S}_{k+1,l-1}$, $\sCurlDagger_{k,l}:\mathcal{S}_{k,l}\rightarrow \mathcal{S}_{k-1,l+1}$ and $\sTwist_{k,l}:\mathcal{S}_{k,l}\rightarrow \mathcal{S}_{k+1,l+1}$ as $$\begin{aligned} (\sDiv_{k,l}\varphi)_{A_1\dots A_{k-1}}{}^{A_1'\dots A_{l-1}'}\equiv{}& \nabla^{BB'}\varphi_{A_1\dots A_{k-1}B}{}^{A_1'\dots A_{l-1}'}{}_{B'},\\ (\sCurl_{k,l}\varphi)_{A_1\dots A_{k+1}}{}^{A_1'\dots A_{l-1}'}\equiv{}& \nabla_{(A_1}{}^{B'}\varphi_{A_2\dots A_{k+1})}{}^{A_1'\dots A_{l-1}'}{}_{B'},\\ (\sCurlDagger_{k,l}\varphi)_{A_1\dots A_{k-1}}{}^{A_1'\dots A_{l+1}'}\equiv{}& \nabla^{B(A_1'}\varphi_{A_1\dots A_{k-1}B}{}^{A_2'\dots A_{l+1}')},\\ (\sTwist_{k,l}\varphi)_{A_1\dots A_{k+1}}{}^{A_1'\dots A_{l+1}'}\equiv{}& \nabla_{(A_1}{}^{(A_1'}\varphi_{A_2\dots A_{k+1})}{}^{A_2'\dots A_{l+1}')}.\end{aligned}$$ 1. [ These operators are all conformally covariant, but the conformal weight differs between the operators. See [@PenRin86 Section 6.7] for details.]{} 2. [The left Dirac-Weyl and Maxwell equations can be written as $(\sCurlDagger_{1,0}\phi)_{A'}=0$ and $(\sCurlDagger_{2,0}\phi)_{AA'}=0$ respectively. Similarly the right equations can be written in terms of the $\sCurl$ operator. ]{} The operator $\sDiv_{k,l}$ only makes sense when $k\geq 1$ and $l\geq 1$. Likewise $\sCurl_{k,l}$ is defined only if $l \geq 1$ and $\sCurlDagger_{k,l}$ only if $k\geq 1$. To make a clean presentation, we will use formulae where invalid operators appear for some choices of $k$ and $l$. However, the operators will always be multiplied with a factor that vanishes for these invalid choices of $k$ and $l$. From the definition it is clear that the complex conjugates of $(\sDiv_{k,l}\varphi)$, $(\sCurl_{k,l}\varphi)$, $(\sCurlDagger_{k,l}\varphi)$ and $(\sTwist_{k,l}\varphi)$ are $(\sDiv_{l,k}\bar\varphi)$, $(\sCurlDagger_{l,k}\bar\varphi)$, $(\sCurl_{l,k}\bar\varphi)$ and $(\sTwist_{l,k}\bar\varphi)$ respectively, with the appropriate indices. The main motivation for the introduction of these operators is the irreducible decomposition of the covariant derivative of a symmetric spinor field. For any $\varphi_{A_1\dots A_k}{}^{A_{1}'\dots A_{l}'}\in \mathcal{S}_{k,l}$, we have the irreducible decomposition $$\begin{aligned} \nabla_{A_1}{}^{A_1'}\varphi{}_{A_2\dots A_{k+1}}{}^{A_2'\dots A_{l+1}'}={}& (\sTwist_{k,l}\varphi){}_{A_1\dots A_{k+1}}{}^{A_1'\dots A_{l+1}'}\nonumber\\ &-\tfrac{l}{l+1}\bar\epsilon^{A_1'(A_2'}(\sCurl_{k,l}\varphi){}_{A_1\dots A_{k+1}}{}^{A_3'\dots A_{l+1}')}\nonumber\\ &-\tfrac{k}{k+1}\epsilon_{A_1(A_2}(\sCurlDagger_{k,l}\varphi){}_{A_3\dots A_{k+1})}{}^{A_1'\dots A_{l+1}'}\nonumber\\ &+\tfrac{kl}{(k+1)(l+1)}\epsilon_{A_1(A_2}\bar\epsilon^{A_1'(A_2'}(\sDiv_{k,l}\varphi){}_{A_3\dots A_{k+1})}{}^{A_3'\dots A_{l+1}')}.\label{eq:IrrDecGeneralDer}\end{aligned}$$ It follows from in [@PenRin84 Proposition 3.3.54] that the irreducible decomposition must have this form. The coefficients are then found by contracting indices and partially expanding the symmetries. With this notation, the Bianchi system takes the form $$\begin{aligned} (\sDiv_{2,2} \Phi)_{AA'}={}&-3 (\sTwist_{0,0} \Lambda)_{AA'},\\ (\sCurlDagger_{4,0} \Psi)_{ABCA'}={}&(\sCurl_{2,2} \Phi)_{ABCA'}.\end{aligned}$$ In the rest of the paper we will use these equations every time the left hand sides appear in the calculations. With the definitions above, a Killing spinor of valence $(k,l)$ is an element $L_{A\cdots F}{}^{ A'\dots F'} \in \ker \sTwist_{k,l}$, a conformal Killing vector is a Killing spinor of valence $(1,1)$, and a trace-less conformal Killing tensor is a Killing spinor of valence $(2,2)$. We further introduce the following operators, acting on a valence $(2,2)$ Killing spinor. For $L_{AB}{}^{ A'B'} \in \ker \sTwist_{2,2}$, define $$\begin{aligned} (\ObstrZero L)_A{}^{A'} \equiv {}&\tfrac{1}{3} \Psi_{ABCD} (\sCurl_{2,2} L)^{BCDA'} + L^{BCA'B'} (\sCurl_{2,2} \Phi)_{ABCB'}\nonumber\\ & + \tfrac{1}{3} \bar\Psi^{A'}{}_{B'C'D'} (\sCurlDagger_{2,2} L)_{A}{}^{B'C'D'} + L_{A}{}^{BB'C'} (\sCurlDagger_{2,2} \Phi)_{B}{}^{A'}{}_{B'C'}. \\ (\ObstrOne L)_{AB}{}^{A'B'} \equiv {}&L{}^{CDA'B'} \Psi_{ABCD} - L{}_{AB}{}^{C'D'} \bar\Psi^{A'B'}{}_{C'D'} \end{aligned}$$ The operators $\ObstrZero$ and $\ObstrOne$ are the right hand sides of and in conditions \[point:A0\] and \[point:A1\] respectively. They will play an important role in the conditions for the existence of symmetry operators. Given a conformal Killing vector $\xi^{AA'}$, we follow [@MR2056970 Equations (2) and (15)], see also [@anco:pohjanpelto:2003RSPSA.459.1215A], and define a conformally weighted Lie derivative acting on a symmetric valance $(2s,0)$ spinor field as follows For $\xi^{AA'} \in \ker \sTwist_{1,1}$, and $\varphi_{A_1\dots A_{2s}}\in \mathcal{S}_{2s,0}$, we define $$\begin{aligned} \hat{\mathcal{L}}_{\xi}\varphi_{A_1\dots A_{2s}}\equiv{}&\xi^{BB'} \nabla_{BB'}\varphi_{A_1\dots A_{2s}}+s \varphi_{B(A_2\dots A_{2s}} \nabla_{A_1)B'}\xi^{BB'} + \tfrac{1-s}{4} \varphi_{A_1\dots A_{2s}} \nabla^{CC'}\xi_{CC'}.\end{aligned}$$ This operator turns out to be important when we describe first order symmetry operators. See Section \[sec:symopsecondDiracfactored\] for further discussion. Commutator relations -------------------- Let $\varphi_{A_1\dots A_k}{}^{A_{1}'\dots A_{l}'}\in \mathcal{S}_{k,l}$ and define the standard commutators $$\square_{AB}\equiv \nabla_{(A|A'|}\nabla_{B)}{}^{A'} \qquad \text{ and }\qquad \square_{A'B'}\equiv \nabla_{A(A'}\nabla^{A}{}_{B')}.$$ Acting on spinors, these commutators can always be written in terms of curvature spinors as described in [@PenRin84 Section 4.9]. \[lemma:commutators\] The operators $\sDiv$, $\sCurl$, $\sCurlDagger$ and $\sTwist$ satisfies the following commutator relations $$\begin{aligned} (\sDiv_{k+1,l-1} &\sCurl_{k,l} \varphi){}_{A_1\dots A_{k}}{}^{A_1'\dots A_{l-2}'}\nonumber\\* ={}&\tfrac{k}{k+1}(\sCurl_{k-1,l-1} \sDiv_{k,l} \varphi){}_{A_1\dots A_{k}}{}^{A_1'\dots A_{l-2}'} -\square_{B'C'}\varphi{}_{A_1\dots A_{k}}{}^{A_1'\dots A_{l-2}'B'C'}, \quad k\geq 0, l\geq 2, \label{eq:DivCurl}\\ (\sDiv_{k-1,l+1} &\sCurlDagger_{k,l} \varphi){}_{A_1\dots A_{k-2}}{}^{A_1'\dots A_{l}'}\nonumber\\* ={}&\tfrac{l}{l+1}(\sCurlDagger_{k-1,l-1} \sDiv_{k,l} \varphi){}_{A_1\dots A_{k-2}}{}^{A_1'\dots A_{l}'} -\square_{BC}\varphi{}_{A_1\dots A_{k-2}}{}^{BCA_1'\dots A_{l}'}, \quad k\geq 2, l\geq 0,\label{eq:DivCurlDagger}\\ (\sCurl_{k+1,l+1}&\sTwist_{k,l} \varphi){}_{A_1\dots A_{k+2}}{}^{A_1'\dots A_{l}'}\nonumber\\* ={}&\tfrac{l}{l+1}(\sTwist_{k+1,l-1} \sCurl_{k,l} \varphi){}_{A_1\dots A_{k+2}}{}^{A_1'\dots A_{l}'} -\square_{(A_1A_2}\varphi{}_{A_3\dots A_{k+2})}{}^{A_1'\dots A_{l}'}, \quad k\geq 0, l\geq 0,\label{eq:CurlTwist}\\ (\sCurlDagger_{k+1,l+1}&\sTwist_{k,l}\varphi){}_{A_1\dots A_{k}}{}^{A_1'\dots A_{l+2}'}\nonumber\\* ={}& \tfrac{k}{k+1}(\sTwist_{k-1,l+1} \sCurlDagger_{k,l} \varphi){}_{A_1\dots A_{k}}{}^{A_1'\dots A_{l+2}'} -\square^{(A_1'A_2'}\varphi{}_{A_1\dots A_{k}}{}^{A_3'\dots A_{l+2}')}, \quad k\geq 0, l\geq 0,\label{eq:CurlDaggerTwist}\\ (\sDiv_{k+1,l+1}&\sTwist_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}\nonumber\\* ={}& -(\tfrac{1}{k+1}+\tfrac{1}{l+1})(\sCurl_{k-1,l+1}\sCurlDagger_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'} +\tfrac{l(l+2)}{(l+1)^2}(\sTwist_{k-1,l-1}\sDiv_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}\nonumber\\* &-\tfrac{l+2}{l+1}\square^B{}_{(A_1}\varphi_{A_2\dots A_k)B}{}^{A_1'\dots A_l'} -\tfrac{l}{l+1}\square^{B'(A_1'}\varphi_{A_1\dots A_k}{}^{A_2'\dots A_l')}{}_{B'}, \quad k\geq 1, l\geq 0,\label{eq:DivTwistCurlCurlDagger}\\ (\sDiv_{k+1,l+1}&\sTwist_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}\nonumber\\* ={}& -(\tfrac{1}{k+1}+\tfrac{1}{l+1})(\sCurlDagger_{k+1,l-1}\sCurl_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'} +\tfrac{k(k+2)}{(k+1)^2}(\sTwist_{k-1,l-1}\sDiv_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}\nonumber\\* &-\tfrac{k}{k+1}\square^B{}_{(A_1}\varphi_{A_2\dots A_k)B}{}^{A_1'\dots A_l'} -\tfrac{k+2}{k+1}\square^{B'(A_1'}\varphi_{A_1\dots A_k}{}^{A_2'\dots A_l')}{}_{B'}, \quad k\geq 0, l\geq 1,\label{eq:DivTwistCurlDaggerCurl}\\ (\sCurl_{k-1,l+1}&\sCurlDagger_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}\nonumber\\* ={}& (\sCurlDagger_{k+1,l-1}\sCurl_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'} +(\tfrac{1}{k+1}-\tfrac{1}{l+1})(\sTwist_{k-1,l-1}\sDiv_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}\nonumber\\* &-\square_{(A_1}{}^{B}\varphi_{A_2\dots A_k)B}{}^{A_1'\dots A_l'} +\square^{B'(A_1'}\varphi_{A_1\dots A_k}{}^{A_2'\dots A_l')}{}_{B'}, \quad k\geq 1, l\geq 1.\label{eq:CurlCurlDagger}\end{aligned}$$ We first observe that and are related by complex conjugation. Likewise and as well as and are also related by complex conjugation. Furthermore, is given by the difference between and . It is therefore enough to prove , and . We consider each in turn. - [We first prove . We partially expand the symmetry, identify the commutator in one term, and commute derivatives in the other. $$\begin{aligned} (\sDiv_{k+1,l-1}& \sCurl_{k,l} \varphi){}_{A_1\dots A_{k}}{}^{A_1'\dots A_{l-2}'}\\ ={}&\nabla^{BB'}\nabla_{(A_1}{}^{C'}\varphi_{A_2\dots A_k B)}{}^{A_1'\dots A_{l-2}'}{}_{B'C'}\\ ={}&\tfrac{1}{k+1}\nabla^{B(B'}\nabla_{B}{}^{C')}\varphi_{A_1\dots A_k}{}^{A_1'\dots A_{l-2}'}{}_{B'C'} +\tfrac{k}{k+1}\nabla^{B(B'}\nabla_{(A_1}{}^{C')}\varphi_{A_2\dots A_k) B}{}^{A_1'\dots A_{l-2}'}{}_{B'C'}\\ ={}&-\tfrac{1}{k+1}\square^{B'C'}\varphi_{A_1\dots A_k}{}^{A_1'\dots A_{l-2}'}{}_{B'C'} +\tfrac{k}{k+1}\epsilon^B{}_{(A_1}\square^{B'C'}\varphi_{A_2\dots A_k) B}{}^{A_1'\dots A_{l-2}'}{}_{B'C'}\\ &+\tfrac{k}{k+1}\nabla_{(A_1}{}^{C'}\nabla^{BB'}\varphi_{A_2\dots A_k) B}{}^{A_1'\dots A_{l-2}'}{}_{B'C'}\\ ={}& \tfrac{k}{k+1}(\sCurl_{k-1,l-1} \sDiv_{k,l} \varphi){}_{A_1\dots A_{k}}{}^{A_1'\dots A_{l-2}'} -\square_{B'C'}\varphi{}_{A_1\dots A_{k}}{}^{A_1'\dots A_{l-2}'B'C'}.\end{aligned}$$ ]{} - [ To prove , we first partially expand the symmetrization over the unprimed indices in the irreducible decomposition and symmetrizing over the primed indices. This gives $$\begin{aligned} \nabla_{A_1}{}^{(A_2'}\varphi_{A_2\dots A_k B}{}^{A_3'\dots A_{l+2}')}={}& (\sTwist_{k,l}\varphi){}_{A_1\dots A_{k}B}{}^{A_2'\dots A_{l+2}'} -\tfrac{1}{k+1}\epsilon_{A_1B}(\sCurlDagger_{k,l}\varphi){}_{A_2\dots A_{k}}{}^{A_2'\dots A_{l+2}'}\nonumber\\ &-\tfrac{k-1}{k+1}\epsilon_{A_1(A_2}(\sCurlDagger_{k,l}\varphi){}_{A_3\dots A_{k})B}{}^{A_2'\dots A_{l+2}'}.\label{eq:Irrdecvarphihelp}\end{aligned}$$ Using the definitions of $\sTwist$ and $\sCurlDagger$, commuting derivatives and using , we have $$\begin{aligned} (\sTwist_{k-1,l+1} \sCurlDagger_{k,l} \varphi){}_{A_1\dots A_{k}}{}^{A_1'\dots A_{l+2}'} ={}& \nabla_{(A_1}{}^{(A_1'}\nabla^{|B|A_2'}\varphi_{A_2\dots A_{k})B}{}^{A_3'\dots A_{l+2}')} \nonumber\\ ={}& \square^{(A_1'A_2'}\varphi{}_{A_1\dots A_{k}}{}^{A_3'\dots A_{l}')} +\nabla^{B(A_1'}\nabla_{(A_1}{}^{A_2'}\varphi_{A_2\dots A_{k})B}{}^{A_3'\dots A_{l+2}')}\nonumber\\ ={}& \square^{(A_1'A_2'}\varphi{}_{A_1\dots A_{k}}{}^{A_3'\dots A_{l}')} +\nabla^{B(A_1'}(\sTwist_{k,l}\varphi){}_{A_1\dots A_{k}B}{}^{A_2'\dots A_{l+2}')}\nonumber\\ &-\tfrac{1}{k+1}\epsilon_{(A_1|B|}\nabla^{B(A_1'}(\sCurlDagger_{k,l}\varphi){}_{A_2\dots A_{k})}{}^{A_2'\dots A_{l+2}')}\nonumber\\ ={}&\square^{(A_1'A_2'}\varphi{}_{A_1\dots A_{k}}{}^{A_3'\dots A_{l}')} +(\sCurlDagger_{k+1,l+1}\sTwist_{k,l}\varphi){}_{A_1\dots A_{k}}{}^{A_1'\dots A_{l+2}'}\nonumber\\ &+\tfrac{1}{k+1}(\sTwist_{k-1,l+1}\sCurlDagger_{k,l}\varphi){}_{A_1\dots A_{k}}{}^{A_1'\dots A_{l+2}'}.\end{aligned}$$ Isolating the $\sCurlDagger\sTwist$-term gives . ]{} - [ Finally to prove , we assume $k\geq 1$ and observe $$\begin{aligned} (\sDiv_{k+1,l+1}&\sTwist_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}\nonumber\\ ={}&-\nabla^B{}_{B'}\nabla_{(B}{}^{(B'}\varphi_{A_1\dots A_k)}{}^{A_1'\dots A_l')}\nonumber\\ ={}& -\tfrac{1}{k+1}\nabla^B{}_{B'}\nabla_{B}{}^{(B'}\varphi_{A_1\dots A_k}{}^{A_1'\dots A_l')} -\tfrac{k}{k+1}\nabla^B{}_{B'}\nabla_{(A_1}{}^{(B'}\varphi_{A_2\dots A_k)B}{}^{A_1'\dots A_l')}\nonumber\\ ={}& \tfrac{1}{k+1}(\sDiv_{k+1,l+1}\sTwist_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'} -\tfrac{k}{(k+1)^2}(\sCurl_{k-1,l+1}\sCurlDagger_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}\nonumber\\ &-\tfrac{k}{k+1}\nabla^B{}_{B'}\nabla_{(A_1}{}^{(B'}\varphi_{A_2\dots A_k)B}{}^{A_1'\dots A_l')},\end{aligned}$$ where we in the last step used the irreducible decomposition on the first term. We can solve for the $\sDiv\sTwist$-term from which it follows that $$\begin{aligned} (\sDiv_{k+1,l+1}&\sTwist_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}\nonumber\\ ={}& -\tfrac{1}{k+1}(\sCurl_{k-1,l+1}\sCurlDagger_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'} -\nabla^B{}_{B'}\nabla_{(A_1}{}^{(B'}\varphi_{A_2\dots A_k)B}{}^{A_1'\dots A_l')}\nonumber\\ ={}& -\tfrac{1}{k+1}(\sCurl_{k-1,l+1}\sCurlDagger_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'} -\tfrac{1}{l+1}\nabla^B{}_{B'}\nabla_{(A_1}{}^{B'}\varphi_{A_2\dots A_k)B}{}^{A_1'\dots A_l'}\nonumber\\ &-\tfrac{l}{l+1}\nabla^B{}_{B'}\nabla_{(A_1}{}^{(A_1'}\varphi_{A_2\dots A_k)B}{}^{A_2'\dots A_l')B'} \nonumber\\ ={}& -\tfrac{1}{k+1}(\sCurl_{k-1,l+1}\sCurlDagger_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'} -\tfrac{1}{l+1}\nabla_{(A_1}{}^{B'}\nabla^{B}{}_{|B'|}\varphi_{A_2\dots A_k)B}{}^{A_1'\dots A_l'}\nonumber\\ & -\tfrac{2}{l+1}\square^B{}_{(A_1}\varphi_{A_2\dots A_k)B}{}^{A_1'\dots A_l'} -\tfrac{l}{l+1}\nabla_{(A_1}{}^{(A_1'}\nabla^{|B|}{}_{|B'|}\varphi_{A_2\dots A_k)B}{}^{A_2'\dots A_l')B'}\nonumber\\ &-\tfrac{l}{l+1}\square^B{}_{(A_1}\varphi_{A_2\dots A_k)B}{}^{A_1'\dots A_l'} -\tfrac{l}{l+1}\square^{B'(A_1'}\varphi_{A_1\dots A_k}{}^{A_2'\dots A_l')}{}_{B'} \nonumber\\ ={}& -(\tfrac{1}{k+1}+\tfrac{1}{l+1})(\sCurl_{k-1,l+1}\sCurlDagger_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'} +\tfrac{l(l+2)}{(l+1)^2}(\sTwist_{k-1,l-1}\sDiv_{k,l}\varphi)_{A_1\dots A_k}{}^{A_1'\dots A_l'}\nonumber\\ &-\tfrac{l+2}{l+1}\square^B{}_{(A_1}\varphi_{A_2\dots A_k)B}{}^{A_1'\dots A_l'} -\tfrac{l}{l+1}\square^{B'(A_1'}\varphi_{A_1\dots A_k}{}^{A_2'\dots A_l')}{}_{B'}.\end{aligned}$$ ]{} The operators $\sDiv$, $\sCurl$, $\sCurlDagger$ and $\sTwist$ together with the irreducible decomposition and the relations in Lemma \[lemma:commutators\] have all been implemented in the *SymManipulator* package version 0.9.0 [@Bae11a]. Integrability conditions for Killing spinors {#sec:integrabilitycond} -------------------------------------------- Here we demonstrate a procedure for obtaining an integrability condition for a Killing spinor of arbitrary valence. Let $\kappa_{A_1\dots A_k}{}^{A'_1\dots A'_l}\in \ker \sTwist_{k,l}$. By applying the $\sCurl$ operator $l+1$ times to the Killing spinor equation, and repeatedly commute derivatives with we get $$\begin{aligned} 0={}& (\underbrace{\sCurl_{k+l+1,1}\sCurl_{k+l,2} \cdots \sCurl_{k+2,l}\sCurl_{k+1,l+1}}_{l+1}\sTwist_{k,l}\kappa)_{A_{1}\dots A_{k+l+2}}\nonumber\\ ={}&\frac{l}{l+1}(\sCurl_{k+l+1,1}\sCurl_{k+l,2} \cdots \sCurl_{k+2,l}\sTwist_{k+1,l-1}\sCurl_{k,l}\kappa)_{A_{1}\dots A_{k+l+2}}+\text{curvature terms}\nonumber\\ ={}&\frac{1}{l+1}(\sCurl_{k+l+1,1}\sTwist_{k+l,0}\sCurl_{k+l-1,1} \cdots \sCurl_{k+1,l-1}\sCurl_{k,l}\kappa)_{A_{1}\dots A_{k+l+2}}+\text{curvature terms}\nonumber\\ ={}&\text{curvature terms}.\end{aligned}$$ Here, the curvature terms have $l-m$ derivatives of $\kappa$ and $m$ derivatives of the curvature spinors, where $0\leq m\leq l$. The main idea behind this is the observation that the commutator acting on a spinor field without primed indices only gives curvature terms. In the same way we can use to get $$\begin{aligned} 0={}& (\underbrace{\sCurlDagger_{1,k+l+1}\sCurlDagger_{2,k+l} \cdots \sCurlDagger_{k,l+2}\sCurlDagger_{k+1,l+1}}_{k+1}\sTwist_{k,l}\kappa)_{A'_{1}\dots A'_{k+l+2}}\nonumber\\ ={}&\text{curvature terms}.\end{aligned}$$ Splitting equations into independent parts {#sec:indepspinors} ------------------------------------------ In our derivation of necessary conditions for the existence of symmetry operators, it is crucial that, at each fixed point in spacetime, we can freely choose the values of the Dirac-Weyl and the Maxwell field and of the symmetric components of any given order of their derivatives. The remaining components of the derivatives to a given order, which involve at least one pair of antisymmetrized indices, can be solved for using the field equations or curvature conditions. See sections \[sec:ReductionDirac\] and \[sec:ReductionMaxwell\] for detailed expressions. In the literature, the condition that the symmetric components can be freely and independently specified but that no other parts can be is referred to as the exactness of the set of fields [@PenRin84 Section 5.10]. The symmetric components of the derivatives are exactly those that can be expressed in terms of the operator $\sTwist$. One can show that, in a globally hyperbolic spacetime, the Dirac-Weyl and Maxwell fields each form exact sets. However, it is not necessary for the spacetime to be globally hyperbolic for this condition to hold. If the spacetime is such that the fields fail to form an exact set, then our methods still give sufficient conditions for the existence of symmetry operators, but they may no longer be necessary. The freedom to choose the symmetric components is used in this paper to show that equations of the type $L^{ABA'}(\sTwist_{1,0}\phi)_{ABA'}+M^{A}\phi_{A}=0$ with $(\sCurl_{1,0}^\dagger\phi)_{A'}=0$ forces $L^{(AB)A'}=0$ and $M^A=0$ because $(\sTwist_{1,0}\phi)_{ABA'}$ and $\phi_A$ can be freely and independently specified at a single point. Similar arguments involving derivatives of up to third order are also used. In several places we will have equations of the form $$\begin{aligned} \label{eq:ProductExample} 0=S^{ABC}{}_{A'}(\sTwist_{1,0}\phi)_{AB}{}^{A'}T_{C},\end{aligned}$$ where $T_{A}$ and $(\sTwist_{1,0}\phi)_{ABA'}$ are free and independent. In particular all linear combinations of the form $(\sTwist_{1,0}\phi)_{AB}{}^{A'}T_{C}$ will then span the space of spinors $W_{ABC}{}^{A'}=W_{(AB)C}{}^{A'}$. As the equation is linear we therefore get $$\begin{aligned} 0=S^{ABC}{}_{A'}W_{ABC}{}^{A'},\end{aligned}$$ for all $W_{ABC}{}^{A'}=W_{(AB)C}{}^{A'}$. We can then make an irreducible decomposition $$\begin{aligned} W_{ABC}{}^{A'}={}&W_{(ABC)}{}^{A'} - \tfrac{2}{3} W_{(A}{}^{D}{}_{|D|}{}^{A'}\epsilon_{B)C},\end{aligned}$$ which gives $$\begin{aligned} 0={}&(- \tfrac{1}{3} S_{B}{}^{C}{}_{CA'} - \tfrac{1}{3} S^{C}{}_{BCA'}) W^{BA}{}_{A}{}^{A'} - S_{ABCA'} W^{(ABC)A'}.\end{aligned}$$ As $W_{ABC}{}^{A'}$ is free, its irreducible components $W_{(ABC)}{}^{A'}$ and $W_{A}{}^{D}{}_{D}{}^{A'}$ are free and independent. We can therefore conclude that $$\begin{aligned} 0={}&S_{B}{}^{C}{}_{CA'}+ S^{C}{}_{BCA'},\\ 0={}& S_{(ABC)A'}.\end{aligned}$$ Observe that we only get the symmetric part in the last equation due to the symmetry of $W_{(ABC)}{}^{A'}$. Instead of introducing a new spinor $W_{ABC}{}^{A'}$ we will in the rest of the paper work directly with the irreducible decomposition of $(\sTwist_{1,0}\phi)_{AB}{}^{A'}T_{C}$ and get $$\begin{aligned} 0={}& (- \tfrac{1}{3} S_{A}{}^{C}{}_{CA'} - \tfrac{1}{3} S^{C}{}_{ACA'}) T_{B} (\sTwist_{1,0} \phi)^{ABA'} - S_{ABCA'} T^{(A}(\sTwist_{1,0} \phi)^{BC)A'}.\end{aligned}$$ The formal computations will be the same, and by the argument above, the symmetrized coefficients for the irreducible parts $T_{B} (\sTwist_{1,0} \phi)^{ABA'}$ and $T^{(A}(\sTwist_{1,0} \phi)^{BC)A'}$ will individually have to vanish. The conformal wave equation {#sec:spin0} =========================== For completeness we give here a detailed description of the symmetry operators for the conformal wave equation. The equation $$\begin{aligned} (\square+4 \Lambda) \phi={}&0,\end{aligned}$$ has a symmetry operator $\phi \rightarrow \chi$ , with order less or equal to two, if and only if there are spinors $L_{AB}{}^{A'B'}=L_{(AB)}{}^{(A'B')}$, $P_{AA'}$ and $Q$ such that $$\begin{aligned} (\sTwist_{2,2} L)_{ABC}{}^{A'B'C'}={}&0,\\ (\sTwist_{1,1} P)_{AB}{}^{A'B'}={}&0,\\ (\sTwist_{0,0} Q)_{A}{}^{A'}={}&\tfrac{2}{5}(\ObstrZero L)_A{}^{A'}.\label{eq:auxcondwave}\end{aligned}$$ The symmetry operator then takes the form $$\begin{aligned} \chi={}&- \tfrac{3}{5} L^{ABA'B'} \Phi_{ABA'B'} \phi + Q \phi + \tfrac{1}{4} \phi (\sDiv_{1,1} P) + \tfrac{1}{15} \phi (\sDiv_{1,1} \sDiv_{2,2} L) + P^{AA'} (\sTwist_{0,0} \phi)_{AA'}\nonumber\\ & + \tfrac{2}{3} (\sDiv_{2,2} L)^{AA'} (\sTwist_{0,0} \phi)_{AA'} + L^{ABA'B'} (\sTwist_{1,1} \sTwist_{0,0} \phi)_{ABA'B'}.\label{eq:wavesymop1}\end{aligned}$$ The existence of $Q$ satisfying is exactly the auxiliary condition \[point:A0\]. The proof can also be carried out using the same technique as in the rest of the paper. The Dirac-Weyl equation {#sec:spin1/2} ======================= The following theorems imply Theorem \[thm:intro:spin1/2\]. \[Thm::SymOpFirstKindDirac\] There exists a symmetry operator of the first kind for the Dirac-Weyl equation $\phi_{A}\rightarrow \chi_{A}$, with order less or equal to two, if and only if there are spinor fields $L_{AB}{}^{A'B'}=L_{(AB)}{}^{(A'B')}$, $P_{AA'}$ and $Q$ such that $$\begin{aligned} (\sTwist_{2,2} L)_{ABC}{}^{A'B'C'}={}&0,\\ (\sTwist_{1,1} P)_{AB}{}^{A'B'}={}&- \tfrac{1}{3} (\ObstrOne L)_{AB}{}^{A'B'},\label{eq:ObstrDiracSecond1}\\ (\sTwist_{0,0} Q)_{A}{}^{A'}={}&\tfrac{3}{10} (\ObstrZero L)_{A}{}^{A'}.\label{eq:ObstrDiracSecond2}\end{aligned}$$ The symmetry operator then takes the form $$\begin{aligned} \chi_{A}={}&- \tfrac{8}{15} L^{BCA'B'} \Phi_{BCA'B'} \phi_{A} + Q \phi_{A} + \tfrac{1}{2} \phi^{B} (\sCurl_{1,1} P)_{AB} + \tfrac{2}{9} \phi^{B} (\sCurl_{1,1} \sDiv_{2,2} L)_{AB} + \tfrac{3}{8} \phi_{A} (\sDiv_{1,1} P)\nonumber\\ & + \tfrac{2}{15} \phi_{A} (\sDiv_{1,1} \sDiv_{2,2} L) + P^{BA'} (\sTwist_{1,0} \phi)_{ABA'} + \tfrac{8}{9} (\sDiv_{2,2} L)^{BA'} (\sTwist_{1,0} \phi)_{ABA'}\nonumber\\ & + \tfrac{2}{3} (\sCurl_{2,2} L)_{ABCA'} (\sTwist_{1,0} \phi)^{BCA'} + L^{BCA'B'} (\sTwist_{2,1} \sTwist_{1,0} \phi)_{ABCA'B'}.\label{eq:diracsymop1}\end{aligned}$$ 1. [ Observe that is the auxiliary condition \[point:A1\] for existence of a symmetry operator of the first kind for Maxwell equation, and is the auxiliary condition \[point:A0\] for existence of a symmetry operator for the conformal wave equation. ]{} 2. [ With $L_{ABA'B'}=0$ the first order operator takes the form $$\begin{aligned} \chi_{A}={}& \hat{\mathcal{L}}_{P}\phi_{A}+ Q \phi_{A}.\end{aligned}$$ ]{} \[Thm::SymOpSecondKindDirac\] There exists a symmetry operator of the second kind for the Dirac-Weyl equation $\phi_{A}\rightarrow \omega_{A'}$, with order less or equal to two, if and only if there are spinor fields $L_{ABC}{}^{A'}=L_{(ABC)}{}^{A'}$ and $P_{AB}=P_{(AB)}$ such that $$\begin{aligned} (\sTwist_{3,1} L)_{ABCD}{}^{A'B'}={}&0,\label{eq:TwistL31}\\ (\sTwist_{2,0} P)_{ABC}{}^{A'}={}&0,\label{eq:TwistPDirac2}\\ 0={}&- \tfrac{9}{8} \Psi_{ABCD} (\sDiv_{3,1} L)^{CD} + \tfrac{9}{5} L_{(A}{}^{CDA'}(\sCurl_{2,2} \Phi)_{B)CDA'}\nonumber\\ & - \tfrac{5}{2} \Psi_{(A}{}^{CDF}(\sCurl_{3,1} L)_{B)CDF} - 2 L^{CDFA'} (\sTwist_{4,0} \Psi)_{ABCDFA'}. \label{eq:DiracSecondKindObstr}\end{aligned}$$ The operator takes the form $$\begin{aligned} \omega_{A'}={}&- \tfrac{1}{2} L_{BCDB'} \Phi^{CD}{}_{A'}{}^{B'} \phi^{B} + \tfrac{2}{3} \phi^{B} (\sCurlDagger_{2,0} P)_{BA'} + \tfrac{1}{4} \phi^{B} (\sCurlDagger_{2,0} \sDiv_{3,1} L)_{BA'} + P^{BC} (\sTwist_{1,0} \phi)_{BCA'}\nonumber\\ & + \tfrac{3}{4} (\sDiv_{3,1} L)^{BC} (\sTwist_{1,0} \phi)_{BCA'} + \tfrac{3}{4} (\sCurlDagger_{3,1} L)_{BCA'B'} (\sTwist_{1,0} \phi)^{BCB'} + L^{BCDB'} (\sTwist_{2,1} \sTwist_{1,0} \phi)_{BCDA'B'}.\end{aligned}$$ The scheme for deriving integrability conditions in Section \[sec:integrabilitycond\] can be used to show that $$\begin{aligned} 0={}&- \tfrac{2}{5} L_{(ABC}{}^{A'}(\sCurl_{2,2} \Phi)_{DFH)A'} + 3 L_{(AB}{}^{LA'}(\sTwist_{4,0} \Psi)_{CDFH)LA'} + 5 \Psi_{(ABC}{}^{L}(\sCurl_{3,1} L)_{DFH)L}\nonumber\\ & + \tfrac{3}{4} \Psi_{(ABCD}(\sDiv_{3,1} L)_{FH)},\end{aligned}$$ follows from . Despite the superficial similarity of this equation to the condition , we conjecture that does not follow from . Reduction of derivatives of the field {#sec:ReductionDirac} ------------------------------------- In our notation, the Dirac-Weyl equation $\nabla^{A}{}_{A'}\phi_{A}=0$, takes the form $(\sCurlDagger_{1,0} \phi)_{A'}=0$. We see that the only remaining irreducible part of $\nabla_{A}{}^{A'}\phi_{B}$ is $(\sTwist_{1,0} \phi)_{AB}{}^{A'}$. By commuting derivatives we see that all higher order derivatives of $\phi_{A}$ can be reduced to totally symmetrized derivatives and lower order terms consisting of curvature times lower order symmetrized derivatives. Together with the Dirac-Weyl equation, the commutators , , give $$\begin{aligned} (\sDiv_{2,1} \sTwist_{1,0} \phi)_{A}={}&-6 \Lambda \phi_{A},\\ (\sCurl_{2,1} \sTwist_{1,0} \phi)_{ABC}={}&- \Psi_{ABCD} \phi^{D},\\ (\sCurlDagger_{2,1} \sTwist_{1,0} \phi)_{AA'B'}={}&- \Phi_{ABA'B'} \phi^{B}.\end{aligned}$$ The higher order derivatives can be computed by using the commutators , , together with the equations above and the Bianchi system to get $$\begin{aligned} (\sDiv_{3,2} \sTwist_{2,1} \sTwist_{1,0} \phi)_{AB}{}^{A'}={}&\tfrac{5}{6} \phi^{C} (\sCurl_{2,2} \Phi)_{ABC}{}^{A'} + \tfrac{10}{3} \Phi_{(A}{}^{CA'B'}(\sTwist_{1,0} \phi)_{B)CB'} - \tfrac{16}{3} \phi_{(A}(\sTwist_{0,0} \Lambda)_{B)}{}^{A'}\nonumber\\ & - 12 \Lambda (\sTwist_{1,0} \phi)_{AB}{}^{A'} + \tfrac{3}{2} \Psi_{ABCD} (\sTwist_{1,0} \phi)^{CDA'},\\ (\sCurl_{3,2} \sTwist_{2,1} \sTwist_{1,0} \phi)_{ABCD}{}^{A'}={}&\Phi_{(AB}{}^{A'B'}(\sTwist_{1,0} \phi)_{CD)B'} + \tfrac{5}{2} \Psi_{(ABC}{}^{F}(\sTwist_{1,0} \phi)_{D)F}{}^{A'}\nonumber\\ & - \tfrac{1}{10} \phi_{(A}(\sCurl_{2,2} \Phi)_{BCD)}{}^{A'} - \tfrac{1}{2} \phi^{F} (\sTwist_{4,0} \Psi)_{ABCDF}{}^{A'},\\ (\sCurlDagger_{3,2} \sTwist_{2,1} \sTwist_{1,0} \phi)_{AB}{}^{A'B'C'}={}&\tfrac{8}{3} \Phi^{C}{}_{(A}{}^{(A'B'}(\sTwist_{1,0} \phi)_{B)C}{}^{C')} - \tfrac{2}{9} \phi_{(A}(\sCurlDagger_{2,2} \Phi)_{B)}{}^{A'B'C'}\nonumber\\ & - \tfrac{2}{3} \phi^{C} (\sTwist_{2,2} \Phi)_{ABC}{}^{A'B'C'} - \bar\Psi^{A'B'C'}{}_{D'} (\sTwist_{1,0} \phi)_{AB}{}^{D'}.\end{aligned}$$ Using irreducible decompositions and the equations above, one can in a systematic way reduce any third order derivative of $\phi_A$ in terms of $\phi_A$, $(\sTwist_{1,0} \phi)_{AB}{}^{A'}$, $(\sTwist_{2,1}\sTwist_{1,0} \phi)_{ABC}{}^{A'B'}$ and $(\sTwist_{3,2}\sTwist_{2,1}\sTwist_{1,0} \phi)_{ABCD}{}^{A'B'C'}$. First kind of symmetry operator for the Dirac-Weyl equation ----------------------------------------------------------- The general second order differential operator, mapping a Dirac-Weyl field $\phi_{A}$ to $\mathcal{S}_{1,0}$ is equivalent to $\phi_{A}\rightarrow \chi_{A}$, where $$\begin{aligned} \chi_{A}={}&N_{A}{}^{B} \phi_{B} + M_{A}{}^{BCA'} (\sTwist_{1,0} \phi)_{BCA'} + L_{A}{}^{BCDA'B'} (\sTwist_{2,1} \sTwist_{1,0} \phi)_{BCDA'B'},\label{eq:chiDiracDef}\\ \intertext{and} L_{ABCD}{}^{A'B'}={}&L_{A(BCD)}{}^{(A'B')},\qquad M_{ABC}{}^{A'}={}M_{A(BC)}{}^{A'}.\label{eq:SymLMDirac1}\end{aligned}$$ Here, we have used the reduction of the derivatives to the $\sTwist$ operator as discussed in Section \[sec:ReductionDirac\]. The symmetries comes from the symmetries of $(\sTwist_{1,0} \phi)_{AB}{}^{A'}$ and $(\sTwist_{2,1}\sTwist_{1,0} \phi)_{ABC}{}^{A'B'}$. To be able to make a systematic treatment of the dependence of different components of the coefficients, we will use the irreducible decompositions $$\begin{aligned} L_{ABCD}{}^{A'B'}={}&\underset{4,2}{L}{}_{ABCD}{}^{A'B'} + \tfrac{3}{4} \underset{2,2}{L}{}_{(BC}{}^{A'B'}\epsilon_{D)A},\\ M_{ABC}{}^{A'}={}&\underset{3,1}{M}{}_{ABC}{}^{A'} + \tfrac{2}{3} \underset{1,1}{M}{}_{(B}{}^{A'}\epsilon_{C)A},\\ N_{AB}={}&\underset{2,0}{N}{}_{AB} - \tfrac{1}{2} \underset{0,0}{N}{} \epsilon_{AB}.\end{aligned}$$ where $$\begin{aligned} \underset{2,2}{L}{}_{AB}{}^{A'B'}\equiv{}&L^{C}{}_{ABC}{}^{A'B'},& \underset{1,1}{M}{}_{A}{}^{A'}\equiv{}&M^{B}{}_{AB}{}^{A'},& \underset{0,0}{N}{}\equiv{}&N^{A}{}_{A},\\ \underset{4,2}{L}{}_{ABCD}{}^{A'B'}\equiv{}&L_{(ABCD)}{}^{A'B'},& \underset{3,1}{M}{}_{ABC}{}^{A'}\equiv{}&M_{(ABC)}{}^{A'},& \underset{2,0}{N}{}_{AB}\equiv{}&N_{(AB)}.\end{aligned}$$ We use the convention that a spinor with underscripts $\underset{k,l}{T}{}$ is a totally symmetric valence $(k,l)$ spinor. Using these spinors, we can rewrite as $$\begin{aligned} \chi_{A}={}&- \tfrac{1}{2} \underset{0,0}{N}{} \phi_{A} - \underset{2,0}{N}{}_{AB} \phi^{B} - \tfrac{2}{3} \underset{1,1}{M}{}^{BA'} (\sTwist_{1,0} \phi)_{ABA'} - \underset{3,1}{M}{}_{ABCA'} (\sTwist_{1,0} \phi)^{BCA'}\nonumber\\ & - \tfrac{3}{4} \underset{2,2}{L}{}^{BCA'B'} (\sTwist_{2,1} \sTwist_{1,0} \phi)_{ABCA'B'} - \underset{4,2}{L}{}_{ABCDA'B'} (\sTwist_{2,1} \sTwist_{1,0} \phi)^{BCDA'B'}.\end{aligned}$$ The condition for the operator $\phi_{A}\rightarrow \chi_{A}$ to be a symmetry operator is $$(\sCurlDagger_{1,0} \chi)_{A'}=0.\label{eq:chiDirac}$$ The definition of the $\sCurlDagger$ operator, the Leibniz rule for the covariant derivative, and the irreducible decomposition allows us to write this equation in terms of the fundamental operators acting on the coefficients and the field. Furthermore, using the results from the previous subsection, we see that this equation can be reduced to a linear combination of the spinors $(\sTwist_{3,2}\sTwist_{2,1}\sTwist_{1,0} \phi)_{ABCD}{}^{A'B'C'}$, $(\sTwist_{2,1}\sTwist_{1,0} \phi)_{ABC}{}^{A'B'}$, $(\sTwist_{1,0} \phi)_{AB}{}^{A'}$ and $\phi_A$. For a general Dirac-Weyl field and an arbitrary point on the manifold, there are no relations between these spinors. Hence, they are independent, and therefore their coefficients have to vanish individually. After the reduction of the derivatives of the field to the $\sTwist$ operator, we can therefore study the different order derivatives in separately. We begin with the highest order, and work our way down to order zero. ### Third order part The third order derivative term of is $$\begin{aligned} \label{eq:thirdorderpartdirac} 0={}&- \underset{4,2}{L}{}^{ABCDB'C'} (\sTwist_{3,2} \sTwist_{2,1} \sTwist_{1,0} \phi)_{ABCDA'B'C'}.\end{aligned}$$ We will now use the argument from Section \[sec:indepspinors\] to derive equations for the coefficients in a systematic way. To get rid of the free index in equation we multiply with an arbitrary spinor field $T^{A'}$ to get $$\begin{aligned} 0={}&- \underset{4,2}{L}{}^{ABCDB'C'}T^{A'} (\sTwist_{3,2} \sTwist_{2,1} \sTwist_{1,0} \phi)_{ABCDA'B'C'}.\end{aligned}$$ From the argument in Section \[sec:indepspinors\] and the observation that $T^{A'} (\sTwist_{3,2} \sTwist_{2,1} \sTwist_{1,0} \phi)_{ABCDA'B'C'}$ is irreducible we conclude that $$\begin{aligned} \underset{4,2}{L}{}_{ABCD}{}^{A'B'}={}&0.\end{aligned}$$ ### Second order part The second order derivative terms of can now be reduced to $$\begin{aligned} 0={}&- \underset{3,1}{M}{}^{ABCB'} (\sTwist_{2,1} \sTwist_{1,0} \phi)_{ABCA'B'} + \tfrac{1}{2} (\sCurl_{2,2} \underset{2,2}{L}{})^{ABCB'} (\sTwist_{2,1} \sTwist_{1,0} \phi)_{ABCA'B'}\nonumber\\ & + \tfrac{3}{4} (\sTwist_{2,2} \underset{2,2}{L}{})_{ABCA'B'C'} (\sTwist_{2,1} \sTwist_{1,0} \phi)^{ABCB'C'}.\end{aligned}$$ Here we again multiply with an arbitrary spinor field $T^{A'}$, but here $(\sTwist_{2,1} \sTwist_{1,0} \phi)^{ABCB'C'}T^{A'}$ is not irreducible. Therefore, we decompose it into irreducible parts and get $$\begin{aligned} 0={}&\tfrac{3}{4} T^{(A'}(\sTwist_{2,1} \sTwist_{1,0} \phi)^{|ABC|B'C')} (\sTwist_{2,2} \underset{2,2}{L}{})_{ABCA'B'C'} \nonumber\\ & + \bigl(\tfrac{1}{2} (\sCurl_{2,2} \underset{2,2}{L}{})^{ABCB'}- \underset{3,1}{M}{}^{ABCB'} \bigr) T^{A'}(\sTwist_{2,1} \sTwist_{1,0} \phi)_{ABCA'B'}.\end{aligned}$$ The argument in Section \[sec:indepspinors\] tells that the coefficients of the different irreducible parts have to vanish individually which gives $$\begin{aligned} (\sTwist_{2,2} \underset{2,2}{L}{})_{ABC}{}^{A'B'C'}={}&0,\label{eq:TwistL22Dirac1}\\ \underset{3,1}{M}{}_{ABC}{}^{A'}={}&\tfrac{1}{2} (\sCurl_{2,2} \underset{2,2}{L}{})_{ABC}{}^{A'}.\end{aligned}$$ ### First order part The first order derivative terms of are $$\begin{aligned} 0={}&- \underset{2,0}{N}{}^{AB} (\sTwist_{1,0} \phi)_{ABA'} + \tfrac{1}{3} (\sCurl_{1,1} \underset{1,1}{M}{})^{AB} (\sTwist_{1,0} \phi)_{ABA'} - \tfrac{1}{2} (\sDiv_{3,1} \underset{3,1}{M}{})^{AB} (\sTwist_{1,0} \phi)_{ABA'}\nonumber\\ & - \tfrac{2}{3} \underset{2,2}{L}{}_{A}{}^{CB'C'} \Phi_{BCB'C'} (\sTwist_{1,0} \phi)^{AB}{}_{A'} - 6 \Lambda \underset{2,2}{L}{}_{ABA'B'} (\sTwist_{1,0} \phi)^{ABB'}\nonumber\\ & + \tfrac{4}{3} \underset{2,2}{L}{}_{A}{}^{C}{}_{B'}{}^{C'} \Phi_{BCA'C'} (\sTwist_{1,0} \phi)^{ABB'} + \tfrac{5}{3} \underset{2,2}{L}{}_{A}{}^{C}{}_{A'}{}^{C'} \Phi_{BCB'C'} (\sTwist_{1,0} \phi)^{ABB'}\nonumber\\ & + \tfrac{3}{4} \underset{2,2}{L}{}^{CD}{}_{A'B'} \Psi_{ABCD} (\sTwist_{1,0} \phi)^{ABB'} + \tfrac{3}{4} \underset{2,2}{L}{}_{AB}{}^{C'D'} \bar\Psi_{A'B'C'D'} (\sTwist_{1,0} \phi)^{ABB'}\nonumber\\ & - (\sCurlDagger_{3,1} \underset{3,1}{M}{})_{ABA'B'} (\sTwist_{1,0} \phi)^{ABB'} + \tfrac{2}{3} (\sTwist_{1,1} \underset{1,1}{M}{})_{ABA'B'} (\sTwist_{1,0} \phi)^{ABB'}.\end{aligned}$$ Here we again multiply with an arbitrary spinor field $T^{A'}$ and decompose $(\sTwist_{1,0} \phi)^{ABB'}T^{A'}$ into irreducible parts. Due to the argument in Section \[sec:indepspinors\] the coefficients of the different irreducible parts have to vanish individually which gives $$\begin{aligned} 0={}&- \underset{2,0}{N}{}_{AB} + \tfrac{1}{3} (\sCurl_{1,1} \underset{1,1}{M}{})_{AB} - \tfrac{1}{4} (\sDiv_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{AB} - \tfrac{1}{2} \underset{2,2}{L}{}_{(A}{}^{CA'B'}\Phi_{B)CA'B'},\\ 0={}&-6 \Lambda \underset{2,2}{L}{}_{AB}{}^{A'B'} + \tfrac{3}{4} \underset{2,2}{L}{}^{CDA'B'} \Psi_{ABCD} + \tfrac{3}{4} \underset{2,2}{L}{}_{AB}{}^{C'D'} \bar\Psi^{A'B'}{}_{C'D'} - \tfrac{1}{2} (\sCurlDagger_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{AB}{}^{A'B'}\nonumber\\ & + 3 \underset{2,2}{L}{}_{(A}{}^{C(A'|C'|}\Phi_{B)C}{}^{B')}{}_{C'} + \tfrac{2}{3} (\sTwist_{1,1} \underset{1,1}{M}{})_{AB}{}^{A'B'}.\end{aligned}$$ Using the commutators and together with , this reduces to $$\begin{aligned} \underset{2,0}{N}{}_{AB}={}&\tfrac{1}{3} (\sCurl_{1,1} \underset{1,1}{M}{})_{AB} - \tfrac{1}{6} (\sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{AB},\\ (\sTwist_{1,1} \underset{1,1}{M}{})_{AB}{}^{A'B'}={}&- \tfrac{3}{8} \underset{2,2}{L}{}^{CDA'B'} \Psi_{ABCD} + \tfrac{3}{8} \underset{2,2}{L}{}_{AB}{}^{C'D'} \bar\Psi^{A'B'}{}_{C'D'} + (\sTwist_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{AB}{}^{A'B'}.\label{eq:TwistM11Dirac1a}\end{aligned}$$ Isolating the $\sTwist$ terms in leads us to make the ansatz $$\begin{aligned} \underset{1,1}{M}{}_{A}{}^{A'}={}&- \tfrac{3}{2} P_{A}{}^{A'} + (\sDiv_{2,2} \underset{2,2}{L}{})_{A}{}^{A'},\label{eq:M11AnsatzDirac1}\end{aligned}$$ where $P_{A}{}^{A'}$ is undetermined. With this ansatz, the first order equations reduce to $$\begin{aligned} (\sTwist_{1,1} P)_{AB}{}^{A'B'}={}&\tfrac{1}{4} \underset{2,2}{L}{}^{CDA'B'} \Psi_{ABCD} - \tfrac{1}{4} \underset{2,2}{L}{}_{AB}{}^{C'D'} \bar\Psi^{A'B'}{}_{C'D'}\nonumber\\ ={}&\tfrac{1}{4} (\ObstrOne \underset{2,2}{L}{})_{AB}{}^{A'B'},\label{eq:TwistP11Dirac1b}\\ \underset{2,0}{N}{}_{AB}={}&- \tfrac{1}{2} (\sCurl_{1,1} P)_{AB} + \tfrac{1}{6} (\sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{AB}\label{eq:N20Dirac1b}.\end{aligned}$$ ### Zeroth order part {#sec:zerothorderDirac1} Using the equations above, the zeroth order derivative terms of are $$\begin{aligned} 0={}&\phi^{A} (-2 \Lambda \underset{1,1}{M}{}_{AA'} + \tfrac{2}{3} \underset{1,1}{M}{}^{BB'} \Phi_{ABA'B'} + \tfrac{1}{4} \Psi_{ABCD} (\sCurl_{2,2} \underset{2,2}{L}{})^{BCD}{}_{A'} - \tfrac{5}{12} \underset{2,2}{L}{}^{BC}{}_{A'}{}^{B'} (\sCurl_{2,2} \Phi)_{ABCB'}\nonumber\\ & - (\sCurlDagger_{2,0} \underset{2,0}{N}{})_{AA'} - \tfrac{1}{6} \underset{2,2}{L}{}_{A}{}^{BB'C'} (\sCurlDagger_{2,2} \Phi)_{BA'B'C'} - \tfrac{8}{3} \underset{2,2}{L}{}_{ABA'B'} (\sTwist_{0,0} \Lambda)^{BB'} + \tfrac{1}{2} (\sTwist_{0,0} \underset{0,0}{N}{})_{AA'}\nonumber\\ & + \tfrac{1}{2} \underset{2,2}{L}{}^{BCB'C'} (\sTwist_{2,2} \Phi)_{ABCA'B'C'}).\end{aligned}$$ Here, there is no reason to multiply with an arbitrary $T^{A'}$ and do an irreducible decomposition of $T^{A'}\phi^A$ because $T^{A'}\phi^A$ is already irreducible. Still the argument in Section \[sec:indepspinors\] gives that the coefficient of $\phi^A$ will have to vanish. With the substitutions and , the vanishing of this coefficient is equivalent to $$\begin{aligned} (\sTwist_{0,0} \underset{0,0}{N}{})_{A}{}^{A'}={}&-6 \Lambda P_{A}{}^{A'} + 2 \Phi_{AB}{}^{A'}{}_{B'} P^{BB'} - \tfrac{1}{2} \Psi_{ABCD} (\sCurl_{2,2} \underset{2,2}{L}{})^{BCDA'}\nonumber\\ & + \tfrac{5}{6} \underset{2,2}{L}{}^{BCA'B'} (\sCurl_{2,2} \Phi)_{ABCB'} + \tfrac{1}{3} \underset{2,2}{L}{}_{A}{}^{BB'C'} (\sCurlDagger_{2,2} \Phi)_{B}{}^{A'}{}_{B'C'} - (\sCurlDagger_{2,0} \sCurl_{1,1} P)_{A}{}^{A'}\nonumber\\ & + \tfrac{1}{3} (\sCurlDagger_{2,0} \sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{A}{}^{A'} + 4 \Lambda (\sDiv_{2,2} \underset{2,2}{L}{})_{A}{}^{A'} - \tfrac{4}{3} \Phi_{AB}{}^{A'}{}_{B'} (\sDiv_{2,2} \underset{2,2}{L}{})^{BB'}\nonumber\\ & + \tfrac{16}{3} \underset{2,2}{L}{}_{AB}{}^{A'}{}_{B'} (\sTwist_{0,0} \Lambda)^{BB'} - \underset{2,2}{L}{}^{BCB'C'} (\sTwist_{2,2} \Phi)_{ABC}{}^{A'}{}_{B'C'}.\label{eq:TwistN00Dirac1a}\end{aligned}$$ To simplify the $\sCurlDagger\sCurl\sDiv$ term, we first commute the innermost operators with . Then the outermost operators are commuted with . After that, we are left with the operator $\sDiv\sCurlDagger\sCurl$, which can be turned into $\sDiv\sTwist\sDiv$ by using and . Finally, the $\sDiv\sTwist\sDiv$ operator can be turned into $\sCurlDagger\sCurl\sDiv$ and $\sTwist\sDiv\sDiv$, again by using , but this time on the outermost operators. In detail $$\begin{aligned} (\sCurlDagger_{2,0} \sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{AA'}={}&- \tfrac{3}{2} \nabla_{BA'}\square_{B'C'}\underset{2,2}{L}{}_{A}{}^{BB'C'} + \tfrac{3}{2} (\sCurlDagger_{2,0} \sDiv_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{AA'}\\* ={}&3 \square_{BC}(\sCurl_{2,2} \underset{2,2}{L}{})_{A}{}^{BC}{}_{A'} - \tfrac{3}{2} \nabla_{BA'}\square_{B'C'}\underset{2,2}{L}{}_{A}{}^{BB'C'} + 3 (\sDiv_{2,2} \sCurlDagger_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{AA'}\\ ={}&3 \square_{BC}(\sCurl_{2,2} \underset{2,2}{L}{})_{A}{}^{BC}{}_{A'} - \tfrac{3}{2} \nabla_{BA'}\square_{B'C'}\underset{2,2}{L}{}_{A}{}^{BB'C'}\nonumber\\* & - 6 \nabla^{BC'}\square_{(A'}{}^{B'}\underset{2,2}{L}{}_{|AB|C')B'} - 3 \nabla^{CB'}\square_{(A}{}^{B}\underset{2,2}{L}{}_{C)BA'B'}\nonumber\\* & + 4 (\sDiv_{2,2} \sTwist_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{AA'}\\ ={}&2 \square_{AB}(\sDiv_{2,2} \underset{2,2}{L}{})^{B}{}_{A'} + 6 \square_{A'B'}(\sDiv_{2,2} \underset{2,2}{L}{})_{A}{}^{B'} + 3 \square_{BC}(\sCurl_{2,2} \underset{2,2}{L}{})_{A}{}^{BC}{}_{A'}\nonumber\\* & - \tfrac{3}{2} \nabla_{BA'}\square_{B'C'}\underset{2,2}{L}{}_{A}{}^{BB'C'} - 6 \nabla^{BC'}\square_{(A'}{}^{B'}\underset{2,2}{L}{}_{|AB|C')B'}\nonumber\\* & - 3 \nabla^{CB'}\square_{(A}{}^{B}\underset{2,2}{L}{}_{C)BA'B'} - 4 (\sCurlDagger_{2,0} \sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{AA'} + 3 (\sTwist_{0,0} \sDiv_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{AA'}.\end{aligned}$$ Isolating the $\sCurlDagger\sCurl\sDiv$ terms, expanding the commutators and using yield $$\begin{aligned} (\sCurlDagger_{2,0} \sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{AA'}={}&- \tfrac{8}{5} \Phi^{BC}{}_{A'}{}^{B'} (\sCurl_{2,2} \underset{2,2}{L}{})_{ABCB'} + \tfrac{6}{5} \Psi_{ABCD} (\sCurl_{2,2} \underset{2,2}{L}{})^{BCD}{}_{A'}\nonumber\\ & - 2 \underset{2,2}{L}{}^{BC}{}_{A'}{}^{B'} (\sCurl_{2,2} \Phi)_{ABCB'} + \tfrac{6}{5} \bar\Psi_{A'B'C'D'} (\sCurlDagger_{2,2} \underset{2,2}{L}{})_{A}{}^{B'C'D'}\nonumber\\ & - \tfrac{8}{5} \Phi_{A}{}^{BB'C'} (\sCurlDagger_{2,2} \underset{2,2}{L}{})_{BA'B'C'} - 2 \underset{2,2}{L}{}_{A}{}^{BB'C'} (\sCurlDagger_{2,2} \Phi)_{BA'B'C'}\nonumber\\ & - 12 \Lambda (\sDiv_{2,2} \underset{2,2}{L}{})_{AA'} + \tfrac{44}{15} \Phi_{ABA'B'} (\sDiv_{2,2} \underset{2,2}{L}{})^{BB'} - \tfrac{64}{5} \underset{2,2}{L}{}_{ABA'B'} (\sTwist_{0,0} \Lambda)^{BB'}\nonumber\\ & + \tfrac{3}{5} \underset{2,2}{L}{}^{BCB'C'} (\sTwist_{2,2} \Phi)_{ABCA'B'C'} + \tfrac{3}{5} (\sTwist_{0,0} \sDiv_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{AA'}.\label{eq:CurlDaggerCurlDivLL22Dirac}\end{aligned}$$ Using this in , and using combined with gives $$\begin{aligned} (\sTwist_{0,0} \underset{0,0}{N}{})_{A}{}^{A'}={}&- \tfrac{8}{15} \Phi^{BCA'B'} (\sCurl_{2,2} \underset{2,2}{L}{})_{ABCB'} + \tfrac{3}{20} \Psi_{ABCD} (\sCurl_{2,2} \underset{2,2}{L}{})^{BCDA'}\nonumber\\ & - \tfrac{1}{12} \underset{2,2}{L}{}^{BCA'B'} (\sCurl_{2,2} \Phi)_{ABCB'} + \tfrac{3}{20} \bar\Psi^{A'}{}_{B'C'D'} (\sCurlDagger_{2,2} \underset{2,2}{L}{})_{A}{}^{B'C'D'}\nonumber\\ & - \tfrac{8}{15} \Phi_{A}{}^{BB'C'} (\sCurlDagger_{2,2} \underset{2,2}{L}{})_{B}{}^{A'}{}_{B'C'} - \tfrac{1}{12} \underset{2,2}{L}{}_{A}{}^{BB'C'} (\sCurlDagger_{2,2} \Phi)_{B}{}^{A'}{}_{B'C'}\nonumber\\ & - \tfrac{16}{45} \Phi_{AB}{}^{A'}{}_{B'} (\sDiv_{2,2} \underset{2,2}{L}{})^{BB'} + \tfrac{16}{15} \underset{2,2}{L}{}_{AB}{}^{A'}{}_{B'} (\sTwist_{0,0} \Lambda)^{BB'}\nonumber\\ & - \tfrac{4}{5} \underset{2,2}{L}{}^{BCB'C'} (\sTwist_{2,2} \Phi)_{ABC}{}^{A'}{}_{B'C'} - \tfrac{3}{4} (\sTwist_{0,0} \sDiv_{1,1} P)_{A}{}^{A'} + \tfrac{1}{5} (\sTwist_{0,0} \sDiv_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{A}{}^{A'}.\label{eq:TwistN00Dirac1c}\end{aligned}$$ To simplify the remaining terms, we define $$\Upsilon\equiv\underset{2,2}{L}{}_{ABA'B'} \Phi^{ABA'B'}.\label{eq:UpsilonDef}$$ Using the gradient of $\Upsilon$ reduces to $$\begin{aligned} (\sTwist_{0,0} \Upsilon)_{AA'}={}&- \tfrac{4}{3} \underset{2,2}{L}{}_{A}{}^{B}{}_{A'}{}^{B'} (\sTwist_{0,0}\Lambda)_{BB'} + \tfrac{2}{3} \Phi^{BC}{}_{A'}{}^{B'} (\sCurl_{2,2} \underset{2,2}{L}{})_{ABCB'}\nonumber\\ & + \tfrac{2}{3} \underset{2,2}{L}{}^{BC}{}_{A'}{}^{B'} (\sCurl_{2,2} \Phi)_{ABCB'} + \tfrac{2}{3} \Phi_{A}{}^{BB'C'} (\sCurlDagger_{2,2} \underset{2,2}{L}{})_{BA'B'C'}\nonumber\\ & + \tfrac{2}{3} \underset{2,2}{L}{}_{A}{}^{BB'C'} (\sCurlDagger_{2,2} \Phi)_{BA'B'C'} + \tfrac{4}{9} \Phi_{ABA'B'} (\sDiv_{2,2} \underset{2,2}{L}{})^{BB'}\nonumber\\ & + \underset{2,2}{L}{}^{BCB'C'} (\sTwist_{2,2} \Phi)_{ABCA'B'C'}.\label{eq:GradUpsilon}\end{aligned}$$ This can be used to eliminate most of the terms in . Together with the definition of the operator $\ObstrZero$, we find that reduces to $$\begin{aligned} (\sTwist_{0,0} \underset{0,0}{N}{})_{A}{}^{A'}={}&\tfrac{9}{20} (\ObstrZero \underset{2,2}{L}{})_{A}{}^{A'} - \tfrac{4}{5} (\sTwist_{0,0} \Upsilon)_{A}{}^{A'} - \tfrac{3}{4} (\sTwist_{0,0} \sDiv_{1,1} P)_{A}{}^{A'} + \tfrac{1}{5} (\sTwist_{0,0} \sDiv_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{A}{}^{A'}.\end{aligned}$$ It is now clear that the ansatz $$\begin{aligned} \underset{0,0}{N}{}={}&-2 Q - \tfrac{4}{5} \Upsilon - \tfrac{3}{4} (\sDiv_{1,1} P) + \tfrac{1}{5} (\sDiv_{1,1} \sDiv_{2,2} \underset{2,2}{L}{}),\end{aligned}$$ with $Q$ undetermined gives $$\begin{aligned} (\sTwist_{0,0} Q)_{A}{}^{A'}={}&- \tfrac{9}{40} (\ObstrZero \underset{2,2}{L}{})_{A}{}^{A'}.\label{eq:TwistQ00Dirac1}\end{aligned}$$ We can now conclude that the only restrictive equations are , and . The other equations give expressions for the remaining coefficients in terms of $\underset{2,2}{L}{}_{AB}{}^{A'B'}$, $P_{AA'}$, and $Q$. For convenience we make the replacement $\underset{2,2}{L}{}_{AB}{}^{A'B'} \rightarrow - \tfrac{4}{3} L_{AB}{}^{A'B'}$. Second kind of symmetry operator for the Dirac-Weyl equation ------------------------------------------------------------ The general second order differential operator, mapping a Dirac-Weyl field $\phi_{A}$ to $\mathcal{S}_{0,1}$ is equivalent to $\phi_{A}\rightarrow \omega_{A'}$, where $$\begin{aligned} \omega_{A'}={}&N^{B}{}_{A'} \phi_{B} + M_{A'}{}^{BCB'} (\sTwist_{1,0} \phi)_{BCB'} + L_{A'}{}^{BCDB'C'} (\sTwist_{2,1} \sTwist_{1,0} \phi)_{BCDB'C'},\end{aligned}$$ where $$\begin{aligned} L_{A'ABCB'C'}={}&L_{A'(ABC)(B'C')},& M_{A'ABB'}={}&M_{A'(AB)B'}.\label{eq:SymLMDirac2}\end{aligned}$$ Here, we have used the reduction of the derivatives to the $\sTwist$ operator as discussed above. The symmetries comes from the symmetries of $(\sTwist_{1,0} \phi)_{AB}{}^{A'}$ and $(\sTwist_{2,1}\sTwist_{1,0} \phi)_{ABC}{}^{A'B'}$. As we did above, we will decompose the coefficients into irreducible parts to more clearly see which parts are independent. The irreducible decompositions of $L^{A'}{}_{ABC}{}^{B'C'}$ and $M^{A'}{}_{AB}{}^{B'}$ are $$\begin{aligned} L^{A'}{}_{ABC}{}^{B'C'}={}&\underset{3,3}{L}{}_{ABC}{}^{A'B'C'} + \tfrac{2}{3} \underset{3,1}{L}{}_{ABC}{}^{(B'}\bar\epsilon^{C')A'},\\ M^{A'}{}_{AB}{}^{B'}={}&\underset{2,2}{M}{}_{AB}{}^{A'B'} - \tfrac{1}{2} \underset{2,0}{M}{}_{AB} \bar\epsilon^{A'B'},\end{aligned}$$ where $$\begin{aligned} \underset{3,1}{L}{}_{ABC}{}^{A'}\equiv{}&L^{B'}{}_{ABC}{}^{A'}{}_{B'},& \underset{2,0}{M}{}_{AB}\equiv{}&M^{A'}{}_{ABA'},\\ \underset{3,3}{L}{}_{ABC}{}^{A'B'C'}\equiv{}&L^{(A'}{}_{ABC}{}^{B'C')},& \underset{2,2}{M}{}_{AB}{}^{A'B'}\equiv{}&M^{(A'}{}_{AB}{}^{B')}.\end{aligned}$$ With these irreducible decompositions, we get $$\begin{aligned} \omega_{A'}={}&N^{B}{}_{A'} \phi_{B} - \tfrac{1}{2} \underset{2,0}{M}{}^{BC} (\sTwist_{1,0} \phi)_{BCA'} - \underset{2,2}{M}{}_{BCA'B'} (\sTwist_{1,0} \phi)^{BCB'} - \tfrac{2}{3} \underset{3,1}{L}{}^{BCDB'} (\sTwist_{2,1} \sTwist_{1,0} \phi)_{BCDA'B'}\nonumber\\ & - \underset{3,3}{L}{}_{BCDA'B'C'} (\sTwist_{2,1} \sTwist_{1,0} \phi)^{BCDB'C'}.\end{aligned}$$ The condition for the operator $\phi_{A}\rightarrow \omega_{A'}$ to be a symmetry operator is $$(\sCurl_{0,1} \omega)_{A}=0.\label{eq:omegaDirac}$$ Using the results from Section \[sec:ReductionDirac\], we see that this equation can be reduced to a linear combination of the spinors $\phi_A$, $(\sTwist_{1,0} \phi)_{AB}{}^{A'}$, $(\sTwist_{2,1}\sTwist_{1,0} \phi)_{ABC}{}^{A'B'}$ and $(\sTwist_{3,2}\sTwist_{2,1}\sTwist_{1,0} \phi)_{ABCD}{}^{A'B'C'}$. As above, we can treat these as independent, and therefore their coefficients have to vanish individually. After the reduction of the derivatives of the field to the $\sTwist$ operator, we can therefore study the different order derivatives in separately. We begin with the highest order, and work our way down to order zero. ### Third order part The third order part of is $$\begin{aligned} 0={}&- \underset{3,3}{L}{}^{BCDA'B'C'} (\sTwist_{3,2} \sTwist_{2,1} \sTwist_{1,0} \phi)_{ABCDA'B'C'}.\end{aligned}$$ Using the argument from Section \[sec:indepspinors\], we see that this implies $$\begin{aligned} \underset{3,3}{L}{}_{ABC}{}^{A'B'C'}={}&0.\end{aligned}$$ ### Second order part The second order part of now takes the form $$\begin{aligned} 0={}&- \underset{2,2}{M}{}^{BCA'B'} (\sTwist_{2,1} \sTwist_{1,0} \phi)_{ABCA'B'} + \tfrac{1}{2} (\sCurlDagger_{3,1} \underset{3,1}{L}{})^{BCA'B'} (\sTwist_{2,1} \sTwist_{1,0} \phi)_{ABCA'B'}\nonumber\\ & + \tfrac{2}{3} (\sTwist_{3,1} \underset{3,1}{L}{})_{ABCDA'B'} (\sTwist_{2,1} \sTwist_{1,0} \phi)^{BCDA'B'}.\end{aligned}$$ Here we multiply with an arbitrary spinor field $T^{A}$ and decompose $(\sTwist_{2,1}\sTwist_{1,0} \phi)^{BCDC'D'}T^{A}$ into irreducible parts. Due to the argument in Section \[sec:indepspinors\] the coefficients of the different irreducible parts have to vanish individually which gives $$\begin{aligned} (\sTwist_{3,1} \underset{3,1}{L}{})_{ABCD}{}^{A'B'}={}&0,\label{eq:TwistLL31}\\ \underset{2,2}{M}{}_{AB}{}^{A'B'}={}&\tfrac{1}{2} (\sCurlDagger_{3,1} \underset{3,1}{L}{})_{AB}{}^{A'B'}\label{eq:M22Eq1}.\end{aligned}$$ ### First order part The first order part of can now be reduced to $$\begin{aligned} 0={}&- N^{BA'} (\sTwist_{1,0} \phi)_{ABA'} + \tfrac{1}{3} (\sCurlDagger_{2,0} \underset{2,0}{M}{})^{BA'} (\sTwist_{1,0} \phi)_{ABA'} - \tfrac{2}{3} (\sDiv_{2,2} \underset{2,2}{M}{})^{BA'} (\sTwist_{1,0} \phi)_{ABA'}\nonumber\\ & + \tfrac{1}{3} \underset{3,1}{L}{}_{BCDB'} \Phi^{CD}{}_{A'}{}^{B'} (\sTwist_{1,0} \phi)_{A}{}^{BA'} - \tfrac{5}{12} \underset{3,1}{L}{}^{CDF}{}_{A'} \Psi_{BCDF} (\sTwist_{1,0} \phi)_{A}{}^{BA'}\nonumber\\ & - 6 \Lambda \underset{3,1}{L}{}_{ABCA'} (\sTwist_{1,0} \phi)^{BCA'} + \tfrac{1}{3} \underset{3,1}{L}{}_{BCDB'} \Phi_{A}{}^{D}{}_{A'}{}^{B'} (\sTwist_{1,0} \phi)^{BCA'}\nonumber\\ & + \tfrac{5}{3} \underset{3,1}{L}{}_{ACDB'} \Phi_{B}{}^{D}{}_{A'}{}^{B'} (\sTwist_{1,0} \phi)^{BCA'} + \tfrac{5}{4} \underset{3,1}{L}{}_{B}{}^{DF}{}_{A'} \Psi_{ACDF} (\sTwist_{1,0} \phi)^{BCA'}\nonumber\\ & + \tfrac{3}{4} \underset{3,1}{L}{}_{A}{}^{DF}{}_{A'} \Psi_{BCDF} (\sTwist_{1,0} \phi)^{BCA'} - (\sCurl_{2,2} \underset{2,2}{M}{})_{ABCA'} (\sTwist_{1,0} \phi)^{BCA'}\nonumber\\ & + \tfrac{1}{2} (\sTwist_{2,0} \underset{2,0}{M}{})_{ABCA'} (\sTwist_{1,0} \phi)^{BCA'}.\end{aligned}$$ Here we again multiply with an arbitrary spinor field $T^{A}$ and decompose $(\sTwist_{1,0} \phi)^{BCC'}T^{A}$ into irreducible parts. Due to the argument in Section \[sec:indepspinors\] the coefficients of the different irreducible parts have to vanish individually which gives $$\begin{aligned} 0={}&- N_{A}{}^{A'} - \tfrac{1}{3} \underset{3,1}{L}{}^{BCDA'} \Psi_{ABCD} + \tfrac{1}{3} (\sCurlDagger_{2,0} \underset{2,0}{M}{})_{A}{}^{A'} - \tfrac{2}{3} (\sDiv_{2,2} \underset{2,2}{M}{})_{A}{}^{A'},\\ 0={}&-6 \Lambda \underset{3,1}{L}{}_{ABC}{}^{A'} - (\sCurl_{2,2} \underset{2,2}{M}{})_{ABC}{}^{A'} + 2 \underset{3,1}{L}{}_{(BC|DB'|}\Phi_{A)}{}^{DA'B'} + 2 \underset{3,1}{L}{}_{(A}{}^{DFA'}\Psi_{BC)DF}\nonumber\\ & + \tfrac{1}{2} (\sTwist_{2,0} \underset{2,0}{M}{})_{ABC}{}^{A'}.\end{aligned}$$ By , the commutator and these reduce to $$\begin{aligned} N_{A}{}^{A'}={}&- \tfrac{1}{3} \underset{3,1}{L}{}_{ABCB'} \Phi^{BCA'B'} + \tfrac{1}{3} (\sCurlDagger_{2,0} \underset{2,0}{M}{})_{A}{}^{A'} - \tfrac{1}{6} (\sCurlDagger_{2,0} \sDiv_{3,1} \underset{3,1}{L}{})_{A}{}^{A'},\\ (\sTwist_{2,0} \underset{2,0}{M}{})_{ABC}{}^{A'}={}&(\sTwist_{2,0} \sDiv_{3,1} \underset{3,1}{L}{})_{ABC}{}^{A'}.\end{aligned}$$ If we make the ansatz $$\begin{aligned} \underset{2,0}{M}{}_{AB}={}&-2 P_{AB} + (\sDiv_{3,1} \underset{3,1}{L}{})_{AB},\end{aligned}$$ these equations reduce to $$\begin{aligned} N_{A}{}^{A'}={}&- \tfrac{1}{3} \underset{3,1}{L}{}_{ABCB'} \Phi^{BCA'B'} - \tfrac{2}{3} (\sCurlDagger_{2,0} P)_{A}{}^{A'} + \tfrac{1}{6} (\sCurlDagger_{2,0} \sDiv_{3,1} \underset{3,1}{L}{})_{A}{}^{A'},\label{eq:N11Dirac2a}\\ (\sTwist_{2,0} P)_{ABC}{}^{A'}={}&0.\label{eq:TwistPDirac2a}\end{aligned}$$ ### Zeroth order part {#zeroth-order-part} The zeroth order part of can now be reduced to $$\begin{aligned} 0={}&-2 \Lambda \underset{2,0}{M}{}_{AB} \phi^{B} + \tfrac{1}{2} \underset{2,0}{M}{}^{CD} \Psi_{ABCD} \phi^{B} - \phi^{B} (\sCurl_{1,1} N)_{AB} - \tfrac{1}{20} \underset{3,1}{L}{}_{B}{}^{CDA'} \phi^{B} (\sCurl_{2,2} \Phi)_{ACDA'}\nonumber\\ & + \tfrac{1}{60} \underset{3,1}{L}{}^{BCDA'} \phi_{A} (\sCurl_{2,2} \Phi)_{BCDA'} - \tfrac{5}{12} \underset{3,1}{L}{}_{A}{}^{CDA'} \phi^{B} (\sCurl_{2,2} \Phi)_{BCDA'}\nonumber\\ & - \tfrac{1}{3} \Phi_{B}{}^{CA'B'} \phi^{B} (\sCurlDagger_{3,1} \underset{3,1}{L}{})_{ACA'B'} - \tfrac{1}{2} \phi_{A} (\sDiv_{1,1} N) - \tfrac{8}{3} \underset{3,1}{L}{}_{ABCA'} \phi^{B} (\sTwist_{0,0} \Lambda)^{CA'}\nonumber\\ & + \tfrac{1}{3} \underset{3,1}{L}{}^{CDFA'} \phi^{B} (\sTwist_{4,0} \Psi)_{ABCDFA'}.\end{aligned}$$ Here we again multiply with an arbitrary spinor field $T^{A}$ and decompose $\phi^{B}T^{A}$ into irreducible parts. Due to the argument in Section \[sec:indepspinors\] the coefficients of the different irreducible parts have to vanish individually which gives $$\begin{aligned} 0={}&- \tfrac{1}{6} \underset{3,1}{L}{}^{ABCA'} (\sCurl_{2,2} \Phi)_{ABCA'} + \tfrac{1}{6} \Phi^{ABA'B'} (\sCurlDagger_{3,1} \underset{3,1}{L}{})_{ABA'B'} - \tfrac{1}{2} (\sDiv_{1,1} N),\label{eq:ZerithOrderDirac2a}\\ 0={}&-2 \Lambda \underset{2,0}{M}{}_{AB} + \tfrac{1}{2} \underset{2,0}{M}{}^{CD} \Psi_{ABCD} - (\sCurl_{1,1} N)_{AB} - \tfrac{7}{15} \underset{3,1}{L}{}_{(A}{}^{CDA'}(\sCurl_{2,2} \Phi)_{B)CDA'}\nonumber\\ & - \tfrac{1}{3} \Phi_{(A}{}^{CA'B'}(\sCurlDagger_{3,1} \underset{3,1}{L}{})_{B)CA'B'} - \tfrac{8}{3} \underset{3,1}{L}{}_{ABCA'} (\sTwist_{0,0} \Lambda)^{CA'} + \tfrac{1}{3} \underset{3,1}{L}{}^{CDFA'} (\sTwist_{4,0} \Psi)_{ABCDFA'}.\label{eq:ZerithOrderDirac2b}\end{aligned}$$ The equation together with the commutator gives . If we substitute in , we get a term with the third order operator $\sCurl\sCurlDagger\sDiv$. To handle this we use the same technique as in Section \[sec:zerothorderDirac1\]. We first commute the innermost operators with . Then the outermost operators are commuted with . After that, we are left with the operator $\sDiv\sCurl\sCurlDagger$, which can be turned into $\sDiv\sTwist\sDiv$ by using and . Finally, the $\sDiv\sTwist\sDiv$ operator can be turned into $\sCurl\sCurlDagger\sDiv$ and $\sTwist\sDiv\sDiv$, again by using , but this time on the outermost operators. $$\begin{aligned} (\sCurl_{1,1} \sCurlDagger_{2,0} \sDiv_{3,1} \underset{3,1}{L}{})_{AB}={}&2 (\sCurl_{1,1} \sDiv_{2,2} \sCurlDagger_{3,1} \underset{3,1}{L}{})_{AB} + 2 \nabla_{(A}{}^{A'}\square^{CD}\underset{3,1}{L}{}_{B)CDA'}\\ ={}&3 \square_{A'B'}(\sCurlDagger_{3,1} \underset{3,1}{L}{})_{AB}{}^{A'B'} + 3 (\sDiv_{3,1} \sCurl_{2,2} \sCurlDagger_{3,1} \underset{3,1}{L}{})_{AB} + 2 \nabla_{(A}{}^{A'}\square^{CD}\underset{3,1}{L}{}_{B)CDA'}\\ ={}&3 \square_{A'B'}(\sCurlDagger_{3,1} \underset{3,1}{L}{})_{AB}{}^{A'B'} + 2 \nabla_{CB'}\square_{A'}{}^{B'}\underset{3,1}{L}{}_{AB}{}^{CA'} - 6 \nabla^{DA'}\square_{(A}{}^{C}\underset{3,1}{L}{}_{BD)CA'}\nonumber\\ & + 3 (\sDiv_{3,1} \sTwist_{2,0} \sDiv_{3,1} \underset{3,1}{L}{})_{AB} + 2 \nabla_{(A}{}^{A'}\square^{CD}\underset{3,1}{L}{}_{B)CDA'}\\ ={}&3 \square_{A'B'}(\sCurlDagger_{3,1} \underset{3,1}{L}{})_{AB}{}^{A'B'} + 2 \nabla_{CB'}\square_{A'}{}^{B'}\underset{3,1}{L}{}_{AB}{}^{CA'} - 6 \nabla^{DA'}\square_{(A}{}^{C}\underset{3,1}{L}{}_{BD)CA'}\nonumber\\ & - 4 (\sCurl_{1,1} \sCurlDagger_{2,0} \sDiv_{3,1} \underset{3,1}{L}{})_{AB} - 6 \square_{(A}{}^{C}(\sDiv_{3,1} \underset{3,1}{L}{})_{B)C} + 2 \nabla_{(A}{}^{A'}\square^{CD}\underset{3,1}{L}{}_{B)CDA'}.\end{aligned}$$ Isolating the $\sCurl\sCurlDagger\sDiv$ terms and expanding the commutators and using yield $$\begin{aligned} (\sCurl_{1,1} \sCurlDagger_{2,0} \sDiv_{3,1} \underset{3,1}{L}{})_{AB}={}&- \Psi_{B}{}^{CDF} (\sCurl_{3,1} \underset{3,1}{L}{})_{ACDF} - \Psi_{A}{}^{CDF} (\sCurl_{3,1} \underset{3,1}{L}{})_{BCDF}\nonumber\\ & - \tfrac{42}{25} \underset{3,1}{L}{}_{B}{}^{CDA'} (\sCurl_{2,2} \Phi)_{ACDA'} - \tfrac{42}{25} \underset{3,1}{L}{}_{A}{}^{CDA'} (\sCurl_{2,2} \Phi)_{BCDA'}\nonumber\\ & - \tfrac{3}{2} \Phi_{B}{}^{CA'B'} (\sCurlDagger_{3,1} \underset{3,1}{L}{})_{ACA'B'} - \tfrac{3}{2} \Phi_{A}{}^{CA'B'} (\sCurlDagger_{3,1} \underset{3,1}{L}{})_{BCA'B'}\nonumber\\ & - 12 \Lambda (\sDiv_{3,1} \underset{3,1}{L}{})_{AB} + \tfrac{21}{10} \Psi_{ABCD} (\sDiv_{3,1} \underset{3,1}{L}{})^{CD} - 12 \underset{3,1}{L}{}_{ABCA'} (\sTwist_{0,0} \Lambda)^{CA'}\nonumber\\ & + \tfrac{2}{5} \underset{3,1}{L}{}^{CDFA'} (\sTwist_{4,0} \Psi)_{ABCDFA'}.\end{aligned}$$ The equation together with the equation above, the commutator and gives $$\begin{aligned} (\sCurl_{1,1} N)_{AB}={}&4 \Lambda P_{AB} - \Psi_{ABCD} P^{CD} - 2 \Lambda (\sDiv_{3,1} \underset{3,1}{L}{})_{AB} + \tfrac{7}{20} \Psi_{ABCD} (\sDiv_{3,1} \underset{3,1}{L}{})^{CD}\nonumber\\ & - \tfrac{17}{75} \underset{3,1}{L}{}_{(A}{}^{CDA'}(\sCurl_{2,2} \Phi)_{B)CDA'} - \tfrac{1}{3} \Phi_{(A}{}^{CA'B'}(\sCurlDagger_{3,1} \underset{3,1}{L}{})_{B)CA'B'}\nonumber\\ & - \tfrac{1}{3} \Psi_{(A}{}^{CDF}(\sCurl_{3,1} \underset{3,1}{L}{})_{B)CDF} - \tfrac{8}{3} \underset{3,1}{L}{}_{ABCA'} (\sTwist_{0,0} \Lambda)^{CA'}\nonumber\\ & + \tfrac{1}{15} \underset{3,1}{L}{}^{CDFA'} (\sTwist_{4,0} \Psi)_{ABCDFA'}.\end{aligned}$$ Due to this, the equation reduces to the auxiliary condition $$\begin{aligned} 0={}&\tfrac{3}{4} \Psi_{ABCD} (\sDiv_{3,1} \underset{3,1}{L}{})^{CD} - \tfrac{6}{5} \underset{3,1}{L}{}_{(A}{}^{CDA'}(\sCurl_{2,2} \Phi)_{B)CDA'}\nonumber\\ & + \tfrac{5}{3} \Psi_{(A}{}^{CDF}(\sCurl_{3,1} \underset{3,1}{L}{})_{B)CDF} + \tfrac{4}{3} \underset{3,1}{L}{}^{CDFA'} (\sTwist_{4,0} \Psi)_{ABCDFA'}.\label{eq:auxcondDirac2}\end{aligned}$$ We can conclude that the only restrictive equations are , and . The other equations express the remaining coefficients in terms of $\underset{3,1}{L}{}_{ABCA'}$ and $P_{AB}$. For convenience we make the replacement $\underset{3,1}{L}{}_{ABC}{}^{A'} \rightarrow - \tfrac{3}{2} L_{ABC}{}^{A'}$. The Maxwell equation {#sec:spin1} ==================== \[Thm::SymOpFirstKind\] There exists a symmetry operator of the first kind $\phi_{AB}\rightarrow \chi_{AB}$, with order less or equal to two, if and only if there are spinor fields $L_{AB}{}^{A'B'}=L_{(AB)}{}^{(A'B')}$, $P_{AA'}$ and $Q$ such that $$\begin{aligned} (\sTwist_{2,2} L)_{ABC}{}^{A'B'C'}={}&0,\label{eq:Lfirstkind}\\ (\sTwist_{1,1} P)_{AB}{}^{A'B'}={}&- \tfrac{2}{3} (\ObstrOne L)_{AB}{}^{A'B'},\label{eq:Qcondfirstkind}\\ (\sTwist_{0,0} Q)_{BA'}={}&0.\end{aligned}$$ The symmetry operator then takes the form $$\begin{aligned} \chi_{AB}={}&Q \phi_{AB}+(\sCurl_{1,1} A)_{AB},\\ \intertext{where} A_{AA'}={}&- P^{B}{}_{A'} \phi_{AB} + \tfrac{1}{3} \phi^{BC} (\sCurl_{2,2} L)_{ABCA'} - \tfrac{4}{9} \phi_{AB} (\sDiv_{2,2} L)^{B}{}_{A'} - L^{BC}{}_{A'}{}^{B'} (\sTwist_{2,0} \phi)_{ABCB'}.\end{aligned}$$ We also note that $$\begin{aligned} (\sCurlDagger_{1,1} A)_{A'B'} ={}& 0.\label{eq:CurlDaggerA}\end{aligned}$$ 1. [ Observe that one can add a gradient of a scalar to the potential $A_{AA'}$ without changing the symmetry operator. Hence, adding $\nabla_{AA'}(\Lambda^{BC}\phi_{BC})$ to $A_{AA'}$ with an arbitrary field $\Lambda_{AB}$ is possible.]{} 2. [ With $L_{ABA'B'}=0$, the first order operator takes the form $$\begin{aligned} \chi_{AB}={}& \hat{\mathcal{L}}_{P}\phi_{AB}+ Q \phi_{AB}.\end{aligned}$$ ]{} \[Thm::SymOpSecondKind\] There exists second order a symmetry operator of the second kind $\phi_{AB}\rightarrow \omega_{A'B'}$, with order less or equal to two, if and only if there is a spinor field $L_{ABCD}=L_{(ABCD)}$ such that $$\begin{aligned} (\sTwist_{4,0} L)_{ABCDF}{}^{A'}={}&0.\label{eq:LSecondKind}\end{aligned}$$ The symmetry operator then takes the form $$\begin{aligned} \omega_{A'B'}={}&(\sCurlDagger_{1,1} B)_{A'B'},\\ \intertext{where} B_{AA'}={}&\tfrac{3}{5} \phi^{BC} (\sCurlDagger_{4,0} L)_{ABCA'} + L_{ABCD} (\sTwist_{2,0} \phi)^{BCD}{}_{A'}.\end{aligned}$$ We also note that $$\begin{aligned} (\sCurl_{1,1} B)_{AB}={}&0.\label{eq:CurlB}\end{aligned}$$ 1. [ Observe that also here we can add a gradient of a scalar to the potential $B_{AA'}$ without changing the symmetry operator. Hence, adding $\nabla_{AA'}(\Lambda^{BC}\phi_{BC})$ to $B_{AA'}$ with an arbitrary field $\Lambda_{AB}$ is possible.]{} 2. [ Due to the equations and , we can use $A_{AA'}+B_{AA'}$ as a potential for both $\chi_{AB}$ and $\omega_{A'B'}$. ]{} Reduction of derivatives of the field {#sec:ReductionMaxwell} ------------------------------------- In our notation, the Maxwell equation $\nabla^{A}{}_{A'}\phi_{AB}=0$, takes the form $(\sCurlDagger_{2,0} \phi)_{A'}=0$. From this we see that the only irreducible part of $\nabla_{A}{}^{A'}\phi_{BC}$ is $(\sTwist_{2,0} \phi)_{ABC}{}^{A'}$. By commuting derivatives we see that all higher order derivatives of $\phi_{AB}$ can be reduced to totally symmetrized derivatives and lower order terms consisting of curvature times lower order symmetrized derivatives. Together with the Maxwell equation, the commutators , , gives $$\begin{aligned} (\sDiv_{3,1} \sTwist_{2,0} \phi)_{AB}={}& -8 \Lambda \phi_{AB} + 2 \Psi_{ABCD} \phi^{CD},\\ (\sCurl_{3,1} \sTwist_{2,0} \phi)_{ABCD}={}&2 \Psi_{(ABC}{}^{F}\phi_{D)F},\\ (\sCurlDagger_{3,1} \sTwist_{2,0} \phi)_{ABA'B'}={}&2 \Phi_{(A}{}^{C}{}_{|A'B'|}\phi_{B)C}\end{aligned}$$ The higher order derivatives can be computed using the commutators , , together with the equations above and the Bianchi system to get $$\begin{aligned} (\sDiv_{4,2} \sTwist_{3,1} \sTwist_{2,0} \phi)_{ABCA'}={}&\tfrac{9}{2} \Phi_{(A}{}^{D}{}_{|A'|}{}^{B'}(\sTwist_{2,0} \phi)_{BC)DB'} + \tfrac{9}{2} \Psi_{(AB}{}^{DF}(\sTwist_{2,0} \phi)_{C)DFA'}\nonumber\\ & - \tfrac{15}{2} \phi_{(AB}(\sTwist_{0,0}\Lambda)_{C)A'} + \tfrac{21}{10} \phi_{(A}{}^{D}(\sCurl_{2,2} \Phi)_{BC)DA'}\nonumber\\ & + \tfrac{3}{2} \phi^{DF} (\sTwist_{4,0} \Psi)_{ABCDFA'} - 15 \Lambda (\sTwist_{2,0} \phi)_{ABCA'},\\ (\sCurl_{4,2} \sTwist_{3,1} \sTwist_{2,0} \phi)_{ABCDFB'}={}&\Phi_{(AB|B'|}{}^{A'}(\sTwist_{2,0} \phi)_{CDF)A'} + 4 \Psi_{(ABC}{}^{H}(\sTwist_{2,0} \phi)_{DF)HB'}\nonumber\\ & - \tfrac{1}{5} \phi_{(AB}(\sCurl_{2,2} \Phi)_{CDF)B'} - \phi_{(A}{}^{H}(\sTwist_{4,0} \Psi)_{BCDF)HB'},\\ (\sCurlDagger_{4,2} \sTwist_{3,1} \sTwist_{2,0} \phi)_{ABC}{}^{A'B'C'}={}&\tfrac{9}{2} \Phi^{D}{}_{(A}{}^{(A'B'}(\sTwist_{2,0} \phi)_{BC)D}{}^{C')} - \tfrac{1}{2} \phi_{(AB}(\sCurlDagger_{2,2} \Phi)_{C)}{}^{A'B'C'}\nonumber\\ & - \tfrac{3}{2} \phi_{(A}{}^{D}(\sTwist_{2,2} \Phi)_{BC)D}{}^{A'B'C'} - \bar\Psi^{A'B'C'}{}_{D'} (\sTwist_{2,0} \phi)_{ABC}{}^{D'}.\end{aligned}$$ These can in a systematic way be used to reduce any derivative up to third order of $\phi_{AB}$ in terms of $\phi_{AB}$, $(\sTwist_{2,0} \phi)_{ABC}{}^{A'}$, $(\sTwist_{3,1} \sTwist_{2,0} \phi)_{ABCD}{}^{A'B'}$ and $(\sTwist_{4,2} \sTwist_{3,1} \sTwist_{2,0} \phi)_{ABCDF}{}^{A'B'C'}$. First kind of symmetry operator for the Maxwell equation -------------------------------------------------------- The general second order differential operator, mapping a Maxwell field $\phi_{AB}$ to $\mathcal{S}_{2,0}$ is equivalent to $\phi_{AB}\rightarrow \chi_{AB}$, where $$\begin{aligned} \chi_{AB}={}&N_{ABCD} \phi^{CD} + M_{ABCDFA'} (\sTwist_{2,0} \phi)^{CDFA'} + L_{ABCDFHA'B'} (\sTwist_{3,1} \sTwist_{2,0} \phi)^{CDFHA'B'},\end{aligned}$$ and $$\begin{aligned} L_{AB}{}^{CDFHA'B'}={}&L_{(AB)}{}^{(CDFH)(A'B')},& M_{AB}{}^{CDFA'}={}&M_{(AB)}{}^{(CDF)A'},& N_{AB}{}^{CD}={}&N_{(AB)}{}^{(CD)}.\end{aligned}$$ Here, we have used the reduction of the derivatives to the $\sTwist$ operator as discussed in Section \[sec:ReductionMaxwell\]. The symmetries comes from the symmetries of $ (\sTwist_{2,0} \phi)_{ABC}{}^{A'}$ and $(\sTwist_{3,1} \sTwist_{2,0} \phi)_{ABCD}{}^{A'B'}$. To be able to make a systematic treatment of the dependence of different components of the coefficients, we will use the irreducible decompositions $$\begin{aligned} L_{AB}{}^{CDFHA'B'}={}&\underset{6,2}{L}{}_{AB}{}^{CDFHA'B'} - \tfrac{4}{3} \epsilon_{(A}{}^{(C}\underset{4,2}{L}{}^{DFH)}{}_{B)}{}^{A'B'} - \tfrac{3}{5} \epsilon^{(C}{}_{(A}\epsilon_{B)}{}^{D}\underset{2,2}{L}{}^{FH)A'B'},\\ M_{AB}{}^{CDFA'}={}&\underset{5,1}{M}{}_{AB}{}^{CDFA'} - \tfrac{6}{5} \epsilon_{(A}{}^{(C}\underset{3,1}{M}{}^{DF)}{}_{B)}{}^{A'} - \tfrac{1}{2} \epsilon^{(C}{}_{(A}\epsilon_{B)}{}^{D}\underset{1,1}{M}{}^{F)A'},\\ N_{AB}{}^{CD}={}&\underset{4,0}{N}{}_{AB}{}^{CD} - \epsilon_{(A}{}^{(C}\underset{2,0}{N}{}^{D)}{}_{B)} - \tfrac{1}{3} \underset{0,0}{N}{} \epsilon_{(A}{}^{(C}\epsilon^{D)}{}_{B)}.\end{aligned}$$ where the different irreducible parts are $$\begin{aligned} \underset{2,2}{L}{}_{AB}{}^{A'B'}\equiv{}&L^{CD}{}_{ABCD}{}^{A'B'},& \underset{1,1}{M}{}_{A}{}^{A'}\equiv{}&M^{BC}{}_{ABC}{}^{A'},& \underset{0,0}{N}{}\equiv{}&N^{AB}{}_{AB},\\ \underset{4,2}{L}{}_{ABCD}{}^{A'B'}\equiv{}&L_{(A}{}^{F}{}_{BCD)F}{}^{A'B'},& \underset{3,1}{M}{}_{ABC}{}^{A'}\equiv{}&M_{(A}{}^{D}{}_{BC)D}{}^{A'},& \underset{2,0}{N}{}_{AB}\equiv{}&N_{(A}{}^{C}{}_{B)C},\\ \underset{6,2}{L}{}_{ABCDFH}{}^{A'B'}\equiv{}&L_{(ABCDFH)}{}^{A'B'},& \underset{5,1}{M}{}_{ABCDF}{}^{A'}\equiv{}&M_{(ABCDF)}{}^{A'},& \underset{4,0}{N}{}_{ABCD}\equiv{}&N_{(ABCD)}.\end{aligned}$$ Now, we want the operator to be a symmetry operator, which means that $$(\sCurlDagger_{2,0}\chi)_{AA'}=0.\label{eq:chiMaxwell}$$ Using the results from the previous subsection, we see that this equation can be reduced to a linear combination of the spinors $(\sTwist_{2,4} \sTwist_{1,3} \sTwist_{0,2} \phi)^{A'B'C'}{}_{ABCDF}$, $(\sTwist_{1,3} \sTwist_{0,2} \phi)^{A'B'}{}_{ABCD}$, $(\sTwist_{0,2} \phi)^{A'}{}_{ABC}$ and $\phi_{AB}$. For a general Maxwell field and an arbitrary point on the manifold, there are no relations between these spinors. Hence, they are independent, and therefore their coefficients have to vanish individually. After the reduction of the derivatives of the Maxwell field to the $\sTwist$ operator, we can therefore study the different order derivatives of $\phi_{AB}$ in separately. ### Third order part The third order derivative terms of are $$\begin{aligned} \label{eq:thirdorderpartmaxwell} 0={}&\tfrac{2}{3} \underset{4,2}{L}{}_{BCDFB'C'} (\sTwist_{4,2} \sTwist_{3,1} \sTwist_{2,0} \phi)_{A}{}^{BCDF}{}_{A'}{}^{B'C'}\nonumber\\ & + \underset{6,2}{L}{}_{ABCDFHB'C'} (\sTwist_{4,2} \sTwist_{3,1} \sTwist_{2,0} \phi)^{BCDFH}{}_{A'}{}^{B'C'}.\end{aligned}$$ We can multiply this with an arbitrary vector field $T^{AA'}$ and split $(\sTwist_{4,2} \sTwist_{3,1} \sTwist_{2,0} \phi)^{ABCDFA'B'C'}T^{H}{}_{A'}$ into irreducible parts. Then we get $$\begin{aligned} 0={}&\underset{6,2}{L}{}_{ABCDFHB'C'} T^{(A|A'|}(\sTwist_{4,2} \sTwist_{3,1} \sTwist_{2,0} \phi)^{BCDFH)}{}_{A'}{}^{B'C'}\nonumber\\ & + \tfrac{2}{3} \underset{4,2}{L}{}_{BCDFB'C'} T_{AA'} (\sTwist_{4,2} \sTwist_{3,1} \sTwist_{2,0} \phi)^{ABCDFA'B'C'}.\end{aligned}$$ The argument in Section \[sec:indepspinors\] gives that the symmetrized coefficients of the irreducible parts $T^{(A|A'|}(\sTwist_{4,2} \sTwist_{3,1} \sTwist_{2,0} \phi)^{BCDFH)}{}_{A'}{}^{B'C'}$ and $T_{AA'} (\sTwist_{4,2} \sTwist_{3,1} \sTwist_{2,0} \phi)^{ABCDFA'B'C'}$ must vanish. This means that is equivalent to the system \[eq:ThirdOrderSystem\] $$\begin{aligned} \underset{6,2}{L}{}_{ABCDFHB'C'}={}&0,\label{eq:L62}\\ \underset{4,2}{L}{}_{BCDFB'C'}={}&0.\label{eq:L42}\end{aligned}$$ The only remaining irreducible component of $L_{AB}{}^{CDFHA'B'}$ is $\underset{2,2}{L}{}_{AB}{}^{A'B'}$. ### Second order part If we use everything above we find that the second order part of reduces to $$\begin{aligned} 0={}&\tfrac{3}{5} \underset{3,1}{M}{}^{BCDB'} (\sTwist_{3,1} \sTwist_{2,0} \phi)_{ABCDA'B'} - \tfrac{2}{5} (\sCurl_{2,2} \underset{2,2}{L}{})^{BCDB'} (\sTwist_{3,1} \sTwist_{2,0} \phi)_{ABCDA'B'}\nonumber\\ & + \tfrac{3}{5} (\sTwist_{2,2} \underset{2,2}{L}{})^{BCD}{}_{A'}{}^{B'C'} (\sTwist_{3,1} \sTwist_{2,0} \phi)_{ABCDB'C'} + \underset{5,1}{M}{}_{ABCDFB'} (\sTwist_{3,1} \sTwist_{2,0} \phi)^{BCDF}{}_{A'}{}^{B'}.\end{aligned}$$ Again contracting with an arbitrary vector $T^{AA'}$ and splitting $(\sTwist_{3,1} \sTwist_{2,0} \phi)^{ABCDA'B'}T^{FC'}$ into irreducible parts we find $$\begin{aligned} 0={}&\underset{5,1}{M}{}_{ABCDFB'} T^{(A|A'|}(\sTwist_{3,1} \sTwist_{2,0} \phi)^{BCDF)}{}_{A'}{}^{B'}\nonumber\\ & - \tfrac{3}{5} T^{A(A'}(\sTwist_{3,1} \sTwist_{2,0} \phi)_{A}{}^{|BCD|B'C')} (\sTwist_{2,2} \underset{2,2}{L}{})_{BCDA'B'C'}\nonumber\\ & + T_{AA'} (\tfrac{3}{5} \underset{3,1}{M}{}_{BCDB'} - \tfrac{2}{5} (\sCurl_{2,2} \underset{2,2}{L}{})_{BCDB'}) (\sTwist_{3,1} \sTwist_{2,0} \phi)^{ABCDA'B'}.\end{aligned}$$ Again using the argument in Section \[sec:indepspinors\] we find \[eq:SecondOrderSystem\] $$\begin{aligned} (\sTwist_{2,2} \underset{2,2}{L}{})_{BCD}{}^{A'B'C'}={}&0,\label{eq:TwistL22}\\ \underset{5,1}{M}{}_{ABCDFB'}={}&0,\label{eq:M51}\\ \underset{3,1}{M}{}_{BCDB'}={}&\tfrac{2}{3} (\sCurl_{2,2} \underset{2,2}{L}{})_{BCDB'}.\label{eq:M31ToL22}\end{aligned}$$ ### First order part Now, after contracting the first order part of with an arbitrary tensor $T^{A}{}_{A'}$, splitting $(\sTwist_{2,0} \phi)_{ABCA'}T_{DB'}$ into irreducible parts, and using the argument in Section \[sec:indepspinors\], we find that the first order part of is equivalent to the system $$\begin{aligned} \underset{2,0}{N}{}_{BC}={}&\tfrac{1}{2} (\sCurl_{1,1} \underset{1,1}{M}{})_{BC} - \tfrac{1}{2} (\sDiv_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{BC} - \tfrac{3}{2} \underset{2,2}{L}{}_{(B}{}^{DB'C'}\Phi_{C)DB'C'},\label{eq:N20eq1}\\ \underset{4,0}{N}{}_{ABCD}={}&\tfrac{1}{5} (\sCurl_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{ABCD} + \tfrac{3}{10} \underset{2,2}{L}{}_{(AB}{}^{B'C'}\Phi_{CD)B'C'},\label{eq:N40eq1}\\ (\sTwist_{1,1} \underset{1,1}{M}{})_{BC}{}^{A'B'}={}&12 \Lambda \underset{2,2}{L}{}_{BC}{}^{A'B'} - \tfrac{9}{5} \underset{2,2}{L}{}^{DFA'B'} \Psi_{BCDF} - \tfrac{6}{5} \underset{2,2}{L}{}_{BC}{}^{C'D'} \bar\Psi^{A'B'}{}_{C'D'}\nonumber\\ & + (\sCurlDagger_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{BC}{}^{A'B'} - 6 \underset{2,2}{L}{}^{D}{}_{(B}{}^{C'(A'}\Phi_{C)D}{}^{B')}{}_{C'},\label{eq:TwistM11Eq1}\\ (\sTwist_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{ABCD}{}^{A'B'}={}&3 \underset{2,2}{L}{}_{(AB}{}^{C'(A'}\Phi_{CD)}{}^{B')}{}_{C'} + 3 \underset{2,2}{L}{}_{(A}{}^{FA'B'}\Psi_{BCD)F}.\label{eq:TwistCurlFirstOrder}\end{aligned}$$ The commutators , and applied to $\underset{2,2}{L}{}{}_{AB}{}^{A'B'}$ yield $$\begin{aligned} (\sDiv_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{AB}={}&\tfrac{2}{3} (\sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{AB} - 2 \underset{2,2}{L}{}_{(A}{}^{CA'B'}\Phi_{B)CA'B'},\label{eq:CommDivCurlL22}\\ (\sCurlDagger_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{AB}{}^{A'B'}={}&-12 \Lambda \underset{2,2}{L}{}_{AB}{}^{A'B'} + \underset{2,2}{L}{}^{CDA'B'} \Psi_{ABCD} + 2 \underset{2,2}{L}{}_{AB}{}^{C'D'} \bar\Psi^{A'B'}{}_{C'D'}\nonumber\\ & - \tfrac{3}{2} (\sDiv_{3,3} \sTwist_{2,2} \underset{2,2}{L}{})_{AB}{}^{A'B'} + 6 \underset{2,2}{L}{}^{C}{}_{(A}{}^{C'(A'}\Phi_{B)C}{}^{B')}{}_{C'}\nonumber\\ & + \tfrac{4}{3} (\sTwist_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{AB}{}^{A'B'},\label{eq:CommCurlDaggerCurlL22}\\ (\sTwist_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{ABCD}{}^{A'B'}={}&\tfrac{3}{2} (\sCurl_{3,3} \sTwist_{2,2} \underset{2,2}{L}{})_{ABCD}{}^{A'B'} + 3 \underset{2,2}{L}{}_{(AB}{}^{C'(A'}\Phi_{CD)}{}^{B')}{}_{C'}\nonumber\\ & + 3 \underset{2,2}{L}{}_{(A}{}^{FA'B'}\Psi_{BCD)F}.\label{eq:CommTwistCurlL22}\end{aligned}$$ It is now clear that is a consequence of and . The commutators and together with can be used to reduce and to $$\begin{aligned} \underset{2,0}{N}{}_{BC}={}&\tfrac{1}{2} (\sCurl_{1,1} \underset{1,1}{M}{})_{BC} - \tfrac{1}{3} (\sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{BC} - \tfrac{1}{2} \underset{2,2}{L}{}_{(B}{}^{DB'C'}\Phi_{C)DB'C'},\label{eq:N20eq2}\\ (\sTwist_{1,1} \underset{1,1}{M}{})_{BCA'B'}={}&- \tfrac{4}{5} \underset{2,2}{L}{}^{DF}{}_{A'B'} \Psi_{BCDF} + \tfrac{4}{5} \underset{2,2}{L}{}_{BC}{}^{C'D'} \bar\Psi_{A'B'C'D'} + \tfrac{4}{3} (\sTwist_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{BCA'B'}.\label{eq:TwistM11Eq2}\end{aligned}$$ Now, in view of the form of we make the ansatz $$\begin{aligned} \underset{1,1}{M}{}_{AA'}={}&2 P_{AA'} + \tfrac{4}{3} (\sDiv_{2,2} \underset{2,2}{L}{})_{AA'},\label{eq:M11Ansatz}\end{aligned}$$ where $P_{AA'}$ is a new spinor field. With this choice and reduce to $$\begin{aligned} \underset{2,0}{N}{}_{BC}={}&(\sCurl_{1,1} P)_{BC} + \tfrac{1}{3} (\sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{BC} - \tfrac{1}{2} \underset{2,2}{L}{}_{(B}{}^{DB'C'}\Phi_{C)DB'C'},\label{eq:N20eq3}\\ (\sTwist_{1,1} P)_{BCA'B'}={}&- \tfrac{2}{5} \underset{2,2}{L}{}^{DF}{}_{A'B'} \Psi_{BCDF} + \tfrac{2}{5} \underset{2,2}{L}{}_{BC}{}^{C'D'} \bar\Psi_{A'B'C'D'}.\label{eq:TwistPeq1}\end{aligned}$$ In conclusion, the third, second and first order parts of vanishes if and only if , , , , and are satisfied. ### Zeroth order part {#zeroth-order-part-1} After making irreducible decompositions of the derivatives, using and contracting the remaining part of with an arbitrary tensor $T^{AA'}$, splitting $T_{AA'} \phi_{CD}$ into irreducible parts, and using the argument in Section \[sec:indepspinors\], we find that the order zero part of is equivalent to the system $$\begin{aligned} 0={}&4 \Lambda P_{BA'} - \tfrac{4}{3} \Phi_{BCA'B'} P^{CB'} + \tfrac{2}{9} \Phi^{CD}{}_{A'}{}^{B'} (\sCurl_{2,2} \underset{2,2}{L}{})_{BCDB'} - \tfrac{8}{15} \Psi_{BCDF} (\sCurl_{2,2} \underset{2,2}{L}{})^{CDF}{}_{A'}\nonumber\\* & + \tfrac{26}{45} \underset{2,2}{L}{}^{CD}{}_{A'}{}^{B'} (\sCurl_{2,2} \Phi)_{BCDB'} + \tfrac{2}{9} \Phi_{B}{}^{CB'C'} (\sCurlDagger_{2,2} \underset{2,2}{L}{})_{CA'B'C'} + \tfrac{2}{45} \underset{2,2}{L}{}_{B}{}^{CB'C'} (\sCurlDagger_{2,2} \Phi)_{CA'B'C'}\nonumber\\* & + \tfrac{2}{3} (\sCurlDagger_{2,0} \sCurl_{1,1} P)_{BA'} + \tfrac{2}{9} (\sCurlDagger_{2,0} \sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{BA'} + \tfrac{8}{3} \Lambda (\sDiv_{2,2} \underset{2,2}{L}{})_{BA'} - \tfrac{20}{27} \Phi_{BCA'B'} (\sDiv_{2,2} \underset{2,2}{L}{})^{CB'}\nonumber\\* & + \tfrac{28}{9} \underset{2,2}{L}{}_{BCA'B'} (\sTwist_{0,0} \Lambda)^{CB'} - \tfrac{1}{3} (\sTwist_{0,0} \underset{0,0}{N}{})_{BA'} - \tfrac{1}{3} \underset{2,2}{L}{}^{CDB'C'} (\sTwist_{2,2} \Phi)_{BCDA'B'C'},\label{eq:OrderZeroEq1}\\ 0={}&P^{D}{}_{A'} \Psi_{ABCD} + 2 \Lambda (\sCurl_{2,2} \underset{2,2}{L}{})_{ABCA'} + \tfrac{1}{5} (\sCurlDagger_{4,0} \sCurl_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{ABCA'} + \tfrac{2}{3} \Psi_{ABCD} (\sDiv_{2,2} \underset{2,2}{L}{})^{D}{}_{A'}\nonumber\\* & - \tfrac{37}{75} \underset{2,2}{L}{}_{(A}{}^{D}{}_{|A'|}{}^{B'}(\sCurl_{2,2} \Phi)_{BC)DB'} + \tfrac{7}{30} \underset{2,2}{L}{}_{(AB}{}^{B'C'}(\sCurlDagger_{2,2} \Phi)_{C)A'B'C'} - \tfrac{5}{3} \underset{2,2}{L}{}_{(AB|A'|}{}^{B'}(\sTwist_{0,0} \Lambda)_{C)B'}\nonumber\\* & + \tfrac{7}{10} \underset{2,2}{L}{}_{(A}{}^{DB'C'}(\sTwist_{2,2} \Phi)_{BC)DA'B'C'} - \Phi_{(AB|A'|}{}^{B'}P_{C)B'} - \tfrac{19}{15} \Phi_{(A}{}^{D}{}_{|A'|}{}^{B'}(\sCurl_{2,2} \underset{2,2}{L}{})_{BC)DB'}\nonumber\\* & + \tfrac{1}{6} \Phi_{(AB}{}^{B'C'}(\sCurlDagger_{2,2} \underset{2,2}{L}{})_{C)A'B'C'} - \tfrac{5}{9} \Phi_{(AB|A'|}{}^{B'}(\sDiv_{2,2} \underset{2,2}{L}{})_{C)B'} - \tfrac{1}{5} \Psi_{(AB}{}^{DF}(\sCurl_{2,2} \underset{2,2}{L}{})_{C)DFA'}\nonumber\\* & + \tfrac{3}{5} \underset{2,2}{L}{}^{DF}{}_{A'}{}^{B'} (\sTwist_{4,0} \Psi)_{ABCDFB'} - \tfrac{1}{2} (\sTwist_{2,0} \sCurl_{1,1} P)_{ABCA'} - \tfrac{1}{6} (\sTwist_{2,0} \sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{ABCA'}.\label{eq:OrderZeroEq2}\end{aligned}$$ Applying the commutators repeatedly we have in general that $$\begin{aligned} (\sCurlDagger_{4,0} &\sCurl_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{ABC}{}^{A'}\nonumber\\ ={}&\tfrac{5}{3} \square^{A'}{}_{B'}(\sCurl_{2,2} \underset{2,2}{L}{})_{ABC}{}^{B'} - \tfrac{4}{3} (\sDiv_{4,2} \sTwist_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{ABC}{}^{A'} - \square_{(A}{}^{D}(\sCurl_{2,2} \underset{2,2}{L}{})_{BC)D}{}^{A'}\nonumber\\ & + \tfrac{5}{4} (\sTwist_{2,0} \sDiv_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{ABC}{}^{A'}\nonumber\\ ={}&\tfrac{5}{3} \square^{A'}{}_{B'}(\sCurl_{2,2} \underset{2,2}{L}{})_{ABC}{}^{B'} - 2 (\sDiv_{4,2} \sCurl_{3,3} \sTwist_{2,2} \underset{2,2}{L}{})_{ABC}{}^{A'} - \square_{(A}{}^{D}(\sCurl_{2,2} \underset{2,2}{L}{})_{BC)D}{}^{A'}\nonumber\\ & - \nabla^{DB'}\square_{(AB}\underset{2,2}{L}{}_{C)D}{}^{A'}{}_{B'} - \nabla^{DB'}\square_{(A|D|}\underset{2,2}{L}{}_{BC)}{}^{A'}{}_{B'} - \tfrac{5}{4} \nabla_{(A}{}^{A'}\square^{B'C'}\underset{2,2}{L}{}_{BC)B'C'}\nonumber\\ & + \tfrac{5}{6} (\sTwist_{2,0} \sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{ABC}{}^{A'}\nonumber\\ ={}&-10 \Lambda (\sCurl_{2,2} \underset{2,2}{L}{})_{ABC}{}^{A'} - 2 \Psi_{ABCD} (\sDiv_{2,2} \underset{2,2}{L}{})^{DA'} - 2 (\sDiv_{4,2} \sCurl_{3,3} \sTwist_{2,2} \underset{2,2}{L}{})_{ABC}{}^{A'}\nonumber\\ & + \tfrac{25}{3} \underset{2,2}{L}{}_{(AB}{}^{A'B'}(\sTwist_{0,0}\Lambda)_{C)B'} + \tfrac{49}{15} \underset{2,2}{L}{}_{(A}{}^{DA'B'}(\sCurl_{2,2} \Phi)_{BC)DB'} + \tfrac{5}{6} \underset{2,2}{L}{}_{(AB}{}^{B'C'}(\sCurlDagger_{2,2} \Phi)_{C)}{}^{A'}{}_{B'C'}\nonumber\\ & - \tfrac{7}{2} \underset{2,2}{L}{}_{(A}{}^{DB'C'}(\sTwist_{2,2} \Phi)_{BC)D}{}^{A'}{}_{B'C'} + \tfrac{19}{3} \Phi_{(A}{}^{DA'B'}(\sCurl_{2,2} \underset{2,2}{L}{})_{BC)DB'}\nonumber\\ & - \tfrac{5}{6} \Phi_{(AB}{}^{B'C'}(\sCurlDagger_{2,2} \underset{2,2}{L}{})_{C)}{}^{A'}{}_{B'C'} + \tfrac{25}{9} \Phi_{(AB}{}^{A'B'}(\sDiv_{2,2} \underset{2,2}{L}{})_{C)B'}\nonumber\\ & + \tfrac{7}{2} \Phi_{(A}{}^{DB'C'}(\sTwist_{2,2} \underset{2,2}{L}{})_{BC)D}{}^{A'}{}_{B'C'} - \Psi_{(AB}{}^{DF}(\sCurl_{2,2} \underset{2,2}{L}{})_{C)DF}{}^{A'}\nonumber\\ & - \underset{2,2}{L}{}^{DFA'B'} (\sTwist_{4,0} \Psi)_{ABCDFB'} + \tfrac{5}{6} (\sTwist_{2,0} \sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{ABC}{}^{A'}.\end{aligned}$$ With this and , the equation reduces to $$\begin{aligned} 0={}&P^{D}{}_{A'} \Psi_{ABCD} + \tfrac{4}{15} \Psi_{ABCD} (\sDiv_{2,2} \underset{2,2}{L}{})^{D}{}_{A'} + \tfrac{4}{25} \underset{2,2}{L}{}_{(A}{}^{D}{}_{|A'|}{}^{B'}(\sCurl_{2,2} \Phi)_{BC)DB'}\nonumber\\* & + \tfrac{2}{5} \underset{2,2}{L}{}_{(AB}{}^{B'C'}(\sCurlDagger_{2,2} \Phi)_{C)A'B'C'} - \Phi_{(AB|A'|}{}^{B'}P_{C)B'} - \tfrac{2}{5} \Psi_{(AB}{}^{DF}(\sCurl_{2,2} \underset{2,2}{L}{})_{C)DFA'}\nonumber\\* & + \tfrac{2}{5} \underset{2,2}{L}{}^{DF}{}_{A'}{}^{B'} (\sTwist_{4,0} \Psi)_{ABCDFB'} - \tfrac{1}{2} (\sTwist_{2,0} \sCurl_{1,1} P)_{ABCA'}.\end{aligned}$$ Using the commutator $$\begin{aligned} (\sTwist_{2,0} \sCurl_{1,1} P)_{ABCA'}={}&2 P^{D}{}_{A'} \Psi_{ABCD} + 2 (\sCurl_{2,2} \sTwist_{1,1} P)_{ABCA'} - 2 \Phi_{(AB|A'|}{}^{B'}P_{C)B'}.\end{aligned}$$ this becomes $$\begin{aligned} 0={}&- (\sCurl_{2,2} \sTwist_{1,1} P)_{ABCA'} + \tfrac{4}{15} \Psi_{ABCD} (\sDiv_{2,2} \underset{2,2}{L}{})^{D}{}_{A'} + \tfrac{4}{25} \underset{2,2}{L}{}_{(A}{}^{D}{}_{|A'|}{}^{B'}(\sCurl_{2,2} \Phi)_{BC)DB'}\nonumber\\ & + \tfrac{2}{5} \underset{2,2}{L}{}_{(AB}{}^{B'C'}(\sCurlDagger_{2,2} \Phi)_{C)A'B'C'} - \tfrac{2}{5} \Psi_{(AB}{}^{DF}(\sCurl_{2,2} \underset{2,2}{L}{})_{C)DFA'} + \tfrac{2}{5} \underset{2,2}{L}{}^{DF}{}_{A'}{}^{B'} (\sTwist_{4,0} \Psi)_{ABCDFB'}.\end{aligned}$$ However, after substituting in this equation, decomposing the derivatives into irreducible parts and using , this equation actually becomes trivial. Doing the same calculations as for the Dirac-Weyl case we see that also holds for the Maxwell case. Directly from the commutators we find $$\begin{aligned} (\sCurlDagger_{2,0} \sCurl_{1,1} P)_{AA'}={}&-6 \Lambda P_{AA'} + 2 \Phi_{ABA'B'} P^{BB'} - (\sDiv_{2,2} \sTwist_{1,1} P)_{AA'} + \tfrac{3}{4} (\sTwist_{0,0} \sDiv_{1,1} P)_{AA'}.\end{aligned}$$ With this, and we can reduce to $$\begin{aligned} 0={}&- \tfrac{2}{15} \Phi^{CD}{}_{A'}{}^{B'} (\sCurl_{2,2} \underset{2,2}{L}{})_{BCDB'} - \tfrac{4}{15} \Psi_{BCDF} (\sCurl_{2,2} \underset{2,2}{L}{})^{CDF}{}_{A'} + \tfrac{2}{15} \underset{2,2}{L}{}^{CD}{}_{A'}{}^{B'} (\sCurl_{2,2} \Phi)_{BCDB'}\nonumber\\ & + \tfrac{4}{15} \bar\Psi_{A'B'C'D'} (\sCurlDagger_{2,2} \underset{2,2}{L}{})_{B}{}^{B'C'D'} - \tfrac{2}{15} \Phi_{B}{}^{CB'C'} (\sCurlDagger_{2,2} \underset{2,2}{L}{})_{CA'B'C'} - \tfrac{2}{5} \underset{2,2}{L}{}_{B}{}^{CB'C'} (\sCurlDagger_{2,2} \Phi)_{CA'B'C'}\nonumber\\ & - \tfrac{4}{45} \Phi_{BCA'B'} (\sDiv_{2,2} \underset{2,2}{L}{})^{CB'} - \tfrac{2}{3} (\sDiv_{2,2} \sTwist_{1,1} P)_{BA'} + \tfrac{4}{15} \underset{2,2}{L}{}_{BCA'B'} (\sTwist_{0,0} \Lambda)^{CB'} - \tfrac{1}{3} (\sTwist_{0,0} \underset{0,0}{N}{})_{BA'}\nonumber\\ & - \tfrac{1}{5} \underset{2,2}{L}{}^{CDB'C'} (\sTwist_{2,2} \Phi)_{BCDA'B'C'} + \tfrac{1}{2} (\sTwist_{0,0} \sDiv_{1,1} P)_{BA'} + \tfrac{2}{15} (\sTwist_{0,0} \sDiv_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{BA'}.\end{aligned}$$ Using and the irreducible decompositions, we find $$\begin{aligned} (\sDiv_{2,2} \sTwist_{1,1} P)_{BA'}={}&- \tfrac{2}{5} \Psi_{BCDF} (\sCurl_{2,2} \underset{2,2}{L}{})^{CDF}{}_{A'} + \tfrac{2}{5} \underset{2,2}{L}{}^{CD}{}_{A'}{}^{B'} (\sCurl_{2,2} \Phi)_{BCDB'}\nonumber\\ & + \tfrac{2}{5} \bar\Psi_{A'B'C'D'} (\sCurlDagger_{2,2} \underset{2,2}{L}{})_{B}{}^{B'C'D'} - \tfrac{2}{5} \underset{2,2}{L}{}_{B}{}^{CB'C'} (\sCurlDagger_{2,2} \Phi)_{CA'B'C'}.\label{eq:DivTwistQ2}\end{aligned}$$ To simplify the remaining terms, we use the same trick as for the Dirac-Weyl case. The definition and the equation can be used together with , to reduce the equation to $$\begin{aligned} 0={}&- \tfrac{1}{3} (\sTwist_{0,0} \underset{0,0}{N}{})_{BA'} - \tfrac{1}{5} (\sTwist_{0,0} \Upsilon)_{BA'} + \tfrac{1}{2} (\sTwist_{0,0} \sDiv_{1,1} P)_{BA'} + \tfrac{2}{15} (\sTwist_{0,0} \sDiv_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{BA'}.\label{eq:OrderZeroEq3}\end{aligned}$$ We therefore make the ansatz $$\begin{aligned} \underset{0,0}{N}{}={}&3 Q - \tfrac{3}{5} \Upsilon + \tfrac{3}{2} (\sDiv_{1,1} P) + \tfrac{2}{5} (\sDiv_{1,1} \sDiv_{2,2} \underset{2,2}{L}{}).\label{eq:N00Ansatz}\end{aligned}$$ Now, becomes $$\begin{aligned} 0={}&(\sTwist_{0,0} Q)_{AA'}.\label{eq:OrderZeroEq4}\end{aligned}$$ ### Potential representation From all this we can conclude that the only equations that restrict the geometry are and . Now, the operator takes the form $$\begin{aligned} \chi_{AB}={}&\tfrac{1}{3} \underset{0,0}{N}{} \phi_{AB} + \underset{4,0}{N}{}_{ABCD} \phi^{CD} - \underset{2,0}{N}{}_{(A}{}^{C}\phi_{B)C} - \tfrac{4}{5} (\sCurl_{2,2} \underset{2,2}{L}{})_{(A}{}^{CDA'}(\sTwist_{2,0} \phi)_{B)CDA'}\nonumber\\ & + \tfrac{1}{2} \underset{1,1}{M}{}_{CA'} (\sTwist_{2,0} \phi)_{AB}{}^{CA'} + \tfrac{3}{5} \underset{2,2}{L}{}^{CDA'B'} (\sTwist_{3,1} \sTwist_{2,0} \phi)_{ABCDA'B'}. \label{eq:chiform1}\end{aligned}$$ where $\underset{0,0}{N}$, $\underset{2,0}{N}{}_{AB}$, $\underset{4,0}{N}{}_{ABCD}$, $\underset{1,1}{M}{}_{AA'}$ are given by , , and respectively. We can in fact simplify this expression by defining the following spinor $$\begin{aligned} A_{AA'}\equiv{}&P_{BA'} \phi_{A}{}^{B} + \tfrac{1}{5} \phi^{BC} (\sCurl_{2,2} \underset{2,2}{L}{})_{ABCA'} + \tfrac{4}{15} \phi_{A}{}^{B} (\sDiv_{2,2} \underset{2,2}{L}{})_{BA'} + \tfrac{3}{5} \underset{2,2}{L}{}_{BCA'B'} (\sTwist_{2,0} \phi)_{A}{}^{BCB'}.\end{aligned}$$ Substituting this onto the following, and comparing with , we find $$\begin{aligned} (\sCurl_{1,1} A)_{AB}={}&- Q \phi_{AB} + \chi_{AB} - \tfrac{1}{15} \phi_{(A}{}^{C}(\sCurl_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{B)C} + \tfrac{1}{10} \phi_{(A}{}^{C}(\sDiv_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{B)C}\nonumber\\ & - \tfrac{1}{10} \underset{2,2}{L}{}_{(A}{}^{CA'B'}\Phi_{|C}{}^{D}{}_{A'B'|}\phi_{B)D} - \tfrac{1}{10} \underset{2,2}{L}{}^{CDA'B'}\Phi_{(A|CA'B'|}\phi_{B)D}\nonumber\\ ={}&- Q \phi_{AB} + \chi_{AB},\label{eq:CurlAeq1}\end{aligned}$$ where the last equality follows from a commutator relation. In fact the coefficients in $A_{AA'}$ were initially left free, and then chosen so all first and second order derivatives of $\phi_{AB}$ where eliminated in . We also get $$\begin{aligned} (\sCurlDagger_{1,1} A)_{A'B'}={}&\tfrac{12}{5} \Lambda \underset{2,2}{L}{}_{ABA'B'} \phi^{AB} - \tfrac{3}{5} \underset{2,2}{L}{}^{CD}{}_{A'B'} \Psi_{ABCD} \phi^{AB} + \tfrac{1}{5} \phi^{AB} (\sCurlDagger_{3,1} \sCurl_{2,2} \underset{2,2}{L}{})_{ABA'B'}\nonumber\\ & - \tfrac{6}{5} \underset{2,2}{L}{}^{AB}{}_{(A'}{}^{C'}\Phi_{|A|}{}^{C}{}_{B')C'}\phi_{BC} - \phi^{AB} (\sTwist_{1,1} P)_{ABA'B'}\nonumber\\ & - \tfrac{3}{5} (\sTwist_{2,2} \underset{2,2}{L}{})_{ABCA'B'C'} (\sTwist_{2,0} \phi)^{ABCC'} - \tfrac{4}{15} \phi^{AB} (\sTwist_{1,1} \sDiv_{2,2} \underset{2,2}{L}{})_{ABA'B'}\nonumber\\ ={}&0.\end{aligned}$$ where we in the last step used , a commutator and To get the highest order coefficient equal to 1 in $A_{AA'}$ and in $\chi_{AB}$, we define a new symmetric spinor, which is just a rescaling of $\underset{2,2}{L}{}_{ABA'B'}$ $$\begin{aligned} L_{ABA'B'}\equiv{}&\tfrac{3}{5} \underset{2,2}{L}{}_{ABA'B'}.\end{aligned}$$ Now, the only equations we have left are $$\begin{aligned} (\sTwist_{2,2} L)_{ABC}{}^{A'B'C'}={}&0,\\ (\sTwist_{1,1} P)_{AB}{}^{A'B'}={}&- \tfrac{2}{3} (\ObstrOne L)_{AB}{}^{A'B'},\\ (\sTwist_{0,0} Q)_{BA'}={}&0,\\ A_{AA'}={}&P_{BA'} \phi_{A}{}^{B} + \tfrac{1}{3} \phi^{BC} (\sCurl_{2,2} L)_{ABCA'} + \tfrac{4}{9} \phi_{A}{}^{B} (\sDiv_{2,2} L)_{BA'}\nonumber\\ & - L^{BC}{}_{A'}{}^{B'} (\sTwist_{2,0} \phi)_{ABCB'}.\end{aligned}$$ Second kind of symmetry operator for the Maxwell equation {#subsection:second_kind} --------------------------------------------------------- For the symmetry operators of the second kind, one can follow the same procedure as above. However, this case was completely handled in [@KalMcLWil92a]. In that paper it was shown that a symmetry operator of the second kind always has the form $\phi_{AB}\rightarrow \omega_{A'B'}$, $$\begin{aligned} \omega_{A'B'}={}&\tfrac{3}{5} \phi^{CD} (\sCurlDagger_{3,1} \sCurlDagger_{4,0} L)_{CDA'B'} - \tfrac{8}{5} (\sCurlDagger_{4,0} L)^{CDF}{}_{(A'}(\sTwist_{2,0} \phi)_{|CDF|B')}\nonumber\\ & + L^{CDFH} (\sTwist_{3,1} \sTwist_{2,0} \phi)_{CDFHA'B'},\end{aligned}$$ where $L_{ABCD}=L_{(ABCD)}$ satisfies $$\begin{aligned} (\sTwist_{4,0} L)_{ABCDF}{}^{A'}={}&0.\label{eq:LSecKillingSpinor}\end{aligned}$$ Hence, the treatment in [@KalMcLWil92a] is satisfactory. However, it is interesting to see if the operator can be written in terms of a potential. Let $$\begin{aligned} B_{AA'}\equiv{}&\tfrac{3}{5} \phi^{BC} (\sCurlDagger_{4,0} L)_{ABCA'} + L_{ABCD} (\sTwist_{2,0} \phi)^{BCD}{}_{A'}.\label{eq:BDef}\end{aligned}$$ Then, from the definition of $\sCurlDagger$, the irreducible decompositions and we get $$\begin{aligned} (\sCurlDagger_{1,1} B)_{A'B'}={}&\tfrac{3}{5} \phi^{AB} (\sCurlDagger_{3,1} \sCurlDagger_{4,0} L)_{ABA'B'} - \tfrac{8}{5} (\sCurlDagger_{4,0} L)^{ABC}{}_{(A'}(\sTwist_{2,0} \phi)_{|ABC|B')}\nonumber\\ & + L^{ABCD} (\sTwist_{3,1} \sTwist_{2,0} \phi)_{ABCDA'B'}\nonumber\\ ={}&\omega_{A'B'}.\label{eq:CurlDaggerB}\end{aligned}$$ The coefficients in where initially left free, and then chosen to get . We also get $$\begin{aligned} (\sCurl_{1,1} B)_{AB}={}&6 \Lambda L_{ABCD} \phi^{CD} - \tfrac{3}{2} L_{AB}{}^{FH} \Psi_{CDFH} \phi^{CD} + \tfrac{3}{5} \phi^{CD} (\sCurl_{3,1} \sCurlDagger_{4,0} L)_{ABCD}\nonumber\\ & + \tfrac{3}{10} \phi_{(A}{}^{C}(\sDiv_{3,1} \sCurlDagger_{4,0} L)_{B)C} - \tfrac{3}{2} L_{(A}{}^{CDF}\Psi_{B)CD}{}^{H}\phi_{FH} - \tfrac{1}{2} L_{(A}{}^{CDF}\Psi_{|CDF|}{}^{H}\phi_{B)H}\nonumber\\ & - (\sTwist_{4,0} L)_{ABCDFA'} (\sTwist_{2,0} \phi)^{CDFA'}\nonumber\\ ={}&- \tfrac{1}{2} \phi^{CD} (\sDiv_{5,1} \sTwist_{4,0} L)_{ABCD} - \tfrac{4}{5} \phi_{B}{}^{C} L_{(A}{}^{DFH}\Psi_{C)DFH} - \tfrac{4}{5} \phi_{A}{}^{C} L_{(B}{}^{DFH}\Psi_{C)DFH}\nonumber\\ & - (\sTwist_{4,0} L)_{ABCDFA'} (\sTwist_{2,0} \phi)^{CDFA'}\nonumber\\ ={}&0.\end{aligned}$$ Here, we have used together with the irreducible decomposition of $L_{AB}{}^{FH} \Psi_{CDFH}$ and the relations $$\begin{aligned} (\sDiv_{3,1} \sCurlDagger_{4,0} L)_{AB}={}&-2 L_{(A}{}^{CDF}\Psi_{B)CDF},\\ (\sCurl_{3,1} \sCurlDagger_{4,0} L)_{ABCD}={}&-10 \Lambda L_{ABCD} - \tfrac{5}{6} (\sDiv_{5,1} \sTwist_{4,0} L)_{ABCD} + 5 L_{(AB}{}^{FH}\Psi_{CD)FH},\\ L_{(B}{}^{DFH}\Psi_{C)DFH}={}&0.\end{aligned}$$ The last equation follows from the integrability condition (cf. Section \[sec:integrabilitycond\]) $$\begin{aligned} L_{(ABC}{}^{L}\Psi_{DFH)L} ={}& - \tfrac{1}{4} (\sCurl_{5,1} \sTwist_{4,0} L)_{ABCDFH}=0,\label{eq:intcondKS4}\end{aligned}$$ as explained in [@KalMcLWil92a]. Factorizations {#sec:factorizations} ============== In this section we will consider special cases for which the auxiliary conditions will always have a solution. We will now prove Proposition \[prop:factorize\], considering each case in turn. The case when $L_{ABA'B'}$ factors in terms of conformal Killing vectors ------------------------------------------------------------------------ If $\xi_{AA'}$ and $\zeta_{AA'}$ are conformal Killing vectors, i.e. $$\begin{aligned} (\sTwist_{1,1}\xi)_{AB}{}^{A'B'}&=0,& (\sTwist_{1,1}\zeta)_{AB}{}^{A'B'}&=0,\label{eq:xizetaCKV}\end{aligned}$$ then we have a solution $$\begin{aligned} \mathfrak{L}_{\xi\zeta AB}{}^{A'B'}\equiv{}&\zeta_{(A}{}^{(A'}\xi_{B)}{}^{B')}\label{eq:Lxizetadef}\end{aligned}$$ to the equation $$\begin{aligned} (\sTwist_{2,2}\mathfrak{L}_{\xi\zeta})_{ABC}{}^{A'B'C'}&=0.\end{aligned}$$ Let $$\begin{aligned} \mathfrak{Q}_{\xi\zeta}\equiv{}&\Lambda \zeta^{AA'} \xi_{AA'} + \tfrac{1}{3} \Phi_{ABA'B'} \zeta^{AA'} \xi^{BB'} + \tfrac{1}{8} (\sCurl_{1,1} \zeta)^{AB} (\sCurl_{1,1} \xi)_{AB}\nonumber\\ & + \tfrac{1}{6} \xi^{AA'} (\sCurl_{0,2} \sCurlDagger_{1,1} \zeta)_{AA'} + \tfrac{1}{6} \zeta^{AA'} (\sCurl_{0,2} \sCurlDagger_{1,1} \xi)_{AA'} + \tfrac{1}{8} (\sCurlDagger_{1,1} \zeta)^{A'B'} (\sCurlDagger_{1,1} \xi)_{A'B'}\nonumber\\ & - \tfrac{1}{32} (\sDiv_{1,1} \zeta) (\sDiv_{1,1} \xi),\label{eq:Qxizetadef}\\ \mathfrak{P}_{\xi\zeta AA'}\equiv{}&\tfrac{1}{4} \xi^{B}{}_{A'} (\sCurl_{1,1} \zeta)_{AB} + \tfrac{1}{4} \zeta^{B}{}_{A'} (\sCurl_{1,1} \xi)_{AB} - \tfrac{1}{4} \xi_{A}{}^{B'} (\sCurlDagger_{1,1} \zeta)_{A'B'} - \tfrac{1}{4} \zeta_{A}{}^{B'} (\sCurlDagger_{1,1} \xi)_{A'B'}.\label{eq:Pxizetadef}\end{aligned}$$ Applying the $\sTwist$ operator to the equation , decomposing the derivatives into irreducible parts and using gives a long expression with the operators $\sDiv$, $\sCurl$, $\sCurlDagger$, $\sCurl\sCurlDagger$, $\sCurlDagger\sCurl$, $\sTwist\sDiv$, $\sTwist\sCurl$, $\sTwist\sCurlDagger$, $\sDiv\sCurl\sCurlDagger$, $\sCurl\sCurl\sCurlDagger$ and $\sCurlDagger\sCurl\sCurlDagger$ operating on $\xi^{AA'}$ and $\zeta^{AA'}$. Using the commutators , , , and on the outermost operators and using , the list of operators appearing can be reduced to the set $\sDiv$, $\sCurl$, $\sCurlDagger$, $\sDiv\sTwist\sCurlDagger$, $\sCurl\sTwist\sCurlDagger$ and $\sCurl\sCurl\sCurlDagger$. Then using the relations and on the innermost operators the list of operators appearing is reduced to $\sCurl$, $\sCurlDagger$, $\sCurl\sTwist\sDiv$, where the latter can be eliminated with on the outer operators. After making an irreducible decomposition of $\xi^{AA'}\zeta^{BB'}$ and identifying the symmetric part though , one is left with $$\begin{aligned} (\sTwist_{0,0} \mathfrak{Q}_{\xi\zeta})_{A}{}^{A'}={}&\mathfrak{L}_{\xi\zeta}{}^{BCA'B'} (\sCurl_{2,2} \Phi)_{ABCB'} + \tfrac{1}{4} \Psi_{ABCD} \xi^{BA'} (\sCurl_{1,1} \zeta)^{CD} + \tfrac{1}{4} \Psi_{ABCD} \zeta^{BA'} (\sCurl_{1,1} \xi)^{CD}\nonumber\\ & + \mathfrak{L}_{\xi\zeta}{}_{A}{}^{BB'C'} (\sCurlDagger_{2,2} \Phi)_{B}{}^{A'}{}_{B'C'} + \tfrac{1}{4} \bar\Psi^{A'}{}_{B'C'D'} \xi_{A}{}^{B'} (\sCurlDagger_{1,1} \zeta)^{C'D'}\nonumber\\ & + \tfrac{1}{4} \bar\Psi^{A'}{}_{B'C'D'} \zeta_{A}{}^{B'} (\sCurlDagger_{1,1} \xi)^{C'D'}.\label{eq:TwistQxizetaEq1}\end{aligned}$$ Applying the $\sTwist$ operator to the equation , decomposing the derivatives into irreducible parts and using gives a expression with the operators $\sCurl\sCurlDagger$, $\sCurlDagger\sCurl$, $\sTwist\sCurl$, $\sTwist\sCurlDagger$ operating on $\xi^{AA'}$ and $\zeta^{AA'}$. Using the commutators , and and using , the entire expression can be reduced to only contain curvature terms. After making an irreducible decomposition of $\xi^{AA'}\zeta^{BB'}$ and identifying the symmetric part though , one is left with $$\begin{aligned} (\sTwist_{1,1} \mathfrak{P}_{\xi\zeta}{})_{AB}{}^{A'B'}={}&\mathfrak{L}_{\xi\zeta}{}^{CDA'B'} \Psi_{ABCD} - \mathfrak{L}_{\xi\zeta}{}_{AB}{}^{C'D'} \bar\Psi^{A'B'}{}_{C'D'}.\label{eq:TwistPxizetaEq1}\end{aligned}$$ Substituting into the definition of $\ObstrZero$, allows us to see that and reduces to $$\begin{aligned} (\sTwist_{0,0} \mathfrak{Q}_{\xi\zeta})_{A}{}^{A'}={}&(\ObstrZero \mathfrak{L}_{\xi\zeta})_{A}{}^{A'},\label{eq:Qxizetaaux}\\ (\sTwist_{1,1} \mathfrak{P}_{\xi\zeta})_{AB}{}^{A'B'}={}&(\ObstrOne \mathfrak{L}_{\xi\zeta})_{AB}{}^{A'B'}.\label{eq:Pxizetaaux}\end{aligned}$$ The actual form of and was obtained by making sufficiently general symmetric second order bi-linear ansätze. The coefficients where then chosen to eliminate as many extra terms as possible in and . The case when $L_{ABA'B'}$ factors in terms of Killing spinors -------------------------------------------------------------- Another way of constructing conformal Killing tensors is to make a product of valence $(2,0)$ and valence $(0,2)$ Killing spinors. It turns out that also this case admits solutions to the auxiliary conditions. In principle we could construct $L_{ABA'B'}$ from two different Killing spinors, but if the dimension of the space of Killing spinors is greater than one, the spacetime has to be locally isometric to Minkowski space. In these spacetimes the picture is much simpler and has been studied before. The auxiliary conditions will be trivial in these cases. We will therefore only consider one Killing spinor. Let $\kappa_{AB}$ be a Killing spinor, i.e. a solution to $$\begin{aligned} (\sTwist_{2,0} \kappa)_{ABCA'}={}&0.\label{eq:kappaKS}\end{aligned}$$ We have a solution $$\begin{aligned} \label{eq:Lkappadef} \mathfrak{L}_{\kappa AB}{}^{A'B'}\equiv{}&\kappa_{AB}\bar\kappa^{A'B'},\end{aligned}$$ to the equation $$\begin{aligned} (\sTwist_{2,2}\mathfrak{L}_{\kappa})_{ABC}{}^{A'B'C'}&=0.\end{aligned}$$ Now, let $$\begin{aligned} \mathfrak{Q}_{\kappa}\equiv{}&\tfrac{2}{3} \Phi_{ABA'B'} \kappa^{AB} \bar{\kappa}^{A'B'} + \tfrac{1}{9} \kappa^{AB} (\sCurl_{1,1} \sCurl_{0,2} \bar{\kappa})_{AB} + \tfrac{4}{27} (\sCurl_{0,2} \bar{\kappa})^{AA'} (\sCurlDagger_{2,0} \kappa)_{AA'}\nonumber\\ & + \tfrac{1}{9} \bar{\kappa}^{A'B'} (\sCurlDagger_{1,1} \sCurlDagger_{2,0} \kappa)_{A'B'},\label{eq:Qkappadef}\\ \mathfrak{P}_{\kappa AA'}\equiv{}&\tfrac{4}{3} \kappa_{AB} (\sCurl_{0,2} \bar{\kappa})^{B}{}_{A'} - \tfrac{4}{3} \bar{\kappa}_{A'B'} (\sCurlDagger_{2,0} \kappa)_{A}{}^{B'}.\label{eq:Pkappadef}\end{aligned}$$ Applying the $\sTwist$ operator to the equation , decomposing the derivatives into irreducible parts and using gives a long expression with the operators $\sCurl$, $\sCurlDagger$, $\sDiv\sCurl$, $\sDiv\sCurlDagger$, $\sCurl\sCurlDagger$, $\sCurlDagger\sCurl$, $\sTwist\sCurl$, $\sTwist\sCurlDagger$, $\sCurl\sCurlDagger\sCurlDagger$, $\sCurlDagger\sCurl\sCurl$, $\sTwist\sCurl\sCurl$ and $\sTwist\sCurlDagger\sCurlDagger$ operating on $\kappa_{AB}$ and $\bar\kappa_{A'B'}$. Using the commutators , , , , and on the outermost operators and using , the list of operators appearing can be reduced to the set $\sCurl$, $\sCurlDagger$, $\sCurl\sTwist\sCurl$, $\sCurlDagger\sTwist\sCurlDagger$, $\sDiv\sTwist\sCurl$ and $\sDiv\sTwist\sCurlDagger$. Then using the relations , , and on the innermost operators the expression will only contain the operators $\sCurl$, $\sCurlDagger$. $$\begin{aligned} (\sTwist_{0,0} \mathfrak{Q}_{\kappa})_{A}{}^{A'}={}&\kappa^{BC} \bar{\kappa}^{A'B'} (\sCurl_{2,2} \Phi)_{ABCB'} - \tfrac{2}{9} \Phi_{BC}{}^{A'}{}_{B'} \kappa^{BC} (\sCurl_{0,2} \bar{\kappa})_{A}{}^{B'} + \tfrac{1}{3} \Psi_{ABCD} \kappa^{CD} (\sCurl_{0,2} \bar{\kappa})^{BA'}\nonumber\\ & + \tfrac{2}{9} \Phi_{BC}{}^{A'}{}_{B'} \kappa_{A}{}^{C} (\sCurl_{0,2} \bar{\kappa})^{BB'} - \tfrac{2}{9} \Phi_{AC}{}^{A'}{}_{B'} \kappa_{B}{}^{C} (\sCurl_{0,2} \bar{\kappa})^{BB'}\nonumber\\ & + \kappa_{A}{}^{B} \bar{\kappa}^{B'C'} (\sCurlDagger_{2,2} \Phi)_{B}{}^{A'}{}_{B'C'} + \tfrac{1}{3} \bar\Psi^{A'}{}_{B'C'D'} \bar{\kappa}^{C'D'} (\sCurlDagger_{2,0} \kappa)_{A}{}^{B'}\nonumber\\ & - \tfrac{2}{9} \Phi_{ABB'C'} \bar{\kappa}^{B'C'} (\sCurlDagger_{2,0} \kappa)^{BA'} + \tfrac{2}{9} \Phi_{ABB'C'} \bar{\kappa}^{A'C'} (\sCurlDagger_{2,0} \kappa)^{BB'}\nonumber\\ & - \tfrac{2}{9} \Phi_{AB}{}^{A'}{}_{C'} \bar{\kappa}_{B'}{}^{C'} (\sCurlDagger_{2,0} \kappa)^{BB'}.\label{eq:TwistQkappaEq1}\end{aligned}$$ Applying the $\sTwist$ operator to the equation , decomposing the derivatives into irreducible parts and using gives an expression with the operators $\sCurl\sCurlDagger$, $\sCurlDagger\sCurl$, $\sTwist\sCurl$ and $\sTwist\sCurlDagger$ operating on $\kappa_{AB}$ and $\bar\kappa_{A'B'}$. Using the commutators , , and and using , the expression reduces to $$\begin{aligned} (\sTwist_{1,1} \mathfrak{P}_{\kappa}{})_{AB}{}^{A'B'}={}&\Psi_{ABCD} \kappa^{CD} \bar{\kappa}^{A'B'} - \bar\Psi^{A'B'}{}_{C'D'} \kappa_{AB} \bar{\kappa}^{C'D'}.\label{eq:TwistPkappaEq1}\end{aligned}$$ Substituting into the definition of $\ObstrZero$, and making an irreducible decomposition of $\kappa_{AB} (\sCurl_{0,2} \bar{\kappa})_{C}{}^{B'}$ and $\bar{\kappa}_{A'B'} (\sCurlDagger_{2,0} \kappa)_{AC'}$, allows us to see that and reduces to $$\begin{aligned} (\sTwist_{0,0} \mathfrak{Q}_{\kappa})_{A}{}^{A'}={}&(\ObstrZero \mathfrak{L}_{\kappa})_{A}{}^{A'},\\ (\sTwist_{1,1} \mathfrak{P}_{\kappa})_{AB}{}^{A'B'}={}&(\ObstrOne \mathfrak{L}_{\kappa})_{AB}{}^{A'B'}.\end{aligned}$$ Example of a conformal Killing tensor that does not factor ---------------------------------------------------------- The following shows that the condition \[point:A0\] is non-trivial. We also see that \[point:A1\] does not imply \[point:A0\]. Unfortunately, we have not found any example of a valence $(1,1)$ Killing spinor which does not satisfy \[point:A1\]. Consider the following Stäckel metric (see [@michel:radoux:silhan:2013arXiv1308.1046M] and [@kalnins:mclenaghan:BelgAcad1984] for more general examples.) $$g_{ab} = dt^2- dz^2 - (x + y)(dx^2 + dy^2)$$ with the tetrad $$\begin{aligned} l^{a}={}&\tfrac{1}{\sqrt{2}}(\partial_t)^{a} + \tfrac{1}{\sqrt{2}}(\partial_z)^{a},& n^{a}={}&\tfrac{1}{\sqrt{2}}(\partial_t)^{a} - \tfrac{1}{\sqrt{2}}(\partial_z)^{a},& m^{a}={}&\frac{(\partial_x)^{a}}{\sqrt{2} (x + y)^{1/2}} + \frac{i (\partial_y)^{a}}{\sqrt{2} (x + y)^{1/2}}.\end{aligned}$$ Expressed in the corresponding dyad $(o_A,\iota_A)$, the curvature takes the form $$\begin{aligned} \Psi_{ABCD}={}&-12 \Lambda o_{(A}o_{B}\iota_{C}\iota_{D)},& \Phi_{ABA'B'}={}&12 \Lambda o_{(A}\iota_{B)} \bar o_{(A'}\bar\iota_{B')},& \Lambda={}& \frac{1}{12 (x + y)^3}.\end{aligned}$$ We can see that the spinor $$\begin{aligned} L_{AB}{}^{A'B'}={}&\tfrac{1}{2} (x + y) (\bar o^{A'} \bar o^{B'} \iota_{A} \iota_{B} + o_{A} o_{B} \bar\iota^{A'} \bar\iota^{B'}) - (x - y) o_{(A}\iota_{B)} \bar o^{(A'}\bar\iota^{B')}\end{aligned}$$ is a trace-free conformal Killing tensor. We trivially have solutions to the auxiliary condition \[point:A1\] because $$\begin{aligned} (\ObstrOne L)_{AB}{}^{A'B'} = {}&L{}^{CDA'B'} \Psi_{ABCD} - L{}_{AB}{}^{C'D'} \bar\Psi^{A'B'}{}_{C'D'} = 0.\end{aligned}$$ If there is a solution to we will automatically have $(\sCurl_{1,1}\ObstrZero L)_{AB}=0$ because $\sCurl_{1,1}\sTwist_{0,0}=0$. However, with the current $L_{AB}{}^{A'B'}$ we get $$\begin{aligned} (\sCurl_{1,1} \ObstrZero L)_{AB}={}&\frac{5i o_{(A}\iota_{B)}}{(x + y)^5}.\end{aligned}$$ This is non vanishing, which means that the auxiliary condition \[point:A0\] does not admit a solution. This example shows that the conditions \[point:A0\] and \[point:A1\] are not equivalent. From the previous two sections, we can also conclude that this $L_{AB}{}^{A'B'}$ can not be written as a linear combination of conformal Killing tensors of the form $\zeta_{(A}{}^{(A'}\xi_{B)}{}^{B')}$ or $\kappa_{AB} \bar{\kappa}_{A'B'}$. For the more general metric in [@kalnins:mclenaghan:BelgAcad1984] we can in fact also construct a valence $(2,2)$ Killing spinor which trivially satisfies condition \[point:A1\], but which in general will not satisfy condition \[point:A0\]. It is interesting to note that in general this metric does not admit Killing vectors, but we can still construct symmetry operators for the Maxwell equation. Auxiliary condition for a symmetry operator of the second kind for the Dirac-Weyl equation ------------------------------------------------------------------------------------------ Let $\kappa_{AB}$ be a Killing spinor, and $\xi^{AA'}$ a conformal Killing vector, i.e. $$\begin{aligned} (\sTwist_{2,0} \kappa)_{ABCA'}={}&0, &(\sTwist_{1,1}\xi)_{AB}{}^{A'B'}&=0.\label{eq:TwistkappaTwistxi}\end{aligned}$$ then we have a solution $$\begin{aligned} \mathfrak{L}_{\kappa\xi ABC}{}^{A'}\equiv{}&\kappa_{(AB}\xi_{C)}{}^{A'}\end{aligned}$$ to the equation $$\begin{aligned} (\sTwist_{3,1}\mathfrak{L}_{\kappa\xi})_{ABCD}{}^{A'B'}&=0.\end{aligned}$$ The auxiliary equation now takes the form $$\begin{aligned} 0={}&\tfrac{3}{4} \Psi_{ABDF} \kappa^{CD} (\sCurl_{1,1} \xi)_{C}{}^{F} + \Psi_{ABCD} \xi^{CA'} (\sCurlDagger_{2,0} \kappa)^{D}{}_{A'} - \tfrac{3}{4} \Psi_{ABCD} \kappa^{CD} (\sDiv_{1,1} \xi)\nonumber\\ & - \tfrac{5}{4} \Psi_{(A}{}^{CDF}\kappa_{B)C}(\sCurl_{1,1} \xi)_{DF} - \tfrac{5}{4} \Psi_{(A}{}^{CDF}\kappa_{|CD|}(\sCurl_{1,1} \xi)_{B)F} + \tfrac{6}{5} \kappa_{(A}{}^{C}\xi^{DA'}(\sCurl_{2,2} \Phi)_{B)CDA'}\nonumber\\ & + \tfrac{3}{5} \kappa^{CD}\xi_{(A}{}^{A'}(\sCurl_{2,2} \Phi)_{B)CDA'} - 2 \kappa^{DF} \xi^{CA'} (\sTwist_{4,0} \Psi)_{ABCDFA'}.\label{eq:auxcondDirac2factored}\end{aligned}$$ Using the technique from Section \[sec:integrabilitycond\] we get that the integrability conditions for are $$\begin{aligned} 0={}&\Psi_{(ABC}{}^{F}\kappa_{D)F},\label{eq:intcondkappaDirac2}\\ 0={}&\tfrac{1}{2} \Psi_{ABCD} (\sDiv_{1,1} \xi) + 2 \Psi_{(ABC}{}^{F}(\sCurl_{1,1} \xi)_{D)F} - \tfrac{4}{5} \xi_{(A}{}^{A'}(\sCurl_{2,2} \Phi)_{BCD)A'} + \xi^{FA'} (\sTwist_{4,0} \Psi)_{ABCDFA'}.\label{eq:intcondxiDirac2}\end{aligned}$$ Applying the operator $\sCurlDagger$ on the condition gives $$\begin{aligned} 0={}&- \tfrac{1}{2} \Psi_{ABCD} (\sCurlDagger_{2,0} \kappa)^{D}{}_{A'} - \tfrac{9}{10} \kappa_{(A}{}^{D}(\sCurl_{2,2} \Phi)_{BC)DA'} + \tfrac{1}{4} \kappa^{DF} (\sTwist_{4,0} \Psi)_{ABCDFA'}.\label{eq:curlintcondkappaDirac2}\end{aligned}$$ Using to elliminate $\Psi_{ABCD} (\sDiv_{1,1} \xi)$ and to elliminate $\kappa^{DF} (\sTwist_{4,0} \Psi)_{ABCDFA'}$, and doing an irreducible decomposition of $\Psi_{ABCF} \kappa_{D}{}^{F}$ we see that reduces to $$\begin{aligned} 0={}&-2 (\sCurl_{1,1} \xi)^{CD} \Psi_{(ABC}{}^{F}\kappa_{D)F},\end{aligned}$$ which is trivially satisfied due to . Factorization of valence $(4,0)$ Killing spinors with aligned matter {#sec:factorValence4} -------------------------------------------------------------------- Assume that the matter field and the curvature are aligned, that is $$\begin{aligned} 0={}&\Psi_{(ABC}{}^{F}\Phi_{D)FA'B'}.\label{eq:AlignmentPsiPhi}\end{aligned}$$ Furthermore, assume that $\Psi_{ABCD}$ does not vanish, and assume that there is a solution $L_{ABCD}$ to $$(\sTwist_{4,0}L)_{ABCDEA'} = 0.\label{eq:KS4}$$ The integrability condition for this equation together with the non-vanishing of the Weyl spinor, gives that $L_{ABCD}$ and $\Psi_{ABCD}$ are proportional (c.f. [@KalMcLWil92a]). This means that $$\begin{aligned} 0={}&L_{(ABC}{}^{F}\Phi_{D)FA'B'},\label{eq:LPhiAlign}\\ 0={}&- L_{(ABCD}(\sTwist_{0,0}\Lambda)_{F)A'} + L_{(ABC}{}^{H}(\sCurl_{2,2} \Phi)_{DF)HA'} + \tfrac{1}{5} \Phi_{(AB|A'|}{}^{B'}(\sCurlDagger_{4,0} L)_{CDF)B'},\label{eq:DerLPhiAlign}\end{aligned}$$ where the second equation is obtained by taking a derivative of the first, decomposing the derivatives into irreducible parts, using the Killing spinor equation, and symmetrizing over all unprimed indices. Split $L_{ABCD}$ into principal spinors $L_{ABCD}=\alpha_{(A}\beta_B\gamma_C\delta_{D)}$. Now, the Killing spinor equation , and the alignment equation gives $$\begin{aligned} 0={}&\alpha^{A} \alpha^{B} \alpha^{C} \alpha^{D} \alpha^{F} (\sTwist_{4,0} L)_{ABCDFA'}= \alpha^{A} \beta_{A} \alpha^{B} \gamma_{B} \alpha^{C} \delta_{C} \alpha^{D} \alpha^{F} \nabla_{FA'}\alpha_{D},\\ 0={}&\alpha^{A} \alpha^{B} \alpha^{C} \alpha^{D} L_{(ABC}{}^{F}\Phi_{D)FA'B'}= \tfrac{1}{4} \alpha^{A} \beta_{A} \alpha^{B} \gamma_{B} \alpha^{C} \delta_{C} \alpha^{D} \alpha^{F}\Phi_{DFA'B'}.\end{aligned}$$ We will first assume that $\alpha^A$ is not a repeated principal spinor of $L_{ABCD}$. This means that $\alpha^A\beta_A\alpha^B\gamma_B\alpha^C\delta_{C} \neq 0$ and hence $\alpha^A\alpha^B\nabla_{A'A}\alpha_{B}=0$, that is $\alpha_A$ is a shear-free geodesic null congruence. We also get $\alpha^{D} \alpha^{F}\Phi_{DFA'B'}=0$. Contracting with $\alpha^A\alpha^B\alpha^C\alpha^D\alpha^F$ we get $$\begin{aligned} 0 ={}& \tfrac{1}{4} \alpha^{A} \beta_{A}\alpha^{B} \gamma_{B}\alpha^{C} \delta_{C} \alpha^{D} \alpha^{F} \alpha^{H} (\sCurl_{2,2} \Phi)_{DFHA'} + \tfrac{1}{5} \Phi_{ABA'}{}^{B'} \alpha^{A} \alpha^{B} \alpha^{C} \alpha^{D} \alpha^{F} (\sCurlDagger_{4,0} L)_{CDFB'}\nonumber\\ ={}&\tfrac{1}{4} \alpha^{A} \beta_{A}\alpha^{B} \gamma_{B}\alpha^{C} \delta_{C} \alpha^{D} \alpha^{F} \alpha^{H} (\sCurl_{2,2} \Phi)_{DFHA'}.\end{aligned}$$ Hence, $\alpha^{A} \alpha^{B} \alpha^{C} (\sCurl_{2,2} \Phi)_{ABCA'}=0$. But the Bianchi equations give $$\begin{aligned} \alpha^{A} \alpha^{B} \alpha^{C} \nabla^{DD'}\Psi_{ABCD}={}&\alpha^{A} \alpha^{B} \alpha^{C} (\sCurl_{2,2} \Phi)_{ABCA'}=0.\end{aligned}$$ It follows from the generalized Goldberg-Sachs theorem that $\alpha^A$ is a repeated principal spinor of $\Psi_{ABCD}$, see for instance [@PenRin86 Proposition 7.3.35]. But $L_{ABCD}$ and $\Psi_{ABCD}$ are proportional, so $\alpha^A$ is a repeated principal spinor of $L_{ABCD}$ after all. Without loss of generality, we can assume that $\gamma^A=\alpha^A$, a relabelling and rescaling of $\beta^{A}$, $\gamma^{A}$ and $\delta^{A}$ can achieve this. Repeating the argument with $\beta^A$, we find that also $\beta^A$ is a repeated principal spinor of $L_{ABCD}$. If $\beta^A\alpha_A=0$, we can repeat the argument again with $\delta^A$ and see that all principal spinors are repeated, i.e. Petrov type N. Otherwise, we have Petrov type D. In conclusion, we have after rescaling $L_{ABCD}=\alpha_{(A}\alpha_{B}\beta_{C}\beta_{D)}$. Now, let $\kappa_{AB}=\alpha_{(A}\beta_{B)}$. First assume that $\alpha^A\beta_A\neq 0$. Contracting with $\alpha^{A} \alpha^{B} \alpha^{C} \alpha^{D} \beta^{F}$, $\alpha^{A} \alpha^{B} \alpha^{C} \beta^{D} \beta^{F}$, $\alpha^{A} \alpha^{B} \beta^{C} \beta^{D} \beta^{F}$, $\alpha^{A} \beta^{B} \beta^{C} \beta^{D} \beta^{F}$ we find $$\begin{aligned} 0={}&\alpha^{A} \alpha^{B} \alpha^{C} (\sTwist_{2,0} \kappa)_{ABCA'},& 0={}&\alpha^{A} \alpha^{B} \beta^{C} (\sTwist_{2,0} \kappa)_{ABCA'},\\ 0={}&\alpha^{A} \beta^{B} \beta^{C} (\sTwist_{2,0} \kappa)_{ABCA'},& 0={}&\beta^{A} \beta^{B} \beta^{C} (\sTwist_{2,0} \kappa)_{ABCA'}.\end{aligned}$$ Hence, $(\sTwist_{2,0} \kappa)_{ABCA'}=0$. If $\alpha^A\beta_A= 0$, we can find a dyad $(o^A, \iota^A)$ so that $\alpha^A=o^A$. Then we have $L_{ABCD}=\upsilon^2 o_{A}o_{B}o_{C}o_{D}$ and $\kappa_{AB}=\upsilon o_{A}o_{B}$. Contracting with $o^{A} o^{B} o^{C} \iota^{D} \iota^{F}\upsilon^{-1}$, $o^{A} o^{B} \iota^{C} \iota^{D} \iota^{F}\upsilon^{-1}$, $o^{A} \iota^{B} \iota^{C} \iota^{D} \iota^{F}\upsilon^{-1}$, $\iota^{A} \iota^{B} \iota^{C} \iota^{D} \iota^{F}\upsilon^{-1}$ we find $$\begin{aligned} 0={}&o^{A} o^{B} o^{C} (\sTwist_{2,0} \kappa)_{ABCA'},& 0={}&o^{A} o^{B} \iota^{C} (\sTwist_{2,0} \kappa)_{ABCA'},\\ 0={}&o^{A} \iota^{B} \iota^{C} (\sTwist_{2,0} \kappa)_{ABCA'},& 0={}&\iota^{A} \iota^{B} \iota^{C} (\sTwist_{2,0} \kappa)_{ABCA'}.\end{aligned}$$ Hence, $(\sTwist_{2,0} \kappa)_{ABCA'}=0$. We can therefore conclude that if the curvature satisfies , $\Psi_{ABCD}$ does not vanish, and we have a valence $(4,0)$ Killing spinor $L_{ABCD}$, then we have a valence $(2,0)$ Killing spinor $\kappa_{AB}$ such that $L_{ABCD}=\kappa_{(AB}\kappa_{CD)}$. The symmetry operators with factorized Killing spinor {#sec:symopfactored} ===================================================== Symmetry operators for the conformal wave equation -------------------------------------------------- Let us now consider special cases of symmetry operators for the conformal wave equation. If we choose $$\begin{aligned} L_{ABA'B'}={}&\mathfrak{L}_{\xi\zeta ABA'B'},& P_{AA'}={}&0,& Q ={}& \tfrac{2}{5} \mathfrak{Q}_{\xi\zeta}.\end{aligned}$$ Then the operator takes the form $$\begin{aligned} \chi={}&\tfrac{1}{2} \hat{\mathcal{L}}_{\zeta}\hat{\mathcal{L}}_{\xi}\phi + \tfrac{1}{2} \hat{\mathcal{L}}_{\xi}\hat{\mathcal{L}}_{\zeta}\phi.\end{aligned}$$ One can also add an arbitrary first order symmetry operator to this. We can also choose $$\begin{aligned} L_{ABA'B'}={}&\mathfrak{L}_{\kappa ABA'B'},& P_{AA'}={}&0,& Q ={}& \tfrac{2}{5} \mathfrak{Q}_{\kappa}.\end{aligned}$$ Substituting these expressions into gives a symmetry operator, but we have not found any simpler form than the one given by . Apart from factorizations, one can in special cases get symmetry operators from Killing tensors. If $K_{AB}{}^{A'B'}$ is a Killing tensor, then we have $$\begin{aligned} (\sTwist_{2,2} L)_{ABCA'B'C'}={}&0, & (\sDiv_{2,2} L)_{AA'} ={}& - \tfrac{3}{4} (\sTwist_{0,0} S)_{AA'},& K^{ABA'B'}={}&L^{ABA'B'} + \tfrac{1}{4} S \epsilon^{AB} \bar\epsilon^{A'B'}\end{aligned}$$ where $L_{AB}{}^{A'B'}=K_{(AB)}{}^{(A'B')}$ and $S=K_{A}{}^{A}{}_{A'}{}^{A'}$. The commutator gives $(\sCurl_{1,1} \sDiv_{2,2} L)_{AB} = 0$. If we also assume vacuum, then the equation gives $$\begin{aligned} (\sTwist_{0,0} \sDiv_{1,1} \sDiv_{2,2} L)_{AA'}={}&-2 \Psi_{ABCD} (\sCurl_{2,2} L)^{BCD}{}_{A'} - 2 \bar\Psi_{A'B'C'D'} (\sCurlDagger_{2,2} L)_{A}{}^{B'C'D'}.\end{aligned}$$ Hence, we can choose $$\begin{aligned} Q ={}& - \tfrac{1}{15} (\sDiv_{1,1} \sDiv_{2,2} L),\end{aligned}$$ to satisfy condition \[point:A0\], and get the well known symmetry operator $$\begin{aligned} \chi={}&-\tfrac{1}{2} (\sTwist_{0,0} S)^{AA'} (\sTwist_{0,0} \phi)_{AA'} + L^{ABA'B'} (\sTwist_{1,1} \sTwist_{0,0} \phi)_{ABA'B'}& ={}&\nabla_{AA'}(K^{ABA'B'} \nabla_{BB'}\phi),\end{aligned}$$ which is valid for vacuum spacetimes. Symmetry operator of the first kind for the Dirac-Weyl equation --------------------------------------------------------------- Let us now consider special cases of symmetry operators of the first kind for the Dirac-Weyl equation. We can choose $$\begin{aligned} L_{ABA'B'}={}&\mathfrak{L}_{\xi\zeta ABA'B'},& P^{AA'}={}&- \tfrac{1}{3} \mathfrak{P}_{\xi\zeta}{}^{AA'},& Q ={}& \tfrac{3}{10} \mathfrak{Q}_{\xi\zeta},\end{aligned}$$ to get a symmetry operator for the Dirac-Weyl equation. The operator then becomes $$\begin{aligned} \chi_{A}={}&\tfrac{1}{2}\hat{\mathcal{L}}_{\xi}\hat{\mathcal{L}}_{\zeta}\phi_{A}+\tfrac{1}{2}\hat{\mathcal{L}}_{\zeta}\hat{\mathcal{L}}_{\xi}\phi_{A}.\end{aligned}$$ We can add any conformal Killing vector to $P^{AA'}$ and any constant to $Q$. Note that if we add the conformal Killing vector $\tfrac{1}{2} (\xi^{BB'} \nabla_{BB'}\zeta^{AA'} - \zeta^{BB'} \nabla_{BB'}\xi^{AA'})$ to $P^{AA'}$, the operator gets the factored form $$\begin{aligned} \chi_{A}={}&\hat{\mathcal{L}}_{\xi}\hat{\mathcal{L}}_{\zeta}\phi_{A}.\end{aligned}$$ We can also choose $$\begin{aligned} L_{ABA'B'}={}&\mathfrak{L}_{\kappa ABA'B'},& P^{AA'}={}&- \tfrac{1}{3} \mathfrak{P}_{\kappa}{}^{AA'},& Q ={}& \tfrac{3}{10} \mathfrak{Q}_{\kappa}.\end{aligned}$$ Substituting these expressions into gives a symmetry operator, but we have not found any simpler form than the one given by . Symmetry operator of the first kind for the Maxwell equation {#sec:symopfirstmaxwellfact} ------------------------------------------------------------ Let us now consider the symmetry operators of the first kind for the Maxwell equation. Let $$\begin{aligned} L_{ABA'B'}={}&\mathfrak{L}_{\xi\zeta ABA'B'},& P^{AA'}={}&- \tfrac{2}{3} \mathfrak{P}_{\xi\zeta}{}^{AA'},& Q ={}&0,\end{aligned}$$ to get a symmetry operator. With this choice the symmetry operator and the potential reduce to $$\begin{aligned} \chi_{AB}={}&\tfrac{1}{2} \hat{\mathcal{L}}_{\zeta}\hat{\mathcal{L}}_{\xi}\phi_{AB} + \tfrac{1}{2} \hat{\mathcal{L}}_{\xi}\hat{\mathcal{L}}_{\zeta}\phi_{AB},\\ A_{AA'}={}&- \tfrac{1}{2} \zeta^{B}{}_{A'}\hat{\mathcal{L}}_{\xi}\phi_{AB} - \tfrac{1}{2} \xi^{B}{}_{A'}\hat{\mathcal{L}}_{\zeta}\phi_{AB} .\end{aligned}$$ A general first order operator can be added to this. If we add an the same commutator as above with an appropriate coefficient to $P^{AA'}$, we get the same kind of factorization of the operator as above. We can also get a solution by setting $$\begin{aligned} L_{ABA'B'}={}&\mathfrak{L}_{\kappa ABA'B'},& P^{AA'}={}&- \tfrac{2}{3} \mathfrak{P}_{\kappa}{}^{AA'},& Q ={}&0,\end{aligned}$$ With this choice the symmetry operator and the potential reduce to $$\begin{aligned} \chi_{AB}={}&(\sCurl_{1,1} A)_{AB},\\ A_{AA'}={}&- \tfrac{1}{3} \Theta_{AB} (\sCurl_{0,2} \bar{\kappa})^{B}{}_{A'} + \bar{\kappa}_{A'B'} (\sCurlDagger_{2,0} \Theta)_{A}{}^{B'},\\ \Theta_{AB}\equiv{}&-2 \kappa_{(A}{}^{C}\phi_{B)C}.\end{aligned}$$ This proves the first part of Theorem \[Thm:SymopMaxwellSimple\]. Symmetry operator of the second kind for the Dirac-Weyl equation {#sec:symopsecondDiracfactored} ---------------------------------------------------------------- Let $$\begin{aligned} L_{ABC}{}^{A'}={}&\mathfrak{L}_{\kappa\xi ABC}{}^{A'},\\ P_{AB} ={}&- \tfrac{1}{2} \hat{\mathcal{L}}_{\xi}\kappa_{AB} + \tfrac{3}{8} \kappa_{AB} (\sDiv_{1,1} \xi)\nonumber\\ ={}&\tfrac{1}{8} \kappa_{AB} (\sDiv_{1,1} \xi) - \tfrac{1}{2} \kappa_{(A}{}^{C}(\sCurl_{1,1} \xi)_{B)C} + \tfrac{1}{3} \xi_{(A}{}^{A'}(\sCurlDagger_{2,0} \kappa)_{B)A'}.\end{aligned}$$ Using the equations , the commutators , , , and the irreducible decompositions of $\Psi_{ABCF} \kappa_{D}{}^{F}$ and $\Phi_{ABA'B'} \xi_{C}{}^{B'}$ we get $$\begin{aligned} (\sTwist_{2,0} P)_{ABCA'}={}&- \tfrac{1}{6} \kappa_{(AB}(\sCurlDagger_{2,0} \sCurl_{1,1} \xi)_{C)A'} - \tfrac{1}{2} \kappa_{(A}{}^{D}(\sTwist_{2,0} \sCurl_{1,1} \xi)_{BC)DA'} + \tfrac{1}{8} \kappa_{(AB}(\sTwist_{0,0} \sDiv_{1,1} \xi)_{C)A'}\nonumber\\ & + \tfrac{1}{6} \xi_{(A|A'|}(\sCurl_{1,1} \sCurlDagger_{2,0} \kappa)_{BC)} + \tfrac{1}{3} \xi_{(A}{}^{B'}(\sTwist_{1,1} \sCurlDagger_{2,0} \kappa)_{BC)A'B'}\nonumber\\ ={}&\xi^{D}{}_{A'} \Psi_{(ABC}{}^{F}\kappa_{D)F}\nonumber\\ ={}&0,\end{aligned}$$ where we in the last step used the integrability condition . Observe that $P_{AB}$ is given by a conformally weighted Lie derivative, but now with a different weight. The operator $\hat{\mathcal{L}}_{\xi}$ has a conformal weight adapted to the weight of the conformally invariant operator $\sCurlDagger$. The operator $\sTwist$ is also conformally invariant, but with a different weight. This explains the extra term in $P_{AB}$. The symmetry operator of the second kind for the Dirac-Weyl equation now takes the form $$\begin{aligned} \omega_{A'}={}&\kappa^{BC} (\sTwist_{1,0} \hat{\mathcal{L}}_{\xi}\phi)_{BCA'}-\tfrac{2}{3} \hat{\mathcal{L}}_{\xi}\phi_{B} (\sCurlDagger_{2,0} \kappa)^{B}{}_{A'}.\end{aligned}$$ Hence, we can conclude that if $L_{ABCA'}$ factors, then one can choose a corresponding $P_{AB}$ so that the operator factors as a first order symmetry operator of the first kind followed by a first order symmetry operator of the second kind. Symmetry operator of the second kind for the Maxwell equation {#sec:symopsecondmaxwellfact} ------------------------------------------------------------- If we let $L_{ABCD}=\kappa_{(AB}\kappa_{CD)}$ with $$\begin{aligned} (\sTwist_{2,0} \kappa)_{ABCA'} ={}& 0,\end{aligned}$$ Then the operator if the second kind now takes the form $$\begin{aligned} \omega_{A'B'}={}&(\sCurlDagger_{1,1}B)_{A'B'},\\ B_{AA'}={}&\kappa_{AB} (\sCurlDagger_{2,0} \Theta)^{B}{}_{A'} + \tfrac{1}{3} \Theta_{AB} (\sCurlDagger_{2,0} \kappa)^{B}{}_{A'},\\ \Theta_{AB}\equiv{}&-2 \kappa_{(A}{}^{C}\phi_{B)C}.\end{aligned}$$ This proves the second part of Theorem \[Thm:SymopMaxwellSimple\]. Acknowledgements {#acknowledgements .unnumbered} ================ The authors would like to thank Steffen Aksteiner and Lionel Mason for helpful discussions. We are particularly grateful to Lionel Mason for his ideas concerning Theorem \[thm:Valence4Factorization\]. Furthermore we would like to thank Niky Kamran and J. P. Michel for helpful comments. LA thanks Shing-Tung Yau for generous hospitality and many interesting discussions on symmetry operators and related matters, during a visit to Harvard University, where some initial work on the topic of this paper was done. This material is based upon work supported by the National Science Foundation under Grant No. 0932078 000, while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the semester of 2013. References {#references .unnumbered} ========== [10]{} url \#1[[\#1]{}]{}urlprefix B 1968 [*Phys. Rev.*]{} [**174**]{} 1559–1571 M and [Penrose]{} R 1970 [*Commun. Math. Phys.*]{} [**18**]{} 265–274 B 1977 [*[Phys. Rev. D]{}*]{} [**16**]{} 3395–3414 Miller Jr W 1977 [*Symmetry and separation of variables*]{} (Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam) ISBN 0-201-13503-5 with a foreword by Richard Askey, Encyclopedia of Mathematics and its Applications, Vol. 4 E G, [Miller]{} Jr W and [Williams]{} G C 1989 [*J Math. Phys.*]{} [**30**]{} 2360–2365 E G, [Miller]{} Jr W and [Williams]{} G C 1989 [*J Math. Phys.*]{} [**30**]{} 2925–2929 E G, [McLenaghan]{} R G and [Williams]{} G C 1992 [*R. Soc. London Proc. Ser. A*]{} [**439**]{} 103–113 R G, [Smith]{} S N and [Walker]{} D M 2000 [*R. Soc. London Proc. Ser. A*]{} [**456**]{} 2629 J P, [Radoux]{} F and [[Š]{}ilhan]{} J 2014 [*SIGMA* ]{} [**10**]{}, 016, 26 pages (*Preprint* ) L and [Blue]{} P 2009 [*ArXiv e-prints*]{} (*Preprint* ) Klainerman S 1985 [*Comm. Pure Appl. Math.*]{} [**38**]{} 321–332 ISSN 0010-3640 S C and [Pohjanpelto]{} J 2003 [*R. Soc. London Proc. Ser. A*]{} [**459**]{} 1215–1239 (*Preprint* ) Andersson L, Bäckdahl T and Blue P “Conserved currents for the [M]{}axwell equation.” *In preparation.* R and [Rindler]{} W 1984 [*“[Spinors and space-time. Vol. 1: Two-spinor calculus and relativistic fields.]{}”*]{} R and [Rindler]{} W 1986 [*“[Spinors and space-time. Vol. 2: Spinor and twistor methods in space-time geometry.]{}”*]{} T 2011-2014 Sym[M]{}anipulator <http://www.xact.es/SymManipulator> Martín-García J M 2002-2014 x[A]{}ct <http://www.xact.es> N and [McLenaghan]{} R G 1985 [*Lett. Math. Phys.*]{} [**9**]{} 65–72 Kress J [*Generalised conformal Killing-Yano tensors: Applications to electrodynamics*]{} Ph.D. thesis University of Newcastle 1997 E G and [Miller]{} Jr W 1983 [*J. Math. Phys.*]{} [**24**]{} 1047–1053 Eastwood M 2005 [*Ann. of Math. (2)*]{} [**161**]{} 1645–1665 ISSN 0003-486X I M and [Kress]{} J M 2004 [*Class. Quantum Grav.*]{} [**21**]{} 427–431 B and [McLenaghan]{} R G 1979 [*[Phys. Rev. D]{}*]{} [**19**]{} 1093–1097 S, [Lina]{} J M and [Vinet]{} L 1988 [*[Phys. Rev. D]{}*]{} [**38**]{} 3837–3839 M and [Kamran]{} N 1990 [*R. Soc. London Proc. Ser. A*]{} [**428**]{} 229–249 J P, 2014 [*ArXiv e-prints*]{} (*Preprint* ) E G, [McLenaghan]{} R G and [Williams]{} G C 1992 “[Symmetry Operators for Maxwell’s Equations on Curved Space-Time]{}” [*4th Canadian Conference on General Relativity and Relativistic Astrophysics*]{} ed [Kunstatter]{} G, [Vincent]{} D E and [Williams]{} J G p 129 Anco S C and Pohjanpelto J 2004 Symmetries and currents of massless neutrino fields, electromagnetic and graviton fields [*Symmetry in physics*]{} ([ *CRM Proc. Lecture Notes*]{} vol 34) (Providence, RI: Amer. Math. Soc.) pp 1–12 N and [McLenaghan]{} R G 1984 [*Acad. Roy. Belg. Bull. Cl. Sci. (5)*]{} [**70**]{} 596–610 [^1]: The use of the terms left and right is explained by noting that spinors of valence $(k,0)$ represent left-handed particles, while spinors of valence $(0,k)$ represent right-handed particles, cf. [@PenRin84 §5.7]. The Dirac equation is the equation for massive, charged spin-1/2 fields, and couples the left- and right-handed parts of the field, see [@PenRin84 §4.4]. We shall not consider the symmetry operators for the Dirac equation here.

--- abstract: '$K_s$-band images of 20 barred galaxies show an increase in the peak amplitude of the normalized $m=2$ Fourier component with the $R_{25}$-normalized radius at this peak. This implies that longer bars have higher $m=2$ amplitudes. The long bars also correlate with an increased density in the central parts of the disks, as measured by the luminosity inside $0.25R_{25}$ divided by the cube of this radius in kpc. Because denser galaxies evolve faster, these correlations suggest that bars grow in length and amplitude over a Hubble time with the fastest evolution occurring in the densest galaxies. All but three of the sample have early-type flat bars; there is no clear correlation between the correlated quantities and the Hubble type.' author: - 'Bruce G. Elmegreen' - Debra Meloy Elmegreen - 'Johan H. Knapen' - 'Ronald J. Buta' - 'David L. Block' - Ivânio Puerari title: Variation of Galactic Bar Length with Amplitude and Density as Evidence for Bar Growth over a Hubble Time --- Introduction {#sect:intro} ============ Bars should slow down and grow over time as bar angular momentum is transferred to the disk (Tremaine & Weinberg 1984) and halo (Kormendy 1979; Sellwood 1980; Little & Carlberg 1991; Hernquist & Weinberg 1992; Debattista & Sellwood 1998, 2000; Valenzuela & Klypin 2003; Athanassoula 2002, 2003). With this growth, the bars should become stronger, longer and thinner (Athanassoula 2003). Pattern speeds are difficult to measure (Knapen 1999) but bar lengths are not (Erwin 2005). To investigate the model predictions, we examined relative bar lengths and intensities in 20 galaxies with conspicuous bars and a range of Hubble types. We consider how these parameters correlate with each other and with the central density of the galaxy. Central luminosity density is used as an indirect measure of the inner angular rotation rate because few galaxies in our sample have observed rotation curves. Galaxies with high central densities should have high central rotation rates and evolve more quickly than galaxies with low central densities. If there is a secular change in bar length or amplitude with angular momentum transfer, then denser galaxies should show the later evolutionary stages. Observations and Analysis ========================= $K_{\rm s}$-band images of barred galaxies were obtained with the Anglo-Australian Telescope (AAT) from 2004 June 28 to July 5. We used the Infrared Imager and Spectrograph (IRIS2) with a $1024\times 1024$ pixel Rockwell HAWAII-1 HgCdTe detector mounted at the AAT’s $f/8$ Cassegrain focus, yielding a pixel scale of 0.447arcsecpx$^{-1}$ and a field of view of 7.7arcmin squared. Exposure times were around one hour in almost all cases and the angular resolution was typically 1.5arcsec. Full details of the observations will be presented in Buta et al. (2007). Images were pre-processed using standard IRAF[^1] routines, and each image was cleaned of foreground stars and background galaxies. Deprojections were derived as follows. For each galaxy, estimates of the orientation parameters were obtained using an ellipse fitting routine, [*sprite*]{}, originally written by W. D. Pence. These fits were either based on the $K_s$-band image itself, or on an optical image if available. Because the bulges may not be as flat as the disks, we used a two-dimensional multi-component decomposition code (Laurikainen, Salo, & Buta 2005) to derive the parameters of the bulges and disks. Images were deprojected, assuming the bulges are spherical, using the IRAF routine IMLINTRAN. This assumption has little impact on our Fourier analyses. The results of the decompositions, as well as the orientation parameters used, will be presented in Buta et al. (2007). Results ======= Bar and spiral arm amplitudes were measured from the $m=2$ Fourier components of azimuthal intensity profiles taken at various radii from polar plots using the deprojected, star-cleaned, background-subtracted images (as in Regan & Elmegreen 1997 and Block et al. 2004). The $m$=2 Fourier intensity amplitude, $I_2$, was normalized to the average intensity, $I_0$, at each radius; $I_2$ is defined to be the amplitude of the sinusoidal fit to the azimuthal profile. Figure \[fig:20gal\] shows this normalized amplitude, $A_2=I_2/I_0$, versus the radius normalized to the standard isophotal radius $R_{25}$ for each galaxy ($R_{25}$ is half the diameter $D_{25}$ of the $\mu_B=25$ mag arcsec$^{-2}$ isophote given by de Vaucouleurs et al. 1991). The 20 profiles have been divided into four panels for clarity. Figure \[fig:20gal\] shows that $A_2$ increases with radius and then decreases. The maximum, $A_2^{max}$, occurs at a radius which we denote by $R_2$. This radius is approximately equal to the bar length determined by eye in all cases. Theory suggests the two lengths should scale together, with $R_2$ slightly less than the visible bar length (Athanassoula & Misiriotis 2002). A correlation may be seen in Figure \[fig:20gal\] in the sense that galaxies with higher $A_2^{max}$ also have larger radii at this peak (the peaks are indicated by the circles; empty circles are flat bars and circles with plus-signs are exponential bars). This correlation is shown in Figure \[fig:correl\] (top left), which plots $A_2^{max}$ versus the normalized radius $R_2/R_{25}$. The dashed line is a bi-variate least squares fit, repeated in the other panels. Longer bars are higher amplitude in relative intensity. This is sensible considering the general exponential decline of disk intensity: longer bars extend further out in the disk, placing their ends where the average background is fainter. For example, each radial interval of $\sim0.25R/R_{25}$ corresponds to about one exponential scale length in most galaxies, which is a factor of 2.7 in disk brightness. This factor is only slightly larger than the increase in Figure \[fig:correl\]. Thus, growing bars can stay somewhat flat in their intensity profile and still increase their relative amplitude along with their length because the surrounding disk is decreasing with radius. Bars apparently grow relative to the disk size even if the disk grows too because of angular momentum transfer from the bar (Valenzuela & Klypin 2003). Figure \[fig:correl\] (top right) includes three previous surveys in which this correlation was present but not noticed. The crosses are from $K$-band images of 8 different barred galaxies studied by Regan & Elmegreen (1997), the circles are from $K_s$-band images of 24 different early type (S0-Sa) barred galaxies in Buta et al. (2006), and the triangles are from 10 $I$-band images of different galaxies in Elmegreen & Elmegreen (1985). Among these three samples, there are only 3 overlapping galaxies and they are only between the 1985 and 1997 surveys. The Regan & Elmegreen $A_2^{max}$ values were multiplied by 2 because they used the standard definition of a Fourier component, which, for example, gives a relative value of 0.5 for an azimuthal profile of $1+\sin(2\theta)$. We and the other references in Figure \[fig:correl\] use twice the Fourier component to reflect the amplitude of the sinusoidal part of the profile. The lower panels of Figure \[fig:correl\] show correlations present in data from two other studies of bar Fourier amplitudes. The lower left panel shows data from Laurikainen et al. (2006), who determined the Fourier amplitudes and bar radii for 28 early type galaxies (S0,Sa, Sab) in $K_s$ band. The lower right panel shows data from Laurikainen et al. (2004), who used the Ohio State Bright Galaxy Survey and 2MASS to measure the H-band properties of 113 galaxies of various Hubble types. Their tabulations give the bar lengths, not the radii at the peak of the Fourier amplitude. Bar length is slightly larger than $R_2$, so the points are shifted to the right of the dashed lines in the figures. Also, $A_2$ is lower for S0 galaxies than other early types, which lowers some of the points in the lower left panel (Laurikainen, Salo, & Buta 2004). The present correlation was not noticed in either study but it is present in the data. Our previous study of $K_s$-band images for 17 barred galaxies (Block et al. 2004) found a length-amplitude correlation related to the present one. There we plotted the bar/interbar intensity contrast at 0.7 bar length versus the deprojected length of the bar (determined by eye). There was no overlap in galaxies with the present or the Buta et al. (2006) samples, and only one overlap each with the Regan & Elmegreen (1997) and Elmegreen & Elmegreen (1985) samples. The bar/interbar intensity contrast was shown by Block et al. to correlate with the relative amplitude of the $m=2$ Fourier component, and with the bar torque parameter, $Q_b$. This previous study discussed the length-amplitude correlation in a different context, however, noting that the long and high-amplitude bars tended to be early Hubble type and flat-profile, while the short and low-amplitude bars tended to be late Hubble type and exponential. This is true in general, but the present result is in addition to that. In the present work, the length-amplitude correlation is present even for the flat bars, and there is no strong correlation with Hubble type because most of our galaxies are flat-barred. The $R_2/R_{25}$ length is plotted versus Hubble type for our sample in Figure \[fig:ht\]. The circled plus-signs are exponential bars, and the rest are flat bars. Most of the galaxies in our current sample are Hubble types Sbc or earlier. The three exponential bars in our sample have slightly weaker Fourier components than the average for the flat bars (Fig. \[fig:correl\]). Evidently, there are two length-amplitude correlations: one discussed by Block et al. differentiating early and late type bars (which is presumably related to different bar resonances; Combes & Elmegreen 1993), and another found here that remains even for early-type, flat bars. The lower right panel of Figure \[fig:correl\] illustrates these two correlations in another way by plotting the various Hubble types with different symbols. The late types tend to be confined to the lower left in the figure, while the early types display the full range of bar lengths and amplitudes. Laurikainen, Salo, & Buta (2004) found no correlation between the peak relative torque, $Q_g$, normalized to the radial force, and the relative radius at the peak of this torque. The relative torque is a combination of the azimuthal bar amplitude, which determines the torque, and the radial force from the bulge, which is used to normalize this torque. Stronger-bulge galaxies have weaker bar torques for the same relative $m=2$ component. Bulges do not affect the peak $A_2$ much because the bulge intensity at the end of the bar is small. On the other hand, bulges do affect $Q_g$ because the radial force from the bulge is still large at the bar end. The central luminosity densities of the galaxies were measured from the $K_s$-band luminosities inside $0.125R_{25}$, $0.25R_{25}$, and $0.5R_{25}$. The $K_s$-band is dominated by old stars and traces the mass fairly well if dark matter is not significant there. Most of the galaxies are early type and centrally condensed so the 3 luminosities measured in this way were all about equal. Because the $R_2/R_{25}$ lengths vary from $\sim0.1R_{25}$ to $\sim0.5R_{25}$, and we want a representative density in the bar region, we use the luminosity inside $0.25R_{25}$. The central density is then taken to be this luminosity divided by the cube of the radius at $0.25R_{25}$, measured in kpc using the distances in Table 1 (from the galactocentric GSR in the NASA/IPAC Extragalactic Database). Figure \[fig:density\] shows the central $K_s$-band density versus the normalized radius at the peak $m=2$ amplitude (plus signs denote exponential bars). There is a correlation in the sense that longer bars occur in denser galaxies. These two correlations provide new information to supplement properties found in other bar correlations. Athanassoula & Martinet (1980) and Martin (1995) found a correlation between the lengths of bars and bulges, and Elmegreen & Elmegreen (1985) found a correlation between bar length, amplitude, and early versus late Hubble types, as mentioned above (see review in Ohta 1996). Discussion ========== We find that among fairly early type galaxies, relative bar length and relative $m=2$ intensity correlate with each other but not obviously with the Hubble subtype. The lengths and amplitudes also correlate with the central luminosity density of the galaxy. These correlations are in the sense expected by numerical simulations which suggest that angular momentum gradually transfers from a bar to the surrounding disk, bulge, and halo (see Athanassoula 2003 and references therein). With a loss of angular momentum, bars should slow down, and this means their corotation radii move outward. The stellar orbits in the bar should also get more elongated as angular momentum is proportional to the orbital area, and this translates to ellipticity for a constant orbital energy. As the orbital ellipticity increases, the stars become more concentrated in the bar and the bar gets stronger. If the orbits also scatter in energy, then their semi-major axes should grow too, following the moving corotation resonance. In this case, bars would grow in length as they get higher relative amplitudes during angular momentum loss. This is apparently what we observe here. The correlation with central density is consistent with angular momentum loss because galaxies with higher central densities evolve faster. In a given galaxy lifetime, the bars which evolve faster will have transferred more of their angular momentum outward and at the present time will have longer and higher-amplitude bars. The correlation with central density could also result from a larger reservoir for bar angular momentum in the larger bulges. An inverse process might be responsible too, where a strong bar forms first and this causes the bulge to grow through accretion (e.g., Athanassoula 1992, 2003). The lack of a correlation between relative bar length and peak relative bar torque $Q_g$ may be understood from our correlations with central density. For a given bulge, angular momentum transfer should increase both the peak amplitude and the peak torque of the bar over time. Galaxies with denser bulges do this faster, so at any given time, the peak amplitude correlates with bulge density. However, denser bulges weaken $Q_g$ because this quantity is normalized to the radial force (Laurikainen, Salo, & Buta 2004). This normalization offsets the increasing bar amplitude that comes from angular momentum transfer. As a result, $Q_g$ does not show the same correlations as the $m=2$ Fourier amplitude. Galaxies with dense bulges should not have bars if bulges prevent bar formation or growth (e.g., Sellwood 1980). However, our data show that high central densities correlate with high-amplitude bars. The observed correlation suggests that bars and bulges grow together, in agreement with Sheth et al. (2007). Conclusions =========== Bars in intermediate and early type spirals have a correlation between their relative lengths and their relative $m=2$ Fourier components, and both increase with the central density. These correlations are consistent with models in which bars lose angular momentum to the surrounding disk, bulge, and halo over long periods of secular evolution. The bars contain very old stars and must have been present for a high fraction of the Hubble time, like the bulges. We thank Emma Allard for help during the observations and with the data reduction, and Stuart Ryder for excellent support at the AAT. We thank Heikki Salo and Eija Laurikainen for useful comments on the manuscript. Helpful comments by the referee are appreciated. DME thanks Vassar College for publication support through a Research Grant. RB acknowledges the support of NSF grant AST 05-07140. I.P. acknowledges support from the Mexican foundation CONACyT under project 35947­.E. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. Athanassoula, E. 1992, MNRAS, 259, 345 Athanassoula, E. 2002, ApJ, 569, L83 Athanassoula, E. 2003, MNRAS, 341, 1179 Athanassoula E., & Martinet L., 1980, A&A, 87, L10 Athanassoula, E., & Misiriotis, A. 2002, MNRAS, 330, 35 Buta, R., Corwin, H. G., & Odewahn, S. C. 2007, The de Vaucouleurs Atlas of Galaxies, Cambridge, Cambridge University Press Buta, R., Laurikainen, E., Salo, H., Block, D.L., & Knapen, J.H. 2006, AJ, 132, 1859 Buta, R. et al. 2007, in preparation Block, D.L., Buta, R., Knapen, J.H., Elmegreen, D.M., Elmegreen, B.G., & Puerari, I. 2004, AJ, 128, 183 Combes, F., & Elmegreen, B. G. 1993, A&A, 271, 391 Debattista V. P., & Sellwood J. A., 1998, ApJ, 493, L5 Debattista V. P., & Sellwood J. A., 2000, ApJ, 543, 704 de Vaucouleurs, G. et al. 1999, Third Reference Catalogue of Bright Galaxies, Springer (RC3) Elmegreen B. G., & Elmegreen D. M., 1985, ApJ, 288, 438 Erwin, P. 2005, MNRAS, 364, 283 Hernquist L., & Weinberg M. D., 1992, ApJ, 400, 80 Knapen , J.H. 1999, ASPC, 187, 72 Kormendy J., 1979, ApJ, 227, 714 Laurikainen, E., Salo, H., Buta, R., & Vasylyev, S. 2004, MNRAS, 355, 1251 Laurikainen, E., Salo, H., & Buta, R. 2004, ApJ, 607, 103 Laurikainen, E., Salo, H., & Buta, R. 2005, MNRAS, 362, 1319 Laurikainen, E., Salo, H., Buta, R., Knapen, J., Speltincx, T., & Block, D. 2006, ApJ, 132, 2634 Little B., & Carlberg R. G., 1991, MNRAS, 250, 161 Martin P., 1995, AJ, 109, 2428 Ohta K., 1996, in Buta R., Crocker D., & Elmegreen B., eds, ASP Conf. Ser. Vol. 91, Barred Galaxies. (San Francisco: Astron. Soc. Pac.), p. 37 Regan, M.W., & Elmegreen, D.M. 1997, AJ, 114, 965 Sellwood, J.A. 1980, A&A, 89, 296 Sheth, S., et al. 2007, ApJ, in press Tremaine S., & Weinberg M. D., 1984, MNRAS, 209, 729 Valenzuela, O., & Klypin, A. 2003, MNRAS, 345, 406 [lccccccc]{} NGC175 &SB(s)ab&53.9&64.1&0.2&0.33\ NGC521 &SB(s)bc& 69.6 & 94.9 & 0.15 & 0.18\ NGC613 &SB(rs)bc& 19.8 & 164.9 & 0.5 & 0.4\ NGC986 &(R$^\prime_1$)SB(rs)b& 25.7 & 116.7 & 0.6 & 0.62\ NGC4593 &(R$^\prime_1$)SB(rs)ab& 35.6 & 116.7 & 0.5 & 0.48\ NGC5101 &(R$_1$R$^\prime_2$)SB(s)a& 23.7 & 161.1 & 0.3 & 0.36\ NGC5335 &SB(r)b& 63.2 & 64.1 & 0.2 & 0.5\ NGC5365 &(R)SB0$^-$& 31.6 & 88.5 & 0.3 & 0.34\ NGC6221 &SB(s)bc pec& 19 & 106.4 & 0.3 & 0.3\ NGC6782 &(R$_1$R$^\prime_2$)SB(r)a& 52.6 & 65.6 & 0.4 & 0.37\ NGC6907 &SAB(s)bc& 44.5 & 99.3 & 0.3 & 0.42\ NGC7155 &SB(r)0$^o$& 26.7 & 65.6 & 0.3 & 0.31\ NGC7329 &SB(r)b& 43.1 & 116.7 & 0.2 & 0.33 &\ NGC7513 &SB(s)b& 21.9 & 94.9 & 0.25 & 0.35 &\ NGC7552 &(R$^\prime_1$)SB(s)ab& 21.7 & 101.7 & 0.55 & 0.6\ NGC7582 &(R$^\prime_1$)SB(s)ab& 21.3 & 150.4 & 0.45 & 0.45\ IC1438 &(R$_1$R$^\prime_2$)SAB(r)a& 20 & 72 & 0.3 & 0.385\ IC4290 &(R$^\prime$)SB(r)a& 64.3 & 47.6 & 0.4 & 0.42\ IC5092 &(R)SB(s)c& 43.3 & 86.5 & 0.2 & 0.33\ UGC10862 &SB(rs)c& 24.8 & 82.6 & 0.2 & 0.31\ [^1]: IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.

--- abstract: 'We present a new method for simulating Markovian jump processes with time-dependent transitions rates, which avoids the transformation of random numbers by inverting time integrals over the rates. It relies on constructing a sequence of random time points from a homogeneous Poisson process, where the system under investigation attempts to change its state with certain probabilities. With respect to the underlying master equation the method corresponds to an exact formal solution in terms of a Dyson series. Different algorithms can be derived from the method and their power is demonstrated for a set of interacting two-level systems that are periodically driven by an external field.' author: - 'Viktor Holubec$^{1}$' - Petr Chvosta$^1$ - 'Mario Einax$^{2}$' - 'Philipp Maass$^{2}$' title: 'Attempt-time Monte Carlo: an alternative for simulation of stochastic jump processes with time-dependent transition rates' --- Introduction {#sec:1} ============ Stochastic jump processes with time-dependent transition rates are of general importance for many applications in physics and chemistry, in particular for describing the kinetics of chemical reactions [@Gibson/Bruck:2000; @Anderson:2007; @Astumian:2007] and the non-equilibrium dynamics of driven systems in statistical mechanics [@Crooks:1999; @Seifert:2005; @Esposito/VandenBroeck:2010]. With respect to applications in interdisciplinary fields they play an important role in connection with queuing theories. In general a system with $N$ states is considered that at random time instants performs transitions from one state to another. In case of a Markovian jump dynamics the probability for the system to change its state in the time interval $[t,t+\Delta t[$ is independent of the history and given by $w_{ij}(t)\Delta t+ o(\Delta t)$, where $j$ and $i\ne j$ are the initial and target state, respectively, and $w_{ij}(t)$ the corresponding transition rate at time $t$ ($w_{jj}(t)=0$). This implies that, if the systems is in the state $j$ at time $t_0$, it will stay in this state until a time $t>t_0$ with probability $\phi_j(t,t_0)=\exp[-\int_{t_0}^t d\tau\,w_j^{\rm tot}(\tau)]$, where $w_j^{\rm tot}(\tau)=\sum_i w_{ij}(\tau)$ is the total escape rate from state $j$ at time $\tau$. The probability to perform a transition to the target state $i$ in the time interval $[t,t+dt[$ then is $w_{ij}(t)\phi_j(t,t_0)dt$, i. e.$$\psi_{ij}(t,t_0)= w_{ij}(t)\exp\left[-\int_{t_0}^td\tau\,w_j^{\rm tot}(\tau)\right] \label{eq:psi}$$ is the probability density for the first transition to state $i$ to occur at time $t$ after the system was in state $j$ at time $t_0$. Any algorithm that evolves the system according to Eq. (\[eq:psi\]) generates stochastic trajectories with the correct path probabilities. The first algorithm of this kind was developed by Gillespie [@Gillespie:1978] in generalization of the continuous-time Monte-Carlo algorithm introduced by Bortz *et al.* [@Bortz/etal:1975] for time-independent rates. We call it the reaction time algorithm (RTA) in the following. The RTA consists of drawing a random time $t$ from the first transition time probability density $\psi_j^{\rm tot}(t,t_0)=\sum_i \psi_{ij}(t,t_0) =w_j^{\rm tot}\phi_j(t,t_0)=-\partial_t \phi_j(t,t_0)$ to any other state $i\ne j$, and a subsequent random selection of the target state $i$ with probability $w_{ij}(t)/w_j^{\rm tot}(t)$. In practice these two steps can be performed by generating two uncorrelated and uniformly distributed random numbers $r_1$, $r_2$ in the unit interval $[0,1[$ with some random number generator, where the first is used to specify the transition time $t$ via $$W_j(t,t_0)=\int_{t_0}^t d\tau w_j^{\rm tot}(\tau)=-\log(1-r_1) \label{eq:t-selection}$$ and the second is used to select the target state $i$ by requiring $$\sum_{k=1}^{i-1}\frac{w_{kj}(t)}{w_j^{\rm tot}(t)} \le r_2 < \sum_{k=1}^i\frac{w_{kj}(t)}{w_j^{\rm tot}(t)} \label{eq:target-selection}$$ Both steps, however, lead to some unpleasant problems in the practical realization. The first step according to Eq. (\[eq:t-selection\]) requires the calculation of $W_j(t,t_0)$ and the determination of its inverse $\tilde W_j(.,t_0)$ with respect to $t$ in order to obtain the transition time $t=\tilde W_j(-\log(1-r_1),t_0)$. While this is always possible, since $w_j^{\rm tot}>0$ and accordingly $W_j(t,t_0)$ is a monotonously increasing function of $t$, it can be CPU time consuming in case $W_j(t,t_0)$ cannot be explicitly given in an analytical form and one needs to implement a root finding procedure. The second step according to Eq. (\[eq:target-selection\]) can be cumbersome in case there are many states ($N$ large) and a systematic grouping of the $w_{ij}(t)$ to only a few classes is not possible. This situation in particular applies to many-particle systems, where $N$ typically grows exponentially with the number of particles, and the interactions (or a coupling to spatially inhomogeneous time-dependent external fields) can lead to a large number of different transitions rates. Moreover, even for systems with simple interactions (as, for example, Ising spin systems), where a grouping is in principle possible, the subdivision of the unit interval underlying Eq. (\[eq:target-selection\]) cannot be strongly simplified for time-dependent rates. A way to circumvent Eq. (\[eq:target-selection\]) is the use of the First Reaction Time Algorithm (FRTA) for time dependent rates [@Jansen:1995], or modifications of it [@Anderson:2007]. In the FRTA one draws random first transition times $t_k$ from the probability densities $\psi_{kj}(t_k,t_0) =w_{kj}(t_k)\exp[-\int_{t_0}^{t_k} d\tau\,w_{kj}(\tau)]$ for the individual transitions to each of the target states $k$ and performs the transition $i$ with the smallest $t_i=\min_k\{t_k\}$ at time $t_i$. This is statistically equivalent to the RTA, since for the given initial state $j$, the possible transitions to all target states are independent of each other. In short-range interacting systems, in particular, many of the random times $t_k$ can be kept for determining the next transition following $i$. In fact, all transitions from the new state $i$ to target states $k$ can be kept for which $w_{ki}(\tau)=w_{kj}(\tau)$ for $\tau>t$ (see Ref. [@Einax/Maass:2009] for details). However, the random times $t_k$ need to be drawn from $\psi_{kj}(t_k,t_0)$ and this unfortunately involves the same problems as discussed above in connection with Eq. (\[eq:t-selection\]). Algorithms {#sec:2} ========== We now present a new “attempt time algorithm” (ATA) that allows one to avoid the problems associated with the generation of the transition time in Eq. (\[eq:t-selection\]). Starting with the system in state $j$ at time $t_0$ as before, one first considers a large time interval $T$ and determines a number $\mu_j^{\rm tot}$ satisfying $$\mu_j^{\rm tot}\ge \max_{t_0\le\tau\le t_0+T}\{w_j^{\rm tot}(\tau)\}\,\,. \label{eq:mutot}$$ In general this can by done easily, since $w_j^{\rm tot}(\tau)$ is a known function. In particular for bounded transition rates it poses no difficulty, as, for example, in the case of Glauber rates or a periodic external driving, where $T$ could be chosen as the time period. If an unlimited growth of $w_j^{\rm tot}$ with time were present (an unphysical situation for long times), $T$ can be chosen self-consistently by requiring that the time $t$ for the next transition to another state $i\ne j$ (see below) must be smaller than $t_0+T$. Next an attempt time interval $\Delta t_1$ is drawn from the exponential density $F_j(\Delta t_1)=\mu_j^{\rm tot} \exp(-\mu_j^{\rm tot}\Delta t_1)$ and the resulting attempt transition time $t_1=t_0+\Delta t_1$ is rejected with probability $p_j^{\rm rej}(t_1)=1-w_j^{\rm tot}(t_1)/\mu_j^{\rm tot}$. If it is rejected, a further attempt time interval $\Delta t_2$ is drawn from $F_j(\Delta t_2)$, corresponding to an attempt transition time $t_2=t_1+\Delta t_2$, and so on until an attempt time $t<t_0+T$ is eventually accepted. Then a transition to a target state $i$ is performed at time $t$ with probability $w_{ij}(t)/w_j^{\rm tot}(t)$, using the target state selection of Eq. (\[eq:target-selection\]). In order to show that this method yields the correct first transition probability density $\psi_{ij}(t,t_0)$ from Eq. (\[eq:psi\]), let us first consider a sequence, where exactly $n\ge0$ attempts at some times $t_1<\ldots<t_n$ are rejected and then the $(n+1)$th attempt leads to a transition to the target state $i$ in the time interval $[t,t+dt[$. The corresponding probability density $\psi_{ij}^{(n)}(t,t_0)$ is given by $$\begin{aligned} \label{eq:ata-proof-1} \psi_{ij}^{(n)}&(t,t_0)= \int_{t_0}^t dt_n\int_{t_0}^{t_{n-1}} dt_{n-1}\ldots\int_{t_0}^{t_{2}} dt_1 \frac{w_{ij}(t)}{w_j^{\rm tot}(t)} \nonumber \\ &\hspace*{-2em} \times \left[1-p_j^{\rm rej}(t)\right]F_j(t-t_n) \prod_{m=1}^n p_j^{\rm rej}(t_m)F_j(t_m-t_{m-1}) \\ &=\frac{w_{ij}(t)e^{-\mu_j^{\rm tot}(t-t_0)}}{n!} \left[\int_{t_0}^td\tau\,\mu_j^{\rm tot}p_j^{{\rm rej}}(\tau)\right]^n\nonumber\\ &=\frac{w_{ij}(t)e^{-\mu_j^{\rm tot}(t-t_0)}}{n!} \left[\mu_j^{\rm tot}(t-t_0)-\int_{t_0}^td\tau\,w_j^{\rm tot}(\tau)\right]^n. \nonumber\end{aligned}$$ Summing over all possible $n$ hence yields $$\psi_j(t,t_0)=\sum_{n=0}^\infty\psi_{ij}^{(n)}(t,t_0)= w_{ij}(t)\exp\left[-\int_{t_0}^td\tau\,w_j^{\rm tot}(\tau)\right] \label{eq:ata-proof-2}$$ from Eq. (\[eq:psi\]). It is clear that for avoiding the root finding of Eq. (\[eq:t-selection\]) by use of the ATA, one has to pay the price for introducing rejections. If the typical number of rejections can be kept small and an explicit analytical expression for $t$ cannot be derived from Eq. (\[eq:t-selection\]), the ATA should become favorable in comparison to the RTA. Moreover, the ATA can be implemented in a software routine independent of the special form of the $w_{ij}(\tau)$ for applicants who are not interested to invest special thoughts on how to solve Eq. (\[eq:t-selection\]). One may object that the ATA still entails the problem connected with the cumbersome target state selection by Eq. (\[eq:target-selection\]). However, as the RTA has the first reaction variant FRTA, the ATA has a first attempt variant. In this first attempt time algorithm (FATA) one first determines, instead of $\mu_j^{\rm tot}$ from Eq. (\[eq:mutot\]), upper bounds for the individual transitions to all target states $k\ne j$ ($\mu_{jj}=0$), $$\mu_{kj}\ge\max_{t_0\le\tau\le t_0+T}\{w_{kj}(\tau)\}\,\,. \label{eq:mukj}$$ Thereon random time intervals $\Delta t_k$ are drawn from $F_{kj}(\Delta t_k)=\mu_{kj}\exp(-\mu_{kj}\Delta t_k)$, yielding corresponding attempt transition times $t_k^{(1)}=t_0+\Delta t_k$. The transition to the target state $k'$ with the minimal $t_{k'}^{(1)}=\min_k\{t_k^{(1)}\}=t_1$ is attempted and rejected with probability $p_{k'k}^{\rm rej}(t_{k'}^{(1)})=1-w_{k'k}(t_{k'}^{(1)})/\mu_{k'k}$. If it is rejected, a further time interval $\Delta t_{k'}^{(2)}$ is drawn from $F_{k'j}(\Delta t_{k'}^{(2)})$, yielding $t_{k'}^{(2)}=t_{k'}^{(1)}+\Delta t_{k'}^{(2)}$, while the other attempt transition times are kept, $t_k^{(2)}=t_k^{(1)}$ for $k\ne k'$ (it is not necessary to draw new time intervals for these target states due to the absence of memory in the Poisson process). The target state $k''$ with the new minimal $t_{k''}^{(2)}=\min_k\{t_k^{(2)}\}=t_2$ is then attempted and so on until eventually a transition to a target state $i$ is accepted at a time $t<t_0+T$. The determination of the minimal times can be done effectively by keeping an ordered stack of the attempt times. Furthermore, as in the FRTA, one can, after a successful transition to a target state $i$ at time $t$, keep the (last updated) attempt times $t_k$ for all target states that are not affected by this transition (i. e. for which $w_{ki}(\tau)=w_{kj}(\tau)$ for $\tau\ge t$). Overall one can view the procedure implied by the FATA as that each state $k$ has a next attempt time $t_k$ (with $t_j=\infty$ if the system is in state $j$) and that the next attempt is made to the target state with the minimal $t_k$. After each attempt, updates of some of the $t_k$ are made as described above in dependence of whether the attempt was rejected or accepted. In order to prove that the FATA gives the $\psi_j(t,t_0)$ from Eq. (\[eq:psi\]), we show that the probability densities $\chi_{ij}(t,t_n)=[w_{ij}(t)/w_j^{\rm tot}(t)](1-p_j^{\rm rej}(t))F_j(t-t_n)=w_{ij}(t)\exp[-\mu_j^{\rm tot}(t-t_n)]$ and $\eta_j(t_m,t_{m-1})= p_j^{\rm rej}(t_m)F_j(t_m-t_{m-1})=[\mu_j^{\rm tot}-w_j^{\rm tot}] \exp[-\mu_j^{\rm tot}(t_m-t_{m-1})]$ appearing in Eq. (\[eq:ata-proof-1\]) are generated, if we set $\mu_j^{\rm tot}=\sum_k \mu_{kj}$ (note that Eq. (\[eq:mutot\]) is automatically satisfied by this choice). These probability densities have the following meaning: $\chi_{ij}(t,t_n)dt$ is the probability that, if the system is in state $j$ at time $t_n$, the next attempt to a target state occurs in the time interval $[t,t+dt[$, the attempt is accepted, and it changes the state from $j$ to $i$; $\eta_j(t_m,t_{m-1})dt_m$ is the probability that, after the attempt time $t_m$, the next attempt occurs in $[t_m,t_m+dt_m[$ with $t_m>t_{m-1}$ and is rejected. In the FATA the probability $\kappa_{lj}(t_m,t_{m-1})dt_m$ that, when starting at time $t_{m-1}$, the next attempt is occurring in $[t_m,t_m+dt_m[$ to a target state $l$ is given by $$\begin{aligned} \kappa_{lj}(t_m,t_{m-1})&= \mu_{lj}\exp[-\mu_{lj}(t_m-t_{m-1})]\nonumber \\ & \hspace*{0.3cm} \times \prod_{k\ne l} \int_{t_m-t_{m-1}}^\infty d\tau\,\mu_{kj}\exp(-\mu_{kj}\tau)\nonumber\\ &=\mu_{lj}\exp[-\mu_j^{\rm tot}(t_m-t_{m-1})]\,\,. \label{eq:kappa}\end{aligned}$$ The product ensures that $t_m$ is the minimal time (the lower bound in the integral can be set equal to $(t_m-t_{m-1})$ for all $k\ne l$ due to the absence of memory in the Poisson process). The probability that this attempted transition is rejected is $p_{lj}^{\rm rej}(t_m)=1-w_{lj}(t_m)/\mu_{lj}$ and accordingly, by summing over all target states $l$, we obtain $$\begin{aligned} \eta_j(t_m,t_{m-1})&=\sum_l p_{lj}^{\rm rej}(t_m)\mu_{lj}\exp[-\mu_j^{\rm tot})(t_m-t_{m-1})]\nonumber\\ &=[\mu_j^{\rm tot}-w_j^{\rm tot}(t_m)]\exp[-\mu_j^{\rm tot}(t_m-t_{m-1})] \label{eq:eta}\end{aligned}$$ in agreement with the expression appearing in Eq. (\[eq:ata-proof-1\]). Furthermore, when starting from time $t_n$, the probability density $\chi_{ij}(t,t_n)$ referring to the joint probability that the next attempted transition occurs in $[t,t+dt[$ to state $i$ and is accepted is given by $$\chi_{ij}(t,t_n)=\frac{w_ {ij}(t)}{\mu_{ij}}\kappa_{ij}(t,t_n)= w_{ij}(t)\exp[-\mu_j^{\rm tot}(t-t_{n})]\,\,. \label{eq:chi}$$ Hence one recovers the decomposition in Eq. (\[eq:ata-proof-1\]) with $\mu_j^{\rm tot}=\sum_k \mu_{kj}$. Before discussing an example, it is instructive to see how the ATA (and RTA) can be associated with a solution of the underlying master equation $$\frac{\partial}{\partial t}\,\mathbb{G}(t,t')=-\mathbb{M}(t)\, \mathbb{G}(t,t')\,\,,\qquad\mathbb{G}(t',t')=\mathbb{I} \label{eq:master}$$ where $\mathbb{G}(t,t')$ is the matrix of transition probabilities $G_{ij}(t,t')$ for the system to be in state $i$ at time $t$ if it was in state $j$ at time $t'\le t$, and $\mathbb{M}(t)$ is the transition rate matrix with elements $M_{ij}(t)=-w_{ij}(t)$ for $i\ne j$ and $M_{jj}(t)=-\sum_{i\ne j} M_{ij}(t)=w_j^{\rm tot}(t)$. Let us decompose $\mathbb{M}(t)$ as $\mathbb{M}(t)=\mathbb{D}+\mathbb{A}(t)$, where $\mathbb{D}=\mathrm{diag}\left\{ \mu_1^{\rm tot},\ldots,\mu_N^{\rm tot}\right\}$. If $\mathbb{A}(t)$ were missing, the solution of the master equation (\[eq:master\]) would be $\mathbb{G}_0(t,t')= \mathrm{diag}\left\{\exp(-\mu_1^{\rm tot}(t-t'), \ldots,\exp(-\mu_N^{\rm tot}(t-t')\right\}$. Hence, when introducing $\tilde{\mathbb{A}}(t,t')=\mathbb{G}_0^{-1}(t,t')\mathbb{A}(t) \mathbb{G}_0(t,t')=\mathbb{G}_0(t',t)\mathbb{A}(t) \mathbb{G}_0(t,t')$ in the “interaction picture”, the solution of the master equation can be written as $$\begin{aligned} \mathbb{G}(t,t')&=\mathbb{G}_0(t,t')\left[ \mathbb{I} +\int_{t'}^t dt_1 \tilde{\mathbb{A}}(t_1,t') \right. \nonumber \\ & \hspace*{0.3cm} + \left.\int_{t'}^t dt_2 \int_{t'}^{t_2} dt_1 \tilde{\mathbb{A}}(t_2,t') \tilde{\mathbb{A}}(t_1,t')+\ldots\right] \label{eq:master-solution-1}\end{aligned}$$ Inserting $\mathbb{I}=\mathbb{D}^{-1}\mathbb{D}$ after each matrix $\tilde{\mathbb{A}}$, one arrives at $$\begin{aligned} &\mathbb{G}(t,t')=\mathbb{G}_0(t,t') +\int_{t'}^t dt_1 \mathbb{G}_0(t,t_1)\mathbb{B}(t_1)F_0(t_1,t') \nonumber \\ &{}+\int_{t'}^t dt_2 \int_{t'}^{t_2} dt_1 \mathbb{G}_0(t,t_2)\mathbb{B}(t_2)\mathbb{F}_0(t_2,t_1) \mathbb{B}(t_1)\mathbb{F}_0(t_1,t') \nonumber \\ &+\ldots \label{eq:master-solution-2}\end{aligned}$$ where $\mathbb{F}_0(t,t')=\mathbb{D}\,\mathbb{G}_0(t,t')= \mathrm{diag}\{\mu_1^{\rm tot}\exp[-\mu_1^{\rm tot}(t-t')] ,\ldots,$ $\mu_N^{\rm tot}\exp[-\mu_N^{\rm tot}(t-t')]\}$, and $\mathbb{B}(t)=\mathbb{A}(t)\mathbb{D}^{-1}$ has the matrix elements $B_{ij}(t)=-w_{ij}(t)/\mu_j^{\rm tot}$ for $i\ne j$ and $B_{jj}(t)=1-w_j^{\rm tot}(t)/\mu_j^{\rm tot}$. Equation (\[eq:master-solution-2\]) resembles the ATA: The transition probabilities $G_{ij}(t,t')$ are decomposed into paths with an arbitrary number $n=0,1,2,\ldots$ of “Poisson points”, where transitions are attempted. The times between successive attempted transitions are exponentially distributed according to the matrix elements of $\mathbb{F}_0$ and the attempted transitions are accepted or rejected according to the probabilities encoded in the diagonal and non-diagonal elements of the $\mathbb{B}$ matrix, respectively. The $\mathbb{G}_0$ entering Eq. (\[eq:master-solution-2\]) takes care that after the last attempt in a path with exactly $n$ attempted transitions no further attempt occurs and the system remains in the target state $i$. The RTA can be associated with an analogous formal solution of the master equation if one replaces $\mathbb{G}_0(t,t')$ by $\mathbb{G}_0^{\rm RTA}(t,t')= \mathrm{diag}\left\{ w_1^{\rm tot}(t)\exp[-\int_{t'}^td\tau\,w_1^{\rm tot}(\tau)] ,\ldots,\right.$ $\left. w_N^{\rm tot}(t)\exp[-\int_{t'}^td\tau\,w_N^{\rm tot}(\tau)]\right\}$ and $\mathbb{B}(t)$ by $\mathbb{B^{\rm RTA}}(t)$ with elements $B_{ij}^{\rm RTA}(t)=(1-\delta_{ij})w_{ij}(t)/w_j^{\rm tot}(t)$ (the diagonal elements are zero since the RTA is rejection-free). Example {#sec:example} ======= Let us now demonstrate the implementation of the FATA in an example. To this end we consider three mutually coupled two-level systems that are periodically driven. For an arbitrary given $i$, $i=1,2,3$, the state $|\,i,\pm\,\rangle$ has the energy $\pm E(t)$. The occupancy of the state $|\,i,\pm\,\rangle$ is specified by the occupation number $n_i=\pm 1$. For example, if $n_i=-1$, the $i$-th two level system resides in the state $|\,i,-\,\rangle$ and it possesses the energy $-E(t)$. The coupling is described by the (positive) interaction parameter $V$. The total energy of the three coupled two-level systems is given by the expression $$H(\mathbf{n},t)=V(n_1n_2+n_1n_3+n_2n_3)+E(t)\sum_{i=1}^{3}n_{i}\,\,, \label{eq:h}$$ where $\mathbf{n}=(n_{1},n_{2},n_{3})$ specifies the microstate of the compound system. The periodic driving is considered to change energies of the individual two-level systems as $$E(t)=\frac{\Delta E}{2}\,\sin(\omega t)\,, \label{eq:e}$$ where $\Delta E>0$ is the amplitude of modulation and $\omega$ its frequency. Due to contact of the compound system with a heat reservoir at temperature $T$, transitions between its microstates occur. Assume that in the initial state $\mathbf{m}$ one and only one occupation number differs from the corresponding occupation number in the final state $\mathbf{n}$. Then instantaneous value of the detailed-balanced Glauber jump rates connecting these two states reads $$\begin{aligned} w(\mathbf{m}\rightarrow\mathbf{n},t)= \frac{\nu}{1+\exp\left\{\beta\left[H(\mathbf{n},t)-H(\mathbf{m},t)\right]\right\}}\,\,. \label{eq:w}\end{aligned}$$ The other pairs of microstates are not connected, that is, the transition rates between them vanish. In the above expression, $\nu$ designates an attempt frequency, and $\beta$ is the inverse temperature. In the following we will use $k_{\rm B}T$ as our energy unit and $\nu^{-1}$ as our time unit. ![Work distributions $p(w)$ as obtained from the FATA for $\Delta E=V=5$ and $\omega=0.1$ (squares, green color), 1 (circles, blue color), and $\omega=10$ (stars, red color).[]{data-label="fig:fig1"}](fig1.eps){width="45.00000%"} In current research of non-equilibrium systems, in particular of processes in small molecular systems, the investigation of distributions of microscopic work receives much attention. Among others, this is largely motivated by questions concerning the optimization of processes, and by the connection of the work distributions to fluctuation theorems. These theorems allow one to obtain equilibrium thermodynamic quantities from the study of non-equilibrium processes and they are useful for getting a deeper insight into the manifestation of the second law of thermodynamics. At the same time, the analytical expressions for the work distribution are rarely attainable (one exception is reported in [@Chvosta:2007]). It is therefore interesting to see how the FATA can be employed for studies in this research field. To be specific, we focus on the stationary state and calculate work distributions within one period of the external driving. For these distributions we check the detailed fluctuation theorem of Crooks [@Crooks:1999], as generalized by Hatano and Sasa [@Hatano/Sasa:2001] to steady states (for a nice summary of different forms of detailed and integral fluctuation theorems, see [@Esposito/VandenBroeck:2010]). In our model, due to the possibility of thermally activated transitions between the eight microstates, the state vector $\mathbf{n}$ must be understood as a stochastic process. We designate it as $\mathbf{n}(t)$, and let $\mathbf{n}^{{\rm tr}}(t)$ denotes its arbitrary fixed realization. The instantaneous energy of the compound system along this realization is then $H(\mathbf{n}^{{\rm tr}}(t),t)$. The work done on the system during the $m$th period $[m\tau,(m+1)\tau]$, $\tau=2\pi/\omega$, if the system evolves along the realization in question, is given by $$\begin{aligned} w^{{\rm tr}}_{m}&=\int_{m\tau}^{(m+1)\tau}dt\,\frac{\partial}{\partial t}H(\mathbf{n}^{{\rm tr}}(t),t) \nonumber\\ &= \frac{\omega\Delta E}{2}\sum_{i=1}^3\int_{m\tau}^{(m+1)\tau}dt\, n^{{\rm tr}}_{i}(t)\cos(\omega t)\,\,. \label{eq:work}\end{aligned}$$ In the stationary limit $m\to\infty$ ($m\gg1$) we can drop the index $m$. According to the detailed fluctuation theorem, the work distribution $p(w)$ should, in our case (time-symmetric situation with respect to the initial microstate distribution for starting forward and backward paths), obey the relation $p(w)\exp(-w)=p(-w)$. Figure \[fig:fig1\] shows the results for $p(w)$ obtained from the FATA for $\Delta E=V=5$, and three different frequencies $\omega=0.1$, 1, and 10. First, we let the system evolve during the $N_{{\rm ini}}=1$ ($\omega=0.1$), $N_{{\rm ini}}=3$ ($\omega=1$), $N_{{\rm ini}}=9$ ($\omega=10$) periods to reach the stationary state. Subsequently, the work values $w^{{\rm tr}}$ according to Eq. (\[eq:work\]) were sampled over $N=10^{4}$ ($\omega=0.1$), $N=10^{5}$ ($\omega=1$), and $N=10^{5}$ ($\omega=10$) periods. With decreasing $\omega$, the maxima of the work distributions in Fig. \[fig:fig1\] shift toward $w=0$, and $\delta$-singularities, marked by the vertical lines, receive less weight. These $\delta-$singularities are associated with stochastic trajectories of the system, where no transitions occur within a period of the driving. For $\omega=0.1$, $p(w)$ is already close to the Gaussian fluctuation regime. In Fig. \[fig:fig2\] we show that the work distributions from Fig. \[fig:fig1\] indeed fulfill the detailed fluctuation theorem. This demonstrates that the FATA successfully generates system trajectories with the correct statistics of the stochastic process. ![Check of the detailed fluctuation theorem $p(w)\exp(-w)=p(-w)$ for the work distributions shown in Fig. \[fig:fig1\]. The same symbols/colors are used for the three different frequencies as in Fig. \[fig:fig1\].[]{data-label="fig:fig2"}](fig2.eps){width="45.00000%"} Summary {#sec:summary} ======= In summary, we have presented new simulation algorithms for Markovian jump processes with time-dependent transition rates, which avoid the often cumbersome or unhandy calculation of inverse functions. The ATA and FATA rely on the construction of a series of Poisson points, where transitions are attempted and rejected with certain probabilities. As a consequence, both algorithms are easy to implement, and their efficiency will be good as long as the number of rejections can be kept small. For complex interacting systems, the FATA has the same merits as the FRTA with respect to the FRA. Both the ATA and FATA generate exact realizations of the stochastic process. Their connection to perturbative solutions of the underlying master equation may allow one to include in future work also non-Markovian features of a stochastic dynamics by letting the rejection probabilities to depend on the history [@Chvosta:1999]. Compared to the RTA and FRTA, the new algorithms should in particular be favorable, when considering periodically driven systems with interactions. Such systems are of much current interest in the study of non-equilibrium stationary states and we thus hope that our findings will help to investigate them more conveniently and efficiently. Support of this work by the Ministry of Education of the Czech Republic (project No. MSM 0021620835), by the Grant Agency of the Charles University (grant No. 143610) and by the project SVV - 2010 - 261 301 of the Charles University in Prague is gratefully acknowledged. [99]{} Gibson, M. A. Bruck, J., *J. Phys. Chem. A*, **104** (2000) 1876. Anderson, D. F., *J. Chem. Phys.*, **127** (2007) 214107. Astumian, R. D., *Procl. Natl. Acad. Sci.*, **104** (2007) 19715. Crooks, G. E., *Phys. Rev. E*, **60** (1999) 2721; *ibid.*, **61** (2000) 2361. Seifert, U., *Phys. Rev. Lett.*, **95** (2005) 040602. Esposito, M. Van den Broeck, C., *Phys. Rev. Lett.*, **104** (2010) 090601. Gillespie, D. T. *J. Comput. Phys.*, **28** (1978) 395. Bortz, A. B., Karlos, M. H. Lebowitz J. L., *J. Comp. Phys.*, **17** (1975) 10. Jansen, A. P. J., *Comp. Phys. Comm.*, **86** (1995) 1. Einax, M. Maass, P., *Phys. Rev. E*, **80** (2009) 020101(R). Hatano, T. Sasa, S. I., *Phys. Rev. Lett.* **86** (2001) 3463. Chvosta, P., Reineker, P. Schulz, M., *Phys. Rev. E*, **75** (2007) 041124. Chvosta, P. Reineker, P., *Physica A* **268** (1999) 103.

--- abstract: 'Motivated by the recent experiments on periodically modulated, two dimensional electron systems placed in large transversal magnetic fields, we investigate the interplay between the effects of disorder and periodic potentials in the integer quantum Hall regime. In particular, we study the case where disorder is larger than the periodic modulation, but both are small enough that Landau level mixing is negligible. In this limit, the self-consistent Born approximation is inadequate. We carry extensive numerical calculations to understand the relevant physics in the lowest Landau level, such as the spectrum and nature (localized or extended) of the wave functions. Based on our results, we propose a qualitative explanation of the new features uncovered recently in transport measurements.' author: - Chenggang Zhou - Mona Berciu - 'R. N. Bhatt' title: Effects of large disorder on the Hofstadter butterfly --- Introduction {#sec1} ============ Two-dimensional electron systems (2DES) placed in a uniform perpendicular magnetic field exhibit a rich variety of phenomena, such as the integer[@IQHE] and fractional[@FQHE] quantum Hall effects. [@Springer] Another well-studied problem is that of a 2DES in a uniform perpendicular magnetic field subjected to a periodic potential. Even before the discovery of the quantum Hall effects, Hofstadter[@Hofst] showed that in this case, the electronic bands split into a remarkable fractal structure of subbands and gaps, the so-called Hofstadter butterfly. Two “asymptotic” regimes are usually considered: (i) if the magnitude of the periodic potential is very large compared to the cyclotron energy and the Zeeman splitting, then one can use lattice models to describe the hopping of electrons between Wannier-like states localized at the minima of the periodic potential, whereas (ii) if the magnitude of the periodic potential is small compared to the cyclotron energy, the periodic potential lifts the degeneracy of each Landau level. In both cases, the resulting butterfly structure is a function only of the ratio between the flux $\phi=B{\cal A}$ of the magnetic field through the unit cell of the periodic lattice, and the elementary magnetic flux $\phi_0=hc/e$. Remarkably, if $\phi/\phi_0$ of the first asymptotic case is equal to $\phi_0/\phi$ of the second case, their electronic structures are solutions of the same Harper’s equation.[@equiv] If the periodic potential is comparable to the cyclotron energy, Landau level mixing must be taken into account; although Landau levels still split into subbands, the structure is no longer universal, but depends also on the ratio of the periodic potential amplitude and the cyclotron energy.[@germans] Experimentally, the case with a small periodic modulation can be realized more easily. This is because the periodic potential is usually imprinted at some distance from the 2DES layer; as a result, its magnitude in the 2DES is considerably attenuated. The interesting cases to study experimentally also correspond to small values of $\phi/\phi_0$ (of order unity), where the butterfly structure shows a small number of subbands separated by large gaps, and should therefore be easier to identify. Periodic modulations have been created using lithography[@Holland; @Weimann; @Weiss] and holographic illumination. [@Wulf] The lattice constants of the resulting square lattices are of order 100 nm. As a result, the condition $\phi/\phi_0 \approx 2$ (for instance) is satisfied for $B\approx 0.8$ T. This is a very low value, in the Shubnikov-de Haas (SdH) regime, not the high-$B$ quantum regime. Significant Landau level mixing and complications from the fact that the Fermi level is inside one of the higher Landau levels for such small $B$-values make the identification of the Hofstadter structure difficult. Recently, a new method for lateral periodic modulation has been developed using a self-organized ordered phase of a diblock copolymer deposited on a GaAs/AlGaAs heterostructure.[@Sorin] The polymer spheres create a 2D triangular lattice with a lattice constant of about 39 nm. The corresponding unit cell area is almost an order of magnitude smaller than those achieved in previous experiments, implying that the condition $\phi/\phi_0 \approx 2$ is now satisfied for very strong magnetic fields, $B\approx 6$ T. At such high magnetic fields the system is in the strong quantum regime, and Landau level mixing can be safely ignored. For the experimental 2DES electron concentrations, the Fermi level is in the spin-down lowest Landau level.[@Sorin] As a result, this experimental setup appears more promising for the successful observation of the butterfly. Nevertheless, one must take into account the disorder which is present in the system (without disorder, there is no integer Quantum Hall Effect – IQHE – to begin with). If the disorder is very small compared to the periodic potential amplitude, one expects that the subbands of the Hofstadter structure are “smeared” on a scale $\hbar/\tau$, where $\tau$ is the scattering time, and $\tau \rightarrow \infty$ as disorder becomes vanishingly small. As a result, the larger gaps in the Hofstadter structure should remain open at the positions predicted in the absence of disorder, and one expects a series of minima in the longitudinal conductivity as the Fermi level traverses such gaps. The experiment indeed shows a very non-trivial modification of the longitudinal resistivity, with many peaks and valleys appearing in what is (in the absence of the periodic modulation) a smooth Lorentz-like peak.[@Sorin] However, the position of the minima in $\rho_{xx}$ do not track the positions of the main gaps in the corresponding Hofstadter butterfly structure. Instead, the data suggests that in this experimental setup, disorder is not small, but rather large compared to the estimated amplitude of the periodic potential. This is not a consequence of poor samples, since these 2DES have high mobilities. It is due to the fact that the periodic modulation is considerably attenuated in the 2DES, leading to a small energy scale for the Hofstadter butterfly spectrum as compared to $\hbar/\tau$. As a result, the Hofstadter structure predicted in the absence of disorder is of little use for interpreting the experimental data. One might expect that in this case the periodic potential should have basically no effect on the disorder-broadened Landau level. This is indeed true for the strongly localized states at the top and bottom of the Landau level. However, states in the center of the Landau level extend over many unit cells of the periodic potential, and, as we demonstrate in the following, are non-trivially modified by its presence. In this paper, we investigate numerically the behavior of a 2DES subject to a perpendicular magnetic field, a periodic potential and a disorder potential, under conditions applicable to the experimental system. The effective electron mass in GaAs is $0.067m_e$ while the magnetic fields of interest are on the order of 10 T. Under these conditions, the cyclotron energy $\hbar \omega_c$, of the order of 200 K, is the largest energy scale in the problem. The Zeeman energy $g^*\mu_B B$ for these fields is roughly 3 K, but electron interaction effects lead to a considerable enhancement of the spin splitting between the (spin polarized) Landau levels, which has been measured to be 20 K.[@Sorin2] The amplitude of the periodic potential’s largest Fourier components is estimated to be of the order of 1 K, and the scattering rate from the known zero field mobility is estimated to be $\hbar/\tau \sim 8$ K. [@Chaikin] As a result of this ordering of energy scales, we neglect Landau level inter-mixing and study non-perturbatively the combined effects of a periodic and a large smooth disorder potential on the electronic structure of the lowest Landau level. Previously, the effects of small disorder on a Hofstadter butterfly have been perturbatively investigated using the self-consistent Born approximation (SCBA), [@MacDonald] and the combined effect of white-noise disorder and periodic modulation on Hall resistance was studied following the scaling theory of IQHE.[@Huckestein] Our results reveal details of the electronic structure not investigated previously. The two-lead geometry we consider is schematically shown in Fig. \[fig1\]: the finite 2DES is assumed to have periodic boundary conditions in the $y$-direction (along which the Hall currents flow), and is connected to metallic leads at the $x=-L_x/2$ and $x=+L_x/2$ edges. In particular, in this paper we study the effects of the periodic potential on the extended states carrying longitudinal currents between the two leads, and identify a number of interesting properties, in qualitative agreement with simple arguments provided by a semi-classical picture. Our main conclusion is that while the beautiful Hofstadter structure is destroyed by large disorder, the system still exhibits very interesting and non-trivial physics. ![ The two-lead geometry considered: the finite-size 2DES has periodic boundary conditions in the $y$-direction, and is attached to metallic leads at the $x=\pm L_x/2$ ends. []{data-label="fig1"}](Fig1.eps){width="0.9\FigWidth"} The paper is organized as follows: in Section \[sec2\] we briefly review the computation of the Hofstadter structure for a small-amplitude periodic potential. In Section \[sec3\] we describe the type of disorder potentials considered. Section \[sec4\] describes the numerical methods used to analyze the spectrum and the nature of the electronic states, with both semi-classical and fully quantum-mechanical formalisms. Results are presented in Section \[sec5\], while Section \[sec6\] contains discussions and a summary of our conclusions. Periodic Potential {#sec2} ================== To clarify our notation, we briefly review the problem of a free electron of charge $-e$ moving in a 2D plane (from now on, the $xy$-plane, of dimension $L_x \times L_y$) in a magnetic field ${{\bf B}}=B{{\bf e}}_z$ perpendicular to the plane, as described by $${\cal H} = { 1 \over 2m} \left({{\bf p}}+{e \over c}{{\bf A}}\right)^2 - {1 \over 2} g\mu_B\vec{\sigma}\cdot {{\bf B}}$$ In the Landau gauge ${{\bf A}} = (0, Bx,0)$, the eigenfunctions of the Schrödinger equation ${\cal H}|n,k_y,\sigma\rangle= E_{n,\sigma}|n,k_y,\sigma\rangle$ are: $${ \label{2.1} } \langle {{\bf r}} | n, k_y, \sigma \rangle = { e^{-ik_yy} \over \sqrt{L_y}} e^{-{1 \over 2}\left( {x \over l} - lk_y\right)^2 } { H_n\left({x \over l} - lk_y\right) \over \sqrt{2^nn!\sqrt{\pi}l}}\chi_\sigma,$$ with eigenenergies $${ \label{2.2} } E_{n,\sigma} = \hbar\omega_c\left( n + { 1 \over2 }\right) - {1 \over2} g \mu_B B\sigma.$$ Here $l= \sqrt{ \hbar c/ eB}$ is the magnetic length, $\omega_c ={eB/ mc}$ is the cyclotron frequency, $H_n(x)$ are the Hermite polynomials and $\chi_{+1}^T =(1\; 0)$, respectively $\chi_{-1}^T =(0\; 1)$ are the eigenspinors of $\sigma_z$: $\sigma_z \chi_\sigma =\sigma \chi_\sigma$. The degeneracy of a Landau level is given by the number of distinct $k_y$ values allowed. Imposing cyclic boundary conditions in the $y$-direction, we find $${ \label{2.3} } k_y = { 2\pi \over L_y} j,$$ where $j$ is an integer. The allowed values for $j$ are found from the condition that the electron wave-functions, which are centered at positions $x_j = l^2 k_y = l^2 2\pi j/L_y $ \[see Eq. (\[2.1\])\] are within the boundary along the $x$-axis, i. e. $-L_x/2 < x_j \le L_x/2$. It follows that the degeneracy of each Landau level is $N = L_xL_y B/ \phi_0$, with $\phi_0 = hc/e$. Consider now the addition of a periodic potential, with a lattice defined by two non-collinear vectors ${{\bf a}}_1$ and $ {{\bf a}}_2$, such that $V({{\bf r}}) = V({{\bf r}} + n {{\bf a}}_1 + m{{\bf a}}_2)$ for any $n, m \in {\cal Z}$. The periodic potential has non-vanishing Fourier components only at the reciprocal lattice vectors ${{\bf g}} = h {{\bf g}}_1 + k {{\bf g}}_2$, where ${{\bf g}}_i\cdot{{\bf a}}_j = 2 \pi \delta_{ij}$ and $h,k$ are integers. Thus: $${ \label{2.4} } V({{\bf r}}) = \sum_{{{\bf g}}} V_{{{\bf g}}} e^{i{{\bf r\cdot g}}}.$$ Further, since $V({{\bf r}})$ is real, it follows that $V_{{{\bf g}}} = V^*_{-{{\bf g}}}$. In the absence of Landau level mixing, the Hofstadter spectrum for both square[@Hofst] $${ \label{2.6} } V_s(x,y) = 2A\left[ \cos{2 \pi \over a}x +\cos{2 \pi \over a}y \right],$$ and triangular[@Wannier] $$V_t(x,y) = -2A\left[\cos {4 \pi \over \sqrt{3}a}x + \cos {2 \pi \over \sqrt{3}a}\left(x -y \sqrt{3} \right) \right.$$ $${ \label{2.7} } \left. + \cos {2 \pi \over \sqrt{3}a}\left(x +y \sqrt{3} \right)\right]$$ periodic potentials, with nonzero Fourier components only for [*the shortest reciprocal lattice vectors*]{}, have been studied extensively in the literature.[@Hofst; @Wannier; @Geisel; @Gerhardts] The parameter defining the spectrum is the ratio between the flux $\phi={{{\bf B}}}\cdot({{\bf a}}_1 \times {{\bf a}}_2)$ of the magnetic field through a unit cell and the elementary flux $\phi_0$. For $\phi/\phi_0=q/p$, where $p$ and $q$ are mutually prime integers, the original Landau level is split into $q$ sub-bands. We would like to emphasize a qualitative difference between the two types of potentials: the square potential in Eq. (\[2.6\]) is particle-hole symmetric, since $V_s(x,y) = -V_s(x+{a\over 2},y+{a \over 2})$. As a result, the sign of its amplitude is irrelevant. On the other hand, the triangular potential does not have this symmetry. With the sign chosen in Eq. (\[2.7\]) and $A>0$, $V_t$ has deep local minima at the sites of the triangular lattice, whereas the maxima are relatively flat and located on a (displaced) honeycomb lattice. As a result, the sign of $V_t$ is highly relevant. The second fact that must be mentioned is that the choice made in Eqs. (\[2.6\]) and (\[2.7\]) is rather simple, since it aligns the periodic potential with the edges of the sample in a very specific way. In general, however, one could consider the case where the periodic lattice is rotated by some finite angle with respect to the sample edges; study of such cases will be discussed in future work. Finally, it may seem that this choice of periodic potentials is very restrictive also because only the shortest lattice vectors have been kept in the Fourier expansion. In fact, the methods we employ can be directly used for potentials with more Fourier components, but their inclusion leads to no new physics. Disorder Potential {#sec3} ================== Real samples always have disorder. The current consensus is that high-quality GaAs/AlGaAs samples exhibit a slowly varying, smooth disorder potential. In a semi-classical picture, the allowed electron trajectories in the presence of such disorder follow its equipotential lines.[@Springer; @Trugman] Closed trajectories imply localized electron states, while extended trajectories connecting opposite edges of the sample are essential for current transport through the sample (for more details, see Sec. \[classical\]). In typical experimental setups,[@Sorin] dopant Si impurities with a concentration of $\sim10^{13}$ cm$^{-2}$ are introduced in a thin layer of 6 nm in thickness, located 20 nm above the GaAs/AlGaAs interface. Typically, up to 10% of the Si atoms are ionized. A small fraction of the ionized electrons migrate to the GaAs/AlGaAs interface where they form the 2D electron gas. The electrostatic potential created by the ionized impurities left behind is the major source of disorder in the 2DES layer. On the length-scale we are interested in, there are $10^4$ to $10^5$ such ionized Si impurities per $\mu m^2$. The resulting disorder potential must be viewed as a collective effect of the density fluctuation of the ionized impurities[@Nixon] rather than a simple summation of the Coulomb potential of a few impurities. The electrostatic potential from Si impurities is compensated and partially screened by other mobile negative charges in the system such as, for example, the surface screening effect by mirror charges considered by Nixon and Davies.[@Nixon] An exact treatment of this problem is difficult, since one should consider the spatial correlation of the ionized impurities. [@Stopa; @DasSarma] One model used to describe such disorder consists of randomly placed Gaussian scatterers.[@Ando] This model captures the main feature of a smooth disorder potential and supports classical trajectories on equipotential contours, but it has no natural energy/length scales associated with it. As a result, here we choose to also investigate a different model of the disorder, which incorporates the smooth character of the Coulomb potential in real space. We generate a realization of the disorder potential in the following way: positive and negative charges, corresponding to a total concentration of $10^{3}\,\mu m^{-3}$ are randomly distributed within a volume $[-L_x/2,L_x/2]\times[-L_y/2,L_y/2]\times[20\mbox{nm}+d,26\mbox{nm}+d]$ above the electron gas which is located in the $z=0$ plane. Here, we choose $d=4$ nm as an extra spacer since the electronic wave-functions are centered about 3-5 nm below the GaAs/AlGaAs interface. Since we are not simulating single impurities but density fluctuations, these charges are not required to be elementary charges. Instead, we use a uniform distribution in the range $[-e,e]$ for convenience (a Gaussian distribution would also be a valid choice), and sum up all Coulomb potentials from these charges, using the static dielectric constant in GaAs $\epsilon = 12.91$.[@Ralph] The resulting disorder potential has energy and length scales characteristic of the real samples. Typical contours for such potentials are shown in Sec. \[sec5\]. In an infinite system, in the quantum Hall regime, the existence of quantum Hall steps implies the existence of critical energies at which the localization length diverges.[@Halperin] This is the quantum analog of the two dimensional percolation problem in a smooth random landscape, for which there exists a single critical energy.[@Trugman] In the case of potentials with electron-hole symmetry $\left<V({{{\bf r}}})\right>=0$, the critical energy lies in the middle of the band ($E_c=0$), leading to percolating path at half filling. For a finite mesoscopic sample, however, not only does the percolating path (critical energy) $E_c$ deviate from this value, but in samples without a periodic boundary condition one need not have a percolating path traversing the system in the desired direction. This arises from the fluctuations near the edge of a mesoscopic system with free boundary conditions. We circumvent such a possibility by adding an extra smooth potential $V'(x,y)$ to the impurity-induced disorder potential $V_i(x,y)$, such that the total potential $V = V_i + V'$ is zero on the opposite edges $ x= \pm L_x/2$ of the sample where the metallic leads are attached. The supplementary contribution $V'(x,y)$ can be thought of as simulating the effect of the leads on the disorder potential, since the metallic leads hold the potential on each edge constant by accumulating extra charges near the interface. Therefore, physically we expect that the extra potential $V'$ decays exponentially over the screening length $\lambda$ inside the sample. This implies: $$V'(x,y) = -{V_i\left(-{L_x/2},y\right)+V_i(L_x/2, y) \over 2 } {\cosh(x/\lambda)\over \cosh(L_x /2\lambda)}$$ $$+{V_i(-L_x/2,y)-V_i(L_x/2,y) \over 2 } {\sinh(x/\lambda)\over\sinh(L_x /2\lambda)}$$ where $\lambda$ is taken to be $100$ nm in our calculation. ![Averaged Fourier amplitudes of two types of disorder potential as a function of wavevector $q = |{{{\bf q}}}|$. For both Coulomb and Gaussian model, $V(q)^2$ is averaged over 116 disorder realizations. The relation between $V(q)$ and $V({{\bf r}})$ and relevant parameters are discussed in the text. The standard deviation, $(L_xL_y)^{-1} \left<\int d{{\bf r}} V^2({{\bf r}})\right>$ for the Coulomb model is $3.2\times10^{-7}$eV$^2$, and $2.1\times10^{-7}$eV$^2$ for the Gaussian model.[]{data-label="fig1.5"}](Fig2.eps){width="\FigWidth"} In Fig. \[fig1.5\], we plot the average of Fourier transform of the magnitude of the random potential $\sqrt{\left< |V(q)|^2\right>}$ versus $q = |{{{\bf q}}}|$ for the Coulomb model and the Gaussian model. The Gaussian model is generated by adding 100 randomly placed Gaussian scatterers on an area of $3\mu m \times 3\mu m$, each contributing $A_d e^{-r^2/d^2}$, where $A_d$ is uniformly distributed in $[-2,2]$ meV, and $d$ is uniformly distributed in $[0,0.2]$ $\mu$m. $V({{\bf q}})$ is related to $V({{\bf r}})$ by $V({{\bf r}}) = \sum_{{{\bf q}}} V({{\bf q}}) e^{i{{{\bf q}} \cdot r}}$, where the summation is over all the wavevectors involved in the fast Fourier transformation. The Gaussian model has an arbitrary energy scale which is fixed by the maximum value of the distribution $A_m$. Here $A_m = 2$ meV. As can be seen, $V(q)$ of both models are decreasing functions of $q$. The trends of decay are exponential at large $q$. At small $q$, the two models behave differently. Despite the difference, both models lead to the same qualitative results, although, as expected, minor quantitative differences are present. This shows that the physics we uncover is independent of the particular type of slowly-varying disorder potential considered, and therefore should be relevant for the real samples. Numerical Calculations {#sec4} ====================== In this section we discuss the numerical methods we use, including derivations of some relevant formulas. As already stated, we focus on the case where the amplitudes of the periodic and disorder potentials are very small compared to the cyclotron energy and the Zeeman splitting, and therefore inter-level mixing is ignored. Semi-classical treatment {#classical} ------------------------ The semi-classical approach is valid[@Trugman] for the integer quantum Hall effect in the presence of a slowly varying, smooth disorder potential and large magnetic fields (such as we consider), so that the magnetic length $l$ which determines the spatial extent of the electron wave-functions is much smaller than the length scale of variation of the smooth disorder potential, $|\nabla V({{\bf r}})| \ll \hbar\omega_c/l$. Then, semi-classically the electron moves along the equipotential contours of the disorder potential $V({{\bf r}})$, in the direction parallel to $\nabla V({{\bf r}}) \times {{\bf B}} $. Since the kinetic energy is quenched in the lowest Landau level, the total energy of the electron simply equals the value of the disorder potential on the equipotential line on which its trajectory is located. As a result, the density of states in the semi-classical approach is directly given by the probability distribution for the disorder potential, which can be calculated by randomly sampling the potential energy and plotting a histogram of the obtained values.[@Trugman; @note1] In Sec. \[sec5\] we compare the results obtained within this semi-classical approach with fully quantum mechanical results. As expected, the agreement is good if only the disorder potential is present. However, if the periodic modulation is also included, the lattice constant $a$ provides a new length-scale which is comparable to the magnetic length $l$, and the semi-classical picture breaks down. Quantum mechanical calculations are absolutely necessary to quantitatively treat this case. Quantum Mechanical Treatment ---------------------------- As shown in Sec. \[sec2\], for a finite sample of size $L_x \times L_y$ at a given magnetic field $B$, the degeneracy of the unperturbed Landau level is $N = L_xL_yB/\phi_0 = L_xL_y/(2\pi l^2)$. Since the disorder varies very slowly, we need to consider systems with $L_x,L_y \gg l$ to properly account for its effects. As a result, the number of states in a Landau level can be as large as $10^4$ in our calculations. Storage of the Hamiltonian as a dense matrix requires considerable amount of computer memory and its direct diagonalization is prohibitively time-consuming. Sparse matrix diagonalization techniques could be employed, but they are less efficient when all eigenvectors are needed, and also have some stability issues. Here we introduce the numerical methods we use to compute densities of states and infer the nature (localized or extended) as well as the spatial distribution of the wave-functions, while avoiding direct diagonalization. ### Matrix elements Since inter-level mixing is ignored, the Hilbert subspaces corresponding to different spin-polarized Landau levels do not hybridize. Each Hilbert subspace $(n,\sigma)$ has a basis described by Eq. (\[2.1\]), containing $N$ orthonormal vectors indexed by different $k_y$ values. In order to compute matrix elements of the total Hamiltonian in such a basis, we use the following identity derived in Ref. (notice their different sign convention for $k_y$. If $\sigma\ne \sigma'$, the overlap is zero): $${ \label{2.5} } {\left< n', k'_y \right|} e^{i{{\bf q\cdot r}}}{\left|n,k_y \right>} = \delta_{k'_y,k_y-q_y} {\cal L}_{n',n}({{\bf q}}) e^{{il^2\over 2}q_x(k'_y+k_y)},$$ where $$\begin{aligned} {\cal L}_{n',n} ({{\bf q}}) = \left(m! \over M! \right)^{1 \over 2}i^{|n'-n|}\left[ q_x+iq_y \over \sqrt{q_x^2+q_y^2}\right]^{n-n'} && \\ \times e^{-{1 \over 2} Q}Q^{{1 \over 2}|n'-n|}L_m^{(|n'-n|)}(Q), &&\end{aligned}$$ with $Q = {1 \over 2}l^2(q_x^2+q_y^2)$, $m$ and $M$ the minimum and the maximum of $n'$ and $n$ respectively, and $L^{(|n'-n|)}_m(Q)$ the associated Laguerre polynomial. When band-mixing is neglected $n = n'$ and ${\cal L}_{n,n} ({{\bf q}}) = e^{-{1 \over 2}Q} L_n(Q)$. For the first Landau level, $L_0(x) =1$. Eq. (\[2.5\]) gives us the matrix elements for the square \[Eq. (\[2.6\])\] or triangular \[Eq. (\[2.7\])\] periodic potentials. In either case, there are Fourier components corresponding to $q_y=\pm 2\pi/a$ and $q_y=0$. Since only basis vectors for which the difference $k_y-k_y'=q_y$ give non-vanishing matrix elements, we must choose the length $L_y$ of the sample to be a multiple integer of $a$, the lattice constant. The matrix elements of the disorder potential are computed in a similar way. We use a grid of dimension $N_x\times N_y$ to cover the sample and generate the values of the disorder potential on this grid. Then, fast Fourier transform (FFT)[@FFT] is used to find the long wavelength components of the disorder potential corresponding to the allowed values $q_{x,y}=0,\pm{2\pi\over L_{x,y}}, ..., \pm \left[{N_{x,y}\over 2}\right]{ 2\pi\over L_{x,y}}$ (proper care is taken to define Fourier components so that $V_{{{\bf q}}} = V^*_{-{{\bf q}}}$). The matrix elements of this discretized disorder potential are then computed using Eq. (\[2.5\]). In principle, finer grids (increased values for $N_x$ and $N_y$) will improve accuracy. However, they also result in longer computation times, since they add extra matrix elements in the sparse matrix, corresponding to large wave-vectors. We have verified that a grid size of dimension $N_x=N_y=72$ is already large enough to accurately capture the landscape of a $3\mu m \times 3\mu m$ sample and the computed quantities have already converged, with larger grids leading to hardly noticeable changes. This procedure is also justified on a physical basis. First, the neglected large wave-vector components describe very short-range spatial features, which are probably not very accurately captured by our disorder models to begin with, and which are certainly not believed to influence the basic physics. Secondly, this procedure insures that the actual disorder potential we use is periodic in the $y$-direction, since each Fourier component retained has this property. This is consistent with our use of a basis of wave-functions which are periodic along $y$. The matrix elements of the Hamiltonian within a given Landau level $(n,\sigma)$ are then $\langle n, k_y,\sigma | {\cal H} | n, k'_y,\sigma\rangle= E_{n,\sigma} + \langle n, k_y | {\cal V} | n, k'_y\rangle$, where $E_{n,\sigma}$ are given by Eq. (\[2.2\]) and the matrix elements of both the periodic and the disorder part of the potential ${\cal V}$ are computed as already discussed. This produces a sparse matrix, which is stored efficiently in a column compressed format. ### Densities of States and Filling Factors {#subs2} A quantity that can be computed without direct diagonalization is the filling factor $\nu_{n,\sigma}(E_F)$ as a function of Fermi energy. The filling factor is defined as: $${ \label{4.6} } \nu_{n,\sigma}(E_F) = { 1 \over N} \sum_{\alpha} \Theta(E_F- E_{n,\alpha,\sigma}),$$ where $\Theta(x)$ is the Heaviside function and $N$ is the total number of states in the $(n,\sigma)$ Landau level. (Since we neglect Landau-level mixing, we can define this quantity for individual levels.) The filling factor tells us what fraction of the states in the given Landau level are occupied at $T=0$, for a given value of the Fermi energy. This corresponds to the average filling factor measured in experiment and is also proportional to the integrated total (as opposed to local) density of states. The filling factor is straightforward to compute if the eigenenergies $E_{n,\alpha,\sigma}$ are known. However, we want to avoid the time-consuming task of numerical brute force diagonalization. The strategy we follow is a generalization to Hermitian matrices of the method used in Ref. . We restate the problem in the following way: assume we have a Hermitian matrix of size $N \times N$ (no Landau level mixing), given by the matrix elements of $M = {\cal H} - E_F {\bf 1} $ in the basis $|n, k_y, \sigma\rangle$ (${\bf 1}$ is the unit matrix). Then, $\nu_{n,\sigma}(E_F)$ is proportional to the number of negative eigenvalues of the matrix $M$. We now generate the quadratic form ${\cal M} = \sum_{i,j = 1}^{N} \zeta_i \zeta_j^* M_{ij}$, and transform it into its standard form ${\cal M} = \sum_{i= 1}^{N} d_i|\chi_i|^2$ using the Jacobian method described below. Here, $d_i$’s are all real numbers, and the $\chi_i$’s are linear combinations of the $\zeta_i$’s. This is a similarity transformation which retains the signature of the matrix. As a result, even though the numbers $d_i$ are not eigenvalues of $M$, the number of negative eigenvalues equals the number of negative $d_i$ values. It follows that $\nu_{n,\sigma}(E_F)$ is obtained by simply counting the number of negative $d_i$ values for the given $E_F$. The Jacobian method is iterative in nature. First, all terms containing $\zeta_1$ and $\zeta_1^*$ are collected and the needed complementary terms are added to form the first total square $d_1|\chi_1|^2$, so that $\zeta_1$ and $\zeta_1^*$ are eliminated from the rest of the quadratic form ${\cal M}$. The procedure is then repeated for all $\zeta_2$ and $\zeta_2^*$ terms (producing $d_2$) etc., until all $N$ values $d_i$ are found. Computationally, this can be done by scanning the lower or upper triangle of the Hermitian matrix $M$ only once. The total number of operations is proportional to the number of nonzero elements of the matrix, meaning that for a dense matrix it scales with $N^2$ (sparse matrices require much fewer operations). As a result, this procedure is much faster than brute force diagonalization which scales with $N^3$ (for us, $N\sim 10^4$). The filling factor $\nu_{n,\sigma}(E)$ is a sum of step-like functions, with steps located at the eigenvalues. By scanning $E$ and identifying the position of these steps we can also find the true eigenvalues $E_{n,\alpha,\sigma}$, with the desired accuracy. Finally, the total density of states is given by $ \rho_{n\sigma}(E) = d \nu_{n,\sigma}(E)/dE$. ### Green’s functions: extended vs. localized states {#subs3} The advanced/retarded Green’s functions are the solutions of the operator equation $${ \label{4.1} } \left(\hbar\omega - {\cal H} \pm i \delta\right) {\hat G}^{R,A}(\omega) = {\bf 1},$$ where $\delta \rightarrow 0^{+}$. (In practice we use a set of small positive numbers, and use the dependence on $\delta$ to obtain results.) If the exact eigenstates and eigenvalues of the total Hamiltonian ${\cal H}$ are known, ${\cal H} | n, \alpha, \sigma\rangle = E_{n,\alpha,\sigma} | n, \alpha, \sigma\rangle, $ (no Landau level mixing), it follows: $${ \label{4.2} } {\hat G}^{R,A}(\omega) = \sum_{n,\alpha,\sigma}^{}{ | n,\alpha,\sigma\rangle\langle n, \alpha, \sigma| \over \hbar\omega - E_{n,\alpha,\sigma} \pm i\delta} = \sum_{n,\sigma}^{} {\hat G}^{R,A}_{n,\sigma}(\omega).$$ The exact eigenstates can be expanded in terms of the basis states $|n, k_y,\sigma\rangle$ as $${ \label{4.3} } | n, \alpha, \sigma\rangle = \sum_{k_y} c_{n,\alpha}(k_y) |n, k_y,\sigma\rangle.$$ Since the states $|n, k_y,\sigma\rangle$ are localized near $x=k_yl^2$ \[see Eq. (\[2.1\])\], the coefficients $c_{n,\alpha}(k_y)$ describe the probability amplitude for an electron in the state $| n, \alpha, \sigma\rangle$. Knowledge of these coefficients allows us to infer whether such states are extended or localized in the $x$-direction, i.e. whether they can carry currents between the leads. However, as already stated, we wish to avoid direct diagonalization. We can still infer whether the Hamiltonian has extended or localized wave-functions near a given energy $\hbar\omega$ in the following way. We introduce the matrix elements: $$G^{R,A}_{n,\sigma}(k_y, k'_y; \omega) = \langle n, k_y,\sigma | {\hat G}^{R,A}(\omega) | n, k'_y,\sigma\rangle$$ $${ \label{4.4} } = \sum_{\alpha}^{} { c_{n,\alpha}(k_y) c^*_{n,\alpha}(k'_y) \over \hbar\omega - E_{n,\alpha,\sigma} \pm i\delta}.$$ If Landau level mixing is neglected, Eq. (\[4.1\]) can be rewritten in the basis $|n, k_y,\sigma\rangle$ as: $$\sum_{k''_y}^{} \left[(\hbar\omega \pm i\delta) \delta_{k_y,k''_y} - \langle n, k_y,\sigma | {\cal H} | n, k''_y,\sigma\rangle \right]$$ $${ \label{4.5} } \times G^{R,A}_{n,\sigma} (k''_y,k'_y; \omega)=\delta_{k_y,k'_y}.$$ We use the popular numerical library SuperLU,[@SuperLU] based on LU decomposition and Gaussian reduction algorithm for sparse matrices, to solve these linear equations. Consider now the matrix element $ G^{R,A}_{n,\sigma}(k_{\min},k_{\max}; \omega) $ corresponding to the smallest $k_y=k_{\min}$ and the largest $k_y=k_{\max}$ values. If all wave-functions with energies close to $\hbar\omega$ are localized in the $x$-direction, it follows that $|G^{R,A}_{n,\sigma}(k_{\min},k_{\max}; \omega)| $ is a very small number, of the order $e^{-L_x/\xi(\omega)}$, where $\xi(\omega)$ is the localization length at the given energy. On the other hand, we expect to see a sharp peak in the value of $|G^{R,A}_{n,\sigma}(k_{\min},k_{\max}; \omega)| $ if $\hbar\omega$ is in the vicinity of an extended state eigenvalue, since \[see Eqs. (\[4.3\],\[4.4\])\] both $c_{n,\alpha}(k_{\min})$ and $c_{n,\alpha}(k_{\max})$ are non-vanishing for an extended wave-function with significant weight near both the $-L_x/2$ and the $L_x/2$ edges. Moreover, the height of this peak scales like $1/\delta$, so by varying $\delta$ we can easily locate the energies of the extended states. ### Green’s functions: local densities of states {#subs4} We can also use Green’s functions techniques to image the local density of states at a given energy $E$. By definition (and neglecting Landau level mixing), the local density of states in the level $(n,\sigma)$ is: $$\rho_{n,\sigma}({{\bf r}};E) = \sum_{\alpha}^{} |\langle {{\bf r}}| n, \alpha, \sigma\rangle |^2 \delta\left(E-E_{n,\alpha,\sigma}\right)$$ $${ \label{5.1} } = { 1 \over \pi} \mbox{Im}\langle {{\bf r}}| {\hat G}^{A}_{n,\sigma}(E)|{{\bf r}}\rangle,$$ where the second equality follows from Eq. (\[4.2\]). This function traces the contours of probability $|\phi_{n,\alpha,\sigma}({{\bf r}})|^2$ for electrons with the given energy $E$. Its direct computation, however, is difficult and very time-consuming. For the rest of this subsection, the discussion is restricted to the Lowest Landau level $n=0$ (the value of $\sigma$ is irrelevant). We know that in the lowest Landau level, electronic wave-functions cannot be localized in any direction over a length-scale shorter that the magnetic length $l$. As a result, it suffices to compute a projected local density of states on a grid with $l\times l$ (or larger) spacings. The projection is made on maximally localized wave-function, defined as follows. Let ${{\bf r}}_0=(x_0,y_0)$ be a point on the grid. We associate it with a vector: $${ \label{5.2} } |x_0, y_0\rangle = \sum_{k_y}^{} |k_y\rangle \langle k_y| x_0,y_0\rangle,$$ where we use the simplified notation $|k_y\rangle \equiv |n=0,k_y,\sigma\rangle$ for the basis states of the first Landau level (see Eq. (\[2.1\])) and we take $${ \label{5.3} } \langle k_y| x_0,y_0\rangle = \sqrt{{2l\pi^{1 \over2}} \over L_y} e^{-{x_0^2 \over 2l^2} - {k_y^2l^2\over 2} + k_y(x_0+iy_0)}.$$ It is then straightforward to show that $${ \label{5.4} } \langle {{\bf r}}| x_0, y_0\rangle = { 1\over \sqrt{2\pi}l} e^{- {(x-x_0)^2 \over 4l^2 } - {(y-y_0)^2 \over 4l^2 }}e^{-{i \over 2l^2}(x+x_0)(y-y_0)}.$$ In other words, $|x_0,y_0\rangle$ is an eigenstate of the first Landau level strongly peaked at ${{\bf r}}= {{\bf r}}_0$. (The phase factor is due to the proper magnetic translation). We then define the projected density of states \[compare with Eq. (\[5.1\])\]: $${ \label{5.5} } \rho_P (x_0,y_0; E) = { 1 \over \pi} \mbox{Im} \langle x_0,y_0|\hat{G}^{A}(E)|x_0,y_0\rangle,$$ and use it to study the spatial distribution of the electron wave-functions at different energies. Strictly speaking, the local density of states defined in Eq. (\[5.1\]) cannot be projected exactly on the lowest Landau level, because the lowest Landau level does not support a $\delta$-function ($\langle {{\bf r}}| n,k_y,\sigma\rangle \ne 0$, $\forall n$). However, the coherent states $|x_0,y_0\rangle$ we select are the maximally spatially-localized wave functions in the lowest Landau level, and have the added advantage that they can be easily stored as sparse vectors, because of their Gaussian profiles \[see Eq. (\[5.3\])\]. Moreover, in the limit $l \rightarrow 0$ ($B \rightarrow \infty$) where $|\langle{{\bf r}} | x_0, y_0\rangle|\rightarrow \delta(x-x_0)\delta(y-y_0)$, the projected density of states $\rho_P(x_0,y_0; E)\rightarrow \rho_{0,\sigma}({{\bf r}}; E)$. Therefore, for the large $B$ values that we consider here, the projected density of states $\rho_P$ should provide a faithful copy of the local density of states. We compute the projected local density of states following the method of Ref. . Let ${{\bf u}}_0$ be the vector with elements $\langle k_y| x_0, y_0\rangle$ obtained from the representation of $|x_0,y_0\rangle$ in the $|k_y\rangle$ basis \[see Eq. (\[5.2\])\], and let $H$ be the matrix of the Hamiltonian ${\cal H}$ in the $|k_y\rangle$ basis. We generate the series of orthonormal vectors ${{\bf u}}_0, {{\bf u}}_1, ...$ using: $$\begin{aligned} {{\bf v}}_1 &=& H {{\bf u}}_0 ,\\ a_0 &=& {{\bf u}}_0^\dagger {{\bf v}}_1 ,\\ {{\bf u}}_1 &=& { {{\bf v}}_1 - a_0{{\bf u}}_1 \over \sqrt{{{\bf v}}_1^\dagger {{\bf v}}_1-a_0^2}},\end{aligned}$$ and for $n \ge 2$, $$\begin{aligned} {{\bf v_n}} &=& H {{\bf u}}_{n-1} ,\\ a_{n-1} &=& {{\bf u}}_{n-1}^\dagger {{\bf v_n}} ,\\ b_{n-2} &=& {{\bf u}}_{n-2}^\dagger {{\bf v_n}} ,\\ {{\bf u}}_n &=& {{{\bf v}}_n - a_{n-1} {{\bf u}}_{n-1} - b_{n-2}{{\bf u}}_{n-2} \over \sqrt{{{\bf v}}_n^\dagger {{\bf v}}_n - a_{n-1}^2 - b_{n-2}^2 }}.\end{aligned}$$ The numbers $a_n$ and $b_n$ can be shown to be real. We do not have a “terminator”[@Haydock] to end this recursive series. Instead, our procedure ends when the orthonormal set of vectors ${{\bf u}}_0, {{\bf u}}_1, ...$ exhausts a subspace of the lowest Landau level containing all states coupled through the disorder and/or periodic potential to the state $|x_0,y_0\rangle$ (i.e., all states that contribute to the projected DOS at this point). In the presence of disorder, this usually includes the entire lowest Landau level. Then, the projected density of states is given by Eq. (\[5.5\]), where the matrix element of the Green’s function is the continued fraction: $$\langle x_0,y_0|G^A(E)|x_0,y_0\rangle=$$ $${ \label{5.6} } \left[E-i\delta-a_0 -b_0^2\left[E-i\delta - a_1 -b_1^2\left[ \ldots\right]^{-1} \right]^{-1} \right]^{-1}$$ Because the Hamiltonian is a sparse matrix, the generation of these orthonormal sets and computation of $\rho_p(E)$ for all the grid points is a relatively fast procedure. Moreover, this computation is ideally suited for parallelization, with different grid points assigned to different CPUs. Numerical Results {#sec5} ================= In this section we present numerical results obtained using these methods. We have analyzed over 20 different disorder realizations for samples of different sizes, and all exhibit the same qualitative physics. Here, we show results for several typical samples. The lattice constant is always $a=39$ nm if periodic potential is present, as defined by the experimental system.[@Sorin] ![\[fig2\] Profile of the disorder potential obtained from our Coulomb model on a $3.11\mu$m$\times 2.96\mu$m sample, without (upper panel) and with (lower panel) the $V'({{\bf r}})$ correction at the $x=\pm L_x/2$ edges. The disorder potential varies between $-3$ meV and 3 meV, on a spatial length-scale much larger than $l=12.03$ nm. The critical region containing extended states is in the vicinity of $E=0.06$ meV. The contours are shown for $E=$0.0575 meV (dashed), 0.17 meV (thick solid) and 0.31 meV (thin solid). These energy values correspond to classical filling factors $\nu$=0.47, 0.58 and 0.68 in the upper panel and $\nu$=0.45, 0.56, 0.66 in the lower panel. The difference is due to the supplementary smooth potential $V'$.](Fig3a_s.eps "fig:"){width="\FigWidth"} ![\[fig2\] Profile of the disorder potential obtained from our Coulomb model on a $3.11\mu$m$\times 2.96\mu$m sample, without (upper panel) and with (lower panel) the $V'({{\bf r}})$ correction at the $x=\pm L_x/2$ edges. The disorder potential varies between $-3$ meV and 3 meV, on a spatial length-scale much larger than $l=12.03$ nm. The critical region containing extended states is in the vicinity of $E=0.06$ meV. The contours are shown for $E=$0.0575 meV (dashed), 0.17 meV (thick solid) and 0.31 meV (thin solid). These energy values correspond to classical filling factors $\nu$=0.47, 0.58 and 0.68 in the upper panel and $\nu$=0.45, 0.56, 0.66 in the lower panel. The difference is due to the supplementary smooth potential $V'$.](Fig3b_s.eps "fig:"){width="\FigWidth"} ![ Semi-classical (dashed line) and quantum (solid line) filling factors for the disorder potential shown in Fig. \[fig2\], but different amplitudes of the triangular periodic potential (a) $A$=0; (b) $A$=0.05meV; (c) $A$=0.5meV and (d) $A$=5 meV. As expected, agreement exists only in the limit $A \rightarrow 0$. []{data-label="fig3"}](Fig4.ps){width="\FigWidth"} ![ Semi-classical (dashed line) and quantum (full line) density of states calculated from corresponding filling factors in Fig. \[fig3\]. We show only the center of the disorder-broadened lowest Landau level, where the density of states is large. []{data-label="fig4"}](Fig5.ps){width="\FigWidth"} For the first sample, we consider $\phi/\phi_0=3/2$ ($B = 4.71$ T). The magnetic length is $l= 12.03$ nm, and we choose a sample size $L_x=3.11\mu$m and $L_y= 76a=2.964\mu$m. With these choices, the Landau level contains $N=10108$ states. The disorder potential obtained with our scheme described in Sec. \[sec3\] is shown in Fig. \[fig2\], both with and without the correction $V'({{\bf r}})$. An extended equipotential line appears, as expected, at $\nu \approx 0.5$. In Figs. \[fig3\] and \[fig4\] we plot the filling factor $\nu(E)$ and the corresponding total density of states $\rho(E)$ as a function of $E$ (computation details were given in Sec. \[subs2\]). These quantities are obtained in the semi-classical limit (dashed line) and with the full, quantum-mechanical treatment (solid line). Results are shown for 4 different cases: (a) only disorder potential and (b, c, d) disorder plus a triangular periodic potential with amplitudes $A=0.05$, 0.5 and 5 meV, respectively. We only plot a relatively small energy interval where the DOS is significant, and ignore the asymptotic regions with long tails of localized states. While the agreement between the semi-classical and quantum-mechanical treatment is excellent in the limit $A\rightarrow 0$, the two methods give more and more different results as the periodic potential amplitude is increased. As already explained, this is a consequence of the fact that the magnetic length $l$ is comparable to the lattice constant $a$, leading to a failure of the semi-classical treatment when this extra length-scale is introduced. In particular, in the case with the largest periodic potential \[panel (d) of Figs. \[fig3\] and \[fig4\]\] we can clearly see the appearance of the 3 subbands expected for the Hofstadter butterfly at $\phi/\phi_0=3/2$, although the disorder leads to broadened and smooth peaks, and partially fills-in the gap between the lower two subbands. This picture \[panel (d)\] is quite similar to the density of states that Ref.  calculated using the self-consistent Born approximation. This is expected since the SCBA approach is valid in the limit of strong periodic potential with weak disorder. However, the SCBA approach is not appropriate in the limit of moderate or strong disorder, where the higher order terms neglected in SCBA are no longer small. For disorder varying on a much longer length-scale than the periodic potential, one still expects that [*locally*]{}, on relative flat regions of disorder, the system exhibits the Hofstadter-type spectrum. However, these spectra are shifted with respect to one another by the different local disorder values. If disorder variations are small, then the total spectrum shows somewhat shifted subbands with partially filled-in gaps, but overall the Hofstadter structure is still recognizable. On the other hand, for moderate and large disorder, the detailed structure of the local density of states from various flat regions are hidden in the total density of states. All one sees are some broadened, weak peaks and gaps superimposed on a broad, continuous density of states. We now analyze the nature of the electronic states for these configurations. We start with the case which has only disorder. In Fig. \[fig5\] we plot $|G^R(k_{\min},k_{\max};E)|^2$ as a function of the energy $E$, for different values of $\delta$ (computation details were given in Sec. \[subs3\]). As already discussed, extended states are indicated by large values of this quantity, as well as a strong (roughly $1/\delta^2$) dependence on the value of the small parameter $\delta$. Figure \[fig5\] reveals that as $\delta$ is reduced, resonant behavior appears in a narrow energy interval $E= 0.02-0.36$ meV. Panel (a) shows that results corresponding to $\delta = 10^{-7}$ eV and $\delta = 10^{-8}$ eV indeed differ by roughly 2 orders of magnitude, with $\delta = 10^{-8}$ eV showing sharper resonance peaks. The difference between results for $\delta = 10^{-8}$ eV and $\delta =10^{-9}$ eV shown in panel (b), is no longer so definite. The reason is simply that for such small $\delta$, the denominator in the Green’s function expression is usually limited by $|E-E_{n,\alpha,\sigma}|$ and not by $\delta$ \[see Eq. (\[4.4\])\], and the dependence on $\delta$ is minimal. Only if $E$ is such that $|E-E_{n,\alpha,\sigma}| <\delta$ can we expect to see a $\delta$ dependence, and indeed this is observed at some energies. Finally, in panel (c) we show the comparison with a larger energy interval. The value of the Green’s function decreases exponentially fast on both sides of the critical region, indicating strongly localized states. Here, data for $\delta = 10^{-6}$ eV is a smooth curve, whose magnitude is much less than that of the other three values even for localized states. This is due to the fact that this $\delta$ is larger than typical level spacings. As a result, several levels contribute significantly to Green’s function at each $E$ value, and the destructive interference of the random phases of different eigenfunctions lead to the supplementary $\delta$-dependence. We conclude that the disorder potential has a critical energy regime of approximately $0.3$ meV width, covering less than 5% (in energy) and 20% (in number of states) of the disorder-broadened band with total width $\sim 6$ meV. The position of the critical energy interval is in agreement with the semi-classical results which suggest an extended state in the vicinity of $E=0.06$meV. By comparison with Fig. \[fig3\], we can also see that this critical regime corresponds to a roughly half-filled band, in agreement with the experiment. The effect of an additional triangular periodic potential is shown in Fig. \[fig6\], where we plot the same quantity shown in Fig. \[fig5\] for a fixed $\delta = 10^{-7}$ eV and different amplitudes $A=0$, 0.05, 0.5 and 5 meV, respectively. These results correspond to a different Coulomb disorder potential (not shown), as can be seen from the different location of its extended states. Here we see how the narrow critical interval of extended states grows gradually as the amplitude of periodic potential is increased and finally exhibits the three well-separated extended subbands expected for $\phi/\phi_0 = 3/2$ in the limit of vanishing disorder. The three subbands can already be resolved for the moderate case $A = 0.5~$meV, although they are very wide and exhibit significant overlap. ![ Semi-log plot of the amplitude of Green’s function matrix element between the two edge states near $x=\pm L_x/2$, as a function of energy. Only the disorder potential of Fig. \[fig2\] is present. (a) comparison between $\delta=10^{-7}$ and $\delta=10^{-8}$ results; (b) comparison between $\delta=10^{-8}$ and $\delta=10^{-9}$ results; (c) comparison between results corresponding to $\delta =\delta=10^{-6},10^{-7},10^{-8}$ and $10^{-9}$. (the last three curves are indistinguishable to the eye on this scale.) All $\delta$ values are in eV units. []{data-label="fig5"}](Fig6_s.eps){width="\FigWidth"} ![ The effect of a triangular periodic potential on the critical energy regime. The disorder potential used here (not shown) supports a narrow interval of extended states centered at about $-0.6$ meV. As the amplitude $A$ of the periodic potential increases, the range of extended states increases dramatically. The left panel shows results for disorder-only and two relatively weak periodic potentials, while the right panel shows two larger periodic potentials where the three-subband structure expected for $\phi/\phi_0 = 3/2$ is clearly seen.[]{data-label="fig6"}](Fig7_s.eps){width="\FigWidth"} Qualitatively similar behavior is obtained if we use the Gaussian scatterers model for disorder. A typical realization of this disorder is shown in Fig. \[fig7bis\]. Results for the Green’s function’s values with such disorder are shown in Fig. \[fig7\], for cases with pure disorder, and also cases with either a triangular or a square periodic potential. The magnetic field has been doubled, such that $\phi/\phi_0=3$. Similar to the case shown in Fig. \[fig6\], the periodic potential leads to a widening of the critical regime. For large periodic potentials, the expected Hofstadter-like three-subband structure emerges again. We conclude that Coulomb and Gaussian disorder models show qualitatively similar behavior. ![A disorder potential of Gaussian type on a roughly $3\mu m \times 3\mu m$ square. The three lines are equipotential contours close to the critical regime, with energies of -0.1 meV (dashed), 0 meV (thick solid) and 0.1 meV (thin solid). Cyclic boundary condition are applied in the $y$ direction.[]{data-label="fig7bis"}](Fig8_s.eps){width="\FigWidth"} ![ Green’s functions for a sample with Gaussian disorder and various periodic potentials. The calculation included 20216 states with $\phi/\phi_0=3$. Similar to results shown in Fig. \[fig7\], we see that the periodic potentials widen the critical region.[]{data-label="fig7"}](Fig9.eps){width="\FigWidth"} We now analyze the projected local density of states $\rho_P(E)$ discussed in Sec. \[subs4\], in order to understand the reason for this substantial widening of the critical region by even small periodic potentials. We consider a smaller sample, of size approximately $1.6\mu$m$\times1.6\mu$m, and compute the projected density of states for 500 equally-spaced energy values, on a $60\times60$ square grid and for a value $\delta = 10^{-8}$ eV. This $\delta$ value is comparable or smaller than the level spacing, so we expect to see sharp resonances from the contribution of individual eigenfunctions as we scan the energy spectrum. Each computation generates a large amount of data (roughly 24M), corresponding to the 500 plots of the local density of states at the 500 values of $E$. Since we cannot show all this data, we select a couple of representative cases and some statistical data to interpret the overall results. Figures \[fig9\] and \[fig10\] show some of our typical results. The two figures are calculated for the same Coulomb-disorder potential, for values of $E=-0.504$ meV (at the bottom of the band) and $E=-0.124$ meV (close to, but below the band center) respectively. Each figure contains four panels, panel (a) shows the profile of the disorder potential as well as an equipotential line (solid black) corresponding to the value $E$ considered; the other three panels show the projected density of states $\rho_P(E)$ for (b) pure disorder; (c) disorder plus triangular periodic potential with $A=0.1$ meV; (d) disorder plus square periodic potential with $A=0.1$ meV. In Fig. \[fig9\], this equipotential line (which traces the semi-classical trajectory of electrons with the same energy $E$) ![ (color online) Projected local density of states $\rho_P({{\bf r}};E)$ for $E=-0.504$ meV. Panel (a) shows the profile of the disorder potential, and the equipotential contour (black line) corresponding to $E=-0.504$ eV. The other three panels show $\rho_P({{\bf r}};E)$ for (b) disorder only; (c) disorder plus triangular periodic potential with $A = 0.1$ meV; (d) disorder plus square periodic potential with $A = 0.1$ meV. The width and length of the sample are both $1.6\,\mu$m, and $\phi/\phi_0=3/2$. Increased brightness corresponds to larger values.[]{data-label="fig9"}](Fig10_s.eps){width="\FigWidth"} surrounds local minima of the disorder potential, suggesting localized electron states at such low energies. Indeed, this is what panels (b), (c) and (d) show. The projected density of states $\rho_P(E)$ is large (bright color) at the positions where electrons of energy $E$ are found with large probabilities. For pure disorder, we observe only closed trajectories (localized states), whose shape is in excellent agreement with the semi-classical trajectory, as expected. If a moderate periodic potential is added, the wave-functions spread over a larger area, and nearby contours sometimes merge together. Instead of sharp lines, as seen in panel (b), the contours now show clear evidence of interference effects of the wave-functions on the periodic potential decorating the electron reservoirs. Some periodic modulations can also be observed in the background of panels (c) and (d), especially for the square potential. These are [*not* ]{} the direct oscillations of the periodic potentials, since the grid we use to compute these figures has a linear size equal to $7/10$ of the period $a=39$ nm of the periodic potential. Capturing detailed behavior inside each unit cell would require a much smaller grid, which is not only time consuming, but also violates the requirement that the grid size be of order $l$ or larger. Figure \[fig10\] for an energy close to the band center, shows the same characteristics. For pure disorder, the electrons at this energy trace a sharp contour very similar ![ (color online) The same as in Fig. \[fig9\], but for an energy $E=-0.124$ meV close to the band center. []{data-label="fig10"}](Fig11_s.eps){width="\FigWidth"} to the corresponding equipotential line shown in panel (a). Electrons are still not delocalized, since this contour does not connect either pair of opposite edges. However, addition of the periodic potential now leads to extended states for both types of periodic potentials \[(c) and (d)\] at roughly $E=-0.124$ meV, demonstrating the widening of the critical region with the addition of a periodic potential. Physically, one can understand this spread of the wave function in the presence of the periodic potential using the semi-classical picture.[@Sorin] If only a smooth disorder potential is present, the equipotential at any energy $E$ must be a smooth, continuous line. However, if a periodic potential with minima $-V_m$ and maxima $V_M$ is superimposed over disorder, the new equipotential line now breaks into a series of small “bubbles” surrounding the disorder-only contour. This happens throughout the area defined by the equipotentials $E-V_M$ and $E+V_m$ of the disorder potential, since the addition of the periodic potential leads new regions in this area to have a total energy $E$. Quantum mechanically, we expect some tunneling inside this wider area and this is indeed what we observe in Figs. \[fig9\] and \[fig10\]. This mechanism suggests [*enhanced delocalization* ]{} on both sides of the critical region as localized wave functions spread out over larger areas, as well as a widening of the critical region itself, in agreement with our numerical results. This spreading of the wave functions in the presence of the periodic potential can also be characterized by counting, at a given energy $E$, the number of grid points ${{\bf r}}$ which have a value $\rho_P({{\bf r}}; E) > \rho_c$, where $\rho_c$ is some threshold value. For sufficiently large $\rho_c$, this procedure counts grid points where electrons with energy $E$ are found with large probabilities, thus, in effect it characterizes the “spatial extent” of the wave functions. The results of such counting are shown in Fig. \[fig11\] for 500 energy values corresponding to the disorder potential analyzed in Figs. \[fig9\] and \[fig10\]. There are a total of $60\times60=3600$ grid points on the sample. For the case of pure disorder (black line) we see that the largest values are found at energies just below 0, where the extended states (the critical region) are found for this particular realization of disorder. Because it is a smooth, sharp line, even the most extended trajectory has significant probabilities at only about 10% of the grid sites. For both higher and lower energies, this number decreases very fast, indicating wave functions localized more and more about maxima or minima of the disorder potential, as expected. Addition of a small periodic potential increases this number substantially, clearly showing the supplementary spreading of the wave functions in the presence of the periodic potential. Figure \[fig11\] shows this effect for three types of periodic potential: triangular lattices with $A > 0$ and $A<0$ (upper panel), and square lattice in the lower panel. All three cases show significant enhancement, as compared to the pure disorder case. In addition, we see that while the square potential gives a fairly symmetrical enhancement, the triangular potential does not, with curves for $\pm A$ not overlapping. This is a consequence of the asymmetric shape of the periodic potential, which has different values for its minima and maxima $|V_m|\ne |V_M|$, as well as different arrangements for the points where minima/maxima appear (triangular lattice vs. honeycomb lattice). Fig. \[fig11\] clearly shows that $A >0$ favors increased delocalization below the critical energy regime, while $A <0$ favors increased delocalization above it. The reason for this different response to the two signs of the triangular potential can be nicely explained within the semi-classical framework. In Fig. \[fig12\] we show the equipotential lines corresponding to filling factors $\nu=0.3$ (well below critical region) and $\nu=0.7$ (well above the critical region) for a realization of Coulomb disorder (not shown) plus a triangular potential with $A>0$. Areas with energy below the equipotential value are shaded. In this case we can clearly see that instead of the continuous, smooth trajectory expected for disorder-only cases, there are also extra “bubbles” regions connecting the areas between such contours. Since the choice $A>0$ leads to deep minima at $-V_m=-6A$ with triangular arrangement and relatively flat maxima at $+V_M=3A$ with honeycomb arrangement \[see Eq. (\[2.7\])\], it follows that the triangular (honeycomb) “bubbles” region appear roughly in the area bounded by the equipotentials $E$ and $E+V_m$ (respectively, $E-V_M$ and $E$) of the pure disorder potential. At low filling factors, the pure disorder $E$ equipotential is a collection of closed contours surrounding local minima \[see panel (d) of Fig. \[fig9\] for an illustration\]. It follows that for the choice $V_m > V_M$, the more extended region with triangular “bubbles” will be found outside these “islands” and will lead to a spread of the wave function over considerably larger areas, as indeed seen in the upper panel of Fig. \[fig12\]. On the other hand, at large filling factors the disorder-only contours are “islands” surrounding the maxima of the disorder potential. In this case, contours between $E$ and $E+V_m$ are inside the $E$ contour, so the triangular “bubbles” region does not help to connect various “islands” as before. The honeycomb “bubbles” regions does this, but because $V_M < V_m$ the extension of the wave function between “islands” is significantly smaller in this case. In the quantum-mechanical case one expects interference (due to tunneling) effects between the small “bubbles” regions, and therefore a wave function which is extended over their entire area, as indeed we observe to be the case in Figs. \[fig9\] and \[fig10\]. In other words, one expects that a triangular potential with $A>0$ will lead to considerable increase of the localization length, and respectively widening of the critical energy region, at filling factors below one-half, whereas $A<0$ will favor delocalization at filling factors above one-half, as seen in Fig. \[fig11\]. This asymmetry is therefore clearly a consequence of the asymmetry of the triangular potential, and is absent for a square potential with only lowest order Fourier coefficients, which possesses electron-hole symmetry. This should have clear implications for the transport properties of the system. Summary and Conclusions {#sec6} ======================= In this study we have investigated the effects of moderate-to-large smooth disorder on the Hofstadter butterfly expected for 2DES in a perpendicular magnetic field and a pure periodic modulation. The parameters of our study are chosen so as to be suitable for the interpretation of recent experiments on a two dimensional electron system in GaAs/AlGaAs heterostructure with periodic modulation provided by a diblock copolymer, [@Sorin] The experiment shows that (i) the longitudinal resistance $R_{xx}$ is still peaked approximately at half filling; (ii) there are many reproducible oscillations in $R_{xx}$, indicating non-trivial electronic structures in the patterned sample; (iii) the distribution of these oscillatory features is asymmetric, with most of them appearing on the high magnetic fields (i.e. low filling factors $\nu < 0.5$) side of the peak of $R_{xx}$; and (iv) the temperature dependence of $R_{xx}$ indicates that the asymmetric off-peak resistance is thermally excited, whereas the central $R_{xx}$ peak (close to half filling) has metallic behavior. These observations cannot be explained on the basis of the Hofstadter structure.[@Sorin] This is not surprising, since one expects that large disorder will modify this structure considerably. Effects of small disorder on the Hofstadter butterfly had been investigated previously using SCBA,[@MacDonald] but this basically perturbational approach is not appropriate for the case of moderate-to-large smooth disorder. Instead, we identify and use a number of techniques which give the exact solution (if electron-electron interactions, as well as inelastic scattering are neglected) while avoiding brute force numerical diagonalizations. Our results demonstrate that while the Hofstadter butterfly is destroyed by large disorder, the effects of the periodic potential are non-trivial for states near the critical regime. Firstly, they lead to a significant increase in localization lengths of the localized states at mesoscopic ($\mu m$) length scale and induce an effective widening of the critical regime near the critical regime. This is achieved through a spreading of the electron wave-function on the flat regions of the slowly varying disorder potential, where their behavior is dominated by the periodic modulation. This regime shows an interesting transition between the pure disorder and the pure periodic potential cases. In the case of pure disorder, the semi-classical approach tells us that at finite filling factors, some areas of the sample are fully occupied by electrons with the maximum possible density of $1/(2\pi l^2)$ (these are the areas where the disorder potential has minima) whereas other areas are fully devoid of electrons (areas where the potential has maxima) and the boundary between such regions is very sharp. On the other hand, for a pure periodic modulation all wave functions have translational invariance with the proper symmetry, and therefore electron densities are uniform over the entire sample (up to small periodic modulations inside each unit cell). When both types of potential are present, with disorder being dominant, our results show [*three*]{} types of areas. There are regions which are fully occupied and regions which are completely devoid of electrons, as in the case of pure disorder. However, the periodic potential leads to a widening of the boundary between the two, where the wave functions interact with several oscillations of the periodic modulation and therefore have some partial local filling. As the critical regime containing wave functions percolating throughout the sample is approached, this spreading of the wave function becomes dominant in establishing the transport properties of the system. An equivalent way to say this is that the main effect of the periodic potential is to provide bridging between the fully-occupied electron “puddles” created by the disorder potential. Since the connecting areas are relatively flat, the local wave functions respond to the local periodic potential, and therefore locally have a Hofstadter-butterfly like structure. If the partial filling factor in such a region is inside the gap of the local Hofstadter butterfly structure, one expects no transport through this local area. This should result in a dip in the longitudinal transport, since in such cases the periodic potential will not transport electrons from one “puddle” to another one. By contrast, if the local filling factor in such a region is inside a subband of a local Hofstadter structure, this area will establish a link between different “puddles” and thus help enhance the transport through the sample. Transport in this regime should show strong thermal activated behavior, in contrast to metallic transport in the critical regime where the wave functions connect opposite edges of the sample. As a result, one expects a series of local minima and maxima in the longitudinal resistivity on either side of the central peak induced by the extended states (critical regime). Furthermore, for an asymmetric triangular potential, this response should be strongly asymmetric, with the effect most visible on one side of the central peak. (One must keep in mind that since tunneling leads to exponential dependencies, even small differences in the extent of the wave functions can have rather large effects on $\rho_{xx}$). Such an asymmetry should also be present in longitudinal conductance of finite but low temperature, e. g. in the hopping regime which is sensitively dependent on the nature of the localized wavefunctions, as is indeed seen experimentally.[@Sorin] To summarize, our qualitative explanation for the various experimental features are as follows: \(i) The $R_{xx}$ peak is roughly at the center of the band because the weak periodic potential cannot establish a Hofstadter-like structure over the whole band. Instead, low and high $\nu$ states are strongly localized and do not transport longitudinal currents. \(ii) New extended states induced by the periodic potential are responsible for the reproducible peaks and valleys appearing in $R_{xx}$. \(iii) The periodic potential also leads to the expansion of localized wave functions, which contribute to the thermally activated conduction at lower filling factors. The detailed structure of the wave functions gives rise to the oscillations of the off-peak $R_{xx}$, similar to conductance fluctuations.[@Shayegan] Finally, \(iv) the asymmetry in $R_{xx}$ is a manifestation of the asymmetry of the triangular potential, which has a stronger effect at low filling factors than at high filling factors for $A>0$. We predict that this asymmetry should be absent for a symmetric square periodic potential. The weak point in our calculation is that we are unable to accurately model the potential in the real samples, because various screening effects have not been properly taken into account. Also, we have no quantitative information about the magnitude of the periodic potential in the 2DES layer, because of the additional strain[@Larkin] contribution induced by the periodic decoration. As a result, we only claim qualitative agreement with the experiment, although our investigations show the same type of behavior for various types of disorder potentials and various (small-to-moderate) strengths of the periodic potential. The most direct check of this work would be an experimental demonstration that thermally activated conduction appears symmetrically on both sides of the $R_{xx}$ peak for a periodic potential with square symmetry and primarily lowest Fourier coefficients. Limited computer resources restrict our calculations to samples no larger than $3 \mu$m$ \times 3 \mu $m, while the sample used in experiment has a size of $20 \mu$m$ \times 20\mu$m. From a theoretical point of view, it is interesting to ask what is the thermodynamic limit. For pure disorder, it is believed that in this limit the typical size of wavefunction diverges at a single critical energy. Since we cannot pursue size-dependent analysis for samples larger than $3 \mu$m$ \times 3 \mu $m, we do not know whether the small periodic potential will lead to a finite size critical regime in the thermodynamic limit, although this seems likely. From an experimental point of view, the interesting question is whether the Hofstadter structure can be observed at all. Our studies suggest that this may be possible for small mesoscopic samples, where the slowly-varying disorder has less effect. Alternatively, one must find a way to boost the strength of the periodic modulations inside the 2DES. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Sorin Melinte, Mansour Shayegan, Paul M. Chaikin and Mingshaw W. Wu for valuable discussions. We also thank Prof. Li Kai’s group in Computer Science Department of Princeton University for sharing their computer cluster with us. This research was supported by NSF grant DMR-213706 (C.Z. and R.N.B.) and NSERC (M.B.). M.B. and R.N.B. also acknowledge the hospitality of the Aspen Center for Physics, where parts of this work were carried out. [99]{} K. von Klitzing, G. Dora and M. Pepper, Phys. Rev. Lett. [**45**]{}, 494 (1980). D. C. Tsui, H. L. St$\ddot o$rmer, and A. C. Gossard, Phys. Rev. Lett. [**48**]{}, 1559 (1982). For a review, see [*“The Quantum Hall Effect”*]{}, edited by R.E. Prange and S.M. Girvin, Graduate Texts in Contemporary Physics (Springer-Verlag, New York, 1987). D. R. Hofstadter, Phys. Rev. B [**14**]{}, 2239 (1976). Dieter Langbein, Phys. Rev. [**180**]{}, 633 (1969); the electronic structure in the asymptotic cases is periodic in $\phi/\phi_0$ or $\phi_0/\phi$, and the equality is meant modulo this periodicity. D. Springsguth, R. Ketzmerick, and T. Geisel, Phys. Rev. B [**56**]{}, 2036 (1997). T. Schlösser, K. Ensslin, J. P. Kottahaus and M. Holland, Europhys. Lett. [**33**]{}, 683 (1996). D. Weiss, M. L. Roukes, A. Menschig, P. Grambow, K. von Klitzing and G. Weimann, Phys. Rev. Lett. [**66**]{}, 2790 (1991). C. Albrecht, J. H. Smet, K. von Klitzing, D. Weiss, V. Umansky and H. Schweizer, Phys. Rev. Lett. [**86**]{}, 147 (2001). Rolf R. Gerhardts, Dieter Weiss and Ulrich Wulf, Phys. Rev. B [**43**]{}, 5192 (1991). S. Melinte, M. Berciu, C. Zhou, E. Tutuc, S. J. Papadakis, C. Harisson, E. P. De Poortere, M. Wu, P. M. Chaikin, M. Shayegan, R. N. Bhatt and R. A. Register, to appear in Phys. Rev. Lett. (cond-mat/0311400). S. Melinte, E. Grivei, V. Bayot, and M. Shayegan, Phys. Rev. Lett. [**82**]{}, 2764 (1999), and references therein. P. Chaikin (private communication). U. Wulf and A. H. MacDonald, Phys. Rev. B [**47**]{}, 6566 (1993). B. Huckstein and R. N. Bhatt, Surface Science [**305**]{}, 438 (1994). F.H. Claro and G.H. Wannier, Phys. Rev. B [**19**]{}, 6068 (1979). D. Springsguth, R. Ketzmerick, and T. Geisel, Phys. Rev. B [**56**]{}, 2036, (1997). D. Pfannkuche and R. R. Gerhardts, Phys. Rev. B [**46**]{}, 12606 (1992). S. A. Trugman, Phys. Rev. B [**27**]{}, 7539 (1983). John A. Nixon and John H. Davies, Phys. Rev. B [**41**]{}, 7929 (1990). M. Stopa, Phys. Rev. B [**53**]{}, 9595 (1996); Physica B [**227**]{}, 61 (1996). S. Das Sarma and S. Kodiyalam, Semiconductor Science and Technology [**13**]{}, A59 (1998). T. Ando, J. Phys. Soc. Japan [**53**]{}, 3101 (1984). Ralph Williams, [*“Modern GaAs Processing Methods”*]{}, (Artech House Publishers, Boston$\cdot$London, 1990). B. I. Halperin, Phys. Rev. B [**25**]{}, 2185 (1982). Allowed trajectories are such that the total flux through the area enclosed by the trajectory is an integer number of elementary fluxes $\phi_0$. This can be understood in the spirit of the Bohr-Sommerfeld quantization rule, since it ensures constructive interference of the wave function around the contour. Except for the most localized states, found at the bottom and the top of the band, all other localized states are such that they enclose large numbers of magnetic fluxes. Imposing the exact quantization condition (which is numerically time consuming) leads to a negligible change in the value of the allowed equipotential value with respect to a randomly chosen value. Since such small changes do not influence the shape of the density of states, we ignore imposing this quantization condition in obtaining the semi-classical densities of states. we use the package FFTW2.1.3 available on-line at http://www.fftw.org. Chenggang Zhou and R. N. Bhatt, Phys. Rev. B [**68**]{}, 045101 (2003). Xiaoye S. Li, M. Baertschy, T. N. Rescigno, W. A. Issacs and C. W. McCurdy, Phys. Rev. A [**63**]{}, 022712 (2001). Details regarding the software also available at http://www.nersc.gov/ xiaoye/SuperLU. Roger Haydock, Phys. Rev. B [**61**]{}, 7953 (2000). J. A. Simmons, H. P. Wei, L. W. Engel, D. C. Tsui and M. Shayegan, Phys. Rev. Lett. [**63**]{}, 1731 (1989). J. H. Davies and I. A. Larkin, Phys. Rev. B [**49**]{}, 4800 (1994).

--- abstract: 'We investigated the electrostatic interaction between two identical dust grains of an infinite mass immersed in homogeneous plasma by employing first-principles N-body simulations combined with the Ewald method. We specifically tested the possibility of an attractive force due to overlapping Debye spheres (ODSs), as was suggested by Resendes et al. (1998). Our simulation results demonstrate that the electrostatic interaction is repulsive and even stronger than the standard Yukawa potential. We showed that the measured electric field acting on the grain is highly consistent with a model electrostatic potential around a single isolated grain that takes into account a correction due to the orbital motion limited theory. Our result is qualitatively consistent with the counterargument suggested by Markes and Williams (2000), indicating the absence of the ODS attractive force.' author: - 'H. Itou' - 'T. Amano' - 'M. Hoshino' title: 'First-principles simulations of electrostatic interactions between dust grains' --- Introduction ============ Dust grains are quite common in astrophysical environments. They are thought to exist in, for example, interstellar molecular clouds, protoplanetary disks, planetary rings, the Earth’s magnetosphere, and tails of comets. In addition, in laboratories, the lattice formation of dust grains, known as Coulomb crystallization, is a well-known phenomenon that has fascinated many researchers. Dust grains immersed in plasmas usually acquire a large amount of charge through several charging processes, such as collisions with plasma particles and photoemission. Such charged grains and the ambient plasma are electromagnetically coupled with each other, forming so-called dusty plasmas or complex plasmas. Dusty plasma has been studied for both industrial and astrophysical applications, largely motivated by the in-situ detection of dust grains in the Solar System and Ikezi’s prediction, and subsequent experimental verification of Coulomb crystallization.[@Goertz89; @Angelis92; @Shukla09; @Shukla01; @text; @Ikezi86] When collisions between dust grains and plasma particles are dominant among the charging processes, dust grains become negatively charged because the thermal velocity of electrons is generally higher than that of ions, resulting in a larger electron current. Therefore, one would expect a repulsive shielded electrostatic Coulomb potential (or Yukawa potential) to exist. In reality, however, forces acting on dust grains may be much more complex because the interaction forces between charged dust grains are mediated by the ambient plasma in a complicated manner. There has been much discussion on forces acting between dust grains, including attractive forces for which the ambient plasma response plays the essential role.[@Shukla09; @Lampe00] It is necessary to understand the nature of such attractive interactions among dust grains because they may play a role in the aggregation or crystallization of dust grains observed in laboratories, as well as the formation of stars and planets in the dense cores of interstellar molecular clouds. One such attractive force acting between two grains, and on which we focus in the present study, is the force due to overlapping Debye spheres (ODSs).[@Resendes98] According to Resendes et al. (1998), when two charged dust grains (each having charge $q$) exist in a plasma, their interaction potential, including the electrostatic energy of ambient plasma particles, may be modified from the simple Yukawa potential. The potential in this case may be written as $$\label{l-j} q{\phi_{{\rm ODS}}}\left(d\right)=\frac{q^{2}}{\lambda_{{\rm D}}}\left(\frac{\lambda_{{\rm D}}}{d}-\frac{1}{2} \right)\exp\left(-\frac{d}{\lambda_{{\rm D}}} \right)+{\rm constant},$$ where $\lambda_{{\rm D}}$ is the Debye length and $d$ is the intergrain distance. This is similar to the Lennard-Jones potential, which is repulsive at short distances and weakly attractive at longer distances. It is clear that a Lennard-Jones-like potential can assist the processes of aggregation and crystallization, and in fact, it has been shown that the attractive force due to ODSs has a drastic effect on aggregation and crystallization in dusty plasmas if indeed effective.[@Hou09] It has also been suggested that the ODS attractive force may enhance gravitational instability and assist the formation of stars and planets in astrophysical environments.[@Shukla06] On the other hand, the derivation of this attractive potential is based on several nontrivial assumptions that need to be verified. For instance, one must assume that the electrostatic potential around a dust grain is given by the Yukawa potential: $$\label{yukawa} q{\phi}\left(r\right)=\frac{q^{2}}{r}\exp\left(-\frac{r}{\lambda_{{\rm D}}}\right).$$ In addition, linear superposition of the potential around two dust grains (with ODSs) should be valid in order for such an attractive force to exist. Since the concept of Debye shielding is the key to understanding the attractive force, one must be careful in adopting these assumptions. Furthermore, the derivation of the ODS attractive ${\it force}$ from Eq. (\[l-j\]) assumes that the force operating between the grains is given by the derivative of Eq. (\[l-j\]) with respect to the intergrain distance $d$. We note that Markes and Williams (2000) pointed out that this assumption is incorrect in that it does not take into account energy exchange with the ambient plasma.[@Markes00] Lampe et al. (2000) also suggested that, on the basis of orbital motion limited (OML) theory, such an attractive force would not exist.[@Lampe00] Nevertheless, those counterarguments are also based on some non-trivial assumptions. Consequently, the existence or nonexistence of the ODS attractive force has yet remained a controversial issue. The purpose of our study is thus to investigate the validity of the theory of the ODS attractive force from first principles. We employ the direct N-body simulation method in which all particle-particle interactions acting through the electrostatic Coulomb force are calculated. This first-principles approach allows us to investigate the electrostatic potential structure of sub-Debye scales without making any assumptions, and thus provides a direct answer to the problem. It is demonstrated herein that the electric field acting on a charged grain actually deviates from the standard Yukawa-type field in general. We find that the electrostatic force acting between two dust grains is repulsive rather than attractive, which may be well explained by OML theory for an isolated test charge. There is no noticeable signature of the net attractive force due to the effect of ODSs around dust grains. Our result is qualitatively consistent with the analysis given by Markes and Williams (2000). Although the simulations were performed within a limited range of plasma parameters, this strongly indicates the ODS attractive force is absent in reality. Simulation method ================= Our N-body simulations are performed in a periodic system (surrounded by a virtual perfectly conducting medium at the infinite distance). The system consists of the simulation box and its replicas, and the box contains many plasma particles (ions and electrons) and two charged dust grains. For the time integration, the Coulomb force acting on each particle must be evaluated by taking the summation over all particles. Since the Coulomb interaction is a long-range interaction, convergence of the summation is very slow and the calculation of contributions from many particles at long distances significantly increases the number of operations required. We thus adopt the Ewald method, which allows us to accelerate the summation by dividing it into two parts: one in real space and the other in wavenumber space. For instance, the electrostatic potential may be calculated as follows: $$\label{ewald} U=U_{{\rm real}}+U_{{\rm wave}}-U_{{\rm self}},$$ $$\label{real} U_{{\rm real}}=\frac{1}{2}\sum_{i,j}\sum_{n}{\frac{q_{i}q_{j}}{r_{ijn}} {\rm erfc}\left(\frac{r_{ijn}}{\sigma} \right)},$$ $$\label{wave} U_{{\rm wave}}=\frac{1}{2}\sum_{i,j}\sum_{{\bm k}\neq 0}{q_{i}q_{j}\frac{\exp\left[{-\pi^{2}\sigma^{2}k^{2}+2\pi i{\bm k}\cdot\left({\bm r}_{i}-{\bm r}_{j}\right)}\right]}{\pi V k^{2}}},$$ $$\label{self} U_{{\rm self}}=\frac{1}{\sqrt{\pi}\sigma}\sum_{i}{q_{i}^{2}}.$$ Here, $n$ represents the labels of boxes, $r_{ijn}$ is the distance between particles $i$ and $j$ in box $n$, $q_{i}$ is the charge of particle $i$, ${\bm k}$ is the wavenumber vector, and $V$ is the volume of the box. The parameter $\sigma$ gives a cut-off radius beyond which the direct summation in real space, Eq. (\[real\]), is replaced by that in wavenumber space, Eq. (\[wave\]). Note that in Eq. (\[real\]), the term $n=0$ has to be excluded for $i=j$. This method approximates long-wavelength modes associated with the long-range nature of the Coulomb interaction in wavenumber space with the aid of the Fourier transform, whereas short-wavelength components arising from close encounters between particles are accurately calculated. The electric field is given by the spatial derivatives of Eqs. (\[real\]) and (\[wave\]) and is calculated in the same way.[@Deserno98-1; @Pollock96] In calculating Eq. (\[real\]), we introduce a small softening parameter $\epsilon$ and rewrite Eq. (\[real\]) as $$\label{ereal} U_{{\rm real}}=\frac{1}{2}\sum_{i,j}\sum_{n}{\frac{q_{i}q_{j}}{\sqrt{r_{ijn}^{2}+\epsilon^{2}}} {\rm erfc}\left(\frac{\sqrt{r_{ijn}^{2}+\epsilon^{2}}}{\sigma} \right)}.$$ With the softening technique, we ignore large-angle scatterings between particles at distances ${\protect\raisebox{-0.5ex}{$\:\stackrel{\textstyle <}{\sim}\:$}}\epsilon$ because resolving such scatterings would require very small time steps. Since we are interested in weakly coupled space and astrophysical plasmas that are defined by a large plasma parameter $\Lambda$ (where small-angle scatterings play the dominant role), we think this technique is reasonable for our purpose. Having calculated the electric fields acting on particles, we can solve the equations of motion for each particle: $$\label{newton1} m_{i}\frac{d}{dt}{\bm v}_{i}=q_{i}{\bm E},$$ $$\label{newton2} \frac{d}{dt}{\bm r}_{i}={\bm v}_{i},$$ where $m_{i}$, ${\bm v}_{i}$, and ${\bm r}_{i}$ are the mass, velocity, and position of particle $i$, respectively, and ${\bm E}$ is the electric field at each particle position ${\bm r}_{i}$. In Eq. (\[newton1\]), assuming nonrelativistic plasma temperatures, $v_{i}/c\ll1$, we ignore the effect of magnetic fields. Throughout the present paper, the masses of ions and electrons are assumed to be equal to allow the system to relax quickly to an equilibrium state. This assumption may be justified because the mass ratio affects only the time scale, and structures of the equilibrium state can be assumed to be independent of the mass ratio. Therefore, we only discuss the properties of equilibrium states. Note that because of the symmetry of ion and electron masses, the sign of the grain charge is irrelevant. Simulations are performed with two identical dust grains of infinite mass in the box. That is to say, the grain mass is so large that the change in positions can be ignored on the simulation time scale, which is typically limited to a few plasma oscillation periods. The effect of finite grain size is also ignored. These assumptions are made to simplify the problem as much as possible for our purpose of investigating the electrostatic interactions between plasma particles and dust grains. simulation result ================= Simulations were initialized with plasma particles distributed randomly in space, and two dust grains placed at fixed distances in the box. The velocity distribution was initialized to a Maxwellian distribution for a given temperature. Time integration was carried out until the system reached an equilibrium state, at which point we measured the properties of the system. The simulation box was a cuboid whose dimensions were $2L$ in the $x$ direction and $L$ in the $y$ and $z$ directions. Throughout this paper, we use a softening parameter of $\epsilon=0.03L$ in simulations. Each grain was located at $\left(y,z\right)=\left(L/2,L/2\right)$, and the intergrain distance along the $x$ axis was varied in each simulation run. By comparing the equilibrium states of different runs, we measured the dependence on the intergrain distance. ![Temporal evolution of total electrostatic potential energy for a run with $d=0.1L$, $q=1000e$, $2L^{3}n_{{\rm e}}=10000$, $2L^{3}n_{{\rm p}}=8000$, $\lambda_{{\rm D}0}\simeq0.11L$, $\lambda_{{\rm D}}\simeq0.12L$, and $\Lambda\simeq16$. The dotted line indicates the equilibrium value.[]{data-label="ene"}](enenene.eps){width="90mm"} The system is characterized by the dust charge $q$ and the intergrain distance $d$. The number densities of electrons and ions are denoted $n_{{\rm e}}$ and $n_{{\rm p}}$, which are chosen so that charge neutrality (including dust charges) is satisfied. In the following, time and space are respectively normalized by the inverse plasma frequency $1/\omega_{p}$, where $\omega_{p}=\left(4\pi n_{{\rm e}} e^{2}/m_{{\rm e}}+4\pi n_{{\rm p}} e^{2}/m_{{\rm p}}\right)^{1/2}$, and the Debye length $\lambda_{{\rm D}}$. Note that the Debye length is defined as $\lambda_{{\rm D}}=\left(4\pi n_{{\rm e}} e^{2}/kT_{{\rm e}}+4\pi n_{{\rm p}} e^{2}/kT_{{\rm p}}\right)^{-1/2}$, including both ion and electron contributions, and the temperatures of the resultant equilibrium states are used. Here, $e$ is the elementary charge, and $m_{{\rm e}}$, $m_{{\rm p}}$, $T_{{\rm e}}$, and $T_{{\rm p}}$ are the electron mass, proton mass, electron temperature, and proton temperature, respectively. Note that we always assumed that the initial electron and proton temperatures were the same for simplicity. In Fig.\[ene\], the time variation of the potential energy integrated over the simulation box is shown for the example of a run with an intergrain distance of $d=0.1L$. The energy is normalized by $q^2/\lambda_{{\rm D}0}$, where $\lambda_{{\rm D}0}$ is the Debye length defined by the initial temperature. In this run, $q=1000e$, $2L^{3}n_{{\rm e}}=10000$, $2L^{3}n_{{\rm p}}=8000$, $\lambda_{{\rm D}0}\simeq0.11L$, $\lambda_{{\rm D}}\simeq0.12L$, and $\Lambda\equiv\left(n_{{\rm e}}+n_{{\rm p}}\right)\lambda_{{\rm D}}^{3}\simeq16$. Generally speaking, the Debye length in the final equilibrium state, denoted $\lambda_{{\rm D}}$, actually differs from $\lambda_{{\rm D}0}$, as explained below. We see from Fig.\[ene\] that the potential energy decreases during the first $\sim1/\omega_{p}$, and then fluctuates around the equilibrium value. This initial decrease in the potential may be explained by the redistribution of plasma particles due to Debye shielding. This decrease in the potential energy is compensated by an increase in the plasma temperature, changing the Debye length from the initial value accordingly. All runs discussed in this paper showed essentially the same trend. We thus assume that the equilibrium was achieved by the time $\omega_{{\rm p}}t\sim8$, and physical quantities averaged after this time were regarded as equilibrium values. ![Summary of simulation results. The normalized electric field acting on the grain multiplied by $d^{2}$ is shown as a function of the intergrain distance $d$. Note that the distance is normalized by the Debye length defined with the kinetic energy measured at the equilibrium states rather than the initial temperature. The red and green lines are the theoretical curves expected from the standard Yukawa potential and the ODS attractive potential, respectively. Blue triangles and magenta diamonds show the results of our simulations with $\Lambda\simeq13$ and $\Lambda\simeq16$, respectively.[]{data-label="res_ele"}](reseses.eps){width="90mm"} Figure \[res\_ele\] summarizes the results of our simulations. Blue triangles show the results for $2L^{3}n_{{\rm e}}=5000$, $2L^{3}n_{{\rm p}}=3000$, $\lambda_{{\rm D}}/L\simeq0.15L$, and $\Lambda\simeq13$. Individual triangles represent intergrain distances of $d=0.2L, 0.4L, 0.5L, 0.6L, 0.8L$. Simulations with a different set of parameters ($2L^{3}n_{{\rm e}}=10000$, $2L^{3}n_{{\rm p}}=8000$, $\lambda_{{\rm D}}/L\simeq0.12L$, and $\Lambda\simeq16$) were also run, and the results are shown by magenta diamonds; in this case, the intergrain distances were $d=0.1L, 0.25L, 0.4L, 0.5L, 0.6L, 0.75L, 0.9L$. In all runs, $q=1000e$ and $\lambda_{{\rm D}0}\simeq0.11L$. Note that $\lambda_{{\rm D}}$, which normalizes the intergrain distances in Fig.\[res\_ele\], was defined at the equilibrium states, and thus not necessarily the same in each run because the self-consistent increase in temperature depends on plasma densities, plasma parameters and intergrain distances $d$. The red and green lines in Fig.\[res\_ele\] show the theoretical curves expected from the standard Yukawa potential and the ODS attractive potential of Resendes et al. (1998), respectively, which are written as $$\label{yukawa-e} qE_{{\rm Yukawa}}\left(d\right)=\frac{q^{2}}{\lambda_{{\rm D}}^{2}}\left[\left(\frac{\lambda_{{\rm D}}}{d}\right)^{2}+\frac{\lambda_{{\rm D}}}{d}\right]\exp\left(-\frac{d}{\lambda_{{\rm D}}}\right)$$ and $$\label{l-j-e} qE_{{\rm ODS}}\left(d\right)=\frac{q^{2}}{\lambda_{{\rm D}}^{2}}\left[\left(\frac{\lambda_{{\rm D}}}{d}\right)^{2}+\frac{\lambda_{{\rm D}}}{d}-\frac{1}{2}\right]\exp\left(-\frac{d}{\lambda_{{\rm D}}}\right).$$ Eq. (\[l-j-e\]) assumes that the force on the grain is given by the derivative of Eq. (\[l-j\]) with respect to $d$. The error bars represent the standard deviation ($1\sigma$) of temporal fluctuations after the system has reached an equilibrium state. Note that when calculating the electric field acting on the grain, we used a softening parameter of $\epsilon=d/12$, which is different from that used in the simulation to reduce the variance of the measured electric fields. That is to say, the softening parameter $\epsilon$ is chosen to be proportional to the intergrain distance, whereas it is constant in all simulations. This choice is mainly motivated by the conjecture that the equilibrium electrostatic structure will not strongly depend on the softening parameter. However, some remarks must be made before discussing the results. First, the effect of softening is not seen even at $d \lesssim 0.03 L\left(\simeq0.2-0.25\lambda_{{\rm D}}\right)$ because the softening parameters used in the calculations are smaller than the simulation value at $d < 0.36L\simeq2.5-3\lambda_{{\rm D}}$. In the region where the softening effect is significant, it is obvious that the potential approaches the Coulomb potential because the softening parameter in the simulations is chosen to be smaller than the mean particle distance. Therefore, this will not change our conclusions. Second, the error bars may be underestimated at $d > 0.36L$ because the softening parameter used in the calculation becomes larger than that in the simulations. (Note that large error bars are caused by close encounters with plasma particles.) In any case, the error bars are so large that it is difficult to extract a physically meaningful argument in this regime. Third, we have confirmed that calculation with a constant softening parameter of $\epsilon=0.03L$ (i.e., consistent with the simulations) does not change the result substantially. Although the error bars in the far regions, i.e., $d > 0.36L$, tend to increase, the average electric fields stay within the error bars shown in Fig.\[res\_ele\]. Based on these discussions, we believe that the simulation results are reliable at least in the intermediate regime, i.e., $1\lambda_{{\rm D}}\lesssim d\lesssim2.5\lambda_{{\rm D}}$. In this region, it is evident from Fig.\[res\_ele\] that the simulation results deviate from the theoretical prediction of the ODS attractive potential beyond $2\sigma$. The result also suggests that the electric fields acting on the grain are even larger than the standard Yukawa potential prediction. Although the large error bars make it difficult to draw conclusions from this result alone, the systematic deviation from the theoretical predictions suggests that the underlying assumptions made in the derivation of (\[yukawa-e\]) and (\[l-j-e\]) may be violated. In the next section, we discuss possible reasons for this discrepancy between the theory and simulations. Discussion ========== Our simulation results show that the force between two dust grains is repulsive and stronger than that predicted by the standard Yukawa potential Eq. (\[yukawa\]). At first, we discuss the validity of Eq. (\[yukawa\]). When the grain radius is negligible, the functional form of the Yukawa potential itself must be correct at large distances, where the shielding is nearly complete and the first-order expansion of the Boltzmann-type density distribution is appropriate. In fact, Poisson’s equation and the linearized Boltzmann distributions give $$\label{long} q\phi\left(r\right)=\alpha\frac{q^{2}}{r}\exp{\left(-\frac{r}{\lambda_{{\rm D}}}\right)}.$$ However, the coefficient $\alpha$ (integration constant) in Eq. (\[long\]) is unknown and must be determined by the inner boundary condition. In standard textbooks, it is determined by assuming that the outer solution smoothly connects to the bare Coulomb potential at $r \rightarrow 0$, which gives $\alpha=1$. On the other hand, according to OML theory, $\alpha\neq1$ in general. In OML theory, when particle absorption by dust grains is ignored, the density distribution of ions around a negatively charged dust grain may be written as [@Lampe00] $$\label{oml} n_{{\rm p}}=n_{0}\left[\exp\left(-\frac{e\phi}{kT_{{\rm p}}}\right){\rm erfc}\left(\sqrt{-\frac{e\phi}{kT_{{\rm p}}}} \right)+\frac{2}{\sqrt{\pi}}\sqrt{-\frac{e\phi}{kT_{{\rm p}}}} \right]$$ instead of the Boltzmann distribution $$\label{boltz-p} n_{{\rm p}}=n_{0}\exp\left(-\frac{e\phi}{kT_{{\rm p}}}\right),$$ whereas the electron density distribution is written as $$\label{boltz-e} n_{{\rm e}}=n_{0}\exp\left(\frac{e\phi}{kT_{{\rm e}}}\right)$$ in both cases. It is easy to show that Eqs. (\[oml\]) and (\[boltz-p\]) give the same dependence on $e\phi/kT$ when expanded to first order in $e\phi/kT\ll1$, meaning that the functional form is the same far from the grain. Since the OML correction given by Eq. (\[oml\]) gives an ion density much lower than that suggested by the Boltzmann distribution given by Eq. (\[boltz-p\]) close to the grain, the shielding of the potential becomes weaker. We may thus expect $\alpha\geq1$ in general if the OML correction is taken into account.[@Lampe00] The parameter $\alpha$ may be determined by the solution in the inner region, where the OML correction may become important. On the other hand, the OML solution must also be connected to the bare Coulomb potential $$\label{short} q\phi\sim \frac{q^{2}}{r},$$ at distances on the order of the mean interparticle distance $a$, which is defined as $$\label{wig2} \frac{a}{\lambda_{{\rm D}}}\equiv \sqrt[3]{\frac{3}{4\pi \Lambda}},$$ where $\Lambda$ is the plasma parameter. While it is difficult to analyze the potential structure analytically in the inner region with the OML correction, we expect $\alpha\left(\Lambda\right)$ to be a decreasing function of $\Lambda$ because a larger $\Lambda$ narrows the region in which the OML correction should be taken into account and strengthens the shielding effect. ![Comparison between simulation results and theoretical models including the OML correction for the electric field acting on the grain. Only results with $\Lambda\simeq16$ are shown.[]{data-label="com"}](comcom.eps){width="90mm"} To determine the value of $\alpha$, Figure \[com\] compares the simulation results and a theoretical electric field around ${\it a}$ ${\it single}$ ${\it isolated}$ ${\it grain}$ including the OML correction. That is to say, the potential $\phi$ was determined by solving Poisson’s equation, $$\label{poi} \nabla^{2}\phi=-4\pi e\left(n_{{\rm p}}-n_{{\rm e}}\right),$$ with the ion and electron densities given by Eqs. (\[oml\]) and (\[boltz-e\]), respectively. $n_{0}$ in Eqs. (\[oml\]) and (\[boltz-e\]) was approximated as $n_{0}=(n_{{\rm e}0}+n_{{\rm p}0})/2$ for simplicity. The plasma parameter was $\Lambda\simeq16$, which is almost the same as that in our simulations. The electric field $E$ was calculated by taking spatial derivatives of $\phi$. As we have already mentioned, the functional form of Eq. (\[long\]) should be valid far from the grain even if the OML correction is included. Therefore, Poisson’s equation was integrated from a large radial distance toward the inner region by taking $\alpha$ as a free parameter. We then tried to find the values of $\alpha$ for which this theoretical solution reasonably matched the simulation results. It is readily seen from Fig.\[com\] that the simulation results are well explained by this model with $\alpha\simeq 1.8-2.0$. Note again that the theoretical curve is for an isolated grain, whereas the simulation results are obtained with two dust grains. This means that the effect of ODSs is not observed, at least to a detectable level beyond the error bars of our simulations. This result is qualitatively consistent with the suggestion by Markes and Williams (2000). They have shown explicitly that the electrostatic force acting between two grains surrounded by a plasma is repulsive by solving Poisson’s equation. The critical assumption in their model is that the ion and electron densities can be written as a function of the local electrostatic potential alone. Although this assumption sounds reasonable for instance in the collisionless limit where OML theory should apply, its validity must be tested carefully. On the other hand, our first principles approach free from such an assumption also demonstrates a repulsive nature for the electrostatic interaction. Furthermore, the fact that the electric field around the grain is consistent with the OML theory indicates the assumption made by Markes and Williams (2000) is indeed reasonable. One might argue that the fact that $\alpha\neq1$ explains the discrepancy between the simulation results and ODS theory, but this is not the case. Assuming that linear superposition of the potential is also possible for $\alpha\neq1$, we can easily calculate the ODS attractive force for this case as well. The resulting attractive potential force may be written as $$\label{ml-j} q{\phi_{{\rm ODS}}}\left(d\right)=\alpha\frac{q^{2}}{\lambda_{{\rm D}}}\left(\frac{\lambda_{{\rm D}}}{d}-\frac{\alpha}{2}\right)\exp\left(-\frac{d}{\lambda_{{\rm D}}} \right),$$ which is shown in Fig.\[mod\] for $\alpha=1, 1.2, 1.4$. It can be easily understood that the potential minimum moves inward and the depth increases as $\alpha$ increases. In fact, an easy analytical calculation confirms this tendency. Clearly, $\alpha\neq1$ does not help to explain the discrepancy. ![Modified ODS attractive potential given by Eq. (\[ml-j\]). Red, green, and blue lines represent $\alpha = 1, 1.2, 1.4$, respectively.[]{data-label="mod"}](modod.eps){width="90mm"} Although it is not easy to analytically determine the value of $\alpha$ in general, we can estimate the upper and lower bounds as follows. We define $r_{{\rm c}}$ as a solution to the equation $$\label{exp-equ} \frac{\alpha}{4\pi\Lambda}\frac{q}{e}\frac{\lambda_{{\rm D}}}{r_{{\rm c}}}\exp{\left(-\frac{r_{{\rm c}}}{\lambda_{{\rm D}}}\right)}=1,$$ where the left-hand side is the normalized outer potential. An analytic solution to this equation is given by $$\label{sol} r_{{\rm c}}=\lambda_{{\rm D}} W\left(\frac{\alpha}{4\pi \Lambda}\frac{q}{e}\right),$$ where $W\left(x\right)$ is the inverse function of $x=W\exp\left(W\right)$, which is also known as the Lambert W-function. The potential at $r=r_{{\rm c}}$ may be approximated by $\phi\left(r_{{\rm c}}\right)=\alpha q \exp{\left(-r_{{\rm c}}/\lambda\right)}/r_{{\rm c}}$ and should be bounded by $q \exp{\left(-a/\lambda\right)}/a$ and the bare Coulomb potential $q/a$, leading to the inequality $1\leq\alpha\leq \exp{\left(r_{{\rm c}}/\lambda\right)}$. Using $\Lambda$ and $q/e$, we can rewrite this inequality as $$\label{ine1} 1\leq\alpha\leq \exp{\left(\frac{1}{4\pi \Lambda} \frac{q}{e}\right)}.$$ This estimate must be modified when $\Lambda$ is much larger than the critical value $\Lambda_{{\rm c}}$ for which the condition $a=r_{{\rm c}}$ is satisfied. When $a\gg r_{{\rm c}}$, $\phi\left(a\right)$ rather than $\phi\left(r_{{\rm c}}\right)$ must be used for a similar comparison, yielding $$\label{ine2} 1\leq\alpha\leq \exp{\left(\sqrt[3]{\frac{3}{4\pi\Lambda}}\right)}.$$ The condition $r_{{\rm c}}=a$ leads to $\Lambda_{{\rm c}}\sim\left(q/e\right)^{3/2}$, which can also be expressed as $kT_{{\rm c}}\sim eq/a$ with a critical temperature $T_{{\rm c}}$. From this, it is clear that when the temperature is above the critical value, the plasma is weakly coupled even with dust grains having relatively large charge. This indicates that the OML correction in this regime is only a minor modification, and essentially, the Yukawa-type potential in the far zone directly connects to the bare Coulomb potential. In our simulations, since we used large dust charges with relatively small numbers of particles, the plasma parameter is smaller than the critical value. Note that the plasma parameter of dusty plasmas in space is usually huge, and so is almost always above the critical value. Our choice of dust charge was motivated by the fact that the theoretical ODS attractive force is proportional to $q$, and the effect is expected to be more pronounced for larger dust charges. As a drawback, we were forced to use sub-critical plasma parameters owing to limited computational resources. Because of this, it was not possible to draw a final conclusion. Nevertheless, the qualitative consistency between our results and the counterargument against the ODS attractive force strongly indicates that the ODS attractive force may not operate in reality. In particular, we believe the assumption that the derivative of the potential energy of the whole system with respect to the intergrain distance provides a net force acting on the grain is incorrect as was pointed out by Markes and Williams (2000). Equation (\[ine2\]) shows that, when the plasma parameter is sufficiently large, $\alpha$ becomes almost unity and the potential structure around the grain approaches Eq. (\[yukawa\]), on which the derivation of the ODS attractive potential is based. Even in this parameter regime, our results suggest that the electric field acting on the grain is given by the spatial derivative of the potential at the grain’s position rather than that of the potential of the whole system with respect to the intergrain distance. In this case, the electrostatic force acting between two dust grains is always repulsive. Of course, our results should apply only to the simplest situation where two infinitely small dust grains remain at rest with respect to an ambient fully ionized collisionless plasma. There has been a lot of discussion on the force acting on dust grains that may be affected by, e.g., finite grain size, relative streaming between the plasma and grains. Comprehensive understanding of the net force due the combined effect of those contributions is needed for, e.g., star and planet formation in astrophysical environments. Conclusion ========== We investigated the electrostatic interaction between dust grains surrounded by a plasma by employing first-principles N-body simulations combined with the Ewald method. It was shown that the interaction between two charged dust grains is repulsive and its magnitude is somewhat larger than that derived from the Yukawa potential. The force acting on the dust grains was explained by OML theory for a single isolated grain quite well. The result is consistent with the analysis given by Markes and Williams (2000). Consequently, we think that the electrostatic force acting between dust grains are always repulsive. Nevertheless, since our simulations have been performed only in a limited parameter range, a final conclusion awaits simulations with much higher plasma parameters, which will be made possible by adopting modern numerical schemes such as particle-particle particle-mesh and special-purpose GRAPE (GRAvity-piPE) computers for N-body simulations. [@text2; @Yamamoto06] Acknowledgement =============== We are grateful to the anonymous referee for his/her critical and constructive comments on the manuscript. [15]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****,  ()](\doibase 10.1029/RG027i002p00271) [****, ()](\doibase 10.1088/0031-8949/45/5/010) [****,  ()](\doibase 10.1103/RevModPhys.81.25) [****,  ()](\doibase 10.1063/1.1343087) @noop [**]{}, Astrophysics and Space Science Library (, , ) @noop [****,  ()]{} [****,  ()](\doibase 10.1063/1.1288910) [****,  ()](\doibase 10.1016/S0375-9601(97)00964-X) [****,  ()](\doibase 10.1016/j.physleta.2008.11.063) @noop [****,  ()]{} @noop [****,  ()]{} @noop [**** ()]{} [****,  ()](\doibase 10.1016/0010-4655(96)00043-4), @noop [ ]{} @noop [****, ()]{}

--- abstract: 'Recognition of grocery products in store shelves poses peculiar challenges. Firstly, the task mandates the recognition of an extremely high number of different items, in the order of several thousands for medium-small shops, with many of them featuring small inter and intra class variability. Then, available product databases usually include just one or a few studio-quality images per product (referred to herein as **reference** images), whilst at test time recognition is performed on pictures displaying a portion of a shelf containing several products and taken in the store by cheap cameras (referred to as **query** images). Moreover, as the items on sale in a store as well as their appearance change frequently over time, a practical recognition system should handle seamlessly new products/packages. Inspired by recent advances in object detection and image retrieval, we propose to leverage on state of the art object detectors based on deep learning to obtain an initial product-agnostic item detection. Then, we pursue product recognition through a similarity search between global descriptors computed on *reference* and cropped *query* images. To maximize performance, we learn an ad-hoc global descriptor by a CNN trained on *reference* images based on an image embedding loss. Our system is computationally expensive at training time but can perform recognition rapidly and accurately at test time.' author: - | Alessio Tonioni\ DISI, University of Bologna\ [[email protected]]{} - | Eugenio Serra\ DISI, University of Bologna\ [[email protected]]{} - | Luigi Di Stefano\ DISI, University of Bologna\ [[email protected]]{} bibliography: - 'biblio.bib' title: A deep learning pipeline for product recognition on store shelves --- Introduction {#sec:introduction} ============ ![Given a *query* image featuring multiple products (a), our system first detects the regions associated with the individual items and then recognizes the product enclosed in each region based on a database featuring only one *reference* image per product (two examples are shown in (b)). All the products are correctly recognized in (a) with bounding boxes colored according to the recognized classes.[]{data-label="fig:teaser"}](tabella.pdf){width="45.00000%"} Recognizing products displayed on store shelves based on computer vision is gathering ever-increasing attention thanks to the potential for improving the customer’s shopping experience (, via augmented reality apps, checkout-free stores, support to the visually impaired …) and realizing automatic store management (, automated inventory, on-line shelf monitoring…). The seminal work on product recognition dates back to [@merler2007recognizing], where Merler highlight the peculiar issues to be addressed in order to achieve a viable approach. First of all, the number of different items to be recognized is huge, in the order of several thousands for small to medium shops, well beyond the usual target for current state-of-the-art image classifiers. Moreover, product recognition can be better described as a hard instance recognition problem, rather than a classification one, as it deals with lots of objects looking remarkably similar but for small details (, different flavors of the same brand of cereals). Then, any practical methodology should rely only on the information available within existing commercial product databases, at most just one high-quality image for each side of the package, either acquired in studio settings or rendered (see -(b)). *Query* images for product recognition are, instead, taken in the store with cheap equipment (, a smart-phone) and featuring many different items displayed on a shelf (see -(a)). Unfortunately, this scenario is far from optimal for state-of-the-art multi-class object detectors based on deep learning [@redmon2016yolo9000; @huang2016speed; @ren2015faster], which require a large corpus of annotated images as similar as possible to the deployment scenario in order to provide good performance. Even acquiring and manually annotating with product labels a huge dataset of in-store images is not a viable solution due to the products on sale in stores, as well as their appearance, changing frequently over time, which would mandate continuous gathering of annotated in-store images and retraining of the system. Conversely, a practical approach should be trained once and then be able to handle seamlessly new stores, new products and/or new packages of existing products (, seasonal packages). To tackle the above issues, we address product recognition by a pipeline consisting of three stages. Given a shelf image, we perform first a class-agnostic object detection to extract region proposals enclosing the individual product items. This stage relies on a deep learning object detector trained to localize product items within images taken in the store; we will refer to this network as to the *Detector*. In the second stage, we perform product recognition separately on each of the region proposal provided by the *Detector*. Purposely, we carry out K-NN (K-Nearest Neighbours) similarity search between a global descriptor computed on the extracted region proposal and a database of similar descriptors computed on the *reference* images available in the product database. Rather than deploying a general-purpose global descriptor (, Fisher Vectors [@perronnin2010large]), we train a CNN using the *reference* images to learn an image embedding function that maps RGB inputs to n-dimensional global descriptors amenable for product recognition; this second network will be referred to as to the *Embedder*. Eventually, to help prune out false detections and improve disambiguation between similarly looking products, in the third stage of our pipeline we refine the recognition output by re-ranking the first K proposals delivered by the similarity search. An exemplary output provided by the system is depicted in -(a)). It is worth pointing out how our approach needs samples of annotated in-store images only to train the product-agnostic *Detector*, which, however, does not require product-specific labels but just bounding boxes drawn around items. In we will show how the product-agnostic *Detector* can be trained once and for all so to achieve remarkable performance across different stores despite changes in shelves disposition and product appearance. Therefore, new items/packages are handled seamlessly by our system simply by adding their global descriptors (computed through the *Embedder*) in the *reference* database. Besides, our system scales easily to the recognition of thousands of different items, as we use just one (or few) *reference* images per product, each encoded into a global descriptor in the order of one thousand float numbers. Finally, while computationally expensive at training time, our system turns out light (memory efficient) and fast at deployment time, thereby enabling near real-time operation. Speed and memory efficiency do not come at a price in performance, as our system compares favorably with respect to previous work on the standard benchmark dataset for product recognition. Related Work {#sec:related} ============ Grocery products recognition was firstly investigated in the already mentioned paper by Merler [@merler2007recognizing]. Together with a thoughtful analysis of the problem, the authors propose a system based on local invariant features. However, their experiments report performance far from conducive to real-world deployment in terms of accuracy and speed. A number of more recent works tried to improve product recognition by leveraging on: a) stronger features followed by classification [@cotter2014hardware], b) the statistical correlation between nearby products on shelves [@advani2015visual; @baz2016context] c) additional information on the expected product disposition [@tonioni2017product] or d) a hierarchical multi-stage recognition pipeline [@franco2017grocery]. Yet, all these recent papers focus on a relatively small-scale problem, recognition of a few hundreds different items at most, whilst usually several thousands products are on sale in a real shop. George [@george2014recognizing] address more realistic settings and propose a multi-stage system capable of recognizing $\sim3400$ different products based on a single model image per product. The authors’ contribution includes releasing the dataset employed in their experiments, which we will use in our evaluation. More recently, [@yoruk2016efficient] has tackled the problem using a standard local feature based recognition pipeline and an optimized Hough Transform to detect multiple object instances and filter out inconsistent matches, which brings in a slight performance improvement. Nowadays, CNN-based systems are dominating object detection benchmarks and can be subdivided into two main families of algorithms based on the number of stages required to perform detection. On one hand, we have the slower but more accurate two stage detectors [@ren2015faster], which decompose object detection into a region proposal followed by an independent classification for each region. On the other hand, fast one stage approaches [@redmon2016yolo9000; @huang2016speed] can perform detection and classification jointly. A very recent work has also addressed the specific domain of grocery products, so as to propose an ad hoc detector [@qiao2017scalenet] that analyzes the image at multiple scales to produce more meaningful region proposals. Besides, deploying CNNs to obtain rich image representations is an established approach to pursue image search, both as a strong off-the-shelf baseline [@sharif2014cnn] and as a key component within more complex pipelines [@gordo2016deep]. Inspiration for our product recognition approach came from [@schroff2015facenet; @bell2015learning]. In [@schroff2015facenet], Schroff train a CNN using triplets of samples to create an embedding for face recognition and clustering while in [@bell2015learning] Bell rely on a CNN to learn an image embedding to recognize the similarity between design products. Similarly, in the related field of fashion items recognition, relying on learned global descriptor rather than classifiers is an established solution shared among many recent works [@hadi2015buy; @wang2014learning; @shankar2017deep]. Proposed Approach {#sec:product_detection_recognition} ================= {width="75.00000%"} shows an overview of our proposed pipeline. In the first step, described , a CNN (*Detector*) extracts region proposals from the *query* image. Then, as detailed in , each region proposal is cropped from the *query* image and sent to another CNN (*Embedder*) which computes an ad-hoc image representation. These will then be deployed to pursue product recognition through a K-NN similarity search in a database of representations pre-computed off-line by the *Embedder* on the *reference* images. Finally, as illustrated in , we combine different strategies to perform a final refinement step which helps to prune out false detections and disambiguate among similar products. Detection {#ssec:detection} --------- Given a *query* image featuring several items displayed in a store shelf, the first stage of our pipeline aims at obtaining a set of bounding boxes to be used as region proposals in the following recognition stage. Ideally, each bounding box should contain exactly one product, fit tightly the visible package and provide a confidence score measuring how much the detection should be trusted. State-of-the-art CNN-based object detectors may fulfill the above requirements for the product recognition scenario, as demonstrated in [@qiao2017scalenet]. Given an input image, these networks can output several accurate bounding boxes, each endowed by a confidence and a class prediction. To train CNN-based object detectors, such as [@redmon2016yolo9000; @huang2016speed; @ren2015faster], a large set of images annotated with the position of the objects alongside with their class labels is needed. However, due to the ever-changing nature of the items sold in stores, we do not train the *Detector* to perform predictions at the fine-grained class level (, at the level of the individual products), but to carry out a product-agnostic item detection. Accordingly, the in-store training images for our *Detector* can be annotated for training just by drawing bounding-boxes around items without specifying the actual product label associated with each bounding-box. This formulation makes the creation of a suitable training set and the training itself easier and faster. Moreover, since the *Detector* is trained only to recognize *generic products* from everything else it is general enough to be deployable across different stores and products. Conversely, training a CNN to directly predict bounding boxes as well as product labels would require a much more expensive and slow image annotation process which should be carried out again and again to keep up with changes of the products/packages to be recognized. This continuous re-training of the *Detector* is just not feasible in any practical settings. Recognition {#ssec:recognition} ----------- Starting from the candidate regions delivered by the *Detector*, we perform recognition by means of K-NN similarity search between a global descriptor computed on each candidate region and a database of similar descriptors (one for each product) pre-computed off-line on the *reference* images. Recent works (, [@sharif2014cnn]) have shown that the activations sampled from layers of pre-trained CNNs can be used as high quality global image descriptors. [@wang2014learning] extended this idea by proposing to train a CNN (, the *Embedder*) to learn a function $E:\mathcal{I}\rightarrow\mathcal{D}$ that maps an input image $i \in I$ into a k-dimensional descriptor $d^k \in \mathcal{D}$ amenable to recognition through K-NN search. Given a set of images with associated class labels, the training is performed sampling triplets of different images, referred to as *anchor* ($i_a$), *positive* ($i_p$) and *negative* ($i_n$), such that $i_a$ and $i_p$ depict the same class while $i_n$ belongs to a different one. Given a distance function in the descriptor space, $d(\mathbf{X},\mathbf{Y})$, with $X,Y\in\mathcal{D}$, and denoted as $E(i)$ the descriptor computed by the the *Embedder* on image $i$, the network is trained to minimize the so called *triplet ranking loss*: $$\begin{gathered} \label{eq:tripletLoss} \mathcal{L} = max( 0,d(E(i_a),E(i_p))-d(E(i_a),E(i_n))+\alpha)\end{gathered}$$ with $\alpha$ a fixed margin to be enforced between the pair of distances. Through minimization of this loss, the network learns to encode into nearby positions within $\mathcal{D}$ the images depicting items belonging to the same class, whilst keeping items of different classes sufficiently well separated. We use the same formulation and cast it for the context of grocery product recognition where different products corresponds to different classes (the two reference images of -(b) corresponds to different classes and could be used as $i_p$ and $i_n$). Unfortunately, we can not sample different images for $i_a$ and $i_p$ due to available commercial datasets featuring just a single exemplar image per product (, per class). Thus, to create the required triplet, at each training iteration we randomly pick two products and use their *reference* images as $i_p$ and $i_n$. Then, we synthesize a new $i_a$ from $i_p$ by a suitable data augmentation function $A:\mathcal{I}\rightarrow\mathcal{I}$, to make it similar to *query* images (, $i_a=A(i_p)$)[^1]. To perform recognition, firstly, the *Embedder* network is used to describe each available *reference* image $i_r$ by a global descriptor $E(i_r)$ and thus create the *reference* database of descriptors associated with the products to be recognized. Then, when a *query* image is processed, the same embedding is computed on each of the candidate regions, $i_{pq}$, cropped from the *query* image, $i_q$, so to get $E(i_{pq})$. Finally, for each $i_{pq}$ we compute the distance in the embedding space with respect to each *reference* descriptor, denoted as $d(E(i_{pq}),E(i_{r}))$, in order to sift-out the first $K$-NN of $E(i_{pq})$ in the *reference* database. These are subject to further processing in the final refinement step. Refinement {#ssec:refinement} ---------- The aim of the final refinement is to remove false detections and re-rank the first K-NN found in the previous step in order to fix possible recognition mistakes. Since the initial ranking is obtained comparing descriptors computed on whole images, a meaningful re-ranking of the first K-NN may be achieved by looking at peculiar image details that may have been neglected while comparing global descriptors and yet be crucial to differentiate a product from others looking very similar. Thus, both the *Query* and each of the first K-NN *reference* images are described by a set of local features ${F_1,F_2,...,F_k}$, each consisting in a spatial position $(x_i,y_i)$ within the image and a compact descriptor $f_i$. Given these features, we look for similarities between descriptors extracted from *query* and *reference* images, to compute a set of matches. Matches are then weighted based on the distance in the descriptor space, $d(f_i,f_j)$ and a geometric consistency criterion relying on the unit-norm vector, $\vec{v}$, from the spatial location of a feature to the image center. In particular, given a match, $M_{ij}=(F^q_i,F^r_j)$, between feature $i$ of the *query* image and feature $j$ of the *reference* image, we compute the following weight: $$W_{ij}=\frac{(\vec{v}^{\,q}_i \cdot \vec{v}^{\,r}_j)+1}{d(f^q_i,f^r_j)+\epsilon}$$ where $\cdot$ marks scalar products between vectors and $\epsilon$ is a small number to avoid potential division by zero. Intuitivelly $W_{ij}$ is bigger for matching features which share the same relative position with respect to the image center (high $(\vec{v}^{\,q}_i \cdot \vec{v}^{\,r}_j)$) and have descriptors close in the feature space (small $d(f^q_i,f^r_j)$). Finally, the first K-NN are re-ranked according to the sum of the weights $W_{ij}$ computed for the matches between the local features. In we will show how good local features can be obtained at zero computational cost as a by-product of our learned global image descriptor. This refinement technique will be referred to as ***+lf***. A simple additional refinement step consists in filtering out wrong recognitions by the *distance ratio* criterion [@lowe2004distinctive] (, by thresholding the ratio of the distances in feature space between the *query* descriptor and its 1-NN and 2-NN). If the ratio is above a threshold, $\tau_{d}$, the recognition is deemed as ambiguous and discarded. In the following, we will denote this refinement technique as ***+th***. Finally, as commercial product databases typically provide a multilevel classification of the items (, at least instance and category level), we propose a re-ranking and filtering method specific to the grocery domain where, as pointed out by [@george2014recognizing], products belonging to the same macro category are typically displayed close one to another on the shelf. In particular, given the candidate regions extracted from the *query* image and their corresponding sets of K-NN, we consider the 1-NN of the region proposals extracted with a high confidence ($>0.1$) by the *Detector* in order to find the main macro category of the image. Then, in case the majority of detections votes for the same macro category, it is safe to assume that the pictured shelf contains almost exclusively items of that category thus filter the K-NN for all candidate regions accordingly. It is worth observing how this strategy implicitly leverages on those products easier to identify (, the high-confidence detections) to increase the chance to correctly recognize the harder ones. We will refer to this refinement strategy as to ***+mc***. Experimental Results {#sec:experimental} ==================== To validate the performance of our product recognition pipeline we take into account two possible use cases dealing with different final users: - **Customer**: the system should be deployed for a guided or partially automated shopping experience (, product localization inside the shop or augmented reality overlays or support to visually impaired). As proposed in [@george2014recognizing], in this use case the goal is to detect *at least one* instance of each visible type of products displayed in a shelf picture. - **Management**: the system will be used to partially automate the management of a shop (, automatic inventory and restocking). Here, the goal is to recognize *all* visible product instances displayed in a shelf picture. Datasets and Evaluation Metrics {#ssec:datasets} ------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![In (a) the system should identify at least one instance for each product type, while in (b) it should find and correctly localize all the displayed product instances.[]{data-label="fig:bboxes"}](bb_zurigo.jpg "fig:"){width="22.00000%"} ![In (a) the system should identify at least one instance for each product type, while in (b) it should find and correctly localize all the displayed product instances.[]{data-label="fig:bboxes"}](bb_iciap.jpg "fig:"){width="22.00000%"} (a)-Customer[@george2014recognizing] (b)-Management[@tonioni2017product] ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- For our experimental evaluation we rely on the publicly available *Grocery Products* dataset [@george2014recognizing], which features more than 8400 grocery products organized in hierarchical classes and with each product described by exactly one *reference* image. The dataset contains also 680 in-store (*query*) images that display items belonging to the *Food* subclass of the whole dataset. The annotations released by the authors for the *query* images allow for evaluating performance in the *Customer* use case, as they consist in bounding boxes drawn around spatial clusters of instances of the same products. To test our system also in the *Management* use case, we deploy the annotations released by the authors of [@tonioni2017product], which consist of boxes around each product instance for a subset of 70 in-store pictures of the *Grocery Products* dataset. shows examples of the two kinds of annotations used to evaluate the system in the two different use cases. To compare our work with previously published results we use the metrics proposed by the authors in [@george2014recognizing]: mean average precision-*mAP* (the approximation of the area under the Precision-Recall curve for the detector) and Product Recall-*PR* (average product recall across all the test image). As for scenario (a), we report also mean average multi-label classification accuracy-*mAMCA*. To train the *Detector* we acquired multiple videos with a tablet mounted on a cart facing the shelves and manually labeled a subset of sampled frames to create a training set of 1247 images. Thus, our videos are acquired in different stores with respect to those depicted in the *Grocery Products* dataset and feature different products on different shelves, vouching for the generalization ability of our system. Implementation Details {#ssec:ideal_study} ---------------------- For all our tests we have used as *Detector* the state-of-the-art one-stage object detector known as yolo\_v2 (shortened in *yolo*) [@redmon2016yolo9000]. We choose this network as it grants real-time performance on a GPU and for the availability of the original implementation. Starting from the publicly available weights[^2], we have fine tuned the network on our 1247 in-store images for 80000 steps keeping the hyperparameters suggested by the original authors. The backbone network for our *Embedder* is a VGG\_16 [@Simonyan14c] pre-trained on the Imagenet-1000 classification task (weights publicly available[^3]). From this network we obtain global image descriptors by computing *MAC* features [@tolias2015particular] on the conv4\_3 convolutional layer and applying L2 normalization to obtain unit-norm embedding vectors. To carry out the comparison between descriptors, both at training and test time, we used as distance function $d(X,Y)=1-X \cdot Y$ with $X,Y \in \mathcal{D}$ (, 1 minus the cosine similarity between the two descriptors). To better highlight the advantage of learning an ad-hoc descriptor in the following we will report experiments using general purpose descriptors obtained without fine tuning the *Embedder* with the suffix $\_gd$ (general descriptor), while descriptors obtained after fine-tuning as described in will be denoted by the suffix $\_ld$ (learned descriptor). To train the *Embedder* we use the *reference* images of *Grocery Products* dealing with the products belonging to the *Food* subclass (, 3288 different product with exactly one training image for each). To enrich the dataset and create the anchor images, $i_a$, we randomly perform the following augmentation functions $A$: blur by a Gaussian kernel with random $\sigma$, random crop, random brightness and saturation changes. This augmentations were engineered so to transform the *reference* images in a way that renders them similar to the proposals cropped from the *query* images. The hyper-parameters obtained by cross validation for the training process are as follows: $\alpha=0.1$ for the triplet loss, learning rate $lr=0.000001$, ADAM optimizer and fine-tuning by 10000 steps with batch size 24. We propose a novel kind of local features for the *+lf* refinement: as *MAC* descriptors are obtained by applying a max-pool operation over all the activations of a convolutional layer, by changing the size and stride of the pool operation it is possible to obtain a set of local descriptor with the associated location being the center of the pooled area reprojected into the original image. By leveraging on this intuition, we can obtain in a single forward computation both a global descriptor for the initial K-NN search () as well as a set of local features to be deployed in the refinement step (). For our test we choose kernel size equal $16$ and stride equals $2$ as to have 64 features per *reference* image. Customer Use Case {#ssec:customer_fine} ----------------- In this sub-section we evaluate the effectiveness of our system in the *Customer* scenario. To measure performance we rely on the annotations displayed in -(a) and score a correct recognition when the product has been correctly identified and its bounding box has a non-empty intersection with that provided as ground-truth. We compare our method with already published work tested on the same dataset: FV+RANSAC (Fisher Vector classification re-ranked with RANSAC) [@george2014recognizing], RF+PM+GA (Category prediction with Random Forests, dense Pixel Matching and Genetic Algorithm) [@george2014recognizing] and FM+HO (local feature matching optimized with Hough) [@yoruk2016efficient]. We report the results obtained by the different methods according to the tree metrics in . As [@yoruk2016efficient] does not provide the mAP figure but only the values of precision and recall, for FM+HO we report an approximate mAP computed by multiplying precision and recall. Using our trained *yolo* network for product detection, in we report the results obtained by either deploying a general purpose VGG-based descriptor ($yolo_{gd}$) or learning an ad-hoc embedding for grocery products $yolo_{ld}$. Moreover, we report the results achieved with the different refinement strategies presented in . shows that our pipeline can provide a higher recall than previous methods even with a general purpose image descriptor (*yolo\_gd*), although with a somehow lower precision, as demonstrated by the slightly inferior mAP score. However, our complete proposal relies on learning an ad-hoc descriptor for grocery products (*yolo\_ld*), which yields a significant performance improvement, as vouched by an average gain of about 6% in terms of both Recall and mAP. Wrongly classified proposals can be discarded to further improve accuracy by the threshold refinement strategy (*yolo\_ld + th* - with $\tau_d=0.9$), thereby increasing the mAMCA from 16.32% to 28.74%. Re-ranking based on the proposed local features (*yolo\_ld+lf*) turns out an effective approach to ameliorate both precision and recall, as demonstrated by a gain of about 5% in mAP and PR with respect to the pipeline without final refinement (*yolo\_ld*). The category-based re-ranking strategy (*yolo\_ld + mc*) seems to fix some of the recognition mistakes and improve the recognition rate with respect to (*yolo\_ld*) , providing gains in all metrics. Finally, by mixing all the refinement strategies to obtain our overall pipeline (*yolo\_ld+lf-mc-th*), we neatly get the best trade-off between precision and recall, as vouched by the 57.07% PR and 36.02% mAP, about 14% and 12.5% better than previously published results, respectively, with a mAMCA turning out nearly on par with the best previous result. ![Precision-recall curves obtained in the *customer* use case by the *yolo\_ld* system when trying to recognize either the individual products or just their category.[]{data-label="fig:catInstance"}](category_instance.pdf){width="40.00000%"} We found that casting recognition as a similarity search through learned global descriptors has the nice property that even when the 1-NN does not correspond to the right product, it usually corresponds to items belonging the correct category (cereals, coffee,…). We believe this behavior being due to items belonging to the same category sharing similar peculiar visual patterns that are effectively captured by the descriptor and help to cluster nearby in descriptor space items belonging to the same categories (, coffee cups often displayed on coffee packages or flowers on infusions). To highlight this generalization property, we perform here an additional test in the Customer scenario by considering a recognition as correct if the category of the 1-NN match is the same as those of the annotated bounding box. Accordingly, we compare the performance of *yolo\_ld* when trying to recognize either the individual products or their category. The results of this experiment are reported as Precision-Recall curves in . The large difference between the two curves proves that very often the system mistakes items at product level though correctly recognizing their category. Eventually, it is worth pointing out that our method not only provides a significant performance improvement with respect to previously published results but turns out remarkably fast. Indeed, our whole pipeline can be run on a GPU in less than one second per image. ![Precision-Recall curves for our full pipeline in the *Management* use case. full@180 denotes performing recognition on the small *reference* database of [@tonioni2017product] ($\sim 180$ entries), full@3200 against all the products in the *Food* category of *Grocery Products* ($\sim 3200$).[]{data-label="fig:zmiePrecRec"}](zmie.pdf){width="40.00000%"} Management Use Case {#ssec:management_fine} ------------------- -------------------------------------------- -------------------------------------------- -------------------------------------------- {width="32.00000%"} {width="32.00000%"} {width="32.00000%"} (a) (b) (c) -------------------------------------------- -------------------------------------------- -------------------------------------------- The experiments presented in this Section concern the *Management* use case, a task requiring correct recognition of all the individual products displayed on shelves. Thus, we rely on the annotations shown in -(b) and consider a recognition as correct when the item has been correctly recognized and the intersection over union (IoU) between the predicted and ground truth bounding boxes is higher than 0.5. To the best of our knowledge, the only published results on the *Grocery Products* dataset that deals with recognition of all the individual products are reported in [@tonioni2017product]. However, in [@tonioni2017product] the authors address a specific task referred to as *Planogram Compliance*, which consists in checking the compliance between the actual product disposition and the planned one. Accordingly, the pipeline proposed in [@tonioni2017product] includes an initial unconstrained product recognition stage, which addresses the same settings as our *Management* use case, followed by a second stage that deploys the exact knowledge on the planned product disposition in order to improve recognition accuracy and detect compliance issues (, missing or misplaced products). Therefore, we compare our proposal to the most effective configuration of the first stage of the pipeline presented in [@tonioni2017product], referred to hereinafter as *FS*, which is based on matching BRISK local features[@leutenegger2011brisk] followed by Hough Voting and pose estimation through RANSAC. To compare our pipeline with respect to *FS*, we use the annotations provided by the authors and perform recognition against the smaller *reference* database of 182 products used in [@tonioni2017product]. The results are reported in . Firstly, it is worth pointing out how, despite the task being inherently more difficult than in the *Customer* use case, we record higher recognition performance. We ascribe this mainly to the smaller subset of in-store images used for testing, (, 70 vs. 680) as well as to these images featuring mainly rigid packaged products, which are easier to recognize than deformable packages. Once again, in our pipeline, the use of a learned descriptor (*yolo\_ld*) provides a substantial performance gain with respect to a general purpose descriptor (*yolo\_gd*), as the mAP improves from 66.95% to 74.32% and the PR from 78.89% to 84.75%. The different refinement strategies provide advantages similar to those discussed in , the best improvement yielded by re-ranking recognitions based on the local features extracted from the *Embedder* network (*yolo\_ld+lf*). The optimal trade-off between precision and recall is achieved again by deploying together all the refinement strategies (*yolo\_ld+lf-mc-th*), which provided a mAP and PR as high as 76.93% and 84.75%, respectively (, both about 10% better than the previously published results on this dataset). reports results aimed at assessing the scalability of the methods with respect to the number of products in the *reference* database. We carried out an additional experiment by performing the recognition of each item detected within the 70 *query* images against all the 3200 products of the “Food” category in *Grocery Products* rather than the smaller subset of 182 products proposed in [@tonioni2017product]. By comparing the values in and , we can observe how, unlike *FS*, our method can scale nicely from few to thousands of different products: our full method *yolo\_ld+lf-mc-th* looses only 3.43% mAP upon scaling-up the *reference* database quite significantly, whilst the performance drop for *FS* is as relevant as 19.05%. In we also plot the precision-recall curves obtained by our full pipeline (*yolo\_ld+lf-mc-th*) using the smaller (full@180) and larger (full@3200) sets of reference products. The curves show clearly how our pipeline can deliver almost the same performance in the two setups, which vouches for the ability of our proposal to scale smoothly to the recognition of thousands of different products. As far as recognition time is concerned, our pipeline can scale fairly well regardless of the size of the *reference* database, due to the NN search, even if extensive, amounting to a negligible fraction of the overall computation: the difference in inference time between recognizing 180 and 3200 product is less than a tenth of a second. Qualitative Results {#sec:qualitative} =================== reports some qualitative results obtained by our pipeline. Picture (a) shows the recognition results on an image taken quite far from the shelf and featuring a lot of different items; (b) deal with some successful recognitions in a close-up *query* image, where only a few items are visible at once. Finally, (c) refers to recognition of products featuring deformable and highly reflective packages, which are quite challenging to recognize due to the appearance of the items within the *query* images turning out significantly different than in the available *reference* images. Yet, in (c) our system was able to find at least one item for each product type (, as required in the *Customer* use case). Conclusion {#sec:conclusion} ========== In this paper we have proposed a fast and effective approach to the problem of recognizing grocery products on store shelves. Our proposal addresses the task by three main steps: class agnostic object detection to identify the individual items appearing on a shelf image, recognition through K-NN similarity search based on a global image descriptor, final refinement to further boost performance. All the three steps deploy modern deep learning techniques, as we detect items by a state-of-the-art CNN (*Detector*), learn the image descriptor by another CNN trained to disentangle the appearance of grocery products (*Embedder*) and extract local cues key to refinement as MAC features computed alongside with the global embedding. The experiments prove that our pipeline compares favourably to the state-of-the-art on the public dataset available for performance assessment while being remarkably fast. Yet, we plan to investigate how to further improve the speed at test time (, to enable execution on a low-cost and/or mobile device). Purposely, we envisage devising a unified CNN architecture acting as both *Detector* and *Embedder*. Furthermore we are currently investigating on the use of generative models (, GANs) to augment the number of samples per product to train the *Embedder*. A generative model could also be trained to render *reference* more similar to proposals cropped from *query* images in order to shrink the gap between the training and testing domains. [^1]: The details concerning the adopted augmentation function are reported in [^2]: <https://github.com/pjreddie/darknet> [^3]: <https://github.com/tensorflow/models/tree/master/research/slim>

--- author: - 'Quanshi Zhang, Ruiming Cao, Ying Nian Wu, and Song-Chun Zhu *Fellow, IEEE*' bibliography: - 'TheBib.bib' title: Mining Interpretable AOG Representations from Convolutional Networks via Active Question Answering --- [Shell : Bare Demo of IEEEtran.cls for Computer Society Journals]{} Introduction ============ Convolutional neural networks [@CNN; @CNNImageNet; @ResNet; @DenseNet] (CNNs) have achieved superior performance in many visual tasks, such as object detection and segmentation. However, in real-world applications, current neural networks still suffer from low interpretability of their middle-layer representations and data-hungry learning methods. Thus, the objective of this study is to mine thousands of *latent patterns* from the mixed representations in conv-layers. Each latent pattern corresponds to a constituent region or a contextual region of an object part. We use an interpretable graphical model, namely an And-Or graph (AOG), to organize latent patterns hidden in conv-layers. The AOG maps implicit latent patterns to explicit object parts, thereby explaining the hierarchical representation of objects. We use very few (*e.g.* 3–20) part annotations to mine latent patterns and construct the AOG to ensure high learning efficiency. As shown in Fig. \[fig:rawMapToModel\], compared to ordinary CNN representations where each filter encodes a mixture of textures and parts (evaluated by [@Interpretability]), we extract clear object-part representations from CNN features. Our weakly-supervised learning method enables people to model objects or object parts on-the-fly, thereby ensuring broad applicability. {width="\linewidth"} **And-Or graph representations:**[` `]{} As shown in Fig. \[fig:rawMapToModel\], the AOG represents a semantic hierarchy on the top of conv-layers, which consists of four layers, *i.e.* the *semantic part*, *part templates*, *latent patterns*, to *CNN units*. In the AOG, AND nodes represent compositional regions of a part, and OR nodes represent a list of alternative template/deformation candidates for a local region. - Layer 1: the top *semantic part* node is an OR node, whose children represent template candidates for the part. - Layer 2: a *part template* in the second layer describes a certain part appearance with a specific pose, *e.g.* a black sheep head from a side view. A part template is an AND node, which uses its children latent patterns to encode its constituent regions. - Layer 3: a *latent pattern* in the third layer represents a constituent region of a part (*e.g.* an eye in the head part) or a contextual region (*e.g.* the neck region *w.r.t.* the head). A latent pattern is an OR node, which naturally corresponds to a group of units within the feature map of a certain CNN filter. The latent pattern selects one of its children *CNN units* as the configuration of the geometric deformation. - Layer 4: terminal nodes are *CNN units*, *i.e.* raw activation units on feature maps of a CNN filter. In this hierarchy, the AOG maps implicit latent patterns in raw CNN feature maps to explicit semantic parts. We can use the AOG to localize object parts and their constituent regions for hierarchical object parsing. The AOG is interpretable and can be used for communications with human users. **Weakly-supervised learning via active question-answering:**[` `]{} We propose a new active learning strategy to build an AOG in a weakly-supervised manner. As shown in Fig. \[fig:QA\], we use an active question-answering (QA) process to mine latent patterns from raw feature maps and gradually grow the AOG. {width="0.99\linewidth"} The input is a pre-trained CNN and its training samples (*i.e.* object images without part annotations). The QA method actively discovers the missing patterns in the current AOG and asks human users to label object parts for supervision. In each step of the QA, we use the current AOG to localize a certain semantic part among all unannotated images. Our method actively identifies object images, which cannot fit well to the AOG. *I.e.* the current AOG cannot explain object parts in these images. Our method estimates the potential gain of asking about each of the unexplained objects, thereby determining an optimal sequence of questions for QA. Note that the QA is implemented based on pre-define ontology, instead of using open-ended questions or answers. As in Fig. \[fig:QA\], the user is asked to provide five types of answers (*e.g.* labeling the correct part position when the AOG cannot accurately localize the part), in order to guide the growth of the AOG. Given each specific answer, our method may either refine the AOG branch of an existing part template or construct a new AOG branch for a new part template. Based on human answers, we mine latent patterns for new AOG branches as follows. We require the new latent patterns - to represent a region highly related to the annotated object parts, - to frequently appear in unannotated objects, - to consistently keep stable spatial relationships with other latent patterns. Similar requirements were originally proposed in studies of pursuing AOGs, which mined hierarchical object structures from Gabor wavelets on edges [@MiningAOG] and HOG features [@OurICCV15AoG]. We extend such ideas to feature maps of neural networks. The active QA process mines object-part patterns from the CNN with fewer human supervision. There are three mechanisms to ensure the stability of weakly-supervised learning. - Instead of learning all representations from scratch, we transfer patterns in a pre-trained CNN to the target object part, which boosts the learning efficiency. Because the CNN has been trained using numerous images, latent patterns in the AOG are supposed to consistently describe the same part region among different object images, instead of over-fitting to part annotations obtained during the QA process. For example, we use the annotation of a specific tiger head to mine latent patterns. The mined patterns are not over-fitted to the head annotation, but represent generic appearances of different tiger heads. In this way, we can use very few (*e.g.* 1–3) part annotations to extract latent patterns for each part template. - It is important to maintain the generality of the pre-trained CNN during the learning procedure. *I.e.* we do not change/fine-tune the original convolutional weights within the CNN, when we grow new AOGs. This allows us to continuously learn new semantic parts from the same CNN, without the model drift. - The active QA strategy reduces the excessive usage of the human labor of annotating object parts that have been well explained by the current AOG. In addition, we use object-level annotations for pre-training, considering the following two facts: 1) Only a few datasets [@SemanticPart; @CUB200] provide part annotations, and most benchmark datasets [@PascalVOC; @ImageNet; @MSCOCO] mainly have annotations of object bounding boxes. 2) More crucially, real-world applications may focus on various object parts on-the-fly, and it is impractical to annotate a large number of parts for each specific task. This paper makes the following three contributions. 1\) From the perspective of object representations, we semanticize a pre-trained CNN by mining reliable latent patterns from noisy feature maps of the CNN. We design an AOG to represent the semantic hierarchy inside conv-layers, which associates implicit neural patterns with explicit semantic parts. 2\) From the perspective of learning strategies, based on the clear semantic structure of the AOG, we present an active QA method to learn each part template of the object sequentially, thereby incrementally growing AOG branches on a CNN to enrich part representations in the AOG. 3\) In experiments, our method exhibits superior performance to other baselines of weakly-supervised part localization. For example, our methods with 11 part annotations outperformed fast-RCNNs with 60 annotations on the Pascal VOC Part dataset. A preliminary version of this paper appeared in [@CNNAoG] and [@DeepQA]. Related work ============ **CNN visualization:**[` `]{} Visualization of filters in a CNN is a direct way of exploring the pattern hidden inside a neural unit. Lots of visualization methods have been used in the literature. Gradient-based visualization [@CNNVisualization_1; @CNNVisualization_2; @CNNVisualization_3] estimates the input image that maximizes the activation score of a neural unit. Dosovitskiy *et al.* [@FeaVisual] proposed up-convolutional nets to invert feature maps of conv-layers to images. Unlike gradient-based methods, up-convolutional nets cannot mathematically ensure the visualization result reflects actual neural representations. In recent years, [@olah2017feature] provided a reliable tool to visualize filters in different conv-layers of a CNN. Zhou *et al.* [@CNNSemanticDeep] proposed a method to accurately compute the image-resolution receptive field of neural activations in a feature map. Theoretically, the actual receptive field of a neural activation is smaller than that computed using the filter size. The accurate estimation of the receptive field is crucial to understand a filter’s representations. Unlike network visualization, our mining part representations from conv-layers is another choice to interpret CNN representations. **Active network diagnosis:**[` `]{} Going beyond “passive” visualization, some methods “actively” diagnose a pre-trained CNN to obtain insight understanding of CNN representations. [@CNNAnalysis_1] explored semantic meanings of convolutional filters. [@CNNAnalysis_2] evaluated the transferability of filters in intermediate conv-layers. [@CNNAnalysis_3; @CNNVisualization_5] computed feature distributions of different categories in the CNN feature space. Methods of [@visualCNN_grad; @visualCNN_grad_2] propagated gradients of feature maps *w.r.t.* the CNN loss back to the image, in order to estimate the image regions that directly contribute the network output. [@trust] proposed a LIME model to extract image regions that are used by a CNN to predict a label (or an attribute). Network-attack methods [@pixelAttack; @CNNInfluence; @CNNAnalysis_1] diagnosed network representations by computing adversarial samples for a CNN. In particular, influence functions [@CNNInfluence] were proposed to compute adversarial samples, provide plausible ways to create training samples to attack the learning of CNNs, fix the training set, and further debug representations of a CNN. [@banditUnknown] discovered knowledge blind spots (unknown patterns) of a pre-trained CNN in a weakly-supervised manner. Zhang *et al.* [@CNNBias] developed a method to examine representations of conv-layers and automatically discover potential, biased representations of a CNN due to the dataset bias. Furthermore, [@wu2007compositional; @yang2009evaluating; @wu2011numerical] mined the local, bottom-up, and top-down information components in a model for prediction. **CNN semanticization:**[` `]{} Compared to the diagnosis of CNN representations, semanticization of CNN representations is closer to the spirit of building interpretable representations. Hu *et al.* [@LogicRuleNetwork] designed logic rules for network outputs, and used these rules to regularize neural networks and learn meaningful representations. However, this study has not obtained semantic representations in intermediate layers. Some studies extracted neural units with certain semantics from CNNs for different applications. Given feature maps of conv-layers, Zhou *et al.* [@CNNSemanticDeep; @CNNSemanticDeep2] extracted scene semantics. Simon *et al.* mined objects from feature maps of conv-layers [@ObjectDiscoveryCNN_2], and learned explicit object parts [@CNNSemanticPart]. Unlike above research, we aim to explore the entire semantic hierarchy hidden inside conv-layers of a CNN. Because the AOG structure [@MumfordAOG; @MiningAOG] is suitable for representing the semantic hierarchy of objects, our method uses an AOG to represent the CNN. In our study, we use semantic-level QA to incrementally mine object parts from the CNN and grow the AOG. Such a “white-box” representation of the CNN also guided further active QA. With clear semantic structures, the AOG makes it easier to transfer CNN patterns to other part-based tasks. **Unsupervised/active learning:**[` `]{} Many methods have been developed to learn object models in an unsupervised or weakly supervised manner. Methods of [@Gpt_WeaklyCNN; @WeaklyMIL; @OurICCV15AoG; @ObjectDiscoveryCNN_2] learned with image-level annotations without labeling object bounding boxes. [@UnsuperCNN; @ChoDiscovery] did not require any annotations during the learning process. [@OnlineMetric] collected training data online from videos to incrementally learn models. [@Language2VideoAlign; @Language2ActionAlign] discovered objects and identified actions from language Instructions and videos. Inspired by active learning [@Active4; @i13; @Active2], the idea of learning from question-answering has been used to learn object models [@KB_Fei_Annotation; @KB_Fei_InteractionLabel; @TuQA]. Branson *et al.* [@ActivePart] used human-computer interactions to label object parts to learn part models. Instead of directly building new models from active QA, our method uses the QA to mine AOG part representations from CNN representations. **AOG for knowledge transfer:** Transferring hidden patterns in the CNN to other tasks is important for neural networks. Typical research includes end-to-end fine-tuning and transferring CNN representations between different categories [@CNNAnalysis_2; @CNNSemantic] or datasets [@UnsuperTransferCNN]. In contrast, we believe that a good explanation and transparent representation of parts will create a new possibility of transferring part features. As in [@AllenAoG; @MiningAOG], the AOG is suitable to represent the semantic hierarchy, which enables semantic-level interactions between human and neural networks. **Modeling “objects” vs. modeling “**parts**” in un-/weakly-supervised learning:**[` `]{} Generally speaking, in the scenario of un-/weakly-supervised learning, it is usually more difficult to model object parts than to represent entire objects. For example, object discovery [@ObjectDiscoveryCNN_1; @ObjectDiscoveryCNN_2; @ObjectDiscoveryCNN_3] and co-segmentation [@InteractiveCoseg] only require image-level labels without object bounding boxes. Object discovery is mainly implemented by identifying common foreground patterns from the noisy background. People usually consider closed boundaries and common object structure as a strong prior for object discovery. In contrast to objects, it is difficult to mine true part parsing of objects without sufficient supervision. Up to now, there is no reliable solution to distinguishing semantically meaningful parts from other potential divisions of object parts in an unsupervised manner. In particular, some parts (*e.g.* the abdomen) do not have shape boundaries to determine their shape extent. **Part localization/detection vs. semanticizing CNN patterns:** There are two key points to differentiate our study from conventional part-detection approaches. First, most detection methods deal with classification problems, but inspired by graph mining [@OurICCV15AoG; @OurSAPPAMI; @OurCVPR14Graph], we mainly focus on a mining problem. *I.e.* we aim to discover meaningful latent patterns to clarify CNN representations. Second, instead of summarizing common knowledge from massive annotations, our method requires very limited supervision to mine latent patterns. Method ====== The overall objective is to sequentially minimize the following three loss terms. $${Loss}={Loss}^{\textrm{CNN}}+{Loss}^{\textrm{QA}}+{Loss}^{\textrm{AOG}} \label{eqn:obj}$$ ${Loss}^{\textrm{CNN}}$ denotes the classification loss of the CNN. ${Loss}^{\textrm{QA}}$ is referred as to the loss for active QA. Given the current AOG, we use ${Loss}^{\textrm{QA}}$ to actively determine a sequence of questions about objects that cannot be explained by the current AOG, and require people to annotate bounding boxes of new object parts for supervision. ${Loss}^{\textrm{AOG}}$ is designed to learn an AOG for the CNN. ${Loss}^{\textrm{AOG}}$ penalizes 1) the incompatibility between the AOG and CNN feature maps of unannotated objects and 2) part-location errors *w.r.t.* the annotated ground-truth part locations. It is essential to determine the optimization sequence for the three losses in the above equation. We propose to first learn the CNN by minimizing ${Loss}^{\textrm{CNN}}$ and then build an AOG based on the learned CNN. We use the active QA to obtain new part annotations and use new part annotations to grow the AOG by optimizing ${Loss}^{\textrm{QA}}$ and ${Loss}^{\textrm{AOG}}$ alternatively. We introduce details of the three losses in the following subsections. Learning convolutional neural networks -------------------------------------- To simplify the story, in this research, we just consider a CNN for single-category classification, *i.e.* identifying object images of a specific category from random images. We use the log logistic loss to learn the CNN. $${Loss}^{\textrm{CNN}}=\mathbb{E}_{I\in{\bf I}}\big[{Loss}(\hat{y}_{I},y^{*}_{I})\big]$$ where $\hat{y}_{I}$ and $y^{*}_{I}$ denote the predicted and ground-truth labels of an image $I$. If the image $I$ belongs to the target category, then $y^{*}_{I}=+1$; otherwise $y^{*}_{I}=-1$. Learning And-Or graphs ---------------------- We are given a pre-trained CNN and its training images without part annotations. We use an active QA process to obtain a small number of annotations of object-part bounding boxes, which will be introduced in Section \[sec:QA\]. Based on these inputs, in this subsection, we focus on the approach for learning an AOG to represent the object part. ### And-Or graph representations Before the introduction of learning AOGs, we first briefly overview the structure of the AOG and the part parsing (inference) based on the AOG. As shown in Fig. \[fig:rawMapToModel\], an AOG represents the semantic structure of a part at four layers. Layer Name Node type ------- ---------------- --------------- 1 semantic part OR node 2 part template AND node 3 latent pattern OR node 4 neural unit Terminal node In the AOG, each OR node encodes a list of alternative appearance (or deformation) candidates as children. Each AND node uses its children to represent its constituent regions. More specifically, the top node is an OR node, which represents a certain semantic part, *e.g.* the head or the tail. The semantic part node encodes some part templates as children. Each part template corresponds to a specific part appearance from a certain perspective. During the inference process, the semantic part (an OR node) selects the best part template among all template candidates to represent the object. The part template in the second layer is an AND node, which uses its children latent patterns to represent a constituent region or a contextual region *w.r.t.* the part template. The part template encodes spatial relationships between its children. The latent pattern in the third layer is an OR node, whose receptive field is a square block within the feature map of a specific convolutional filter. The latent pattern takes neural units inside its receptive field as children. Because the latent pattern may appear at different locations in the feature map, the latent pattern uses these neural units to represent its deformation candidates. During the inference process, the latent pattern selects the strongest activated child unit as its deformation configuration. Given an image $I$[^1], we use the CNN to compute feature maps of all conv-layers on image $I$. Then, we can use the AOG for hierarchical part parsing. *I.e.* we use the AOG to semanticize the feature maps and localize the target part and its constituent regions in different layers. The parsing result is illustrated as red lines in Fig. \[fig:rawMapToModel\]. From a top-down perspective, the parsing procedure 1) identifies a part template for the semantic part; 2) parses an image region for the selected part template; 3) for each latent pattern under the part template, it selects a neural unit within a specific deformation range to represent this pattern. **OR nodes:** Both the top semantic-part node and latent-pattern nodes in the third layer are OR nodes. The parsing process assigns each OR node $u$ with an image region $\Lambda_{u}$ and an inference score $S_{u}$. $S_{u}$ measures the fitness between the parsed region $\Lambda_{u}$ and the sub-AOG under $u$. The computation of $\Lambda_{u}$ and $S_{u}$ for all OR nodes shares the same paradigm. $$S_{u}=\max_{v\in Child(u)}S_{v},\qquad\Lambda_{u}=\Lambda_{\hat{v}}$$ where let $u$ have $m$ children nodes $Child(u)=\{v_{1},v_{2},\ldots,v_{m}\}$. $S_{v}$ denotes the inference score of the child $v$, and $\Lambda_{v}$ is referred to as the image region assigned to $v$. The OR node selects the child with the highest score $\hat{v}={\arg\!\max}_{v\in Child(u)}S_{v}$ as the true parsing configuration. Node $\hat{v}$ propagates its image region to the parent $u$. More specifically, we introduce detailed settings for different OR nodes. - The OR node of the top semantic part contains a list of alternative part templates. We use $top$ to denote the top node of the semantic part. The semantic part chooses a part template to describe each input image $I$. - The OR node of each latent pattern $u$ in the third layer naturally corresponds to a square deformation range within the feature map of a convolutional filter of a conv-layer. All neural units within the square are used as deformation candidates of the latent pattern. For simplification, we set a constant deformation range (with a center $\overline{{\bf p}}_{u}$ and a scale of $\frac{h}{3}\times\frac{w}{3}$ in the feature map where $h$ and $w$ ($h=w$) denote the height and width of the feature map) for each latent pattern. $\overline{{\bf p}}_{u}$ is a parameter that needs to be learned. Deformation ranges of different patterns in the same feature map may overlap. Given parsing configurations of children neural units as input, the latent pattern selects the child with the highest inference score as the true deformation configuration. **AND nodes:** Each part template is an AND node, which uses its children (latent patterns) to represent its constituent or contextual regions. We use $v$ and $Child(v)=\{u_{1},u_{2},\ldots,u_{m}\}$ to denote the part template and its children latent patterns. We learn the average displacement from $\Lambda_{u}$ to $\Lambda_{v}$ among different image, denoted by $\Delta{\bf p}_{u}$, as a parameter of the AOG. Given parsing results of children latent patterns, we use the image region of each child node $\Lambda_{u}$ to infer the region for the parent $v$ based on its spatial relationships. Just like a deformable part model, the parsing of $v$ can be given as $$S_{v}\!=\!\!\!\!\!\!\!\sum_{u\in Child(v)}\!\!\!\!\!\!\!\big[S_{u}\!+\!S^{\textrm{inf}}(\Lambda_{u}|\Lambda_{v})\big],\;\;\Lambda_{v}\!=\!f(\Lambda_{u_{1}},\ldots,\Lambda_{u_{m}})\!$$ where we use parsing results of children nodes to infer the parent part template $v$. $S^{\textrm{inf}}(\Lambda_{u}|\Lambda_{v})$ denotes the spatial compatibility between $\Lambda_{u}$ and $\Lambda_{v}$ *w.r.t.* their average displacement $\Delta{\bf p}_{u}$. Please see the appendix for details of $S^{\textrm{inf}}(\Lambda_{u}|\Lambda_{v})$. For the region parsing of the part template $v$, we need to estimate two terms, *i.e.* the center position ${\bf p}_{v}$ and the scale $scale_{v}$ of $\Lambda_{v}$. We learn a fixed scale for each part template, which will be introduced in Section \[sec:learnAOG\]. In this way, we can simply implement region parsing by computing the region position that maximizes the inference score ${\bf p}_{v}=f(\Lambda_{u_{1}},\Lambda_{u_{2}},\ldots,\Lambda_{u_{m}})={\arg\!\max}_{{\bf p}_{v}}S_{v}$. **Terminal nodes (neural units):** Each terminal node under a latent pattern represents a deformation candidate of the latent pattern. The terminal node has a fixed image region, *i.e.* we propagate the neural unit’s receptive field back to the image plane as its image region. We compute a neural unit’s inference score based on both its neural response value and its displacement *w.r.t.* its parent latent pattern. Please see the appendix for details. Based on the above node definitions, we can use the AOG to parse each given image $I$ by dynamic programming in a bottom-up manner. ### Learning And-Or graphs {#sec:learnAOG} The core of learning AOGs is to distinguish reliable latent patterns from noisy neural responses in conv-layers and select reliable latent patterns to construct the AOG. **Training data:**[` `]{} Let ${\bf I}^{\textrm{obj}}\subset{\bf I}$ denote the set of object images of a target category. During the active question-answering, we obtain bounding boxes of the target object part in a small number of images, ${\bf I}^{\textrm{ant}}\!=\!\{I_1,I_2,\ldots,I_{M}\}\subset{\bf I}^{\textrm{obj}}$ among all objects. The other images without part annotations are denoted by ${\bf I}^{\textrm{unant}}={\bf I}^{\textrm{obj}}\setminus{\bf I}^{\textrm{ant}}$. In addition, the question-answering process collects a number of part templates. Thus, for each image $I\in{\bf I}^{\textrm{ant}}$, we annotate $(\Lambda_{top}^{*},v^{*})$, where $\Lambda_{top}^{*}$ denotes the ground-truth bounding box of the part in $I$, and $v^{*}\in Child(top)$ specifies the ground-truth template for the part. **Which AOG parameters to learn:**[` `]{} We can use human annotations to define the first two layers of the AOG. If human annotators specify a total of $m$ different part templates during the annotation process, correspondingly, we can directly connect the top node with $m$ part templates as children. For each part template $v\in Child(top)$, we fix a constant scale for its region $\Lambda_{v}$. *I.e.* if there are $n$ ground-truth part boxes that are labeled for $v$, we compute the average scale among the $n$ part boxes as the constant scale $scale_{v}$. Thus, the key to AOG construction is to mine children latent patterns for each part template $v$. We need to mine latent patterns from a total of $K$ conv-layers. We select $n_{k}$ latent patterns from the $k$-th ($k=1,2,\ldots,K$) conv-layer, where $K$ and $\{n_{k}\}$ are hyper-parameters. Let each latent pattern $u$ in the $k$-th conv-layer correspond to a square deformation range, which is located in the $D_{u}$-th slice of the conv-layer’s feature map. $\overline{\bf p}_{u}$ denotes the center of the range. As analyzed in the appendix, we only need to estimate the parameters of $D_{u},\overline{\bf p}_{u}$ for $u$. **How to learn:**[` `]{} Just like the pattern pursuing in Fig. \[fig:rawMapToModel\], we mine the latent patterns by estimating their best locations $D_{u},\overline{\bf p}_{u}\in{\boldsymbol\theta}$ that maximize the following objective function, where ${\boldsymbol\theta}$ denotes the parameter set of the AOG. $$\begin{split} {Loss}^{\textrm{AOG}}=\mathbb{E}_{I\in{\bf I}^{\textrm{ant}}}\big[-S_{top}+L(\Lambda_{top},\Lambda_{top}^{*})\big]\qquad\\ +\lambda^{\textrm{unant}}\mathbb{E}_{I\in{\bf I}^{\textrm{obj}}}\big[-S^{\textrm{unant}}_{\textrm{AOG}}+L^{\textrm{unant}}({\boldsymbol\Lambda}_{\textrm{AOG}})\big] \end{split} \label{eqn:LossAOG}$$ First, let us focus on the first half of the equation, which learns from part annotations. $S_{top}$ and $L(\Lambda_{top},\Lambda_{top}^{*})$ denote the final inference score of the AOG on image $I$ and the loss of part localization, respectively. Given annotations $(\Lambda_{top}^{*},v^{*})$ on $I$, we get $$\begin{split} &S_{top}=\max_{v\in Child(top)}S_{v}\approx S_{v^{*}}\\ &L(\Lambda_{top},\Lambda_{top}^{*})=-\lambda_{v^{*}}\Vert{\bf p}_{top}-{\bf p}^{*}_{top}\Vert \end{split}$$ where we approximate the ground-truth part template $v^{*}$ as the selected part template. We ignore the small probability of the AOG assigning an annotated image with an incorrect part template to simplify the computation. The part-localization loss $L(\Lambda_{top},\Lambda_{top}^{*})$ measures the localization error between the parsed part region ${\bf p}_{top}$ and the ground truth ${\bf p}^{*}_{top}={\bf p}(\Lambda_{top}^{*})$. The second half of Equation (\[eqn:LossAOG\]) learns from objects without part annotations. $$\begin{split} S^{\textrm{unant}}_{\textrm{AOG}}&={\sum}_{u\in Child(v^{*})}S^{\textrm{unant}}_{u}\\ L^{\textrm{unant}}({\boldsymbol\Lambda}_{\textrm{AOG}})&={\sum}_{u\in Child(v^{*})}\lambda^{\textrm{close}}\Vert\Delta{\bf p}_{u}\Vert^2 \end{split} \label{sec:unsuper}$$ where the first term $S^{\textrm{unant}}_{\textrm{AOG}}$ denotes the inference score at the level of latent patterns without ground-truth annotations of object parts. Please see the appendix for the computation of $S^{\textrm{unant}}_{u}$. The second term $L^{\textrm{unant}}({\boldsymbol\Lambda}_{\textrm{AOG}})$ penalizes latent patterns that are far from their parent $v^{*}$. This loss encourages the assigned neural unit to be close to its parent latent pattern. We assume that 1) latent patterns that frequently appear among unannotated objects may potentially represent stable part appearance and should have higher priorities; and that 2) latent patterns spatially closer to their parent part templates are usually more reliable. When we set $\lambda_{v^{*}}$ to a constant $\lambda^{\textrm{inf}}\sum_{k=1}^{K}n_{k}$, we can transform the learning objective in Equation (\[eqn:LossAOG\]) as follows. $$\forall v\in Child(top), \quad\max_{{\boldsymbol\theta}_{v}}{\bf L}_{v},\quad {\bf L}_{v}\!=\!\!\!\!\!\!\!\!\sum_{u\in Child(v)}\!\!\!\!\!\!\!Score(u) \label{eqn:subAOG}$$ where [$Score(u)\!=\!\mathbb{E}_{I\in{\bf I}_{v}}[S_{u}+S^{\textrm{inf}}(\Lambda_{u}|\Lambda^{*}_{v})]$ $+\mathbb{E}_{I'\in{\bf I}^{\textrm{obj}}}$ $\lambda^{\textrm{unant}}[S^{\textrm{unant}}_{u}-\lambda^{\textrm{close}}\Vert\Delta{\bf p}_{u}\Vert^2]$]{}. ${\boldsymbol\theta}_{v}\subset{\boldsymbol\theta}$ denotes the parameters for the sub-AOG of the part template $v$. We use ${\bf I}_{v}\subset{\bf I}^{\textrm{ant}}$ to denote the subset of images that are annotated with $v$ as the ground-truth part template. **Learning the sub-AOG for each part template:**[` `]{} Based on Equation (\[eqn:subAOG\]), we can mine the sub-AOG for each part template $v$, which uses this template’s annotations on images $I\in{\bf I}_{v}\subset{\bf I}^{\textrm{ant}}$, as follows. 1\) We first enumerate all possible latent patterns corresponding to the $k$-th CNN conv-layer ($k=1,\ldots,K$), by sampling all pattern locations *w.r.t.* $D_{u}$ and $\overline{\bf p}_{u}$. 2\) Then, we sequentially compute $\Lambda_{u}$ and $Score(u)$ for each latent pattern. 3\) Finally, we sequentially select a total of $n_{k}$ latent patterns. In each step, we select $\hat{u}\!=\!{\arg\!\max}_{u\in Child(v)}\Delta{\bf L}_{v}$. *I.e.* we select latent patterns with top-ranked values of [$Score(u)$]{} as children of part template $v$. Learning via active question-answering {#sec:QA} -------------------------------------- We propose a new learning strategy, *i.e.* active QA, which is more efficient than conventional batch learning. The QA-based learning algorithm actively detects blind spots in feature representations of the model and ask questions for supervision. In general, blind spots in the AOG include 1) neural-activation patterns in the CNN that have not been encoded in the AOG and 2) inaccurate latent patterns in the AOG. The unmodeled neural patterns potentially reflect new part templates, while inaccurate latent patterns correspond to sub-optimized part templates. As an interpretable representation of object parts, the AOG can represent blind spots using linguistic description. We design five types of answers to project these blind spots onto semantic details of objects. Our method selects and asks a series of questions. We then collect answers from human users, in order to incrementally grow new AOG branches to explain new part templates and refine existing AOG branches of part templates. Our approach repeats the following QA process. As shown in Fig. \[fig:QA\], at first, we use the current AOG to localize object parts on all unannotated objects of a category. Based on localization results, the algorithm selects and asks about the object $I$, from which the AOG can obtain the most information gain. A question [$q\!=\!(I,\hat{v},\Lambda_{\hat{v}})$]{} requires people to determine whether our approach predicts the correct part template $\hat{v}$ and parses a correct region $\Lambda_{top}=\Lambda_{\hat{v}}$ for the part. Our method expects one of the following answers. **Answer 1:** the part detection is correct. **Answer 2:** the current AOG predicts the correct part template in the parse graph, but it does not accurately localize the part. **Answer 3:** neither the part template nor the part location is correctly estimated. **Answer 4:** the part belongs to a new part template. **Answer 5:** the target part does not appear in the image. In particular, in case of receiving Answers 2–4, our method will ask people to annotate the target part. In case of getting Answer 3, our method will require people to specify its part template and whether the object is flipped. Our method uses new part annotations to refine (for Answers 2–3) or create (for Answer 4) an AOG branch of the annotated part template based on Equation (\[eqn:LossAOG\]). ### Question ranking The core of the QA-based learning is to select a sequence of questions that reduce the uncertainty of part localization the most. Therefore, in this section, we design a loss function to measure the incompatibility between the AOG and real part appearances in object samples. Our approach predicts the potential gain (decrease of the loss) of asking about each object. Objects with large gains usually correspond to not well explained CNN neural activations. Note that annotating a part in an object may also help localize parts on other objects, thereby leading to a large gain. Thus, we use a greedy strategy to select a sequence of questions [$\Omega=\{q_{i}|i=1,2,\ldots\}$]{}, *i.e.* asking about the object that produces the most gain in each step. For each object image $I$, we use [${\bf P}(y|I)$]{} and [${\bf Q}(y|I)$]{} to denote the prior distribution and the estimated distribution of an object part on $I$, respectively. A label [$y\in\{+1,-1\}$]{} indicates whether $I$ contains the target part. The AOG estimates the probability of object $I$ containing the target part as [${\bf Q}(y\!=\!+1|I)\!=\!\frac{1}{Z}\exp[\beta S_{top}]$]{}, where $Z$ and $\beta$ are parameters for scaling (see Section \[sec:implement\] for details); [${\bf Q}(y=-1|I)\!=\!1-{\bf Q}(y=+1|I)$]{}. Let [${\bf I}^{\textrm{ant}}$]{} denote the set of objects without being asked during previous QA. For each asked object [$I\in{\bf I}^{\textrm{ant}}$]{}, we set its prior distribution [${\bf P}(y=+1|I)=1$]{} if $I$ contains the target part; [${\bf P}(y=+1|I)=0$]{} otherwise. For each un-asked object [$I\in{\bf I}^{\textrm{unant}}$]{}, we set its prior distribution based on statistics of previous answers, [${\bf P}(y=+1|I)=\mathbb{E}_{I'\in{\bf I}^{\textrm{ant}}}{\bf P}(y=+1|I')$]{}. Therefore, we formulate the loss function as the KL divergence between the prior distribution [${\bf P}$]{} and the estimated distribution [${\bf Q}$]{}. $$\begin{split} \!\!{Loss}^{\textrm{QA}}\!\!\!=\!{\bf KL}({\bf P}\Vert{\bf Q})\!=\!&\sum_{I\in{\bf I}^{\textrm{obj}}}\sum_{y}{\bf P}(y,I)\log\frac{{\bf P}(y,I)}{{\bf Q}(y,I)}\!\!\!\!\!\!\!\\ =&\lambda\sum_{I\in{\bf I}^{\textrm{obj}}}\sum_{y}{\bf P}(y|I)\log\frac{{\bf P}(y|I)}{{\bf Q}(y|I)} \end{split}$$ where [${\bf P}(y,I)\!=\!{\bf P}(y|I)P(I)$; ${\bf Q}(y,I)\!=\!{\bf Q}(y|I)P(I)$; $\lambda=P(I)\!=\!1/\vert{\bf I}^{\textrm{obj}}\vert$]{} is a constant prior probability for $I$. We keep modifying both the prior distribution ${\bf P}$ and the estimated distribution ${\bf Q}$ during the QA process. Let the algorithm select an unannotated object [$\tilde{I}\in{\bf I}^{\textrm{unant}}={\bf I}^{\textrm{obj}}\setminus{\bf I}^{\textrm{ant}}$]{} and ask people to label its part. The annotation would encode part representations of $\tilde{I}$ into the AOG and significantly change the estimated distribution for objects that are similar to $\tilde{I}$. For each object $I'\in{\bf I}^{\textrm{obj}}$, we predict its estimated distribution after a new part annotation as $$\begin{split} \tilde{\bf Q}(y=+1|I')=&\frac{1}{Z}\exp[\beta S_{top,I'}^{\textrm{new}}|_{\tilde{I}}]\\ S_{top,I'}^{\textrm{new}}|_{\tilde{I}}=&S_{top,I'}+\Delta S_{top,\tilde{I}}e^{-\alpha\cdot dist(I',\tilde{I})}\!\!\!\!\!\!\!\!\!\! \end{split} \label{eqn:predict}$$ where $S_{top,I'}$ indicates the current AOG’s inference score of $S_{top}$ on image $I'$. $S_{top,I'}^{\textrm{new}}|_{\tilde{I}}$ denotes the predicted inference score of $I'$ when people annotate $\tilde{I}$. We assume that if object $I'$ is similar to object $\tilde{I}$, the inference score of $I'$ will have an increase similar to that of $\tilde{I}$. [$\Delta S_{top,\tilde{I}}\!=\!\mathbb{E}_{I\in{\bf I}^{\textrm{ant}}}S_{top,I}-S_{top,\tilde{I}}$]{} denotes the score increase of $\tilde{I}$. $\alpha$ is a scalar weight. We formulate the appearance distance between $I'$ and $\tilde{I}$ as [$dist(I',\tilde{I})\!=\!1-\frac{\phi(I')^{T}\phi(\tilde{I})}{\vert\phi(I')\vert\cdot\vert\phi(\tilde{I})\vert}$]{}, where [$\phi(I')\!=\!{\bf M}\,{\bf f}_{I'}$]{}. ${\bf f}_{I'}$ denotes features of $I'$ at the top conv-layer after ReLU operation, and ${\bf M}$ is a diagonal matrix representing the prior reliability for each feature dimension[^2]. In addition, if $I'$ and $\tilde{I}$ are assigned with different part templates by the current AOG, we set an infinite distance between $I'$ and $\tilde{I}$ to achieve better performance. Based on Equation (\[eqn:predict\]), we can predict the changes of the KL divergence after the new annotation on $\tilde{I}$ as $$\Delta{\bf KL}(\tilde{I})=\lambda{\sum}_{I\in{\bf I}^{\textrm{obj}}}{\sum}_{y}{\bf P}(y|I)\log\frac{\tilde{\bf Q}(y|I)}{{\bf Q}(y|I)} \label{eqn:delta}$$ Thus, in each step, our method selects and asks about the object that decreases the KL divergence the most. $$\hat{I}={\arg\!\max}_{I\in{\bf I}^{\textrm{unant}}}\Delta{\bf KL}(I) \label{eqn:select}$$ **QA implementations:**[` `]{} In the beginning, for each object $I$, we initialize [${\bf P}(y\!=\!+1|I)\!=\!1$]{} and [${\bf Q}(y\!=\!+1|I)\!=\!0$]{}. Then, our approach selects and asks about an object $\hat{I}$ based on Equation (\[eqn:select\]). We use the answer to update ${\bf P}$. If a new object part is labeled during the QA process, we apply Equation (\[eqn:LossAOG\]) to update the AOG. More specifically, if people label a new part template, our method will grow a new AOG branch to encode this template. If people annotate a part for an old part template, our method will update its corresponding AOG branch. Then, we compute the new distribution [${\bf Q}$]{} based on the new AOG. In this way, the above QA procedure gradually grows the AOG. Experiments =========== Implementation details {#sec:implement} ---------------------- We used a 16-layer VGG network (VGG-16) [@VGG], which was pre-trained for object classification using 1.3M images in the ImageNet ILSVRC 2012 dataset [@ImageNet]. Then, for each testing category, we further fine-tune the VGG-16 using object images in this category to classify target objects from random images. We selected the last nine conv-layers of VGG-16 as valid conv-layers. We extracted neural units from these conv-layers to build the AOG. **Active question-answering:** Three parameters were involved in our active-QA method, *i.e.* $\alpha$, $\beta$, and $Z$. Because most objects of the category contained the target part, we ignored the small probability of ${\bf P}(y=-1|I)$ in Equation (\[eqn:delta\]) to simplify the computation. As a result, $Z$ was eliminated in Equation (\[eqn:delta\]), and the constant weight $\beta$ did not affect object-selection results in Equation (\[eqn:select\]). We set $\alpha=4.0$ in our experiments. **Learning AOGs:** Multiple latent patterns corresponding to the same convolutional filter may have similar positions $\overline{\bf p}_{u}$, and their deformation ranges may highly overlap. Thus, we selected the latent pattern with the highest $Score(u)$ within a small range of $\epsilon\times\epsilon$ in the filter’s feature map and removed other nearby patterns to obtain a spare AOG. Besides, for each part template $v$, we estimated $n_{k}$ latent patterns in the $k$-th conv-layer. We assumed that scores of all latent patterns in the $k$-th conv-layer follow the distribution of [$Score(u)\sim\alpha\exp[-(\xi\cdot{rank})^{0.5}]+\gamma$]{}, where $rank$ denotes the score rank of [$u$]{}. We set [$n_{k}=\lceil0.5/\xi\rceil$]{}, which learned the best AOG. Datasets -------- Because evaluation of part localization requires ground-truth annotations of part positions, we used the following three benchmark datasets to test our method, *i.e.* the PASCAL VOC Part Dataset [@SemanticPart], the CUB200-2011 dataset [@CUB200], and the ILSVRC 2013 DET Animal-Part dataset [@CNNAoG]. Just like in [@SemanticPart; @CNNAoG], we selected animal categories, which prevalently contain non-rigid shape deformation, for testing. *I.e.* we selected six animal categories—*bird, cat, cow, dog, horse*, and *sheep*—from the PASCAL Part Dataset. The CUB200-2011 dataset contains 11.8K images of 200 bird species. We followed [@ActivePart; @CNNSemanticPart; @CNNAoG] and used all these images as a single bird category for learning. The ILSVRC 2013 DET Animal-Part dataset [@CNNAoG] contains part annotations of 30 animal categories among all the 200 categories in the ILSVRC 2013 DET dataset [@ImageNet]. {width="\linewidth"} The 2nd column shows the number of part annotations for training. The 3rd column indicates whether the baseline used all object-box annotations in the category to pre-fine-tune a CNN before learning the part (*object-box annotations are more than part annotations*). Baselines --------- We used the following thirteen baselines for comparison. The first two baselines were based on the Fast-RCNN [@FastRCNN]. We fine-tuned the fast-RCNN with a loss of detecting a single class/part for a fair comparison. The first baseline, namely *Fast-RCNN (1 ft)*, fine-tuned the VGG-16 using part annotations to detect parts on well-cropped objects. To enable a more fair comparison, we conducted the second baseline based on two-stage fine-tuning, namely *Fast-RCNN (2 fts)*. This baseline first fine-tuned the VGG-16 using numerous object-box annotations in the target category, and then fine-tuned the VGG-16 using a few part annotations. The third baseline was proposed in [@CNNSemanticPart], namely *CNN-PDD*. *CNN-PDD* selected a filter in a CNN (pre-trained using ImageNet ILSVRC 2012 dataset) to represent the part on well-cropped objects. Then, we slightly extended [@CNNSemanticPart] as the fourth baseline *CNN-PDD-ft*. *CNN-PDD-ft* first fine-tuned the VGG-16 using object bounding boxes, and then applied [@CNNSemanticPart] to learn object parts. The strongly supervised DPM (*SS-DPM-Part*) [@SSDPM] and the approach of [@PLDPM] (*PL-DPM-Part*) were the fifth and sixth baselines. These methods learned DPMs for part localization. The graphical model proposed in [@SemanticPart] was selected as the seventh baseline, namely *Part-Graph*. The eighth baseline was the interactive learning for part localization [@ActivePart] (*Interactive-DPM*). Without lots of training samples, “simple” methods are usually insensitive to the over-fitting problem. Thus, we designed the last four baselines as follows. We first fine-tuned the VGG-16 using object bounding boxes, and collected image patches from cropped objects based on the selective search [@SelectiveSearch]. We used the VGG-16 to extract *fc7* features from image patches. The two baselines (*i.e.* *fc7+linearSVM* and *fc7+RBF-SVM*) used a linear SVM and an RBF-SVM, respectively, to detect object parts. The other baselines *VAE+linearSVM* and *CoopNet+linearSVM* used features of the VAE network [@VAE] and the CoopNet [@CoopNet], respectively, instead of *fc7* features, for part detection. The last baseline [@CNNAoG] learned AOGs without QA (*AOG w/o QA*). We randomly selected objects and annotated their parts for training. Both object annotations and part annotations are used to learn models in all the thirteen baselines (including those without fine-tuning). *Fast-RCNN (1 ft)* and *CNN-PDD* used the cropped objects as the input of the CNN; *SS-DPM-Part*, *PL-DPM-Part*, *Part-Graph*, and *Interactive-DPM* used object boxes and part boxes to learn models. *CNN-PDD-ft*, *Fast-RCNN (2 fts)*, and methods based on *fc7* features used object bounding boxes for fine-tuning. Evaluation metric ----------------- As discussed in [@SemanticPart; @CNNAoG], a fair evaluation of part localization requires removing factors of object detection. Thus, we used ground-truth object bounding boxes to crop objects as testing images. Given an object image, some competing methods (*e.g.* *Fast-RCNN (1 ft)*, *Part-Graph*, and *SS-DPM-Part*) estimate several bounding boxes for the part with different confidences. We followed [@CNNSemanticPart; @SemanticPart; @ObjectDiscoveryCNN_1; @CNNAoG] to take the most confident bounding box per image as the part-localization result. Given part-localization results of a category, we applied the *normalized distance* [@CNNSemanticPart] and the *percentage of correctly localized parts* (PCP) [@fineGrained1; @fineGrained2; @fineGrained3] to evaluate the localization accuracy. We measured the distance between the predicted part center and the ground-truth part center, and then normalized the distance using the diagonal length of the object as the normalized distance. For the PCP, we used the typical metric of “$IoU\geq0.5$” [@FastRCNN] to identify correct part localizations. See Table \[tab:imgnet\] for the introduction of the 2nd and 3rd columns. The 4th column shows the number of questions for training. The 4th column indicates whether the baseline used all object annotations (*more than part annotations*) in the category to pre-fine-tune a CNN before learning the part. The 3rd and 4th columns show the number of part annotations and the average number of questions for training. Experimental results -------------------- We learned AOGs for the head, the neck, and the nose/muzzle/beak parts of the six animal categories in the Pascal VOC Part dataset. For the ILSVRC 2013 DET Animal-Part dataset and the CUB200-2011 dataset, we learned an AOG for the head part[^3] of each category. It is because all categories in the two datasets contain the head part. We did not train human annotators. Shape differences between two part templates were often very vague, so that an annotator could assign a part to either part template. {width="\linewidth"} {width="\linewidth"} ![Image patches corresponding to different latent patterns.[]{data-label="fig:patches"}](patches.pdf){width="0.8\linewidth"} Table \[tab:stat\] shows how the AOG grew when people annotated more parts during the QA process. Given AOGs learned for the PASCAL VOC Part dataset, we computed the average number of children for each node in different AOG layers. The AOG mainly grew by adding new branches to represent new part templates. The refinement of an existing AOG branch did not significantly change the node number of the AOG. Fig. \[fig:energyCurve\] analyzes activation states of latent patterns in AOGs that were learned with different numbers of part annotations. Given a testing image $I$ for part parsing, we only focused on the inferred latent patterns and neural units, *i.e.* latent patterns and their inferred neural units under the selected part template. Let ${\bf V}$ and ${\bf V'}\subset{\bf V}$ denote all units in a specific conv-layer and the inferred units, respectively. $a_{v}$ denotes the activation score of $v\in{\bf V}$ after the ReLU operation. $a_{v}$ is also normalized by the average activation level of $v$’s corresponding feature maps *w.r.t.* different images. Thus, in Fig. \[fig:energyCurve\](left), we computed the ratio of the inferred activation energy as $\frac{\sum_{v\in{\bf V'}}a_{v}}{\sum_{v\in{\bf V}}a_{v}}$. For each inferred latent pattern $u$, $a_{u}$ denotes the activation score of its selected neural unit[^4]. Fig. \[fig:energyCurve\](middle) measures the relative magnitude of the inferred activations, which was measured as $\frac{\mathbb{E}_{u\in{\bf U}}[a_{u}]}{\mathbb{E}_{v\in{\bf V}}[a_{v}]}$. Fig. \[fig:energyCurve\](right) shows the ratio of the latent patterns being strongly activated. We used a threshold $\tau=\mathbb{E}_{v\in{\bf V}}[a_{v}]$ to identify strong activations, *i.e.* computing the activation ratio as $\mathbb{E}_{u\in{\bf U}}[{\bf 1}(a_{u}>\tau)]$. Curves in Fig. \[fig:energyCurve\] were reported as the average performance using images in the CUB200-2011 dataset. Fig. \[fig:visualization\] visualizes latent patterns in the AOG based on the technique of [@FeaVisual]. More specifically, Fig. \[fig:patches\] lists images patches inferred by different latent patterns in the AOG with high inference scores. It shows that each latent pattern corresponds to a specific part shape through different images. Fig. \[fig:results\] shows part localization results based on AOGs. Tables \[tab:imgnet\], \[tab:VOC\], and \[tab:cub200\] compare the part-localization performance of different baselines on different benchmark datasets using the evaluation metric of the normalized distance. Tables \[tab:VOC\], and \[tab:cub200\] show both the number of part annotations and the number of questions. Fig. \[fig:curve\] shows the performance of localizing the head part on objects in the PASCAL VOC Part Dataset, when people annotated different numbers of parts for training. Table \[tab:pcp\] lists part-localization performance, which was evaluated by the PCP metric. In particular, the method of *Ours+fastRCNN* combined our method and the fast-RCNN to refine part-localization results[^5]. Our method learned AOGs with about $1/6$–$1/2$ part annotations, but exhibited superior performance to the second best baseline. ![Part localization performance on the Pascal VOC Part dataset.[]{data-label="fig:curve"}](curve.pdf){width="\linewidth"} Justification of the methodology -------------------------------- We have three reasons to explain the good performance of our method. First, **generic information**: the latent patterns in the AOG were pre-fine-tuned using massive object images in a category, instead of being learned from a few part annotations. Thus, these patterns reflected generic part appearances and did not over-fit to a few part annotations. Second, **less model drifts:** Instead of learning new CNN parameters, our method just used limited part annotations to mine the related patterns to represent the part concept. In addition, during active QA, Equation (\[eqn:predict\]) usually selected objects with common poses for QA, *i.e.* choosing objects sharing common latent patterns with many other objects. Thus, the learned AOG suffered less from the model-drift problem. Third, **high QA efficiency:** Our QA process balanced both the commonness and the accuracy of a part template in Equation (\[eqn:predict\]). In early steps of QA, our approach was prone to asking about new part templates, because objects with un-modeled part appearance usually had low inference scores. In later QA steps, common part appearances had been modeled, and our method gradually changed to ask about objects belonging to existing part templates to refine the AOG. Our method did not waste much labor of labeling objects that had been well modeled or had strange appearance. Summary and discussion ====================== In this paper, we have proposed a method to bridge and solve the following three crucial issues in computer vision simultaneously. - Removing noisy representations in conv-layers of a CNN and using an AOG model to reveal the semantic hierarchy of objects hidden in the CNN. - Enabling people to communicate with neural representations in intermediate conv-layers of a CNN directly for model learning, based on the semantic representation of the AOG. - Weakly-supervised transferring of object-part representations from a pre-trained CNN to model object parts at the semantic level, which boosts the learning efficiency. Our method incrementally mines object-part patterns from conv-layers of a pre-trained CNN and uses an AOG to encode the mined semantic hierarchy. The AOG semanticizes neural units in intermediate feature maps of a CNN by associating these units with semantic parts. We have proposed an active QA strategy to learn such an AOG model in a weakly-supervised manner. We have tested the proposed method for a total of 37 categories in three benchmark datasets. Our method has outperformed other baselines in the application of weakly-supervised part localization. For example, our method with 11 part annotations performed better than fast-RCNN with 60 part annotations on the ILSVRC dataset in Fig. \[fig:curve\]. Acknowledgments {#acknowledgments .unnumbered} =============== Acknowledgment {#acknowledgment .unnumbered} ============== This work is supported by ONR MURI project N00014-16-1-2007, DARPA XAI Award N66001-17-2-4029, and NSF IIS 1423305. [Quanshi Zhang]{} received the B.S. degree in machine intelligence from Peking University, China, in 2009 and M.S. and Ph.D. degrees in center for spatial information science from the University of Tokyo, Japan, in 2011 and 2014, respectively. In 2014, he went to the University of California, Los Angeles, as a post-doctoral associate. Now, he is an associate professor at the Shanghai Jiao Tong University. His research interests include computer vision, machine learning, and robotics. [Ruiming Cao]{} received the B.S. degree in computer science from the University of California, Los Angeles, in 2017. Now, he is a master student at the University of California, Los Angeles. His research mainly focuses on computer vision. [Ying Nian Wu]{} received a Ph.D. degree from the Harvard University in 1996. He was an Assistant Professor at the University of Michigan between 1997 and 1999 and an Assistant Professor at the University of California, Los Angeles between 1999 and 2001. He became an Associate Professor at the University of California, Los Angeles in 2001. From 2006 to now, he is a professor at the University of California, Los Angeles. His research interests include statistics, machine learning, and computer vision. [Song-Chun Zhu]{} Song-Chun Zhu received a Ph.D. degree from Harvard University, and is a professor with the Department of Statistics and the Department of Computer Science at UCLA. His research interests include computer vision, statistical modeling and learning, cognition and AI, and visual arts. He received a number of honors, including the Marr Prize in 2003 with Z. Tu et. al. on image parsing,the Aggarwal prize from the Int’l Association of Pattern Recognition in 2008, twice Marr Prize honorary nominations in 1999 for texture modeling and 2007 for object modeling with Y.N. Wu et al., a Sloan Fellowship in 2001, the US NSF Career Award in 2001, and the US ONR Young Investigator Award in 2001. He is a Fellow of IEEE. And-Or graph representations {#and-or-graph-representations-1 .unnumbered} ============================ Parameters for latent patterns {#parameters-for-latent-patterns .unnumbered} ------------------------------ We use the notation of ${\bf p}_{u}$ to denote the central position of an image region $\Lambda_{u}$. For simplification, all position variables ${\bf p}_{u}$ are measured based on the image coordinates by propagating the position of $\Lambda_{u}$ to the image plane. Each latent pattern $u$ is defined by its location parameters $\{L_{u},D_{u},\overline{\bf p}_{u},\Delta{\bf p}_{u}\}\subset{\boldsymbol\theta}$, where ${\boldsymbol\theta}$ is the set of AOG parameters. It means that a latent pattern $u$ uses a square within the $D_{u}$-th channel of the $L_{u}$-th conv-layer’s feature map as its deformation range. The center position of the square is given as $\overline{\bf p}_{u}$. When latent pattern $u$ is extracted from the $k$-th conv-layer, $u$ has a fixed value of $L_{u}=k$. $\Delta{\bf p}_{u}$ denotes the average displacement from $u$ and $u$’s parent part template $v$ among various images, and $\Delta{\bf p}_{u}$ is used to compute $S^{\textrm{inf}}(\Lambda_{u}|\Lambda_{v})$. Given parameter $\overline{\bf p}_{u}$, the displacement $\Delta{\bf p}_{u}$ can be estimated as $$\Delta{\bf p}_{u}=\overline{\bf p}^{*}_{v}-\overline{\bf p}_{u}\nonumber$$ where $\overline{\bf p}^{*}_{v}$ denotes the average position of all ground-truth parts that are annotated for part template $v$. As a result, for each latent pattern $u$, we only need to learn its channel $D_{u}\in{\boldsymbol\theta}$ and central position $\overline{\bf p}_{u}\in{\boldsymbol\theta}$. Scores of terminal nodes {#scores-of-terminal-nodes .unnumbered} ------------------------ The inference score for each terminal node $v^{\textrm{unt}}$ under a latent pattern $u$ is formulated as $$\begin{aligned} &S_{v^{\textrm{unt}}}=S_{v^{\textrm{unt}}}^{\textrm{rsp}}+S_{v^{\textrm{unt}}}^{\textrm{loc}}+S_{v^{\textrm{unt}}}^{\textrm{pair}}\nonumber\\ &S_{v^{\textrm{unt}}}^{\textrm{rsp}}=\left\{\begin{array}{ll}\lambda^{\textrm{rsp}}X(v^{\textrm{unt}}),& X(v^{\textrm{unt}})>0\\ \lambda^{\textrm{rsp}}S_{none},& X(v^{\textrm{unt}})\leq0\end{array}\right.\nonumber\\ &S_{v^{\textrm{unt}}}^{\textrm{pair}}=-\lambda^{\textrm{pair}}\!\!\!\!\!\!\!\!\underset{u_{\textrm{upper}}\in\!\textrm{Neighbor}(u)}{\mathbb{E}}\!\!\!\!\!\!\Vert[{\bf p}_{v^{\textrm{unt}}}-{\bf p}_{u_{\textrm{upper}}}]-[\overline{\bf p}_{u_{\textrm{upper}}}-\overline{\bf p}_{u}]\Vert\nonumber\end{aligned}$$ The score of $S_{v^{\textrm{unt}}}$ consists of the following three terms: 1) $S_{v^{\textrm{unt}}}^{\textrm{rsp}}$ denotes the response value of the unit $v^{\textrm{unt}}$, when we input image $I$ into the CNN. $X(v^{\textrm{unt}})$ denotes the normalized response value of $v^{\textrm{unt}}$; $S_{none}=-3$ is set for non-activated units. 2) When the parent $u$ selects $v^{\textrm{unt}}$ as its location inference (*i.e.* $\Lambda_{u}\leftarrow\Lambda_{v^{\textrm{unt}}}$), $S_{v^{\textrm{unt}}}^{\textrm{loc}}$ measures the deformation level between $v^{\textrm{unt}}$’s location ${\bf p}_{v^{\textrm{unt}}}$ and $u$’s ideal location $\overline{\bf p}_{u}$. 3) $S_{v^{\textrm{unt}}}^{\textrm{pair}}$ indicates the spatial compatibility between neighboring latent patterns: we model the pairwise spatial relationship between latent patterns in the upper conv-layer and those in the current conv-layer. For each $v^{\textrm{unt}}$ (with its parent $u$) in conv-layer $L_{u}$, we select 15 nearest latent patterns in conv-layer $L_{u}+1$, *w.r.t.* $\Vert\overline{\bf p}_{u}-\overline{\bf p}_{u_{\textrm{upper}}}\Vert$, as the neighboring latent patterns. We set constant weights $\lambda^{\textrm{rsp}}=1.5,\lambda^{\textrm{loc}}=1/3,\lambda^{\textrm{pair}}=10.0$, $\lambda^{\textrm{unant}}=5.0$, and $\lambda^{\textrm{close}}=0.4$ for all categories. Based on the above design, we first infer latent patterns corresponding to high conv-layers, and use the inference results to select units in low conv-layers. During the learning of AOGs, we define $S^{\textrm{unant}}_{u}=S_{\hat{v}^{\textrm{unt}}}^{\textrm{rsp}}+S_{\hat{v}^{\textrm{unt}}}^{\textrm{loc}}$ to measure the latent-pattern-level inference score in Equation (5), where $\hat{v}^{\textrm{unt}}$ denotes the neural unit assigned to $u$. Scores of AND nodes {#scores-of-and-nodes .unnumbered} ------------------- $$S^{\textrm{inf}}(\Lambda_{u}|\Lambda_{v})=-\lambda^{\textrm{inf}}\min\{\Vert{\bf p}(\Lambda_{u})+\Delta{\bf p}_{u}-{\bf p}(\Lambda_{v})\Vert^2,d^2\}\nonumber$$ where we set $d=37$ pixels and $\lambda^{\textrm{inf}}=5.0$. [^1]: Because the CNN has demonstrated its superior performance in object detection, we assume that the target object can be well detected by the pre-trained CNN. As in [@SemanticPart], we regard object detection and part localization as two separate processes for evaluation. Thus, to simplify the learning scenario, we crop $I$ only to contain the object, resize it to the image size for CNN inputs, and just focus on the part localization task to simplify the scenario of learning for part localization. [^2]: ${\bf M}_{ii}\!\propto\!\exp[\mathbb{E}_{I\in{\bf I}}S_{v^{\textrm{unt}}_{i}}]$, where $v^{\textrm{unt}}_{i}$ is the neural unit corresponding to the $i$-th element of ${\bf f}_{I'}$. [^3]: It is the “forehead” part for birds in the CUB200-2011 dataset. [^4]: Two latent patterns may select the same neural unit [^5]: We used part boxes annotated during the QA process to learn a fast-RCNN for part detection. Given the inference result $\Lambda_{v}$ of part template $v$ on image $I$, we define a new inference score for localization refinement $S_{v}^{\textrm{new}}(\Lambda_{v}^{\textrm{new}})=S_{v}+\lambda_1\Phi(\Lambda^{\textrm{new}}_{v})+\lambda_2\frac{\Vert{\bf p}_{v}-{\bf p}_{v}^{\textrm{new}}\Vert}{2\sigma^2}$, where $\sigma=70$ pixels, $\lambda_1=5$, and $\lambda_2=10$. $\Phi(\Lambda^{\textrm{new}}_{v})$ denotes the fast-RCNN’s detection score for the patch of $\Lambda^{\textrm{new}}_{v}$.

--- abstract: 'Multileaved comparison methods generalize interleaved comparison methods to provide a scalable approach for comparing ranking systems based on regular user interactions. Such methods enable the increasingly rapid research and development of search engines. However, existing multileaved comparison methods that provide reliable outcomes do so by degrading the user experience during evaluation. Conversely, current multileaved comparison methods that maintain the user experience cannot guarantee correctness. Our contribution is two-fold. First, we propose a theoretical framework for systematically comparing multileaved comparison methods using the notions of *considerateness*, which concerns maintaining the user experience, and *fidelity*, which concerns reliable correct outcomes. Second, we introduce a novel multileaved comparison method, , that performs comparisons based on document-pair preferences, and prove that it is *considerate* and has *fidelity*. We show empirically that, compared to previous multileaved comparison methods, is more *sensitive* to user preferences and *scalable* with the number of rankers being compared.' author: - Harrie Oosterhuis - Maarten de Rijke bibliography: - 'cikm2017-lambda-multileave.bib' title: | Sensitive and Scalable Online Evaluation\ with Theoretical Guarantees --- [1]{} **Acknowledgments.** This research was supported by Ahold Delhaize, Amsterdam Data Science, the Bloomberg Research Grant program, the Criteo Faculty Research Award program, the Dutch national program COMMIT, Elsevier, the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement nr 312827 (VOX-Pol), the Microsoft Research Ph.D. program, the Netherlands Institute for Sound and Vision, the Netherlands Organisation for Scientific Research (NWO) under project nrs 612.001.116, HOR-11-10, CI-14-25, 652.002.001, 612.001.551, 652.001.003, and Yandex. All content represents the opinion of the authors, which is not necessarily shared or endorsed by their respective employers and/or sponsors.

--- abstract: 'We introduce the notion of homotopy inner products for any cyclic quadratic Koszul operad $\mathcal O$, generalizing the construction already known for the associative operad. This is done by defining a colored operad $\widehat{\mathcal O}$, which describes modules over $\mathcal O$ with invariant inner products. We show that $\widehat{\mathcal O}$ satisfies Koszulness and identify algebras over a resolution of $\widehat{\mathcal O}$ in terms of derivations and module maps. As an application we construct a homotopy inner product over the commutative operad on the cochains of any Poincaré duality space.' address: - 'Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, Piazzale Aldo Moro, 2 I-00185 Roma, Italy' - 'College of Technology of the City University of New York, Department of Mathematics, 300 Jay Street, Brooklyn, NY 11201, USA' author: - Riccardo Longoni - Thomas Tradler title: Homotopy Inner Products for Cyclic Operads --- Introduction ============ In [@GeK], the notion of cyclic operads and invariant inner product for such operads was defined. A homotopy version of these inner products for the associative operad was given in [@Tr] and the starting point for a similar version for the commutative operad was considered in [@Ginot]. It is natural to ask for a generalization of these constructions applicable to any cyclic operad. This is what is done in this paper. Starting with a cyclic operad $\mathcal O$, we use the notion of colored operads to incorporate the cyclic structure of $\mathcal O$ into the colored operad $\widehat{\mathcal O}$. Algebras over the colored operad $\widehat{\mathcal O}$ consist of pairs $(A,M)$, where $A$ is an algebra over $\mathcal O$ and $M$ is an $\mathcal O$-module over $A$ which has an invariant inner product. Section \[cyclic-op\] is devoted to explicitly defining $\widehat{\mathcal O}$, and in the case that $\mathcal O$ is given by quadratic generators and relations, we give a description of $\widehat{\mathcal O}$ in terms of generators and relations coming from those of $\mathcal O$. A major tool in the theory of operads is the notion of Koszul duality. Let us recall, that a (colored) operad $\mathcal P$ is called Koszul if there is a quasi-isomorphism of operads $\mathbf{D}(\mathcal P^!)\to \mathcal P$, where $\mathbf{D}(\mathcal P^!)$ denotes the dual operad (in the sense of [@GK (3.2.12)]) on the dual quadratic operad $\mathcal P^!$. This implies that the notion of algebras of $\mathcal P$ has a canonical infinity version given by algebras over $\mathcal P_\infty:=\mathbf{D}(\mathcal P^!)$. Our main theorem states, that the Koszulness property is preserved when going from $\mathcal O$ to $\widehat{\mathcal O}$. Let $\mathcal O$ be a cyclic quadratic operad. If $\mathcal O$ is Koszul, then so is $\widehat{\mathcal O}$, i.e. we have a resolution $\widehat{\mathcal O}_\infty:=\mathbf{D}(\widehat{\mathcal O^!})$ of $\widehat{\mathcal O}$. The proof of this theorem will be given in section \[quadrat-koszul\]. Theorem \[O\_hat\_Koszul\] justifies the concept of the infinity version of algebras and modules with invariant inner products over cyclic operads $\mathcal O$ as algebras over the operad $\widehat{\mathcal O}_\infty=\mathbf{D}(\widehat{\mathcal O^!})$. The concept of algebras over the operad $\widehat{\mathcal O}_\infty$ will be investigated in more detail in section \[homotop-ip\]. In particular, we explicitly reinterpret in proposition \[O\_hat\_algebras\] algebras over $\widehat{\mathcal O}_\infty$ in terms of derivations and module maps over free $\mathcal O^!$-algebras and modules. Recall that the associative operad $\mathcal Assoc$, the commutative operad $\mathcal Comm$ and the Lie operad $\mathcal Lie$ are all cyclic quadratic Koszul operads, so that theorem \[O\_hat\_Koszul\] may be applied to all these cases. As a particularly application of infinity inner products, we consider the examples of the associative operad $\mathcal Assoc$ and the commutative operad $\mathcal Comm$, which have an interesting application to the chain level of a Poincaré duality space $X$. In [@TZ], it was shown that the simplicial cochains $C^\ast(X)$ on $X$ with rational coefficients form an algebra over the operad $\widehat{\mathcal Assoc}_\infty$. This structure was then used in [@Tr2] and [@TZ2] to obtain string topology operations on the Hochschild cohomology, respectively the Hochschild cochain complex, of $C^\ast(X)$. Since the method of constructing the $\widehat{\mathcal Assoc}_\infty$ algebra on $C^\ast (X)$ easily transfers to the commutative case, we will show in section \[Comm-section\] that $C^\ast(X)$ also forms a $\widehat{\mathcal Comm}_\infty$-algebra. We expect that this stronger algebraic structure should induce even more string topology operations, which take into account the commutative nature of the cochains of the space $X$. A first step in this direction was done in [@TZ3], where the string topology operations for algebras over $\widehat{\mathcal Assoc}$ and $\widehat{\mathcal Comm}$ were investigated. We are grateful to Dennis Sullivan for many valuable suggestions and illuminating discussions. We also thank Domenico Fiorenza, Martin Markl, Jim Stasheff and Scott Wilson for useful comments and remarks regarding this topic. The second author was partially supported by the Max-Planck Institute in Bonn. $\widehat{\mathcal Comm}_\infty$ structure for Poincaré duality spaces {#Comm-section} ====================================================================== Before going into the details of the construction of homotopy inner products over a general cyclic quadratic operad $\mathcal O$, we give an application for the case of the commutative operad $\mathcal Comm$. More precisely, we show how a homotopy $\mathcal Comm$-inner product arises on the chain level of a Poincaré duality space $X$. In fact, the construction for the homotopy $\mathcal Comm$-algebra is taken from R. Lawrence and D. Sullivan’s paper [@S] on the construction of local infinity structures. In [@TZ], M. Zeinalian and the second author construct homotopy $\mathcal Assoc$-inner products on a Poincaré duality space $X$. The same reasoning may in fact be used to construct homotopy $\mathcal Comm$-inner products on $X$. The proof of the next proposition will be a sketch using these arguments. \[prop:comm-pd\] Let $X$ be a closed, finitely triangulated Poincaré duality space, such that the closure of every simplex is contractible. Denote by $C=C_\ast(X)$ the simplicial chains on $X$. Then its dual space $A:=C^*=Hom(C_*(X),k)$ has the structure of a $\widehat{\mathcal Comm}_\infty$ algebra, such that the lowest multiplication is the symmetrized Alexander-Whitney multiplication and the lowest inner product is given by capping with the fundamental cycle $\mu\in C$. Let $\mathcal Lie$ denote the Lie-operad, $F_{\mathcal Lie}V=\bigoplus_{n\geq 1} (\mathcal Lie(n)\otimes V^{\otimes n})_{S_n}$ denote the free Lie algebra generated by $V$, and $F_{\mathcal Lie,V}W=\bigoplus_{n\geq 1} (\bigoplus_{k+l=n-1}\mathcal Lie(n)\otimes V^{\otimes k}\otimes W\otimes V^{\otimes l})_{S_n}$ the canonical module over $F_{\mathcal Lie}V$. We will see in proposition \[O\_hat\_algebras\] and example \[exa-comm\], that the required data for a homotopy $\mathcal Comm$-inner product consists of, - a derivation $d\in \mathrm{Der}(F _{\mathcal Lie}\,C[1])$ of degree $1$, with $d^2=0$, - a derivation $g\in \mathrm{Der}_d (F_{\mathcal Lie,\,C[1]}C[1])$ over $d$ of degree $1$, with $g^2=0$, which imduces a derivation $h\in \mathrm{Der}_d (F_{\mathcal Lie,\,C[1]}C^*[1])$ over $d$ with $h^2=0$, - a module map $f\in \mathrm{Mod}(F_{\mathcal Lie, C[1]}C^*, F_{\mathcal Lie,C[1]}C[1])$ of degree $0$ such that $f\circ h = g \circ f$. In order to construct the derivation $d\in \mathrm{Der}(F _{\mathcal Lie}\,C[1])$ with $d^2=0$, let $F_{\mathcal Lie}C[1]=L_1\oplus L_2\oplus\dots$, where $L_n=(\mathcal Lie(n)\otimes C[1]^{\otimes n})_{S_n}$, be the decomposition of $F_{\mathcal Lie}C[1]$ by the monomial degree in $C[1]$. Then, $d:F_{\mathcal Lie}C[1]\to F_{\mathcal Lie}C[1]$ is determined by maps $d=d_1+d_2+\dots$, where $d_i:C[1]\to L_i$ is lifted to $F_{\mathcal Lie}(C[1])$ as a derivation. Let $d_1$ be the differential on $C[1]$, and $d_2$ be the symmetrized Alexander-Whitney comultiplication. For the general $d_i$, we use the inductive hypothesis that $d_1$, …, $d_{i-1}$ are local maps so that $\nabla_i:=d_1+ \dots+d_{i-1}$ has a square $\nabla_i^2$ mapping only into higher components $L_{i}\oplus L_{i+1}\oplus \dots$. Here, “local” means that every simplex maps into the sub-Lie algebra of its closure. Now, by the Jacobi-identity, it is true that $0=[\nabla_i, [\nabla_i,\nabla_i]]= [d_1,e_i]\text{+ higher terms}$, where $e_i:C[1]\to L_i$ is the lowest term of $[\nabla_i,\nabla_i]$. Thus $e_i$ is $[d_1,.]$-closed and thus, using the contractibility hypothesis of the proposition, also locally $[d_1,.]$-exact. These exact terms can be put together to give a map $d_i$, so that $[d_1,d_i]$ vanishes on $L_1\oplus \dots \oplus L_{i-1}$ and equals $- 1/2\cdot e_i$ on $L_i$. In other words, $(d_1+ \dots+d_i)^2=1/2\cdot [d_1+ \dots+d_i,d_1+ \dots+d_i]=1/2\cdot [\nabla_i ,\nabla_i ]+ [d_1,d_i]+\text{higher terms}$, maps only into $L_{i+1}\oplus L_{i+2}\oplus\dots$. This completes the inductive step, and thus produces the wanted derivation $d$ on $F_{\mathcal Lie}(C[1])$. In a similar way, we may produce the derivation $g$ of $F_{\mathcal Lie,\,C[1]}C[1]$ over $d$, by decomposing $F_{\mathcal Lie, C[1]} C[1]=L'_1\oplus L'_2\oplus\dots$, where $L'_n$ is given by the space $\left(\bigoplus_{k+l=n-1} \mathcal Lie(n) \otimes C[1]^{\otimes k} \otimes C[1] \otimes C[1]^{\otimes l}\right)_{S_{n}}$. With this notation, $g$ is written as a sum $g=g_1+g_2+\dots$, where $g_i:C[1]\to L'_i$ is lifted to $F_{\mathcal Lie, C[1]} C[1]$ as a derivation over $d$, and $(g_1+\dots+g_i)^2$ only maps into $L'_{i+1}\oplus L'_{i+2}\oplus\dots$. Using a slight variation of the above method, we may also construct the wanted homotopy $\mathcal Comm$-inner product, i.e. the module map $f$ stated above. More precisely, we build a map $\chi:C[1]\to Mod(F_{\mathcal Lie, C[1]}C^*[1],F_{\mathcal Lie, C[1]}C[1])$, so that $\chi$ is a chain map under the differential $d_1$ on $C[1]$, and the differential $\delta(f)=f\circ h - (-1)^{|f|} g\circ f$ on $Mod(F_{\mathcal Lie, C[1]}C^*[1],F_{\mathcal Lie, C[1]}C[1])$. Since a module map is given by the components $M_n=\bigoplus_{k+l=n-2} \mathcal Lie(n)\otimes C[1]^{\otimes k}\otimes C[1]\otimes C[1]^{\otimes l}\otimes C[1]$, it is enough to construct $\chi$ as a sum $\chi=\chi_2+\chi_3+\dots$, where $\chi_i:C[1]\to M_i$. Now, the lowest component $\chi_2:C[1]\to C[1]\otimes C[1]$ is defined to be the symmetrized Alexander-Whitney comultiplication. For the induction, we assume that $\Upsilon_i:=\chi_2+\dots+\chi_{i-1}$ are local maps such that $D(\Upsilon_i):=\Upsilon_i\circ d_1-\delta\circ \Upsilon_i$ maps only into higher components $M_i\oplus M_{i+1}\oplus\dots$. Let $\epsilon_i:C[1]\to M_i$ be the lowest term of $D(\Upsilon_i)$. Since $D^2=0$ and $\delta$ has $d_1$ as its lowest component, we see that $\epsilon_i$ is $[d_1,.]$-closed, and by the hypothesis of the proposition also locally $[d_1,.]$-exact. These exact terms can be put together as before to produce a map $\chi_i$, so that now $\Upsilon_{i+1}:=\chi_2+\dots+\chi_i$ only maps into $M_{i+1}\oplus M_{i+2}\oplus\dots $. We therefore obtain the chain map $\chi$, and with this, we define the homotopy $\mathcal Comm$-inner product as $f:=\chi(\mu)\in Mod(F_{\mathcal Lie, C[1]}C^*[1],F_{\mathcal Lie, C[1]}C[1]) $, where $\mu\in C$ denotes the fundamental cycle of the space $X$. Since $\mu$ is $d_1$-closed, it follows that $f\circ h-g\circ f=0$. The operad $\widehat{\mathcal O}$ {#cyclic-op} ================================= In this section, we define for any cyclic operad $\mathcal O$ the colored operad $\widehat{\mathcal O}$. In the case that $\mathcal O$ is cyclic quadratic, we give an explicit description of $\widehat{\mathcal O}$ in terms of generators and relations coming from generators and relations in $\mathcal O$. We assume that the reader is familiar with the notion of operads, colored operads and cyclic operads. For a good introduction to operads, we refer to [@Ad], [@GK] and [@MSS], for cyclic operads we recommend [@GeK] and [@MSS]. Colored operads were first introduced in [@BV] and appeared in many other places, see e.g. [@L] and [@BM]. Since in our case, we only need a special type of colored operad, it will be convenient to setup notation with the following definition. As in [@GK (1.2.1)] and [@GeK (1.1)], we assume throughout this paper that $k$ is a field of characteristic $0$. Note however, that for certain operads such as e.g. the associative operad, a more general setup is possible. \[0/1-operad\] Let $\mathcal P$ be a 3-colored operad in the category of (differential graded) vector spaces, where we use the three colors “full", “dashed" and “empty", in symbols written ${ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}},\varnothing$. This means that to each finite sequence of symbols $x_1, \dots, x_n, x\in \{{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}},\varnothing \}$, we have (differential graded) vector spaces $\mathcal P(x_1,\dots, x_n; x_0)$ over $k$, which label the operations with $n$ inputs with colors $x_1,\dots, x_n$, and one output with color $x$. The operad $\mathcal P$ comes with maps $\circ_i: \mathcal P(x_1,\dots,x_n;x) \otimes \mathcal P(y_1,\dots,y_m;x_i) \to \mathcal P(x_1,\dots,x_{i-1},y_1,\dots, y_m,x_{i+1},\dots, x_n;x)$ which label the composition in $\mathcal P$, and with an action of the symmetric group $S_n$, which, for $\sigma\in S_n$, maps $\mathcal P(x_1,\dots, x_n;x)\to \mathcal P(x_{\sigma(1)},\dots,x_{\sigma(n)};x)$. These maps have to satisfy the usual associativity and equivariance axioms of colored operads. $\mathcal P$ is called a 0/1-operad if the color $\varnothing$ can appear only as an output, and the only nontrivial spaces with one input are $\mathcal P({ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}};{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}})=k$ and $\mathcal P({ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}};{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}})=k$. We assume furthermore, that there are fixed generators of the spaces $\mathcal P({ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}};{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}})=k$ and $\mathcal P({ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}};{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}})=k$. Graphically, we represent $\mathcal P(x_1,\dots,x_n;x)$ by a tree with $n$ inputs and one output of the given color. Since the color $\varnothing$ cannot appear as an input, we may use the following convention: we represent the output $\varnothing$ with a blank line, i.e., with no line, and we say that the operation “has no output”. $$\begin{pspicture}(0,.5)(4,4) \psline[linestyle=dashed, arrowsize=0.1, arrowinset=0](2,2)(1.2,3) \psline[arrowsize=0.1, arrowinset=0](2,2)(1.6,3) \psline[linestyle=dashed, arrowsize=0.1, arrowinset=0](2,2)(2,3) \psline[linestyle=dashed, arrowsize=0.1, arrowinset=0](2,2)(2.4,3) \psline[arrowsize=0.1, arrowinset=0](2,2)(2.8,3) \rput[b](2,.5){$\mathcal P ({ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}};\varnothing)$} \rput[b](2,3.2){$1\,\,\, 2\,\,\, 3\,\,\, 4\,\,\, 5$} \end{pspicture} \quad \quad \quad \begin{pspicture}(0,0)(4,3.6) \psline[arrowsize=0.1, arrowinset=0](2,2)(1.2,3) \psline[linestyle=dashed, arrowsize=0.1, arrowinset=0](2,2)(1.6,3) \psline[linestyle=dashed, arrowsize=0.1, arrowinset=0](2,2)(2,3) \psline[arrowsize=0.1, arrowinset=0](2,2)(2.4,3) \psline[arrowsize=0.1, arrowinset=0](2,2)(2.8,3) \psline[linestyle=dashed, arrowsize=0.1, arrowinset=0](2,2)(2,1) \rput[b](2,0){$\mathcal P ({ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}};{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}})$} \rput[b](2,3.2){$1\,\,\, 2\,\,\, 3\,\,\, 4\,\,\, 5$} \end{pspicture}$$ The canonical example of a 0/1-operad is the endomorphism 0/1-operad given for $k$-vector spaces $A$ and $M$ by $$\begin{aligned} {\mathcal E\!nd}^{A,M}(\vec X;{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}})=& Hom(\text{tensor products of $A$ and $M$},A)\\ {\mathcal E\!nd}^{A,M}(\vec X;{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}})=& Hom(\text{tensor products of $A$ and $M$},M)\\ {\mathcal E\!nd}^{A,M}(\vec X;\varnothing)=& Hom(\text{tensor products of $A$ and $M$},k).\end{aligned}$$ With this notation $(A,M,k)$ is an algebra over the 0/1-operad $\mathcal P$ if there exists a 0/1-operad map $\mathcal P \to {\mathcal E\!nd}^{A,M}$. By slight abuse of language we will also call the tuple $(A,M)$ an algebra over $\mathcal P$. It is our aim to define for each cyclic operad $\mathcal O$, the associated 0/1-operad $\widehat{ \mathcal O}$, which incorporates the cyclic structure as a colored operad. Before doing so, let us briefly recall the definition of a cyclic operad from [@GeK Theorem (2.2)]. Let $\mathcal O$ be a operad, i.e we have vector spaces $\mathcal O(n)$ for $n\geq 1$, composition maps $\circ_i:\mathcal O(n)\otimes \mathcal O(m)\to \mathcal O(n+m-1)$, and an $S_n$-action on $\mathcal O(n)$ for each $n$, satisfying the usual axioms, see [@GK (1.2.1)]. $\mathcal O$ is called [*cyclic*]{} if there is an action of the symmetric group $S_{n+1}$ on $\mathcal O(n)$, which extends the given $S_n$-action, and satisfies, for $1\in\mathcal O(1)$, $\alpha\in \mathcal O(m)$, $\beta\in \mathcal O(n)$ the following relations: $$\begin{aligned} \label{compos_cyclic1} \tau_2(1)&=&1,\\ \label{compos_cyclic2} \tau_{m+n}(\alpha\circ_k \beta)&=&\tau_{m+1}(\alpha)\circ_{k+1} \beta,\quad\quad\quad \text{ for } k<m \\ \label{compos_cyclic3} \tau_{m+n}(\alpha\circ_m \beta)&=&\tau_{n+1}(\beta)\circ_1 \tau_{m+1}(\alpha),\end{aligned}$$ where $\tau_{j}\in S_{j}$ denotes the cyclic rotation of $j$ elements $\tau_{j} :=1\in{\mathbb{Z}}_{j}\subset S_{j}$. \[def\_O\_hat\] Let $\mathcal O$ be a cyclic operad with $\mathcal O(1)=k$. For a sequence of $n$ input colors $\vec X=(x_1,\dots, x_n)$ and the output color $x$, where $x_1, \dots, x_n, x\in\{{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}},\varnothing\}$, let $$\widehat{\mathcal O}(\vec X;x):= \begin{cases} \mathcal O(n) & \text{if } x \text{ is ``full'', and } \vec X=({ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}},\ldots,{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}}),\\ \mathcal O(n) & \text{if } x \text{ is ``dashed'', and $\vec X$ has exactly one ``dashed'' input}\\ \mathcal O(n-1) & \text{if } x=\varnothing \text{ and $\vec X$ has exactly two ``dashed' inputs,} \\ \{0\} & \text{otherwise}. \end{cases}$$ The definition of $\widehat{\mathcal O}(\vec X,\varnothing)$ is motivated by the idea that one considers trees with $n-1$ inputs and one output, and then uses the $S_{n+1}$ action to turn this output into a new input: $$\begin{pspicture}(0,0.8)(4,3.4) \psline[linestyle=dashed](2,2)(1.4,3) \psline(2,2)(1.8,3) \psline(2,2)(2.2,3) \psline(2,2)(2.6,3) \psline[linestyle=dashed](2,2)(2,1) \rput[b](2,3.2){$1\,\,\, 2\,\,\, 3\,\,\, 4$} \rput[b](4,2){$\rightsquigarrow$} \end{pspicture} \begin{pspicture}(0,0.8)(4,3.4) \psline[linestyle=dashed](2,2)(1.2,3) \psline(2,2)(1.6,3) \psline(2,2)(2,3) \psline(2,2)(2.4,3) \psline[linestyle=dashed](2,2)(2.8,3) \rput[b](2,3.2){$1\,\,\, 2\,\,\, 3\,\,\, 4\,\,\, 5$} \end{pspicture}$$ We define the $S_n$-action on $\widehat{\mathcal O}(\vec X;{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}})$ and $\widehat{\mathcal O}(\vec X;{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}})$ as before by using the $S_n$-action on $\mathcal O(n)$, and the $S_n$-action on $\widehat{\mathcal O} (\vec X;\varnothing)$ by using the $S_n$-action on $\mathcal O(n-1)$ given by the cyclicity of $\mathcal O$. Diagrams with different positions of the two “dashed” inputs can be mapped to each other using the action of the symmetric group. In fact, as each $\sigma\in S_{n+1}$ induces an isomorphism which preserves all the structure, any statement about diagrams with a fixed choice of position of “dashed” inputs immediately carries over to any other choice of positions of “dashed” inputs. We therefore often restrict our attention to the choice where the two “dashed” inputs are at the far left and the far right, as shown in the above picture. It is left to define the composition. On $\widehat{\mathcal O}(\vec X;{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}})$ and $\widehat{\mathcal O}(\vec X;{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}})$, the composition is simply the composition in $\mathcal O(n)$, so that it clearly satisfies associativity and equivariance. If $|\vec X|=n+1$, then on $\widehat{\mathcal O}(\vec X;\varnothing)=\mathcal O(n)$, the composition is predetermined on the first $n$ components by the usual composition in $\mathcal O$. As for the last component, we define $$\label{def_cyclic_compos} \alpha\circ_{m+1} \beta:=\tau_{n+m} (\tau^{-1}_{m+1}(\alpha)\circ_m \beta)\stackrel{\eqref{compos_cyclic3}}{=} \tau_{n+1}(\beta) \circ_1 \alpha$$ $$\begin{pspicture}(1,1.6)(10.5,5.6) \psline[linestyle=dashed](2,2)(1.2,2.9) \psline(2,2)(1.6,2.9) \psline(2,2)(2,2.9) \psline(2,2)(2.4,2.9) \psline[linestyle=dashed](2,2)(2.8,2.9) \rput[b](3,2){$\alpha$} \rput[b](2.4,3.4){$\beta$} \psline[linestyle=dashed](2.8,3)(2.8,3.5) \psline[linestyle=dashed](2.8,3.5)(3,4) \psline(2.8,3.5)(2.8,4) \psline(2.8,3.5)(2.6,4) \rput[b](3.5,3){$:=$} \psline[linestyle=dashed](5,2)(4.2,2.9) \psline[linestyle=dashed](4.2,3)(4.3,3.1)(5,3.25)(5.7,3.4)(5.8,3.5) \psline(5,2)(4.6,2.9) \psline(4.6,3)(4.2,3.5) \psline(5,2)(5,2.9) \psline(5,3)(4.6,3.5) \psline(5,2)(5.4,2.9) \psline(5.4,3)(5,3.5) \psline[linestyle=dashed](5,2)(5.8,2.9) \psline[linestyle=dashed](5.8,3)(5.4,3.5) \rput[b](6,2){$\alpha$} \rput[b](6.4,3.1){$\tau_{m+1}^{-1}$} \psline[linestyle=dashed](5.4,3.6)(5.4,4.1) \psline[linestyle=dashed](5.4,4.1)(5.6,4.5) \psline(5.4,4.1)(5.4,4.5) \psline(5.4,4.1)(5.2,4.5) \rput[b](5.2,3.8){$\beta$} \psline[linestyle=dashed](5.8,3.6)(5.8,4.5) \psline(4.2,3.6)(4.2,4.5) \psline(4.6,3.6)(4.6,4.5) \psline(5,3.6)(5,4.5) \psline(4.2,4.6)(4.4,5.3) \psline(4.6,4.6)(4.8,5.3) \psline(5 ,4.6)(5.2,5.3) \psline(5.2,4.6)(5.4,5.3) \psline(5.4,4.6)(5.6,5.3) \psline[linestyle=dashed](5.6,4.6)(5.8,5.3) \psline[linestyle=dashed](5.8,4.6)(5.6,4.8)(5,4.95)(4.4,5.1)(4.2,5.3) \rput[b](6.4,4.7){$\tau_{n+m}$} \rput[b](7.3,3){$=$} \psline[linestyle=dashed](8.55,2)(8.1,2.9) \psline(8.55,2)(8.4,2.9) \psline(8.55,2)(8.7,2.9) \psline[linestyle=dashed](8.55,2)(9,2.9) \rput[b](9.7,2){$\tau_{n+1}(\beta)$} \psline[linestyle=dashed](8.1,3)(8.1,3.5) \psline[linestyle=dashed](8.1,3.5)(7.8,4) \psline(8.1,3.5)(8 ,4) \psline(8.1,3.5)(8.2,4) \psline(8.1,3.5)(8.4,4) \rput[b](8.5,3.5){$\alpha$} \end{pspicture}$$ where the last equality follows form equation , $\alpha\in \widehat{\mathcal O}(\vec X;\varnothing) =\mathcal O(m)$ has $m+1$ inputs, and $\beta\in\widehat{\mathcal O}(\vec Y;{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}})=\mathcal O(n)$ (or similarly $\beta\in\widehat{\mathcal O}(\vec Y;{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}})$) has $n$ inputs. It is clear that this will satisfy equivariance, since equivariance was just used to define the composition. The next lemma establishes the final property for $\widehat{\mathcal O}$ being a 0/1-operad. The composition in $\widehat{\mathcal O}$ satisfies the associativity axiom. By definition the composition is just the usual composition in $\mathcal O(n)$, except for inserting trees in the last input of elements in $\widehat{\mathcal O}(\vec X;\varnothing)$. Thus, except for composition in the last spot, associativity of $\widehat{\mathcal O}$ follows from the associativity of $\mathcal O$. Now, let $\alpha\in \widehat{\mathcal O}(\vec X;\varnothing)\cong \mathcal O(m)$, $\beta\in \widehat{\mathcal O}(\vec Y;y)\cong \mathcal O(n)$, and $\gamma\in \widehat{\mathcal O} (\vec Z;z) \cong \mathcal O(p)$, where $y,z\in\{{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}}\}$. Then, associativity is satisfied, because for $1\leq j\leq m$, it is $$\begin{gathered} (\alpha\circ_{m+1}\beta)\circ_{j} \gamma \stackrel{\mathit{\eqref{def_cyclic_compos}}}{=} (\tau_{n+1}(\beta) \circ_1 \alpha)\circ_j \gamma \stackrel{\mathit{op.comp}}{=}\\ =\tau_{n+1}(\beta) \circ_1 (\alpha\circ_j \gamma ) \stackrel{\mathit{\eqref{def_cyclic_compos}}}{=} (\alpha\circ_j \gamma )\circ_{m+p}\beta,\end{gathered}$$ and for $m< j< m+n$, it is $$\begin{gathered} (\alpha\circ_{m+1}\beta)\circ_{j} \gamma \stackrel{\mathit{\eqref{def_cyclic_compos}}}{=} (\tau_{n+1}(\beta) \circ_1 \alpha)\circ_j \gamma \stackrel{\mathit{op.comp}}{=} (\tau_{n+1}(\beta) \circ_{j-m+1} \gamma)\circ_1 \alpha \stackrel{\eqref{compos_cyclic2}}{=}\\ = \tau_{n+p}(\beta \circ_{j-m}\gamma)\circ_1\alpha \stackrel{\mathit{ \eqref{def_cyclic_compos}}}{=} \alpha\circ_{m+1} (\beta\circ_{j-m}\gamma),\end{gathered}$$ while $$\begin{gathered} (\alpha\circ_{m+1}\beta)\circ_{m+n} \gamma \stackrel{\mathit{\eqref{def_cyclic_compos}}}{=} (\tau_{n+1}(\beta) \circ_1 \alpha)\circ_{m+n} \gamma \stackrel{\mathit{ \eqref{def_cyclic_compos}}}{=}\\ = \tau_{p+1}(\gamma)\circ_1 (\tau_{n+1}(\beta) \circ_1 \alpha) \stackrel{\mathit{op.comp}}{=} (\tau_{p+1}(\gamma)\circ_1 \tau_{n+1}(\beta)) \circ_1 \alpha \stackrel{\eqref{compos_cyclic3}}{=}\\ = \tau_{n+p}(\beta \circ_{n}\gamma) \circ_1 \alpha \stackrel{\mathit{ \eqref{def_cyclic_compos}}}{=} \alpha\circ_{m+1} (\beta \circ_{n}\gamma).\end{gathered}$$ We end this section by giving a presentation of $\widehat{\mathcal O}$ in terms of generators and relations, when $\mathcal O$ is given by quadratic generators and relations. Let us first recall the notion of operads given by generators and relations. Fix a set of colors $C$. Then let $E=\{E^{x,y}_z\}_{x,y,z\in C}$ be a collection of $k$-vector spaces, together with an $S_2$-action compatible with the colors. We want $E$ to be the binary generating set of a colored operad, where $x$, $y$ and $z$ correspond to the colors of the edges of a binary vertex, i.e. a vertex with exactly two incoming and one outgoing edge. Let $T$ be a rooted, colored tree where each vertex is binary, and let $v$ be a vertex with colors $(x,y;z)$ in $T$. Then, we define $E(v):= \left( E^{x,y}_z \oplus E^{y,x}_z \right)_{S_2}$, and with this, we set $E(T):=\bigotimes_{\text{vertex }v\text{ of }T} E(v)$. We define the free colored operad $\mathcal F(E)$ generated by $E$ to be $$\mathcal F(E)(\vec X;z):= \bigoplus_{ \text{binary trees $T$ of type }(\vec X;z) } E(T).$$ The $S_n$-action is given by an obvious permutation of the leaves of the tree using the $S_2$-action on $E$, and the composition maps are given by attaching trees. This definition can readily be seen to define a colored operad. An ideal $\mathcal I$ of a colored operad $\mathcal P$ is a collection of $S_n$-sub-modules $\mathcal I(\vec X;z)\subset \mathcal P(\vec X;z)$ such that $f\circ_{i} g$ belongs to the ideal whenever $f$ or $g$ or both belong to the ideal. A colored operad $\mathcal P$ is said to be quadratic if $\mathcal P=\mathcal F(E)/(R)$ where $\mathcal F(E)$ is the free colored operad on some generators $E$, and $(R)$ is the ideal in $\mathcal F(E)$ generated by a subspace with 3 inputs, called the relations, $R\subset \bigoplus_{w,x,y,z\in C} \mathcal F(E)(w,x,y;z)$. We recall from [@GeK (3.2)], that an operad $\mathcal O$ is called cyclic quadratic if it is quadratic, with generators $E$ and relations $R$, so that the $S_2$-action on $E$ is naturally extended to a $S_3$-action via the sign-representation $sgn:S_3\to S_2$, and $R\subset\mathcal F(E)(3)$ is an $S_4$-invariant subspace. In this case, $\mathcal O$ becomes a cyclic operad, see [@GeK (3.2)]. The following lemma is straight forward to check. \[O\_hat\_quadratic\] Let $\mathcal O$ be cyclic quadratic with generators $E$ and relations $R\subset\mathcal F(E)(3)$. Then $\widehat{\mathcal O}$ is generated by $\widehat{E}:=\widehat{E} ^{{ \begin{pspicture}(0,0)(0.2,0.2) \psline[linewidth=1pt](0.1,0)(0.1,0.2) \end{pspicture}},{ \begin{pspicture}(0,0)(0.2,0.2) \psline[linewidth=1pt](0.1,0)(0.1,0.2) \end{pspicture}}}_{{ \begin{pspicture}(0,0)(0.2,0.2) \psline[linewidth=1pt](0.1,0)(0.1,0.2) \end{pspicture}}}\oplus \widehat{E} ^{{ \begin{pspicture}(0,0)(0.2,0.2) \psline[linewidth=1pt](0.1,0)(0.1,0.2) \end{pspicture}},{ \begin{pspicture}(0,0)(0.2,0.2) \psline[linewidth=1pt, linestyle=dashed,dash=3pt 2pt](0.1,0)(0.1,0.2) \end{pspicture}}}_{{ \begin{pspicture}(0,0)(0.2,0.2) \psline[linewidth=1pt, linestyle=dashed,dash=3pt 2pt](0.1,0)(0.1,0.2) \end{pspicture}}} \oplus \widehat{E} ^{{ \begin{pspicture}(0,0)(0.2,0.2) \psline[linewidth=1pt, linestyle=dashed,dash=3pt 2pt](0.1,0)(0.1,0.2) \end{pspicture}},{ \begin{pspicture}(0,0)(0.2,0.2) \psline[linewidth=1pt](0.1,0)(0.1,0.2) \end{pspicture}}}_{{ \begin{pspicture}(0,0)(0.2,0.2) \psline[linewidth=1pt, linestyle=dashed,dash=3pt 2pt](0.1,0)(0.1,0.2) \end{pspicture}}} \oplus \widehat{E} ^{{ \begin{pspicture}(0,0)(0.2,0.2) \psline[linewidth=1pt, linestyle=dashed,dash=3pt 2pt](0.1,0)(0.1,0.2) \end{pspicture}},{ \begin{pspicture}(0,0)(0.2,0.2) \psline[linewidth=1pt, linestyle=dashed,dash=3pt 2pt](0.1,0)(0.1,0.2) \end{pspicture}}}$, defined as $$\begin{aligned} \widehat{E} ^{{ \begin{pspicture}(0,0)(0.2,0.2) \psline[linewidth=1pt](0.1,0)(0.1,0.2) \end{pspicture}},{ \begin{pspicture}(0,0)(0.2,0.2) \psline[linewidth=1pt](0.1,0)(0.1,0.2) \end{pspicture}}}_{{ \begin{pspicture}(0,0)(0.2,0.2) \psline[linewidth=1pt](0.1,0)(0.1,0.2) \end{pspicture}}}:=E&\subset \widehat {\mathcal O}({ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}};{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}}),\\\widehat{E} ^{{ \begin{pspicture}(0,0)(0.2,0.2) \psline[linewidth=1pt](0.1,0)(0.1,0.2) \end{pspicture}},{ \begin{pspicture}(0,0)(0.2,0.2) \psline[linewidth=1pt, linestyle=dashed,dash=3pt 2pt](0.1,0)(0.1,0.2) \end{pspicture}}}_{{ \begin{pspicture}(0,0)(0.2,0.2) \psline[linewidth=1pt, linestyle=dashed,dash=3pt 2pt](0.1,0)(0.1,0.2) \end{pspicture}}}:=E&\subset \widehat {\mathcal O}({ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}};{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}}) ,\\\widehat{E} ^{{ \begin{pspicture}(0,0)(0.2,0.2) \psline[linewidth=1pt, linestyle=dashed,dash=3pt 2pt](0.1,0)(0.1,0.2) \end{pspicture}},{ \begin{pspicture}(0,0)(0.2,0.2) \psline[linewidth=1pt](0.1,0)(0.1,0.2) \end{pspicture}}}_{{ \begin{pspicture}(0,0)(0.2,0.2) \psline[linewidth=1pt, linestyle=dashed,dash=3pt 2pt](0.1,0)(0.1,0.2) \end{pspicture}}}:= E&\subset \widehat {\mathcal O}({ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}};{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}}),\\\widehat{E}^{{ \begin{pspicture}(0,0)(0.2,0.2) \psline[linewidth=1pt, linestyle=dashed,dash=3pt 2pt](0.1,0)(0.1,0.2) \end{pspicture}},{ \begin{pspicture}(0,0)(0.2,0.2) \psline[linewidth=1pt, linestyle=dashed,dash=3pt 2pt](0.1,0)(0.1,0.2) \end{pspicture}}}:=k&\subset \widehat {\mathcal O}({ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}};\varnothing),\end{aligned}$$ and has relations $$\begin{aligned} R\subset\mathcal F(E)(3)\cong\mathcal F(\widehat{E})({ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}};{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}}), \\ R\subset\mathcal F(E)(3)\cong\mathcal F(\widehat{E})({ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}};{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}}), \\ R\subset\mathcal F(E)(3)\cong\mathcal F(\widehat{E})({ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}};{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}}), \\ R\subset\mathcal F(E)(3)\cong\mathcal F(\widehat{E})({ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}};{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}}).\end{aligned}$$ together with the relations $$\begin{aligned} G\subset \mathcal F(\widehat{E})({ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}};\varnothing), \\ G\subset \mathcal F(\widehat{E})({ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}};\varnothing), \\ G\subset \mathcal F(\widehat{E})({ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}};\varnothing),\end{aligned}$$ where $G$ corresponds for a given coloring to the space $$G:= span \left< \begin{pspicture}(0,0.2)(1,1) \psline(0.5,0)(0.3,0.5) \psline(0.5,0)(0.7,0.5) \psline(0.3,0.5)(0.3,1) \psline(0.3,0.7)(0.1,1) \psline(0.3,0.7)(0.5,1) \rput[b](0.7,0.8){$\alpha$} \end{pspicture} - \begin{pspicture}(0,0.2)(2,1) \psline(0.5,0)(0.3,0.5) \psline(0.5,0)(0.7,0.5) \psline(0.7,0.5)(0.7,1) \psline(0.7,0.7)(0.9,1) \psline(0.7,0.7)(0.5,1) \rput[b](1.5,0.6){$\tau_3(\alpha)$} \end{pspicture} \text{ , for all } \alpha\in E^{{ \begin{pspicture}(0,0)(0.2,0.2) \psline[linewidth=1pt](0.1,0)(0.1,0.2) \end{pspicture}},{ \begin{pspicture}(0,0)(0.2,0.2) \psline[linewidth=1pt](0.1,0)(0.1,0.2) \end{pspicture}}}_{{ \begin{pspicture}(0,0)(0.2,0.2) \psline[linewidth=1pt](0.1,0)(0.1,0.2) \end{pspicture}}} \right>.$$ Koszulness of $\widehat{\mathcal O}$ {#quadrat-koszul} ==================================== This section is concerned with our main theorem, that Koszulness for $\mathcal O$ implies Koszulness for $\widehat{\mathcal O}$. To set up notation, we briefly recall the notion of quadratic dual, cobar dual and Koszulness of a (colored) operad. Recall that if the vector space $V$ is an $S_n$-module and $sgn_n$ is the sign representation, then we defined $V^\vee$ to be $V^*\otimes sgn_n$, where $V^*=Hom(V,k)$ denotes the dual space. Let $C$ be a set of colors. For every quadratic colored operad $\mathcal P$, we define the quadratic dual colored operad $\mathcal P^!:=\mathcal F(E)^\vee/(R^\perp)$, where $(R^\perp)$ is the ideal in $\mathcal F(E)^\vee$ generated by the orthogonal complement $R^\perp$ of $R$ as a subspace of $\left(\bigoplus_{w,x,y,z\in C} \mathcal F(E)(w,x,y;z) \right) ^\vee$. Notice that $\mathcal F(E)(\vec X;x)^\vee = \mathcal F(E^\vee) (\vec X;x)$, so that $\mathcal P^!$ is generated by $E^\vee$ with relations $R^\perp$, see [@GK (2.1.9)]. Now if $\mathcal P=\mathcal F(E)/(R)$ is a quadratic colored operad, then we can follow the definition from [@GK (3.2.12)], and define the cobar dual colored operad $\textbf{D} (\mathcal P)$ of $\mathcal P$, to be given by the complexes $\textbf{D} (\mathcal P)(\vec X;x)$ concentrated in non-positive degree, $\textbf{D} (\mathcal P)(\vec X;x):=$ $$\begin{gathered} \bigoplus_{ \substack{ \text{trees $T$ of}\\ \text{type $(\vec X;x)$},\\ \text{no internal edge}}} \mathcal P(T)^*\otimes {\mathrm{Det}}(T)\stackrel{{\partial}}{{{\longrightarrow}}} \bigoplus_{ \substack{ \text{trees } T \text{ of}\\ \text{type }(\vec X;x),\\ \text{1 internal edge}}} \mathcal P(T)^*\otimes {\mathrm{Det}}(T)\stackrel{{\partial}}{{{\longrightarrow}}}\\ {{\longrightarrow}}\dots\stackrel{{\partial}}{{{\longrightarrow}}} \bigoplus_{ \substack{ \text{trees } T \text{ of}\\ \text{type }(\vec X;x),\\ \text{binary tree}}} \mathcal P(T)^*\otimes {\mathrm{Det}}(T).\end{gathered}$$ Here, $\mathcal P(T)^*$ denotes the dual of $\mathcal P(T)$, and ${\mathrm{Det}}(T)$ denotes the top exterior power on the space $k^{Ed(T)}$, where $Ed(T)$ is the space of edges of the tree $T$. By definition, we let the furthest right space whose sum is over binary trees, be of degree zero, and all other spaces be in negative degree. In general, the zero-th homology of this complex is always canonically isomorphic to the quadratic dual $\mathcal P^!$, i.e. $H^0(\textbf{D}(\mathcal P)(\vec X;x))\cong \mathcal P^!(\vec X;x)$, see [@GK (4.1.2)]. The quadratic operad $\mathcal P$ is then said to be Koszul if the cobar dual on the quadratic dual $\textbf{D}(\mathcal P^!)(\vec X;x)$ is quasi-isomorphic to $\mathcal P(\vec X;x)$, i.e., by the above, $\textbf{D}(\mathcal P^!)(\vec X;x)$ has homology concentrated in degree zero. We now state our main theorem. \[O\_hat\_Koszul\] If $\mathcal O$ is cyclic quadratic and Koszul, then $\widehat{\mathcal O}$ has a resolution, which for a given sequence $\vec X$ of colors, with $|\vec X|=n$, is given by the quasi-isomorphisms $$\begin{aligned} \textbf{D}(\widehat{\mathcal O^!}) (\vec X;{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}})&\to& \widehat{ \mathcal O}(\vec X;{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}})=\mathcal O(n) \\ \textbf{D}(\widehat{\mathcal O^!}) (\vec X;{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}})&\to& \widehat{ \mathcal O}(\vec X;{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}})=\mathcal O(n) \\ \textbf{D}(\widehat{\mathcal O^!}) (\vec X;\varnothing) &\to& \widehat{ \mathcal O}(\vec X;\varnothing)=\mathcal O(n-1)\end{aligned}$$ Koszulness of $\mathcal O$ means exactly that the first and second maps are quasi-isomorphisms. The proof that the third map is also a quasi-isomorphism will concern the rest of this section. We need to show that the homology of $\textbf{D}(\widehat{\mathcal O^!}) (\vec X; \varnothing)$ is concentrated in degree $0$: $$\begin{aligned} \label{H0} H_0 \left(\textbf{D}(\widehat{\mathcal O^!}) (\vec X;\varnothing)\right )&=&\widehat{ \mathcal O}(\vec X;\varnothing)\\ \label{Hi<0} H_{i<0} \left(\textbf{D}(\widehat{\mathcal O^!}) (\vec X;\varnothing)\right)&=&\{0\}\end{aligned}$$ As mentioned in definition \[def\_O\_hat\], it is enough to restrict attention to the case where $\vec X = ({ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}},\ldots,{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}})$. Let us first show the validity of equation . The map $\textbf{D} (\widehat{\mathcal O^!}) (\vec X;\varnothing)\to \widehat{ \mathcal O}(\vec X;\varnothing)$ expanded in degrees of $\textbf{D}(\widehat{\mathcal O^!})(\vec X;\varnothing)$ is written as $\cdots \stackrel{{\partial}}{{{\longrightarrow}}} \textbf{D}(\widehat{\mathcal O^!}) (\vec X;\varnothing)^{-1} \stackrel{{\partial}}{{{\longrightarrow}}} \textbf{D}(\widehat{\mathcal O^!}) (\vec X;\varnothing)^0 \stackrel{proj}{{\longrightarrow}}\widehat{\mathcal O}(\vec X;\varnothing)$. As $\mathcal O$ and thus $\mathcal O^!$ are quadratic, we have the following identification, using the language and results of lemma \[O\_hat\_quadratic\]: $$\begin{aligned} \widehat{\mathcal O}(\vec X;\varnothing)&=&\mathcal F( \widehat{E})/(R,G) (\vec X;\varnothing) \\ \textbf{D}(\widehat{\mathcal O^!}) (\vec X;\varnothing)^0&=& \bigoplus_{ \substack{ \text{binary trees }T \\ \text{of type }(\vec X;\varnothing) }} (\widehat{E^\vee}(T))\otimes {\mathrm{Det}}(T) =\mathcal F(\widehat{E})(\vec X;\varnothing)\\ \textbf{D}(\widehat{\mathcal O^!}) (\vec X;\varnothing)^{-1}&=&\bigoplus_{ \substack{ \text{trees }T\text{ of type }(\vec X;\varnothing), \\ \text{binary vertices, except}\\ \text{one ternary vertex} }} \left(\widehat{\mathcal O^!}(T)\right)\otimes {\mathrm{Det}}(T) \\ &=& \left\{ \begin{array}{c} \text{space of relations in $\mathcal F(\widehat{E})(\vec X;\varnothing)$}\\ \text{generated by $R$ and $G$} \end{array} \right\}\end{aligned}$$ The last equality follows, because the inner product relations for $\mathcal O^!$, which are the relation space $G$ for the cyclic quadratic operad $\mathcal O^!$ from lemma \[O\_hat\_quadratic\], are the orthogonal complement of the inner product relations for $\mathcal O$. Hence, the map $proj$ is surjective with kernel ${\partial}\big(\textbf{D}(\widehat{\mathcal O^!}) (\vec X; \varnothing )^{-1}\big)$. This implies equation . As for equation (\[Hi&lt;0\]), we will use an induction that shows that every closed element in $\textbf{D}(\widehat{\mathcal O^!}) (\vec X;\varnothing)^{-r}$, for $r\geq 1$ is also exact. The argument will use an induction which slides all of the “full” inputs from one of the two “dashed” inputs to the other. As a main ingredient of this, we will employ the products $*$ and $\#$ defined below, which are used to uniquely decompose an element in $\textbf{D}(\widehat {\mathcal O^!})(\vec X;\varnothing)$ as a sum of products of $*$ and $\#$. We need the following definition. Given two decorated trees $\varphi\in \textbf{D}(\mathcal O^!)(k), \psi \in \textbf{D}(\mathcal O^!)(l)$, we define new elements $\varphi * \psi$ and $\varphi \# \psi$ in $\textbf{D}(\widehat{ \mathcal O^!})({ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}},\ldots,{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}};\varnothing)$ as follows. First, for $\varphi * \psi$ take the outputs of $\varphi$ and $\psi$ and insert them into the unique inner product decorated by the generator $1\in \widehat{\mathcal O^!}({ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}};\varnothing)$: $$\begin{pspicture}(-3,0)(6,4) \psline[linestyle=dashed](2.5,0.5)(0,3) \psline[linestyle=dashed](2.5,0.5)(5,3) \rput(2.7,0.3){\tiny $1$} \pscircle[linestyle=dotted](4,2.2){1.4} \rput(-0.2,1){$\varphi$} \psline(1.5,1.5)(2,3) \psline(1.5,1.5)(1,3) \psline(1.3,2.1)(1.5,3) \psline(0.4,2.6)(0.4,3) \psline(0.7,2.3)(0.7,3) \rput(1.4,1.3){\tiny $\alpha_1$} \rput(0.2,2.4){\tiny $\alpha_2$} \rput(0.6,2.1){\tiny $\alpha_3$} \rput(1,2.3){\tiny $\alpha_4$} \pscircle[linestyle=dotted](1,2.2){1.4} \rput(5.2,1){$\psi$} \psline(3.5,1.5)(3,3) \psline(3.5,1.5)(4,3) \psline(4.5,2.5)(4.2,3) \psline(3.3,2.1)(3.3,3) \psline(3.3,2.1)(3.6,3) \rput(3.6,1.2){\tiny $\beta_1$} \rput(3,2){\tiny $\beta_2$} \rput(4.6,2.2){\tiny $\beta_3$} \rput(-1.5,2){$\varphi * \psi =$} \end{pspicture}$$ As for $\varphi\# \psi$, we assume, that $\varphi\in \textbf{D}(\mathcal O^!)(k)$ with $k\geq 2$. Then identify $\varphi$ with an element in $\textbf{D} (\widehat{ \mathcal O^!})({ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}},\ldots,{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}};\varnothing)$ by interpreting the lowest decoration $\alpha_1\in O^!(m)$ of $\varphi$, as an inner product $\alpha_1\in \widehat{\mathcal O^!}({ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}},\ldots,{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt](0.1,0)(0.1,0.4) \end{pspicture}},{ \begin{pspicture}(0,0.1)(0.2,0.4) \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4) \end{pspicture}};\varnothing)=\mathcal O^!(m)$. $\varphi\# \psi$ is defined by attaching $\psi$ to the dashed input labeled by $\alpha_1$: $$\begin{pspicture}(-3,0)(6,4) \psline[linestyle=dashed](2.5,0.5)(0,3) \psline[linestyle=dashed](2.5,0.5)(5,3) \rput(2.7,0.3){\tiny $\alpha_1$} \pscircle[linestyle=dotted](4,2.2){1.4} \rput(-0.2,1){$\varphi$} \psline(2.5,0.5)(2,3) \psline(2.5,0.5)(1.3,3) \psline(1.7,2.2)(1.7,3) \psline(0.4,2.6)(0.4,3) \psline(0.7,2.3)(0.7,3) \rput(0.2,2.4){\tiny $\alpha_2$} \rput(0.6,2.1){\tiny $\alpha_3$} \rput(1.4,2.1){\tiny $\alpha_4$} \psccurve[linestyle=dotted](0.5,0.8)(-0.5,3)(1,3.6)(2.4,3)(2.4,2)(2.7,1.2)(3.1,0.2)(2.4,0) \rput(5.2,1){$\psi$} \psline(3.5,1.5)(3,3) \psline(3.5,1.5)(4,3) \psline(4.5,2.5)(4.2,3) \psline(3.3,2.1)(3.3,3) \psline(3.3,2.1)(3.6,3) \rput(3.6,1.2){\tiny $\beta_1$} \rput(3,2){\tiny $\beta_2$} \rput(4.6,2.2){\tiny $\beta_3$} \rput(-1.5,2){$\varphi \# \psi =$} \end{pspicture}$$ Both $\varphi$ and $\psi$ are elements of $\textbf{D} (\mathcal O^!)$ and thus uncolored. The colorations for $\varphi *\psi$ and $\varphi\#\psi$ are uniquely determined by having the first and last entry dashed. The operations $*$ and $\#$ are extended to $\textbf{D} (\mathcal O^!)\otimes \textbf{D} (\mathcal O^!)\to \textbf{D} (\widehat{\mathcal O^!})(\vec X; \varnothing)$ by bilinearity. It is important to notice that every labeled tree, whose first and last inputs are “dashed", can uniquely be written as a product of $*$ or $\#$. To be more precise, suppose $\varphi$ and $\psi$ are labeled trees with $k+1$ and $l+1$ inputs respectively. We restrict our attention to the case of elements in $\textbf{D} (\widehat{\mathcal O^!})(\vec X; \varnothing)$ whose “dashed” inputs are labeled to be the first and the last input, and thus appear in the planar representation on the far left and the far right. Let $\sigma$ be a $(k,l)$-shuffle, i.e. a permutation of $\{1,\dots,k+l\}$ such that $\sigma(1)<\dots <\sigma(k)$ and $\sigma(k+1)<\dots <\sigma(k+l)$. Then define $\phi *_\sigma \psi$, resp. $\phi \#_\sigma \psi$, as the composition of $*$, resp. $\#$, with $\sigma$ applied to the “full” leaves of the resulting labeled tree. The “dashed” inputs remain far left and far right. With these notations it is now clear, that every labeled tree in $\textbf{D}(\widehat {\mathcal O^!})(\vec X;\varnothing)$ with “dashed” first and last inputs, can uniquely be written in the form $\varphi *_\sigma\psi$ or $\varphi\#_\sigma\psi$ for some $\varphi,\psi$ and $\sigma$. For each $r\geq 1$, we show that the $(-r)$th homology of $\textbf{D}(\widehat{\mathcal O^!})(\vec X;\varnothing)$ vanishes by decomposing every element in $\textbf{D}(\widehat{\mathcal O^!})(\vec X;\varnothing)^{-r}$ as a sum of terms of the form $\varphi *_\sigma\psi$ or $\varphi\#_\sigma\psi$, and then performing and induction on the degree of $\psi$. More precisely, let us define the order of $\varphi*_\sigma \psi$ or $\varphi\#_\sigma \psi$ to be the degree of $\psi$ in $\textbf{D}(\mathcal O^!)$. Then for $s\in \mathbb N$, we claim the following statement: - Let $\chi\in \textbf{D}(\widehat{\mathcal O^!}) (\vec X;\varnothing)^{-r}$ be a closed element ${\partial}(\chi)=0$. Then $\chi$ is homologous to a sum $\sum_i \sum_\sigma \varphi_i *_\sigma \psi_i+\sum_j \sum_\sigma \varphi'_j \#_\sigma\psi'_j$, where the order of each term is less or equal to $(-s)$. Rather intuitively this means, that for smaller $(-s)$, $\chi$ is homologous to decorated trees whose total degree is more and more concentrated on the right branch of the tree. It is easy to the that the above is true for $s=1$. As for the inductive step, let $\chi=\sum_i \sum_\sigma \varphi_i *_\sigma \psi_i+\sum_j\sum_\sigma \varphi'_j \#_\sigma\psi'_j$, where we assume an expansion so that $\{\varphi_i\}_i$ are linear independent, $\{\varphi'_j \}_j$ are linear independent, but the $\psi_i$ and the $\psi'_j$ are allowed to be linear combinations in $\textbf{D}(\mathcal O^!)$. We claim that those elements $\psi_i$ and $\psi'_j$, which are of degree $-s$, are closed in $\textbf{D} (\mathcal O^!)$. This follows from ${\partial}(\chi)=0$ and the inductive hypothesis, because the only terms of the boundary ${\partial}(\chi)$, which are of order $-s+1$, are terms of the form $\varphi_i *_\sigma {\partial}(\psi_i)$ and $\varphi'_j \#_\sigma {\partial}(\psi'_j)$. But then, $\psi_i$ and $\psi'_j$ are necessarily of degree $-s$. The exactness of $\textbf{D} (\mathcal O^!)$ at the degree $-s$ implies that $\psi$s of the degree $-s$ are exact, i.e. there are elements $\Psi_i, \Psi'_j\in \textbf{D}(\mathcal O^!)$ such that $\psi_i={\partial}(\Psi_i)$, $\psi'_j ={\partial}(\Psi'_j)$. Then, take $S:={\partial}\left( \sum_i\sum_\sigma \varphi_i *_\sigma \Psi_i+\sum_j \sum_\sigma \varphi'_j \#_\sigma\Psi'_j\right)$, where the sum is over those $i$’s and $j$’s which have the constructed $\Psi_i$’s and $\Psi'_j$’s. As the total degree of the $\Psi$’s is $-s-1$, we see that the only terms of $S$ with degree greater or equal to $-s$ are the terms $$\sum_i \sum_\sigma \varphi_i *_\sigma {\partial}(\Psi_i) + \sum_j\sum_\sigma \varphi'_j \#_\sigma{\partial}(\Psi'_j) =\sum_i \sum_\sigma \varphi_i *_\sigma \psi_i+ \sum_j \sum_\sigma \varphi'_j \#_\sigma\psi'_j.$$ It follows that $\chi-S$ only contains terms of degree less or equal to $-s-1$, and $\chi$ is homologous to $\chi-S$. This concludes the inductive step. We complete the proof of the theorem by noticing that $\textbf{D}(\mathcal O^!) (l)$ is concentrated in finite degrees, so that the $\psi_i$s and $\psi'_j$s eventually have to be identically $0$. As a consequence every closed element $\chi$ is eventually homologous to $0$. It follows that the complex $ \textbf{D}(\widehat{\mathcal O^!}) (\vec X;\varnothing)$ has no homology in degrees $r\neq 0$. Algebras over $\widehat{\mathcal O}_\infty$ {#homotop-ip} =========================================== In this section, we investigate algebras over $\widehat{\mathcal O}_\infty:=\textbf{D} (\widehat{\mathcal O^!})$. The particular cases of the associative operad $\mathcal Assoc$ and the commutative operad $\mathcal Comm$ will be considered. Before looking at $\widehat{\mathcal O}_\infty$, let us first consider algebras over $\widehat{\mathcal O}$. These are given by “algebra maps” $\mathcal O(n) \otimes A^{\otimes n}\to A$, “module maps” $\bigoplus_{r+s=n-1} \mathcal O(n) \otimes A^{\otimes r}\otimes M\otimes A^{\otimes s}\to M$, and since $\widehat{\mathcal O}(\vec X;\varnothing) =\mathcal O(n-1)$ for $|\vec X|=n$, we also have “inner product maps” $ \mathcal O(n-1) \to Hom(A^{\otimes i-1} \otimes M \otimes A ^{\otimes j-i-1} \otimes M \otimes A^{\otimes n-j} ,k)$. Notice that in the lowest case $n=2$, the “inner product map" $\mathcal O(1)\to Hom(M\otimes M,k)$ determines a map $<.,.>:M\otimes M\to k$, given by the image of the unit $1_k\in k=\mathcal O(1)$. Note that $<.,.>$ is invariant under the module maps mentioned above, and using the composition and the $S_n$-action of $\widehat{\mathcal O}$, all the higher inner product maps are determined by $<.,.>$ together with the module maps. We now describe algebras over the operad $\textbf{D} (\widehat{\mathcal O^!})$. This will be given in terms of derivations and maps of free $\mathcal O$-algebras and free $\mathcal O$-modules. Recall from [@GK (1.3.4)], that a free $\mathcal P$-algebra is generated by the $k$-vector space $A$ given by $F_{\mathcal P} A:=\bigoplus_{n\geq 1}(\mathcal P(n)\otimes A^{\otimes n})_{S_n}$. $F_{\mathcal P} A$ is an algebra over $\mathcal P$, i.e., there are maps $ \gamma:\mathcal P(n)\otimes (F_{\mathcal P} A )^{\otimes n} \to F_{\mathcal P} A$ coming from the composition in $\mathcal P$ and the tensor products, which satisfy the required compatibility conditions, see [@GK (1.3.2)]. An algebra derivation $d\in \mathrm{Der}(F_{\mathcal P}A)$ is defined to be a map from $F_{\mathcal P} A$ to itself, making the following diagram commute: $$\begin{CD} \mathcal P(k) \otimes (F_{\mathcal P}A)^{\otimes k}& @>\gamma>> & F_{\mathcal P}A\\ @V\sum_i \mathrm{id}\otimes \mathrm{id}^{\otimes i} \otimes d \otimes \mathrm{id}^{\otimes (k-i-1)}VV & & @VVdV\\ \mathcal P(k) \otimes (F_{\mathcal P}A)^{\otimes k}& @>\gamma>> & F_{\mathcal P}A \end{CD}$$ Similarly, if $A$ and $M$ are $k$-vector spaces, we define the free module $M$ over $A$ to be $F_{\mathcal P,A} M := \bigoplus_{n\geq 1} \left(\bigoplus_{r+s=n-1} \mathcal P(n) \otimes A^{\otimes r} \otimes M \otimes A^{\otimes s}\right)_{S_{n}}$. Then $F_{\mathcal P,A} M$ is a module over $F_{\mathcal P}A$ over $\mathcal P$, which means that there are maps $\gamma^M:\bigoplus_{r+s=n-1} \mathcal P(n)\otimes (F_{\mathcal P} A )^{\otimes r} \otimes F_{\mathcal P,A} M \otimes (F_{\mathcal P} A )^{\otimes s} \to F_{\mathcal P,A} M$ given by composition of elements of the operad and tensor product of elements of $A$. These maps satisfy the required module axioms, see [@GK (1.6.1)]. A module derivation $g\in \mathrm{Der}_d (F_{\mathcal P, A}M)$ over $d\in \mathrm{Der}(F_{\mathcal P}A)$, is define to be a map from $F_{\mathcal P, A}M$ to itself, making the following diagram commutative: $$\begin{CD} \mathcal P(k) \otimes (F_{\mathcal P}A)^{\otimes (k-1)}\otimes F_{\mathcal P, A}M & @>\gamma^M>> & F_{\mathcal P, A}M\\ @V\sum_{i<k} \mathrm{id}\otimes \mathrm{id}^{\otimes i} \otimes d \otimes \mathrm{id}^{\otimes (k-i-1)}+ \mathrm{id}\otimes \mathrm{id}^{\otimes (k-1)} \otimes g VV & & @VVgV\\ \mathcal P(k) \otimes (F_{\mathcal P}A)^{\otimes (k-1)}\otimes F_{\mathcal P, A}M & @>\gamma^M>> & F_{\mathcal P, A}M \end{CD}$$ Finally, a module map $f\in \mathrm{Mod}(F_{\mathcal P, A}M, F_{\mathcal P,A}N)$ is defined to be a map from $F_{\mathcal P,A}M$ to $F_{\mathcal P,A}N$ making the following diagram commutative: $$\begin{CD} \mathcal P(k) \otimes (F_{\mathcal P}A)^{\otimes (k-1)}\otimes F_{\mathcal P, A}M & @>\gamma^M>> & F_{\mathcal P, A}M\\ @V\mathrm{id}\otimes \mathrm{id}^{\otimes (k-1)} \otimes f VV & & @VVfV\\ \mathcal P(k) \otimes (F_{\mathcal P}A)^{\otimes (k-1)}\otimes F_{\mathcal P, A}N & @>\gamma^N>> & F_{\mathcal P, A}N \end{CD}$$ If $\mathcal P$ is a cyclic operad, then we can use this extra datum to associate to every derivation $g$ over a free module $M$ a derivation $h$ over the free module on the dual space $M^\ast=Hom(M,k)$. \[dual-module\] Let $\mathcal P$ be a cyclic operad, and let $A$ and $M$ be a vector space over $k$, which are finite dimensional in every degree. Assume furthermore, that there are derivations $d\in \mathrm{Der} (F_{\mathcal P}\,A)$, and $g\in \mathrm{Der}_d (F_{\mathcal P,A}M)$ over $d$. The maps $d$ and $g$ are determined by maps $$\begin{aligned} d_n:\,A&\to& \bigoplus_n \mathcal P (n)\otimes A^{\otimes n},\\ g_n:\,M&\to& \bigoplus_{k+l=n-2} \mathcal P (k+l+1)\otimes A^{\otimes k} \otimes \,M \otimes A^{\otimes l}.\end{aligned}$$ Then there is an induced derivation on $M^*$ over $d$ in the following way. Define $h\in \mathrm{Der}_d (F_{\mathcal P,A}M^*)$ by maps $h_n:\,M^* \to \bigoplus_{k+l=n-2} \mathcal P(k+l+1)\otimes A^{\otimes k} \otimes M^* \otimes A^{\otimes l} $, which are given by its application to $a^\ast_1,\dots,a^\ast_{k+l} \in \,A^\ast$, $m^*_1\in \,M^*$, $m_2\in \,M$ as $$\begin{gathered} h_n(m^*_1) (a^*_1,\dots,a^*_{k},m_2,a^*_{k+1},\dots,a^*_{k+l}) \\ :=(-1)^\epsilon \cdot \tau_{k+l+2}^{l+1}(g_n(m_2)(a^*_{k+1},\dots,a^*_{k+l},m^*_1,a^*_1,\dots, a^*_{k})) \in\mathcal P(k+l+1).\end{gathered}$$ $$\begin{pspicture}(0,-1)(5,2.5) \psline(2,1)(0.8,0) \psline(2,1)(1.2,0) \psline(2,1)(1.6,0) \psline(2,1)(2.4,0) \psline(2,1)(2.8,0) \psline(2,1)(3.2,0) \psline[arrowsize=0.2, arrowinset=0, linestyle=dashed](2,2)(2,1) \rput[l](2.5,1.35){$h$} \rput[m](2,2.2){$m^*_1$} \psline[linestyle=dashed](2,0)(2,1) \rput(2,-0.4){$a^*_{k+l}\dots a^*_{k+1}$ $m_2$ $a^*_{k} \dots a^*_{1}$\quad} \psline[linearc=0.3, arrowsize=0.15]{->}(1.6,1.2)(1.9,1.5)(2.4,1)(1.9,0.5)(1.5,0.9) \rput(4.5,1.5){$:=(-1)^\epsilon \cdot$} \end{pspicture} \begin{pspicture}(0,-1)(4,2.5) \psline(2,1)(0.8,0) \psline(2,1)(1.2,0) \psline(2,1)(1.6,0) \psline(2,1)(2.4,0) \psline(2,1)(2.8,0) \psline(2,1)(3.2,0) \psline[arrowsize=0.2, arrowinset=0, linestyle=dashed](2,2)(2,1) \rput[l](2.5,1.35){$\left(\tau_{k+l+2}^{l+1}\otimes id\right) (g)$} \rput[m](2,2.2){$m_2$} \psline[linestyle=dashed](2,0)(2,1) \rput(2,-0.4){\quad $a^*_{k} \dots a^*_{1}$ $m^*_1$ $a^*_{k+l} \dots a^*_{k+1}$} \psline[linearc=0.3, arrowsize=0.15]{->}(1.6,1.2)(1.9,1.5)(2.4,1)(1.9,0.5)(1.5,0.9) \end{pspicture}$$ Here $\tau_{k+l+2}^{l+1}$ denotes the $(l+1)$st iteration of $\tau_{k+l+2}$, and $\epsilon=(|m^*_1|+|a^*_1|+\dots+|a^*_k|)\cdot (|m_2|+|a^*_{k+1}|+\dots+|a^*_{k+l}|)$. A straightforward check shows that if $d^2=0$ and $g^2=0$, then it is also $h^2=0$. The following proposition identifies algebras over $\textbf{D} (\widehat{\mathcal O^!})$. Its proof is similar to that in [@GK Proposition (4.2.14)]. \[O\_hat\_algebras\] Let $\mathcal O$ be a cyclic quadratic operad, and let $A$ and $M$ be graded vector spaces over $k$, which are finite dimensional in every degree. Then giving $(A,M)$ the structure of an algebra over $\textbf{D} (\widehat{\mathcal O^!})$ is equivalent to the following data: - a derivation $d\in \mathrm{Der}(F _{\mathcal O^!}\,A^*[1])$ of degree $1$, with $d^2=0$, - a derivation $g\in \mathrm{Der}_d (F_{\mathcal O^!,\,A^*[1]}M^*[1])$ over $d$ of degree $1$, with $g^2=0$, which by definition \[dual-module\] also implies a derivation $h\in \mathrm{Der}_d (F_{\mathcal O^!,\,A^*[1]}M[1])$ over $d$ with $h^2=0$, - a module map $f\in \mathrm{Mod}(F_{\mathcal O^!, A^*[1]}M[1], F_{\mathcal O^!,A^*[1]}M^*[1])$ of degree $0$ such that $f\circ h = g \circ f$, and satisfying the following symmetry condition: let $f$ be given by maps $f_n:\,M[1]\to \bigoplus_{k+l=n-2} \mathcal O^!(k+l+1)\otimes A^*[1]^{\otimes k}\otimes \,M^*[1] \otimes A^*[1]^{\otimes l}$, then $$\begin{gathered} \label{eq:symm} f_n(m_2)(\alpha^*;a_1,\dots,a_i,m_1,a'_1,\dots,a'_j)=\\ = (-1)^\epsilon f_n(m_1)(\alpha^*\circ \tau_{i+j+2}^{j+1} ;a'_1,\dots,a'_j,m_2,a_1,\dots,a_i).\end{gathered}$$ where $\epsilon=(|m_2|+|a_1|+\dots+|a_i|+i+1)\cdot (|m_1|+ |a'_1|+\dots+ |a'_j|+j+1)$. If $\mathcal O$ is cyclic quadratic and Koszul, then, by theorem \[O\_hat\_Koszul\], $\textbf{D}( \widehat{\mathcal O^!} )\cong \widehat{\mathcal O}$ and we call $(A,M)$ a homotopy $\mathcal O$-algebra with homotopy $\mathcal O$-module and homotopy $\mathcal O$-inner product if there are derivations $d$ and $g$ together with a module map $f$ satisfying the conditions from proposition \[O\_hat\_algebras\]. \[assoc\] Let $\mathcal Assoc$ be the associative operad, see [@GK (1.3.7)] and [@GeK (2.3)]. In this case, $\mathcal Assoc^!=\mathcal Assoc$, and therefore the free $\mathcal Assoc^!$-algebra is given by the tensor algebra on the underlying vector space, i.e. $F_{\mathcal Assoc^!} A^*[1]=T(A^*[1])=\bigoplus_{n} A^*[1]^{\otimes n}$. Similarly, $F_{\mathcal Assoc^!, A^*[1]}M^*[1]=T^{A^*[1]}{M^*[1]}:=\bigoplus_{k,l} M^*[1]^{\otimes k}\otimes A^*[1]\otimes M^*[1]^{\otimes l}$. We see after dualizing, that $(A,M)$ is a homotopy $\mathcal Assoc$-algebra with homotopy $\mathcal Assoc$-module and homotopy $\mathcal Assoc$-inner product if there are coderivations $D:TA[1]\to TA[1]$ and $G:T^{A[1]} M[1]\to T^{A[1]} M[1]$ and a comodule map $F:T^{A[1]} M[1]\to T^{A[1]} M^*[1]$ satisfying $D^2=0, G^2=0, F\circ H=G\circ F$ and the symmetry condition , where $H$ is the induced coderivation on the dual space $M^*[1]$. Thus, we recover exactly the concept of $\infty$-inner product over an $A_\infty$-algebra as defined in [@Tr], which additionally satisfies the symmetry condition , coming from switching the factors of $M$. Furthermore, one can see from [@Tr2 Lemma 2.14], that this additional symmetry implies that the $\infty$-inner product is symmetric in the sense of [@Tr2 Definition 2.13]. In [@Tr2], it was shown that if a symmetric $\infty$-inner product is also non-degenerate, then it induces a BV-structure on the Hochschild-cohomology of the given $A_\infty$-algebra. It would be interesting to have a generalization of this result to any cyclic operad $\mathcal O$, which amounts to look for a similar “$\mathcal O$-BV-structure” on the homology of the chain complex $ C_n^{\mathcal O}(A)=\left(\mathcal O^!(n)^{\vee}\otimes A^{\otimes n} \right)_{S_n} $ of an $\mathcal O$-algebra $A$ from [@GK (4.2.1)]. \[exa-comm\] Let $\mathcal Comm$ be the commutative operad, see [@GK (1.3.8)] and [@GeK (3.9)]. It is well known, that $\mathcal Comm^!=\mathcal Lie$ is the Lie operad. Thus, we get $F_{\mathcal Comm^!} A^*[1]=L(A^*[1]):=\bigoplus_n \left(\mathcal Lie(n)\otimes A^*[1]^{\otimes n} \right)_{S_n}$ is the free Lie algebra generated by $A^*[1]$, and $F_{\mathcal Comm^!, A^*[1]}M^*[1]= \bigoplus_n \left(\bigoplus_{k+l=n-1} \mathcal Lie(n) \otimes A^*[1]^{\otimes k} \otimes M^*[1] \otimes A^*[1]^{\otimes l}\right)_{S_{n}}$. It is worth noting, that in [@Ginot Section 2], these spaces were already considered as $C_\infty$-algebras with $C_\infty$-modules. We thus have the concept of homotopy $\mathcal Comm$-inner products as module maps between $C_\infty$-modules $M$ and $M^*$. [99]{} J. F. Adams, “Infinite loop spaces”, Annals of Mathematics Studies, 90. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo (1978) C. Berger, I. Moerdijk, “Resolution of coloured operads and rectification of homotopy algebras", Categories in algebra, geometry and mathematical physics, 31–58, Contemp. Math., 431, Amer. Math. Soc., Providence, RI (2007) J. M. Boardman, R. M. Vogt, “Homotopy Invariant Algebraic Structures on Topological Spaces” , Springer Verlag, (1973) E. Getzler, M. M. Kapranov, “Cyclic Operads and Cyclic Homology” Geometry, Topology, and Physics for Raoul Bott, International Press (1994) G. Ginot, “(Co)homology of $C_\infty$-algebras and applications to string topology", to be published in workshop proceedings “Algebraic and Geometric Deformation Spaces" V. Ginzburg, M. M. Kapranov, “Koszul duality for operads”, , no. 1, 203-272 (1994) P. van der Laan, “Coloured Koszul duality and strongly homotopy operads” `math.QA/0312147` R. Lawrence, D. Sullivan, “A free differential Lie algebra for the interval" `arXiv:math/0610949v2` M. Markl, S. Shnider, J. Stasheff, “Operads in Algebra, Topology and Physics” AMS Mathematical Surveys and Monographs [**96**]{} (2002) T. Tradler, “Infinity-inner-products on A-infinity-algebras”, to appear in Homotopy and Related Structures T. Tradler, “The BV algebra on Hochschild Cohomology induced by Infinity-inner products” to appear in Annales de L’institut Fourier T. Tradler, M. Zeinalian, “On the Cyclic Deligne Conjecture" , J. Pure Appl. Algebra 204, no. 2, p. 280-299, 2006 T. Tradler, M. Zeinalian, “Algebraic String Operations", K-Theory, vol. 38, no. 1, 2007 T. Tradler, M. Zeinalian, D. Sullivan, “Infinity Structure of Poincare Duality Spaces”, Algebraic and Geometric Topology, vol. 7, p. 233-260, 2007

--- abstract: 'We present observations with the Cosmic Origins Spectrograph onboard the [*Hubble Space Telescope*]{} of eight compact star-forming galaxies at redshifts $z$ = 0.02811 – 0.06540, with low oxygen abundances 12 + log(O/H) = 7.43 – 7.82 and extremely high emission-line flux ratios O$_{32}$ = \[O [iii]{}\]$\lambda$5007/\[O [ii]{}\]$\lambda$3727 $\sim$ 22 – 39, aiming to study the properties of Ly$\alpha$ emission in such conditions. We find a diversity in Ly$\alpha$ properties. In five galaxies Ly$\alpha$ emission line is strong, with equivalent width (EW) in the range 45 – 190Å. In the remaining galaxies, weak Ly$\alpha$ emission with EW(Ly$\alpha$) $\sim$ 2 – 7Å is superposed on a broad Ly$\alpha$ absorption line, indicating a high neutral hydrogen column density $N$(H [i]{}) $\sim$ (1 – 3)$\times$10$^{21}$ cm$^{-2}$. We examine the relation between the Ly$\alpha$ escape fraction $f_{\rm esc}$(Ly$\alpha$) and the Lyman continuum escape fraction $f_{\rm esc}$(LyC), using direct measures of the latter in eleven low-redshift LyC leakers, to verify whether $f_{\rm esc}$(Ly$\alpha$) can be an indirect measure of escaping LyC radiation. The usefulness of O$_{32}$, of the Ly$\alpha$ equivalent width EW(Ly$\alpha$) and of the Ly$\alpha$ peak separation $V_{\rm sep}$ as indirect indicators of Ly$\alpha$ leakage is also discussed. It is shown that there is no correlation between O$_{32}$ and $f_{\rm esc}$(Ly$\alpha$). We find an increase of $f_{\rm esc}$(Ly$\alpha$) with increasing EW(Ly$\alpha$) for EW(Ly$\alpha$)$<$100Å, but for higher EW(Ly$\alpha$)$\ga$150Å  the $f_{\rm esc}$(Ly$\alpha$) is nearly constant attaining the value of $\sim$0.25. We find an anticorrelation between $f_{\rm esc}$(Ly$\alpha$) and $V_{\rm sep}$, though not as tight as the one found earlier between $f_{\rm esc}$(LyC) and $V_{\rm sep}$. This finding makes $V_{\rm sep}$ a promising indirect indicator of both the Ly$\alpha$ and ionizing radiation leakage.' author: - | Y. I. Izotov$^{1,2}$, D. Schaerer$^{3,4}$, G. Worseck$^{5}$, A. Verhamme$^{3}$, N. G. Guseva$^{1,2}$,  T. X. Thuan$^{6}$, I. Orlitová$^{7}$ & K. J. Fricke$^{8,2}$\ $^{1}$Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, 14-b Metrolohichna str., Kyiv, 03143, Ukraine,\ E-mail: [email protected], [email protected]\ $^{2}$Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany\ $^{3}$Observatoire de Genève, Université de Genève, 51 Ch. des Maillettes, 1290, Versoix, Switzerland,\ E-mail: [email protected], [email protected]\ $^{4}$IRAP/CNRS, 14, Av. E. Belin, 31400 Toulouse, France\ $^{5}$ Institut für Physik und Astronomie, Universität Potsdam, Karl-Liebknecht-Str. 24/25, D-14476 Potsdam, Germany,\ E-mail: [email protected]\ $^{6}$Astronomy Department, University of Virginia, P.O. Box 400325, Charlottesville, VA 22904-4325, USA,\ E-mail: [email protected]\ $^{7}$Astronomical Institute, Czech Academy of Sciences, Bočn[í]{} II 1401, 141 00, Prague, Czech Republic,\ E-mail: [email protected]\ $^{8}$Institut für Astrophysik, Göttingen Universität, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany,\ E-mail: [email protected] date: 'Accepted XXX. Received YYY; in original form ZZZ' title: 'Diverse properties of Ly$\alpha$ emission in low-redshift compact star-forming galaxies with extremely high \[O [iii]{}\]/\[O [ii]{}\] ratios' --- \[firstpage\] (cosmology:) dark ages, reionization, first stars — galaxies: abundances — galaxies: dwarf — galaxies: fundamental parameters — galaxies: ISM — galaxies: starburst Introduction {#intro} ============ The extremely high O$_{32}$ = \[O [iii]{}\]$\lambda$5007/\[O [ii]{}\]$\lambda$3727 ratio in spectra of low-redshift dwarf star-forming galaxies (SFGs) indicates the presence of either an ionization-bounded compact H [ii]{} region with intense ionizing radiation from a very young star-forming region or a density-bounded H [ii]{} region losing its ionizing radiation, or both [e.g. @JO13; @NO14; @S15; @I17]. The ionizing photon production efficiency $\xi_{\rm ion}$ = $N_{\rm LyC}$/$L_\nu$ in these galaxies could be very high, where $N_{\rm LyC}$ is the Lyman continuum photon production rate and $L_\nu$ is the monochromatic UV luminosity at $\lambda$ = 1500Å. Therefore, they are considered as the local counterparts of the dwarf SFGs at high redshifts, which are likely the main source of the last phase transition of the matter in the Universe, namely its reionization, which ends at $z$ $\sim$ 6 [@O09; @WC09; @K11; @J13; @M13; @Y11; @B15a; @Sm18; @N18; @St18; @K19]. An alternative hypothesis is the dominant role of active galactic nuclei (AGN) in the reionization of the Universe, as proposed e.g. by @Madau15. However, in recent years, growing evidence for a small contribution of AGN to reionization has accumulated [e.g. @H18; @M18; @P18; @Mat18; @Ku19]. On the other hand, SFGs can be considered as an important source of ionization only if the product $\xi_{\rm ion}$$\times$$f_{\rm esc}$(LyC) is sufficiently high, where $f_{\rm esc}$(LyC) is the escape fraction of Lyman continuum (LyC), i.e. the fraction of ionizing radiation emitted into the intergalactic medium (IGM). It has been estimated that $f_{\rm esc}$(LyC) should not be less than 10–20 per cent on average in order to accomplish reionization [e.g. @O09; @R13; @D15; @Robertson15; @K16]. Currently, direct observations of the LyC are available for only a limited number of galaxies at high and low redshifts. Searches for and confirmation of LyC emission from high redshift galaxies are difficult due to their faintness, contamination by lower-redshift interlopers and attenuation by residual H [i]{} in the highly ionized IGM. [e.g., @V10; @V12; @Inoue14; @Gr15]. They have resulted in the discovery of approximately three dozens of LyC leakers with reliably high escape fraction: [*Ion2*]{} [@Va15; @B16], Q1549-C25 [@Sh16], A2218-Flanking [@B17], [*Ion3*]{} [@Va18], SunBurst Arc [@RT19], 15 SFGs studied by @St18, and a dozen of galaxies at $z$ = 3.1 with strong \[O [iii]{}\] $\lambda$5007 emission [@F19]. At low redshifts, $z$ $\sim$ 0.3 - 0.4, @I16 [@I16b; @I18; @I18b] have detected LyC leaking emission in eleven compact SFGs with high O$_{32}$ $\sim$ 5 - 28 and derived $f_{\rm esc}$(LyC) ranging between 2 and 72 per cent. These values are higher than $f_{\rm esc}$(LyC) of $\la$ 1 per cent derived by @B14 and @C17 in SFGs with lower O$_{32}$, or by @He18 in SFGs with lower equivalent widths of the H$\beta$ emission line. There is hope that the number of low-$z$ LyC leakers will considerably be increased after the completion in 2020 of the [*HST*]{} Large Program GO 15626 (P.I. A. Jaskot), which includes more than 70 targets. The [*Hubble Space Telescope*]{} ([*HST*]{})/Cosmic Origins Spectrograph (COS) sensitivity curve limits LyC observations to $z \ga 0.25$. At lower redshifts, only very bright targets are accessible [e.g. @L16]. Therefore, for low-redshift SFGs, reliable indirect indicators of LyC leakage are needed. @NO14 proposed that high O$_{32}$ can be an indication of density-bounded H [ii]{} region and thus of LyC leakage. However, O$_{32}$ depends on some other parameters, such as ionization parameter and hardness of radiation which in turn depend on metallicity. @F16 has found a correlation between $f_{\rm esc}$(LyC) and O$_{32}$, using a small sample of galaxies which includes low-$z$ LyC leakers with O$_{32}$ $<$ 8. On the other hand, no correlation has been found by @I18b for eleven confirmed LyC leakers with O$_{32}$ in a wide range of $\sim$ 5 - 28. A similar conclusion was obtained by @N18 and @B19 [see also @R17]. However, this conclusion is based mainly on observations of galaxies with O$_{32}$ $\la$ 16. Only one observed galaxy has a higher value of O$_{32}$. It is thus important to observe LyC emission in more galaxies with higher O$_{32}$. The He [i]{} emission-line ratios in optical spectra of galaxies can also be used to estimate the column density of neutral gas and thus to indirectly estimate $f_{\rm esc}$(LyC) [@I17]. However, deep spectra in the optical range are needed since these lines are weak with intensities $\la$10 per cent of the H$\beta$ emission line. Finally, the properties of the Ly$\alpha$ emission line can serve as indirect indicators of escaping LyC emission. Its profile is double-peaked in the spectra of most of LyC leakers, with the separation between the peaks $V_{\rm sep}$ decreasing with decreasing neutral hydrogen column density $N$(H [i]{}) [e.g. @I16; @I16b; @I18; @I18b]. In rare cases, the Ly$\alpha$ profile is more complex and consists of three and more peaks [@RT17; @I18b; @Va18]. @V15 and @I18b have shown that there is a tight relation between the LyC escape fraction $f_{\rm esc}$(LyC) and the separation between Ly$\alpha$ peaks, making $V_{\rm sep}$ the most reliable indicator of escaping ionizing radiation. Furthermore, @V17 have shown for a limited sample of LyC leakers that the Ly$\alpha$ escape fractions $f_{\rm esc}$(Ly$\alpha$) are larger than the LyC escape fractions $f_{\rm esc}$(LyC), in accord with theoretical predictions of @D16. In this paper we present new [*HST*]{}/COS observations of the Ly$\alpha$ in eight low-redshift ($z$ $<$ 0.07) compact SFGs with the highest O$_{32}$ $\sim$ 22 – 39 found in the Sloan Digital Sky Survey (SDSS), aiming to examine the properties of Ly$\alpha$ emission in these galaxies and to verify possible indirect indicators of escaping Ly$\alpha$ and LyC emission. The studied galaxies have lower stellar masses than previously observed Ly$\alpha$ emitting galaxies such as the “green pea” (GP) galaxies at $z$ $\sim$ 0.1 – 0.3 studied e.g. by @J17, @Y17, @MK19, and the confirmed LyC leakers at $z$ $\sim$ 0.3 – 0.4 [@I16; @I16b; @I18; @I18b]. The selection criteria and the properties of the selected galaxies derived from observations in the optical range are presented in Section \[sec:select\]. The [*HST*]{} observations and data reduction are described in Section \[sec:obs\]. The surface brightness profiles in the UV range are discussed in Section \[sec:sbp\]. Ly$\alpha$ emission is considered in Section \[sec:lya\]. In Section \[discussion\], our results are compared with the Ly$\alpha$ escape fractions obtained for other galaxies in some recent studies. We summarize our findings in Section \[summary\]. Selection criteria and the integrated properties of selected galaxies {#sec:select} ====================================================================== Our galaxies were selected from the SDSS Data Release 14 (DR14) [@A18]. We adopted the following selection criteria for each galaxy: 1\) it has a very high O$_{32}$ $>$ 20, which is an indication of either a density-bounded H [ii]{} region or a high ionization parameter. 2\) it has a compact structure in the SDSS image in order to observe the total flux of the galaxy with the small 2.5 arcsec COS aperture. 3\) it is isolated from other sources to avoid contamination in the [*HST*]{}/COS spectra. 4\) it is sufficiently bright in the UV range ([*GALEX*]{} NUV magnitude brighter than 22 mag) to be acquired with COS. In total, we selected the 8 brightest galaxies satisfying the above conditions. The SDSS spectra of the selected galaxies reveal considerably lower oxygen abundances, in the range 12 + log(O/H) = 7.43 – 7.82 (see below), as compared to the confirmed LyC leakers [@I16; @I16b; @I18; @I18b]. They show comparably high rest-frame equivalent widths EW(H$\beta$) = 258 – 483 Å  (Table \[tab1\]), indicating very recent star formation. These galaxies are located at the extreme upper tip of the SFG branch in the Baldwin-Phillips-Terlevich (BPT) diagram (filled circles in Fig. \[fig2\]a) implying the presence of high-excitation H [ii]{} regions. They are considerably more extreme than the confirmed LyC leakers shown by open circles. It is seen in Fig. \[fig2\]b, which shows the O$_{32}$ – R$_{23}$ diagram, that the selected SFGs (filled circles) have the highest O$_{32}$ ratios among all SDSS SFGs, and that they are offset to low R$_{23}$ values compared to the main sequence of SFGs (grey dots) because of their lower metallicities. On the other hand, most of the confirmed LyC leakers with higher metallicities are located in the upper part of the main sequence (open circles). The SDSS spectra were used to derive the interstellar extinction from the hydrogen Balmer decrement and ionized gas metallicity using the prescriptions of @ITL94, @I06 and @G13. The extinction-corrected emission-line flux densities, the extinction coefficients $C$(H$\beta$), the rest-frame equivalent widths of H$\beta$, \[O [iii]{}\]$\lambda$5007 and H$\alpha$ emission lines, and the observed H$\beta$ flux densities are shown in Table \[taba1\]. We note that the H$\alpha$ emission line in the SDSS spectra of J0820$+$5431 and J1032$+$4919 and the \[O [iii]{}\]$\lambda$5007 emission line in the SDSS spectrum of J1032$+$4919 are clipped. Therefore, for J1032$+$4919, $I$(\[O [iii]{}\]$\lambda$5007) = 3$\times$$I$(\[O [iii]{}\]$\lambda$4959) is used to derive O$_{32}$, the electron temperature $T_{\rm e}$(O [iii]{}) and oxygen abundance. Furthermore, the H$\alpha$ emission line in this galaxy and in J0820$+$5431 is excluded in the calculation of extinction. The He [ii]{} $\lambda$4686Å emission line is detected in seven out of eight galaxies implying hard ionizing radiation. The flux densities from Table \[taba1\] and the direct $T_{\rm e}$ method are used to derive the electron temperature, electron number density and the element abundances in the H [ii]{} regions. These quantities are shown in Table \[taba2\]. The oxygen abundances of our galaxies are on average $\sim$ 0.2 - 0.3 dex lower than those in known low-redshift LyC leakers [@I16; @I16b; @I18; @I18b]. Four galaxies from our sample have been observed with the Large Binocular Telescope (LBT) by @I17. Those authors derived oxygen abundances that are in good agreement with the present determinations from SDSS spectra. The neon, sulfur and argon to oxygen abundance ratios (Fig. \[fig3\]) are similar to those in dwarf emission-line galaxies [e.g. @I06; @G13] and in confirmed LyC leakers, with similar small dispersions. On the other hand, the N/O ratios are scattered over a large range (Fig. \[fig3\]a) which can not be explained by N/O errors (Table \[taba2\]). Compared to SDSS SFGs (shown by grey dots), they tend to have systematically higher values both for the galaxies in our sample and for the confirmed LyC leakers. The emission-line luminosities and stellar masses of our galaxies were obtained adopting a luminosity distance [NED, @W06] with the cosmological parameters $H_0$=67.1 km s$^{-1}$Mpc$^{-1}$, $\Omega_\Lambda$=0.682, $\Omega_m$=0.318 [@P14]. Using these parameters and apparent magnitudes, we also derived absolute magnitudes $M_g$ and $M_{\rm FUV}$ in the SDSS $g$ and [*GALEX*]{} FUV bands (Table \[tab3\]), assuming negligible bandpass corrections and neglecting dust extinction. The H$\beta$ luminosity $L$(H$\beta$) and corresponding star-formation rates SFR shown in Table \[tab3\] were obtained from the extinction-corrected H$\beta$ flux densities using the relation by @K98 and adopting the value of 2.75 for the H$\alpha$/H$\beta$ flux ratio. They are one order of magnitude lower than the respective values for LyC leakers. Specific star formation rates sSFR = SFR/$M_\star$ of $\sim$ 100 Gyr$^{-1}$ are similar to sSFRs for LyC leakers [@I16; @I16b; @I18; @I18b] and are among the highest known for GPs and luminous compact galaxies [@Ca09; @I11], and dwarf SFGs at $z < 1$ [@I16c]. On the other hand, they are several orders of magnitude higher than the sSFRs of $\sim$ 0.01 - 1 Gyr$^{-1}$ of the SDSS main sequence galaxies. The extinction-corrected SDSS spectra of our galaxies were used to fit the spectral energy distribution (SED) of both the stellar and nebular components and to derive their stellar masses, starburst ages and the modelled intrinsic absolute magnitudes $M_{\rm FUV}^{\rm SED}$ in the FUV range. The fitting method, using a two-component model, is described in @I18. We find that the stellar masses and FUV luminosities of our galaxies are $\sim$ 2 orders of magnitude lower compared to the confirmed LyC leakers (Table \[tab3\]), but their extinction-corrected absolute FUV magnitudes are similar to those of the faintest detected galaxies at $z$ = 6 – 8 [e.g. @L17; @Bo17b]. [*HST*]{}/COS observations and data reduction {#sec:obs} ============================================= [*HST*]{}/COS spectroscopy of the eight galaxies was obtained in program GO 15136 (PI: Y. I. Izotov) during the period January 2018 – February 2019. The observational details are presented in Table \[tab4\]. The galaxies were acquired by COS near ultraviolet (NUV) imaging. Since our targets are compact but faint, as based on shallow [*GALEX*]{} imaging, a considerable fraction of one orbit was spent per object for deep NUV imaging and reliable acquisition. The NUV-brightest region of each target was centered in the 2.5arcsec diameter spectroscopic aperture (Fig. \[fig4\]). We note that acquisition observations failed for J1242$+$4851. We do not know the reason for that. The acquisition image has a flag raised in the header: EXPFLAG=‘INVALID CHECK IMAGE’. Possibly, something went wrong during one of the two acquisition exposures. Therefore, the imaging data for J1242$+$4851 are not present in this paper. However, the spectroscopic observations were not affected. For the other galaxies, some structure with an extended low-surface-brightness (LSB) component is seen. However, their sizes are smaller than the central unvignetted $0.8$arcsec diameter region of the spectroscopic aperture. Hence, the galaxy quantities derived from the COS spectra here do not require corrections for vignetting. In fact, most of the NUV continuum in these galaxies is concentrated in much smaller central regions, with full widths at half maximum $\la$ 0.1 arcsec. We obtained spectra of the Ly$\alpha$ emission lines with the COS G130M grating positioned at the central wavelength 1222Å and a small wavelength coverage of $\sim$ 300Å at COS Detector Lifetime Position 4, yielding a spectral resolving power $\lambda/\Delta\lambda\simeq 14,000$ in the wavelength range of interest. All four focal-plane offset positions were employed to correct grid wire shadows and detector blemishes. The individual exposures were reduced with the CALCOS pipeline v3.3.4, followed by accurate background subtraction and co-addition with custom software [@W16]. Apart from adjustments of the rectangular extraction aperture due to the wide spatial profile of the G130M 1222Å setup (57 pixels) and adjustments in the detector pulse-height thresholds to reflect the current detector state, the reduction procedures were analogous to @I16. We verified in the two-dimensional spectra that the extraction aperture contains all visible Ly$\alpha$ flux and preserves the spectrophotometry of the continuum. The estimated background uncertainties of a few per cent do not affect our analysis. The homogeneous reduction enables a fair comparison to our previous results [@I16; @I16b; @I18; @I18b]. Surface brightness distribution in the NUV range {#sec:sbp} ================================================ Using the COS NUV acquisition images and the routine [*ellipse*]{} in [iraf]{}[^1]/[stsdas]{}[^2] we obtain the surface brightness (SB) profiles of our galaxies in the UV continuum. The [*GALEX*]{} NUV magnitudes were used to express the profiles in the units mag arcsec$^{-2}$. For this, we have scaled the total NUV flux of the galaxy measured in the acquisition image to the flux corresponding to the [*GALEX*]{} NUV magnitude. The shape of the SB profiles (Fig. \[fig5\]) in our galaxies is similar to those of confirmed LyC leakers [@I16; @I16b; @I18; @I18b] with a sharp increase in the central part because of the presence of the bright star-forming region(s) and a linear decrease (in magnitudes) in the outer part, reminiscent of a disc structure. However, the scale lengths $\alpha$ of our galaxies in the range $\sim$ 0.09 – 0.48 kpc (Fig. \[fig5\]) are considerably lower than $\alpha$ = 0.6 – 1.8 kpc in LyC leakers [@I16; @I16b; @I18; @I18b], presumably because of their considerably lower stellar masses. The scale lengths of our galaxies are among the lowest found for local blue compact dwarf galaxies [@P02]. On the other hand, the corresponding surface densities of SFR in the studied galaxies $\Sigma$ = SFR/($\pi \alpha^2$) are comparable to those of LyC leakers. Because of the compactness of the bright star-forming regions, the half-light radii $r_{50}$ of our galaxies in the NUV are considerably smaller than $\alpha$ (see Table \[tab3\]). Adopting $r_{50}$ as a measure of the size of these galaxies (Table \[tab3\]), the corresponding $\Sigma$s are typically two orders of magnitude larger, and comparable to those found for low-redshift LyC leakers and SFGs at high redshifts [@CL16; @PA18; @Bo17]. Ly$\alpha$ emission {#sec:lya} =================== The LyC emission can not be directly observed in our galaxies because of their low redshifts. Therefore, we wish to use the Ly$\alpha$ profiles in their spectra to derive information about possible leaking LyC radiation for galaxies with much lower stellar masses than those of confirmed LyC leakers. @D16 and @V17 have proposed that the presence of a double-peaked Ly$\alpha$ profile, with a small peak separation, would be a good indicator of LyC leakage. According to the models of @V15, the peak separation decreases with decreasing column density of the neutral gas. This in turn would result in a higher escape fraction of the LyC radiation. Additionally, the role of the blue Ly$\alpha$ peak in the LyC escape has been discussed by e.g. @H15 and @O18. They found that the escape fraction of the Ly$\alpha$ radiation increases when the blue peak velocity decreases. The Ly$\alpha$ profiles in the medium-resolution spectra of the 8 observed galaxies are shown in Fig. \[fig6\]. In the upper panel for each object, both the flux and the wavelength are scaled so that the continuum behaviour can be seen over a broader wavelength range around the Ly$\alpha$ line, while in the lower panel both the flux and the wavelength scales are adjusted to examine the profile of the Ly$\alpha$ emission line. A strong Ly$\alpha$ $\lambda$1216Å emission-line without any evidence for an absorption profile is detected in five galaxies (Fig. \[fig6\]b,c,f,g,h), similarly to the low-$z$ LyC leakers [fig. 7 in @V17]. In the remaining 3 galaxies, the weak Ly$\alpha$ emission line is superposed on a broad absorption component (Fig. \[fig6\]a,d,e). The fraction of galaxies with a detected Ly$\alpha$ absorption line is similar to that obtained by @MK19. However, the intensity at the bottom of the absorption lines is above zero in these galaxies, indicating that the covering factor of the H [i]{} cloud surrounding the ionizing source is less than unity, allowing for some Ly$\alpha$ emission and some continuum emission in this wavelength range to escape. The full widths at zero intensity of the Ly$\alpha$ emission lines in all our galaxies are FWZI = 3 – 4Å  (Fig. \[fig6\]). On the other hand, the widths of the absorption lines at the bottom of their profiles are considerably broader, $\sim$ 10 – 20Å in the three galaxies J0007$+$0226, J0926$+$4504 and J1032$+$4919. Therefore, it is unlikely that the emission above zero at the bottom of the absorption profiles is due to the Ly$\alpha$ emission line. Six out of eight observed galaxies show double-peaked Ly$\alpha$ emission profiles with the peak separations much larger than the nominal COS spectral resolution of $\sim$ 20 km s$^{-1}$ for the G130M grating. The Ly$\alpha$ emission line in J0926$+$4504 is too weak to definitely derive its profile. The two-peak shape is similar to that observed in known LyC leakers [@I16; @I16b; @I18; @I18b; @V17] and in GP galaxies [@JO14; @H15; @Y17; @MK19]. The Ly$\alpha$ emission-line profile of J1032$+$4919 is more complex and consists of three peaks, similar to that of one low-$z$ LyC leaker [@I18b], and two $z$ $>$ 2 LyC leakers [@Va18; @RT17]. However, at variance with J1032$+$4919, all other triple-peaked spectra in the literature do not show clear signs of underlying Ly$\alpha$ absorption. Some parameters of the Ly$\alpha$ emission profile are presented in Table \[tab5\]. The Ly$\alpha$ emission-line fluxes in the spectra of galaxies with detected broad Ly$\alpha$ absorption are measured placing the continuum at the bottom of the absorption profiles. For J1032$+$4919, two separations (blue peak - centre peak and centre peak - red peak) are given. It is seen that the separation between the peaks is in a narrow range $\sim$ 240 – 310 km s$^{-1}$, indicating low neutral hydrogen column densities $N$(H [i]{}) and implying considerable fractions of escaping LyC emission. However, in the spectra of three galaxies, J0007$+$0226, J0926$+$4504 and J1032+4919, strong Ly$\alpha$ absorption is present as well. The flux in the central part of absorption profiles is above zero, although their shape indicates that they are clearly saturated. This appearance is in contrast to expectations for a single source surrounded by an uniform neutral gas cloud with high $N$(H [i]{}). We derive $N$(H [i]{}) using the relation $$I(\lambda)=I_{\rm cont} \exp{[-\tau(\lambda)]}, \label{profile}$$ where $I_{\rm cont}$ and $I(\lambda)$ are the flux of the continuum and the flux in the line at the wavelength $\lambda$, respectively, after subtraction of the flux at the bottom of the Ly$\alpha$ absorption line. The optical depth $\tau$($\lambda$) in the Ly$\alpha$ profile at the wavelength $\lambda$ is defined by the relation of @B75 $$\tau(\lambda) = 4.26\times 10^{-20} \frac{N({\rm H~I})}{6.04\times 10^{-10} + (\lambda - \lambda_0)^2}, \label{Lya}$$ where $N$(H [i]{}) is in cm$^{-2}$, $\lambda$ is in Å, $\lambda_0$ = 1215.67 Å, Using Eqs. \[profile\] and \[Lya\] we find $N$(H [i]{}) = 1.8$\times$10$^{21}$ cm$^{-2}$, 1.0$\times$10$^{21}$ cm$^{-2}$ and 3.0$\times$10$^{21}$ cm$^{-2}$ from the fit of wings of absorption Ly$\alpha$ profiles in the spectra of J0007$+$0226, J0926$+$4504 and J1032$+$4919, respectively (grey lines in the upper panels of Fig. \[fig6\]a, \[fig6\]d, and \[fig6\]e). We note that only the blue damped wing was used for fitting because the red wing is contaminated by the stellar N [v]{} $\lambda$1240Å  line with a P Cygni profile. Then the superposition of the Ly$\alpha$ absorption and emission lines could be explained by the emission of two star-forming regions, where the fainter one, with a flux in the continuum corresponding to the one at the bottom of the absorption profile, is surrounded by neutral gas with low $N$(H [i]{}), and the brighter one is surrounded by a neutral optically thick cloud. The flux of the continuum for the brighter region is determined by the difference between the observed flux outside the absorption profile and the flux at its bottom. The presence of Ly$\alpha$ emission in galaxies with broad Ly$\alpha$ absorption could also be explained by low-column-density holes in a thick cloud of neutral hydrogen. Discussion ========== Comparison with other galaxy samples ------------------------------------ We compare our galaxies with those studied in other recent papers. All galaxies from the sample in this paper are dwarf systems with SDSS $g$ absolute magnitudes $M_g$ $\sim$ –16 to –18 mag. Their absolute FUV magnitudes $M_{\rm FUV}$, in the range –14 to –16, compare well with the average $M_{\rm FUV}$ $\sim$ –15 mag of the faintest spectroscopically confirmed LAEs at redshifts 2.9 - 6.7, similar to those of the sources believed to reionize the Universe [@Ma18]. Our galaxies are much more compact in the NUV continuum (Fig. \[fig5\]) and have lower metallicities (Table \[tab1\]) than most Ly$\alpha$ emitters (LAEs) and LyC leakers. They have high ionizing photon production efficiencies $\xi_{\rm ion}$ (Table \[tab3\]), above the canonical value in log scale of $\log\xi_\mathrm{ion}\sim 25.2$ in units of log Hz erg$^{-1}$ adopted in high-$z$ studies [e.g. @R13] and similar to the values in LAEs at $z$ $\sim$ 3 - 7 [@Ha18; @Na18]. These high values are a natural consequence of a very young starburst age (Table \[tab3\]). High $\xi_{\rm ion}$ in our galaxies are also similar to those in confirmed low-$z$ LyC leakers [e.g. @S16]. If applicable to all high-$z$ galaxies, such $\xi_{\rm ion}$ values would suffice to reionize the Universe with a constant escape fraction $f_{\rm esc}$(LyC) $\sim$ 10 per cent [e.g. @O09; @R13; @D15; @Robertson15; @K16]. However, we note that the $\xi_{\rm ion}$ values in Table \[tab3\] should be considered only as lower limits, as including possible escaping LyC radiation, which is not taken into account, would somewhat increase these values. The high-$z$ LAEs show a tight relation between the Ly$\alpha$ luminosity $L$(Ly$\alpha$) and the absolute magnitude $M_{\rm FUV}$ at $\lambda$ = 1500Å(small open grey circles in Fig. \[fig7\]a). The confirmed LyC leakers [open black circles, @I16; @I16b; @I18; @I18b] follow the same relation. On the other hand, GPs at redshifts $\la$ 0.3 (grey asterisks) are offset to lower $L$(Ly$\alpha$) by one order of magnitude. This offset can be caused by aperture effects because GPs are on average closer to us and have larger angular sizes. Furthermore, the EW(Ly$\alpha$)s in GPs are considerably lower than in low-$z$ LyC leakers (Fig. \[fig7\]a), indicating older burst ages and thus lower Ly$\alpha$ luminosities. We note that for the LyC leakers and GPs we do not use the published values for Ly$\alpha$ in @I16 [@I16b; @I18; @I18b], @JO14, @H15, @J17 and @Y17. Instead for the sake of data homogeneity, we measured Ly$\alpha$ fluxes and equivalent widths using the public [*HST*]{} data, correcting the fluxes for Milky Way extinction and converting them to luminosities. As for the absolute FUV magnitudes, we derived them from the apparent [*GALEX*]{} FUV magnitudes. Our galaxies with extreme O$_{32}$ extend the range of $M_{\rm FUV}$ to appreciably lower FUV brightesses. Their low Ly$\alpha$ luminosities ($L$(Ly$\alpha$)$\la$10$^{42}$ erg s$^{-1}$) make the $L$(Ly$\alpha$)-$M_{\rm FUV}$ relation bend downwards at low $M_{\rm FUV}$ (Fig. \[fig7\]a)). We note that LAEs with low Ly$\alpha$ luminosities, similar to those of our galaxies, are also detected in the high-$z$ Universe [e.g. @L19]. A notable feature of the four galaxies with the highest EW(Ly$\alpha$) and highest $L$(Ly$\alpha$) in our sample is that their Ly$\alpha$ luminosities are lower by a factor $>$ 2 than the value derived from the extrapolation of the relation for high-$z$ galaxies to fainter absolute FUV magnitudes (dashed line). While they are consistent with the extrapolation of the trend defined by the GPs to fainter absolute UV magnitudes, the Ly$\alpha$ equivalent widths of the galaxies with extreme O$_{32}$ are similar to those of the low-$z$ LyC leakers [@I16; @I16b; @I18; @I18b] and are much higher than those of GPs. It was shown e.g. by @Wi16 [@Wi18] that high-$z$ galaxies at $z$ $>$ 3 are surrounded by extended diffuse Ly$\alpha$-emitting haloes tracing neutral gas up to several tens of kpc and contributing 40 – 90 per cent of the total Ly$\alpha$ flux. These extended haloes are seen even in low-mass SFGs with stellar masses down to 10$^{8}$ – 10$^{9}$ M$_\odot$ [@E18; @E19]. @Y17b have shown that Ly$\alpha$ emission in Ly$\alpha$-emitting galaxies at lower redshifts $z$ $\la$ 0.3 is spatially more extended compared to that of the UV continuum. The difference between our high-EW(Ly$\alpha$) and high-$z$ galaxies may possibly be caused by a low-intensity diffuse Ly$\alpha$ emission in our galaxies extending outside the extraction aperture used to obtain the one-dimensional spectrum. In this case, an aperture correction is needed for comparison of the Ly$\alpha$ flux with fluxes of other hydrogen emission lines. However, we do not expect a large aperture correction for $L$(Ly$\alpha$) ($>$ 2) because the adopted aperture for the extraction of one-dimensional spectra includes nearly all light from our galaxies. Additionally, we have compared Ly$\alpha$ fluxes in one-dimensional spectra obtained with different extraction apertures and reduced with the same custom and pipeline data reduction packages. We find that they differ by not more than 5 per cent. Ly$\alpha$ imaging of the lowest-mass SFGs, with stellar masses $\la$ 10$^{8}$ M$_\odot$, is needed to verify the importance of aperture corrections. Other factors may play a role in reducing $L$(Ly$\alpha$) in our low-mass galaxies, e.g. unaccounted contribution of underlying Ly$\alpha$ absorption, enhanced UV extinction due to a larger fraction of dust and/or a steeper reddening law in the UV range than in LyC leakers and high-$z$ galaxies. But these factors are unlikely to be enough to explain the observed large offset of our galaxies from the extrapolated value (dashed line in Fig. \[fig7\]a). We note that the H$\beta$ luminosities in our galaxies are more than one order of magnitude smaller than in the LyC leakers, corresponding to a smaller number of massive stars by the same factor. Probably, dynamical processes such as large-scale outflows due to the evolution of massive stars in our galaxies are considerably less energetic and may result in a reduced escaping Ly$\alpha$ emission. The distribution of the Ly$\alpha$ rest-frame equivalent widths EW(Ly$\alpha$) for the same samples of galaxies as in Fig. \[fig7\]a is shown in Fig. \[fig7\]b. The range of EW(Ly$\alpha$) for our objects is larger than that for LyC leakers with four galaxies having EW(Ly$\alpha$) smaller than those for LyC leakers, and four galaxies with EW(Ly$\alpha$) greater than the average value for LyC leakers. On the other hand, GPs from the literature have in general considerably lower EW(Ly$\alpha$). Comparing to high-$z$ galaxies we note that the EW(Ly$\alpha$)s of our galaxies are several times lower than of high-$z$ galaxies, due to the same reasons (i.e possible aperture effects) as for Ly$\alpha$ luminosities. @Ha17 found a trend of increasing EW(Ly$\alpha$) with decreasing UV luminosity. They attributed this trend to the lower metallicities of lower-luminosity galaxies. However, the distribution of our galaxies which extend the EW(Ly$\alpha$)-$M_{\rm FUV}$ relation to lower UV luminosities does not support the existence of this trend. Although a dependence of EW(Ly$\alpha$) on metallicity is not excluded, the relation in Fig. \[fig7\]b can also be explained by increasingly higher EW(Ly$\alpha$) errors at their high values for high-$z$ galaxies and by the fact that only galaxies with high EW(Ly$\alpha$) at high redshifts can be selected. The presence of many low-$z$ GPs with low EW(Ly$\alpha$) favours this selection effect. The Ly$\alpha$ escape fraction ------------------------------ An important quantity to consider is the escape fraction $f_{\rm esc}$(Ly$\alpha$). In this paper we define $f_{\rm esc}$(Ly$\alpha$) as the ratio of the extinction-corrected Ly$\alpha$ to H$\beta$ flux ratio to its Case B value of 23.3, corresponding to that at a low electron number density, $N_{\rm e}$ $\sim$ 10$^2$ cm$^{-3}$ [@SH95]: $$f_{\rm esc}({\rm Ly}\alpha) = \frac{1}{23.3}\frac{I({\rm Ly}\alpha)}{I({\rm H}\beta)}, \label{fesc}$$ where $I$(Ly$\alpha$) is the flux density corrected for the Milky Way extinction, and $I$(H$\beta$) is the flux density corrected for both the Milky Way and internal galaxy extinction. At higher electron densities, for example at $N_{\rm e}$ $\sim$ 10$^3$ cm$^{-3}$, the Case B Ly$\alpha$/H$\beta$ ratio is higher by $\sim$ 10 per cent, resulting in a lower $f_{\rm esc}$(Ly$\alpha$). Using Eq. \[fesc\], we calculate $f_{\rm esc}$(Ly$\alpha$) for the eleven LyC leakers from @I16 [@I16b; @I18; @I18b] and compare them with their directly measured $f_{\rm esc}$(LyC) in Fig. \[fig8\]. It is seen that there is a general trend of increasing $f_{\rm esc}$(LyC) with increasing $f_{\rm esc}$(Ly$\alpha$). That trend can be approximated by the maximum likelihood relation (solid line) $$\log f_{\rm esc}({\rm LyC})=(2.67\pm 0.51)\times \log f_{\rm esc}({\rm Ly}\alpha) + (0.40\pm 0.29), \label{LyC-Lya}$$ but with large uncertainties. The distribution of the data points in Fig. \[fig8\] is somewhat different from that of the dusty, clumpy interstellar medium models of @D16 (shaded region in Fig. \[fig8\]). Their models predict $f_{\rm esc}$(Ly$\alpha$) $\ga$ 2$\times$$f_{\rm esc}$(LyC) and a wide range of $f_{\rm esc}$(Ly$\alpha$) at fixed $f_{\rm esc}$(LyC), depending on the covering factor of the neutral gas. The observational data show a much narrower range of $f_{\rm esc}$(Ly$\alpha$) at fixed $f_{\rm esc}$(LyC), with some galaxies lying outside the model prediction regions. A relatively small dispersion of the data points in the observed relation in Fig. \[fig8\] gives hope that $f_{\rm esc}$(Ly$\alpha$) can be used to indirectly derive $f_{\rm esc}$(LyC). However, the number of confirmed LyC leakers is still very small and larger statistics are needed to verify the relation given by Eq. \[LyC-Lya\]. There is also the possibility that there exist sources with significant Ly$\alpha$ escape but with no leaking LyC radiation which would flatten the relation shown by the solid line. The filled circles in Fig. \[fig8\] show the galaxies from our sample with the exclusion of the three galaxies with Ly$\alpha$ absorption. For these galaxies, $f_{\rm esc}$(LyC) is derived using the velocity separation between the peaks of Ly$\alpha$ emission lines. It is seen that, for a given $f_{\rm esc}$(LyC), the galaxies are offset from the confirmed LyC leakers to lower $f_{\rm esc}$(Ly$\alpha$). Possible aperture corrections for Ly$\alpha$ emission would be too small to account for those offsets. On the other hand, corrections for unaccounted Ly$\alpha$ absorption would increase $f_{\rm esc}$(Ly$\alpha$). Overall, these corrections do not appear to be enough to account for the offsets. Alternatively, parts of the offsets could be due to an overestimation of the indirect value of $f_{\rm esc}$(LyC), as determined by eq. 2 of @I18b. It was suggested by @NO14 that the fraction of escaping ionizing radiation from galaxies correlates with the O$_{32}$ ratio. However, @I18b found that the O$_{32}$ – $f_{\rm esc}$(LyC) correlation for the confirmed LyC leakers is rather weak and a high O$_{32}$ does not garantee a high $f_{\rm esc}$(LyC). Adopting the correlation between the Ly$\alpha$ and LyC escape fractions (Fig. \[fig8\]), we investigate whether O$_{32}$ can be used as indirect indicator of escaping ionizing radiation for galaxies with the highest O$_{32}$. We show in Fig. \[fig9\]a the O$_{32}$ – $f_{\rm esc}$(Ly$\alpha$) diagram for the same galaxies as in Fig. \[fig7\], excluding the high-$z$ LAEs. The galaxies discussed in this paper (filled circles) extend the range of O$_{32}$ up to $\sim$ 40, a factor of two higher than in previous studies. However, no correlation is found. Thus, we confirm and strengthen the conclusion of @I18b that O$_{32}$ is not a reliable indicator of escaping Ly$\alpha$ emission and likely also of escaping LyC radiation. This conclusion is in agreement with the models of @B19 who have shown that galaxies with various metallicities and ionization parameters fill all the $f_{\rm esc}$(LyC) – O$_{32}$ space. However, we note that $f_{\rm esc}$(Ly$\alpha$) for our galaxies (filled circles in Fig. \[fig9\]) could be somewhat higher because of unaccounted possible aperture corrections for Ly$\alpha$ emission and underlying absorption. We discuss other possible indicators of $f_{\rm esc}$(Ly$\alpha$). A possible candidate is the rest-frame equivalent width EW(Ly$\alpha$). The diagram EW(Ly$\alpha$) – $f_{\rm esc}$(Ly$\alpha$) is shown in Fig. \[fig9\]b. This diagram is somewhat puzzling. At low EW(Ly$\alpha$), the Ly$\alpha$ escape fraction $f_{\rm esc}$(Ly$\alpha$) increases with increasing EW(Ly$\alpha$). However, no such dependence is found at high EW(Ly$\alpha$) $\ga$ 60Å for four of our galaxies and three confirmed LyC leakers with the highest EW(Ly$\alpha$). One possible explanation for that behavior has been proposed by @SM19. They have shown that the EW(Ly$\alpha$) – $f_{\rm esc}$(Ly$\alpha$) relation flattens considerably in the presence of dust absorption and is nearly independent of EW(Ly$\alpha$), given a varying ionizing photon production efficiency $\xi_{\rm ion}$. In particular, our data, in the flat part of the diagram at EW(Ly$\alpha$) $\ga$ 150Å, can be reproduced adopting $E(B-V)$ $\sim$ 0.2 mag, corresponding to $C$(H$\beta$) $\sim$ 0.3, given that log $\xi_{\rm ion}$ increases from $\sim$ 25.0 to $\sim$ 25.7 with increasing EW(Ly$\alpha$) (solid line in Fig. \[fig9\]b). This value of $C$(H$\beta$) is somewhat higher than the values derived from the SDSS spectra (Table \[tab5\]), but is consistent with the values obtained from the high signal-to-noise ratio LBT spectra for J0159$+$0751, J1032$+$4919, J1205$+$4551 and J1355$+$4651 [@I17]. Another indirect indicator of escaping radiation is the velocity separation between Ly$\alpha$ peaks $V_{\rm sep}$. @I18b have found a tight correlation between $f_{\rm esc}$(LyC) and $V_{\rm sep}$. Fig. \[fig9\]c shows a possible correlation between the peak separation $V_{\rm sep}$ and the Ly$\alpha$ escape fraction $f_{\rm esc}$(Ly$\alpha$), described by the maximum likelihood relation (solid line in Fig. \[fig9\]c): $$f_{\rm esc}({\rm Ly}\alpha)=6.56\times 10^{-7} V_{\rm sep}^2 -1.25\times 10^{-3}V_{\rm sep}+0.607. \label{fesc1}$$ The large scatter of the data around the relation found between an integrated property of Ly$\alpha$ ($f_{\rm esc}$(Ly$\alpha$)) and a characteristic of the Ly$\alpha$ profile ($V_{\rm sep}$) is probably to be expected, since several physical properties, such as dust content, H [i]{} column density, H [i]{} kinematics and geometry affect the Ly$\alpha$ radiative transfer [see e.g. @S11; @V15]. Finally, we consider whether He [i]{} emission lines in the optical range can be indicators of Ly$\alpha$ emission. @I17 have argued that He [i]{} emission line ratios are promising diagnostics because some of these lines can be optically thick, indicating a high column density of the neutral gas. The He [i]{} $\lambda$3889Å emission line is most sensitive to this effect. This line is blended with the hydrogen H8 $\lambda$3889Å emission line, and its intensity can be obtained by subtracting the H8 intensity equal to 0.107$\times$$I$(H$\beta$). The recombination intensity of the He [i]{} $\lambda$3889Å emission line is $\sim$ 0.1$\times$$I$(H$\beta$). This line can be enhanced by collisions with electrons but weakened if its optical depth is high. Therefore, the condition $I$(He [i]{} $\lambda$3889)/$I$(H$\beta$) $<$ 0.1, corresponding to $I$(H8 + He [i]{} $\lambda$3889)/$I$(H$\beta$) $<$ 0.2, would be an indication of high optical depth and high $N$(H [i]{}). It is seen in Table \[taba1\] that $I$(H8 + He [i]{} $\lambda$3889)/$I$(H$\beta$) is considerably lower than 0.2 in only two galaxies, J0926$+$4504 and J1032$+$4919. Both of these show broad Ly$\alpha$ absorption line in their spectra (Fig. \[fig6\]d,e), indicating high $N$(H [i]{}). For the remaining galaxies, this technique predicts lower $N$(H [i]{}) and thus stronger Ly$\alpha$ in emission. The technique fails for only one galaxy, J0007$+$0226, which shows Ly$\alpha$ absorption in its spectrum (Fig. \[fig6\]a), but has $I$(H8 + He [i]{} $\lambda$3889)/$I$(H$\beta$) $\sim$0.2. Thus we conclude that the He [i]{} $\lambda$3889Å emission line is generally a good indicator of $N$(H [i]{}), and hence of Ly$\alpha$ emission. Conclusions {#summary} =========== We present here new [*HST*]{}/COS observations of eight low-redshift ($z$ $<$ 0.07) compact star-forming galaxies (SFG) with extremely high O$_{32}$ in the range $\sim$ 22 – 39. All studied objects are compact low-mass (log ($M_\star$/M$_\odot$) = 5.8 – 8.6) and low-metallicity (12$+$log(O/H)= 7.43 – 7.82) galaxies with strong nebular emission lines in their spectra (EW(H$\beta$) = 258 - 483Å), indicating very young starburst ages of 0.5 – 3.3 Myr. The Ne/O, S/O and Ar/O abundance ratios in our galaxies are similar to the values for the bulk of compact SDSS SFGs. On the other hand, similarly to confirmed LyC leakers, the N/O abundance ratios in our galaxies are higher than those of the most compact SDSS SFGs. We also study the Ly$\alpha$ emission and indirect indicators of the escaping Ly$\alpha$ and Lyman continuum (LyC) radiation of these SFGs. Our main results are summarized as follows: 1\. A strong Ly$\alpha$ emission line with two peaks was observed in the spectra of five galaxies, while a Ly$\alpha$ emission line with a low equivalent width on top of a broad Ly$\alpha$ absorption profile is present in the remaining three galaxies. Using the damped wings of these absorption profiles, we derive neutral hydrogen column densities $N$(H [i]{}) in the range (1 – 3)$\times$10$^{21}$ cm$^{-2}$. 2\. We discuss various indirect indicators of escaping Ly$\alpha$ and ionizing radiation, such as the O$_{32}$ ratio, EW(Ly$\alpha$) and the velocity separation between Ly$\alpha$ emission line peaks $V_{\rm sep}$, assuming that the strength of the Ly$\alpha$ emission line is a measure of LyC leakage. We found that there is no correlation between O$_{32}$ and the Ly$\alpha$ escape fraction $f_{\rm esc}$(Ly$\alpha$). The dependence of $f_{\rm esc}$(Ly$\alpha$) on EW(Ly$\alpha$) is such that at low EW(Ly$\alpha$) $\la$ 100Å, there is a linear increase of $f_{\rm esc}$(Ly$\alpha$), while at high EW(Ly$\alpha$) $>$ 150Å, $f_{\rm esc}$(Ly$\alpha$) is nearly constant with a value $\sim$ 0.25. This behavior may be explained by dust absorption. We find a trend of increasing of $f_{\rm esc}$(Ly$\alpha$) with decreasing of $V_{\rm sep}$. However, even galaxies showing double-peaked Ly$\alpha$ profiles exhibit a large scatter between the Ly$\alpha$ escape fraction and the separation of their peaks $V_{\rm sep}$. The latter quantity has been shown to correlate more strongly with the LyC escape fraction [@I18b]. 3\. Bright compact star-forming regions are seen in the COS near ultraviolet (NUV) acquisition images. The surface brightness profile at the outskirts can be approximated by an exponential disc profile, with a scale length of $\sim$ 0.09 – 0.48 kpc. These scale lengths are several times lower than those of confirmed LyC leakers and are among the lowest ones found for local blue compact dwarf galaxies. The global properties of the selected galaxies are very similar to those of the lowest-mass high-$z$ galaxies. They are thus ideal nearby laboratories for investigating the mechanisms responsible for the escape of Ly$\alpha$ and ionizing radiation from galaxies during the epoch of the reionization of the Universe. Acknowledgements {#acknowledgements .unnumbered} ================ We thank the anonymous referee for valuable comments. These results are based on observations made with the NASA/ESA [*Hubble Space Telescope*]{}, obtained from the data archive at the Space Telescope Science Institute. STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555. Support for this work was provided by NASA through grant number HST-GO-15136.002-A from the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555. Y.I.I. and N.G.G. acknowledge support from the National Academy of Sciences of Ukraine (Project No. 0116U003191) and by its Program of Fundamental Research of the Department of Physics and Astronomy (Project No. 0117U000240) of the National Academy of Sciences of Ukraine. G.W. has been supported by the Deutsches Zentrum für Luft- und Raumfahrt (DLR) through grant number 50OR1720. I.O. acknowledges grants GACR 14-20666P and 17-06217Y of the Czech National Foundation. Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org. SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration. GALEX is a NASA mission managed by the Jet Propulsion Laboratory. This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space AdministrationGALEX is a NASA mission managed by the Jet Propulsion Laboratory. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. Emission-line fluxes in SDSS spectra and element abundances =========================================================== \[lastpage\] [^1]: [iraf]{} is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. [^2]: [stsdas]{} is a product of the Space Telescope Science Institute, which is operated by AURA for NASA.

--- abstract: 'We study a reaction diffusion system of the activator-inhibitor type with inhomogeneous reaction terms showing spatiotemporal chaos. We analyze the topological properties of the unstable periodic orbits in the slow chaotic dynamics appearing, which can be embedded in three dimensions. We perform a bi-orthogonal decomposition analyzing the minimum number of modes necessary to find the same organization of unstable orbits.' address: - 'Grupo de Física Estadística, Centro Atómico Bariloche (CNEA) and Instituto Balseiro (CNEA and UNC), 8400-San Carlos de Bariloche, Argentina' - 'Departament de Física, Universitat de les Illes Balears and IMEDEA (CSIC-UIB), E-07122 Palma de Mallorca, Spain' - 'Departamento de Física, FCEN, UBA, Ciudad Universitaria, Pab. I, C.P.(1428), Buenos Aires, Argentina' author: - 'S. Bouzat,' - 'H.S Wio ,' - 'G.B. Mindlin' title: Characterization of spatiotemporal chaos in an inhomogeneous active medium --- Active Media ,Spatiotemporal Chaos ,Bi-Orthogonal Decomposition 05.45.-a ,47.54.+r ,47.52.+j **Introduction** ================ Spatiotemporal chaos [@general1] has been extensively studied within the context of coupled maps, the complex Ginzburg–Landau equation, the Kuramoto–Shivashinsky equation and other related equations [@chaos1]. However, studies of spatiotemporal chaos in reaction–diffusion models closely connected to experimental systems are scarce. Here we analyze the characteristics of the chaotic dynamics recently found in a simple inhomogeneous reaction–diffusion system [@PLAinh] of the the type used to describe chemical reactions in gels [@gels] and patterns in coupled electrical circuits [@purw]. Among the main issues in the study of spatiotemporal chaos we can select those related to clarifying some aspects of the relation between pattern formation and chaos as well as the low dimensional description of the chaotic behavior. The latter aspect, that is understanding that physically continuous systems with an infinity of degrees of freedom (spatially extended systems) usually show temporal behavior that can be well described by models with few degrees of freedom, is of extreme relevance. In this context there arise some questions. For instance, in low dimensional dynamical systems, chaotic solutions coexist with unstable periodic orbits which constitute the backbone of the strange attractor: could some orbits be extracted from the time series of our extended system? and, is the complex time evolution of the system of a dimensionality small enough to be understood in terms of simple stretching and folding mechanisms? In order to investigate these questions within reaction–diffusion systems, we have analyzed the same simple, inhomogeneous, activator-inhibitor model discussed in Ref. [@PLAinh]. It is worth remembering here that reaction–diffusion models of the activator–inhibitor type have provided a useful theoretical framework for describing pattern formation phenomena with applications ranging from physics to chemistry, biology and technology [@murray; @mik; @general2]. In Ref. [@PLAinh] by introducing spatial dependence of the parameters of the activator–inhibitor equations, a system in which different parts of the media do not share the same reaction properties was modelled. The case considered corresponds to a finite one dimensional oscillatory medium with an immersed bistable spot. In that system, in addition to stationary, Hopf–like and Turing–like patterns, quasiperiodic inhomogeneous oscillations and spatiotemporal chaos were also found. In Ref. [@PREinh], different generalizations of the system (bidimensional versions) have been studied. In the present paper we analyze the dynamics of the same one dimensional system in the quasiperiodic and chaotic regions. More specifically, one of our main aims is to understand the topological properties of the chaotic dynamics. The model is given by the reaction–diffusion equations $$\begin{aligned} \label{sys} \dot{u}&=&\partial^2_x u-u^3+u-v \nonumber \\ \dot{v}&=&D_v \partial^2_x v + u -\gamma v,\end{aligned}$$ which describe a bistable medium for $\gamma>1$ and an oscillatory one for $\gamma<1$. In order to model the inhomogeneous situation of a bistable domain immersed in an oscillatory medium, a spatial dependence of this parameter is introduced setting $\gamma=\gamma(x)\equiv .9+5\,\exp(-10 \, x^4)$ [@coment1]. This leads to $\gamma \simeq .9 < 1$ for $|x|>.8$ (oscillatory medium) and $\gamma>1$ for $|x|<.8$ (bistable medium). As was done in [@PLAinh], we here consider a finite one–dimensional domain ($-L \le x \le L$) with non–flux boundary conditions in $\pm L$ and homogeneous initial conditions belonging to the homogeneous limit cycle that exists for the case $\gamma=.9$. This choice of the initial state corresponds to the description of an initially homogeneous oscillatory medium whose reaction properties are suddenly modified in a localized region. In the central bistable region the fields converge rapidly to values close to those corresponding to one of the two natural states of the bistable medium ($u_{\pm}\simeq\pm .8, v_{\pm}\simeq\pm .14$) (chosen depending on the initial condition), and continue performing small amplitude oscillations around those values. Hence, there is a spontaneous symmetry breaking which is “inherited” from the properties of the (uncoupled) bistable medium. Note that the equations of the model are symmetric under the simultaneous changes $u \to -u$, $v \to -v$. The rest of the system evolves to different asymptotic behaviors depending on the parameters $L$ and $D_v$ as indicated in Figure 1, and described in detail in [@PLAinh]. All the numerical calculations have been done as follows. First, the system of partial differential equations was approximated by a system of coupled ordinary differential equations, obtained by a finite difference scheme. Then the resulting equations were solved by a Runge–Kutta method of order $2$. Different space and time discretization schemes were employed in order to check the results. The organization of the paper is the following. In the next Section we show that, in the quasiperiodic and chaotic regimes, there are two dynamical time scales, a fast and a slow one. We show that it is possible to find segments of the time series of the slow dynamics which approximate unstable periodic orbits and study the organization of the orbits. In Section III, we present the biorthogonal decomposition of the spatiotemporal time series, and show that it is possible to capture the main features of the chaotic dynamics by considering a small number of modes. In the last Section we present our conclusions. Characterization of the slow chaotic dynamics: analysis of the unstable periodic orbits. ======================================================================================== In the non–stationary regions of the phase diagram shown in Figure 1, the time evolution of the fields $u$ and $v$ is classified as periodic, quasiperiodic or chaotic [@PLAinh]. Here, we analyze the transition from periodic oscillations to chaotic behavior along the line indicated with a vertical dotted segment in Figure 1. We fix $L=72$, for which the dynamics corresponds to inhomogeneous periodic oscillations for $D_v<1$, quasiperiodic oscillations for $1<D_v<1.3$, and spatiotemporal chaos for $1.3<D_v<2$. (For $D_v>2$ the quasiperiodic and periodic behaviors appear again and for $D_v>2.3$ stationary Turing patterns arise.) To begin, we will mainly focus our attention on the chaotic region. As a measure of chaoticity, in [@PLAinh], the sensibility to initial conditions was computed. It is important to notice that the time series displayed a common feature: a fast oscillation of the field (at the natural frequency of the oscillatory medium), eventually modulated by a slow varying amplitude. It is the dynamics of this amplitude what we will analyze here. In order to study this slow dynamics we record the times $t_n$ ($n=1,2,...$) at which the $u$–field at $x=L$ reaches a local maximum as function of $t$ (i.e. when $ \partial_t u(x,t)|_{x=L}=0$ and $ \partial^2_t u(x,t)|_{x=L}<0$ holds simultaneously), and analyze the values of $u$ for these times at different spatial positions. This is equivalent to taking a Poincare section, and is a way of averaging the fast time scales. The difference $t_n-t_{n-1}$ is of the order of the natural period of the oscillatory medium ($\tau_0=14.6$), but slightly larger (in general, it fluctuates between $\tau_0$ and $20$) as the oscillations are slowed down by the presence of the bistable inhomogeneity. In the periodic region, $t_n-t_{n-1}$ converges to the period of the motion as $n \to \infty$. Typical time series are shown in Figure 2. In Figure 2a (c), the time evolution of $u(L)$ is displayed for a parameter value at which the system behaves quasiperiodically (chaotically). The slow varying amplitude is shown in Figure 2b (d), where we have plotted the values of the maxima of $u(L)$ as a function of $t$ (i.e. $u(L)$ measured at times $t_n$). In general, in low dimensional dynamical systems, chaotic solutions coexist with unstable periodic orbits which constitute the backbone of the strange attractor. We will see that, in our system, it is possible to extract approximations of periodic unstable orbits from the time series of the mentioned slow dynamics, and that the analysis of the organization of these orbits shows that the chaotic dynamics is low–dimensional. We begin by defining as [*reconstructed periodic orbits*]{} the segments of the time series which can be used as surrogates of the unstable periodic orbits of the system. These segments are chosen if they pass a close return test [@mind92]. More precisely, if $y(i)$ represents the data, a close return is a segment of $p$ points beginning at the $i^{th}$ position of the file, for which $y(i+k) \approx y(i+k+p)$ for $k=1,2,...$. In this notation, $p$ is called the period. We have looked for unstable orbits at the whole time series of the slow dynamics of the $u$ field (data taken at times $t_n$) at four different positions: $x_0=0$, $x_1=14$ (approximately one Turing wavelength away the bistable domain), $x\simeq x_2=L/2=36$, and $x=L$. In Figure 3a(c) we display a segment of period 2(4) taken from a time series corresponding to data at $x=L$. An embedding of the data (a multivariate environment created using time delays) is shown in Figure 3b(d). In Figure 4a and 4b we show the embedding of two different reconstructed segments of periods two and three respectively, coming from data at $x=x_1$. It is worth mentioning that the unstable periodic orbits do not have properties corresponding to the inversion symmetry of Eqs. (\[sys\]) because of the symmetry breaking of the solutions and also because of the “stroboscopic” observation of the dynamics. In Figure 4c, we show a more complex reconstructed periodic orbit coming also from data at $x=x_1$. Since we have no elements to conjecture that the chaotic dynamics can live in three dimensions, it could be argued that embedding the segments in a three dimensional space might not be useful. Yet, if the reconstructed shows some kind of geometrical organization it would be a most valuable indication of the geometric process taking place in a small dimensional manifold within the available phase space. It is possible to see that the orbits of Figures 3b and 3d wind around each other as expected if they were related by a period doubling bifurcation. The topological organization of the orbits is quantitatively described in terms of their relative rotation rates and self relative rotation rates. These numbers aim at describing the way in which the orbits wind around each other [@gilm00]. In order to do so, the curves are given an orientation, and in a two dimensional projection, a record is made of which segments pass over which in the original embedded orbits. In terms of these indices, the relative rotation rates are computed as explained in [@gilm00]. For the period two orbit of Figure 4a, the self relative rotation rate is $srrr=-\frac{1}{2},0$, for the period three orbit of Figure 4b, it is $srrr=(-\frac{1}{3})^2,0$, and the relative rotation rate between the orbits of period two and three was found to be $rrr=-\frac{1}{3}$. Notice that this organization is compatible with a horseshoe mechanism [@gilm00], and that this mechanism includes the signature of period doubling. A challenge exists in order to find a simple geometrical mechanism responsible for the creation of the orbit displayed in Figure 4c. This orbit can not be placed in a horseshoe template. Yet, recently, a classification of templates was proposed for covering the Smale horseshoe [@LetGil]. We have observed that the orbit of Figure 4c can be placed in one of such geometric objects, which is one of the four inequivalent four-branched [*double covers*]{} with rotation symmetry of the Smale. More specifically, the one identified with topological indices $(n_0,n_1)=(1,0)$ [@LetGil]. This template can also hold any orbit of the Smale horseshoe template. However, it can not be expected that such template correctly describes the whole dynamics of the slow varying amplitude. This is because it is not possible that a rotation symmetry appear when using a delay embedding. Hence, the embedded attractor must have a different symmetry or not symmetry at all, and it is expected that other unstable periodic orbits exist (different to the ones we have found and with no rotation symmetry). Note that, when observing the $u$–field at times $t_n$, it is found that the scales over which it varies are quite different at the four studied positions ($x_0,x_1,x_2$ and $x_L$): at $x=L$, $u(t_n)$ oscillates between $.2$ and $.35$ (since we are watching only the times at which $u(L,t)$ is maximum); at $x=x_1$ and $x=x_2$, $u(t_n)$ take values in a more or less symmetric way between $\pm .35$ (in the whole range of the free limit cycle); at $x=x_0$ (in the bistable domain) the oscillations are of much smaller amplitude (typically two orders of magnitude), and are not centered at zero. We remark that, in spite of these differences in the metrical properties of the dynamics at the several positions, the organization of the unstable periodic orbits that we have found is the same everywhere. (In the four positions we find the same kind of orbits, including the one of Figure 4c.) However, there are some differences in the frequency of occurrence of the orbits: note that, we have orbits with the “small curl” upward (as in Figure 3b) or downward (as in Figure 4a). The same two possibilities appear for orbits of periods three and four. For the cases of the signal taken at $x_0,x_1$ and $x_2$, the orbits of period two and three occur preferably with the small curl downward, while, for $x=L$ they occur (almost always) with the small curl upward. This is found independently of whether the fields in the bistable domain converge to negative or positive values. The observation that the organization of unstable periodic orbits is more complex, but some how related to the one of the Smale horseshoe, gives a hint of what kind of periodic orbits can eventually be found as parameters are changed. For example, it suggest that a period doubling sequence may occur in the transition from the periodic regime to the chaotic regime. With this in mind, we revisited in detail the transition zone in the phase diagram of the system going from the periodic region to the chaotic one along the segment indicated in Figure 1. A period doubling of the slow dynamics can be clearly identified, as can be seen in Figure 5. The analysis made of the slow dynamics of our extended system showed that the high dimension of the phase space is not fully explored. On the contrary, an important collapse of dimensionality takes place. In the next section we investigate the minimum number of spatial (linear) modes approximating the spatiotemporal dynamics that are required in order to recover the topological organization of unstable periodic orbits observed in the slow dynamics. Biorthogonal decomposition. =========================== It is not easy to know a priori which is the number of spatial modes which are activated as the dynamics becomes non trivial. In our problem, we only know that at least three modes should be active in order to account for the complex behavior described in the previous section. A method exists to unveil the active structures in a spatiotemporal problem: the biorthogonal decomposition (BOD) [@bod1; @bod2]. This is the most efficient linear decomposition scheme, in the sense that there is no other linear decomposition able to capture, with a smaller number of modes, the same degree of approximation. In our system, the BOD for the spatiotemporal signal $(u(x,t),v(x,t))$ is given by $$\label{bod} (u(x,t),v(x,t))=\sum_{k=1}^{\infty} \alpha_k \psi_k(t) \vec\phi_k(x),$$ where the $\alpha^2_k$ (with $\alpha_1 > \alpha_2 >...> 0$) are the eigenvalues of the temporally–averaged two point correlation matrix [@bod1], the $\vec\phi_k(x)=(\phi_{u k}(x),\phi_{v k}(x))$ are the corresponding eigenfunctions (called topos), and the $\psi_k(t)$ (called chronos) are given by $$\psi_k(t)=\frac{1}{\alpha_k}\int_{0}^L \left( u(x,t) \phi_u(x)+v(x,t) \phi_v(x) \right) dx.$$ We have observed that, for the system (\[sys\]), the main differences in the BOD corresponding to the three dynamical regimes appear in the chronos and that the spatial modes are similar in all the three cases. However, we have neither studied in detail the BOD along the transition to chaos nor analyzed the question of modes’ competition. Our analysis was mainly focused on finding the number of modes that are necessary to recover the topological organization of unstable orbits for the chaotic situation presented in the previous section. In Figure 6a we show the eigenvalues of the BOD computed for three different points along the transition line indicated in Figure 1: a periodic case, a quasiperiodic case and the chaotic situation studied in the previous section. In Figure 6b we show the first four topos for the chaotic case. No significant differences are observed in the spatial modes corresponding to the three regimes. We have observed that, in all the three regimes, the $i-th$ mode has $i-1$ spatial nodes (that is, the spatial points where $u(x)=0$), see Figure 6b. Also, in all the three cases, the second mode is quasi–stationary and it mainly contributes to the formation of the fields’ profiles in the (quasi–stationary) bistable region. (Note that the topo 2 in Figure 6b contributes only around the bistable region ($x\sim 0$)). In the case of periodic motion, in order to reconstruct the trivial topology of a single periodic orbit, only the first mode is necessary (which gives a quasi–homogeneous periodic oscillation). Moreover, in this region, we have observed that the whole spatiotemporal dynamics (that is, the periodic wave propagation phenomenon) can be highly accurately described by considering an expansion with only three modes (\[bod\]), as it is suggested by Figure 6a. Regarding the description of quasiperiodic and chaotic motion, it requires a higher number of modes, as can be inferred from Figure 6a. In these regimes all the chronos seems to be non periodic (excepting the second, which is constant up to a good approximation). Finally, for the chaotic case analyzed in the previous section, we have reconstructed the dynamics of the system using different numbers of modes. The main result of our analysis is that the minimum number of modes required to recover the topological organization of orbits is five. This means that using five modes (and not four) we were able to recover the orbits presented in the previous section. There seems to be neither something special in this number, nor a way of having predicted it a priori. However, it is important to point out that the extended system under study can in principle display an infinite dimensional dynamics, and yet, it dynamically collapses to a five dimensional system which describe the dynamics of the amplitudes of the linear modes. Furthermore, it is remarkable that the fact that five modes are active does not imply that the dimensionality of the observed strange attractor is larger than four. On the contrary, the topological organization of the approximated unstable periodic orbits clearly suggest a lower dimensionality. Conclusions =========== In this work we studied the spatiotemporal solutions of a reaction-diffusion system of the activator-inhibitor type. Despite the infinite number of possible degrees of freedom, we have found that the complex dynamics that emerges can be described in terms of a small number of modes. The activated modes are coherent structures which were computed from the simulations of this extended problem. By separating the dynamics over two time scales, we observed that the origin of the chaoticity lies on the behavior of the slow time scale dynamics. The study of these time series showed not only that the system behaves as a small dimensional dynamical system, but also suggest that this dynamics may be understood in terms of simple geometrical process related to the Smale horseshoe. In fact, a branched manifold recently described in the literature can hold all the approximated unstable orbits that we were able to reconstruct. However, symmetry reasons indicate that the true mechanism should not be exactly the one corresponding to that template. The description of the dynamics in terms of a simple geometric structure not only highlights the collapse of dimensionality, but it also allowed us to predict the existence of specific solutions for unexplored regions of parameter space, such as the reported period doubling sequence. [**ACKNOWLEDGMENTS:\ **]{} To Verónica Grunfeld for reading the manuscript. Partial support from ANPCyT and UBACyT, Argentine agencies, is also acknowledged. SB and GB thanks for the kind hospitality extended to them during their stays at the DF (UBA) and IB-CAB respectively. HSW thanks the MECyD, Spain, for an award within the [*Sabbatical Program for Visiting Professors*]{}, and to the Universitat de les Illes Balears for the kind hospitality extended to him. [99]{} M.C. Cross and P.C. Hohenberg, Rev. Mod. Phys. [**65**]{}, 851 (1993). K. Kaneko, Physica D [**34**]{}, 1 (1989); P. C. Hohenberg and B. I. Shraiman, Physica D [**37**]{}, 109 (1989); B. I. Shraiman, et al., Physica D [**57**]{}, 241 (1992); A. De Wit, G. Dewel and P. Borckmans, Phys. Rev. E [**48**]{}, R4191 (1993); R. Montagne, et al., Phys. Rev. E [**56**]{}, 151 (1997). See also Sect. VII of Ref.[@general1]. S. Bouzat and H.S. Wio, Phys. Lett. A 268 (2000) 323. J. J. Tyson and P. C. Fife, J. Chem. Phys. [**73**]{}(5), 2224 (1980); S. Kádár, J. Wang and K. Showalter, Nature [**391**]{} 770 (1998); A. Horváth, M. Dolnik, A. P. Muñuzuri, A. Zhabotinsky and I. Epstein, Phys. Rev. Lett. [**83**]{}, 2950 (1999). G. Heidemann, M. Bode and H.G. Purwins, Phys. Lett. A [**177**]{} (1993) 225. J.D. Murray, [*Mathematical Biology*]{} (Springer-Verlag, 1985). A.S. Mikhailov, [*Foundations of Synergetics I*]{}, (Springer-Verlag, 1990). D. Walgraef, [*Spatio-temporal pattern formation*]{}, (Springer-Verlag, New York , 1997); E. Meron, Phys. Rep. [**218**]{}, 1 (1992) S. Bouzat and H.S. Wio, Physical Review E [**63**]{}, 056213 (10 pags.) (May 2001). The exact shape of this function is unimportant, as was discussed in [@PLAinh] and [@PREinh]. G. B. Mindlin and R. Gilmore, Physica D [**58**]{} 229 (1992). R. Gilmore, Rev. Mod. Phys. [**70**]{}, No. 4, (1998). C. Letellier and R. Gilmore, Phys Rev E, [**63**]{} 016206 (2000). N. Aubry, R. Guyonnet and R. Lima, Jour. of Stat. Phys. [**64**]{}, 683 1991. P Holmes, J. L. Lumely, G. Berkooz, [*Turbulence, Coherent structures dynamical systems and symmetry*]{}, Cambridge University Press, 1997

--- abstract: 'Automated detection of cervical cancer cells or cell clumps has the potential to significantly reduce error and increase productivity in cervical cancer screening. However, most traditional methods rely on the success of accurate cell segmentation and discriminative hand-crafted features extraction. Recently there are emerging deep learning-based methods which train convolutional neural networks (CNN) to classify image patches, but they are computationally expensive. In this paper we propose to exploit contemporary object detection methods for cervical cancer detection. To deal with the limited size of training samples, we develop the comparison classifier into the state-of-the-art two-stage object detection method based on the comparison with the reference images of each category. In addition, we propose to learn the reference images of the background from the data instead of manually choosing them by some heuristic rules. This architecture, called the Comparison detector, shows significant improvement for small size dataset, achieving a mean Average Precision (mAP) 26.3% and an Average Recall (AR) 35.7%, both improving about **20** points compared to baseline model. Moreover, Comparison detector achieves same mAP performance as the current state-of-the-art model when training on the medium size dataset, and improves AR by **4** points. Our method is promising for the development of automation-assisted cervical cancer screening systems.' author: - 'Yixiong Liang, Zhihong Tang, Meng Yan, Jialin Chen, and Yao Xiang [^1] [^2] [^3]' bibliography: - 'IEEEabrv.bib' - 'paper.bib' title: 'Comparison Detector: Convolutional Neural Networks for Cervical Cell Detection' --- Cervical cancer screening, object detection, convolutional neural networks, prototype-based classification Introduction ============ Cervical cytology is the most common and effective screening method for cervical cancer and premalignant cervical lesions [@davey2006effect], which is performed by a visual examination of cytopathological analysis under the microscope of the collected cells that have been smeared on a glass slide and stained and finally giving a diagnosis report according to the descriptive diagnosis method of the Bethesda system (TBS)[@nayar2015bethesda]. Currently in developed countries, it has been widely used and has significantly reduced the number of deaths caused by related diseases, but it is still unavailable for population-wide screening in the developing countries [@saslow2012american], partly due to the fact that it is labor-intensive, time-consuming and expensive [@bengtsson2014screening]. In addition, it is subjective and therefore has motivated lots of automated methods for the automation of cervical screening based on the image analysis techniques. Over the past 30 years extensive research has attempted to develop automation-assisted screening methods [@birdsong1996automated; @chankong2014automatic; @zhang2014automation; @phoulady2016automatic]. Most of them try to classify a single cell into various stages of carcinoma, which often consists three steps: cell (cytoplasm and nuclei) segmentation, feature extraction and classification. The performance of these methods, however, heavily depends on the accuracy of the segmentation and the effectiveness of the hand-crafted features. With the overwhelming success in a broad range of applications such as image classification [@krizhevsky2012imagenet; @he2016deep], semantic segmentation [@long2015fully], object detection [@ren2015faster; @lin2017feature] and medical imaging analysis [@litjens2017survey; @gulshan2016development; @esteva2017dermatologist], CNN has also been applied to the segmentation and classification of cervical cell [@tareef2017optimizing; @song2017accurate; @song2015accurate; @zhang2017combining; @lu2017evaluation; @zhang2017deeppap; @jith2018deepcerv; @gautam2018considerations]. The majority of them [@tareef2017optimizing; @song2015accurate; @zhang2017combining] are trying to take advantage of CNN to improve the segmentation accuracy of cytoplasm and nuclei, but they do not provide the needed segmentation accuracy [@lu2017evaluation; @jith2018deepcerv], whereas once the segmentation error are taken into account, the classification accuracy would drop [@zhang2017deeppap]. To avoid the dependence on accurate segmentation, the patch-based methods try to use CNN to classify the image patches [@jith2018deepcerv; @gautam2018considerations]. However, the extraction of such patches still requires the segmentation of nuclei. The recent work [@zhang2017deeppap] also adopt the patch-based strategy but during the inference the random-view aggregation and multiple crop testing are needed to produce the final prediction results and thereby is time-consuming. In this paper, we propose an efficient and effective strategy to apply CNN for cervical cancer screening, *without* any pre-segmentation step. Specifically, we exploit the contemporary CNN-based object detection methods [@ren2015faster; @lin2017feature] to detect the cervical cytological abnormalities directly. It is straightforward and has been successfully applied for other medical image analysis [@liang2018end; @liang2018object], but we are not aware of any works try to apply CNN-based object detection for automated cervical cytology. We attribute this to the lack of the right cervical cancer microscopic image dataset for the detection task. CNN-based object detection methods often need sufficient annotated data to obtain good generalization, but for cervical cytological abnormalities detection, collecting the large amounts of data with careful and accurate annotation is difficult partially due to the limitation by laws, the scarcity of positive samples and especially the unanimous agreement between cytopathologists [@stoler2001interobserver]. To alleviate the small training dataset size problem, we propose the named *Comparison Detector*, which migrate the idea of *comparison* in one/few-shot learning for image classification [@koch2015siamese; @vinyals2016matching; @snell2017prototypical; @yang2018learning] into CNN-based object detection, for cervical cancer detection. Specifically, we choose the state-of-the-art object detection method, Faster R-CNN [@ren2015faster] with FPN [@lin2017feature], as our baseline model and replace the original parameter classifier with a non-parametric one based on the idea of comparison with the reference images of each category. Furthermore, instead of manually choosing the reference images of the background category by some heuristic rules, we propose to learn them from the data. We also investigate several important factors including generating prototype representations of categories and the design of head model for cervical cell detection. Our algorithm directly operate on the whole image rather than the extracted patches based on the nuclei and hereby only need one forward propagation for each image, making the inference very efficient. In addition, the proposed method is *flexible* to be intergraded into other proposal-based methods. We collect a small size dataset $D_s$ and a medium size dataset $D_f$ which are directly dedicated to cervical cell/clumps detection, on which we evaluate the performance of the proposed Comparison Detector. When the model is learned from the small size dataset, the performance of our method is significantly better than the one of baseline model, i.e. Comparison Detector has an mAP 26.3% and an AR 35.7% but the baseline model only gains an mAP 6.6% and an AR 12.9%. When the model is learn from the medium size dataset, our Comparison Detector achieves almost the same performance with a mAP of 45.3%, but improves nearly 4 points comparing to baseline model with AR. We summarize our contributions as follows: 1) To the best of our knowledge, this is the first application of the CNN-based object detection methods to cervical cancer detection; 2) We propose Comparison Detector method to deal with the small training sample size problem in cervical cell detection; 3) We propose a strategy to directly learn the background reference images and 4) Our method performs much better than the baseline on both small size and medium size dataset and has the potential applications to the real automation-assisted cervical cancer screening systems. Related work ------------ **Cervical cell segmentation and classification.** Traditional cytological criteria for classifying cervical cell abnormalities are based on the changes in nucleus to cytoplasm ratio, nuclear size, irregularity of nuclear shape and membrane, therefore there are numerous works focusing on the segmentation of cell or cell components (nuclei, cytoplasm) [@chankong2014automatic; @zhang2014segmentation; @gencctav2012unsupervised; @chen2014semi; @song2015accurate; @song2017accurate; @zhang2017graph; @lu2015improved; @lee2016segmentation; @li2012cytoplasm; @tareef2018multi]. Although significant progress has been achieved recently, the segmentation of cell or cell components remains an open problem due to the large shape and appearance variation between cells, the poor contrast of cytoplasm boundaries and the overlap between cells [@zhang2017deeppap; @lee2016segmentation; @tareef2018multi; @lu2017evaluation]. On the other hand, cervical cell classification methods try to differentiate the dysplastic cells from the norm cells and classify them into various stages of carcinoma. According to TBS rules [@nayar2015bethesda], a large number of hand-crafted features are designed to describe the shape, texture and appearance characteristics of the nucleus and cytoplasm [@gencctav2012unsupervised; @chen2014semi; @marinakis2009pap; @chankong2014automatic; @phoulady2016automatic; @bora2017automated]. The resulting features are often further organized by feature selection or dimensionality reduction and then are fed into various classifiers (e.g. random forests, SVM, softmax regression, neural network, etc.) to perform the final classification. However, as mentioned above, the extraction of those engineered features depends on the accurate segmentation of cell or cell components. Furthermore, it is also limited by the current understanding of cervical cytology [@zhang2017deeppap]. To reduce the dependency on the accurate segmentation, the CNN are used to learn the features from data recently [@jith2018deepcerv; @gautam2018considerations], but an approximate segmentation or (region of interest) ROI detection is still necessary. Although the DeepPap [@zhang2017deeppap] is claimed totally segmentation-free, it still needs the nucleus centroid information for training and the random-view aggregation and multiple crop testing during the inference stage, which are very time-consuming. There are a handful public available microscopic image datasets dedicated to cervical cell segmentation such as ISBI-14[^4], ISBI-15 [^5], but to our best knowledge for cervical cell classification the only public available microscopic image dataset is the Herlev benchmark dataset [@marinakis2009pap], which consists of 917 single cell images corresponding to four categories of abnormal cell with different severity (namely light dysplastic, moderate dysplastic, severe dysplastic and carcinoma in situ) and three categories of normal cells (normal columnar, normal intermediate and normal superficial). The limited annotated data prevents the applications of traditional object detection methods such as Viola-Jones detector [@viola2004robust] or contemporary CNN-based detectors [@liu2018deep] to cervical cancer screening. **CNN-based object detection.** The Overfeat [@sermanet2013overfeat] made the earliest efforts to apply CNN for object detection and has achieved a significant improvement of more than 50% mAP when compared to the best methods at that time which were based on the hand-crafted features. Since then, a lot of CNN-based methods [@ren2015faster; @liu2016ssd; @girshick2014rich; @girshick2015fast; @he2017mask; @li2017light; @lin2018focal; @singh2018r; @redmon2016you; @redmon2017yolo9000; @zhang2018single; @redmon2018yolov3] have been proposed for high-quality object detection, which can be roughly classified into two categories: object proposal-based and proposal-free. The road-map of proposal-based methods starts from the notable R-CNN [@girshick2014rich] and is improved by Fast-RCNN [@girshick2015fast] in an end-to-end manner and by Faster R-CNN [@ren2015faster] to quickly generate object regions, which has motivated a lot of follow-up improvements [@lin2017feature; @he2017mask; @li2017light; @lin2018focal; @singh2018r] in terms of accuracy and speed. The proposal-free methods [@liu2016ssd; @redmon2016you; @redmon2017yolo9000; @redmon2018yolov3] directly predict the bounding boxes without the proposal generation step. Generally, the proposal-free methods are conceptually simpler and much faster than the proposal-based methods, but the detection accuracy is usually behind that of the proposal-based methods [@zhang2018single]. Here we choose the Faster R-CNN [@ren2015faster] with FPN [@lin2017feature] as our baseline model but our method is compatible with other proposal-based methods.  **One/few-shot learning.** One/few-shot learning is a task of learning from just one or a few training samples per class and has been extensively discussed in the context of image recognition and classification [@koch2015siamese; @vinyals2016matching; @snell2017prototypical]. Recently significant progress has been made for one/few-shot learning tackled by meta-learning or learning-to-learn strategy, which can be roughly divided into three categories: metric-based, memory-based and optimization-based. The metric-based methods [@koch2015siamese; @vinyals2016matching; @snell2017prototypical; @yang2018learning] learn to compare the query image with support set images. The memory-based methods [@santoro2016one] exploit the memory-augmented neural network to quickly store and retrieve sufficient information for each classification task, while the optimization-based methods [@ravi2016optimization; @finn2017model] aim to learning a base-model which can be fine-tuned quickly for a new classification task. All these works only tackle image classification tasks. **Object detection with limited-data.** Most prior works on object detection with limited labels use semi-/weakly-supervised methods or few-example learning [@dong2018few] to make use of abundant unlabeled data, whereas in limited-data regime there are few work focus on using few-shot learning to address this problem [@schwartz2018repmet; @kang2018few]. Kang et al. [@kang2018few] decompose the training into base-model learning and meta-model learning and train a meta-model to reweight the features extracted by the base-model to assist novel object detection. However, the training of base model still needs abundant annotated data for base classes. RepMet [@schwartz2018repmet] introduces a metric learning-based sub-network architecture to learn the embedding space and distribution of the training categories without using external data. However, RepMat involves an alternating optimization between the external class distribution module learning and net parameters updating, whereas our solution is a clean, single-step training framework. Comparison Detector {#section3} =================== Basic Architecture ------------------ Our proposed comparison detector is based on proposal-based detection framework consisting of a region proposal network (RPN) for proposal generating, a backbone network for feature extraction and a head for the proposal classification and bounding box regression. Here we choose the Faster R-CNN with FPN [@lin2017feature] as our baseline. Then we decouple the regression and classification in the head and replace the original parameter classifier with our comparison classifier. Our no-parameters classifier introduces a inductive bias, namely the within-class distance is less than the between-class in the embedding space, into the model and henceforth mitigates the small sample size issue to some extent [@battaglia2018relational]. The framework of the proposed Comparison detector is depicted in Fig.\[fig:1\], which is divided into three stages to describe. At the first stage, as shown in Fig. \[fig:1\](a), the features of both the reference and the object images are computed by backbone network with FPN [@lin2017feature], without using any extra models to encode the reference images. The only difference is there are no RPN operation on the reference images. Assuming that there are $n$ samples per category with $t$ levels pyramid feature in the reference images. Let $F_i^l$ be the $i$-th categories’ prototype representation of the $l$-level pyramid features, which can be computed by average operation as follows $$F_i^l = \frac{1}{n} \sum_jF^l (R_{ij}), \label{equ:1}$$ where $F^l(\cdot)$ and $R_{ij}$ denote the $l$-th level feature extraction function and the $j$-th reference image of class $i$, respectively. The second stage is to generate the prototype representations of each category from the reference images’ pyramid features. We need to find a map function $S(\cdot)$ which use all level pyramid features each category as input to compute the final prototype representation $F_i$ for class $i$ $$F_i = S (\{F_i^l\}). \label{equ:3}$$ The third stage is the design of the head model for classification and bounding box regression (Fig. \[fig:1\](b)), consisting of a few convolutional ($Conv$) and fully connected ($FC$) layers. Let $d(P_m, F_i)$ be a metric function to compute the distance between the feature of $m$-th proposal $P_m$ and prototype representation of the $i$-th category $F_i$. It is important to note that $P_m$ and $F_i$ have the same size. Each proposal’s classification $p_i$ and bounding box regression $b_i$ can be obtained by $$p_i = \frac{e^{-d(P_m, F_i)}}{\sum_ke^{-d(P_m, F_k)}}, \label{equ:4}$$ $$b_i = B(P_m, F_i), \label{equ:5}$$ where $B(\cdot,\cdot)$ denotes the box regression function. The rest of the model is the same as Faster R-CNN with FPN model [@lin2017feature]. ![ The block for learning prototype representations of background class[]{data-label="fig:2"}](figure2.pdf) Learning the reference background --------------------------------- There are many negative proposals generated by RPN, so the R-CNN [@girshick2014rich] adds a background category to represent them. In our Comparison detector, we need to select a number of reference images for each category and therefore we also need to choose reference images for the background category. Due to the overwhelming diversity, selecting background reference is very difficult. Notice that a region is considered to a the proposal indicating that it has certain similarity with categories. Therefore, it can be inferred that its features are a combination of different categories in the most case. So we propose to learn it by combining the prototype representations of all the categories in the reference samples, which can be implemented by a simple $1\times1$ convolution operation, as shown in Fig. \[fig:2\]. Generating prototype representations of categories -------------------------------------------------- As shown in Eq. \[equ:4\], the Comparison detector uses metric function to measure distance or the dissimilarity between the prototype representation of categories and the features of the proposal, then obtains the label of the proposals based on the dissimilarity. Features of proposal may come from any of the four level pyramids, and the prototype representation of the categories is obtained according to Eq. \[equ:3\]. For simplicity, we directly resize the each feature pyramid which is generated by reference images to a fixed size, and then calculate prototype representation by averaging operation, i.e. $$\label{pr_avg} S(\{F_i^l\}) = \frac{1}{t}\sum_{l}r(F_i^l, s),$$ where $t$ is the total number of level feature pyramids, $r(\cdot, \cdot)$ is resize function and $s$ is the size of final features. Different levels pyramid features of the category are resized into fixed size, and then getting the prototype representation by simply averaging them. The head for classification and regression {#section:3.3} ------------------------------------------ As shown in Fig. \[fig:3\](a), the structure of the baseline model’s head is to transform the proposal feature firstly and then one branch is used for classification, and another is used to predict the offset of the bounding box. For our Comparison detector, due to the introduction of the reference images, we need to re-organise the head. The are two choices according to whether the reference images are involved in the box regression branch. One is that the reference prototypes are only used for classification, as shown in Fig. \[fig:1\](b). Unlike the baseline model, the comparison classifier and bounding box regressor in the head of Comparison detector are independent. And the bounding box regressor only uses the features of ROI to predict the offset of the bounding box. It is equivalent to $$\begin{aligned} d(P_m, F_i) & = FC(F(P_m, F_i)), \\ B(P_m, F_i) & = FC(FC(FC(P_m))),\end{aligned}$$ where $F(P_m, F_i) = Conv_3(Conv_1(|F_i - P_m|^2))$. Another choice is to use the reference prototypes for both classification and regression, as shown in Fig. \[fig:3\](b), which means $$B(P_m, F_i) = FC(F(P_m, F_i)).$$ We call this method as shared module. They all achieve good performance in our experiments but the independent module performs slightly better (see Table \[tab:1\]). Reference images sampling ------------------------- In our Comparison detector, we also need to choose the reference images for each category. A intuitive way is to select them according to the Bethesda atlas [@nayar2015bethesda]. However, there are very significant difference between the given atlas and our data due to the variations of the preparation and digitization of slide. Hence we resort to other feasible data-driven alternatives. We randomly select about 150 instances of each category from the training sets. The shortest side of these instances is greater than 16 pixels. Therefore we get a total of 1560 instances and from them, we can select suitable instances in these objects as our reference images. There are two possible way. The first is to randomly choose several instances of each category as the reference images. The second is to first map all 1560 objects into the feature space through the pre-trained model and get the features of each object and then use t-SNE [@maaten2008visualizing] for feature dimension reduction (Fig. \[fig:4\]). Based on the results of t-SNE, we select the most representative samples in 3D space as our reference images. Experiment and Result ===================== Materials and experiments {#sect:4.1} ------------------------- Since there are no established benchmarks for cervical cell object detection in the community, we first establish a database consisting of 7,086 cervical microscopical images and based on which 48,587 object instance bounding boxes were labeled by experienced pathologists. Conforming to TBS categories [@nayar2015bethesda], we divide these objects into 11 categories, namely ASC-US (ascus), ASC-H (asch), low-grade squamous intraepithelial lesion (lsil), high-grade squamous intraepithelial lesion (hsil), squamous-cell carcinoma (scc), atypical glandular cells (agc), trichomonas (trich), candida (cand), flora, herps, actinomyces (actin). Figure \[fig:5\] shows some examples of each category in our database. Then we divide the dataset into training set $D_f$ which contains 6667 images, test set which contains 419 images for experiment. We randomly choose 762 images from the training dataset to form a small dataset of $D_s$. The number of categories in each dataset is shown in the Fig. \[fig:6\] {width=".7\textwidth"}  In all experiments, we used ResNet50 as backbone network with ImageNet pre-trained model. For reference images, we re-scale them such that their side is $w=h=224$ which is coincident with pre-trained model. The initial learning rate is 0.001, and then decreased by a factor of 10 at 35-th and 50-th epoch. Training is stopped after 60 epochs and the other parameters are the same as FPN [@lin2017feature]. The experiment is firstly trained on the $D_f$ to evaluate the performance on sufficient data. In our setting, the reference images are fixed in each training iteration for the stability of the training model. And test stage is the same. As for the cervical cell images, annotators are prone to take a higher threshold when label the objects due to the low discrimination of them. At the same time, multiple nearby objects with the same category will be marked as one, so the performance of the model can not be well reflected by mAP. Therefore, the performance of the model is evaluated by using mAP and AR as a complement on test set. If the mAP does not decrease and the AR improves, it surely signifies the performance is improved. Herein, the results are reported in both mAP and AR. A summary of results can be found in Table \[tab:1\] and some detection results on the test set are shown in Fig.\[fig:7\]. . [|c|c|c|c|c|c|c|c|]{} ------------------------------------------------------------------------ **model & ****** ------------ learning background ------------ & **independent mode & ****** ------------------ using all pyramid features ------------------ & **refine box & **balance loss & **mAP & **AR\ ******** ------------------------------------------------------------------------ `A` &$\surd$ &$\surd$ &$\surd$ &$\surd$ & &34.1 &53.3\ ------------------------------------------------------------------------ `B` & &$\surd$ &$\surd$ &$\surd$ & &31.4 &49.3\ ------------------------------------------------------------------------ `C` &$\surd$ &$\surd$ & &$\surd$ & &32.7 &50.8\ ------------------------------------------------------------------------ `D` &$\surd$ & &$\surd$ &$\surd$ & &41.0 &51.3\ ------------------------------------------------------------------------ `E` &$\surd$ & & &$\surd$ & &38.9 &49.8\ ------------------------------------------------------------------------ `F` &$\surd$ &$\surd$ &$\surd$ & & &37.7 &51.1\ ------------------------------------------------------------------------ `G` &$\surd$ &$\surd$ &$\surd$ & &$\surd$ &38.8 &52.3\ ------------------------------------------------------------------------ `H` &$\surd$ & &$\surd$ &$\surd$ &$\surd$ &43.5 &58.9\ ------------------------------------------------------------------------ `I` &$\surd$ &$\surd$ &$\surd$ &$\surd$ &$\surd$ &**43.7** &**60.7**\ ----------------------------------------------------------------------------------- -- -- -- **comparator & **$\ell_2$-distance &**parameterized $\ell_2$-distance & **concat\ mAP &34.1(43.7)& 38.2(**44.5**) & 40.7(42.5)\ AR &53.3(60.7)& 56.8(**61.6**) & 49.1(58.1)\ ******** ----------------------------------------------------------------------------------- -- -- -- Reference background -------------------- We first evaluate our scheme to learn the background reference. Experiment shows our method is feasible (See model `A` in Table \[tab:1\]). It should be noted that because the prototype representation of the background category is learned from the prototype representation of other categories, the gradient propagation will also have some effect on the optimization of other prototype representation. In order to make sure whether this effect is beneficial, we stop gradient propagation at the fork position in Fig. \[fig:2\]. The performance of the model has declined with an mAP of 33.0% and a AR of 52.6%. In order to compare the effect of learning background, we randomly select some background samples from the proposals to obtain the features of background. The result is model `B` in Table \[tab:1\]. Besides generating the prototype of background, we try to remove it but the training fails to converge. Prototype representations of categories --------------------------------------- In our approach, as shown in Eq. \[pr\_avg\], we use all pyramid features to generate prototype representation of categories. Another choice is to only use the last level pyramid feature as the category of prototype, i.e. $S (\{F_i^l\}) = F^5_i$. As shown the results of model `A` comparing to model `C` and model `D` comparing to model `E` in Table \[tab:1\], using all pyramid features performs much better. Because it can combine features from multiple level pyramids, which not only have rich semantics but also take into account objects of different size. We also fuse different levels of pyramid features by using LSTM [@hochreiter1997long], but the speed is greatly reduced. Head model {#section:4.3} ---------- As mentioned before, in independent module, the box regression function $B(\cdot,\cdot)$ is the same as baseline model because experiment found that removing one layer will make the result worse. The results show shared module (model `D`) performs much better than independent module (model `A`). Furthermore, we drop the operation of refining bounding box from the head. As shown in Table \[tab:1\], it’s weird that model `F` is better than model `A` which goes against common sense that fine-tuning bounding box twice is often better than just once. We conjecture that the importance of classification should be more important than bounding box regression in our model [@liang2018object]. So we add a weight coefficient $\lambda$ to balance the classification loss and bounding box regression loss. Here we select $\lambda=5$. The results of model `G`, `H` and `I` in Table \[tab:1\] confirm our analysis. By analyzing model `H` and model `I`, we find that the difference between them is not only classification and bbox regression is independent, but also the comparison classifier of model `H` is semi-parameter. After changing the comparison classifier of model `I` into semi-parameter(Fig. \[fig:1\] (b)), the results show that it is better than model `H`. ---------------------------------------------------- -- -- -- **method & **fixed mode & **random mode & **t-SNE\ mAP &44.5 &42.8&**45.3**\ AR & 61.6 &61.0& **62.8**\ ******** ---------------------------------------------------- -- -- -- : Different way of selecting the reference images.[]{data-label="tab:3"} {height="3.2cm" width="4.3cm"} {height="3.2cm" width="4.3cm"} {height="3.2cm" width="4.3cm"} {height="3.2cm" width="4.3cm"} {height="3.2cm" width="4.3cm"} {height="3.2cm" width="4.3cm"} {height="3.2cm" width="4.3cm"} {height="3.2cm" width="4.3cm"} {height="3.2cm" width="4.3cm"} {height="3.2cm" width="4.3cm"} {height="3.2cm" width="4.3cm"} {height="3.2cm" width="4.3cm"} [|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|]{} **method & **dataset & **AR & **mAP & **ascu &**asch &**lsil &**hsil &**scc & **agc & **trich & **cand & **flora & **herps&**actin\ ****************************** ---------- baseline model ---------- & $D_s$ &12.9&6.6&11.0&2.0&23.7&21.6&0.0&3.5&0.0&11.5&0.0&0.0&0.0\ ------------ Comparison detector ------------ &$D_s$ &**35.7**&**26.3**&10.5&1.7&42.8&32.3&0.8&40.5&37.5&24.1&6.9&45.0&46.6\ ---------- baseline model ---------- &$D_f$ &58.9&45.2&27.2&6.7&41.7&35.3&18.6&57.3&46.7&72.2&57.3&83.0&51.4\ ------------ Comparison detector ------------ &$D_f$ &**62.8**&**45.3**&29.1&7.8&43.0&37.8&19.3&56.2&50.4&62.2&59.4&64.4&68.3\ Optimizing comparison classifier -------------------------------- We evaluate three distance metrics in the comparison classifier. The first is $\ell_2$-distance which means $d(P_m, F_i) = M(|F_i-P_m |^2) $. $M(\cdot)$ represents averaging function for tensor. The second is the parameterized $\ell_2$-distance, such as $d(P_m, F_i) = Conv_7(|F_i - P_m|^2)$. Similar to [@yang2018learning], we also try to make the model to learn the metric function instead of the predefined ones. According to the result of Table \[tab:2\], our model ultimately adopts parameterized $\ell_2$-distance. When $\lambda =5$, the result is shown in brackets. Combining with the results shown in Table \[tab:1\], it is universal that the balance trick can improve performance in our model. So we adopt this trick in all the next experiments. Reference images sampling ------------------------- We first evaluate the scheme of randomly choosing reference image which includes two tests. The first is to randomly choose 3 instances of each category (this number is limited by GPU’s memory) as the reference images (`fixed mode`). The second one is to randomly select 5 candidates of each category in those objects. Then the model randomly selected three of the five candidates as templates during training, but five in testing (`random mode`). The results are listed in Table \[tab:3\], which shows that compared with `random mode`, `fixed mode` performs better. Therefore when evaluate the scheme of choosing reference image by applying t-SNE, we also adopt the `fixed mode`. During the training of t-SNE, we adopt the following parameters setting, i.e. the hyper-parameters are 30 for perplexity, 1 for learning rate, and 10 for label supervision. Table \[tab:3\] shows that the selection reference image via t-SNE performs the best. Performance on training dataset $D_f$ and $D_s$ ----------------------------------------------- As shown in Table \[tab:4\], Comparison detector has the almost same mAP as the baseline model when training on the $D_f$ dataset, but improves the AR by near 4 points. Due to the special annotating situation as described in Section \[sect:4.1\], some correct predictions may be identified as false positives. Therefore, there is an significant increase in AR, but little improvement in mAP. When training on the $D_s$, Comparison detector is completely superior to baseline model. It achieves a top result on the test set with a mAP of 26.3% compared to 6.6%, which indicates our method alleviates the over fitting problem to some extent. Prototype representation in this model is generated by reference images, however, it can be generated by other way, such as external memory. In the future work, We expect a better solution for the generation of prototype representations. Conclusion ========== In this work, we propose to apply contemporary CNN-based object detection methods for automated cervical cancer detection. To deal with the limited size of training samples, we develop the comparison classifier into the state-of-the-art two-stage object detection method based on the comparison with the reference images of each category. Instead of manually choosing the reference images of the background by some heuristic rules, we present a scheme to learn them form the data directly. We also investigate several important ingredients including the generation of prototype representations of each class and the design of head model for cervical cell detection. Experimental results show that compared with the baseline, our method improves the mAP by **19.7** points and the AR by **22.8** when trained on the small size training data, and achieves almost the same mAP but improves the AR by **3.9** when trained on the medium size training data. It should be noticed that our algorithm directly operate on the whole image rather than the extracted patches based on the nuclei and hereby only need one forward propagation for each image, making the inference very efficient. In addition, the proposed method is *flexible* to be intergraded into other proposal-based methods. [^1]: This research was partially supported by the National Natural Science Foundation of China under Grant No. 61602522, the Natural Science Foundation of Hunan Province, China under Grant No.14JJ2008 and the Fundamental Research Funds of the Central Universities of Central South University under Grant No. 2018zzts577. (*Corresponding author: Yao Xiang*.) [^2]: Y. Liang, Z. Tang, M. Yan, J. Chen and Y. Xiang are with the School of Computer Science and Engineering, Central South University, Hunan 410083, China. E-mail: {yxliang,yao.xiang}@csu.edu.cn. [^3]: This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. [^4]: <https://cs.adelaide.edu.au/~carneiro/isbi14_challenge/index.html> [^5]: <https://cs.adelaide.edu.au/~zhi/isbi15_challenge/index.html>